Modeling, Reachability and Observability of Linear

ARISTOTLE UNIVERSITY OF THESSALONIKI DEPARTMENT OF MATHEMATICS

DOCTORAL DISSERTATION

Modeling, Reachability and Observability of Linear Multivariable Discrete Time Systems

LAZAROS MOYSIS

Thessaloniki, May 2017

ARISTOTLE UNIVERSITY OF THESSALONIKI DEPARTMENT OF MATHEMATICS

DOCTORAL DISSERTATION

Modeling, Reachability and Observability of Linear Multivariable Discrete Time Systems

LAZAROS MOYSIS

Supervisor:

Nicholas Karampetakis Professor, AUTH

Supervising committee:

N. Karampetakis, Professor, AUTH. E. Antoniou, Associate Professor, ATEITH A. I. Vardulakis, Professor Emeritus, AUTH.

Approved by the seven member examination board on May 2017. …………………………

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N. Karampetakis Professor, AUTH

E. Antoniou Associate Professor, ATEITH

A.I. Vardulakis Professor Emeritus, AUTH

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A. Pantelous Reader, Un. Of Liverpool

P. Seferlis Associate Professor, AUTH

E. Kappos Associate Professor, AUTH

……………………………. O. Kosmidou Associate Professor, DUTH

ΑΡΙΣΤΟΤΕΛΕΙΟ ΠΑΝΕΠΙΣΤΗΜΙΟ ΘΕΣΣΑΛΟΝΙΚΗΣ ΤΜΗΜΑ ΜΑΘΗΜΑΤΙΚΩΝ ΔΙΔΑΚΤΟΡΙΚΗ ΔΙΑΤΡΙΒΗ

Μοντελοποίηση, Εφικτότητα και Παρατηρησιμότητα Γραμμικών Πολυμεταβλητών Συστημάτων Διακριτού Χρόνου

ΛΑΖΑΡΟΣ ΜΩΥΣΗΣ

Επιβλέπων: Νικόλαος Καραμπετάκης Καθηγητής Α.Π.Θ. Τριμελής επιτροπή: Ν. Καραμπετάκης, Καθηγητής Α.Π.Θ. Ε. Αντωνίου, Αν. Καθηγητής, Α.Τ.Ε.Ι.Θ. Α. Ι. Βαρδουλάκης, Ομότιμος Καθηγητής Α.Π.Θ. Εγκρίθηκε από την επταμελή εξεταστική επιτροπή τον Μάιο, 2017. …………………………

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Ν. Καραμπετάκης Καθηγητής Α.Π.Θ.

Ε. Αντωνίου Αν. Καθηγητής Α.Τ.Ε.Ι.Θ.

Α. Ι. Βαρδουλάκης Ομότιμος Καθηγητής Α.Π.Θ.

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Α. Παντελους Reader, Un. Of Liverpool

Π. Σεφερλής Αν. Καθηγητής Α.Π.Θ.

Ε. Κάππος Αν. Καθηγητής Α.Π.Θ.

…………………………… Ο. Κοσμίδου Αν. Καθηγήτρια, Δ.Π.Θ.

Copyright © Lazaros Moysis, 2017. Με επιφύλαξη παντός δικαιώματος. All rights reserved. Απαγορεύεται η αντιγραφή, αποθήκευση και διανομή της παρούσας εργασίας, εξ ολοκλήρου ή τμήματος αυτής, για εμπορικό σκοπό. Επιτρέπεται η ανατύπωση, αποθήκευση και διανομή για σκοπό μη κερδοσκοπικό, εκπαιδευτικής ή ερευνητικής φύσης, υπό την προϋπόθεση να αναφέρεται η πηγή προέλευσης και να διατηρείται το παρόν μήνυμα. Ερωτήματα που αφορούν τη χρήση της εργασίας για κερδοσκοπικό σκοπό πρέπει να απευθύνονται προς τον συγγραφέα. Οι απόψεις και τα συμπεράσματα που περιέχονται σε αυτό το έγγραφο εκφράζουν τον συγγραφέα και δεν πρέπει να ερμηνευτεί ότι εκφράζουν τις επίσημες θέσεις του Α.Π.Θ.

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Preface I would like to express my sincere thanks to everyone who has helped me during these years. Above all, I would like to thank my parents, my family and friends for their constant support. The completion of this dissertation would not be possible without your encouragement. I am thankful to Professor Karampetakis for his valuable supervision that has guided me through every problem that arose during these years. It has been a pleasure working with him and I feel glad that I had the opportunity to work with such a great man. Sincere thanks also go the other two members of my supervising committee. It has been great working with Professor Antoniou, who has given me a lot of valuable feedback that greatly improved my work and also helped me answer a lot of questions that arose during this dissertation. I also thank Professor Vardulakis for his interest in my dissertation and also for being the one that sparked my interest for the theory of control systems, back in my undergraduate years. I am also grateful that I had the opportunity to work with Professor Pantelous, who has helped me expand my knowledge on the theory of dynamical systems. I thank him for all the support and feedback he has given me. I would also like to thank Professor Azar from Benha University in Egypt for his collaboration and work we had together. In addition, many thanks go to my colleagues Ioannis, Michalis, Nikos and Maria, who I had fun working with. I would also like to acknowledge the scholarship committee of the Aristotle University of Thessaloniki for awarding me with a scholarship for the year 2014.

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Περίληψη Αντικείμενο της παρούσας διδακτορικής διατριβής είναι η μελέτη γραμμικών συστημάτων εξισώσεων διαφορών και αλγεβρικών εξισώσεων διακριτού χρόνου, που περιγράφονται από την παρακάτω σχέση Aq β(k + q) + ... + A1 β(k + 1) + A0 β(k) = Bq u(k + q) + ... + B1 u(k + 1) + B0 u(k) (1αʹ) ξ(k) = Cq β(k + q) + ... + C1 β(k + 1) + C0 β(k)

(1βʹ)

όπου k ∈ N, Ai ∈ Rr×r , Bi ∈ Rr×m , Ci ∈ Rp×r . Οι συναρτήσεις διακριτού χρόνου u(k) ∶ N → Rm , β(k) ∶ N → Rr και ξ(k) ∶ N → Rp ορίζουν την είσοδο, την κατάσταση και την έξοδο του συστήματος αντίστοιχα. Χρησιμοποιώντας τον τελεστή σ με σ i β(k) = β(k + i), το παραπάνω σύστημα μπορεί να γραφτεί ισοδύναμα ως A(σ)β(k) = B(σ)u(k)

(2αʹ)

ξ(k) = C(σ)β(k)

(2βʹ)

όπου A(σ) = A0 + A1 σ + ... + Aq σ q ∈ R[σ]r×r

(3)

B(σ) = B0 + B1 σ + ... + Bq σ q ∈ R[σ]r×m

(4)

C(σ) = C0 + C1 σ + ... + Cq σ q ∈ R[σ]p×r

(5)

πολυωνυμικοί πίνακες με det A(σ) ≡/ 0. Τα θέματα με τα οποία ασχολούμαστε στην παρούσα διατριβή είναι το πρόβλημα της μοντελοποίησης του ομογενούς συστήματος A(σ)β(k) = 0 σε πεπερασμένο χρονικό ορίζοντα, και η μελέτη της εφικτότητας και της παρατηρησιμότητας του μη ομογενούς συστηματος (1) σε άπειρο χρονικό ορίζοντα. Πιο αναλυτικά, το πρόβλημα της μοντελοποίησης μελετά τη δυνατότητα κατασκευής ενός συστήματος της μορφής A(σ)β(k) = 0 το οποίο να έχει μια επιθυμητή δοσμένη συμπεριφορά. Στην περίπτωση του πεπερασμένου χρονικού ορίζοντα, το σύστημα μελετάται σε ένα κλειστό διάστημα [0, N ] και οι λύσεις του μπορεί να είναι είτε προς τα εμπρός κινούμενες στο χρόνο (ξεκινώνας από κάποιες αρχικές συνθήκες) είτε προς τα πίσω κινούμενες (ξεκινώντας από κάποιες τελικές συνθήκες). Ζήτημα λοιπόν είναι, έχοντας σαν δεδομένο ένα σύνολο από συναρτήσεις, να κατασκευάσουμε ένα σύστημα που να έχει ως λύσεις αυτό το σύνολο συναρτήσεων. Επειδή οι λύσεις ενός τέτοιου συστήματος συνδέονται με την δομή των πεπερασμένων μηδενικών του A(σ) και τη δομή των πόλων και

9 μηδενικών του στο άπειρο, δείχνουμε ότι το πρόβλημα της μοντελοποίησης συνδέεται με το πρόβλημα της κατασκευής ενός πολυωνυμικού πίνακα A(σ) με συγκεκριμένη δομή στο C και στο άπειρο. Η εφικτότητα είναι θεμελιώδης ιδιότητα των δυναμικών συστημάτων και αναφέρεται στην ύπαρξη κατάλληλης εισόδου η οποία να οδηγεί το σύστημα από την ηρεμία, σε μια επιθυμητή τελική κατάσταση z σε πεπερασμένο χρόνο. Το πρόβλημα έχει μελετηθεί εκτενώς για τα συστήματα του χώρου των καταστάσεων (state space systems) και τα ιδιόμορφα συστήματα (singular or descriptor systems). ΄Ενα σημαντικό πρόβλημα που εμφανίζεται στα ιδιόμορφα συστήματα αλλά και στα συστήματα ανώτερης τάξης της μορφής (1α΄), είναι η ύπαρξη ενός συνόλου από περιορισμούς που οι αρχικές τιμες της κατάστασης και της εισόδου του συστήματος πρέπει να ικανοποιούν, ώστε το σύστημα να έχει λύση. Το σύνολο των αρχικών τιμών που ικανοποιούν αυτούς τους περιορισμούς ονομάζεται σύνολο παραδεκτών αρχικών τιμών. Επομένως, πέρα από το πρόβλημα της περιγραφής του υποχώρου εφικτότητας, δηλαδή του υποχώρου των καταστάσεων πού είναι προσβάσιμες από την ηρεμία, τίθεται επιπλέον το πρόβλημα της κατασκευής μιας κατάλληλης εισόδου που να οδηγεί το σύστημα στην επιθυμητή θέση, ικανοποιώντας ταυτόχρονα τις αρχικές παραδεκτές συνθήκες. Μια ιδιότητα που συνδέεται με αυτή της εφικτότητας είναι αυτή της ελεγξιμότητας, η οποία αναφέρεται στην ύπαρξη εισόδου που να οδηγεί το σύστημα από μια μη μηδενική κατάσταση στην ηρεμία, σε πεπερασμένο χρόνο. Μάλιστα η εφικτότητα συνεπάγεται την ελεγξιμότητα, χωρίς το αντίστροφο να ισχύει πάντοτε. Η παρατηρησιμότητα αποτελεί επίσης θεμελιώδη ιδιότητα των δυναμικών συστημάτων και αναφέρεται στη δυνατότητα προσδιορισμού του διανύσματος αρχικών τιμών του συστήματος, έχοντας γνώση της εισόδου και της εξόδου του συστήματος για ένα πεπερασμένο χρονικό διάστημα. ΄Εχει επίσης μελετηθεί εκτενώς για τα συστήματα του χώρου των καταστάσεων και τα ιδιόμορφα συστήματα. Μια ιδιότητα που συνδέεται με αυτή της παρατηρησιμότητας είναι η κατασκευασιμότητα, που αναφέρεται στη δυνατότητα προσδιορισμού της παροντικής κατάστασης του συστήματος, έχοντας γνώση των παροντικών και προηγούμενων τιμών της εισόδου και της εξόδου του συστήματος. Η παρατηρησιμότητα συνεπάγεται την κατασκευασιμότητα, χωρίς το αντίστροφο να ισχύει πάντοτε. Μάλιστα, στα συστήματα του χώρου των καταστάσεων, οι έννοιες της εφικτότητας και της παρατηρησιμότητας, και οι έννοιες της ελεγξιμότητας και κατασκευασιμότητας θεωρούνται δυικές.

10 Αναλυτικά, η παρούσα διατριβή δομείται ως εξής: Στο Κεφάλαιο 1 παρουσιάζεται το μαθηματικό υπόβαθρο που είναι απαραίτητο για την παρούσα διατριβή. Παρουσιάζονται οι πολυωνυμικοί και οι ρητοί πίνακες και δίνονται σημαντικές ιδιότητες τους οι οποίες θα χρησιμεύσουν στα επόμενα κεφάλαια. Συγκεκριμένα, ορίζουμε τους αριστερά (δεξιά) ισοδύναμους πίνακες (left (right) equivalent) και τους αριστερά (δεξιά) πρώτους πίνακες (left (right) coprime), τον δυικό ενός πίνακα και τον βαθμό M c−M illan του. Ορίζουμε τη Smith μορφή ενός πίνακα και τη Smith μορφή του στο άπειρο, και μέσω αυτών ορίζουμε τα αναλοίωτα πολυώνυμα (invariant polynomials) ενός πολυωνυμικού πίνακα, τα πεπερασμένα μηδενικά (finite zeros), τους πεπερασμένους στοιχειώδεις διαιρέτες (finite elementary divisors), καθώς επίσης και τους πόλους και μηδενικά στο άπειρο (poles/zeros at infinity) και τους άπειρους στοιχειώδεις διαιρέτες (infinite elementary divisors). Ορίζουμε επίσης την W eierstrass κανονική μορφή για πρωτοβάθμιους πολυωνυμικούς πίνακες. Στη συνέχεια, δίνουμε πληροφορίες για την πραγμάτωση και την ελάχιστη πραγμάτωση ρητών πινάκων, το ανάπτυγμα Laurent ενός ρητού πίνακα στο άπειρο, και τη σχέση που συνδέει τους πίνακες συντελεστές του αναπτύγματος Laurent με τους πίνακες που δίνουν την πραγμάτωση του πίνακα αυτού. Ξεχωριστή προσοχή δίνεται στους πρωτοβάθμιους πίνακες και στις ιδιότητες που ικανοποιούν οι πίνακες συντελεστές του αναπτύγματος Laurent. Στο Κεφάλαιο 2 γίνεται μελέτη των ιδιόμορφων συστημάτων διακριτού χρόνου. Με τον όρο αυτό αναφερόμαστε σε συστήματα τα οποία περιγράφονται από τη σχέση Ex(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)

(6)

όπου ο πίνακας E δεν είναι αντιστρέψιμος. Αρχικά, παρουσιάζεται η λύση του συστήματος σε πεπερασμένο και άπειρο χρονικό ορίζοντα, συναρτήσει των συντελεστών του αναπτύγματος Laurent του πίνακα (σE − A)−1 . ΄Επιπλέον, χρησιμοποιώντας τη W eierstrass κανονική μορφή, χωρίζουμε το σύστημα σε δύο υποσυστήματα, το αιτιατό και το μηαιτιατό και η λύση του κάθε συστήματος δίνεται ξεχωριστά. Επίσης, περιγράφονται τα σύνολα παραδεκτών αρχικών τιμών για το αρχικό σύστημα και για τα δύο του υποσυστήματα και δείχνεται πως οι παραδεκτές συνθήκες για τις δύο αυτές περιπτώσεις είναι ισοδύναμες. ΄Επειτα, μελετώνται οι ιδιότητες της εφικτότητας και της παρατηρησιμότητας. Η εφικτότητα των ιδιόμορφων συστημάτων είχε μελετηθεί από τους Karampetakis & Gregoriadou (2014) όπου περιγράφτηκε ο υποχώρος εφικτότητας και δόθηκε μια μέθο-

11 δος κατασκευής μιας παραδεκτής εισόδου που οδηγεί το σύστημα από την ηρεμία, στην επιθυμητή εφικτή κατάσταση. Αναλύοντας το σύστημα στα δύο προαναφερθέντα υποσυστήματα, περιγράφουμε με παρόμοιο τρόπο τον υποχώρο εφικτότητας των δύο συστημάτων και δίνουμε μια μέθοδο κατασκευής μιας παραδεκτής εισόδου που να ικανοποιεί τις αρχικές συνθήκες και να οδηγεί την κατάσταση του κάθε υποσυστήματος από την ηρεμία, στην επιθυμητή εφικτή θέση. ΄Επειτα, δίνονται κριτήρια εφικτότητας για το σύστημα που εξαρτώνται είτε από τους συντελεστές του αναπτύγματος Laurent του (σE − A)−1 , είτε από τους πίνακες των δύο υποσυστημάτων. Η παρατηρησιμότητα μελετάται επίσης μέσω της διάσπασης σε δύο υποσυστήματα. Δείχνουμε πως το μη-αιτιατό υποσύστημα είναι πάντοτε παρατηρήσιμο, επομένως η παρατηρησιμότητα του ιδιόμορφου συστήματος είναι ισοδύναμη με την παρατηρησιμότητα του αιτιατού υποσυστήματος. Κριτήρια παρατηρησιμότητας δίνονται που εξαρτώνται είτε από τους συντελεστές του αναπτύγματος Laurent του (σE −A)−1 , είτε από τους πίνακες του αιτιατού υποσυστήματος. Επίσης, καταλήγουμε στο συμπέρασμα πως το σύστημα είναι παρατηρήσιμο αν και μόνο αν οι πίνακες (σE − A) και C είναι δεξιά πρώτοι. Τέλος, μελετούμε την παρατηρησιμότητα μιας ειδικής κατηγορίας ιδιόμορφων συστημάτων όπου η έξοδος εξαρτάται όχι μόνο από την κατάσταση x(k), αλλά και από την x(k + 1), δηλαδή y(k) = C1 x(k) + C2 x(k + 1). Αυτή η περίπτωση έχει ξεχωριστή σημασία γιατί θα μας βοηθήσει να μελετήσουμε την παρατηρησιμότητα στα συστήματα υψηλότερης τάξης της μορφής (1), αφού όπως θα δείξουμε στο 5ο Κεφάλαιο μπορούμε να μετασχηματίσουμε ένα σύστημα της μορφής (1) σε ένα ιδιόμορφο σύστημα με αυτή τη μορφή. Στο Κεφάλαιο 3 μελετάται το πρόβλημα της μοντελοποίησης ομογενών συστημάτων της μορφής A(σ)β(k) = 0 σε πεπερασμένο χρονικό ορίζοντα. Αρχικά ορίζονται τα πεπερασμένα ζεύγη Jordan ενός πολυωνυμικού πίνακα και δίνεται ένας τρόπος κατασκευής τους μέσω της Smith μορφής. Μέσω των πεπερασμένων ζευγών Jordan κατασκευάζουμε τον προς τα εμπρός χώρο λύσεων του συστήματος. ΄Επειτα, ορίζουμε τα ζεύγη Jordan στο άπειρο ενός πολυωνυμικού πίνακα και δίνεται ένας τρόπος κατασκευής τους μέσω της Smith μορφής του δυικού πίνακα στο μηδέν. Μέσω των ζευγών Jordan στο άπειρο κατασκευάζουμε τον προς τα πίσω χώρο λύσεων του συστήματος. ΄Εχοντας παρουσιάσει τον πλήρη χώρο λύσεων του συστήματος, γίνεται μελέτη του αντίστροφου προβλήματος, δηλαδή της κατασκευής ενός συστήματος το οποίο να έχει έναν δοσμένο χώρο λύσεων. Η λύση στο πρόβλημα αυτό δίνεται με δύο διαφορετικές μεθοδολογίες. Αρχικά, το πρόβλημα

12 αυτό είχε απαντηθεί από τους Gohberg et al., (2009) αλλά μονάχα για την περίπτωση του προς τα εμπρός χώρου λύσεων. Βασιζόμενοι σε αυτή την υπάρχουσα μέθοδο, γίνεται επέκταση της, ώστε να συμπεριληφθεί η περίπτωση των προς τα πίσω λύσεων. Η επέκταση αυτή γίνεται μέσω της αντιστοίχισης των προς τα εμπρός και προς τα πίσω λύσεων του συστήματος με προς τα εμπρός λύσεις του δυικού συστήματος, το οποίο μας επιτρέπει να εφαρμόσουμε απευθείας τη μέθοδο των Gohberg et al., (2009) στο δυικό σύστημα, κατασκευάζοντας έτσι τον επιθυμητό πολυωνυμικό πίνακα A(σ). Με αυτό τον τρόπο, γίνεται επίσης μια πλήρη αντιστοίχιση μεταξύ των μηδενικών (πεπερασμένων και άπειρων) του πολυωνυμικού πίνακα A(σ) και των λύσεων στις οποίες αυτές αντιστοιχούν για το αρχικό σύστημα και για το δυικό του. Στη δεύτερη προσέγγιση, δείχνεται πως το πρόβλημα της κατασκευής ενός συστήματος με συγκεκριμένη επιθυμητή συμπεριφορά, είναι ισοδύναμο με την επίλυση ενός γραμμικού συστήματος εξισώσεων, με αγνώστους τους συντελεστές A0 , ..., Aq του πίνακα. ΄Ετσι, καταφέρνουμε να απλοποιήσουμε το πρόβλημα σε μια επίλυση ενός απλού γραμμικού συστήματος. Σε σύγκριση με την πρώτη μέθοδο, αυτή η μέθοδος είναι πιο εύκολη στην κατανόηση και την υλοποίηση, και επιπλέον προσφέρει το μεγάλο πλεονέκτημα πως μπορεί να εφαρμοσθεί και σε μη τετράγωνα συστήματα. Τέλος, και για τις δύο μεθόδους, δείχθηκε πως αναλόγως του πλήθους των λύσεων που θα δοθούν (μετρώντας τις πολλαπλότητες τους), είναι πιθανόν το σύστημα που θα κατασκευασθεί να έχει επιπλέον λύσεις, γραμμικά ανεξάρτητες από τις δοσμένες. Αυτό το αποτέλεσμα προκύπτει από μια θεμελιώδη σχέση που συνδέει το πλήθος των πεπερασμένων και άπειρων στοιχειωδών διαιρετών με τη διάσταση r και το βαθμό q του συστήματος. Περαιτέρω διερεύνηση μπορεί να γίνει για αν το σύστημα μπορεί να κατασκευασθεί έτσι ώστε οι ανεπιθύμητες λύσεις να ικανοποιούν ορισμένες ιδιότητες, όπως για παράδειγμα η ευστάθεια. Στο Κεφάλαιο 4 γίνεται μελέτη της εφικτότητας του μη ομογενούς συστήματος (1α΄). Αρχικά, δίνεται μια νέα φόρμουλα για τη λύση του συστήματος συναρτήσει των πινάκων που δίνουν μια ελάχιστη πραγμάτωση του A(σ)−1 . ΄Επειτα, δίνεται το σύνολο των παραδεκτών αρχικών τιμών και περιγράφεται ο υποχώρος εφικτότητας του συστήματος. Παρόμοια με τη μέθοδο που παρουσιάστηκε στο Κεφάλαιο 2 και από τους Karampetakis & Gregoriadou (2014), δίνεται μια μέθοδος κατασκευής μιας παραδεκτής εισόδου που οδηγεί το σύστημα από την ηρεμία στην επιθυμητή εφικτή κατάσταση. Εξάγονται έτσι κριτήρια για τον έλεγχο της εφικτότητας του συστήματος που εξαρτώνται από τους συντελεστές του αναπτύγματος Laurent, που αποτελούν επεκτάσεις των γνωστών κριτηρίων για τα

13 ιδιόμορφα συστήματα και τα συστήματα του χώρου των καταστάσεων. Στο Κεφάλαιο 5, γίνεται μελέτη της παρατηρησιμότητας του συστήματος (1). Αρχικά παρουσιάζεται μια φόρμουλα για την έξοδο του συστήματος. ΄Επειτα, χρησιμοποιώντας τη μέθοδο που παρουσιάστηκε από τους Karampetakis & Vologiannidis (2003), το σύστημα (1) μετασχηματίζεται σε ένα ιδιόμορφο σύστημα με έξοδο που εξαρτάται από τα x(k) και x(k + 1), το οποίο είχαμε μελετήσει στο Κεφάλαιο 2. Για δύο συστήματα αυτά, αν και δεν είναι ισοδύναμα με την έννοια της αυστηρής ισοδυναμίας κατά Rosenbrock (1970), υπάρχει 1-1 απεικόνιση μεταξύ των διανυσμάτων εισόδων και εξόδων τους. Επιπλέον όπως δείχνουμε, τα σύνολα των παραδεκτών αρχικών συνθηκών για τα δύο συστήματα ταυτίζονται. Αυτό μας επιτρέπει να μελετήσουμε την παρατηρησιμότητα του ιδιόμορφου συστήματος και να βγάλουμε συμπεράσματα για το αρχικό σύστημα υψηλότερης τάξης (1). Αναλύοντας επιπλέον το ιδιόμορφο σύστημα στα δύο υποσυστήματά του, δείχνουμε πως η παρατηρησιμότητα του αρχικού συστήματος (1) είναι ισοδύναμη με την παρατηρησιμότητα του αιτιατού υποσυστήματος. Κριτήρια παρατηρησιμότητας εξάγονται που εξαρτώνται είτε από τους συντελεστές του αναπτύγματος Laurent του πίνακα A(σ)−1 , είτε από τους συντελεστές του αναπτύγματος Laurent του πίνακα του αντίστοιχου ιδιόμορφου συστήματος, είτε από τους πίνακες του αιτιατού υποσυστήματος, και δείχνεται πως τα τρία αυτά κριτήρια είναι ισοδύναμα. Επιπλέον, καταλήγουμε στο συμπέρασμα πως το σύστημα είναι παρατηρήσιμο αν και μόνο αν οι πίνακες A(σ) και C(σ) είναι δεξιά πρώτοι, κριτήριο το οποίο είναι ισοδύναμο με το να είναι δεξιά πρώτοι οι πίνακες του αντίστοιχου ιδιόμορφου συστήματος σE − A και C1 + C2 σ, ή του αιτιατού υποσυστήματος. Στην τελευταία ενότητα δίνονται τα συμπεράσματα της διατριβής. Γίνεται μια ανακεφαλαίωση των αποτελεσμάτων και των επεκτάσεων που μπορούν να προκύψουν.

14

Abstract The aim of this doctoral dissertation is the study of linear discrete time systems of algebraic and difference equations, that can be described by the following system of equations Aq β(k + q) + ... + A1 β(k + 1) + A0 β(k) = Bq u(k + q) + ... + B1 u(k + 1) + B0 u(k) ξ(k) = Cq β(k + q) + ... + C1 β(k + 1) + C0 β(k)

(1a) (1b)

where k ∈ N, Ai ∈ Rr×r , Bi ∈ Rr×m , Ci ∈ Rp×r . The discrete time functions u(k) ∶ N → Rm , β(k) ∶ N → Rr and ξ(k) ∶ N → Rp define the input, state and output vectors of the system. Using the forward shift operator σ, with σ i β(k) = β(k + i), the above system can be rewritten as A(σ)β(k) = B(σ)u(k)

(2a)

ξ(k) = C(σ)β(k)

(2b)

A(σ) = A0 + A1 σ + ... + Aq σ q ∈ R[σ]r×r

(3)

B(σ) = B0 + B1 σ + ... + Bq σ q ∈ R[σ]r×m

(4)

C(σ) = C0 + C1 σ + ... + Cq σ q ∈ R[σ]p×r

(5)

where

are polynomial matrices with det A(σ) ≡/ 0. The subjects that are studied in this dissertation is the problem of modeling the homogeneous system A(σ)β(k) = 0 over a finite time horizon, and the study of reachability and observability of the non-homogeneous system (1) over an infinite time horizon. In detail, the problem of modeling concerns the construction of a system of the form A(σ)β(k) = 0, that has a desired prescribed behavior. In the case of a finite time horizon, this system is studied over a closed interval [0, N ] and its solutions can either be forward propagating in time (starting from some initial values) or backward propagating in time (starting from some final values). So the problem is to construct a system that has a given set of vector valued functions as its solutions. Since the solutions of such a system are connected to the structure of the finite zeros of A(σ) and the structure of its poles and zeros at infinity, it is shown that the modeling

15 problem is connected to the problem of constructing a polynomial matrix A(σ) with specific algebraic structure. Reachability is a fundamental property of dynamical systems and refers to the existence of an appropriate input that can drive the system from the origin to another desired state z, over a finite time. This property has been extensively studied for state space systems and singular or descriptor systems. A notable problem that arises in descriptor systems, as well as higher order systems of the form (1a), is the existence of constraints that the initial values of the state and input need to satisfy, in order for the system to have a solution. The set of initial values that satisfy these constraints is called the set of consistent (or admissible) initial values. Thus, in addition to the problem of giving a description of the reachable subspace, that is, the subspace of all the states that are reachable from the origin, the problem of constructing a suitable input that drives the system to the desired state and also satisfies the admissible initial conditions is also addressed. Another property that is closely related to reachability is controllability, which refers to the existence of an appropriate input that can drive the system from a non zero state to the origin, over a finite time. Reachability implies controllability, but the inverse does not always hold. Observability is another fundamental property of dynamical systems and refers to the determination of the initial value of the system’s state, by knowledge of its input and output values over a finite time interval. Observability has been extensively studied for state space and descriptor systems. Another property that is closely related to observability is constructibility, which refers to the determination of the present state of the system from present and past output and inputs. Observability implies contructibility, but the inverse does not always hold. In addition, for state space systems, the properties of reachability, observability and the properties of controllability, constructibility, are considered as dual concepts. Analytically, this dissertation is structured as follows: In Chapter 1, the mathematical preliminaries that are necessary for this dissertation are presented. Important results from the theory of polynomial and rational matrices are presented, that shall be of use in following chapters. Specifically, we define left and right equivalent matrices, left and right coprime polynomial matrices, the Mc-Millan degree of a polynomial matrix and the dual of a polynomial matrix. The Smith form

16 (in C) and the Smith form at infinity of a matrix are defined, its invariant polynomials, finite zeros and finite elementary divisors, as well as the poles and zeros at infinity and the infinite elementary divisors. The Weierstrass canonical form for matrix pencils is also defined. The subject of realization and minimal realization of rational matrices is also studied, and the Laurent expansion of a rational matrix at infinity is given. Attention is given to matrix pencils and the properties that the coefficients of the Laurent expansion of the inverse of a matrix pencil satisfy. In Chapter 2, discrete time descriptor systems are studied. By this term, we refer to systems that are described by Ex(k + 1) = Ax(k) + Bu(k) y(k) = Cx(k)

(6)

where the matrix E is not invertible. Initially, the solution of the system over a finite or an infinite time horizon is presented, given in terms of the coefficients of the Laurent expansion of (σE − A)−1 . In addition, using the Weierstrass canonical form, the system is decomposed into two subsystems, the causal and the noncausal, and the solution of each separate system is given. The sets of consistent (or admissible) initial values for the descriptor system and the causal/noncausal subsystems are given and it is shown that the two sets are equivalent. Thereafter, the properties of reachability and observability are studied. The reachability for descriptor systems has been studied by Karampetakis & Gregoriadou (2014), where the reachable subspace was described and a method was proposed for constructing an admissible input that can drive the system from the origin, to a desired state in the reachable subspace. By decomposing the system into its two subsystems, we describe in a similar way the reachable subspace of the two subsystems and provide a method for constructing an admissible input that satisfies the initial constraints and drives each subsystem from the origin to a desired reachable state. Then, reachability criteria are given, based on the rank of the coefficients of the Laurent expansion of (σE−A)−1 , or the rank of the subsystems’ matrices. Observability is also studied by decomposing the system into its subsystems. It is shown that the noncausal system is always observable, so the observability of the descriptor system is equivalent to the observability of the causal subsystem. Observability criteria are then proposed based on the rank of the coefficients of the Laurent expansion of (σE−A)−1 , or the rank of the causal subsystem’s matrices. In addition, it is shown that the system is

17 observable iff the matrices (σE − A) and C are right coprime. Lastly, the observability of a special case of descriptor systems is considered, where the output depends not only on the state x(k) but also on x(k + 1), that is y(k) = C1 x(k) + C2 x(k + 1). This case has special interest since it will help in the study of observability of higher order systems. It will be shown on Chapter 5 that it is possible to transform a system of the form (1) into a descriptor system of the above special form. In Chapter 3, the problem of modeling homogeneous systems of the form A(σ)β(k) = 0 over a finite time horizon is studied. Initially, the finite Jordan pairs of a polynomial matrix are defined, and a method to construct them from the Smith form is given. Using the finite Jordan pairs of the matrix the forward solution space of the system is constructed. Then, the infinite Jordan pairs of a polynomial matrix are defined, and a method to construct them from the Smith form of the dual matrix at zero is given. Using the infinite Jordan pairs the backward solution space of the system is constructed. Having presented the complete solution space of the system, the inverse problem is then studied, that is, how to construct a system with a prescribed solution space. Two different approaches are given for the solution of this problem. Initially, this modeling problem was solved by Gohberg et al., (2009), but only for the case of forward solutions. Based on this method, an extension is made to also include the case of the backward solutions. This is done by corresponding the forward and backward solutions of the original system to forward solutions of the dual system, which allows us to use the method proposed by Gohberg et al., (2009) on the dual system, which then gives the desired matrix A(σ). This way, there is a complete correspondence between between the zeros (finite and infinite) of the polynomial matrix A(σ) and the solutions to which they correspond for the system A(σ)β(k) = 0 and its dual. In the second approach, it is shown that the problem of constructing a system with desired behavior is equivalent to solving a linear system of equations, in terms of the unknown coefficients A0 , ..., Aq of A(σ). So the problem is simplified to the solution of a simple linear system of equations. In comparison to the first proposed method, this approach is much simpler to understand and implement, and can also be used for constructing non-regular systems. Finally, for both methods, it is shown that depending on the number of solutions that are given (counting multiplicities), it is possible for the constructed system to have additional undesired solutions, linearly independent from the

18 given ones. This result is derived from a fundamental relation connecting the sum of the finite and infinite elementary divisors of a matrix A(σ) with its dimension r and lag q. A subject for further research would be the study of the conditions under which the undesired solutions satisfy certain properties, like stability. In Chapter 4, the reachability of the nonhomogeneous system (1a) is studied. Initially, an alternative formula for the solution of the system is provided, in terms of the matrices that give a minimal realization of A(σ)−1 . Then, the set of admissible initial values is provided and the reachable subspace of the system is given. Similarly to the method presented in Chapter 2, and also by Karampetakis & Gregoriadou (2014), a method is provided for constructing a consistent input that can drive the system from the origin to any desired state in the reachable subspace. Criteria for the reachability of the system are given, based on the Laurent coefficients of A(σ)−1 , that are generalisations of the known criteria for state space and descriptor systems. In Chapter 5, the observability of (1) is studied. First, a formula for the output of (1) is given. Then, using the method presented in Karampetakis & Vologiannidis (2003), the system (1) is transformed into a descriptor system where the output depends on x(k) and x(k + 1). Such systems were studied in Chapter 2. For these two systems, although they are not equivalent in a strict system equivalence sense (see Rosenbrock (1970)), there is a 1-1 correspondence between their input and output vectors. In addition, it is shown that the sets of consistent initial values for the two systems are the same. This makes it possible to study the observability of the descriptor system in order to derive information for the observability of the original higher order system (1). By further decomposing the descriptor system into its two subsystems, it is shown that the observability of the original system is equivalent to the observability of the causal subsystem. Observability criteria are derived, based on the coefficients of the Laurent expansion of A(σ)−1 , or the coefficients of the Laurent expansion of the corresponding descriptor system, or the matrices of the causal subsystem and it is shown that all these criteria are equivalent. In addition, it is shown that the original system is observable iff the matrices A(σ) and C(σ) are right coprime, which is equivalent to the right coprimeness of the matrices (σE − A) and (C1 + C2 σ) of the descriptor system, or the right coprimeness of the matrices of the causal subsystem. In the last Chapter, the results of this dissertation are summarised. The conclusions,

19 as well as further research topics are also outlined.

20

Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

Περίληψη . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8

Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

14

1 Mathematical preliminaries

23

1.1

Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23

1.2

Polynomial Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

27

1.3

Matrix Pencils . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

36

1.4

Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

37

1.5

Realization of Rational Matrices . . . . . . . . . . . . . . . . . . . . . . . .

41

1.6

Laurent Expansion of a Rational Matrix . . . . . . . . . . . . . . . . . . .

44

2 Descriptor Systems

49

2.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

49

2.2

Solution of Descriptor Systems . . . . . . . . . . . . . . . . . . . . . . . . .

51

2.3

Reachability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57

2.4

Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

66

2.5

Observability of Descriptor Systems with State Lead in the Output . .

72

2.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

78

3 Modeling of Linear Homogeneous Systems of Algebraic and Difference Equations

81

3.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81

3.2

Finite Jordan Pairs and the Forward Solution Space . . . . . . . . . . . .

84

3.3

Infinite Jordan Pairs and the Backward Solution Space . . . . . . . . . .

87

21

22

CONTENTS 3.4

Construction of a Homogeneous System with Prescribed Forward/Backward Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

91

3.4.1

First method: An extension of the method in [33] . . . . . . . . .

91

3.4.2

Second method: Reduction to a linear system of equations . . . 104

3.5

Notes on the Power of a model . . . . . . . . . . . . . . . . . . . . . . . . . 123

3.6

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

4 Reachability of Linear Systems of Algebraic and Difference Equations133 4.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

4.2

Solution of ARMA representations . . . . . . . . . . . . . . . . . . . . . . 135

4.3

Reachable Subspace . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139

4.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154

5 Observability of Linear Systems of Algebraic and Difference Equations

157

5.1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

5.2

Transformation of higher order systems to descriptor form . . . . . . . . 160

5.3

Observability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

5.4

Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176

Conclusions

179

Bibliography

181

Index

190

Chapter 1 Mathematical preliminaries In this chapter some basic definitions and theorems regarding the theory of matrices will be presented. The purpose of this chapter is to serve as a reference to the reader and provide all the necessary background for later chapters. A brief presentation of fundamental results from the theory of polynomial and rational matrices is presented. Polynomial and rational matrices are closely related to the analysis of dynamical systems and have been the subject of extensive research, starting from the early works of [31, 33]. Later works include [1, 10, 24, 25, 28, 34, 43, 57, 77, 85, 86, 100, 106, 116, 119, 121–123, 132] and the references therein.

1.1

Matrices

Let Rn×m be the set of all n-by-m matrices, with elements in R. The above set, equipped with matrix addition and scalar multiplication, is a vector space over R. The set of all column matrices (or column vectors) is denoted by Rn and is also a vector space over R. Let A ∈ Rn×m . The element in the i-th row and j-th column shall be denoted by Ai,j or ai,j . If the matrix is a single row or a single column, we only use a single subscript ai . Definition 1.1. [103] Elementary row and column operations on any matrix A ∈ Rn×m are defined as: 1. Interchange of any two rows or columns of A. 23

24

CHAPTER 1. MATHEMATICAL PRELIMINARIES 2. Multiplication of any row or column of A by a non-zero scalar c ∈ R. 3. Addition of a scalar multiple of any row (column) of A to another row (column). A matrix that is the result of performing any of the above row operations on the

identity matrix In is called an elementary matrix . In general, any elementary row or column operation on a matrix A can be accomplished by multiplying the given matrix on the left (for rows) or on the right (for columns) by an elementary matrix. Theorem 1.2. [103] Two matrices A, B ∈ Rn×m are equivalent if and only if either one can be obtained from the other by a series of elementary row and column operations. Equivalently, we can say that two matrices are equivalent if there exist invertible matrices U, V such that A = U BV

(1.1)

Definition 1.3. [103] Let A ∈ Rn×m . The subspace of Rm spanned by the rows of A is called the row space of A and the subspace of Rn spanned by the columns of A is called the column space of A. The dimensions of these spaces are called the row rank and column rank of the matrix. The row and column rank of a matrix A are always equal. This number is called the rank of the matrix A and is denoted by rankA. The following relation holds: rankA ≤ min{n, m}

(1.2)

Lemma 1.4. [103] Let A ∈ Rn×m . Then elementary row or column operations do not affect the rank of the matrix. Two important features of a matrix are its range and its null space. Definition 1.5. [10] Let A ∈ Rn×m . The range or image of A is defined as Im A = {y ∣ y ∈ Rn , ∃x ∈ Rm ∶ y = Ax} ⊆ Rn

(1.3)

Is is easily seen that the image of A is the column span of A Im A = span{a1 , a2 , ..., am }

(1.4)

In light of the above definition, we can see that the dimension of the image of a matrix A is actually its rank, i.e. rankA = dim Im A

(1.5)

1.1. MATRICES

25

Definition 1.6. [10] Let A ∈ Rn×m . The nullspace or kernel of a matrix A is defined as ker A = {x ∈ Rm ∣ Ax = 0} ⊆ Rm

(1.6)

The dimension of the nullspace of A is called the defect of A, that is def A = dim ker A

(1.7)

Similarly to the definition of the kernel, we can define the left kernel of a matrix. Definition 1.7. [63] Let A ∈ Rn×m . The left nullspace or left kernel of a matrix A is defined as left ker A = {w ∈ Rn ∣ wT A = 0} ⊆ Rn

(1.8)

and is equivalent to the nullspace of AT , i.e. left ker A ≡ ker AT

(1.9)

The rank and the defect of a matrix are connected by the following relation rankA + def A = m

(1.10)

For a square and invertible matrix A ∈ Rn×n the following properties hold: rankA = n

def A = 0

(1.11)

Note also that the rank of a matrix is not affected by elementary row or column operations. Theorem 1.8. [10] Let A ∈ Rr×n and B ∈ Rn×m . Then rankAB ≥ rankA + rankB − n Theorem 1.9. [10] Let A ∈ Rn×n . The following relations hold 1. Im A = Im AAT 2. ker A = kerAAT

(1.12)

26

CHAPTER 1. MATHEMATICAL PRELIMINARIES

Definition 1.10. [103] Let A ∈ Rn×n . We define the characteristic polynomial of A as φ(s) = det(sIn − A)

(1.13)

The roots λi of the characteristic polynomial are called the eigenvalues of A. An eigenvector xi corresponding to the eigenvalue λi is a column vector that satisfies Axi = λi xi

(1.14)

The set of all eigenvectors associated with a given eigenvalue λi , together with the zero vector, forms a subspace of Cn , called the eigenspace of A. Definition 1.11. [43][103] Let λ1 , λ2 , ..., λ` be the ` distinct eigenvalues of A ∈ Rn×n . We say that the eigenvalue λi has algebraic multiplicity ni if φ(λi ) = φ(1) (λi ) = ⋯ = φ(ni −1) (λi ) = 0 and φ(ni ) (λi ) ≠ 0

(1.15)

where φ(i) denotes the i-th derivative of φ(s). The geometric multiplicity of the eigenvalue λi is defined as the dimension of the eigenspace of λi . The following constitutes one of the most fundamental theorems in linear algebra. Theorem 1.12 (Cayley-Hamilton). [43] Every square matrix satisfies its own characteristic polynomial, i.e. φ(A) = 0

(1.16)

A special case of matrices that we will encounter in later chapters are nilpotent matrices. Definition 1.13. [63] A square matrix A is called nilpotent if there exists k ∈ R such that Ai = 0 ∀i ≥ k and Ak−1 ≠ 0

(1.17)

The number k is called the index of nilpotency of A. We will encounter nilpotent matrices in later chapters, when we will talk about infinite Jordan pairs of polynomial matrices.

1.2. POLYNOMIAL MATRICES

1.2

27

Polynomial Matrices

Let R be the field of reals, R [s] the ring of polynomials with coefficients from R and R(s) the field of rational functions i.e. the functions of the form R(s) = {f (s) =

n(s) , d(s)

n(s), d(s) ∈ R[s] with d(s) ≡/ 0}

(1.18)

By R[s]p×m we denote the sets of p × m polynomial matrices with real coefficients. we denote the sets of p × m rational and proper rational Similarly, by R(s)p×m , R(s)p×m pr matrices with real coefficients. Recall that a rational function is called proper when deg(n(s)) ≤ deg(d(s)) and strictly proper when deg(n(s)) < deg(d(s)). We say that a rational function is of standard form if and only if the polynomials n(s),d(s) are relatively prime, i.e. their greatest common divisor is 1 and d(s) is monic. Let A(s) ∈ R[s]r×r . If q is the highest degree among the polynomial elements of A(s), then the matrix can be written as A(s) = Aq sq + Aq−1 sq−1 + ... + A1 s + A0

(1.19)

The number q is called the lag or the order of the matrix. If Aq = Ir , then A(s) is called a monic polynomial matrix . .6 Definition 1.14. [116] The degree of a polynomial matrix A(s) ∈ R[s]r×m is defined as the maximum degree among the degrees of all its maximum order nonzero submatrices. The reader should not not confuse the two terms degree and lag, since they represent two different properties of a polynomial matrix.1 Definition 1.15. [116] The rank of a rational matrix A(s) ∈ R(s)r×m is the maximum number of its linearly independent rows or columns and it is denoted by rankR(s) A(s). Example 1.16. The matrix A(s) =

⎛s2 + s + 1 2s ⎞ 2 ⎝ 5 s + 1⎠

(1.20)

is a polynomial matrix and can be written as A(s) =

⎛1 0⎞ 2 ⎛1 2⎞ ⎛1 0⎞ s + s+ ⎝0 1⎠ ⎝0 0⎠ ⎝5 1⎠ ´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶ A2

1

A1

(1.21)

A0

In some books, like [43], the term degree is used to denote the order of a matrix. Here, we prefer

to use the terms order or lag, to avoid confusion.

28

CHAPTER 1. MATHEMATICAL PRELIMINARIES r×r

Definition 1.17. [31, 116] A square polynomial matrix A(s) ∈ R[s]

is called uni-

r×r ¯ ¯ modular if there exists a A(s) ∈ R[s] such that A(s)A(s) = Ir . A polynomial matrix

is unimodular if and only if det A(s) = c ∈ R, c ≠ 0

(1.22)

If det A(s) ≡ 0 or A(s) is not square, the matrix is called non-regular or singular. r×r

Definition 1.18. [116] A square polynomial matrix A(s) ∈ R[s]

is called regular if

rankAq = r

(1.23)

As with matrices A ∈ Rr×m , we define elementary row and column operations in a similar fashion. Definition 1.19. [116] Elementary row and column operations on any polynomial r×n

matrix A(s) ∈ R[s]

are defined as:

1. Interchange of any two rows or columns of A(s). 2. Multiplication of a row or column by a nonzero element x ∈ R. 3. Addition to a row (or column) of another row (or column) multiplied by any nonzero polynomial p(s) ∈ R[s]. Definition 1.20. [116] Let A(s) ∈ R[s]r×m , B(s) ∈ R[s]r×p , C(s) ∈ R[s]p×m be three polynomial matrices that satisfy A(s) = B(s)C(s)

(1.24)

The matrix B(s) is called a left divisor of A(s) and the matrix C(s) a right divisor of A(s). Furthermore, the matrix A(s) is called a left multipl e of C(s) and a right multiple of B(s). Definition 1.21. [116] Let T1 (s) ∈ R[s]r×m , T2 (s) ∈ R[s]r×l be two polynomial matrices with the same number of rows. The polynomial matrix TL (s) ∈ R[s]r×r is called a common left divisor of T1 (s), T2 (s) if T1 (s) = TL (s)T 1 (s)

T2 (s) = TL (s)T 2 (s)

(1.25)


29

where T 1 (s) ∈ R[s]r×m , T 2 (s) ∈ R[s]r×l . Furthermore, if TL (s) is a right multiple of every common left divisor, i.e. TL (s) = T L (s)T3 (s)

(1.26)

then TL (s) is a greatest common left divisor of T1 (s), T2 (s). In the same fashion, we can define a (greatest) common right divisor for matrices with the same number of columns. Definition 1.22. [43, 116] Two polynomial matrices T1 (s), T2 (s) ∈ R[s]r×m are said to be equivalent, if there exist unimodular matrices U1 (s) ∈ R[s]r×r , U2 (s) ∈ R[s]m×m such that T2 (s) = U1 (s)T1 (s)U2 (s)

(1.27)

if the matrices are connected by T2 (s) = U1 (s)T1 (s)

(1.28)

they are called left equivalent and if they are connected by T2 (s) = T1 (s)U2 (s)

(1.29)

they are called right equivalent. In the following we give the definition of the Smith form of a polynomial matrix. The Smith form will play a crucial role in the analysis of higher order systems in later Chapters. Theorem 1.23. [31, 116] Let A(s) ∈ R[s]r×r as in (1.19). There exist unimodular matrices UL (s) ∈ R[s]

r×r

r×r

,UR (s) ∈ R[s]

such that

C SA(s) (s) = UL (s)A(s)UR (s) = diag (1, ..., 1, fz (s), fz+1 (s), ..., fr (s))

(1.30)

C with 1 ≤ z ≤ r and fj (s)∣fj+1 (s) j = z, z + 1, ..., r. SA(s) (s) is called the Smith form

of A(s) (in C) where fj (s) ∈ R [s] are the invariant polynomials of A(s). The zeros λi ∈ C of fj (s), j = z, z + 1, ..., r are called finite zeros of A(s). Assume that A(s) has ` distinct zeros. The partial multiplicities of each zero λi ∈ C, i = 1, ..., ` satisfy 0 ≤ ni,z ≤ ni,z+1 ≤ ... ≤ ni,r

(1.31)

30


with ni,j

fj (s) = (s − λi )

fˆj (s)

(1.32)

n j = z, ..., r and fˆj (λi ) ≠ 0 The terms (s − λi ) i,j are called finite elementary divisors of

A(s) at λi . Denote by n the sum of the degrees of the finite elementary divisors of A(s) r

`

r

n ∶= deg det A(s) = deg [∏ fj (s)] = ∑ ∑ ni,j j=z

(1.33)

i=1 j=z

Similarly, we can find UL (s) ∈ R(s)r×r , UR (s) ∈ R(s)r×r having no poles and zeros at s = λ0 such that n

n

λ0 SA(s) (s) = UL (s)A(s)UR (s) = diag (1, ..., 1, (s − λ0 ) z , ..., (s − λ0 ) r )

(1.34)

λ0 SA(s) (s) is called the Smith form of A(s) at the local point λ0 .

Remark 1.24. Numerical methods for computing the Smith form of a polynomial matrix have been presented in [85, 86, 106, 122, 123]. The notion of zeros and elementary divisors for a polynomial matrix also includes the case of constant matrices. Theorem 1.25. [43] Consider a matrix A ∈ Rr×r and its corresponding polynomial matrix (sIr − A) ∈ R[s]r×r . Then C S(sI (s) = diag(f1 (s), ..., fr (s)) r −A)

(1.35)

where fi (s) = (s − λ1 )ni,1 ⋯(s − λq )ni,q

i = 1, ..., r

(1.36)

and λ1 , ..., λq are the eigenvalues of A. The terms (s − λj )ni,j , i = 1, ..., r, j = 1, ...q are the elementary divisors of A. Definition 1.26. [77] Let λi be a zero of A(s). The algebraic multiplicity of λi , denoted by ni , is the sum of the degrees of the finite elementary divisors (s − λi ) r

ni = ∑ ni,j , i = 1, ..., `

(1.37)

j=z

while the geometric multiplicity of λi , denoted by [λi ] is the number of non-zero elements of the set {ni,z , ni,z+1 , ..., ni,r } or equivalently the number of distinct entries of (s−λi ) in the Smith form of A(s). The sequence of the degrees of λi {ni,z , ni,z+1 , ..., ni,r } is called the partial multiplicity sequence.


31

Example 1.27. Let A(s), B(s) ∈ R[s]3×3 with Smith forms C SA(s) (s)

0 ⎞ ⎛1 0 =⎜ 0 ⎟ ⎜0 1 ⎟ ⎝0 0 (s + 1)2 ⎠

C SB(s) (s)

0 0 ⎞ ⎛1 =⎜ 0 ⎟ ⎜0 (s + 1) ⎟ ⎝0 0 (s + 1)⎠

(1.38)

For both of these matrices, the algebraic multiplicity of the zero λ1 = −1 is n1 = 2. On the other hand, for the matrix A(s) the geometric multiplicity for the zero is [λ1 ] = 1 while for the matrix B(s) its [λ1 ] = 2, because the zero λ1 = −1 appears in two entries of the Smith form. Remark 1.28. [43] The zeros of A(s) can be equivalently defined as those values of the variable s for which the matrix loses rank. Remark 1.29. [116] Another simple method to compute the Smith form of a polynomial matrix is the following. Let mi (s) be the greatest common divisor of all minors (i.e. the determinants of the submatrices) of A(s) of order i. Then the invariant polynomials fi (s) are given by fi (s) =

mi (s) mi−1 (s)

(1.40)

where m0 (s) = 1 Remark 1.30. A numerical algorithm for the computation of the Smith form of a polynomial matrix based on the previous method can be found in [85, 86], along with a comparison between previous existing methods. Other construction methods can be found in [106, 122, 123]. Example 1.31. The matrix A(s) =

⎛s2 + s s + 1⎞ ⎝ s s + 2⎠

(1.41)

has the following Smith form C SA(s) (s) =

⎛1 ⎞ ⎛ −1 0 1 ⎞ ⎛s2 + s s + 1⎞ ⎛0 1 ⎞ = ⎝0 s(s + 1)2 ⎠ ⎝s + 2 −1 − s⎠ ⎝ s s + 2⎠ ⎝1 s2 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¶ UL (s)

(1.42)

UR (s)

where det UL (s) = −1 and det UR (s) = −1. The above matrix A(s) has two zeros, one at λ1 = 0 with algebraic multiplicity n1,2 = 1 and another at λ2 = −1 with algebraic multiplicity n2,2 = 2

32


Algorithm 1 [6] An algorithm for the construction of the Smith form of a polynomial matrix. 1. Using row and column elementary operations, transfer the element of least degree in the matrix A(s) to the (1,1) position. 2. By elementary row operations, make all entries in the first column below (1,1) equal to zero. 3. By column operations, make all entries in the first row zero except (1,1). 4. If nonzero entries have reappeared in the first column, repeat the above steps until all entries in the first column and row are zero except for the (1,1) entry. 5. If the (1,1) element does not divide every other entry in the matrix, use polynomial division and row and column interchanges to bring a lower degree element to the (1,1) position. Repeat the above steps until all other elements in the first column and row are zero and the (1,1) entry divides every other entry in the matrix, that is: ⎛ f1 (s) 01×(r−1) ⎞ ⎝0(r−1)×1 E1 (s) ⎠

(1.39)

where f1 (s) divides all entries of E1 (s). 6. Repeat the above steps on E1 (s) and on other such terms, if necessary, to obtain the Smith form of A(s).

A special case of a polynomial matrix is the following. Definition 1.32. [43] Let A(s) ∈ R[s]

r×r

. The polynomial matrix A(s) is called simple

if and only if it has only one invariant polynomial distinct from the identity polynomial 1. The Smith form of a simple matrix is C SA(s) (s) = diag(1, ..., 1, fr (s))

(1.43)

Example 1.33. The following polynomial matrix 0 ⎞ ⎛4 + 4s + s2 0 ⎜ A(s) = ⎜ 2 + s −s −2 + s⎟ ⎟ ⎝ 0 s2 1 ⎠

(1.44)


33

is simple, since its Smith form is C SA(s) (s)

0 ⎛1 0 ⎞ ⎜ ⎟ = ⎜0 1 0 ⎟ ⎝0 0 s(s − 1)2 (s + 2)2 ⎠

(1.45)

On the other hand, the matrix ⎛1 + s 0 0 ⎞ ⎟ B(s) = ⎜ ⎜2 + s s 0 ⎟ ⎝ 0 0 s2 ⎠

(1.46)

is not simple, since its Smith form is C SB(s) (s)

0 ⎛1 0 ⎞ ⎜ ⎟ = ⎜0 s 0 ⎟ ⎝0 0 s2 (1 + s)⎠

(1.47)

Now that we have presented the finite algebraic structure of a polynomial matrix, we shall continue with its structure at infinity. Theorem 1.34. [119] Let A(s) as in (1.19). There exist biproper matrices UL (s), UR (s) ∈ R(s)r×r pr (see Definition 1.50 ) such that r−u

⎛ ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ · ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ⎞ 1 1 ⎟ 1 ⎜ ∞ SA(s) (s) = UL (s)A(s)UR (s) = diag ⎜sq1 , ..., squ , qû+1 , qû+2 , ..., qˆr ⎟ ⎜´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ s s s ⎟ u ⎠ ⎝

(1.48)

with q1 ≥ . . . ≥ qu ≥ 0,

(1.49)

qˆr ≥ qˆr−1 ≥ . . . ≥ qû+1 > 0,

(1.50)

∞ and 1 ≤ u ≤ r. SA(s) (s) is called the Smith form of A(s) at infinity.

If p∞ is the number of qi ’s in (1.49) with qi > 0, then we say that A(s) has p∞ poles at infinity, each one of order qi > 0. Also, if z∞ is the number of qî ’s in (1.50), then we say that A(s) has z∞ zeros at infinity, each one of order qî > 0. Lemma 1.35. [116] For A(s), it holds that that q1 = q. Definition 1.36. [116] The dual polynomial matrix of A(s) is defined as 1 ˜ A(s) ∶= sq A ( ) = A0 sq + A1 sq−1 + ... + Aq s

(1.51)

34


r×r ˜ Theorem 1.37. [116] Let A(s) as in (1.51). There exist matrices U˜L (s) ∈ R[s] , r×r U˜R (s) ∈ R[s] having no poles or zeros at s = 0, such that 0 µ1 µr ˜ ˜ ˜ SA(s) ˜ (s) = UL (s)A(s)UR (s) = diag (s , . . . , s )

(1.52)

0 µj are the finite elementary ˜ SA(s) ˜ (s) is the local Smith form of A(s) at s = 0. The terms s

˜ divisors of A(s) at zero and are called the infinite elementary divisors (i.e.d.) of A(s). The connection between the Smith form at infinity of A(s) and the Smith form at zero of the dual matrix is given in [34, 116]:

0 SA(s) ˜ (s)

=s

q

⎛ ⎞ q−q2 q−qu q+ˆ qu+1 q+ˆ qr ⎟ ⎜ = diag (s , ..., s ) = diag ⎜1, s , . . . , s ,s ,...,s ⎟ s ´¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶⎠ ⎝

1 ∞ SA(s) ( )

µ1

µr

i.p.e.d.

i.z.e.d.

(1.53) where by i.p.e.d. and i.z.e.d. we denote the infinite pole and infinite zero elementary divisors respectively. Therefore, the orders of the infinite elementary divisors are given by q=q

µ1 = q − q1 = 1 0

(1.54a)

µj = q − qj

j = 2, 3, ..., u

(1.54b)

µj = q + qˆj

j = u + 1, ..., r

(1.54c)

We denote by µ the sum of the degrees of the infinite elementary divisors of A(s) i.e. r

µ ∶= ∑ µj

(1.55)

j=1

Definition 1.38. [116] Let A(s) as in (1.19) with Smith form at infinity as in (1.48). The Mc-Millan degree δM (A(s)) of a polynomial matrix A(s) is defined as u

δM (A) = ∑ qi

(1.56)

i=1

and is equal to the sum of the orders of its poles at infinity. Lemma 1.39. [4, 33, 117] Let A(s) as in (1.19). Let also n, µ be the sum of degrees of the finite and infinite elementary divisors of A(s), as defined in (1.33),(1.55). Then n + µ = r × q.

(1.57)


35

The above relation shall be of fundamental importance in the following chapters, since it will help determining whether or not the construction of a system with prescribed behavior is possible. It should be noted that in the case where the matrix A(s) is non-regular, the algebraic structure of A(s) is connected with additional invariants due to the left and right null space of A(s), see [53, 100]. Definition 1.40. [24] The collection of all the zeros of a polynomial matrix, finite and infinite, is called the spectrum of the matrix. The collection of all the elementary divisors, both finite and infinite, including repetitions, constitutes the spectral structure of the matrix. Definition 1.41. [116] Let T1 (s) ∈ R[s]r×m , T2 (s) ∈ R[s]r×l be two polynomial matrices with the same number of rows. These matrices are called left comprime if their greatest common left divisor TGL (s) is unimodular. Similarly, let A1 (s) ∈ R[s]r×m , T2 (s) ∈ R[s]w×m be two polynomial matrices with the same number of columns. These matrices are called right comprime if their greatest common right divisor TGR (s) is unimodular. Theorem 1.42. [116] Let T1 (s) ∈ R[s]r×m , T2 (s) ∈ R[s]r×l with p ∶= m + l ≥ r = rankR(s) (T1 (s) T2 (s)). The following statements are equivalent. 1. T1 (s), T2 (s) are left coprime. 2. The polynomial matrix T (s) = (T1 (s) T2 (s)) has no finite zeros. 3. There exists unimodular matrix TR (s) ∈ Rp×p such that (T1 (s) T2 (s)) TR (s) = (Ir 0r,p−r ) = STC(s) (s)

(1.58)

rank (T1 (s0 ) T2 (s0 )) = r, ∀s0 ∈ C

(1.59)

4.

Similarly, let A1 (s) ∈ R[s]r×m , T2 (s) ∈ R[s]w×m . The following statements are equivalent. 1. A1 (s), A2 (s) are right coprime. T

2. The polynomial matrix A(s) = (A1 (s)T , A2 (s)T ) has no finite zeros.

36

CHAPTER 1. MATHEMATICAL PRELIMINARIES 3. C SA(s) (s) =

⎛ Im ⎞ ⎝0p−m,0 ⎠

(1.60)

T

where p = r + w ≥ m = rankR(s) (A1 (s)T , A2 (s)T ) . 4. rank

1.3

⎛A1 (s0 )⎞ = m, ∀s0 ∈ C ⎝A2 (s0 )⎠

(1.61)

Matrix Pencils

The following definition and theorem for regular matrices (E, A) will be of great importance in the analysis of descriptor systems. Definition 1.43. [31] Let E, A ∈ Rr×m . The matrix (sE − A) is called regular if 1. r = m, i.e. the matrices are square. 2. det(sE − A) ≡/ 0. In all other cases, (r ≠ m or n = m but det(sE − A) = 0) the matrix (sE − A) is called non-regular or singular. Theorem 1.44. [10, 43] Let E, A ∈ Rr×r be square regular matrices i.e. det(sE − A) ≠ 0

(1.62)

Then there exist invertible matrices P, Q ∈ Rr×r such that the matrix (sE − A) can be decomposed as (sE − A) = P

⎛In s − A1 0 ⎞ Q ⎝ 0 sN − Iµ ⎠

(1.63)

where A1 ∈ Rn×n , N ∈ Rµ×µ , with n = deg det(sE − A)

(1.64)

n+µ=r

(1.65)

where A1 is in Jordan form and N is a nilpotent matrix with index h. The above decomposition of the matrix (sE − A) is called the Weierstrass Decomposition. The Weierstrass decomposition is useful for the analysis of singular state space systems, as we shall see in later chapters.

1.4. RATIONAL MATRICES

1.4

37

Rational Matrices

Definition 1.45. [43] Let f (s) =

n(s) d(s)

be a rational function. Then the zeros of the

numerator polynomial n(s) are called the finite zeros of f (s) and the zeros of the denominator polynomial d(s) are called the finite poles of f (s). Example 1.46. The matrix A(s) = is a rational matrix while A(s) =

⎛ s+1 s2 ⎝ 1s

⎛ s+1 s2 ⎝ 1s

0 ⎞ s3 +5 ⎠ s+1

0 ⎞ s+1 ⎠ s2 +s+1

(1.66)

(1.67)

is a strictly proper matrix. Definition 1.47. [43] Consider a rational matrix A(s) ∈ R(s)p×m . Let the monic polynomial m(s) be the least common multiple of all the denominators of the entries of A(s). Then we can write the matrix in the following form: A(s) =

N (s) m(s)

(1.68)

where N (s) ∈ R[s]p×m . If N (s) ≠ 0 for all the zeros of m(s), then the matrix A(s) is called irreducible and the above formula is called the standard form of A(s). If N (λi ) = 0p,m for some λi then all the entries of the matrix N (s) are divisible by (s−λi ) and the matrix A(s) is reducible by (s − λi ). Example 1.48. The following matrix ⎛ 1 A(s) = s+1 ⎝ 0

s2 (s+1)(s+2) ⎞ 1 ⎠ (s+2)

(1.69)

can be written in standard form as A(s) =

⎛s + 2 s2 ⎞ 1 (s + 1)(s + 2) ⎝ 0 s + 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ m(s)

(1.70)

N (s)

and since N (s) ≠ 0, for s = −1, −2 the matrix is irreducible. Definition 1.49. [116] Two rational matrices T1 (s), T2 (s) ∈ R(s)p×m are called equivalent if there exist unimodular matrices TL (s) ∈ R[s]p×p , TR ∈ R[s]m×m such that TL (s)T1 (s)TR (s) = T2 (s)

(1.71)

38


Definition 1.50. [116] A square rational matrix A(s) ∈ R(s)r×r pr is called biproper ¯ ¯ rational matrix if there exists A(s) ∈ R(s)r×r pr such that A(s)A(s) = Ir or equivalently if lims→∞ A(s) = E ∈ Rr×r , with rankE = r. Remark 1.51. [119] A square rational matrix A(s) ∈ R(s)r×r pr is biproper if and only if det A(s) ∈ Rpr (s) is a biproper rational function, i.e. deg d(s) = deg n(s). Definition 1.52. Two rational matrices T1 (s), T2 (s) ∈ R(s)p×m are called equivalent m×m such at infinity if there exist biproper rational matrices TL (s) ∈ R(s)p×p pr , TR ∈ R(s)pr

that: TL (s)T1 (s)TR (s) = T2 (s)

(1.72)

Theorem 1.53. [116] Let A(s) ∈ R(s)p×m with rankR(s) A(s) = r, r ≤ min{p, m}. Then C A(s) is equivalent to a diagonal matrix SA(s) (s) having the form r (s) C ) SA(s) (s) = diag ( ψ11(s) (s) , ..., ψr (s) , 0p−r,m−r

(1.73)

where i (s), ψi (s) ∈ R[s] are monic and coprime and satisfy i (s)∣i+1 (s) , i = 1, ..., r − 1 ψi+1 ∣ψi (s)

(1.74)

C SA(s) (s) is called the Smith-McMillan form of A(s) in C. The rational functions

fi (s) =

i (s) ψi (s)

are called the invariant rational functions of A(s). The zeros of A(s) are

defined as the zeros of the polynomials i (s) and the poles are defined as the zeros of ψi (s). Remark 1.54. A method of obtaining the Smith-McMillan form of a rational matrix is through the use of Remark 1.29. By writing the matrix in its standard form A(s) =

1 N (s) d(s)

(1.75)

we can compute the Smith form of N (s) and get C SA(s) (s) =

1 C S (s) d(s) N (s)

(1.76)

which will give us the Smith-McMillan form of A(s) after all the cancellations have been carried out.

1.4. RATIONAL MATRICES

39

Example 1.55. The rational matrix s

⎛ (1+s)(s2 −4) A(s) = ⎜ 0 ⎜ ⎝ 0

s2 +2s s+2 1 s+2

1

1 s

⎞ 0 ⎟ ⎟ 1 ⎠

(1.77)

1+s

has the following Smith-Mcmillan Form 1 ⎛ (−2+s)s(1+s)(2+s) ⎜ C SA(s) (s) = ⎜ 0 ⎜ ⎝ 0

0 1 (1+s)(2+s)

0

0⎞ ⎟ 0⎟ ⎟ s2 ⎠

(1.78)

Theorem 1.56. [116] Let T (s) ∈ R(s)p×m be a rational matrix with rankR(s) T (s) = r ≤ min{p, m}. There exist non-unique polynomial matrices A1 (s) ∈ R[s]p×p , B1 (s) ∈ R[s]p×m and non-unique polynomial matrices A2 (s) ∈ R[s]m×m , B2 (s) ∈ R[s]p×m such that T (s) = A1 (s)−1 B1 (s) = B2 (s)A2 (s)−1

(1.79)

This representation is called a left (right) polynomial matrix fraction description (MFD) of T (s). If in addition the matrices A1 (s), B1 (s) are left coprime and the matrices A2 (s), B2 (s) are right coprime, then the representation is called a coprime left (right) MFD. The above factorization has the advantage of separating the pole and zero structure of a rational matrix. Theorem 1.57. [116] Let T (s) ∈ R(s)p×m be a rational matrix and T (s)=A1 (s)−1 B1 (s)= B2 (s)A2 (s)−1 be a left (right) coprime MFD of T (s). Then 1. The zero structure of T (s) is the zero structure of B1 (s) or B2 (s). 2. The pole structure of T (s) is the zero structure of A1 (s) or A2 (s). Theorem 1.58. [116] Let A(s) ∈ R(s)p×m be a rational function with rankR(s) A(s) = r, C r ≤ min{p, m}. Then A(s) is equivalent at infinity to a diagonal matrix SA(s) (s) having

the form ∞ 1 SA(s) (s) = diag (sq1 , ..., sqk , sqˆk+1 , ..., s1qˆr , 0p−r,m−r )

(1.80)

q 1 ≥ q 2 ≥ ⋯ ≥ qk ≥ 0

(1.81)

with

40

CHAPTER 1. MATHEMATICAL PRELIMINARIES qˆr ≥ qˆr−1 ≥ ⋯ ≥ qˆk+1 ≥ 0

(1.82)

∞ The matrix SA(s) (s) is called the Smith-McMillan form at infinity of A(s). The terms

sq1 , ..., sqk are the poles at infinity and the terms sqˆk+1 , ..., sqˆr are the zeros at infinity of A(s). Definition 1.59. [116] The Mc-Millan degree δM (T ) of a rational matrix T (s) is equal to the total number of its poles, finite ones and those at infinity, with orders accounted for. For any rational function f (s) =

n(s) d(s)

define the map δ∞ (⋅) ∶ R(s) → Z ∪ {+∞} as:

δ∞ (f (s)) = deg d(s) − deg n(s)

f (s) ≠ 0

(1.83)

δ∞ (f (s)) = +∞

f (s) ≡ 0

(1.84)

The above map offers great help in the analysis of rational matrices2 . For example, it can be of use in the computation of the Smith form at infinity. Lemma 1.60. [116] Let A(s) ∈ R(s)p×m be a rational function with rankR(s) A(s) = r. Denote by ξi (A) the least δ∞ (⋅) among the δ∞ (⋅) of all submatrices of A(s) of dimension i, with ξ0 (A) = 0 and define the rational functions: f1 (s) = sξ0 (A)−ξ1 (A) f2 (s) = sξ1 (A)−ξ2 (A) ⋮ fr (s) = sξr−1 (A)−ξr (A)

(1.85)

The Smith form at infinity of A(s) is ∞ SA(s) (s) = diag(f1 , f2 , ..., fr , 0p−r,m−r )

Definition 1.61. [43] Let f (s) =

n(s) d(s)

(1.86)

be a rational function and deg n(s) = n, deg d(s) =

d. If δ∞ (f (s)) = d − n < 0 then we say that f (s) has a pole at infinity of multiplicity d − n and if δ∞ (f (s)) = d − n > 0 we say that f (s) has a zero at infinity of multiplicity d − n. Remark 1.62. [43] Proper and strictly proper functions have no poles at infinity. 2

In some books, like [43] the above map is called the order of a rational function. We will discard

this terminology in order to avoid confusion with the order of a polynomial matrix.

1.5. REALIZATION OF RATIONAL MATRICES

1.5

41

Realization of Rational Matrices

This section presents results from the realization theory of rational matrices. Definition 1.63. [116] Let T (s) ∈ Rpr (s)m×r . Then a quadruple of matrices (C, J, B, E) with C ∈ Rm×p , J ∈ Rp×p , B ∈ Rp×r , E ∈ Rm×r , q ∈ Z+ such that T (s) = C(sIp − J)−1 B + E

(1.87)

is called a realization of T (s). The realization is called minimal if J has the minimal possible size p, or equivalently if p = δM (T ). Theorem 1.64. [116] Let T (s) = A1 (s)−1 B1 (s) = B2 (s)A2 (s)−1 = C(sIn − J)−1 B + E ∈ Rpr (s)m×r be a proper rational matrix, with A1 (s) ∈ Rm×m , B1 (s) ∈ R[s]m×r left coprime and B2 (s) ∈ R[s]m×r , A2 (s) ∈ R[s]r×r right coprime and (C, J, B, E) a realization of T (s), with C ∈ Rm×n , J ∈ Rn×n , B ∈ Rn×r , E ∈ Rm×r . The realization (C, J, B, E) is a minimal realization of T (s) if and only if the following equivalent conditions hold: 1. n = δM (T ) = deg det A1 (s) = deg det A2 (s). 2. The pairs (J, B) and (J, C) are respectively controllable and observable. 3. ⎛ C ⎞ ⎜ CJ ⎟ ⎜ ⎟ ⎟=n rank (B, JB, ...J n−1 B) = rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝CJ n−1 ⎠

(1.88)

The following theorem gives a formula for the realization of the inverse of a polynomial matrix A(s) in terms of its finite and infinite Jordan Pairs. The theory of Jordan Pairs, their properties, as well as a procedure to compute them, will be analytically presented in later chapters. For that matter, we only present now a short list of their properties that are useful for defining a realization of A(s)−1 . Theorem 1.65. [33] Let (CF , JF ) and (C∞ , J∞ ) be the finite and infinite Jordan Pairs of A(s), with CF ∈ Rr×n , JF ∈ Rn×n , C∞ ∈ Rr×µ , J∞ ∈ Rµ×µ , and µ = rq − n. These pairs satisfy the following properties: 1. deg det A(s) = n

42

CHAPTER 1. MATHEMATICAL PRELIMINARIES 2. det sq A(s−1 ) has a zero at λ = 0 with multiplicity µ. i =0 3. ∑qi=0 Ai CF JFi = 0, ∑qi=0 Aq−i C∞ J∞

4. ⎛ CF ⎞ ⎜ CF JF ⎟ ⎜ ⎟ ⎟ = n, rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝CF JFq−1 ⎠

⎛ C∞ ⎞ ⎜ C∞ J∞ ⎟ ⎜ ⎟ ⎟=µ rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ q−1 ⎠ ⎝C∞ J∞

(1.89)

In addition, a realization of A(s)−1 is given by −1

A(s)

−1

= (CF

⎞ ⎛ BF ⎞ ⎛sIn − JF 0 C∞ ) ⎝ 0 sJ∞ − Iµ ⎠ ⎝B∞ ⎠

(1.90)

with BF ∈ Rn×r , B∞ ∈ Rµ×r and −1

⎛ BF ⎞ ⎛In = ⎝B∞ ⎠ ⎝ 0

q−2 C∞ J ∞ ⎞ ⎛ CF ⎟ ⎜ ⋮ ⋮ ⎟ 0 ⎞⎜ ⎟ ⎜ q−2 ⎟ ⎜ q−1 ⎜ CF J C ∞ ⎟ F J∞ ⎠ ⎜ ⎟ q−1 ⎜ q−1 q−1−i ⎟ A C J − A C J ∑ i ∞ ∞ ⎠ ⎝ q F F

⎛0⎞ ⎜⋮⎟ ⎜ ⎟ ⎝Ir ⎠

(1.91)

i=0

The above realization is not minimal, since the realization of the polynomial part of A(s)−1 given by (C∞ , J∞ , B∞ ) does not have the minimal possible dimension, which qi + 1) ≤ µ. The following theorem presents is shown in [116] to be equal to µ ˆ = ∑ri=u+1 (ˆ a formula for a minimal realization of A(s)−1 . Theorem 1.66. [116] The inverse of A(s) can be decomposed as ˆ∞ , A(s)−1 = Hsp (s) + Hpol (s) = C(sIn − J)−1 B + Cˆ∞ (Iµˆ − sJˆ∞ )−1 B

(1.92)

ˆ∞ ) are the minimal realizations of the where the matrix triple (C, J, B) and (Cˆ∞ , Jˆ∞ , B strictly proper and polynomial parts of A(s)−1 respectively, given by Hsp (s) = C(sIn − J)−1 B

(1.93)

ˆ∞ s−1 Hpol (s−1 ) = Cˆ∞ (sIµˆ − Jˆ∞ )−1 B

(1.94)

ˆ∞ Hpol (s) = Cˆ∞ (Iµˆ − sJˆ∞ )−1 B

(1.95)

and

or equivalently

1.5. REALIZATION OF RATIONAL MATRICES

43

ˆ∞ ∈ Rµˆ×r , where the matrix with C ∈ Rr×n , J ∈ Rn×n , B ∈ Rn×r , Cˆ∞ ∈ Rr×ˆµ , Jˆ∞ ∈ Rµˆ×ˆµ , B J can be assumed to be in Jordan form and Jˆ∞ = blockdiag(J∞,r , J∞,r−1 , ⋯, J∞,u+1 )

(1.96)

and J∞,i are nilpotent matrices of the form ⎛0 ⎜0 ⎜ J∞,i = ⎜ ⎜0 ⎜ ⎝0

1 0 0 0

⋯ ⋱ ⋱ ⋯

0⎞ ⋮⎟ ⎟ ⎟ ∈ R(ˆqi +1)×(ˆqi +1) , i = u + 1, ..., r, 1⎟ ⎟ 0⎠

(1.97)

qi + 1). It holds that and µ ˆ = ∑ri=u+1 (ˆ (sIn − J)

= s−1 I + s−2 J + s−3 J 2 + . . .

(1.98a)

−1 qˆr 2 (Iµˆ − sJˆ∞ ) = I + sJˆ∞ + s2 Jˆ∞ . + ... + sqˆr Jˆ∞

(1.98b)

−1

There are many available techniques for obtaining a minimal realization of a strictly ˆ∞ ) proper matrix that can be used to compute the matrices (C, J, B) and (Cˆ∞ , Jˆ∞ , B from the formulas (1.93), (1.94), see for example [29], [47, Chapter 8], [116, Section 1.11]. For the minimal realization of the strictly proper part Hsp (s) we prove the following result. Theorem 1.67. Given a minimal realization of the strictly proper part of A(s)−1 , (C, J, B) and the realization (CF , JF , BF ) as in Theorem 1.65, it holds that ⎛ C ⎞ ⎜ CJ ⎟ ⎜ ⎟ ⎟=n rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝CJ q−1 ⎠

(1.99)

Proof. We consider two cases. If n ≤ q, then (1.99) is straightforward from (1.88), by using the Cayley-Hamilton Theorem. Now, let n > q. From Theorems 1.65 and 1.66, the strictly proper part of A(s)−1 is given by −1 −1 A(s)−1 sp = CF (sIn − JF ) BF = C(sIn − J) B

(1.100)

it holds that CJ i B = CF JFi BF ,

i = 0, 1, 2, . . .

(1.101)

44


In view of (1.101) it is easy to verify that ⎛ C ⎞ ⎛ CF ⎞ ⎜ CJ ⎟ ⎜ CF JF ⎟ ⎜ ⎜ ⎟ ⎟ n−1 ⎜ ⎟ (B, JB, ...J n−1 B) = ⎜ ⎟ (B )⇔ ⎜ ⋮ ⎟ ⎜ ⋮ ⎟ F , JF BF , ...JF BF ⎜ ⎟ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ⎜ ⎟ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ Ln ˆn ⎝CJ n−1 ⎠ ⎝CF JFn−1 ⎠ L ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ Mn

(1.102)

ˆn M

ˆ nL ˆ n. Mn Ln = M

(1.103)

Now, since from (1.88), rankLn = n, it follows that Ln LTn is square and invertible. Post-multiplying (1.103) by LTn and in turn by (Ln LTn )−1 , yields ˆ nW Mn = M

(1.104)

ˆ n LTn (Ln LTn )−1 ∈ Rn×n . It is easy to check that if x ∈ Rn is chosen such that where W = L W x = 0, then Mn x=0, which implies that x = 0, since from (1.88), rankMn = n. Thus, W is invertible. Now, since n > q, keeping only the q first block rows of (1.104) reads ⎛ C ⎞ ⎛ CF ⎞ ⎜ ⋮ ⎟ = ⎜ ⋮ ⎟W ⎟ ⎟ ⎜ ⎜ ⎝CJ q−1 ⎠ ⎝CF J q−1 ⎠ F ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ Mq

(1.105)

ˆq M

ˆq = Taking into account (1.89) and the invertibility of W it follows that rankMq = rankM n.

1.6

Laurent Expansion of a Rational Matrix

In the present section the Laurent expansion of a rational matrix at infinity is presented. We first give the form of the Laurent expansion for a rational function. Definition 1.68. [116] Let f (s) =

n(s) d(s)

be a rational function. Applying long division

between the numerator and denominator, we get 1 1 f (s) = fv sv + fv−1 sv−1 + ... + f1 v + f0 + f−1 + f−2 2 + ... s s

(1.106)

The above formula for f (s) is called the Laurent expansion at infinity of the rational function, where fi ∈ R and fv ≠ 0. It is proven that v = −δ∞ (f (s))

(1.107)

1.6. LAURENT EXPANSION OF A RATIONAL MATRIX Remark 1.69. [43] If the rational function f (s) =

n(s) d(s)

45

is proper, the Laurent expansion

at infinity is 1 1 f (s) = f0 + f−1 + f−2 2 + ... s s

(1.108)

with f0 ≠ 0. If it is strictly proper, then f0 = 0. In a similar fashion, we can define the Laurent expansion at infinity of the inverse of a polynomial matrix. Lemma 1.70. [116] Let A(s) ∈ R[s]r×r as in (1.19). Its inverse matrix A(s)−1 ∈ R(s)r×r is a rational matrix, with Laurent expansion at infinity is given by A(s)−1 = Hpol (s) + Hsp (s) = Hqˆr sqˆr + ⋅ ⋅ ⋅ + H1 s + H0 + H−1 s−1 + H−2 s−2 + . . . ,

(1.109)

where qˆr is the maximum order amongst the orders of zeros at infinity of A(s) as in (1.48). The matrix sequence Hi ∈ Rr×r is known as the (forward) fundamental matrix sequence of A(s) . Lemma 1.71. By equating the coefficients of the powers of si in the two expressions for A(s)−1 given in (1.92) and (1.109), taking into account (1.98), we obtain i ˆ Hi = Cˆ∞ Jˆ∞ B∞

H−i = CJ i−1 B

i = 0, 1, 2, ..., qˆr

(1.110a)

i = 1, 2, . . .

(1.110b)

Remark 1.72. The fundamental matrix sequence Hi of A(s) can be effectively computed using the technique proposed in [28]. The Laurent expansion is also defined for matrix pencils. Lemma 1.73. Let (sE − A) ∈ R[s]r×r . Its inverse matrix (sE − A)−1 ∈ R(s)r×r is a rational matrix, with Laurent expansion at infinity given by ∞

(sE − A)−1 = s−1 ∑ Φi s−i = Φ−h sh−1 + ... + Φ−1 s0 + Φ0 s−1 + Φ1 s−2 + ...

(1.111)

i=−h

where h ≤ rankE −deg det(sE −A)+1 is the index of nilpotency of the pair (E, A). The sequence of matrices Φi ∈ Rr×r is known as the (forward) fundamental matrix sequence of (sE − A) . Remark 1.74. The fundamental matrix sequence Φi of (sE − A) can be computed by the methods presented in [15, 41, 82, 105].

46

CHAPTER 1. MATHEMATICAL PRELIMINARIES The matrices Φi satisfy the following properties.

Theorem 1.75. [62] Let (sE−A) regular and Φi as in (1.111). The following properties hold Φi E − Φi−1 A = Iδi

(1.112a)

EΦi − AΦi−1 = Iδi

(1.112b)

Φi = 0, i < −h ⎧ (Φ0 A)i Φ0 , i≥0 ⎪ ⎪ ⎪ ⎪ i Φi = ⎨ Φ0 (AΦ0 ) , i>0 ⎪ ⎪ ⎪ −i−1 ⎪ ⎩ (−Φ−1 E) Φ−1 , i < 0 Φi EΦj = Φj EΦi , ∀i, j ⎧ −Φi+j , i < 0, j < 0 ⎪ ⎪ ⎪ ⎪ Φi EΦj = ⎨ Φi+j , i ≥ 0, j ≥ 0 ⎪ ⎪ ⎪ ⎪ otherwise ⎩ 0, ⎧ −Φi+j+1 , i < 0, j < 0 ⎪ ⎪ ⎪ ⎪ Φi AΦj = ⎨ Φi+j+1 , i ≥ 0, j ≥ 0 ⎪ ⎪ ⎪ ⎪ otherwise ⎩ 0, ⎧ ⎪ ⎪ Φi , i ≥ 0 Φ0 EΦi = ⎨ ⎪ ⎪ ⎩ 0, i < 0 ⎧ ⎪ i≥0 ⎪ 0, −Φ−1 EΦi = ⎨ ⎪ Φi−1 , i < 0 ⎪ ⎩ ⎧ ⎪ ⎪ Φi+1 , i ≥ 0 Φ0 AΦi = ⎨ ⎪ i 0 and an admissible control sequence u(0), ..., u(k + h − 1) that can drive the system from the origin x(0) = 0 to z, that is, x(k) = z. Definition 2.10. [67] The descriptor system (2.1) is reachable from the origin if every state z ∈ Rr is reachable. Definition 2.11. [55] Denote by R(0) be the set of all reachable states, that is R(0) = {z ∈ Rr ∣ ∃u(0), ..., u(k + µ − 1) such that x(0) = 0, x(k) = z}

(2.49)

Definition 2.12. [55] The causal reachability Grammian is defined as k−1

Wc (0, k) = ∑ Φk−i−1 BB T ΦTk−i−1 = (Φ0 B ... Φk−1 B) (Φ0 B ... Φk−1 B)

T

(2.50)

i=0

The noc-ncausal reachability Grammian is defined as −1

Wnc (−1, −h) = ∑ Φi BB T ΦTi = (Φ−h B ... Φ−1 B) (Φ−h B ... Φ−1 B)

T

(2.51)

i=−h

and the reachability Grammian is defined as W (−h, k) = Wnc (−1, −h) + Wc (0, k)

(2.52)

58

CHAPTER 2. DESCRIPTOR SYSTEMS

Lemma 2.13. [55] Introduce the following notation ⟨Φ/ Im B⟩ = Φn−1 Im B + Φn−2 Im B + ... + Φ0 Im B

(2.53)

¯ Im B⟩ = Φ1 Im B + Φ2 Im B + ... + Φ−h Im B ⟨Φ/

(2.54)

for the above subspaces, the following hold ⟨Φ/ Im B⟩ = Im Wc (0, k), k ≥ n

(2.55)

¯ Im B⟩ = Im Wnc (−1, −h) ⟨Φ/

(2.56)

now we can describe the reachable subspace of (2.1) in terms of the matrices Φi . Theorem 2.14. [55] The reachable subspace of (2.1) is given by ¯ Im B⟩ R(0) = ⟨Φ/ Im B⟩ ⊕ ⟨Φ/

(2.57)

where ⊕ denotes the direct sum of two vector spaces. Moreover, an admissible input that can drive the system from the origin x(0) = 0 to a vector z in the reachable subspace in n + h steps is given by ⎧ B T ΦT−i−1 w0 0≤i≤h−1 ⎪ ⎪ ⎪ ⎪ T T u(i) = ⎨ B Φn+h−i−1 w1 h ≤ i ≤ n + h − 1 ⎪ ⎪ ⎪ T T ⎪ ⎩ B Φn+h−i−1 w2 n + h ≤ i ≤ n + 2h − 1

(2.58)

where wi are appropriate vectors. The system will be reachable if the reachable subspace coincides with Rr , that is R(0) ≡ Rr . This brings us to the following result. Theorem 2.15. The system (2.1) will be reachable iff R(0) ≡ Rr

(2.59)

or equivalently considering (2.50), (2.51) and (2.52), iff rankRdescriptor = rank (Φ−h B ⋯ Φn−1 B) = r

(2.60)

rankW (−h, n) = r

(2.61)

or

2.3. REACHABILITY

59

Example 2.16. [55] Consider the following system ⎛2 1 0⎞ ⎛−4 −2 0⎞ ⎛1⎞ ⎜0 0 1⎟ x(k + 1) = ⎜ 3 1 1⎟ x(k) + ⎜−1⎟ u(k) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 2⎠ ⎝ 6 2 1⎠ ⎝2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ² E

(2.62)

B

A

with r = 3, n = deg det(σE − A) = 1, h = 2 = rankE − deg det(σE − A) + 1. The matrices Φi are ⎛0 −2 1 ⎞ ⎛0 1 −1⎞ ⎛−1 0 0⎞ ⎟ ⎜ ⎟ ⎜ ⎟ Φ−2 = ⎜ ⎜0 4 −2⎟ , Φ−1 = ⎜0 −2 2 ⎟ , Φ0 = ⎜ 3 0 0⎟ ⎝0 0 0 ⎠ ⎝0 −2 1 ⎠ ⎝ 0 0 0⎠

(2.63)

⎛−(−2)i 0 0⎞ i ⎟ Φi = ⎜ ⎜ 3(−2) 0 0⎟ , i > 0 ⎝ 0 0 0⎠

(2.64)

The reachability grammians are −1

Wnc (−1, −2) = ∑ Φi BB

T

ΦTi

i=−2

⎛1

0

Wc (0, 1) = ∑ Φ1−i−1 BB i=0

⎛ 25 −50 −12⎞ ⎟ =⎜ ⎜−50 100 24 ⎟ ⎝−12 24 16 ⎠

T

ΦT1−i−1 ⎜ ⎜−3 ⎝0

−3 0⎞ 9 0⎟ ⎟ 0 0⎠

(2.65)

(2.66)

and ⎛ 26 −53 −12⎞ ⎟ rankW (−2, 1) = rank (Wnc (−1, −2) + Wc (0, 1)) = rank ⎜ ⎜−53 109 24 ⎟ = 3 ⎝−12 24 16 ⎠

(2.67)

so the system is reachable. The admissible initial values of the input need to satisfy Φ−2 Bu(1) + Φ−1 Bu(0) = 0

(2.68)

A consistent input that drives the system to any vector z = (z1 , z2 , z3 )T ∈ R3 in 3 steps is given by 3 1 3 1 u(0) = 0, u(1) = 0, u(2) = 2z1 + z2 , u(3) = z3 , u(4) = z1 + z2 + z3 4 4 4 16

(2.69)

and so the state at k = 3 is x(3) = Φ2 Bu(0) + Φ1 Bu(1) + Φ0 Bu(2) + Φ−1 Bu(3) + Φ−2 Bu(4) = z

(2.70)

60

CHAPTER 2. DESCRIPTOR SYSTEMS Similarly to the causal and noncausal reachability Grammians, we can define the

reachability Grammians for the causal and noncausal subsystems. Definition 2.17. The causal subsystem reachability Grammian is defined as k−1

T

k−1 Wcausal (0, k) = ∑ Ak−i−1 B1 B1T Ak−i−1 = (Ak−1 1 1 1 B1 ⋯ B1 ) (A1 B ⋯ B1 )

T

(2.71)

i=0

The reachability Grammian for the noncausal subsystem is defined as −1

T

Wnoncausal (−1, −h) = ∑ N −i−1 B2 B2T N −i−1 = (N h−1 B2 ⋯ B2 ) (N h−1 B2 ⋯ B2 )

T

i=−h

(2.72) Theorem 2.18. For the causal reachability Grammian it holds Im Wcausal (0, k) = Im Wcausal (0, n), k ≥ n

(2.73)

Proof. Since Im Wcausal (0, n) is spanned by the independent columns of w0,n = (An−1 1 B ⋯ B1 )

(2.74)

T ker Wcausal (0, k)T = ker w0,n , k≥n

(2.75)

So it suffices to show that

Now, since Wcausal (0, k) is symmetric, Wcausal (0, k) = Wcausal (0, k)T , the above is equivalent to n−1

T

T ker Wcausal (0, k) = ker w0,n ⇔ ker Wcausal (0, k) = ker ⋂ B1T Ai1 , k ≥ n

(2.76)

i=0

n−1

First, we show that ker Wcausal (0, k) ⊆ ker ⋂ B1T Ai1 for k ≥ n. Let x ∈ ker Wcausal (0, k), i=0

k ≥ n with x ≠ 0. It holds k−1

k−1

T

T

xT Wcausal (0, k)x = 0 ⇒ xT ∑ Ak−i−1 B1 B1T A1k−i−1 x = 0 ⇒ ∑ ∣∣B1T Ak−i−1 x∣∣2 = 0 (2.77) 1 1 i=0

i=0

for k ≥ n. Now, for k = n the above is equal to n−1

T

T

x∣∣ = 0 ⇒ B1T An−i−1 x, ∑ ∣∣B1T An−i−1 1 1

i = 0, ..., n − 1

(2.78)

i=0 n−1

n−1

T

T

This implies that x ∈ ker ⋂ B1T Ai1 and thus ker Wcausal (0, k) ⊆ ker ⋂ B1T Ai1 for k ≥ n. i=0

i=0

n−1

For the reverse inclusion, let x ∈ ker ⋂

i=0

T B1T Ai1 .

This is equal to T

B1T x = B1T AT1 x = ... = B1T An−1 x=0 1

(2.79)

2.3. REACHABILITY

61

Now, it is known according to the Cayley-Hamilton theorem that each matrix satisfies its characteristic polynomial. So for the matrix A1 with characteristic polynomial p(λ) = det(λIn − A1 ) = λn + pn−1 λn−1 + ... + p0

(2.80)

p(A1 ) = 0 ⇒ An1 = −pn−1 An−1 − ... − p0 In 1

(2.81)

it holds

or equivalently T

An1 T = −pn−1 An−1 − ... − p0 In 1

(2.82)

multiplying the above equation from the left by B1T and from the right by x we obtain T

B1T An1 T x = −pn−1 B1T An−1 x − ... − p0 B1T x = 0 1

(2.83)

so x ∈ ker B1T An1 T . By first multiplying (2.82) by A1 , it can be shown in the same way T

that x ∈ ker B1T Ak1 , k > n. So overall, k−1

T

B1 B1T A1k−i−1 x = 0 Wcausal (0, k)x = ∑ Ak−i−1 1

(2.84)

i=0 n−1

T

and ker Wcausal (0, k) ⫆ ker ⋂ B1T Ai1 . i=0

Using the above results, we can derive a result for the reachable subspace for the causal and noncausal subsystems. Theorem 2.19. The reachable subspace of the causal subsystem is given by Rcausal (0) = Im Wcausal (0, n)

(2.85)

and the reachable subspace of the noncausal subsystem is given by Rnoncausal (0) = Im Wnoncausal (−1, −h)

(2.86)

⎛ Wcausal (0, n) ⎞ ⎝Wnoncausal (−1, −h)⎠

(2.87)

or equivalently Rc−nc (0) = Im

T

Proof. Initially, we proove that Rc−nc (0) ⊆ Im (Wcausal (0, n)T Wnoncausal (−1, −h)T ) . Let z ∈ Rr and assume that z ∈ Rc−nc (0). This means that there exists an admissible input sequence u(0), ..., u(k0 + h) and k0 such that z=

⎛z1 ⎞ ⎛x˜1 (k0 )⎞ = ⎝z2 ⎠ ⎝x˜2 (k0 )⎠

(2.88)

62


with x˜1 (0) = 0, x˜2 (0) = 0. This means that k0 −1

z1 = x˜1 (k0 ) = ∑

Ak10 −i−1 B1 u(i)

=

(Ak10 −1 B1

i=0

h−1

z2 = x˜2 (k0 ) = − ∑ N B2 u(k0 + i) = − (B2 i

i=0

⎛ u(0) ⎞ ⎟ ⋯ B1 ) ⎜ ⋮ ⎜ ⎟ ⎝u(k0 − 1)⎠

(2.89)

⎛ u(k0 ) ⎞ ⎟ (2.90) ⋯ N h−1 B2 ) ⎜ ⋮ ⎜ ⎟ ⎝u(k0 + h − 1)⎠

From the above formulas, it is clear that z1 ∈ Im Wcausal (0, k0 ) = Im Wcausal (0, n) and T

z2 ∈ Im Wnoncausal (−1, −h). Thus Rc−nc (0) ⊆ Im (Wcausal (0, n)T Wnoncausal (−1, −h)T ) . Now to prove the reverse induction, Rc−nc (0) ⊇ Im (Wcausal (0, n)T Wnoncausal (−1, −h)T ) let z=

⎛z1 ⎞ ⎛ Wcausal (0, n) ⎞ ∈ Im ⎝z2 ⎠ ⎝Wnoncausal (−1, −h)⎠

(2.91)

with z1 ∈ Im Wcausal (0, n) and z2 ∈ Im Wnoncausal (−1, −h). We need to construct an admissible input such that for a time k0 , x˜1 (k0 ) = z1 and x˜2 (k0 ) = z2 , with x˜1 (k) = 0, x˜2 (0) = 0. We will show that this is possible in n + h steps, as shown in Theorem 2.14 and [55]. First, we need to construct an input that satisfies the consistency condition h−1

x ˜2 (0)=0

x˜2 (0) = − ∑ N i B2 u(i) = −B2 u(0) − N B2 u(1) − .... − N h−1 B2 u(k − 1) ⇒ (2.92) i=0

0 = − (B2

⎛ u(0) ⎞ ⎟ ⋮ N B2 ⋯ N h−1 B2 ) ⎜ ⎟ ⎜ ⎝u(h − 1)⎠

(2.93)

Let w0 ∈ ker (B2 N B2 ⋯ N h−1 B2 ),where ⎛ w0,0 ⎞ ⎟ w0 = ⎜ ⎜ ⋮ ⎟ ⎝w0,h−1 ⎠

(2.94)

by setting u(i) = w0,i ,

i = 0, ..., h − 1

(2.95)

the consistency condition is satisfied. Now consider the noncausal subsystem. The state for k0 = n + h is given by n+h−1

x˜1 (n + h)= ∑ An+h−i−1 B1 u(i) = 1 i=0

= An+h−1 B1 u(0) + ... + An1 B1 u(h − 1) + An−1 1 1 B1 u(h) + ... + B1 u(n + h − 1)

T

2.3. REACHABILITY

63

h−1

n+h−1

i=0

i=h

= ∑ An+h−i−1 B1 u(i) + ∑ An+h−1−i B1 u(i) 1 1

(2.96)

n+h−i−1 B1 u(i). From Theorem 2.18 and the Cayley-Hamilton theNow, let γ = ∑h−1 i=0 A1

orem, it holds that γ ∈ Im Wcausal (0, n) and thus z1 − γ ∈ Im Wcausal (0, n). Thus, there exists a vector w1 ∈ Rn such that n−1

z1 − γ = ∑ An−i−1 B1 B1T (A1n−i−1 )w1 1

(2.97)

i=0

Based on the above relation, by setting T

u(i) = B1T A1n+h−i−1 w1 ,

i = h, ..., n + h − 1

(2.98)

we get n+h−1

T

x˜1 (n + h) = γ + ∑ A1n+h−1−i B1 B1T An+h−i−1 w1 = γ + z1 − γ = z1 1

(2.99)

i=h

and thus Rcausal (0) ⊆ Im Wcausal (0, n). Now for the noncausal subsystem, the output is h−1

x˜2 (n + h) = − ∑ N i B2 u(n + h + i) = −B2 u(n + h) + ... + N h−1 B2 u(n + 2h − 1) (2.100) i=0

since z2 ∈ Im Wnoncausal (−1, −h), there exists w2 ∈ Rµ such that −1

− z2 = ∑ N i−1 B2 B2T (N −i−1 )T w2

(2.101)

i=−h

thus, by setting T

u(i) = B2T N i−n−h w2 ,

i = n + h, ..., n + 2h − 1

(2.102)

we get h−1

T

x˜2 (n + h) = − ∑ N i B2 B2T N i w2 = z2

(2.103)

i=0

and thus Rnoncausal (0) ⊆ Wnoncausal (−1, −h), and the reverse induction holds. Overall, the admissible input that transfers the causal and noncausal subsystems from the origin to any state in the reachable subspace is given by ⎧ w0,i i = 0, ..., h − 1 ⎪ ⎪ ⎪ ⎪ T n+h−i−1 T u(i) = ⎨ B1 A1 w1 i = h, ..., n + h − 1 ⎪ ⎪ T ⎪ T i−n−h w ⎪ i = n + h, ..., n + 2h − 1 2 ⎩ B2 N

(2.104)

64


Theorem 2.20. The system (2.1) will be reachable iff Rc−nc (0) ≡ Rr

(2.105)

or equivalently considering (2.71), (2.72) iff rankRcausal = rank (An−1 1 B1 ⋯ B1 ) = n

(2.106)

rankRnoncausal = rank (N h−1 B2 ⋯ B2 ) = µ

(2.107)

and

Finally, we give the connection between the reachability criteria of the original descriptor system and its causal and noncausal subsystems. Theorem 2.21. The reachability criteria for the descriptor system and its corresponding causal and noncausal subsystems are equivalent. Proof. Consider the reachability matrix of the descriptor system Rdescriptor . Taking into account the transformation (2.13), the matrix Rdescriptor is equal to (2.13)

Rdescriptor = (Φ−h B ⋯ Φn−1 B) =

⎛ ⎛B1 ⎞ ⎛B1 ⎞ ⎛B1 ⎞ ⎛B1 ⎞⎞ (1.113) = Φ−h P −1 ⋯ Φ−1 P −1 Φ0 P −1 ⋯ Φn−1 P −1 = ⎝ ⎝B2 ⎠ ⎝B2 ⎠ ⎝B2 ⎠ ⎝B2 ⎠⎠ =Q

⎛⎛0n ⎛0n 0 ⎞⎛B1 ⎞ ⎛In 0 ⎞⎛B1 ⎞ ⎛A1n−1 0 ⎞ ⎛B1 ⎞⎞ 0 ⎞ ⎛B1 ⎞ ⋯ ⋯ ⎝⎝ 0 −N h−1 ⎠ ⎝B2 ⎠ ⎝ 0 −Iµ ⎠⎝B2 ⎠ ⎝ 0 0µ ⎠⎝B2 ⎠ ⎝ 0 0µ ⎠ ⎝B2 ⎠⎠

=Q

⎛ 0 ⋯ 0 B1 ⋯ An−1 1 B1 ⎞ ⎝−N h−1 B2 ⋯ −B2 0 ⋯ 0 ⎠

(2.108)

From the above, since Q is invertible, it is clear that the rank conditions rankRdescriptor = r and rankRcausal = n, rankRnoncausal = µ are equivalent. Example 2.22. Consider again the system from Example 2.16 ⎛2 1 0⎞ ⎛−4 −2 0⎞ ⎛1⎞ ⎜0 0 1⎟ x(k + 1) = ⎜ 3 1 1⎟ x(k) + ⎜−1⎟ u(k) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 2⎠ ⎝ 6 2 1⎠ ⎝2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ² E

A

(2.109)

B

with r = 3, n = deg det(σE − A) = 1, h = 2 = rankE − deg det(σE − A) + 1. The matrices that give the decomposition into the causal and noncausal subsystems are 1 1 ⎛3 0 0 ⎞ ⎛− 3 − 2 0 ⎞ ⎟ ⎟ ⎜ P =⎜ ⎜0 2 −2⎟ , Q = ⎜ 1 1 0 ⎟ ⎝0 −4 2 ⎠ ⎝ 0 0 −1⎠ 2

(2.110)

2.3. REACHABILITY

65

with ⎛ 1 0 0 P EQ = ⎜ ⎜ 0 0 1 ⎝ 0 0 0

⎞ ⎛ −2 0 0 ⎟ , P AQ = ⎜ 0 1 0 ⎟ ⎜ ⎠ ⎝ 0 0 1

⎞ ⎛3⎞ ⎟ , P B = ⎜−6⎟ ⎟ ⎜ ⎟ ⎠ ⎝8⎠

(2.111)

and the resulting subsystems are x˜1 (k + 1) = −2 x˜1 (k) + 3 u(k) ¯ ® A1

(2.112)

B1

⎛−6⎞ ⎛0 1⎞ u(k) x˜2 (k + 1) = x˜2 (k) + ⎝8⎠ ⎝0 0⎠ ² ´¹¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¶

(2.113)

B2

N

The reachability matrices for each subsystem are Rcausal = B1 = 3 Rnoncausal = (N B2 B2 ) =

(2.114) ⎛8 −6⎞ ⎝0 8 ⎠

(2.115)

with rankRcausal = 1, rankRnoncausal = 2. Thus, the system is reachable, as expected. The reachability Grammians are Wcausal (0, 1) = B1 B1T = 9 Wnoncausal (−1, −2) = (N B2

⎛B T N T ⎞ B2 ) 2 T ⎝ B2 ⎠

(2.116) =

⎛ 100 −48⎞ ⎝−48 64 ⎠

(2.117)

Now, consider a vector z ∈ R3 with ⎛z1 ⎞ ⎟ z=⎜ ⎜z2 ⎟ ⎝z3 ⎠

(2.118)

with z1 ∈ R, (z2 , z3 )T ∈ R2 . To construct an input that drives the system to this desired state in k = n + h = 3 steps, first we must choose an input that satisfies the consistency constraint 0 = −B2 u(0) − N B2 u(1)

(2.119)

By simply choosing u(0) = u(1) = 0, the above condition is satisfied. Now the state of each system becomes 2

x˜1 (3) = ∑ A2−i 1 B1 u(i) = B1 u(2)

(2.120)

i=0

x˜2 (3) = −B2 u(3) + −N B2 u(4)

(2.121)

66


Thus, by solving the systems z1 = B1 B1T w1

(2.122)

⎛z2 ⎞ = − (B2 B2T + N B2 B2T N T ) w2 ⎝z3 ⎠

(2.123)

we find z1 w1 = , 9

z2 3z3 ⎞ ⎛ − 64 − 256 w2 = 3z2 25z3 ⎠ ⎝− 256 − 1024

(2.124)

and thus, the input sequence u(0) = u(1) = 0, u(3) = B2T w2 = − z83 ,

u(2) = B1T w1 = z31 , u(4) = B2T N T w2 =

1 32 (−4z2

(2.125)

− 3z3 )

will drive the causal and noncausal subsystems to any desired state in 3 steps.

2.4

Observability

As mentioned in the introduction, observability refers to the determination of the initial value of the system’s state, by knowledge of its input and output values over a finite interval. By using the causal/noncausal decomposition of the descriptor system (2.20)(2.21), we will show that the problem of observability for the descriptor system reduces to the observability of the causal subsystem that is in state space form. To begin with, it is well understood that the state response of (2.1a) is the sum of its homogeneous response due to initial value x(0) and its dynamic response due to the input u(k), that is x(k) = xhom (k)+xdynamic (k). The dynamic response xdynamic (k) is always known, since it depends on the input u(k). Thus, since we are interested in determining the intial value of the system (2.1a) we can always define a new system with x¯(k) = x(k) − xdynamic (k) = xhom (k) and study it instead of the original. As a result, we will focus our attention to the homogeneous case of (2.1a), that is Ex(k + 1) = Ax(k)

(2.126a)

y(k) = Cx(k)

(2.126b)

The state and output of (2.126) are given by x(k) = Φk Ex(0)

(2.127)

2.4. OBSERVABILITY

67 y(k) = CΦk Ex(0)

(2.128)

and the set of consistent initial values is r Had descriptor = {x(0) ∈ R ∣ x(0) = Φ0 Ex(0)}

(2.129)

hom

The causal and noncausal subsystems become x˜1 (k + 1) = A1 x˜1 (k)

(2.130a)

y1 (k) = C1 x˜1 (k)

(2.130b)

N x˜2 (k + 1) = x˜2 (k)

(2.131a)

y2 (k) = C2 x˜2 (k)

(2.131b)

y(k) = y1 (k) + y2 (k)

(2.132)

and

with output

From (2.23) and (2.27), it is easy to derive that the causal and noncausal subsystems for the homogeneous case have solutions x˜1 (k) = Ak1 x1 (0)

(2.133)

x˜2 (k) = 0

(2.134)

so in this case the noncausal subsystem has only the zero solution. This can also be seen by considering (2.131a) for an arbitraty initial value x˜2 (0): N h =0

x˜2 (0) = N x˜2 (1) = N 2 x˜2 (2) = ⋯ = N h x˜2 (h) = 0

(2.135)

The set of consistent initial values for the causal and noncausal subsystems for the homogeneous case is Had x1 (0) ∈ Rn , x˜2 (0) ∈ Rµ ∣ x˜2 (0) = 0} c−nc = {˜

(2.136)

hom

ad and similar to Theorem (2.8), it can be shown that Had descriptor and H c−nc are equivalent. hom

hom

ad Theorem 2.23. The set of consistent initial values Had descriptor and H c−nc for the homohom

geneous system are equivalent.

hom

68


Proof. Using the transformation (2.15), the condition x(0) = Φ0 Ex(0) equal to Q

⎛x˜1 (0)⎞ (2.11) ⎛x˜1 (0)⎞ ⇒ = Φ0 EQ ⎝x˜2 (0)⎠ ⎝x˜2 (0)⎠

(2.137)

Q

⎛x˜1 (0)⎞ ⎛In 0 ⎞ ⎛x˜1 (0)⎞ = Φ0 P −1 ⇒ ⎝x˜2 (0)⎠ ⎝ 0 N ⎠ ⎝x˜2 (0)⎠

(2.138)

⎛x˜1 (0)⎞ ⎛ x˜1 (0) ⎞ (1.113) = Q−1 Φ0 P −1 ⇒ ⎝x˜2 (0)⎠ ⎝N x˜2 (0)⎠

(2.139)

⎛x˜1 (0)⎞ ⎛In 0 ⎞ ⎛ x˜1 (0) ⎞ ⇒ = ⎝x˜2 (0)⎠ ⎝ 0 0µ ⎠ ⎝N x˜2 (0)⎠

(2.140)

⎛x˜1 (0)⎞ ⎛x˜1 (0)⎞ = ⎝x˜2 (0)⎠ ⎝ 0µ ⎠

(2.141)

which is the desired result.

Now, we proceed with the definition of observability Definition 2.24. The first order descriptor system (2.126) is completely observable if the initial value x(0) and consequently the vector Ex(0) can be uniquely determined from knowledge of the output y(k) over a finite time interval. Since for the discrete time case we only consider systems with consistent initial values, from (2.129), it can be seen that knowledge of Ex(0) gives us x(0). Thus, for discrete time desciptor and higher order systems, the notions of complete and weak observability coincide. Since the noncausal subsystem has only the zero solution, from (2.134) it is seen that the output of the noncausal subsystem is y2 (k) = 0, and the output (2.170) of the system is equal to the output of the causal subsystem, y(k) = y1 (k). This means that the initial state of the noncausal subsystem is always known and thus the noncausal subsystem is always observable. This leads us to the following result. Theorem 2.25. The system (2.126) is completely observable if and only if the causal subsystem is observable, or equivalently if rank

⎛σIn − A1 ⎞ =n ⎝ C1 ⎠

(2.142)

2.4. OBSERVABILITY

69

or ⎛ C1 ⎞ ⎜ C1 A1 ⎟ ⎜ ⎟ ⎟=n rankOcausal = rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ n−1 ⎝C1 A1 ⎠

(2.143)

where Ocausal is the observability matrix of the causal subsystem. We have provided a criterion for the observability connecting it with the causal subsystem’s matrices. Now we will connect this to the fundamental matrix sequence of (σE − A)−1 . Theorem 2.26. The observability criterion proposed in (2.143) is equivalent to

rankOdescriptor

⎛ CΦ0 ⎞ ⎜ CΦ1 ⎟ ⎟ ⎜ ⎟=n =⎜ ⎜ ⋮ ⎟ ⎟ ⎜ ⎝CΦn−1 ⎠

(2.144)

Proof. Multiplying the matrix in (2.144) from the right by the nonsingular matrix P −1 we obtain the following 0⎞ ⎛ In ⎜ 0 0µ ⎟ ⎜ ⎟ ⎜ ⎟ ⎛ CΦ0 ⎞ ⎜ ⎟ A 0 CQ ⎟ 1 ⎞⎜ ⎛ ⎜ CΦ1 ⎟ ⎜ ⎟ (2.14) ⎜ ⎟ −1 (1.113) ⎜ ⎟⎜ 0 ⎟ ⎜ ⎟P = ⎜ 0 ⋱ µ⎟ = = ⎟ ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝ CQ⎠ ⎜ ⋮⎟ ⎜ ⋮ ⎟ ⎝CΦn−1 ⎠ ⎜ n−1 ⎟ ⎜A1 ⎟ 0 ⎜ ⎟ ⎝ 0 0µ ⎠

⎛C1 =⎜ ⎜ ⋮ ⎝0

⎛ In 0 ⎞ ⎜0 0 ⎟ ⎜ µ⎟ ⎜ ⎟ ⎛ C1 0⎞ ⎜ C2 ⋯ 0 ⎞ ⎜ A1 0 ⎟ ⎟ ⎜ ⎜ ⎟ C1 A1 0⎟ ⎟ ⎜ 0 0µ ⎟ = ⎜ ⎜ ⎟ ⋱ ⋱ ⋮ ⎟ ⎟⎜ ⎟ ⎜ ⎟ ⋮ ⎜ ⎟ ⎜ ⎟ ⎟ ⋯ C1 C2 ⎠ ⎜ ⋮ ⋮ ⎜ ⎟ ⎝C An−1 0⎠ ⎜ n ⎟ 1 1 ⎜A1 0 ⎟ ⎜ ⎟ ⎝ 0 0µ ⎠

(2.145)

(2.146)

and the first column is the observability matrix Ocausal of the causal subsystem. So far, we have given matrix criteria for the observability of the system, in terms of the fundamental matrix sequence of (σE − A) and in terms of the causal subsystem matrices C1 , A1 . Now we will connect the observability of (2.126) with the system’s compound matrix.

70


Theorem 2.27. The system (2.126) is observable if and only if the matrices (σE − A) and C are right coprime, or equivalently rank

⎛σE − A⎞ =r ⎝ C ⎠

(2.147)

Proof. Since the matrices P, Q are nonsingular, we can multiply the above matrix by them without affecting its rank, so ⎛A1 0 ⎞⎞ ⎛σIn − A1 ⎛⎛In 0 ⎞ ⎛P 0 ⎞ ⎛σE − A⎞ ⎜⎝ 0 N ⎠ σ − ⎝ 0 I ⎠⎟ ⎜ Q=⎜ 0 µ ⎟= ⎜ ⎜ ⎟ ⎝ 0 Ip ⎠ ⎝ Cσ ⎠ (C1 C2 ) ⎝ ⎠ ⎝ C1

0 ⎞ σN − Iµ ⎟ ⎟ C2 ⎠

(2.148)

The second column block of the above matrix always has full column rank equal to µ. A similar result was showcased in [129] and is easy to explain. Since N is nilpotent, it is similar to an upper triangular matrix with zeros in its diagonal. Thus, the matrix σN − Iµ is similar to a matrix with -1 on its diagonal and therefore it has full column rank equal to µ. In addition, the first column of the above matrix is the compound observability matrix of the causal subsystem. Thus, the compound matrix (2.147) has full column rank if and only if the causal subsystem is observable. All of the above results regarding observability are summarized in the following theorem. Theorem 2.28. The following statements are equivalent. 1. The descriptor system (2.126) is completely observable. 2. rankOdescriptor = n. 3. The matrices (σE − A) and C are right coprime. 4. The causal subsystem (2.130) is observable. 5. rankOcausal = n. 6. The matrices (σIn − A1 ) and C1 are right coprime.

2.4. OBSERVABILITY

71

Example 2.29. Consider the following system ⎛1 0 0 0⎞ ⎛−1 −1 −2 0 ⎞ ⎜0 0 0 0⎟ ⎜ 0 1 0 −1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x(k + 1) = ⎜ ⎟ x(k) ⎜2 0 1 0⎟ ⎜ 0 0 −1 −1⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 1 0 0⎠ ⎝0 0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ E

(2.149a)

A

y(k) =

⎛0 2 0 1⎞ x(k) ⎝0 0 0 2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(2.149b)

C

The matrices that give the decomposition into the causal and noncausal subsystems are

⎛−1 −1 21 0⎞ ⎛ 0 12 0 21 ⎞ ⎜ 2 0 0 0⎟ ⎜−2 −1 1 0 ⎟ ⎟ ⎜ ⎟ ⎜ ⎟, Q = ⎜ ⎟ P =⎜ ⎜ 0 1 0 0⎟ ⎜−2 −1 2 1 ⎟ ⎟ ⎜ ⎟ ⎜ ⎝ 2 0 0 1⎠ ⎝ 0 −1 0 0 ⎠

(2.150)

with ⎛1 ⎜0 ⎜ P EQ = ⎜ ⎜0 ⎜ ⎝0

0 1 0 0

0⎞ ⎛1 ⎜0 ⎟ 0⎟ ⎜ ⎟ , P AQ = ⎜ ⎜0 ⎟ 0⎟ ⎜ ⎝0 0⎠

0 0 1 0

CQ =

⎛6 0 0 ⎝4 0 0

0 1 0 0

0 1 1 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠

1⎞ 2⎠

(2.151)

(2.152)

and the resulting subsystems are

⎛1 0 0⎞ ⎟ x˜1 (k + 1) = ⎜ ⎜0 1 1⎟ x˜1 (k), ⎝0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

y1 (k) =

⎛6 0 0⎞ x˜1 (k) ⎝4 0 0⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(2.153a)

C1

A1

and 0 = x˜2 (k), ®

y2 (k) =

N

⎛1⎞ x˜2 (k) = 02×1 ⎝2⎠ ±

(2.154a)

C2

with y(k) = y1 (k) + y2 (k) and the transformation x(k) = Q

⎛x˜1 (k)⎞ ⎝x˜2 (k)⎠

(2.155)

72


ad = {˜ x1 (0) ∈ R3 , x˜2 (0) ∈ R ∣ x˜2 (0) = with x˜1 (k) ∈ R3 , x˜2 (k) ∈ R and admissibility set Hc−nc

0}. The observability matrix of the causal subsystem is ⎛6 ⎜4 ⎜ ⎛ C1 ⎞ ⎜ ⎜6 ⎟=⎜ Ocausal = ⎜ C A 1 1 ⎜ ⎟ ⎜ ⎜4 2 ⎝C1 A1 ⎠ ⎜ ⎜ ⎜6 ⎜ ⎝4

0 0 0 0 0 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎠

(2.156)

and has rankOcausal = 1, so the causal subsystem is not observable and thus the complete system is not observable. So the matrices (σE − A), C are not coprime, since ⎛1 ⎜0 ⎜ ⎜ ⎜0 ⎜ C S (σ) = ⎜ ⎜0 ⎛(σE − A)⎞ ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ⎜0 ⎝ ⎠ C ⎜ ⎝0

2.5

0 1 0 0 0 0

0 0 ⎞ ⎟ 0 0 ⎟ ⎟ ⎟ 1 0 ⎟ ⎟ 2 0 1 − 2σ + σ ⎟ ⎟ ⎟ ⎟ 0 0 ⎟ ⎠ 0 0

(2.157)

Observability of Descriptor Systems with State Lead in the Output

In this section we will study a special case of descriptor systems where there is a state lead in the output. In this special case, the output depends not only on x(k) but also on x(k + 1). Such systems are the discrete time analog of continuous time descriptor systems with a derivative in the output, that have been studied in [108]. Such systems are described by Ex(k + 1) = Ax(k)

(2.158a)

y(k) = C1 x(k) + C2 x(k + 1)

(2.158b)

with E, A ∈ Rr×r , C1 , C2 ∈ Rp×r . As established in the previous section, to study the observability we should simply consider the homogeneous system. The reason for considering this special case of descriptor systems is that in later chapters where the observability of higher order systems will be studied, it will be shown that the higher order system in question can be transformed into a descriptor system of the form (2.158).

2.5. DESCRIPTOR SYSTEMS WITH STATE LEAD IN THE OUTPUT

73

As with the descriptor system (2.1), following [31, 108, 129], there exist nonsingular matrices P, Q ∈ Rr×r such that P EQ = diag(In , N )

(2.159)

P AQ = diag(A1 , Iµ )

(2.160)

C1 Q = (F1 , F2 )

(2.161)

C2 Q = (D1 , D2 )

(2.162)

where A1 ∈ Rn×n , N ∈ Rµ×µ , where µ = r − n, n = deg det(σE − A), F1 , D1 ∈ Rp×n , F2 , D2 ∈ Rp×µ and N is nilpotent with index of nilpotency h. Applying the transformation (2.15) on (2.158) and multiplying the first equation from the left by P we get

P EQ

⎛x˜1 (k)⎞ ⎛x˜1 (k + 1)⎞ = P AQ ⎝x˜2 (k)⎠ ⎝x˜2 (k + 1)⎠

y(k) = C1 Q

⎛x˜1 (k)⎞ ⎛x˜1 (k + 1)⎞ + C2 Q ⎝x˜2 (k)⎠ ⎝x˜2 (k + 1)⎠

(2.163) (2.164)

which is equivalent to

⎛In 0 ⎞ ⎛x˜1 (k + 1)⎞ ⎛A1 0 ⎞ ⎛x˜1 (k)⎞ = ⎝ 0 N ⎠ ⎝x˜2 (k + 1)⎠ ⎝ 0 Iµ ⎠ ⎝x˜2 (k)⎠ ⎛x˜1 (k)⎞ ⎛x˜1 (k + 1)⎞ + (D1 , D2 ) = ⎝x˜2 (k)⎠ ⎝x˜2 (k + 1)⎠ = F1 x˜1 (k) + D1 x˜1 (k + 1) + F2 x˜2 (k) + D2 x˜2 (k + 1) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

y(k) = (F1 , F2 )

y1 (k)

(2.165) (2.166) (2.167)

y2 (k)

So the system is decomposed as

x˜1 (k + 1) = A1 x˜1 (k) y1 (k) = F1 x˜1 (k) + D1 x˜1 (k + 1)

(2.168a)

= (F1 + D1 A1 )˜ x1 (k)

(2.168a) (2.168b)

and N x˜2 (k + 1) = x˜2 (k)

(2.169a)

74

CHAPTER 2. DESCRIPTOR SYSTEMS y2 (k) = F2 x˜2 (k) + D2 x˜2 (k + 1)

(2.169b)

y(k) = y1 (k) + y2 (k)

(2.170)

with output

As with the previous section, the following results hold. Remark 2.30. The noncausal subsystem (2.169a) has only the zero solution. In ad addition, the consistent initial values Had descriptor in (2.129) and H c−nc in (2.136) for the hom

hom

homogeneous system are equivalent. A with the descriptor system (2.126), since the noncausal subsystem has only the zero solution, from (2.169b) it is seen that the output of the noncausal subsystem is y2 (k) = 0. This means that the initial state of the noncausal subsystem is always known and thus the noncausal subsystem is always observable. This leads us to the following result. Theorem 2.31. The system (2.158) is completely observable if and only if the causal subsystem is observable, or equivalently if rank

⎛ σIn − A1 ⎞ =n ⎝F1 + D1 A1 ⎠

(2.171)

or ⎛ F1 + D1 A1 ⎞ ⎜ (F1 + D1 A1 )A1 ⎟ ⎜ ⎟ ⎟=n rankOcausal = rank ⎜ ⎟ ⎜ ⋮ ⎜ ⎟ ⎠ ⎝(F1 + D1 A1 )An−1 1

(2.172)

where Ocausal is the observability matrix of the causal subsystem. Theorem 2.32. The observability criterion proposed in (2.172) is equivalent to

rankOdescriptor

⎛ Φ0 ⎞ ⎛ C1 C2 ⋯ 0 ⎞ ⎜ ⎟ ⎟ ⎜ Φ1 ⎟ ⎟=n = rank ⎜ ⎜ ⋮ ⋱ ⋱ ⋮ ⎟⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝ 0 ⋯ C1 C2 ⎠ ⎝Φn ⎠

(2.173)

Proof. Multiplying the matrix Odescriptor in (2.173) from the right by the nonsingular


75

matrix P −1 we obtain the following

⎛ Φ0 ⎞ ⎛C1 C2 ⋯ 0 ⎞ ⎜ ⎟ ⎛ C1 Φ (1.113) 1 ⎜ ⎟ −1 ⎜ ⋮ ⋱ ⋱ ⋮ ⎟⎜ ⎟P = ⎜ ⎜ ⎟⎜ ⎟ ⎜ ⋮ ⋮ ⎟ ⎜ ⎝ 0 ⋯ C1 C2 ⎠ ⎝0 ⎝Φn ⎠

⎛ In 0 ⎞ ⎜0 0 ⎟ ⎜ µ⎟ ⎜ ⎟ ⎟ ⎜ A 0 C2 ⋯ 0 ⎞ ⎛Q ⎟ 1 ⎞⎜ ⎟ ⎜ ⎜ ⎟ ⎜ 0 0µ ⎟ = (2.174) ⋱ ⋱ ⋮ ⎟ ⋱ ⎟ ⎟⎜ ⎟⎜ ⎜ ⎟ ⎠ ⎝ ⎠ ⎜ Q ⎜ ⋮ ⋮⎟ ⋯ C1 C2 ⎟ ⎟ ⎜ n ⎜A1 0 ⎟ ⎟ ⎜ ⎝ 0 0µ ⎠

⎛ In 0 ⎞ ⎜0 0 ⎟ ⎜ µ⎟ ⎟ ⎜ ⎜ 0 ⎞ ⎜ A1 0 ⎟ ⎟ ⎛C1 Q C2 Q ⋯ ⎜ ⎟ (2.14) ⎜ ⎟ ⎜ =⎜ ⋮ = ⋱ ⋱ ⋮ ⎟ ⎜ 0 0µ ⎟ ⎟ (2.162) ⎟ ⎜ ⎝ 0 ⎟ ⋯ C1 Q C2 Q⎠ ⎜ ⎜ ⋮ ⋮⎟ ⎟ ⎜ n ⎜A1 0 ⎟ ⎟ ⎜ ⎝ 0 0µ ⎠

⎛F1 F2 D1 D2 ⋯ =⎜ ⎜ ⋮ ⋱ ⋱ ⋱ ⋱ ⎝ 0 ⋯ F1 F2 D1

(2.175)

⎛ In 0 ⎞ ⎜0 0 ⎟ ⎜ µ⎟ ⎟ ⎛ F1 + D1 A1 ⎜ 0⎞ ⎜ 0 ⎞ ⎜ A1 0 ⎟ ⎟ ⎜ ⎟ ⎜ (F1 + D1 A1 )A1 0⎟ ⎟ ⎟ ⎜ 0 0µ ⎟ = ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎜ ⎟ ⋮ ⎟ ⎜ ⎟ ⎜ ⎟ D2 ⎠ ⎜ ⋮ ⋮ ⎟ ⎝(F + D A )An−1 0⎠ ⎜ ⎟ ⎜ n 1 1 1 1 ⎜A1 0 ⎟ ⎜ ⎟ ⎝ 0 0µ ⎠

(2.176)

and the first column is the observability matrix Ocausal of the causal subsystem. Now we will connect the observability of (2.158) with the system’s compound matrix. Theorem 2.33. The system (2.158) is observable if and only if the matrices (σE − A) and (C1 + C2 σ) are right coprime, or equivalently rank

⎛ σE − A ⎞ =r ⎝C1 + C2 σ ⎠

(2.177)

Proof. Since the matrices P, Q are nonsingular, we can multiply the above matrix by them without affecting its rank, so ⎛A1 0 ⎞ ⎞ ⎛ σIn − A1 ⎛ ⎛In 0 ⎞ ⎛P 0 ⎞⎛ σE − A ⎞ ⎜⎝ 0 N⎠ σ − ⎝ 0 I ⎠⎟ ⎜ Q=⎜ 0 µ ⎟=⎜ ⎜ ⎟ ⎝ 0 Im ⎠⎝C1 + C2 σ ⎠ ⎝ ⎝(F1 F2 ) + (D1 D2 ) σ ⎠ F1 + D1 σ

0 ⎞ σN − Iµ ⎟ ⎟ (2.178) F 2 + D2 σ ⎠

76


The second column block of the above matrix always has full column rank equal to µ, as shown in Theorem 2.27. For the first column, if we multiply the first row by −D1 from the left and add it to the third column, we get ⎛ In 0 ⎞ ⎛ σIn − A1 ⎞ ⎛ σIn − A1 ⎞ ⇒ ⎝−D1 Im ⎠ ⎝F1 + D1 σ ⎠ ⎝F1 + D1 A1 ⎠

(2.179)

and the above matrix has full column rank iff the causal subsystem is observable. Thus, the matrix (2.177) has full column rank iff the system is observable. All of the above results regarding observability of descriptor systems with a lead in the output are summarized in the following theorem. Theorem 2.34. The following statements are equivalent. 1. The descriptor system (2.158) is completely observable. 2. rankOdescriptor = n. 3. The matrices (σE − A) and (C1 + C2 σ) are right coprime. 4. The causal subsystem (2.168) is observable. 5. rankOcausal = n. 6. The matrices (σIn − A1 ) and F1 + D1 A1 are right coprime. Remark 2.35. It is worth noting that using the solution formula (2.7), x(k + 1) can be rewritten as x(k + 1) = Φk+1 Ex(0)

(1.112f)

=

Φ1 EΦk Ex(0) = Φ1 Ex(k)

(2.180)

and so the output descriptor system (2.158) can be written as y(k) = C1 x(k) + C2 x(k + 1) = (C1 + C2 Φ1 E)x(k)

(2.181)

and so the observability results of Theorem 2.34 could also be derived using the results of Section 2.4.


77

Example 2.36. Consider the following system ⎛1 0 0 0⎞ ⎛−1 −1 −2 0 ⎞ ⎜0 0 0 0⎟ ⎜ 0 1 0 −1⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ x(k + 1) = ⎜ ⎟ x(k) ⎜2 0 1 0⎟ ⎜ 0 0 −1 −1⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 1 0 0⎠ ⎝0 0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ E

y(k) =

(2.182a)

A

⎛1 0 0 0⎞ ⎛0 2 0 1⎞ x(k + 1) x(k) + ⎝0 1 1 0⎠ ⎝0 0 0 2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(2.182b)

˜2 C

C1


⎛−1 −1 21 0⎞ ⎛ 0 21 0 21 ⎞ ⎜ 2 0 0 0⎟ ⎜−2 −1 1 0 ⎟ ⎜ ⎜ ⎟ ⎟ ⎟, Q = ⎜ ⎟ P =⎜ ⎜ 0 1 0 0⎟ ⎜−2 −1 2 1 ⎟ ⎜ ⎜ ⎟ ⎟ ⎝ 2 0 0 1⎠ ⎝ 0 −1 0 0 ⎠

(2.183)

with ⎛1 ⎜0 ⎜ P EQ = ⎜ ⎜0 ⎜ ⎝0 C1 Q =

0 1 0 0

⎛6 0 0 ⎝4 0 0

0 0 1 0

0⎞ ⎛1 ⎜0 ⎟ 0⎟ ⎜ ⎟ , P AQ = ⎜ ⎜0 ⎟ 0⎟ ⎜ ⎝0 ⎠ 0

0 1 0 0

0 1 1 0

⎛−1 −1 1/2 1⎞ , C2 Q = ⎝2 1 2⎠ 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠ 0⎞ 0⎠

(2.184)

(2.185)


⎛1 0 0⎞ ⎟ x˜1 (k + 1) = ⎜ ⎜0 1 1⎟ x˜1 (k), ⎝0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

y1 (k) =

⎛5 −1 − 12 ⎞ x˜1 (k) ⎝6 1 1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(2.186a)

(F1 +D1 A1 )

A1

and 0 = x˜2 (k), ® N

y2 (k) =

⎛1⎞ ⎛1⎞ x˜2 (k) + 02×1 x˜2 (k + 1) = x˜2 (k) = 02×1 (2.187a) ⎝2⎠ ⎝2⎠ ± D2 ± F2


⎛x˜1 (k)⎞ ⎝x˜2 (k)⎠

(2.188)

78



0}. The observability matrix of the causal subsystem is ⎛5 −1 − 21 ⎞ ⎜6 1 1 ⎟ ⎜ ⎟ ⎟ ⎛ F1 + D1 A1 ⎞ ⎜ ⎜5 −1 − 3 ⎟ ⎜ 2⎟ ⎟ ⎜ ⎟ Ocausal = ⎜(F1 + D1 A1 )A1 ⎟ = ⎜ ⎜6 1 2 ⎟ ⎜ ⎟ ⎝(F1 + D1 A1 )A21 ⎠ ⎜ ⎟ ⎜5 −1 − 5 ⎟ ⎜ 2⎟ ⎝6 1 3 ⎠

(2.189)

and has rankOcausal = 3, so the causal subsystem is observable and thus the compete system is observable. In addition, the matrices (σE − A), (C1 + C2 σ) are right coprime.

2.6

Conclusions

The solution of discrete time descriptor systems has been presented, and the properties of reachability and observability have been studied. A formula for the solution space over a finite or an infinite horizon was given, in terms of the Laurent expansion of the matrix pencil. Then, by decomposing the system into its causal and noncausal subsystems, solution formulas were given for each subsystem. Furthermore, the reachability of descriptor systems was studied. Special attention was given in taking into account the consistent initial conditions that need to be satisfied and constructing a consistent input that can drive the state from the origin into any point in the reachable subspace. Methods for constructing such an input were given by working either on the descriptor system or with its causal and noncausal subsystems and reachability criteria were established in terms of the Laurent coefficients Φi and in terms of the causal/noncausal subsystem matrices. Lastly, the observability of descriptor systems was studied. By using the causal/noncausal subsystem decomposition, it was shown that observability of the system depends only on the observability of its causal subsystem, since the noncausal subsystem is always observable. Again, observability criteria were established in terms of the Laurent coefficients Φi and in terms of the causal subsystem matrices. In addition, the observability of a special case of descriptor systems with a state lead in the output was studied, since it will be show in later chapters that higher order systems can be rewritten as a descriptor system of this form.

2.6. CONCLUSIONS

79

These results on reachability and observability for descriptor systems will be expanded to higher order systems in the following chapters.

80


Chapter 3 Modeling of Linear Homogeneous Systems of Algebraic and Difference Equations 3.1

Introduction

In this Chapter, we shall consider linear homogeneous systems of higher order discrete time algebraic and difference equations that are described by the matrix equation Aq β(k + q) + Aq−1 β(k + q − 1) + ... + A0 β(k) = 0,

(3.1)

where k = 0, 1, ..., N − q, A0 , ..., Aq ∈ Rr×r with Aq ≠ 0, A0 ≠ 0 and β(k) ∶ [0, N ] → Rr , is the state of the system,. Using the forward shift operator σ with σ i β(k) = β(k + i), the system can be rewritten as A(σ)β(k) = 0,

(3.2)

A(σ) = Aq σ q + ... + A1 σ + A0 ∈ R[σ]r×r ,

(3.3)

where

is a regular polynomial matrix with det [A(σ)] ≠ 0. The number of maximum time shifts q is called the lag of the system [80]. Systems described by (3.2) are called AutoRegressive (AR) representations. It is easy to see that (2.1a) is a special case of the AR representation (3.2) for A(σ) = Eσ − A. 81

82

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS Such systems and their continuous time analogues often appear in system’s theory,

since they can model many natural or artificial systems, like economic or biological continuous and discrete time phenomena. Such examples include the population growth model in biology and the Leontief multisector economy model in economics [15], as well as other applications in biology [72] engineering [2, 11, 51, 93, 99], social sciences and medicine for positive systems [44] or 2D systems [45]. Systems of the form (3.2) can be defined over a finite or an infinite time horizon. In the present Chapter, we will consider the finite horizon case, so the system is defined over the time interval [0, N ], where N ∈ N. In this case, the solution space of (3.2) consists of both forward and backward solutions and is denoted as B ∶= {β(k) ∣ (3.2) is satisfied ∀k ∈ [0, N − q]} .

(3.4)

Forward solutions are defined in the sense that the initial conditions are given and β(k) is to be determined in a forward fashion from its previous values. Backward solutions are defined in the sense that the final conditions are given and β(k) is to be determined in a backward fashion from its future values. The solution of such systems and their continuous time analogues has been previously studied by various authors in [4, 6, 23, 33, 51, 53, 118]. A method for constructing the forward (resp. backward) solution space of (3.2), based on the finite (resp. infinite) elementary divisor structure of A(σ) has been presented in [33] (resp. [4]) whereas the results have been extended for non-regular systems by [53]. An interesting problem that we face in this chapter is the inverse problem, that is: Given a certain forward/backward solution space, find a system of algebraic and difference equations with the prescribed solution space. A partial solution to this problem has been described in [33], where only the smooth behavior for continuous time and the forward behavior for discrete time regular systems was studied. More specifically, [33] proposed a formula that connects the form of the unknown polynomial matrix A(σ) and a finite Jordan pair that can be constructed from the prescribed forward solution space. This method was later extended in [54] for continuous time systems, to include both the smooth and impulsive behavior. The main aim of this work is to extend the results presented in [33] for the case where except from the given forward behavior, a backward behavior is also provided.

3.1. INTRODUCTION

83

We will use two different approaches for solving this problem. In our first approach, we apply a method similar to the one used in [54], to connect the backward solution space of (3.2) with the forward solution space of its dual system A0 β(k + q) + A1 β(k + q − 1) + ... + Aq β(k) = 0,

(3.5)

and combine this with the results given in [33] to construct a system with a prescribed forward and backward behavior. We also give a connection between the finite and ˜ infinite zeros of the original matrix A(σ) and its dual A(σ). A disadvantage of this approach is that it relies on the computation of the Jordan Pairs of A(σ) and cannot be applied to non regular systems, that is, systems with A(σ) ∈ R[σ]r×m and r ≠ m or with A(σ) ∈ R[σ]r×r and det A(σ) = 0. In addition, this method is much less versatile in handling the free parameters of the matrices A0 , ..., Aq and requires a deep understanding of the structure of polynomial matrices. In the second approach to this problem, we develop a novel method for constructing systems with prescribed forward/backward solutions that can also be used to construct non regular systems, a case that was not addressed in [33, 54]. The core of this method lies in the fact that the vectors that consist a solution of the system (forward or backward), actually satisfy a certain system of equations, which we are going to solve in terms of the unknown coefficients of A(σ), in order to obtain the original system. Thus, the problem of constructing a system of linear algebraic and difference equations is reduced to solving a linear system of equations. Another issue that is addressed, following the behavioral framework of [5, 80, 115, 124–126, 133–135], is whether or not the constructed system is the most powerful . As power of a model is defined the ability of the constructed model to describe the given behavior, i.e. the given data, but as little else as possible. This means that the aim is to construct a system whose solution space must include the prescribed set of vector valued functions, but no other functions linearly independent from the prescribed. To better understand this concept, consider the following discrete time function β(k) = 3k

(3.6)

The following two discrete time difference equations x(k + 1) − 3x(k) = 0 ⇔ (σ − 3)x(k) = 0

(3.7)

84

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS y(k + 2) − 5y(k + 1) + 6y(k) = 0 ⇔ (σ 2 − 5σ + 6)y(k) = 0

(3.8)

with solutions x(k) = 3k and y(k) = 3k +2k both have the function β(k) in their solution space. Yet, we can say that the system described by (3.7) is more powerful than the system described by (3.8), since the system (3.7) has no other functions in its solution space while the solution space of (3.8) is given by the linear combination of β(k) and 2k . In addition, the system (3.7) has a lower lag.

3.2

Finite Jordan Pairs and the Forward Solution Space

In the present section we begin by defining the Jordan Pairs of a polynomial matrix, along with a method to construct them from its Smith form, and connect them to the forward solution space of (3.2). Assume that A(σ) has ` finite, distinct zeros λ1 , ..., λ` where λi ∈ C, i = 1, ..., ` and consider its Smith form C SA(σ) (σ) = UL (σ)A(σ)UR (σ) = diag (1, ..., 1, fz (σ), fz+1 (σ), ..., fr (σ))

(3.9)

Assume that the partial multiplicities of the zeros λi ∈ C are 0 ≤ ni,z ≤ ni,z+1 ≤ ... ≤ ni,r

(3.10)

with r

`

r

n ∶= deg det A(s) = deg [∏ fj (s)] = ∑ ∑ ni,j j=z

(3.11)

i=1 j=z

r

ni = ∑ ni,j

(3.12)

j=z

as defined in (1.33),(1.37). r

Theorem 3.1. [33, 53] Let uj (σ) ∈ R[σ] , be the columns of UR (σ). Define the vectors i βj,φ ∶=

1 (φ) φ! uj (λi )

i = 1, 2, ..., ` j = z, z + 1, . . . , r φ = 0, 1, ..., ni,j − 1

(3.13)

and construct the matrices i i i Ci,j ∶= (βj,0 ) ∈ Cr×ni,j βj,1 ... βj,n i,j −1

(3.14)

3.2. FINITE JORDAN PAIRS AND THE FORWARD SOLUTION SPACE

Ji,j

⎛λi 1 ... 0 ⎞ ⎜ 0 λi ⋱ ⋮ ⎟ ⎜ ⎟ ⎟ ∈ Cni,j ×ni,j ∶= ⎜ ⎜ ⋮ ⋱ ⋱ 1⎟ ⎜ ⎟ ⎝ 0 ... 0 λi ⎠

85

(3.15)

where i = 1, 2, ..., `, j = z, z + 1, . . . , r. Now combine them as CF,i ∶= (Ci,z Ci,z+1 ⋯ Ci,r ) ∈ Cr×ni

(3.16)

JF,i ∶= blockdiag (Ji,z Ji,z+1 ⋯ Ji,r ) ∈ Cni ×ni

(3.17)

The matrix pair (CF,i , JF,i ) is called a Jordan pair of A(σ) corresponding to λi and satisfies the following properties: 1. det A(σ) has a zero at λi of multiplicity ni . 2. q−1

k )k=0 = ni rankcol (CF,i JF,i

(3.18)

3. q

k =0 ∑ Ak CF,i JF,i

(3.19)

k=0

Taking the Jordan pairs for all the zeros λi , i = 1, ..., ` of A(σ) and combining them as CF ∶= (CF,1 ⋯ CF,` ) ∈ Cr×n

(3.20)

JF ∶= blockdiag (JF,1 ⋯ JF,` ) ∈ Cn×n

(3.21)

we define the finite Jordan Pair or finite spectral pair of A(σ). The finite Jordan Pair (CF , JF ) satisfies the following properties deg(det A(σ)) = n q−1

rankcol (CF JF k )k=0 = n

(3.22) (3.23)

q

k ∑ Ak CF JF = 0

(3.24)

k=0

Theorem 3.2. [3, 4, 53] Define the vector valued functions k k−φ i F i i βi,j,φ (k) ∶= λki βj,φ + kλk−1 βj,0 for λi ≠ 0 i βj,φ−1 + ... + ( )λi φ F i i i βi,j,φ (k) ∶= δ(k)βj,φ + δ(k − 1)βj,φ−1 + ... + δ(k − φ)βj,0 for λi = 0

(3.25a) (3.25b)

86

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS

where i = 1, 2, ..., `, j = z, z + 1, . . . , r, φ = 0, 1, ..., ni,j − 1 and δ(k) or δk denotes the KroF necker delta. The vector valued functions βi,j,φ (k) are solutions of (3.2). In addition,

the complete forward solution space of (3.2) is given by the column span of k

BF = ⟨CF (JF ) ⟩

(3.26)

with dimension dim BFD = n. i defined in (3.13) for i = 1, 2, ..., `, j = z, z + 1, . . . , r, Lemma 3.3. [116] The vectors βj,φ

φ = 0, 1, ..., ni,j −1 form Jordan chains for A(σ) corresponding to the zero λi and satisfy the following system of equations i A(λi )βj,0 =0 i i (1) A (λi )βj,0 + A(λi )βj,1 =0 ⋮ 1 (ni,j −1) (λ )β i + ... + A(λ )β i i i j,0 j,ni,j −1 = 0 (ni,j −1)! A

(3.27)

In the following Theorem we prove a general version of the above Lemma for the i vector functions (3.25), without the assumption that the vectors βj,φ are Jordan chains

of A(σ). F Theorem 3.4. The vector valued functions βi,j,φ (k) defined in (3.25), are solutions of i i (3.2) if and only if the vectors βj,0 , ..., βj,n satisfy (3.27). i,j −1 F Proof. We show that βi,j,φ (k) are solutions of (3.2) if and only if (3.27) are satisfied. F i First, consider the general case where λi ≠ 0. For βi,j,0 (k) = λki βj,0 , we have: i A(σ)λki βj,0 =0⇔

(3.28a)

i k+1 i k i Aq λk+q i βj,0 + ... + A1 λi βj,0 + A0 λi βj,0 = 0 ⇔

(3.28b)

λi ≠0

i (Aq λqi + ... + A1 λi + A0 )λki βj,0 =0 ⇔

(3.28c)

i A(λi )βj,0 =0

(3.28d)

i k i F so the first equation in (3.27) is proven. Now, letting βi,j,1 (k) = kλk−1 i βj,0 + λi βj,1 , we

obtain: i k i A(σ)(kλk−1 i βj,0 + λi βj,1 ) = 0 ⇔

(3.29a)

i k i A(σ)kλk−1 i βj,0 + A(σ)λi βj,1 ⇔

(3.29b)

k+q i i i k i Aq (k + q)λk+q−1 βj,0 + ... + A0 kλk−1 i βj,0 + Aq λi βj,1 + ... + A0 λi βj,1 = 0 i

(3.29c)

3.3. INFINITE JORDAN PAIRS AND THE BACKWARD SOLUTION SPACE 87 i and taking into account that A(λi )βj,0 = 0, the above equation is written as i i (qAq λq−1 + .... + A1 )λki βj,0 + (Aq λqi + ... + A1 λi + A0 )λki βj,1 =0⇔ i

(3.30a)

i i A(1) (λi )βj,0 + A(λi )βj,1 =0

(3.30b)

so the second equation in (3.27) holds true. Continuing in the same way, the rest of the equations in (3.27) can be proven. F i For the case where λi = 0, letting βi,j,0 (k) = δ(k)βj,0 we obtain: i A(σ)δ(k)βj,0 =0⇔

(3.31a)

i i i Aq δ(k + q)βj,0 + ... + A1 δ(k + 1)βj,0 + A0 δ(k)βj,0 =0

(3.31b)

i i F and for k = 0 this equation becomes A0 βj,0 = A(0)βj,0 = 0. Now, letting βi,j,1 (k) = i i δ(k)βj,1 + δ(k − 1)βj,0 we obtain: i i A(σ)(δ(k)βj,1 + δ(k − 1)βj,0 )=0⇔ i i i i Aq δ(k + q)βj,1 + ... + A0 δ(k)βj,1 + Aq δ(k + q − 1)βj,0 + ... + A0 δ(k − 1)βj,0 =0

(3.32a) (3.32b)

i i i i which for k = 0 yields A1 βj,0 + A0 βj,1 = A(1) (0)βj,0 + A(0)βj,1 = 0. Continuing in the

same fashion, the rest of the equations in (3.27) can be proven for λi = 0.

3.3

Infinite Jordan Pairs and the Backward Solution Space

In this section we define the infinite Jordan Pairs of a polynomial matrix along with a method to construct them from its Smith form and connect them to the backward ˜ solution space of (3.2). Consider the Smith form of the dual matrix A(s) (see (1.51)) ˜ = 0. at λ 0 ˜ U˜R (σ) = diag (σ µ1 , ..., σ µr ) SA(σ) (σ) = U˜L (σ)A(σ) ˜

(3.33)

with orders µi as in (1.54) q=q

µ1 = q − q1 = 1 0

(3.34a)

µj = q − qj

(3.34b)

j = 2, 3, ..., u

88

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS µj = q + qˆj

j = u + 1, ..., r

(3.34c)

and µ as in (1.55) r

µ ∶= ∑ µj

(3.35)

j=1

Theorem 3.5. [33, 53] Let u˜j (σ) ∈ Rr be the columns of U˜R (σ). Define the vectors xj,φ ∶=

1 (φ) ˜j (0) φ! u

φ = 0, 1, ..., µj − 1 j = 2, ..., r

(3.36)

let C∞,j = (xj,0 xj,1 ⋯ xj,µj −1 ) ∈ Rr×µj ⎛0 ⎜0 ⎜ ∶= ⎜ ⎜⋮ ⎜ ⎝0

⋯ ⋱ ⋱ 0

(3.37)

0⎞ ⋮⎟ ⎟ ⎟ ∈ Rµj ×µj 1⎟ ⎟ 0⎠

(3.38)

C∞ ∶= (C∞,2 ⋯ C∞,r ) ∈ Rr×µ

(3.39)

J∞ ∶= blockdiag (J∞,2 ⋯ J∞,r ) ∈ Rµ×µ

(3.40)

J∞,j

1 0 ⋱ ⋯

and combine them as

˜ The matrix pair (C∞ , J∞ ) is a finite Jordan Pair of A(s) corresponding to the zero ˜ = 0 and is called an Infinite Jordan Pair of A(s). The infinite Jordan Pair satisfies λ the following properties ˜ = 0 of multiplicity µ ˜ det A(σ) has a zero at λ q−1

rankcol (C∞ J∞ k )k=0 = µ q

∑ Ak C∞ J∞

q−k

=0

(3.41) (3.42) (3.43)

k=0

Notice that the matrix pairs (C∞,j , J∞,j ) start from j = 2, ...r, since µ1 = 0. Theorem 3.6. [3, 4, 53] Define the vector valued functions B βj,φ (k) ∶= xj,φ δ(N − k) + ... + xj,0 δ(N − (k + φ))

(3.44)

B where j = 2, ..., r, φ = 0, ..., µj − 1. The vector valued functions βj,φ (k) defined in (3.44)

are solutions of (3.2). In addition, the complete backward solution space of (3.2) is spanned by the columns of: N −k

BB = ⟨C∞ (J∞ ) and has dimension dim BBD = µ.

⟩

(3.45)

3.3. INF. JORDAN PAIRS AND THE BACKWARD SOLUTION SPACE

89

Lemma 3.7. [54, 116] The vectors xj,φ , defined in (3.36) for φ = 0, 1, . . . , µj − 1 and ˜ i = 0 with ˜ j = 2, ..., r form Jordan chains for A(σ) corresponding to the eigenvalue λ lengths µj , as in (1.54). Thus, they satisfy the following systems of equations: Aq xj,0 = 0 Aq−1 xj,0 + Aq xj,1 = 0 ⋮ A0 xj,0 + A1 xx,1 + ... + Aq xj,q = 0

j = u + 1, ..., r

(3.46a)

⋮ A0 xj,ˆqj −1 + A1 xj,ˆqj + ... + Aq xj,q+ˆqj −1 = 0 for the case of infinite zero elementary divisors (i.z.e.d.), i.e. µj = q + qˆj , j = u + 1, ..., r, or Aq xj,0 = 0 Aq−1 xj,0 + Aq xj,1 = 0 ⋮

j = 2, ..., u

(3.46b)

Aqj +1 xj,0 + ... + Aq xj,q−qj −1 = 0 for the case of infinite pole elementary divisors (i.p.e.d), i.e. µj = q − qj ,

j = 2, ..., u.

Proof. The proof of (3.46) that corresponds to the infinite zero elementary divisors was provided in [116]. The procedure can be easily extended with no significant changes to include the case of infinite pole elementary divisors (3.46b). Remark 3.8. The equations (3.46) can be rewritten in matrix form as ⎛ Aq 0 0 ⋯ ⋯ 0 ⎞ ⎜ ⋮ A 0 ⋯ ⋯ 0⎟ ⎜ ⎟ q ⎜ ⎟ ⎜A0 ⋮ ⋱ ⋱ ⋱ ⋮ ⎟ ⎛ xj,0 ⎞ ⎛0⎞ ⎜ ⎟⎜ ⎟ = ⎜⋮⎟ ⎜ ⎟ ⋮ ⎟ ⎜ ⎟ ⎜ 0 A0 ⋱ ⋱ ⋱ ⋮ ⎟ ⎜ ⎜ ⎟ ⎝x ⎜ ⎟ j,q+ˆqj −1 ⎠ ⎝0⎠ ⎜ ⋮ ⎟ ⋱ ⋱ ⋱ ⋱ 0 ⎜ ⎟ ⎝ 0 ⋯ 0 A0 ⋯ Aq ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.47a)

Q

for the case of infinite zero elementary divisors and as ⎛ Aq ⋯ 0 ⎞ ⎛ xj,0 ⎞ ⎛0⎞ ⎜ ⋮ ⎜ ⎟ = ⎜⋮⎟ ⋱ ⋮ ⎟ ⋮ ⎜ ⎟⎜ ⎟ ⎜ ⎟ ⎝Aqj +1 ⋯ Aq ⎠ ⎝xj,q−qj −1 ⎠ ⎝0⎠

(3.47b)

90


for the case of infinite pole elementary divisors. The general version of Lemma 3.7 is the following, where there is no assuption made ˜ about the vectors xj,i being Jordan chains of A(σ). B (k) defined in (3.44), are solutions of Theorem 3.9. The vector valued functions βj,φ

(3.2) if and only if the vectors xj,0 , ..., xj,µj −1 in (3.36) satisfy the system of equations (3.46). B Proof. We show that βj,φ (k) are solutions of (3.2) if and only if (3.46) are satisfied. B To begin, let βj,0 (k) = xj,0 δ(N − k) be a solution of (3.2). Thus, it satisfies:

A(σ)xj,0 δ(N − k) = 0 ⇔ Aq δ(N − k − q)xj,0 + ... + A1 δ(N − k − 1)xj,0 + A0 δ(N − k)xj,0 = 0 and since (3.2) is satisfied for k ∈ [0, N − q], setting k = N − q we obtain Aq xj,0 = 0 B For the converse, let Aq xj,0 = 0 hold. Then the vector function βj,0 (k) = xj,0 δ(N − k)

will be a solution of (3.2) for k ∈ [0, N − q]. Thus, the first equation in (3.46) is proven. B Now, let βj,1 (k) = xj,1 δ(N − k) + xj,0 δ(N − k − 1) be a solution of (3.2). We obtain

A(σ) (xj,1 δ(N − k) + xj,0 δ(N − k − 1)) = 0 ⇔ Aq δ(N − k − q)xj,1 + ... + A0 δ(N − k)xj,1 + +Aq δ(N − k − q − 1)xj,0 + Aq−1 δ(N − k − q)xj,0 + ... + A0 δ(N − k − 1)xj,0 = 0 Again, taking k = N − q − 1, we get Aq xj,0 = 0 as before and taking k = N − q, we get Aq−1 xj,0 + Aq xj,1 = 0 which is second equation in (3.46). For the converse, let Aq xj,0 = 0 and Aq−1 xj,0 +Aq xj,1 = B 0 hold. Then the vector function βj,1 (k) = xj,1 δ(N −k)+xj,0 δ(N −k−1) will be a solution

of (3.2) for k ∈ [0, N − q]. Thus the second equation in (3.46) is proven. Continuing in the same fashion, the rest of the equations in (3.46) can be proven, either for the case of i.z.e.d. or for the case of i.p.e.d.

3.4. CONSTRUCTION OF A HOMOGENEOUS SYSTEM WITH PRESCRIBED FORWARD/BAC

3.4

Construction of a Homogeneous System with Prescribed Forward/Backward Behavior

In the previous sections we have provided a method for constructing the complete forward and backward solution space of (3.2), by constructing the finite and infinite Jordan pairs of the matrix A(σ). In the following sections we study the inverse problem, that is, given a specific forward or backward behavior, how we can construct a polynomial matrix A(σ) and its corresponding homogenous system A(σ)β(k) = 0 that will satisfy the given behavior. We provide two different solutions to this problem. First, by extending the method proposed in [33] to include the backward behavior as well, and secondly by constructing a linear system of equations, whose solution will give us the matrices A0 , ..., Aq of the desired matrix A(σ).

3.4.1

First method: An extension of the method in [33]

An answer to this modeling problem was first proposed by [33], for the case of the forward solution space. We first present these results and then we extend them in order to include the case of the backward solution space as well. Then, we will combine these results into a single algorithm for modelling both the forward and the backward solution space of a system. r

Suppose that a finite number of vector valued functions βi (k)∶ [0, N ] → R , of the form βi (k) ∶= λki βi,ni −1 + kλk−1 i βi,ni −2 + ... + (

k k−(ni −1) )λ βi,0 , ni − 1 i

(3.48)

for λi ≠ 0, i = 1, 2, ..., ` or βi (k) = δ(k)βi,ni −1 + ... + δ (k − (ni − 1)) βi,0 ,

(3.49)

for λi = 0, are given, where δ(k) denotes the discrete Kronecker delta and βi,0 , ..., βi,ni −1 ∈ Cr . We want to construct a system of algebraic and difference equations with (3.48), (3.49) as its solutions. In analogy to the continuous time case studied by [33, Proposition 1.9], if (3.48),(3.49) are solutions of the system, then the vectors βi,0 , ..., βi,ni −1 are

92


generalized eigenvectors of A(σ) corresponding to λi and the matrices (Ci , Ji ), with

Ci = (βi,0

⎛λi 1 ⋯ 0 ⎞ ⎜ 0 λi ⋱ ⋮ ⎟ ⎜ ⎟ ⎟, . . . βi,ni −1 ) , Ji = ⎜ ⎜ ⋮ ⋱ ⋱ 1⎟ ⎜ ⎟ ⎝ 0 ⋯ 0 λi ⎠

(3.50)

constitute a Jordan pair of A(σ). Thus, the vector functions βi,0 kλk−1 + βi,1 λki , ... i

βi,0 λki ,

(3.51)

for λi ≠ 0 and βi,0 δ(k),

βi,0 δ(k − 1) + βi,1 δ(k), ...

(3.52)

respectively for λi = 0 are also solutions of the system [33, Section 8.3]. The above set of vector valued functions can be written in matrix form as Ci Jik , where we make use of the formulas k−(n −1) ⋯ (nik−1)λi i ⎞ k−(n −2) ⋯ (nik−2)λi i ⎟ ⎟ ⎟ , λi ≠ 0 ⎟ ⋱ ⋮ ⎟ k ⎠ ⋯ λi

k ⎛λki (1)λik−1 ⎜0 λki ⎜ Jik = ⎜ ⎜⋮ ⎜ ⎝0 ⋯

⎛δ(k) δ(k − 1) ⎜ 0 δ(k) ⎜ Jik = ⎜ ⎜ ⋮ ⎜ ⎝ 0 ⋯

⋯ δ(k − (ni − 1))⎞ ⋯ δ(k − (ni − 2))⎟ ⎟ ⎟ , λi = 0 ⎟ ⋱ ⋮ ⎟ ⎠ ⋯ δ(k)

(3.53)

(3.54)

with Ci ∈ Cr×ni , Ji ∈ Cni ×ni . Define C ∶= (C1 ⋯ C` ) ∈ Cr×n ,

(3.55)

J ∶= blockdiag (J1 , ⋯, J` ) ∈ Cn×n ,

(3.56)

`

where n ∶= ∑ ni . Then we have the following Theorem. i=1

Theorem 3.10. [33] Let a be a complex number other than λi and define A(σ) = Ir − C(J − aIn ) ((σ − a)Vq + ... + (σ − a)q V1 ) , −q

(3.57)

where q = ind(C, J) is the least integer such that the matrix ⎛ C ⎞ ⎜ CJ ⎟ ⎜ ⎟ ⎟ ∈ Crq×n Sq−1 = ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝CJ q−1 ⎠

(3.58)

3.4. CONSTRUCTION OF A SYSTEM WITH PRESCRIBED BEHAVIOR

93

has full column rank, and (V1 , ⋯, Vq ) is the generalized inverse of

S1−q

C ⎛ ⎞ ⎜ C(J − aIn )−1 ⎟ ⎜ ⎟ ⎟ ∈ Crq×n . =⎜ ⎜ ⎟ ⋮ ⎜ ⎟ ⎝C(J − aIn )1−q ⎠

(3.59)

Then βi (k) are solutions of (3.2). Furthermore, q is the minimal possible lag of any polynomial matrix with this property. Remark 3.11. Different choices of a ≠ λi will lead to the construction of matrices that are left equivalent (see Definition 1.22). That is, if A1 (σ) and A2 (σ) are any two matrices that are constructed for different values of the parameter a, there exists a unimodular matrix U (σ) such that A1 (σ) = U (σ)A2 (σ).

(3.60)

Thus, since multiplication by U (σ) does not alter the algebraic structure of A(σ) the forward behavior of the corresponding system remains the same (see [127]). So, the choice of the parameter a does not impose any limitations. Notice that since the time sequences in the form of (3.48) constitute solutions of (3.2), they satisfy (3.2) for all k, regardless of how we consider time propagation. This means that solutions in the form of (3.48) can either be regarded as solutions propagating forward in time for initial conditions β(0), β(1), . . . , β(q − 1), or solutions moving backward for final conditions β(N ), β(N − 1), . . . , β(N − q + 1). Remark 3.12. In case where a backward propagating time sequence is given in the form of βi (k) = Ci JiN −k xN , corresponding to a zero λi ≠ 0, with final conditions ⎛ βi (N ) ⎞ ⎛ Ci ⎞ ⎜ ⎟ = ⎜ ⋮ ⎟ xN , ⋮ ⎜ ⎟ ⎜ ⎟ ⎝βi (N − q + 1)⎠ ⎝Ci J q−1 ⎠ i

(3.61)

k

it can be rewritten as βi (k) = Ci (Ji−1 ) JiN xN and it can be clearly seen that it corre² x0

sponds to the finite Jordan Pair (C, J −1 ) of A(σ) for initial conditions Ci ⎛ βi (0) ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ x0 . ⋮ ⋮ ⎜ ⎟=⎜ ⎟ ⎝βi (q − 1)⎠ ⎝Ci (J −1 )q−1 ⎠ i

(3.62)

94


So in order to construct the system (3.2) with βi (k) as its solution, one can follow Theorem 3.10 using the pair (C, J −1 ). On the other hand, time sequences in the form of (3.49) can only be considered forward solutions for the system. Overall, the finite elementary divisors are connected to either forward/backward propagating solutions of the form (3.48) for a nonzero λi ≠ 0, or to strictly forward propagating solutions of the form (3.49) for λi = 0. In the following, we will show how the infinite elementary divisors are connected to the backward solutions of the system. Remark 3.13. Although we have created an Auto-Regressive representation for the given forward solution space, if equation n + µ = r × q is not satisfied for µ = 0, then the above algorithm will give rise to an AR-representation which will include an extra forward/backward behavior. Examples showcasing this are provided in the last section of this chapter. Example 3.14. Assume that the following vector valued function is given ⎛2⎞ ⎛4⎞ ⎛2⎞ k−1 k ⎜ ⎟ k(k − 1) 2k−2 ⎜ ⎟ ⎟ β1 (k) = ⎜ ⎜3⎟ 2 + ⎜1⎟ k2 + ⎜0⎟ 2 ⎝0⎠ ⎝1⎠ ⎝2⎠ ± ± ± β1,2

(3.63)

β1,0

β1,1

Our aim is to create a system A(σ)β(k) = 0 with a solution space given by β1 (k). We define the matrices C = (β1,0 β1,1

⎛2 4 2⎞ ⎟ β1,2 ) = ⎜ ⎜0 1 3⎟ ⎝0 1 2⎠

⎛2 1 0⎞ ⎟ J =⎜ ⎜0 2 1⎟ ⎝0 0 2⎠

(3.64)

(3.65)

For q = 1, we have det C = −2 ≠ 0. Therefore, the polynomial matrix A (σ) is of lag q = 1, i.e. A (σ) = A1 σ + A0 and ⎛ 2 3 −5⎞ −1 ⎟ V1 = C = ⎜ ⎜ 0 −2 3 ⎟ ⎝ 0 1 −1⎠ 1

(3.66)


95

For a = 1, we get, ⎛2 − σ −2σ + 2 4σ − 4 ⎞ A(σ) = I3 − C(J − I3 ) (σ − 1) V1 = ⎜ 1 1−σ ⎟ ⎜ 0 ⎟ ⎝ 0 σ − 1 −2σ + 3⎠ −1

(3.67)

Indeed, A(σ)β1 (k) = 0 and C SA(σ) (σ) =

⎛I2 0 ⎞ ⎝ 0 (σ − 2)3 ⎠

Note that n + µ = rq ⇔ 3 + µ = 3 ⇔ µ = 0. Therefore, there exists no backward solution space due to the infinite elementary divisor structure of A(σ). Now we will provide a theorem for the backward solution space. The following theorem showcases the connection between the solutions of (3.2) and the solutions of the dual system (3.5). Theorem 3.15. [3, Lemma 4.3],[4] If β(k) is a solution of (3.2) for k = 0, ..., N , then ˜ β(k) = β(N − k) is a solution of (3.5). Based on the above, we can conclude to the following result. B Theorem 3.16. Let βj,φ (k), where j = 2, ..., r, φ = 0, ..., µj − 1. be the vector valued

functions defined in (3.44) that are the backward solutions of (3.2). Then the vector valued functions B β˜j,φ (k) = βj,φ (N − k) = xj,0 δ(k − φ) + ⋯ + xj,φ δ(k)

(3.68)

˜ β˜j (k) = 0. are solutions of the dual system (3.5) i.e. A(σ) The previous theorem showcases that the problem of finding an AR-representation of the form (3.2) with the backward behavior (3.44) is equivalent to the problem of finding an AR-representation of the form (3.5) satisfying the forward behavior (3.68). However this problem can easily be solved using Theorem 3.10. This is showcased in the following theorem. Theorem 3.17. Suppose that the following l vector valued functions φ

B βj,φ (k) = ∑ xj,φ−w δ(N − w − k) w=0

(3.69)

96


are given, where j = 1, ..., l, l < r, φ = 0, ..., µj − 1, xj,0 , ..., xj,µj −1 ∈ Cr . Define Cj = (xj,0 ⋯ xj,µj −1 ) ∈ Rr×µj ,

(3.70)

⋯ ⋱ ⋱ 0

(3.71)

⎛0 ⎜0 ⎜ Jj = ⎜ ⎜⋮ ⎜ ⎝0

1 0 ⋱ ⋯

0⎞ ⋮⎟ ⎟ ⎟ ∈ Rµj ×µj , 1⎟ ⎟ 0⎠

and construct the matrices C = (C1 ⋯ Cl ) ∈ Rr×µ ,

(3.72)

J = blockdiag (J1 , . . . , Jl ) ∈ Rµ×µ ,

(3.73)

l

with µ = ∑ µj . Let a ≠ 0 and define j=1

−q ˜ A(σ) = Ir − C(J − aIµ ) ((σ − a)Vq + ... + (σ − a)q V1 ) ,

(3.74)

where q = ind(C, J) is the least integer such that the matrix ⎛ C ⎞ ⎜ CJ ⎟ ⎟ ⎜ ⎟ Sq−1 = ⎜ ⎜ ⋮ ⎟ ⎟ ⎜ ⎝CJ q−1 ⎠

(3.75)

has full column rank and V = (V1 , ..., Vq ) is the generalized inverse of

S1−q

C ⎞ ⎛ ⎜ C(J − aIn )−1 ⎟ ⎟ ⎜ ⎟. =⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎝C(J − aIn )1−q ⎠

(3.76)

B ˜ 1 ). Furthermore, q is the minimal Then βj,i (k) are solutions of (3.2), where A(σ) = σ q A( σ

possible lag of any r × r polynomial matrix with this property. Example 3.18. We want to find a polynomial matrix A(σ), such that the system A(σ)β(k) = 0 has the backward solution β1 (k) =

⎛0⎞ ⎛−1⎞ ⎛1⎞ ⎛1⎞ δ(N − k) + δ(N − k − 1) + δ(N − k − 2) + δ(N − k − 3) ⎝1⎠ ⎝0⎠ ⎝1⎠ ⎝0⎠ ± ± ± ² x1,3

x1,2

x1,1

x1,0

(3.77)


97

We begin by creating the matrices

C = (x1,0 x1,1 x1,2

⎛0 ⎜0 ⎛1 1 −1 0⎞ ⎜ , J =⎜ x1,3 ) = ⎜0 ⎝0 1 0 1⎠ ⎜ ⎝0

1 0 0 0

0 1 0 0

0⎞ 0⎟ ⎟ ⎟. 1⎟ ⎟ 0⎠

(3.78)

Then, we start assuming values for q. Starting from q = 1, the matrix S1−1 = S 0 = C does not have full column rank. For q = 2 the matrix ⎛1 ⎛C⎞ ⎜ ⎜0 =⎜ S2−1 = S1 = ⎝CJ ⎠ ⎜ ⎜0 ⎝0

1 −1 0 ⎞ 1 0 1⎟ ⎟ ⎟ 1 1 −1⎟ ⎟ 0 1 0⎠

(3.79)

has full column rank, and thus A (σ) has lag q = 2. Let a = 1 and

V = ( V1 V2 ) =

⎛ ⎞ C −1 ⎝C(J − I4 ) ⎠

−1

⎛ 1 −2 ⎜ −1 3 ⎜2 2 = ⎜ −1 −1 ⎜ ⎜2 2 ⎝ 1 −1 2

2

0 −1 2 −1 2 1 2

−1⎞ 1⎟ ⎟ ⎟. 0⎟ ⎟ −1⎠

(3.80)

So ⎛ σ 2σ 2 − σ − 1⎞ −2 2 ˜ A(σ) = I2 − C(J − I4 ) ((σ − 1)V2 + (σ − 1) V1 ) = (, 3.81) 3 2 1 ⎝− 21 σ 2 + 12 σ ⎠ 2σ − 2 with Smith form at zero 0 SA(σ) (σ) = ˜

⎛1 0 ⎞ . ⎝0 σ 4 ⎠

(3.82)

Thus, the matrix that we are looking for is ⎛ σ 1 A(σ) = σ 2 A˜ ( ) = 1 σ ⎝ 2 σ − 12

−σ 2 − σ + 2⎞ , − 12 σ 2 + 23 ⎠

(3.83)

with det A(σ) = 1 and Smith form at infinity ⎛σ 2 1 ∞ 0 SA(σ) (σ) = σ 2 SA(σ) ( ) = ˜ σ ⎝0

0⎞ . 1 ⎠ σ2

(3.84)

So far, we have presented methods for constructing a system with given either a forward or a backward behavior. In the last case, we have constructed a dual polynomial ˜ matrix A(σ) with a forward behavior resulting from a given backward behavior. Our aim now is to derive a similar result for the forward behavior as well, i.e. connect it to

98


a corresponding forward behavior in the dual system. This is different from the result derived in Theorem 3.15, were a solution of the system is connected to the backward solution of its dual. Our motivation behind this is to reduce the problem of finding a system with given forward-backward behavior to the problem of finding its dual system which exhbits only a forward behavior. The following Theorem showcases the connection between a forward behavior of (3.2) and a corresponding forward behavior of the dual system (3.5). Theorem 3.19. If βi (k) = Ci Jik x0 is a solution of (3.2), where (Ci ∈ Cr×ni , Ji ∈ Cni ×ni ) is k a finite Jordan pair corresponding to the zero λi ≠ 0 of A(σ), then β˜i (k) = Ci Ji−1 (Ji−1 ) x0

is a solution of (3.5). Proof. Since (Ci , Ji ) is a finite Jordan Pair of the polynomial matrix A(σ), by Theorem 3.1 it satisfies the equation Aq Ci Jiq + ... + A1 Ci Ji + A0 Ci = 0.

(3.85)

Now, substituting β˜i (k) into (3.5) we get −1 −1 k ˜ β˜i (k) = A(σ)C ˜ A(σ) i Ji (Ji ) x0 = k+q+1

= A0 Ci (Ji−1 )

k+1

x0 + ... + Aq Ci (Ji−1 )

x0 =

k+q+1

= (Aq Ci Jiq + ... + A1 Ci Ji + A0 Ci ) (Ji−1 )

(3.85)

x0 = 0.

So β˜i (k) is a solution of the dual system (3.5). In the above theorem, we managed to match a certain forward solution of (3.2) to a forward solution of (3.5), a result that we will utilize in the following. Since the matrix Ji−1 is not in Jordan form, we can find a nonsingular constant matrix U ∈ Rni ×ni such that Ji−1 = U J˜i U −1 where J˜i is in Jordan form. With this ˜ ˜ β(k) change, the solution of A(σ) = 0 can also be written as follows: k k k ˜ β(k) = Ci Ji−1 (Ji−1 ) x0 = Ci U J˜i U −1 U (J˜i ) U −1 x0 = C˜i (J˜i ) (U −1 x0 ) ,

(3.86)

where C˜i = Ci U J˜i . So we see that instead of using the matrix pair (Ci Ji−1 ∈ Cr×ni , Ji−1 ∈ Cni ×ni )

(3.87)


99

where the matrix Ji−1 is not in Jordan form, we can use the matrix pair (C˜j = Ci U J˜i ∈Cr×ni , J˜i = U −1 Ji−1 U ∈ Cni ×ni )

(3.88)

Overall, from Remark 3.12 and Theorems 3.16 and 3.19, the connection between the finite and infinite elementary divisors of A(σ) and the solutions of (3.2) and its dual system (3.5) are be summarized in Table 3.1. Table 3.1: Connection between the solutions of (3.2) and (3.5). ˜ A(σ)β(k) = 0 A(σ)β(k) =0 forward/backward solutions (λi ≠ 0)

⇔

˜i = forward/backward solutions (λ

⇔ ⇔

≠ 0) ˜ i = 0) strictly backward solutions (λ strictly forward solutions (λi = 0) 1 λi

strictly forward solutions (λi = 0) ˜ i = 0) strictly backward solutions (λ

Summarizing our results, in order to construct an AR-representation for a certain forward/backward behavior, one must follow Algorithm 2. Algorithm 2 Construction of an AR-Representation with given forward and backward behavior (for λi ≠ 0). Step 1. Transform the finite Jordan pairs Ci ∈ Cr×ni , Ji ∈ Cni ×ni that correspond to solutions of the form β(k) = Ci Jik x0 (for λi ≠ 0), to the finite Jordan Pairs (3.88), that k ˜ correspond to solutions of the form β(k) = C˜i (J˜i ) (U −1 x0 ) of the dual system that we

are looking for. Step 2. Construct infinite Jordan pairs of the matrix A(σ) as in Theorem 3.17. These correspond to the finite Jordan pairs of the dual system at σ = 0. ˜ Step 3. Construct the polynomial matrix A(σ) using the method presented in Theorem 3.10. ˜ 1 ) and thus the AR representation Step 4. Get the polynomial matrix A(σ) = σ q A( σ (3.2) that we are looking for.

It should be mentioned again that A(σ) is not the only polynomial matrix satisfying the given behavior. Different choices of a will result in different matrices that are left unimodular equivalent, as mentioned in Remark 3.11.

100


Example 3.20. We want to construct an AR-representation with the following forward and backward solutions: β1 (k) =

⎛ 1 ⎞ k ⎛2⎞ k−1 2 + k2 , ⎝−1⎠ ⎝0⎠ ± ² β1,0

β1,1

β2 (k) =

(3.89)

⎛0⎞ ⎛−1⎞ δ(N − k − 1). δ(N − k) + ⎝3⎠ ⎝−1⎠ ± ²

(3.90)

x1,0

x1,1

First, define the matrix pairs C1 = (β1,0 β1,1 ) = J1 −1 =

⎛ 21 ⎝0

⎛2 ⎝0

1 ⎞ , −1⎠

J1 =

⎛2 1⎞ , ⎝0 2⎠

− 41 ⎞ ⎛1 0 ⎞ ⎛ 21 1 ⎞ ⎛1 0 ⎞ = , 1 1 1 ⎠ ⎝ ⎠ ⎝ ⎠ ⎝ ⎠ 0 0 −4 0 − 2 2 4 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(3.91) (3.92)

U J˜1 U −1

⎛1 0⎞ , C˜1 = C1 U J˜1 = ⎝0 2⎠ C2 = (x1,0 x1,1 ) =

(3.93)

⎛0 −1⎞ ⎛0 1⎞ , J2 = . ⎝3 −1⎠ ⎝0 0⎠

(3.94)

The complete matrix pair is C = (C˜1 C2 ) =

⎛J˜1 J= ⎝0

⎛1 0 ⎝0 2

1 ⎛2 1 1 0⎞ ⎜ ⎜0 =⎜ 2 J2 ⎠ ⎜ ⎜0 0 ⎝0 0

0 −1⎞ , 3 −1⎠ 0 0 0 0

0⎞ 0⎟ ⎟ ⎟. 1⎟ ⎟ 0⎠

(3.95)

(3.96)

For q = 1, the matrix S0 = C does not have full column rank. For q = 2, the matrix ⎛1 ⎛C⎞ ⎜ ⎜0 S1 = = ⎜1 ⎝CJ ⎠ ⎜ ⎜2 ⎝0

0 2 1 1

0 −1⎞ 3 −1⎟ ⎟ ⎟, 0 0⎟ ⎟ 0 3⎠

(3.97)

has full column rank, so the resulting matrix will be A(σ) = A0 + A1 σ + A2 σ 2 ∈ R[σ]2×2 . Let a = 1, and compute the matrix −2 2 ˜ A(σ) = I2 − C(J − aI4 ) {(σ − a)V2 + (σ − a) V1 } ,

(3.98)

3.4. CONSTRUCTION OF A SYSTEM WITH PRESCRIBED BEHAVIOR where −1

( V1 V2 ) =

⎛ ⎞ C ⎝C(J − aI4 )−1 ⎠

⎛ 57 − 25 ⎜− 3 1 ⎜ = ⎜ 85 10 2 ⎜ ⎜ 15 15 ⎝ 2 −2 5 5

1 5 3 − 10 4 15 1 5

101

− 52 ⎞ 1 ⎟ 10 ⎟ ⎟. − 51 ⎟ ⎟ 2⎠ −

(3.99)

5

The resulting matrix is ⎛ 1 (6 − 23σ + 22σ 2 ) ˜ A(σ) = 53 ⎝ 5 (1 − 3σ + 2σ 2 )

− 25 ( −σ + σ 2 )⎞ . 1 2 ⎠ 5 (σ + 4σ )

(3.100)

Therefore, the matrix that we are looking for is ⎛ 15 (6σ 2 − 23σ + 22) 1 ˜ A(σ) = σ A ( ) = σ ⎝ 35 (σ 2 − 3σ + 2) 2

2 5 1 5

(σ − 1)⎞ . (σ + 4)⎠

(3.101)

Indeed, the vector functions β1 (k), β2 (k) are solutions of the system, i.e. they satisfy A(σ)βi (k) = 0, for i = 1, 2. For a different choice of the parameter a, for example a = −1, the resulting matrix is A1 (σ) =

1 ⎛ 15 (6 − 7σ + 2σ 2 ) ⎝ 51 (−2 − σ + σ 2 )

2 ⎞ 45 (1 + σ) 16+σ ⎠ 15

(3.102)

and as expected, the two matrices are left equivalent, that is A(σ) =

⎛ 51 − 45 ⎞ 5 A1 (σ), 3 ⎠ ⎝ 18 5 5 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(3.103)

U

with U unimodular. The above algorithm may display computational difficulties in the case where the forward behavior has pure polynomial vector valued functions (since these are connected with the finite elementary divisors of A(σ) at zero). In that case, the Jordan matrix J, will have determinant equal to zero and thus, will not be invertible. This problem can easily be surpassed by replacing σ with σ + b in A(σ), where b does not correspond to a zero of the polynomial matrix. This replacement moves all possible zeros of A(σ) to non-zero places. The following Remark indicates this solution. Remark 3.21. [33, Proof of Theorem 7.3] Let A(σ) be a polynomial matrix. a) If (C, J) is a finite Jordan Pair of A(σ) then (C, J + bIn ) is a finite Jordan Pair of A(σ − b). b) If (C∞ , J∞ ) is an infinite Jordan Pair of A(σ) then (C∞ , J∞ (Iµ + bJ∞ )−1 ) is an infinite Jordan Pair of A(σ − b).

102


The use of the above Lemma is showcased in the following example Example 3.22. We want to construct an AR-representation of the form (3.2) with the following forward and backward behavior β1 (k) =

⎛1⎞ ⎛1⎞ δ(k − 1), δ(k) + ⎝0⎠ ⎝−1⎠ ± ² β1,0

β1,1

β2 (k) =

⎛−1⎞ ⎛−1⎞ δ(N − k) + δ(N − k − 1). ⎝−1⎠ ⎝1⎠ ² ² x1,1

(3.104)

(3.105)

x1,0

First, define the matrix pairs C1 = (β1,0 β1,1 ) = C2 =

⎛1 1 ⎞ ⎛0 1⎞ , J1 = , ⎝0 −1⎠ ⎝0 0⎠

⎛−1 −1⎞ ⎛0 1⎞ , J2 = . ⎝ 1 −1⎠ ⎝0 0⎠

(3.106) (3.107)

We observe that the matrix J1 is not invertible. So in order to move on, we set b = 2 and use the pairs C1∗ = C1 = (β1,0 β1,1 ) = J1∗ = J1 + bI2 =

⎛1 1 ⎞ , ⎝0 −1⎠

⎛2 1 ⎞ ∗ ⎛−1 −1⎞ , C 2 = C2 = , ⎝0 2 ⎠ ⎝ 1 −1⎠

J2∗ = J2 (I2 + bJ2 )−1 =

⎛0 1⎞ = J2 . ⎝0 0⎠

(3.108) (3.109) (3.110)

Working with the above pairs we construct the matrix A(σ − b) = A(σ − 2). Let also J1∗−1

⎛ 21 − 41 ⎞ ⎛1 0 ⎞ ⎛ 12 1 ⎞ ⎛1 0 ⎞ = = , ⎝ 0 12 ⎠ ⎝0 −4⎠ ⎝ 0 12 ⎠ ⎝0 −4⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ U

J˜1

(3.111)

U −1

⎛1 1 ⎞ ⎛1 0 ⎞ ⎛ 12 1 ⎞ ⎛ 12 −1⎞ ˜ ˜ C1 = C1 U J1 = = . ⎝0 −1⎠ ⎝0 −4⎠ ⎝ 0 21 ⎠ ⎝ 0 2 ⎠

(3.112)

The complete matrix pair is C = (C˜1 C2 ) =

⎛ 12 −1 ⎝0 2

−1 −1⎞ , 1 −1⎠

(3.113)


⎛J˜1 J= ⎝0

1 ⎛2 1 1 0⎞ ⎜ ⎜0 =⎜ 2 J2 ⎠ ⎜ ⎜0 0 ⎝0 0

0 0 0 0

0⎞ 0⎟ ⎟ ⎟. 1⎟ ⎟ 0⎠

103

(3.114)

Now start assuming values for q. For q = 1 the matrix S0 = C does not have full column rank. For q = 2, the matrix 1 ⎛ 2 −1 −1 −1⎞ ⎟ ⎛C⎞ ⎜ ⎜ 0 2 1 −1⎟ ⎟ S2 = = ⎜1 ⎟ ⎝CJ ⎠ ⎜ ⎜ 4 0 0 −1⎟ ⎝0 1 0 1 ⎠

has det(S2 ) =

1 4

(3.115)

≠ 0 so it has full rank. So the resulting matrix will be A(σ) = A0 + A1 σ + A2 σ 2 ∈ R[σ]2×2

(3.116)

Let a = 1, and compute the matrix A˜∗ (σ) as −2 2 A˜∗ (σ) = I2 − C(J − aI4 ) ((σ − a)V2 + (σ − a) V1 ) ,

where −1

(V1 ∣ V2 ) =

⎛ ⎞ C −1 ⎝C(J − aI4 ) ⎠

⎛−8 −2 ⎜1 0 ⎜ =⎜ ⎜−4 0 ⎜ ⎝−2 −1

(3.117)

−5

1⎞ − 21 ⎟ ⎟ ⎟. −2 1 ⎟ ⎟ −1 0 ⎠ 1 2

(3.118)

The resulting matrix is ⎛ −σ + 2σ 2 −σ + σ 2 ⎞ A˜∗ (σ) = . ⎝−1 + 3σ − 2σ 2 −1 + σ + σ 2 ⎠

(3.119)

The matrix A(σ − 2) is the dual of the above matrix ⎛ 2−σ 1−σ ⎞ 1 A(σ − 2) = σ 2 A˜∗ ( ) = σ ⎝−2 + 3σ − σ 2 1 + σ − σ 2 ⎠

(3.120)

and the matrix that we were initially searching for is given by substituting σ by σ + 2 and is equal to A(σ) =

⎛ −σ −1 − σ ⎞ , ⎝−σ − σ 2 −1 − 3σ − σ 2 ⎠

(3.121)

with C SA(σ) (σ) =

⎛1 0 ⎞ , ⎝0 σ 2 ⎠

⎛1 1 ∞ 0 ( ) = σ2 SA(σ) (σ) = σ 2 SA(σ) ˜ σ ⎝0

0 ⎞ ⎛σ 2 0⎞ = . 1 ⎠ ⎝ 0 1⎠ σ2

As expected, it holds that A(σ)βi (k) = 0, for i = 1, 2.

(3.122) (3.123)

104


3.4.2

Second method: Reduction to a linear system of equations

The first approach to the modeling problem was through an extension of the method in [33]. Although this method is successful in constructing a system of the form (3.2) with the prescibed behavior, it has a number of disadvantages. It only works for regular systems and in addition, it is difficult to understand how the formula (3.57) provided in [33] is derived, since it requires a deep understanding of the underlying theory of matrix polynomials. Contrary to the first method presented in subection 3.4.1, the second approach to the modeling problem is much more straightforward. We will take advantage of the results of Theorems 3.4 and 3.9 which state that in order for a set of vector valued functions to be a solution (forward or backward) of (3.2), their coefficients must satisfy a specific set of linear equations that involve the matrices A0 , ..., Aq . Thus, we can simply solve this system in terms of the matrices Ai and obtain the desired polynomial matrix A(σ). This method is much more easy to understand and implement and it can also be used for the construction of non-regular systems. First, let’s consider the forward solutions of (3.2). As stated in Theorem 3.4, in F order for the vector valued function βi,j,φ (k) in (3.25) to be a solution of A(σ)β(k) = 0, i i the vectors βj,0 , ..., βj,n need to satisfy (3.27). This system of equations can be i,j −1

rewritten in matrix form as (ni,j −1)

( A (n

(λi )

i,j −1)!

i ⋯ 0r×1 ⎞ ⎛ βj,0 ⎜ ⋱ ⋮ ⎟ ⋯ A(λi )) ⎜ ⋮ ⎟ = 0r×ni,j i i ⎝βj,n −1 ⋯ βj,0 ⎠ i,j ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.124)

Wi,j

Solving the above system of equations, we can obtain the matrices, A(ni,j −1) (λi ) ..., A′ (λi ), A(λi ), that represent the values of A(σ) and its derivatives at λi . Thus, the evaluation of A(σ) is reduced to a Hermite interpolation problem. Alternatively, using the relation q ε+1 ε A(ε) (λi ) = ( )Aq λi q−ε + ... + ( )Aε+1 λi + ( )Aε ⇒ ε ε! ε ε q q−ε ⎛(ε)λi Ir ⎞ A(ε) (λi ) ⎟ = (Aq ⋯ Aε ) ⎜ ⋮ ⎜ ⎟ ε! ⎝ ⎠ Ir

(3.125)

(3.126)


105

for ε = 0, ..., ni,j − 1 we rewrite (3.124) as follows: (Aq ⋯ A0 ) Qi,j Wi,j = 0r×ni,j

(3.127)

where

Qi,j

q q−(ni,j −1) Ir ⎛(ni,j −1)λi ⎜ ⋮ ⎜ ⎜ ni,j ⎜ (ni,j −1)λi Ir ⎜ =⎜ ⎜ Ir ⎜ ⎜ ⎜ ⋮ ⎜ ⎝ 0r

q ⋯ qλq−1 i Ir λi Ir ⎞ ⋱ ⋮ ⋮ ⎟ ⎟ ⎟ ⋱ ⋮ ⋮ ⎟ ⎟ ⎟ ∈ Cr(q+1)×rni,j 2 ⎟ 2λi Ir λi Ir ⎟ ⎟ ⋱ Ir λi Ir ⎟ ⎟ ⋯ 0r Ir ⎠

(3.128)

with i = 1, 2, ..., `, j = z, z + 1, . . . , r. In case where ni,j > q, the derivatives of A(σ) of order higher than q in (3.124) will be equal to zero. In this case, the matrices Qi,j , Wi,j in (3.127) take the following simplified form (Aq

q i i ⎞ ⎛ Ir ⋯ λi Ir ⎞ ⎛βj,ni,j −q−1 ⋯ βj,0 ⎟ ⎜ ⎟ = 0r×ni,j ⋮ ⋱ ⋯ A0 ) ⎜ ⋮ ⎟ ⎜⋮ ⋱ ⎟⎜ ⎟ ⎜ i ⎝0r ⋯ Ir ⎠ ⎝ β i ⋯ ⋯ βj,0 ⎠ j,ni,j −1

(3.129)

Thus, our problem has been reduced to solving a linear system of equations over R. F That is, given a time sequence in the form of βi,j,φ (k), we can solve (3.127) in terms

of the unknowns A0 , A1 , ..., Aq in order to construct A(σ). For the solution of such linear systems, numerous numerical methods exist. The most commonly used are the Singular Value Decomposition (SVD) and the QR Decomposition [110]. These results give rise to Algorithm 3 for the construction of a system that satisfies a desired forward behavior. As with the Algorithm 2 presented in the previous subsection, the following holds. Remark 3.23. In case where in Step 1, there exists no q such that n = rq, the resulting F matrix A(σ) will describe a system of algebraic/difference equations with βi,j,φ (k) as

part of its solution space, which will include additional vector valued functions linearly F independent from βi,j,φ (k). (This holds true for all the algorithms presented in this

subsection) Similar to the Remark 3.11 in the previous subsection, all the matrices constructed by Algorithm 3 are left equivalent (see Definition 1.22).

106


Algorithm 3 Construction of a system with given forward behavior. Suppose that a finite number of vector valued functions of the form k k−φ i F i i βi,j,φ (k) ∶= λki βj,φ + kλk−1 βj,0 i βj,φ−1 + ... + ( )λi φ F i i i βi,j,φ (k) ∶= δ(k)βj,φ + δ(k − 1)βj,φ−1 + ... + δ(k − φ)βj,0

(3.130) (3.131)

are given, with i = 1, 2, ..., `, j = z, z + 1, . . . , r, φ = 0, 1, ..., ni,j − 1. `

r

Step 1 Define n = ∑ ∑ nij . i=1 j=z

If r∣n, then q=

n r

(3.132)

else n q =[ ]+1 r

(3.133)

where [⋅] denotes the integer part of nr . Step 2 Construct the matrices Qi,j , Wi,j , defined in (3.124) and (3.128). Step 3 Construct the combined matrices Qi = (Qi,z ⋯ Qi,r ) ∈ Cr(q+1)×rni Wi = (Wi,z ⋯ Wi,r ) ∈ Crni ×ni

i = 1, ..., `

(3.134)

where ni = ∑rj=z ni,j and Q = (Q1 ⋯ Q` ) ∈ C(q+1)r×nr

(3.135)

W = blockdiag (W1 , . . . , W` ) ∈ Cnr×n

(3.136)

Step 4 Solve the system of equations (Aq ⋯ A0 )QW = 0r×n

(3.137)

over R in terms of the unknown matrices Ai . Step 5 Choose the free entries aij of each matrix Ai so that det A(σ) ≠ 0.

Remark 3.24. Every matrix that is left equivalent to the polynomial matrix A(σ) constructed in Algorithm 3 gives rise to a model with exactly the same forward behavior


107

with (3.2). That is, all matrices A1 (σ) = U (σ)A(σ)

(3.138)

where U (σ) is unimodular, satisfy A1 (σ)βj,i (k) = 0. This is because multiplication by U (σ) does not alter the finite zero structure of A(σ) and thus, the forward behavior of the corresponding system remains the same (see [127]). Example 3.25. Let the following vector valued functions β1 (k) =

⎛ 2 ⎞ k ⎛ 4 ⎞ k−1 ⎛ 2 ⎞ k(k − 1) k−2 k2 + 2 + 2 2 ⎝ 0 ⎠ ⎝ 1 ⎠ ⎝ 3 ⎠

(3.139a)

⎛ 1 ⎞ k 3 ⎝ −1 ⎠

(3.139b)

β2 (k) =

We want to construct an AR-representation A(σ)β(k) = 0 that has the prescribed functions in its solution space. Step 1 These vector valued functions correspond to the zeros λ1 = 2, λ2 = 3, with multiplicities n1 = 3, n2 = 1. We have n = n1 + n2 = 3 + 1 = 4, µ = 0 and r = 2. From (1.57), it holds that n + 0 = 2q ⇒ q = 2. So the matrix A(σ) is A(σ) = A2 σ 2 + A1 σ + A0 ∈ R[σ]2×2

(3.140)

with expected Smith form C SA(σ) (σ) =

⎛1 ⎞ 0 3 ⎝0 (σ − 3)(σ − 2) ⎠

(3.141)

Thus, the functions β1 (k), β2 (k), following the notation of Algorithm (3), are identified as F β1,2,2 (k) =

⎛2⎞ k ⎛4⎞ k−1 ⎛2⎞ k(k − 1) k−2 2 + k2 + 2 , λ1 = 2 2 ⎝3⎠ ⎝1⎠ ⎝0⎠ ± ± ± 1 1 1 β2,2

β2,1

(3.142a)

β2,0

F β2,2,0 (k) =

⎛1⎞ k 3 , ⎝−1⎠ ² 2

λ2 = 3

(3.142b)

β2,0

F Of course, since we desire the vector β1,2,2 (k) to be a solution of (3.2), the vectors F β1,2,0 (k) =

⎛2⎞ k 2 , ⎝0⎠ ± 1 β2,0

F β1,2,1 (k) =

⎛2⎞ k−1 ⎛4⎞ k k2 + 2 ⎝0⎠ ⎝1⎠ ± ± 1 1 β2,0

β2,1

(3.142c)

108

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS will also be solutions of the system. Yet, only the vector function (3.142a) is required to construct the matrices Q1 , W1 .

Step 2 Construct the matrices ⎛1 ⎜0 ⎜ 22 I2 ⎞ ⎜ ⎜0 ⎜ ⎟ = 2I2 ⎟ ⎜ ⎜0 I2 ⎠ ⎜ ⎜ ⎜0 ⎜ ⎝0

⎛I2 2 ⋅ 2I2 Q1 = ⎜ I2 ⎜02 ⎝02 02

4 0 2 0 1 0

0 0 2 0 4 1

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 2⎟ ⎟ 0⎠

(3.144)

⎛1⎞ ⎝−1⎠

(3.145)

W = blockdiag (W1 , W2 )

(3.146)

1 02×1 02×1 ⎛ β2,0 1 1 W1 = ⎜ ⎜ β2,1 β2,0 02×1 1 1 1 ⎝ β2,2 β2,1 β2,0

⎛ Q2 = ⎜ ⎜ ⎝

⎛9 ⎜0 ⎜ 32 I2 ⎞ ⎜ ⎜3 ⎜ = 3I2 ⎟ ⎟ ⎜ ⎜0 I2 ⎠ ⎜ ⎜ ⎜1 ⎜ ⎝0

0⎞ 9⎟ ⎟ ⎟ 0⎟ ⎟ ⎟, 3⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠

4 0 1 0 0 0

0⎞ 4⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 2⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠

0 4 0 1 0 0

0 1 0 0 0 0

⎛2 ⎜0 ⎜ ⎞ ⎜ ⎜4 ⎟=⎜ ⎟ ⎜ ⎜1 ⎠ ⎜ ⎜ ⎜2 ⎜ ⎝3

2 W2 = β2,0 =

(3.143)

Step 3 Construct the combined matrices Q = ( Q1 Q2 ) , Step 4 Solve the system ( A2 A1 A0 ) QW = 02×4 where Ai =

⎛ai1 ai2 ⎞ , ⎝ai3 ai4 ⎠

i = 0, 1, 2

(3.147)

(3.148)

The resulting matrices are A0 =

⎛− 5a412 − 9a222 − 11a4 12 − 15a2 22 ⎞ ⎝− 5a414 − 9a224 − 11a4 14 − 15a2 24 ⎠

(3.149)

⎛ 7a812 + 11a4 22 a12 ⎞ ⎝ 7a814 + 11a4 24 a14 ⎠

(3.150)

A1 =

3.4. CONSTRUCTION OF A SYSTEM WITH PRESCRIBED BEHAVIOR A2 =

⎛− a812 − a422 a22 ⎞ ⎝− a814 − a424 a24 ⎠

109

(3.151)

Step 5 Now we can choose values for a12 , a14 , a22 , a24 , such that Ai ∈ Rr×r and det A(σ) ≠ 0, since the wrong choice of the free variables may lead to linear dependence of the columns of A(σ). For example, let a12 = 1, a14 = a22 = a24 = −1 and the resulting matrix is σ ⎛ 13 − 15σ 8 + 8 A(σ) = 234 29σ 2 ⎝ 4 − 8 + 3σ8 2

19 4 41 4

+ σ − σ2⎞ − σ − σ2⎠

(3.152)

with det A(σ) = 41 (σ − 3)(σ − 2)3 . We can easily verify that the vector valued functions β1 (k), β2 (k) are solutions of the system, i.e. A(σ)βi (k) = 0, i = 1, 2. It should be clear by Remark 3.24 that different choices of the free parameters, such that det A(σ) ≠ 0, will lead to a matrix that is left equivalent to the previous one constructed. For example, choosing a14 = 4, a22 = 2, a12 = a24 = −0, we get the matrix σ ⎛−9 + 11σ 2 − 2 A1 (σ) = σ2 ⎝ −5 + 7σ 2 − 2

2

and it holds A1 (σ) =

−15 + 2σ 2 ⎞ −11 + 4σ ⎠

⎛−1 −1⎞ A(σ) ⎝ 2 −2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.153)

(3.154)

U (σ)

where U (σ) ≡ U is unimodular. Alternatively, we solve the same problem by using Hermite Interpolation. To begin with, we are going to solve (3.124) over R for λ1 = 2 with n1 = 3 and λ2 = 3 with n2 = 1. The corresponding systems are (

A′′ (2) 2

1 0 0 ⎛ β2,0 1 1 A′ (2) A(2)) ⎜ ⎜ β2,1 β2,0 0 1 1 1 ⎝ β2,2 β2,1 β2,0

⎞ ⎟ = 02×3 ⎟ ⎠

(3.155)

and 2 A(3)β2,0 = 02×1

(3.156)

Solving the above systems, we get the matrices A(2) =

⎛0 6a12 − 2a22 ⎞ , ⎝0 6a32 − 2a42 ⎠

A′ (2) =

⎛a22 − 3a12 2a22 − 8a12 ⎞ ⎝a42 − 3a32 2a42 − 8a32 ⎠

(3.157)

110

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS A′′ (2) ⎛a12 a22 ⎞ = , 2 ⎝a32 a42 ⎠

A(3) =

⎛a22 − 2a12 a22 − 2a12 ⎞ ⎝a42 − 2a32 a42 − 2a32 ⎠

(3.158)

where a12 , a22 , a32 , a42 ∈ R are free parameters. The table of divided differences is the following σ

A(σ)

First divided dif- Second divided dif- Third divided differences ferences

z0 = 2 z1 = 2 z2 = 2 z3 = 3

A(2) A(2) A(2) A(3)

ferences

A′ (2) A′ (2)

A′′ (2) 2 A(3)−A(2)−A′ (2) 3−2

A(3)−A(2) 3−2

′

(2)−A ( A(3)−A(2)−A 3−2

′′ (2)/2

)

and the Hermite Interpolation Polynomial [13] is: A(σ) = A(2) + A′ (2)(σ − 2) +

A′′ (2) A(3) − A(2) − A′ (2) − A′′ (2)/2 (σ − 2)2 + (σ − 2)3 ⇒ 2 3−2

10a12 − 2a22 − (7a12 − a22 )σ + a12 σ 2 22a12 − 2a22 − (8a12 + 2a22 )σ + a22 σ 2 ) 10a32 − 2a42 − (7a32 − a42 )σ + a32 σ 2 22a32 − 2a42 − (8a32 + 2a42 )σ + a42 σ 2

A(σ) = (

(3.159) with determinant det A(σ) = (σ − 3)(σ − 2)3 (a12 a42 − a22 a32 )

(3.160)

As expected, the Hermite polynomial is of order q = 2, with Smith Form C SA(σ) (σ) =

⎛1 ⎞ 0 3 ⎝0 (σ − 3)(σ − 2) ⎠

and A(σ)βi (k) = 0

i = 1, 2

(3.161)

As a last example of a system with a forward behavior, we describe the case where the matrix A(σ) has complex zeros. Example 3.26. Let the following vector valued function √ π ⎞ ⎛ 2 cos( 2π 3 k + 4) β(k) = π ⎝ 2 cos( 2π ⎠ 3 k + 6)

(3.162)

We want to construct an AR-representation A(σ)β(k) = 0 that has the prescribed functions in its solution space. This vector valued function can be equivalently written as

3.4. CONSTRUCTION OF A SYSTEM WITH PRESCRIBED BEHAVIOR √ k √ k ⎛ 12 − 12 i ⎞ 1 ⎛ 21 + 12 i ⎞ 1 3 3 β(k) = √3 1 (− + i) + √3 1 (− − i) 2 2 ⎝ 2 − 2 i⎠ 2 ⎝ 2 + 2 i⎠ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ β1 (k)

111

(3.163)

β2 (k)

So instead of β(k), we should equivalently consider the following two complex vector valued functions β1 (k) and β2 (k). Step 1 These vector valued functions correspond to the zeros λ1 = − 21 + − 21 −

√ 3 2 i,

√ 3 2 i,

λ2 =

with multiplicities n1 = 1, n2 = 1. Since n = n1 + n2 = 2, µ = 0 and r = 2,

it holds that n + 0 = 2q ⇒ q = 1. So A(σ) = A1 σ + A0 ∈ R[σ]2×2

(3.164)

⎞ ⎛1 0 ⎝0 1 + σ + σ 2 ⎠

(3.165)

with expected Smith form C SA(σ) (σ) =

Thus, the functions β1 (k), β2 (k), following the notation of Algorithm 3.3, are identified as √ k ⎛ 21 + 21 i ⎞ 1 3 = √3 1 (− + i) , 2 ⎝ 2 + 2 i⎠ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

√ 1 3 λ1 = − + i 2 2

√ k 1 1 ⎞ ⎛ − i 1 3 F β2,2,0 (k) = √23 21 (− − i) , 2 ⎝ 2 − 2 i⎠ 2 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

√ 1 3 λ2 = − − i 2 2

F β1,2,0 (k)

1 β2,0

(3.166a)

(3.166b)

2 β2,0

Step 2 Construct the matrices

√

⎛(− 12 + 23 i) I2 ⎞ ⎛ 1 + 1i ⎞ 1 Q1 = , W1 = β2,0 = √23 21 ⎝ ⎠ ⎝ 2 + 2 i⎠ I2

(3.167)

√

⎛(− 12 − 23 i) I2 ⎞ ⎛ 1 − 1i ⎞ 2 Q2 = , W2 = β2,0 = √23 21 ⎝ ⎠ ⎝ 2 − 2 i⎠ I2

(3.168)

Step 3 Construct the combined matrices Q = (Q1 Q2 )

W = blockdiag(W1 , W2 )

(3.169)

112


Step 4 Solve the system (A1 A0 ) QW = 02×2

(3.170)

where Ai =

⎛ai1 ai2 ⎞ ∈ R2×2 , i = 0, 1, 2 ⎝ai3 ai4 ⎠

(3.171)

Since in (3.170), the matrices Q, W are complex, this is equivalent to solving the two following systems R [(A1 A0 ) QW ] = 02×2

(3.172a)

I [(A1 A0 ) QW ] = 02×2

(3.172b)

where R, I denote the real and imaginary parts of the expression respectively. The resulting matrices are √ √ ⎛ − (1 + 3) (a11 + 3a12 ) √ √ A0 = ⎝− (1 + 3) (a13 + 3a14 ) A1 =

1 2 1 2

√ √ (3 + 3) a11 + (2 + 3) a12 ⎞ √ √ (3 + 3) a13 + (2 + 3) a14 ⎠

⎛a11 a12 ⎞ ⎝a13 a14 ⎠

(3.173) (3.174)

and the matrix A(σ) is

√ √ ⎛−a11 − 3a11 − 3a12 − 3a12 + a11 σ √ √ ⎝−a13 − 3a13 − 3a14 − 3a14 + a13 σ

A(σ) = 3a11 2 3a13 2

√ √ 11 + 3a + 2a + 3a12 + a12 σ ⎞ 12 √2 √ 3a13 + 2 + 2a14 + 3a14 + a14 σ ⎠

(3.175)

with det A(σ) = −(a12 a13 − a11 a14 )(1 + σ + σ 2 )

(3.176)

We can easily verify that the vector valued functions β1 (k), β2 (k) are solutions of the system, that is A(σ)βi (k) = 0, i = 1, 2. So far we have shown that the problem of constructing a system with a given forward behavior can be reduced to solving a linear system of equations, in terms of the matrices A0 , ..., Aq . A similar result can be derived for the backward behavior of a system as well. As shown in Theorem 3.9, in order for the vector valued function B βj,φ (k) defined in (3.44) to be solutions of(3.2), the vectors xj,0 , ..., xj,µj −1 in (3.36)


113

need to satisfy the system of equations (3.46). Equations (3.46a) and (3.46b) can be rewritten as

(Aq

⎛xj,0 xj,1 ⋯ xj,q ⋯ xj,q+ˆqj −1 ⎞ ⎜ 0 xj,0 ⋯ ⋮ ⎟ ⋮ ⋮ ⎜ ⎟ ⎟ = 0r×(q+ˆqj ) ⋯ A0 ) ⎜ ⎜ ⋮ ⎟ ⋱ ⋯ ⋮ ⋮ ⋮ ⎜ ⎟ ⎝ 0 ⋯ 0 xj,0 ⋯ xj,ˆqj −1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.177a)

z QB j

z r(q+1)×(q+ˆ qj ) , for the case of i.z.e.d. (j = k + 1, ..., r) and with QB j ∈R

(Aq

⎛xj,0 xj,1 ⋯ xj,q−qj −1 ⎞ ⎜ 0 xj,0 ⋯ ⎟ ⋮ ⎜ ⎟ ⎟ = 0r×(q−qj ) ⋯ Aqj +1 ) ⎜ ⎜ ⋮ ⎟ ⋱ ⋯ ⋮ ⎜ ⎟ ⎝ 0 ⋯ 0 xj,0 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.177b)

Bp

Qj B

Qj p ∈ Rr(q+1)×(q−qj ) , for the case of i.p.e.d. (j = 2, ..., k). The above system of equations can be used to solve the inverse problem. That B is, given a time sequence in the form of βj,φ (k), we can always solve the system of

linear equations (3.177) in terms of the unknown matrices A0 , A1 , ..., Aq and therefore construct the AR-Representation (3.2). These results give rise to Algorithm 4 for the construction of a system that satisfies a desired backward behavior. As was showcased in the previous sections the infinite elementary divisors of A(σ), that generate the backward solutions of (3.2), are connected to the finite elementary divisors of the dual matrix at λ = 0, that in turn generate forward solutions for the ˜ dual system A(σ)β(k) = 0. The connection between these two behaviors was explicitly given in Theorems 3.15 and 3.16. Under the above consideration that the backward solutions of (3.2) give rise to forward solutions of its dual system, Remark 3.24 can also be applied here, as follows. Remark 3.27. Every polynomial matrix A1 (σ) whose dual is left equivalent to the dual of the polynomial matrix A(σ) constructed in Algorithm 4 gives rise to a model with exactly the same backward behavior with (3.2). That is, all matrices A1 (σ) such that: ˜ A˜1 (σ) = U (σ)A(σ) B where U (σ) is unimodular, satisfy A1 (σ)βj,φ (k) = 0.

(3.183)

114


Algorithm 4 Construction of a system with given backward behavior. Suppose that a finite number of functions of the form B βj,φ (k) ∶= xj,φ δ(N − k) + ... + xj,0 δ(N − (k + φ))

(3.178)

are given, where j = 2, ..., r, φ = 0, ..., µj − 1. r

Step 1 Define µ ∶= ∑ µj . j=2

If r∣µ, then q=

µ r

(3.179)

else µ q =[ ]+1 r

(3.180)

where [⋅] denotes the integer part of µr . B

p z Step 2 Construct the matrices QB j and/or Qj defined in (3.177).

Step 3 Construct the matrix r(q+1)×µ QB = (QB ⋯ QB r )∈R 2

(3.181)

B

p z that can be a combination of the matrices QB j and Qj , depending on the form

B of βj,φ (k) that are given.

Step 4 Solve the system of equations ( Aq ⋯ A0 ) QB = 0r×µ

(3.182a)

( Aq ⋯ Aqk +1 ) QB = 0r×µ

(3.182b)

or

in terms of the unknown matrices Ai . Step 5 Choose the free entries aij of each matrix Ai so that det A(σ) ≠ 0.

Example 3.28. Let the following vector valued function ⎛1⎞ ⎛1⎞ ⎛1⎞ ⎜ ⎟ ⎜ ⎟ ⎟ β1 (k) = ⎜0⎟δ(N − k) + ⎜−1⎟δ(N − k − 1) + ⎜ ⎜ 0 ⎟δ(N − k − 2) ⎝1⎠ ⎝1⎠ ⎝−1⎠

(3.184)


115

We want to construct an AR-representation A(σ)β(k) = 0 that has the prescribed function in its solution space. Step 1 This vector valued function corresponds to an infinite elementary divisor of order 3, so µ = 3. Since n = 0, r = 3, it holds µ + 0 = 3q ⇒ q = 1. So A(σ) = A1 σ + A0 ∈ R[σ]3×3

(3.185)

The Smith form of the dual matrix has the form 0 ⎞ ⎛σ µ1 0 0 µ ⎟ SA(σ) (σ) = ⎜ ˜ ⎜ 0 σ 2 0 ⎟ ⎝ 0 0 σ µ3 ⎠

(3.186)

with µ1 = 0 and µ = µ1 + µ2 + µ3 = 3. The possible values of µ2 , µ3 are ( µ2 = 1, µ3 = 2) or (µ2 = 0, µ3 = 3). Since β1 (k) corresponds to a zero of multiplicity 3 of ˜ A(σ), only the second case is accepted. In addition, µ3 corresponds either to an infinite pole or to an infinite zero elementary divisor. Since q=1, we have either µ3 = q − q3 = 3 ⇒ 1 − q3 = 3 ⇒ q3 = −2 which is rejected, or µ3 = q + qˆ3 = 3 ⇒ 1 + qˆ3 = 3 ⇒ qˆ3 = 2 which is accepted. So µ3 corresponds to an infinite zero elementary divisor of A(σ), and 0 SA(σ) (σ) ˜

⎛1 0 0 ⎞ ⎟ =⎜ ⎜0 1 0 ⎟ ⎝0 0 σ 3 ⎠

(3.187)

Thus, following the notation of Algorithm 4, β1 (k) is identified as

B β3,2 (k)

⎛1⎞ ⎛1⎞ ⎛1⎞ ⎜ ⎟ ⎜ ⎟ ⎟ = ⎜0⎟ δ(N − k) + ⎜−1⎟ δ(N − k − 1) + ⎜ ⎜ 0 ⎟ δ(N − k − 2) ⎝1⎠ ⎝1⎠ ⎝−1⎠ ± ² ² x3,2

x3,1

(3.188)

x3,0

Step 2 & 3 By using equation (3.182b), construct the matrix

QB =

⎛ x3,0 ⎝03×1

x3,1 x3,0

⎛1 ⎜0 ⎜ ⎜ x3,2 ⎞ ⎜ ⎜−1 =⎜ x3,1 ⎠ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0

1 −1 1 1 0 −1

1⎞ 0⎟ ⎟ ⎟ 1⎟ ⎟ ⎟ 1⎟ ⎟ ⎟ −1⎟ ⎟ 1⎠

(3.189)

116


Step 4 Solve the system ( A1 A0 ) QB = 03×3

(3.190)

⎛ai1 ai2 ai3 ⎞ ⎟ Ai = ⎜ ⎜ai4 ai5 ai6 ⎟ ⎝ai7 ai8 ai9 ⎠

(3.191)

⎛a01 2a03 + a12 a03 ⎞ ⎟ A0 = ⎜ ⎜a04 2a06 + a15 a06 ⎟ ⎝a07 2a09 + a18 a09 ⎠

(3.192)

1 1 1 1 1 1 ⎛− 2 a01 + 2 a03 + 2 a12 a12 − 2 a01 + 2 a03 + 2 a12 ⎞ 1 1 1 1 1 1 ⎟ A1 = ⎜ ⎜− 2 a04 + 2 a06 + 2 a15 a15 − 2 a04 + 2 a06 + 2 a15 ⎟ ⎝− 1 a07 + 1 a09 + 1 a18 a18 − 1 a07 + 1 a09 + 1 a18 ⎠ 2 2 2 2 2 2

(3.193)

where

The resulting matrices are

Step 5 Choosing the values for the parameters of the matrices Ai , so that det A(σ) ≠ 0 we get σ 2 1 + σ2 ⎞ ⎛ 2 σ σ ⎟ A(σ) = ⎜ ⎜ − 2 −1 − σ − 2 ⎟ ⎝1 − σ 0 − σ2 ⎠ 2

(3.194)

˜ with det A(σ) = 1, det A(σ) = det (A0 σ + A1 ) = σ 3 and Smith forms C SA(σ) (σ) = I3 ,

⎛1 0 0 ⎞ 0 ⎟ SA(σ) (σ) = ⎜ ˜ ⎜0 1 0 ⎟ ⎝0 0 σ 3 ⎠

(3.195)

Again, it is easy to confirm that A(σ)β1 (k) = 0. The dual matrix of A(σ) is 1 2σ σ + 21 ⎞ ⎛ 2 1 1 ⎟ ˜ A(σ) =⎜ ⎜ − 2 −1 − σ − 2 ⎟ ⎝σ − 1 0 − 12 ⎠ 2

(3.196)

˜ and taking a matrix that is left unimodularly equivalent to A(σ), like σ 0⎞ ⎛ 1 ⎜ ˜ ˜ A1 (σ) = ⎜1 + σ σ −1⎟ ⎟ A(σ) ⎝ 0 1 0⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ U (σ)

(3.197)


117

with U (σ) unimodular, we end up with 2

2

σ σ σ σ ⎛− 2 + 2 −1 + σ ⎞ 2 + 2 2 2 ⎜ A1 (σ) = ⎜−σ + σ 1+σ 1+σ+σ ⎟ ⎟ 2 σ2 2 ⎝ −σ ⎠ −σ − σ − 2 2

(3.198)

which also satisfies A1 (σ)β1 (k) = 0. So far, we provided a method for constructing a system with backward behavior of the form (3.44), that is connected to the infinite elementary divisors of the system. These are strictly backward propagating solutions of the system. In addition to these, we may have solutions due to the finite elementary divisors of A(σ) that are given in the form N −(k−1) i βj,φ−1

i B + kλi (k) ∶= λiN −k βj,φ βi,j,φ

k N −(k−φ) i + ... + ( )λi βj,0 , λi ≠ 0 φ

(3.199)

which indicates that they can be considered as backward propagating solutions due to λi ≠ 0, as outlined in Table 3.1. Yet, as mentioned in Theorem 3.19 through the simple reformulation B (k) ∶= ( βi,j,φ

1 k N i 1 k−1 k 1 k−φ N i i + ... + ( ) ( ) λi βj,φ + k ( ) λN β ) λi βj,0 i j,φ−1 φ λi λi λi

(3.200)

we can transform the backward solutions (3.199) into their equivalent forward form, and apply Algorithm 3 to construct the desired system. So far we have constructed two separate algorithms, Algorithm 3 and 4 for constructing a system (3.2) with a given forward, or a backward behavior. By combining them together, we can construct Algorithm 5 for computing a system (3.2) with a given forward and backward behavior. Remark 3.29. In the final step of Algorithms 3, 4 and 5, the resulting matrices may have a large number of independent entries aij . If there are no other requirements on the system’s structure, then the only constraint in choosing the free parameters is that the resulting matrix will have a nonzero determinant. Nonetheless, several other structural properties may be required for the constructed system, like the polynomial matrix having symmetric (Ai = ATi ), skew-symmetric (Ai = −ATi ), or alternating (Ai = (−1)i ATi or Ai = (−1)i+1 ATi ) coefficients. Systems with such structure often appear in continuous time, in the modelling of mechanical systems, see for example [81, 109].

118


Algorithm 5 Construction of a system with given forward and backward behavior. Suppose that a finite number of functions of the form k k−φ i F i i βi,j,φ (k) ∶= λki βj,φ + kλk−1 βj,0 i βj,φ−1 + ... + ( )λi φ F i i i βi,j,φ (k) ∶= δ(k)βj,φ + δ(k − 1)βj,φ−1 + ... + δ(k − φ)βj,0 ˜ β˜j,Bφ˜(k) ∶= x˜j,φ˜δ(N − k) + ... + x˜j,0 δ(N − (k + φ))

(3.201) (3.202) (3.203)

are given, with i = 1, 2, ..., `, j = z, z + 1, . . . , r, φ = 0, 1, ..., ni,j − 1, ˜j = 2, ..., r, φ˜ = 0, ..., µ˜j − 1.. `

r

r

Step 1 Define n = ∑ ∑ nij and µ ∶= ∑ µ˜j . i=1 j=z

˜ j=2

If r∣(n + µ), then

n+µ r

(3.204)

n+µ ]+1 r

(3.205)

q= else q=[ where [⋅] denotes the integer part of

n+µ r .

Step 2 Construct the matrices Q and W defined in (3.135), (3.136), according to Algorithm 3 and QB defined in (3.181), according to Algorithm 4. Step 3 Solve the system of equations (Aq ⋯ A0 )QW = 0r×n

(3.206)

and (Aq ⋯ A0 )QB = 0r×µ or (Aq ⋯ Aqk +1 )QB = 0r×µ

(3.207)

in terms of the unknown matrices Ai . Step 4 Choose the free entries aij of each matrix Ai so that det A(σ) ≠ 0.

Combining the results of Remarks 3.24 and 3.27, we conclude to the following. Remark 3.30. Every polynomial matrix A1 (σ) which is left equivalent to the polynomial matrix A(σ) constructed in Algorithm 5 and its dual matrix A˜1 (σ) is left ˜ equivalent to A(σ), gives rise to a model with exactly the same forward and backward


119

behavior. Let Ai (σ)β(k) = 0, i = 1, 2 be two systems having the same forward and backward behavior. This means that the two matrices will have the same number of finite and infinite elementary divisors n and µ. Since these systems have also the same dimension r, from (1.57) it is derived that the two systems will also have the same lag q. In the following theorem, we argue that these systems are connected by a nonsingular transformation matrix U (σ) = U ∈ Rr×r . Theorem 3.31. Two systems of the form A1 (σ)β(k) = 0

(3.208)

A2 (σ)β(k) = 0

(3.209)

with the same lag q give rise to the same forward and backward behavior, if and only if their respective polynomial matrices A1 (σ), A2 (σ) are connected by a left constant transformation matrix U ∈ Rr×r , with U invertible. Proof. First, assume that the systems (3.208) and (3.209) give rise to the same forward and backward behavior. Then, according to Remark 3.30, it holds that A1 (σ) = U (σ)A2 (σ)

(3.210)

A˜1 (σ) = V (σ)A˜2 (σ)

(3.211)

where U (σ), V (σ) ∈ R[σ]r×r are unimodular matrices. From (3.210), it holds σ→ σ1

A1 (σ) = U (σ)A2 (σ) ⇒ 1 1 1 ×σq A1 ( ) = U ( ) A2 ( ) ⇒ σ σ σ 1 1 1 σ q A1 ( ) = σ q U ( ) A2 ( ) ⇒ σ σ σ 1 A˜1 (σ) = U ( ) A˜2 (σ) σ

(3.212) (3.213) (3.214) (3.215)

Combining (3.211) with (3.215), we have 1 (V (σ) − U ( )) A˜2 (σ) = 0 σ

(3.216)

So the matrix (V (σ) − U ( σ1 )) must belong to the left kernel of A˜2 (σ), which since A˜2 (σ) is nonsingular, is equal to the zero vector. So 1 V (σ) = U ( ) σ

(3.217)

120


and since V (σ) ∈ R[σ]r×r and U ( σ1 ) ∈ Rpr (σ)r×r , it holds that V (σ) = U ( σ1 ) = U = V ∈ Rr×r . To prove the inverse, assume that A1 (σ), A2 (σ) are left unimodularly equivalent and A1 (σ) = U A2 (σ). Let β1 (k) be any solution (forward or backward) of A1 (σ)β(k) = 0. It holds that A1 (σ)β1 (k) = 0 ⇒ U A2 (σ)β1 (k) = 0 ⇒ A2 (σ)β1 (k) = 0

(3.218)

So every solution of (3.208) is a solution of (3.209). In the same fashion, let β2 (k) be any solution (forward or backward) of A2 (σ)β(k) = 0. It holds that A2 (σ)β2 (k) = 0 ⇒ U −1 A1 (σ)β2 (k) = 0 ⇒ A1 (σ)β2 (k) = 0

(3.219)

So every solution of (3.209) is a solution of (3.208) and thus, systems (3.208),(3.209) have exactly the same solutions. Example 3.32. Let the following vector valued functions ⎛ 1 ⎞ ⎛ 3 ⎞ k + ⎝ 1 ⎠ ⎝ 1 ⎠

(3.220a)

⎛ 2 ⎞ k ⎛ 4 ⎞ k−1 2 + k2 ⎝ 0 ⎠ ⎝ 1 ⎠

(3.220b)

β1 (k) = β2 (k) = β3 (k) =

⎛−1⎞ ⎛−1⎞ ⎛1⎞ δ(N − k) + δ(N − k − 1) + δ(N − k − 2) + ⎝0⎠ ⎝−1⎠ ⎝1⎠ +

⎛1⎞ δ(N − k − 3) ⎝0⎠

(3.220c)

We want to construct an AR-representation A(σ)β(k) = 0 that has the prescribed functions in its solution space. Step 1 These vector valued functions correspond to the zeros λ1 = 1, λ2 = 2 with multiplicities n1 = 2, n2 = 2 and to an infinite elementary divisor of order µ2 = 4 (since µ1 = 0). Overall, µ = µ1 + µ2 = 4, n = n1 + n2 = 2 + 2 = 4 and r = 2, so from (1.57) we have n + µ = 4 + 4 = 8 = rq ⇒ q = 4

(3.221)

A(σ) = A4 σ 4 + A3 σ 3 + A2 σ 2 + A1 σ + A0 ∈ R[σ]2×2

(3.222)

So


121

with expected Smith forms C SA(σ) (σ) =

⎛1 ⎞ 0 , 2 2 ⎝0 (σ − 1) (σ − 2) ⎠

0 SA(σ) (σ) = ˜

⎛1 0 ⎞ ⎝0 σ 4 ⎠

(3.223)

Thus, the functions β1 (k), β2 (k), β3 (k), following the notation of Algorithm 5, are identified as F β1,2,1 (k) =

⎛ 1 ⎞ ⎛ 3 ⎞ k, + ⎝ 1 ⎠ ⎝ 1 ⎠ ² ² 1 1

⎛ 2 ⎞ k ⎛ 4 ⎞ k−1 2 + k2 , ⎝ 0 ⎠ ⎝ 1 ⎠ ² ² 2 2 β2,1

B β2,3 (k) =

(3.224a)

β2,0

β2,1

F β2,2,1 (k) =

λ1 = 1

λ2 = 2

(3.224b)

β2,0

⎛−1⎞ ⎛−1⎞ ⎛1⎞ δ(N − k) + δ(N − k − 1) + δ(N − k − 2) + ⎝0⎠ ⎝−1⎠ ⎝1⎠ ± ² ² x2,3

x2,1

x2,2

+

⎛1⎞ δ(N − k − 3) ⎝0⎠ ±

(3.224c)

x2,0

Step 2 From the coefficients of β1 (k) and β2 (k), construct the matrices 3 4 ⎛ 4λi I2 λi I2 ⎞ ⎜ 3λ 2 I λ 3 I ⎟ ⎜ i 2 i 2 ⎟ ⎟ ⎜ 2 ⎟ Qi = ⎜ ⎜ 2λi I2 λi I2 ⎟ , ⎟ ⎜ ⎜ I2 λi I2 ⎟ ⎟ ⎜ ⎝ 02 I2 ⎠

Wi =

i ⎛ β2,0 02×1 ⎞ i i ⎝ β2,1 ⎠ β2,0

i = 1, 2

(3.225)

16 0 ⎞ 0 16 ⎟ ⎟ ⎟ 8 0 ⎟ ⎟ ⎟ 0 8 ⎟ ⎟ ⎟ 4 0 ⎟ ⎟ ⎟ 0 4 ⎟ ⎟ 2 0 ⎟ ⎟ ⎟ 0 2 ⎟ ⎟ ⎟ 1 0 ⎟ ⎟ 0 1 ⎠

(3.226)

and combine them

Q = ( Q1

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ Q2 ) = ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

4 0 3 0 2 0 1 0 0 0

0 4 0 3 0 2 0 1 0 0

1 0 1 0 1 0 1 0 1 0

0 1 0 1 0 1 0 1 0 1

32 0 0 32 12 0 0 12 4 0 0 4 1 0 0 1 0 0 0 0

122

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS ⎛3 ⎜1 ⎜ ⎜ ⎜1 ⎜ ⎜ ⎜1 W = blockdiag(W1 , W2 ) = ⎜ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝0

0 0 3 1 0 0 0 0

0 0 0 0 4 1 2 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ ⎟ 4⎟ ⎟ 1⎠

(3.227)

B From the coefficients of β3 (k) (β2,3 (k)), since q = 4 and µ1 = 0, we have that

µ2 = q − q2 = 4 ⇒ 4 − q2 = 4 ⇒ q2 = 0 so µ2 corresponds to an infinite pole elementary divisor. So we will use (3.182b).

⎛ ⎜ ⎜ QB = ⎜ ⎜ ⎜ ⎝

x2,0 02×1 02×1 02×1

x2,1 x2,0 02×1 02×1

x2,2 x2,1 x2,0 02×1

x2,3 x2,2 x2,1 x2,0

⎛1 ⎜0 ⎜ ⎜ ⎜0 ⎞ ⎜ ⎜ ⎟ ⎜0 ⎟ ⎜ ⎟=⎜ ⎟ ⎜0 ⎟ ⎜ ⎠ ⎜ ⎜0 ⎜ ⎜ ⎜0 ⎜ ⎝0

1 1 1 0 0 0 0 0

−1 −1 1 1 1 0 0 0

−1⎞ 0⎟ ⎟ ⎟ −1⎟ ⎟ ⎟ −1⎟ ⎟ ⎟ 1⎟ ⎟ ⎟ 1⎟ ⎟ ⎟ 1⎟ ⎟ 0⎠

(3.228)

Step 3 & 4 Solving the system ( A4 A3 A2 A1 A0 ) QW = 02×4

(3.229)

( A4 A3 A2 A1 ) QB = 02×4

(3.230)

and choosing values for the parameters of the matrices Ai , so that det A(s) ≠ 0, the resulting matrix is 3σ 2 ⎛ 53 − 11σ 2σ − 6σ5 − 4σ5 + 3σ5 ⎞ 5 + 2σ − 5 A(σ) = 1 29σ 7σ2 3σ3 2 11σ 3 3σ 4 ⎝ 10 − 20 + 4 − 5 1 + σ − 29σ ⎠ 20 − 20 + 5 3

2

3

4

(3.231)

It is easily checked that the vector functions βi (k), i = 1, 2, 3 are solutions of the system, i.e. A(σ)βi (k) = 0. Again, we can find a polynomial matrix A1 (σ) and unimodular matrices U (σ), V (σ) such that A1 (σ) = U (σ)A(σ)

(3.232)

3.5. NOTES ON THE POWER OF A MODEL

123

˜ A˜1 (σ) = V (σ)A(σ)

(3.233)

so that A1 (σ) satisfies A(σ)βi (k) = 0. An example is the matrix 7 15σ 6σ 27σ 6σ ⎞ ⎛ 10 − 73σ 1 + 3σ − 53σ 20 + 4 − 5 20 − 20 + 5 A1 (σ) = 13 117σ 23σ2 9σ3 2 43σ 3 9σ 4 ⎝ 10 − 20 + 4 − 5 1 + 5σ − 77σ ⎠ 20 − 20 + 5 2

3

with U (σ) = U = V (σ) = V =

3.5

2

⎛1 1⎞ ⎝2 1⎠

3

4

(3.234)

(3.235)

Notes on the Power of a model

The notion of power in modeling has been introduced by [124] and later studied in [126, 133]. As power of a model is defined the ability of the constructed model to describe the given behavior, i.e. the given data, but as little else as possible. So if we define as B the behavior of the system we have constructed, that is, the complete set of vector valued functions that satisfy it: B = {w ∶ [0, N ] → Rr ∣ A(σ)w(k) = 0} ,

(3.236)

B = ker A(σ),

(3.237)

or equivalently

we do not simply desire this behavior to include the given functions. This should obviously be the aim of the modelling procedure, but the optimal goal for the contructed model is to have no other behavior, linearly independent from the prescribed. So for any other model with behavior B1 we want {B more powerful than B1 } ⇔ {B ⊆ B1 }.

(3.238)

Now, as mentioned previously, for a given number of vector valued functions, the system created by the proposed algorithms may still include extra forward/backward behavior if equation n + µ = rq is not satisfied. In this case the system model is not the most powerful model (and no such model can be created for a square and regular system matrix). Theorem 3.33. Given the following vector valued functions βj (k) = λkj βj,qj −1 + ... + (

k k−q )λj j βj,0 , qj − 1

124

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS βi (k) = δ(k)βi,pi −1 + ... + δ (k − (pi − 1)) βi,0 , βp (k) = xp,µp −1 δ(N − k) + ... + xp,0 δ(N − k − µp + 1), m0

m1

m2

j=1

i=1

p=1

for j = 1, ..., m0 , i = 1, ..., m1 and p = 1, ..., m2 . Let n = ∑ qi + ∑ pi , and µ = ∑ µp . The system A(σ)β(k) = 0 constructed by the proposed Algorithm 2, corresponding to the behavior B = ker A(σ) is the most powerful model that describes the above vector functions if and only if there exists q ∈ N such that (1.57) holds. An example where the constructed system is not the most powerful model is given below. Example 3.34. We want to construct an AR-representation with the following forward and backward solutions ⎛ 1 ⎞ ⎛2⎞ ⎟ ⎜ ⎟ β1 (k) = ⎜ ⎜−1⎟ + ⎜0⎟ k ⎝ 1 ⎠ ⎝0⎠ ² ± β1,1

(3.239)

β1,0

⎛0⎞ ⎛−1⎞ ⎛−1⎞ ⎟ ⎟ ⎜ ⎟ ⎜ β2 (k) = ⎜−1⎟ δ(N − k) + ⎜ 1 ⎟ δ(N − k − 1) + ⎜ ⎜3⎟ δ(N − k − 2) ⎝3⎠ ⎝1⎠ ⎝0⎠ ± ² ² x2,0

x2,1

x2,2

(3.240)

First, define the matrix pairs

C1 = (β1,0

J1 −1 =

⎛2 1 ⎞ ⎛1 1 ⎞ ⎟ , β1,1 ) = ⎜ ⎜0 −1⎟ , J1 = ⎝ 0 1⎠ ⎝0 1 ⎠

⎛1 −1⎞ ⎛−1 0⎞ ⎛1 1⎞ ⎛−1 0⎞ = , ⎝0 1 ⎠ ⎝ 0 1⎠ ⎝0 1⎠ ⎝ 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(3.241)

(3.242)

U J˜1 U −1

C2 = (x1,0

⎛−2 −1⎞ ⎟ ˜ ˜ C1 = C1 U J 1 = ⎜ ⎜ 0 −1⎟ , ⎝0 1⎠

(3.243)

⎛0 −1 −1⎞ ⎛0 1 0⎞ ⎟ ⎜ ⎟ x1,1 ) = ⎜ ⎜3 1 −1⎟ , J2 = ⎜0 0 1⎟ . ⎝3 1 0 ⎠ ⎝0 0 0⎠

(3.244)

The complete matrix pair is C = (C˜1 C2 ), J = blockdiag(J˜1 , J2 ).


125

Before we start assuming values for q, we see that for the functions given above, we have n = 2, µ = 3 and since we want to create a square regular system, r = 3. However, we easily observe that there is no q that satisfies n + µ = rq ⇒ 5 = 3q.

(3.245)

So we expect the final system to include undesired solutions. For q = 1, the matrix S0 = C does not have full column rank. For q = 2, the matrix T

S1 = (C T (CJ)T ) ∈ R6×5 has full column rank. So the matrix A(σ) is A(σ) = A0 + A1 σ + A2 σ 2 ∈ R[σ]3×3

(3.246)

−2 2 ˜ A(σ) = I3 − C(J − aI5 ) {(σ − a)V2 + (σ − a) V1 } ,

(3.247)

Let a = 2, and

where (V1 V2 ) is the generalized inverse of −1

V = (V1

⎛ ⎞ C . V2 ) = −1 ⎝C (J − aI5 ) ⎠

(3.248)

˜ Computing A(σ) and after some simplifications we obtain the matrix we are looking for:

2 ⎛1084 − 1996σ + 912σ 2 =⎜ ⎜714 − 2142σ + 1428σ ⎝ −102 + 306σ − 204σ 2

1 A(σ) = σ 2 A˜ ( ) = σ 562 − 1590σ + 932σ 2 −714 + 1894σ − 932σ 2 ⎞ 347 − 775σ + 1418σ 2 −271 + 1251σ − 1418σ 2 ⎟ ⎟ 2 2 ⎠ 399 − 1235σ + 874σ −51 + 1167σ − 874σ

(3.249)

with Smith forms C SA(σ) (σ)

0 ⎛1 0 ⎞ ⎛1 0 0 ⎞ 0 ⎟, S ⎟ =⎜ (σ) = ⎜ 0 ˜ ⎜0 1 ⎟ A(σ) ⎜0 1 0 ⎟ ⎝0 0 (σ − 1)2 (277σ − 60)⎠ ⎝0 0 σ 3 ⎠

(3.250)

As expected, the final system matrix A(σ) has an extra finite elementary divisor at σ=

60 277 .

This gives rise to a third solution vector for the system, linearly independent

from β1 (k), β2 (k). Using the method proposed in [53] we find that the third solution is

⎛ − 59149 ⎞ 60 k 477249619 ⎟ β3 (k) = ⎜ ( ⎜ 1409047478 ⎟ 277 ) . ⎝− 654963505 ⎠ 21296

1409047478

(3.251)

126


So the above system A(σ)β(k) = 0 is not the most powerful model describing the vectors β1 (k) and β2 (k). A construction of a non-regular system that satisfies β1 (k), β2 (k) is still possible, using the method of Subsection 3.4.2. Using this approach, we can create a non-regular system, by letting r = 1, so n + µ = rq ⇒ 5 = q. and we can construct a matrix of the form A(σ) = A5 σ 5 + A4 σ 4 + A3 σ 3 + A2 σ 2 + A1 σ + A0 ∈ R[σ]1×3

(3.252)

First, construct the matrices

⎛5I3 ⎜4I ⎜ 3 ⎜ ⎜3I3 ⎜ Q=⎜ ⎜2I3 ⎜ ⎜ ⎜ I3 ⎜ ⎝ 03

⎛5 ⎜0 ⎜0 ⎜ ⎜4 ⎜ ⎜0 ⎜ ⎜0 5 1 I3 ⎞ ⎜ ⎜ ⎜3 ⎟ 4 1 I3 ⎟ ⎜ ⎟ ⎜ ⎜0 13 I3 ⎟ ⎟ ⎜ ⎜0 ⎟ = ⎜2 12 I3 ⎟ ⎟ ⎜ ⎟ ⎜ ⎜0 1I3 ⎟ ⎟ ⎜ ⎜0 1I3 ⎠ ⎜ ⎜1 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎜0 ⎝0

0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 0 0

0 0 5 0 0 4 0 0 3 0 0 2 0 0 1 0 0 0

⎛ x2,0 Q =⎜ ⎜03×1 ⎝03×1 B

1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0

0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0

x2,1 x2,0 03×1

0⎞ 0⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ ⎛ ⎟ 0⎟ ⎜ ⎟ ⎜ 0⎟ ⎜ ⎟ ⎜ 1⎟ ⎟, W = ⎛β1,0 03×1 ⎞ = ⎜ ⎜ 0⎟ ⎝β1,1 β1,0 ⎠ ⎜ ⎟ ⎜ ⎜ 0⎟ ⎟ ⎜ ⎜ 1⎟ ⎟ ⎟ ⎝ 0⎟ ⎟ 0⎟ ⎟ 1⎟ ⎟ 0⎟ ⎟ 0⎟ 1⎠

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ x2,2 ⎞ ⎜ ⎜ ⎜ = x2,1 ⎟ ⎟ ⎜ ⎜ x2,0 ⎠ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

0 −1 −1 ⎞ 3 1 −1 ⎟ ⎟ ⎟ 3 1 0 ⎟ ⎟ ⎟ 0 0 −1 ⎟ ⎟ ⎟ 0 3 1 ⎟ ⎟ 0 3 1 ⎟ ⎟ ⎟ 0 0 0 ⎟ ⎟ ⎟ 0 0 3 ⎟ ⎟ 0 0 3 ⎠

2 0 0 1 −1 1

0 0 0 2 0 0

⎞ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎠

(3.253)

(3.254)

and solve the systems (A5 A4 A3 A2 A1 A0 ) QW = 01 × 2

(3.255)

(A5 A4 A3 ) QB = 01×3

(3.256)


127

where Ai = (ai1 ai2 ai3 )

(3.257)

A0 =

(3.258)

The resulting matrices are

(− a202 +

a03 2

−

a12 2

+

a13 2

+ a21 −

a22 2

+

a23 2

+ 2a31 +

17a32 2

+

19a33 2

+

11a42 2

+

13a43 2

+ 4a53 a02 a03 )

A1 = ( a202 −

a03 2

+

a12 2

−

a13 2

− 2a21 +

a22 2

−

a23 2

− 3a31 −

23a32 2

(3.259) −

25a33 2

−

13a42 2

−

15a43 2

− 5a53 a12 a13 )

A2 = (a21 a22 a23 )

(3.260)

A3 = (a31 a32 a33 )

(3.261)

A4 = (3a32 + 3a33 − 2a42 − 2a43 + a53 a42 a43 )

(3.262)

A5 = (3a42 + 3a43 −a53 a53 )

(3.263)

By choosing for example a02 = 1, a03 = a12 = a13 = a21 = a2 = a23 = a31 = a32 = a33 = a42 = a43 = 0, a53 = 1, we obtain the matrix A(σ) = ( 27 − 92 σ + σ 4 1 − σ 5 σ 5 )

(3.264)

which is non-regular and satisfies A(σ)βi (k) = 0. What must be noted though is that solutions of non-regular systems can either be connected to their f.e.ds and i.e.ds or the structure of the right null space, since non-regular systems have an infinite number of forward and backward propagating solutions, due to the right null space of A(σ), as [53] showcases. So in this case too, the constructed system includes additional undesired behavior. Example 3.35. Let, as in Example 3.25, the following vector valued function F β1,2,2 (k) =

⎛ 2 ⎞ k ⎛ 4 ⎞ k−1 ⎛ 2 ⎞ k(k − 1) k−2 2 + k2 + 2 , 2 ⎝ 3 ⎠ ⎝ 1 ⎠ ⎝ 0 ⎠ ² ² ² 1 1 1 β2,2

β2,1

λ1 = 2

(3.265)

β2,0

We want to construct an AR-representation A(σ)β(k) = 0 that has the prescribed function in its solution space. Step 1 Since n = n1 = 3, µ = 0 and r = 2, from (1.57) we have n + 0 = 2q ⇒ q = 3/2. So set q = [ 23 ] + 1 = 2 and the matrix A(σ) is A(σ) = A2 σ 2 + A1 σ + A0 ∈ R[σ]2×2

(3.266)

128


Step 2 & 3 Construct the matrices Q1 and W1 , which are the same as in Example 3.25 Step 4 Solve the system ( A2 A1 A0 ) Q1 W1 = 02×3 where Ai =

⎛ai1 ai2 ⎞ , ⎝ai3 ai4 ⎠

i = 0, 1, 2

(3.267)

(3.268)

Step 5 The resulting matrices are ⎛−a02 − 4(a12 + 3a22 ) a02 ⎞ ⎝−a04 − 4(a14 + 3a24 ) a04 ⎠

(3.269)

⎛ 23 a02 + 5a12 + 14a22 a12 ⎞ A1 = 3 ⎝ 2 a04 + 5a14 + 14a24 a14 ⎠

(3.270)

A0 =

A2 =

⎛ 12 (−a02 − 3a12 − 8a22 ) a22 ⎞ ⎝ 21 (−a04 − 3a14 − 8a24 ) a24 ⎠

(3.271)

and the matrix A(σ) has determinant det A(σ) = (σ − 2)3 (a04 a12 −a02 a14 +3(a04 a22 − a02 a24 )+(a04 a22 +3a14 a22 − a02 a24 − 3a12 a24 )σ) which is a polynomial of degree equal to 4. So it is obvious that the matrix has an extra zero and thus an extra solution, as it was expected. For example, by choosing a22 = a12 = a02 = 1, a14 = a24 = 0, a04 = 2 we get the matrix A(σ) =

2 1 + σ + σ2⎞ ⎛−17 + 41σ 2 − 6σ ⎠ ⎝ −2 + 3σ − σ 2 2

with Smith form C SA(σ) (σ) =

⎛1 ⎞ 0 ⎝0 (σ + 4)(σ − 2)3 ⎠

(3.272)

(3.273)

So A(σ) has the additional finite elementary divisor (σ + 4). Following the procedure described in Subsection 3.2, we find that this divisor gives rise to the solution β2 (k) =

⎛1⎞ (−4)k ⎝15⎠

which is linearly independent from β1 (k).

(3.274)


129

On the other hand, one may assume that by choosing the appropriate values of the free parameters in order to eliminate the coefficient of σ in the extra polynomial of the determinant, while still keeping det A(σ) ≠ 0, will give a simple solution to the problem of undesired behavior. This is not the case, since this will lead to undesired backward behavior. For example, by choosing a12 = 1, a02 = a14 = a22 = a24 = 0, a04 = 2 we get ⎛−4 + 5σ − 3σ2 A(σ) = ⎝ −2 + 3σ − σ 2

2

with C SA(σ) (σ) =

⎛1 0 ⎞ , ⎝0 (σ − 2)3 ⎠

σ⎞ 2⎠

0 SA(σ) (σ) = ˜

(3.275)

⎛1 0 ⎞ ⎝0 σ ⎠

(3.276)

The Smith form of the dual matrix at σ = 0 implies that in this case the matrix A(σ) has an additional infinite elementary divisor. The existence of an infinite elementary divisor implies the existence of additional backward behavior for the above system. Thus, we see that no matter what the values of the free variables aij will be, the system will exhibit additional behavior. As an alternative though, one may proceed to construct a non-regular system that satisfies the prescribed behavior. Step 1 Under the assumption that the constructed system can be non-square, taking r = 1 and n = 3, we find n = rq ⇒ q = 3. So A(σ) = A3 σ 3 + A2 σ 2 + A1 σ + A0 ∈ R[σ]1×2

(3.277)

Step 2 & 3 For the above system, the matrix W1 remains the same, while Q1 is

⎛3 ⋅ 2I2 3 ⋅ 22 I2 ⎜ I2 2 ⋅ 2I2 ⎜ Q1 = ⎜ ⎜ 02 I2 ⎜ ⎝ 02 02

⎛6 ⎜0 ⎜ ⎜ 23 I2 ⎞ ⎜ ⎜1 ⎜ ⎟ 2 2 I2 ⎟ ⎜ ⎜0 ⎟=⎜ ⎜ 2I2 ⎟ ⎟ ⎜0 ⎜ I2 ⎠ ⎜ ⎜0 ⎜ ⎜0 ⎜ ⎝0

0 6 0 1 0 0 0 0

12 0 0 12 4 0 0 4 1 0 0 1 0 0 0 0

8 0 4 0 2 0 1 0

0⎞ 8⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 4⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 2⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠

(3.278)

Step 4 Solve the system (A3 A2 A1 A0 )Q1 W1 = 01×3

(3.279)

130

CHAPTER 3. MODELING OF AUTO-REGRESSIVE REPRESENTATIONS where Ai = (ai1 ai2 )

(3.280)

A0 = (−a02 − 4a12 − 12a22 − 8a31 − 32a32 a02 )

(3.281)

A1 = ( 3a202 + 5a12 + 14a22 + 12a31 + 36a32 a12 )

(3.282)

A2 = (− a202 − 3a212 − 4a22 − 6a31 − 10a32 a22 )

(3.283)

A3 = (a31 a32 )

(3.284)

Step 5 The resulting matrices are

and the constructed matrix A(σ) satisfies A(σ)β(k) = 0. So far, it has been made clear that when the number of given vector valued functions (counting multiplicity) does not satisfy (1.57), the system is always going to have some undesired behavior. In this case, it may be possible to study the conditions under which the additional solutions will have certain properties, like being stable. This is showcased in the following example. Example 3.36. Consider the following vector valued function F β1,2,2 (k) =

⎛ 2 ⎞ 1 k ⎛ 4 ⎞ 1 k−1 ⎛ 2 ⎞ k(k − 1) 1 k−2 ( ) + ( ) , k( ) + 2 2 ⎝ 3 ⎠ 2 ⎝ 1 ⎠ 2 ⎝ 0 ⎠ ² ² ² 1 1 1 β2,2

β2,1

λ1 =

1 2

(3.285)

β2,0

Construct an AR-representation A(σ)β(k) = 0 that has the prescribed function in its solution space. In addition, study the conditions under which: 1. The constructed system is stable 2. The constructed matrix has positive zeros. Step 1 Since n = n1 = 3, µ = 0 and r = 2, from (1.57) we have n + 0 = 2q ⇒ q = 3/2. So set q = [ 23 ] + 1 = 2 and the matrix A(σ) is A(σ) = A2 σ 2 + A1 σ + A0 ∈ R[σ]2×2

(3.286)


131

Step 2 & 3 For the above system, the matrices Q and W are

1 ⎛I2 2 ⋅ ( 2 ) I2 Q=⎜ I2 ⎜02 ⎝02 02

⎛1 ⎜0 ⎜ 1 2 ( 2 ) I2 ⎞ ⎜ ⎜0 ⎜ ( 12 ) I2 ⎟ ⎟=⎜ ⎜0 I2 ⎠ ⎜ ⎜ ⎜0 ⎜ ⎝0

1 02×1 02×1 ⎛ β2,0 1 1 W =⎜ ⎜ β2,1 β2,0 02×1 1 1 1 ⎝ β2,2 β2,1 β2,0

0 1 0 0 0 0

1 0 1 0 0 0

⎛2 ⎜0 ⎜ ⎞ ⎜ ⎜4 ⎟=⎜ ⎟ ⎜ ⎜1 ⎠ ⎜ ⎜ ⎜2 ⎜ ⎝3

0 0 2 0 4 1

1 4

0 1 0 1 0 0

0 1 2

0 1 0 0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 0⎟ ⎟ ⎟ 2⎟ ⎟ 0⎠

0⎞ 1⎟ 4⎟ ⎟ 0⎟ ⎟ ⎟ 1⎟ 2⎟ ⎟ 0⎟ ⎟ 1⎠

(3.287)

(3.288)

Step 4 Solve the system (A3 A2 A1 A0 )QW = 02×3

(3.289)

where ⎛ai1 ai2 ⎞ ⎝ai3 ai4 ⎠

(3.290)

⎛ a802 − a1612 − 3a3222 a02 ⎞ ⎝ a804 − a1614 − 3a3224 a04 ⎠

(3.291)

⎛ a212 + a222 a12 ⎞ ⎝ a214 + a224 a14 ⎠

(3.292)

Ai = Step 5 The resulting matrices are A0 =

A1 =

⎛− a202 − 3a412 − 5a822 a22 ⎞ A2 = ⎝− a204 − 3a414 − 5a824 a24 ⎠

(3.293)

and the constructed matrix A(σ) satisfies A(σ)β(k) = 0. The determinant of A(σ) is given by det A(σ) =

1 (−1 + 2σ)3 g(σ) 32

(3.294)

where g(σ) = 2a04 a12 − 2a02 a14 + 3a04 a22 − 3a02 a24 + (2a04 a22 + 3a14 a22 − 2a02 a24 − 3a12 a24 )σ (3.295)

132


To answer the questions 1,2 we need to find the zeros of g(σ). For the first question, the constructed system will be stable if the zero λ of g(σ) satisfies ∣λ∣ < 1. The zero λ of g(σ) is given by λ=

(−2a04 a12 + 2a02 a14 − 3a04 a22 + 3a02 a24 ) (2a04 a22 + 3a14 a22 − 2a02 a24 − 3a12 a24 )

(3.296)

so, if the parameters of A0 , A1 , A2 satisfy ∣

(−2a04 a12 + 2a02 a14 − 3a04 a22 + 3a02 a24 ) ∣0 (2a04 a22 + 3a14 a22 − 2a02 a24 − 3a12 a24 )

(3.298)

the constructed system will have positive zeros.

3.6

Conclusions

We have proposed two methods for constructing an AR-representation that satisfies a given forward and backward behavior. The first method is an extension of [33] and is the discrete time analog of the work by [54] where the problem was formulated for continuous time AR-representations and the modelling of the smooth and impulsive behavior of a system. In the second method, the modeling problem was reduced to solving a linear system of equations, and can be used to also construct non-regular systems. This method can be extended with minor adjustments to the case of continuous time systems as well. For both methods presented in this chapter, it was shown that when the number of vector valued functions provided (counting multiplicities) does not satisfy (1.57), the constructed system will include additional solutions, linearly independent from the ones that are given. In this case, a subject for further research would be the study of the conditions under which the undesired solutions satisfy certain properties, like stability. Moreover, another subject is the study of the conditions under which the construction of the desired system A(σ) will be possible.

Chapter 4 Reachability of Linear Systems of Algebraic and Difference Equations 4.1

Introduction

In this Chapter, we will consider nonhomogeneous systems of higher order systems of linear discrete time algebraic and difference equations that are described by the matrix equation Aq β(k + q) + ... + A1 β(k + 1) + A0 β(k) = Bq u(k + q) + ... + B1 u(k + 1) + B0 u(k) (4.1) with Ai ∈ Rr×r , Bi ∈ Rr×m , det A(σ) ≡/ 0 and at least one of Aq , Bq is a non zero matrix. The discrete time functions u(k) ∶ N → Rm and β(k) ∶ N → Rr define the input and output vectors of the system. Using the forward shift operator σ with σ i β(k) = β(k +i), the system (4.1) can be rewritten as A(σ)β(k) = B(σ)u(k)

(4.2)

A(σ) = A0 + A1 σ + ... + Aq σ q ∈ R[σ]r×r

(4.3)

B(σ) = B0 + B1 σ + ... + Bq σ q ∈ R[σ]r×m

(4.4)

with

Systems described by (4.2) are also known as ARMA (Auto-Regressive Moving Average) representations. It is easily seen that such representations can be considered as a generalisation of the known descriptor systems presented in Chapter 2. 133

134

CHAPTER 4. REACHABILITY OF ALGEBRAIC/DIFFERENCE EQUATIONS

Various methods of approach have been established to compute their solution and study properties like controllability and observability. A straightforward approach has been used in [30, 37, 52, 116, 118] for the continuous time case and in [4, 40, 53, 56, 58, 89] for the discrete time case, through the use of the Jordan Pairs and the Laurent expansion at infinity of the polynomial matrix A(σ). A different method for continuous time has also been used in [107]. Another approach is to first transform the higher order system into a state-space or descriptor system, as in [6, 58, 64, 78]. The descriptor system can then be further decomposed using the Weierstrass decomposition of the matrix pencil involved, into multiple subsystems, which can then be studied separately, as in [20, 51, 99]. Yet, as [6] comments, the transformation of the system (3.2) into an equivalent representation may not always be desirable, since it involves a change of the internal variables. This may be inconvenient, since it can lead to the loss of the physical meaning of the original variables. This paper utilises a more direct approach, through the use of the minimal realization of A(σ)−1 and its Laurent expansion. As was also mentioned in the introduction of Chapter 2, the concept of reachability refers to the existence of an input that can drive the system’s state β(k) from the origin to a final state z = x(k), over a finite time k. Reachability has been thoroughly examined by numerous authors, initially for state space systems in [6, 35, 47]. These results have been extended to continuous and discrete time descriptor systems in [8, 17, 19, 26, 55, 60, 67, 79, 129], in [64, 74] for second order descriptor systems, in [38, 83, 84] for rectangular descriptor systems, in [18, 41, 42, 113, 114] for positive systems and in [30, 52, 78] for continuous time ARMA systems. In contrast to the state-space case though, the following issue is encountered. Despite the numerous papers studying the concepts of reachability and controllability, there is no in depth analysis regarding the input that drives the system to a desired state. This question goes in pair with the problem of consistent initial conditions, i.e. the equations that the initial values of the state and input need to satisfy, in order for the system to be impulse free for the continuous time case, and have a solution for the discrete time case. Although consistent initial conditions have been studied by many authors in [12, 19, 56, 67, 118], their effect on the choice of the appropriate input that drives the system to the desired state has been left out of question, up until recently in [55].

4.2. SOLUTION OF ARMA REPRESENTATIONS

135

There, reachability criteria for discrete time descriptor systems have been derived in a constructive way, by providing a consistent input that drives the system to a desired state, taking into account the consistency of initial conditions. As an example, consider again Example 2.16 from Chapter 2 ⎛2 1 0⎞ ⎛−4 −2 0⎞ ⎛1⎞ ⎜0 0 1⎟ x(k + 1) = ⎜ 3 1 1⎟ x(k) + ⎜−1⎟ u(k) ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎝0 0 2⎠ ⎝ 6 2 1⎠ ⎝2⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ² E

(4.5)

B

A

where the authors of [55] proposed the input sequence 3 1 3 1 u(0) = 0, u(1) = 0, u(2) = 2z1 + z2 , u(3) = z3 , u(4) = z1 + z2 + z3 4 4 4 16

(4.6)

that drives the state of the system from the origin to any arbitrary state z = (z1 , z2 , z3 )T ∈ R3 in three time steps, that is x(3) = z and in addition, it satisfies the initial conditions of consistency Φ−2 Bu(1) + Φ−1 Bu(0) = 0. In the present chapter, the result of [55] will be expanded for the general case of higher order systems of algebraic and difference equations in the form of (4.2). Hence, the main aim of this work is twofold. Firstly, to present the solution of (4.2) in terms of the matrices that give a minimal realization of A(σ)−1 and secondly, to construct the reachable subspace from the zero state, i.e. R(0) of (4.2) by proposing the consistent inputs that drive the system from the origin to any vector in R(0). This constructive method will yield a simple criterion for the reachability of the system, which is a generalised version of the already known results for state space and descriptor systems.

4.2

Solution of ARMA representations

Consider the system (4.2), with det A(σ) ≡/ 0. Remember from Chapter 1, Sections 1.2, 1.5, 1.6 that the Smith form of A(σ) is given by C SA(σ) (σ) = diag (1, ..., 1, fz (σ), fz+1 (σ), ..., fr (σ))

(4.7)

where fj (σ) the invariant polynomials of A(σ). Its Smith form at infinity is given by r−u

⎛ ³¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹·¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ µ⎞ 1 1 1 ⎟ ⎜ ∞ SA(σ) (σ) = diag ⎜σ q1 , ..., σ qu , qû+1 , qû+2 , ..., qˆr ⎟ ⎜´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ σ σ σ ⎟ u ⎝ ⎠

(4.8)

136


with 0 SA(σ) (σ) ˜

=σ

q

⎛ ⎞ q−q2 q−qu q+ˆ qu+1 q+ˆ qr ⎟ ⎜ = diag (σ , ..., σ ) = diag ⎜1, σ ,...,σ ,σ ,...,σ ⎟ σ ⎝ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶⎠

1 ∞ SA(σ) ( )

µ1

µr

i.p.e.d.

i.z.e.d.

(4.9) The inverse of the polynomial matrix A(σ)−1 is given by its Laurent expansion A(σ)−1 = Hpol (σ) + Hsp (σ) = Hqˆr σ qˆr + ⋅ ⋅ ⋅ + H1 σ + H0 + H−1 σ −1 + H−2 σ −2 + . . . , (4.10) The inverse of A(σ) also has the representation ˆ∞ A(σ)−1 = Hsp (σ) + Hpol (σ) = C(σIn − J)−1 B + Cˆ∞ (Iµˆ − σ Jˆ∞ )−1 B

(4.11)

ˆ∞ ) are the minimal realizations of where the matrix triples (C, J, B) and (Cˆ∞ , Jˆ∞ , B the strictly proper and polynomial parts of A(σ)−1 respectively, given by Hsp (σ) = C(σIn − J)−1 B

(4.12)

ˆ∞ σ −1 Hpol (σ −1 ) = Cˆ∞ (σIµˆ − Jˆ∞ )−1 B

(4.13)

ˆ∞ Hpol (σ) = Cˆ∞ (Iµˆ − σ Jˆ∞ )−1 B

(4.14)

and

or equivalently

ˆ∞ ∈ Rµˆ×r , Jˆ∞ nilpotent with C ∈ Rr×n , J ∈ Rn×n , B ∈ Rn×r , Cˆ∞ ∈ Rr×ˆµ , Jˆ∞ ∈ Rµˆ×ˆµ , B with index of nilpotency qˆr + 1 and n = deg det A(σ)

(4.15)

r

µ ˆ = ∑ (ˆ qi + 1)

(4.16)

i=u+1

Since (σIn − J)

= σ −1 I + σ −2 J + σ −3 J 2 + . . .

(4.17a)

−1 2 qˆr (Iµˆ − σ Jˆ∞ ) = I + σ Jˆ∞ + σ 2 Jˆ∞ + ... + σ qˆr Jˆ∞ .

(4.17b)

−1

the connection between the minimal realizations and the coefficients of the Laurent expansion of A(σ)−1 is given by i ˆ Hi = Cˆ∞ Jˆ∞ B∞

H−i = CJ i−1 B

i = 0, 1, 2, ..., qˆr

(4.18a)

i = 1, 2, . . .

(4.18b)

Now, we can give a formula for the solution of (4.2)

4.2. SOLUTION OF ARMA REPRESENTATIONS

137

Theorem 4.1. [39, 40, 56] The general solution of (4.2) is given by

β(k) = (H−k−q

⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟⎜ ⎟+ ⋯ H−k−1 ) ⎜ ⋮ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠ + (H−k

u(0) ⎛B0 ⋯ Bq ⋯ 0 ⎞ ⎛ ⎞ ⎜ ⎟ ⎜ ⎟ (4.19) ⋯ Hqˆr ) ⎜ ⋮ ⋱ ⋱ ⋱ ⋮ ⎟ ⎜ ⋮ ⎟ ⎝ 0 ⋯ B0 ⋯ Bq ⎠ ⎝u(k + q + qˆr )⎠

Based on the above formula, we can derive a similar formula for the solution Theorem 4.2. The solution of (4.2) is given by k ⎛ ∑ J i−1 Ωu(k − i) ⎞ q−1 ⎜ ⎟ ⎟ + ∑ (CΦi+1 + C∞ Zi ) u(k + i)+ β(k) = (C Cˆ∞ ) ⎜ qˆri=1 ⎜ î ˜ ⎟ i=0 J Ωu(k + q + i) ∑ ∞ ⎝i=0 ⎠ q−1

− CJ k ∑ Φi+1 u(i) + CJ k βf (0) (4.20) i=0

where ⎛Aq ⋯ 0 ⎞⎛ β(0) ⎞ ⎟ ⎟⎜ βf (0) = (J q−1 B ⋯ B)⎜ ⋮ ⎟ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎝A1 ⋯ Aq ⎠⎝β(q − 1)⎠

(4.21)

Ω = J q BBq + J q−1 BBq−1 + ... + JBB1 + BB0

(4.22)

q ˆ ˜ =B ˆ∞ Bq + Jˆ∞ B ˆ∞ Bq−1 + ... + Jˆ∞ Ω B∞ B0

(4.23)

q−j

Φj = ∑ J i BBi+j i=0

q

i ˆ B∞ Bq−j−i Zq−j = ∑ Jˆ∞

(4.24)

i=0

with j = 1, 2, ..., q and Bk ≡ 0 for k < 0. Proof. By replacing in (4.19) the fundamental matrix sequences Hi given in and (1.110a), (1.110b) and taking into account that ⎛Bq ⎞ ⎟ Ω = (J q B ⋯ JB B) ⎜ ⎜ ⋮ ⎟ ⎝B0 ⎠

qˆr ˜ ˜ Jˆ∞ Ω ˜ ⋯ Jˆ∞ ˆ∞ (Ω Ω) = (B

⎛Bq ⎞ ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ qˆr ˆ ⎜ ⎟, ˆ ⋯ J∞ B∞ ) ⎜B0 ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⋱ ⋱ ⎜ ⎟ ⎝ B0 . . . Bq ⎠

(4.25)

q ≤ qˆr (4.26)

138

CHAPTER 4. REACHABILITY OF ALGEBRAIC/DIFFERENCE EQUATIONS ⎛ Bq ⎞ qˆr ˜ qˆr ˆ ⎜ ⎟, ˜ ˆ ˜ ˆ ˆ ˆ (Ω J∞ Ω ⋯ J∞ Ω) = (B∞ ⋯ J∞ B∞ ) ⎜ ⋮ ⋱ ⎟ ⎝Bq−ˆqr ⋯ Bq ⎠ (Φ1 ⋯ Φq ) =

(J q−1 B

ˆ∞ ⋯ (Z0 ⋯ Zq−1 ) = (B

q > qˆr

⎛Bq ⎞ ⎜ ⎟ ⋯ B) ⎜ ⋮ ⋱ ⎟ ⎝B1 ⋯ Bq ⎠ ⎛B0 ⋯ Bq−1 ⎞ ⋱ ⋮ ⎟ ⎟ ⎝ B0 ⎠

q−1 ˆ Jˆ∞ B∞ ) ⎜ ⎜

(4.27)

(4.28)

(4.29)

we obtain after a number of calculations the formula given in (4.20).

The last formula for the general solution of the system, offers a great convenience. It showcases which parts of the input are connected with the finite or infinite Jordan Pairs. So we can see that for an arbitrary k, the first k inputs, from u(0) to u(k − 1) are connected to the matrices of the realization of the strictly proper part of A(σ)−1 , the next u(k) to u(k + q − 1) are connected to both the matrices of the strictly proper and polynomial part A(σ)−1 and the last u(k + q) to u(k + q + qˆr ) are connected only to the matrices of the realization of the polynomial part of A(σ)−1 . This separation will be of utmost importance in the creation of the reachable subspace. As discussed in Chapter 2 for descriptor systems, in order for the above formula to be a solution of (4.2), it needs to satisfy (4.2) for k = 0, 1, ..., q − 1. This brings us to the following definition and theorem. Definition 4.3. [56] The set of all {u(k), β(k)} that satisfy (4.2) for k = 0, 1, ..., q − 1 is called the set of Consistent (or admissible) Initial values of the system and is denoted ad by HARM A.

ad Theorem 4.4. [56] The set of all HARM A is given by

ad HARM A

⎧ ⎪ ⎪ ∶= ⎨β(k), k = 0, ..., q − 1 and u(k), k = 0, ..., 2q + qˆr − 1 ∶ ⎪ ⎪ ⎩ ⎛A0 ⋯ Aq−1 ⎞ ⎛ H0 ⋯ Hq−1 ⎞ ⎛A0 ⋯ Aq−1 ⎞ ⎛ β(0) ⎞ ⎜ ⋮ ⋱ ⎜ ⎜ ⎜ ⎟= ⋮ ⎟ ⋮ ⎟ ⋮ ⎟ ⋮ ⎜ ⎟⎜ ⋮ ⎟⎜ ⋮ ⋱ ⎟⎜ ⎟ ⎝ 0 ⋯ A0 ⎠ ⎝H−q+1 ⋯ H0 ⎠ ⎝ 0 ⋯ A0 ⎠ ⎝β(q − 1)⎠

4.3. REACHABLE SUBSPACE

139

⎫ Hqˆr ⋯ 0 ⎞⎛B0 ⋯ Bq ⋯ 0 ⎞⎛ u(0) ⎛A0 ⋯ Aq−1 ⎞⎛ H0 ⋯ ⎞⎪ ⎪ ⎪ ⎪ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ =⎜ ⋮ ⋱ ⋮ ⎟⎜ ⋮ ⋮ ⋱ 0 ⎟⎜ ⋮ ⋱ ⋱ ⋮ ⎟⎜ ⋮ ⎟⎬ ⎪ ⎪ ⎝ 0 ⋯ A0 ⎠⎝H−q+1 ⋯ Hqˆr −q+1 ⋯ Hqˆr ⎠⎝ 0 ⋯ B0 ⋯ Bq ⎠⎝u(2q + qˆr − 1)⎠⎪ ⎪ ⎭ (4.30)

4.3

Reachable Subspace

In this section the Reachability of system (4.2) is considered. We present a constructive proof to showcase the reachable subspace of the system and provide a criterion to test when a system is reachable. The results are illustrated by an example. Definition 4.5. We say that a vector z ∈ Rr is reachable (controllable from the origin) if there exists an admissible input u(k) that transfers β(k) from the origin T

(β(0)T ⋯ β(q − 1)T ) = 0 to z in some finite time k0 ∈ N i.e. β(k0 ) = z

(4.31)

ad The set of all reachable states β(k) from 0 ∈ HARM A is denoted by R(0).

Definition 4.6. The system (4.2) is called reachable (controllable from the origin) if every point z∈ Rr is reachable. In mathematical terms, a system is reachable iff R(0) ≡ Rr

(4.32)

To describe R(0), we first need to introduce the following formulas C ⟨J/ Im Ω⟩ ∶= C Im Ω + CJ Im Ω + ... + CJ n−1 Im Ω

(4.33)

qˆr ˜ ∶= Cˆ∞ Im Ω ˜ + Cˆ∞ Jˆ∞ Im Ω ˜ + ... + Cˆ∞ Jˆ∞ ˜ Cˆ∞ ⟨Jˆ∞ / Im Ω⟩ Im Ω

(4.34)

Im Ω ∶= {x/x ∈ Rn , ∃u ∈ Rm ∶ x = Ωu} ⊂ Rn

(4.35)

˜ ∶= {x/x ∈ Rµˆ , ∃u ∈ Rm ∶ x = Ωu} ˜ ⊂ Rµˆ Im Ω

(4.36)

where n = deg det (A(σ)) is the sum of the finite elementary divisors of A(σ) and µ ˆ = ∑ri=u+1 (ˆ qi + 1). n

Lemma 4.7. Define W (1, n) ∶= ∑ CJ i−1 ΩΩT (J T )

i−1

C T . Then

i=1

Im W (1, n) = C ⟨J/ Im Ω⟩

(4.37)

140


Proof. It is clear that C ⟨J/ Im Ω⟩ is produced by the linearly independent columns of ⎞ ⎛Bq ⎟ ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ n−1 q+n−1 ⎜ Q = (CJ Ω ⋯ CJΩ CΩ) = (CJ B ⋯ CB) ⎜B0 Bq ⎟ ⎟ ⎟ ⎜ ⎜ ⋱ ⋮ ⎟ ⎟ ⎜ ⎝ B0 ⎠

(4.38)

It is sufficient to show that n−1

i

ker W T (1, n) = ker QT ⇔ ker W (1, n) = ⋂ ker ΩT (J T ) C T

(4.39)

i=0

since W (1, n) is symmetric, i.e. W T (1, n) = W (1, n). n−1

i

First, we prove that ker W (1, n) ⊆ ⋂ ker ΩT (J T ) C T . Let x ∈ ker W (1, n), then i=0

W (1, n) x = 0 ⇒ xT W (1, n) x = 0 ⇒

n

∑ xT CJ i−1 ΩΩT (J T )

i−1

i=1

ΩT C T x n−1

= 0,

n

i−1

C T x = 0 ⇒ ∑ ∥ΩT (J T ) i=1

ΩT J T C T x

= 0, . . . , ΩT (J T )

2

C T x∥ = 0 ⇒

n−1

CT x = 0

i

So x ∈ ⋂ ker ΩT (J T ) C T , therefore i=0

n−1

i

ker W (1, n) ⊆ ⋂ ker ΩT (J T ) C T

(4.40)

i=0

n−1

n−1

i

i

Now we shall prove that ⋂ ker ΩT (J T ) C T ⊆ ker W (1, n). Let x ∈ ⋂ ker ΩT (J T ) C T , i=0

i=0

then n

i−1

W (1, n) x = ∑ CJ i−1 ΩΩT (J T )

CT x =

i=1 n−1

CΩ Ω C x +CJΩ Ω J C x +... + CJ q−1 Ω ΩT (J T ) C T x = 0 ⇒ x ∈ ker W (1, n) ´¹¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ T

T

T

0

n−1

T

T

0

0

i

Thus ⋂ ker ΩT (J T ) C T ⊆ ker W (1, n). i=0

Lemma 4.8. Im W (1, n) = Im W (1, k) for k ≥ n. Proof. We know according to the Cayley-Hamilton theorem that each matrix satisfies its characteristic polynomial. So the matrix J, which can also be assumed to be in block n−1

diagonal form, has the characteristic polynomial p(λ) = det (λIn − J) = λn + ∑ pi λi . i=0

Hence, J satisfies its characteristic equation i.e. p(J) = J n + pn−1 J n−1 + ... + p0 In = 0

(4.41)


141

This means that the matrix J n = −pn−1 J n−1 − ... − p0 In is linearly dependent to the matrices J n−1 , J n−2 ,..., J, In . Multiplying the above equation from the right by J and replacing J n , we get that the same holds for J n+1 . Following this procedure, we get that all J k , k ≥ n are linearly dependent to J n−1 ,J n−2 ,. . . , J, In . As a result CIm (Ω JΩ ⋯ J n−1 Ω) = CIm (Ω JΩ ⋯ J k−1 Ω) ⇔

(4.42)

ImW (1, n) = ImW (1, k)

(4.43)

for k ≥ n. qˆr

i T . Then i Ω T) C ˆ∞ ˜ (0, qˆr ) ∶= ∑ Cˆ∞ Jˆ∞ ˜Ω ˜ T (Jˆ∞ Lemma 4.9. Define W i=0

˜ (0, qˆr ) = Cˆ∞ ⟨Jˆ∞ /ImΩ⟩ ˜ ImW

(4.44)

˜ is generated by the linearly independent columns Proof. It is clear that Cˆ∞ ⟨Jˆ∞ /ImΩ⟩ of qˆr ˜ ˜ = (Cˆ∞ Ω ˜ Cˆ∞ Jˆ∞ Ω ˜ ⋯ Cˆ∞ Jˆ∞ Q Ω)

(4.45)

It is sufficient to show that qˆr

T

i

T T ˜ T (Jˆ∞ ˜ (0, qˆr ) = ker Q ˜ T ⇔ ker W ˜ (0, qˆr ) = ⋂ ker Ω ) Cˆ∞ ker W

(4.46)

i=0

˜ (0, qˆr ) is symmetric, so W ˜ (0, qˆr )T = W ˜ (0, qˆr ). since W qˆr

i T) C T . Let x ∈ ker W ˜ T (Jˆ∞ ˆ∞ ˜ (0, qˆr ), then ˜ (0, qr ) ⊆ ⋂ ker Ω First, we prove that ker W i=0

˜ (0, qˆr ) x = 0 ⇒ xT W ˜ (0, qˆr ) x = 0 ⇒ W qˆr

i

i ˜ ˜ T ˆT T ΩΩ (J∞ ) Cˆ∞ x=0⇒ ∑ xT Cˆ∞ Jˆ∞

(4.47) (4.48)

i=0 qr

2

T i ˆT ˜ T (Jˆ∞ ) C∞ x∥ = 0 ⇒ ∑ ∥Ω

(4.49)

i=0 T T ˆT T qˆr ˆ T ˜ T Cˆ∞ ˜ T Jˆ∞ ˜ T (Jˆ∞ ) C∞ x = 0 Ω x = 0, Ω C∞ x = 0, ⋯, Ω qˆr

qˆr

i=0

i=0 qˆr

i i T) C T and ker W T) C T. ˜ T (Jˆ∞ ˆ∞ ˜ (0, qˆr ) ⊆ ⋂ ker Ω ˜ T (Jˆ∞ ˆ∞ So x ∈ ⋂ ker Ω i

T) C T , then ˜ T (Jˆ∞ ˆ∞ Now we shall prove the reverse. Let x ∈ ⋂ ker Ω i=0

qˆr

i i Ω T) C Tx=0⇒ ˜Ω ˜ T (Jˆ∞ ˆ∞ ˜ (0, qr ) x = ∑ Cˆ∞ Jˆ∞ W i=0

˜ (0, qˆr ) x ∈ ker W qˆr

i

T ) ⊆ ker W ˜ (0, qˆr ). ˜ T (Jˆ∞ So ⋂ ker Ω i=0

(4.50)

142


Now we proceed to the main theorem of this chapter. Theorem 4.10. The reachable subspace R(0) of (4.2) is q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) R(0) = C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩

(4.51)

i=0

Proof. Firstly, we prove that q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) R(0) ⊆ C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩

(4.52)

i=0

Let β(k) ∈ R(0). This means that the initial value vector is ⎛ β(0) ⎞ ⎛0⎞ ⎜ ⎟ = ⎜⋮⎟ ⋮ ⎜ ⎟ ⎜ ⎟ ⎝β(q − 1)⎠ ⎝0⎠

(4.53)

and the above vector, along with the first 2q + qˆr input values u(0) ⎞ ⎛ ⎟ ⎜ ⋮ ⎟ ⎜ ⎝u(2q + qˆr − 1)⎠

(4.54)

need to satisfy the equation of consistency (4.30). The simplest choice of initial input values, for (4.30) to hold true is u(i) = 0,

i = 0, . . . , 2q + qˆr − 1

(4.55)

Alternatively, let w0 ∈ ker Λ, where Hqˆr ⋯ 0 ⎞⎛B0 ⋯ Bq ⋯ 0 ⎞ ⎛A0 ⋯ Aq−1 ⎞⎛ H0 ⋯ ⎜ ⎜ ⎜ ⎟ Λ=⎜ ⋮ ⋱ ⋱ ⋮ ⎟ ⋮ ⋱ 0 ⎟ ⋮ ⎟⎜ ⋮ ⎟ ⎟⎜ ⋮ ⋱ ⎝ 0 ⋯ A0 ⎠⎝H−q+1 ⋯ Hqˆr −q+1 ⋯ Hqˆr ⎠⎝ 0 ⋯ B0 ⋯ Bq ⎠

(4.56)

and partition w0 as ⎛ w0,0 ⎞ ⎟ w0 = ⎜ ⋮ ⎜ ⎟ ⎝w0,2q+ˆqr −1 ⎠

(4.57)

So we can choose the admissible initial values of the output as u(i) = w0,i ,

i = 0, . . . , 2q + qˆr − 1

(4.58)

For an arbitrary k0 > q the output of the system becomes q−1

k0 ⎛∑ J i−1 Ωu(k0 − i) − J k0 ∑ Φi+1 u(i)⎞ q−1 ⎟ ⎜ i=0 ⎟ + ∑ (CΦi+1 + Cˆ∞ Zi ) u(k0 + i) β(k0 ) = (C Cˆ∞ ) ⎜i=1 qˆr ⎜ ⎟ i=0 i ˆ ˜ ∑ J∞ Ωu(k0 + q + i) ⎝ ⎠ i=0 (4.59)


143

and by taking into account (4.28), (4.25), (4.38) it is easy to see that k0

q−1

i=1

i=0

C ∑ J i−1 Ωu(k0 − i) − CJ k0 ∑ Φi+1 u(i) ∈ ImW (1, k0 ) ⊆ C ⟨J/ImΩ⟩

(4.60)

qˆr

i ˜ ˜ ˆ ∞ ⟨Jˆ∞ /ImΩ⟩ ˜ Cˆ∞ ∑ Jˆ∞ Ωu(k0 + q + i) ∈ ImW(0, q ˆr ) = C

(4.61)

i=0 q−1

q−1

i=0

i=0

∑ (CΦi+1 + Cˆ∞ Zi ) u(k0 + i) ∈ ∑ Im (CΦi+1 + Cˆ∞ Zi )

(4.62)

and thus q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) R(0) ⊆ C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩

(4.63)

i=0

Now we shall prove the opposite, that q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) R(0) ⊇ C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩

(4.64)

i=0

Let q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) z ∈ C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩

(4.65)

i=0

i.e. z = β1 + β2 + β3

(4.66) q−1

˜ and β3 ∈ ∑ Im (CΦi+1 + C∞ Zi ). We need to where β1 ∈ C ⟨J/ImΩ⟩, β2 ∈ Cˆ∞ ⟨J∞ / Im Ω⟩ i=0

construct a consistent input u(k), so that for a certain k, we have β(k) = z

(4.67)

We shall show that this control sequence can be constructed for k = 2q + qˆr + n. For this choice of k the output is β(2q + qˆr + n) =

2q+ˆ qr +n

q−1

i=1

i=0

∑ CJ i−1 Ωu(2q + qˆr + n − i) − CJ 2q+ˆqr +n ∑ Φi+1 u(i)+

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ A

q−1

qˆr

i=0

i=0

i ˜ + ∑ (CΦi+1 + Cˆ∞ Zi ) u(2q + qˆr + n + i) +∑ Cˆ∞ Jˆ∞ Ωu(3q + qˆr + n + i) (4.68)

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ B

G

where A=

2q+ˆ qr +n

∑

i=1

q−1

CJ i−1 Ωu(2q + qˆr + n − i) − CJ 2q+ˆqr +n ∑ Φi+1 u(i) = i=0

144

CHAPTER 4. REACHABILITY OF ALGEBRAIC/DIFFERENCE EQUATIONS n

2q+ˆ qr +n

i=1

i=n+1

= ∑ CJ i−1 Ωu(2q + qˆr + n − i) +

∑

q−1

CJ i−1 Ωu(2q + qˆr + n − i) − CJ 2q+ˆqr +n ∑ Φi+1 u(i) i=0

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ γ

(4.69)

corresponds to the input sequence u(0),. . . , u(2q + qˆr + n − 1), and γ depends on the inputs u(0), ..., u(2q + qˆr − 1), that must be chosen to satisfy the admissible initial conditions. q−1

B = ∑ (CΦi+1 + Cˆ∞ Zi ) u(2q + qˆr + n + i)

(4.70)

i=0

corresponds to the input sequence u(2q + qˆr + n), . . . , u(3q + qˆr + n − 1). qˆr

i ˜ G = ∑ Cˆ∞ Jˆ∞ Ωu(3q + qˆr + n + i)

(4.71)

i=0

corresponds to the input sequence u(3q + qˆr + n), . . . , u(3q + 2ˆ qr + n). Let γ =

2q+ˆ qr +n

∑

i=n+1

q−1

CJ i−1 Ωu(2q + qˆr + n − i) − CJ 2q+ˆqr +n ∑ Φi+1 u(i). From (4.28), we i=0

have that

q−1

Im CJ 2q+ˆqr +n ∑ Φi+1 u(i) = i=0

⎞⎛ u(0) ⎞ ⎛Bq ⎟ ∈ Im W (1, n)(4.72) ⎟⎜ = Im (CJ 3q+ˆqr +n−1 B ⋯ CJ 2q+ˆqr +n B)⎜ ⋮ ⎟ ⎟⎜ ⎜ ⋮ ⋱ ⎝B1 ⋯ Bq ⎠⎝u(q − 1)⎠ So γ ∈ Im W (1, n). Since β1 ∈ C ⟨J/ Im Ω⟩ ≡ Im W (1, n), there exists a vector w1 ∈ Rr such that n

i−1

β1 −γ = ∑ CJ i−1 ΩΩT (J T )

n−1

C T w1 = CΩΩT C T w1 +...+CJ n−1 ΩΩT (J T )

C T w1 (4.73)

i=1

and since n

∑ CJ i−1 Ωu(2q + qˆr + n − i) = CΩu(2q + qˆr + n − 1) + ... + CJ n−1 Ωu(2q + qˆr )

(4.74)

i=1

by choosing u(i) = ΩT (J T )

2q+ˆ qr +n−1−i

C T w1 for i = 2q + qˆr , ..., 2q + qˆr +n−1 and u(i) = w0,i ,

i = 0, . . . , 2q + qˆr − 1 as in (4.57), in order to satisfy the admissible initial conditions, we get A=

2q+ˆ qr +n

q−1

i=1

i=0

∑ CJ i−1 Ωu(2q + qˆr + n − i) − CJ 2q+ˆqr +n ∑ Φi+1 u(i) =


145

n

= ∑ CJ i−1 Ωu(2q + qˆr + n − i) + γ = β1 − γ + γ = β1 (4.75) i=1 q−1

Since β3 ∈ ∑ Im (CΦi+1 + C∞ Zi ), there exist vectors β3i ∈ Rm such that i=0

q−1

β3 = ∑ (CΦi+1 + Cˆ∞ Zi )β3i

(4.76)

i=0

By choosing u(2q + qˆr + n) = β3,0 u(2q + qˆr + n + 1) = β3,1 ⋮ u(3q + qˆr + n − 1) = β3,q−1

(4.77)

we get q−1

q−1

i=0

i=0

B = ∑ (CΦi+1 + Cˆ∞ Zi ) u(2q + qˆr + n + i) = ∑ (CΦi+1 + Cˆ∞ Zi )β3i = β3

(4.78)

˜ ≡ Im W ˜ (0, qr ), there exists a vector v ∈ Rr such that Since β2 ∈ Cˆ∞ ⟨J∞ / Im Ω⟩ qˆr

qˆr ˜ ˜ T ˆT qˆr ˆ T T i ˜ ˜ T ˆT i ˆ T ˜Ω ˜ T Cˆ∞ ΩΩ (J∞ ) C∞ v v + ... + Cˆ∞ Jˆ∞ ΩΩ (J∞ ) C∞ v = Cˆ∞ Ω β2 = ∑ Cˆ∞ Jˆ∞

(4.79)

i=0

and since qˆr

qˆr ˜ i ˜ ˜ Ωu(3q + 2ˆ qr + n) (4.80) Ωu(3q + qˆr + n + i) = Cˆ∞ Ωu(3q + qˆr + n) + ... + Cˆ∞ Jˆ∞ ∑ Cˆ∞ Jˆ∞ i=0 T v for i = 3q + q ˜ T (J∞ T )i−3q−ˆqr −n Cˆ∞ ˆr + n, ..., 3q + 2ˆ qr + n we get by choosing u(i) = Ω qˆr

qˆr

i=0

i=0

i

i ˜ i ˜ ˜ T ˆT T C = ∑ Cˆ∞ Jˆ∞ Ωu(3q + qˆr + n + i) = ∑ Cˆ∞ Jˆ∞ ΩΩ (J∞ ) Cˆ∞ v = β2

Overall, by choosing as an input ⎧ ⎪ w0,i ⎪ ⎪ ⎪ ⎪ 2q+ˆ qr +n−1−k T ⎪ ⎪ C w1 ⎪ ΩT (J T ) u(k) = ⎨ ⎪ β3,k−2q−ˆqr −n ⎪ ⎪ ⎪ k−3q−ˆ qr −n ⎪ ⎪ T) Tv ˜ T (Jˆ∞ ⎪ Ω Cˆ∞ ⎪ ⎩

k = 0, ..., 2q + qˆr − 1 k = 2q + qˆr , ..., 2q + qˆr + n − 1 k = 2q + qˆr + n, ..., 3q + qˆr + n − 1

(4.81)

(4.82)

k = 3q + qˆr + n, ..., 3q + 2ˆ qr + n

we end up with β(2q + qˆr + n) = β1 + β2 + β3 = z

(4.83)

which proves that q−1

˜ + ∑ Im (CΦi+1 + Cˆ∞ Zi ) R(0) ⊇ C⟨J/ImΩ⟩ + Cˆ∞ ⟨Jˆ∞ /ImΩ⟩ i=0

(4.84)

146


In the above proof, it was shown that any desired output can be reached within 2q + qˆr + n time steps. However, an admissible input sequence could possibly be constructed in less steps, as will be shown in the illustrative example that follows. So far we have managed to describe the reachable subspace R(0) by the use of the minimal realization matrices of A(σ)−1 and find an input sequence that will drive the system to a desired reachable state. Since the system is reachable when R(0) = Rr we can easily conclude to the following theorem that provides a reachability test for the system. Theorem 4.11. The system (4.2) is reachable from the origin, iff rank (Q1 Q2 Q3 ) = r

(4.85)

Q1 = (CJ n−1 Ω ⋯ CΩ)

(4.86a)

where

Q2 = (CΦ1 + Cˆ∞ Z0 ⋯

CΦq + Cˆ∞ Zq−1 )

qˆr ˜ ˜ ⋯ Cˆ∞ Jˆ∞ Q3 = (Cˆ∞ Ω Ω)

(4.86b) (4.86c)

By expanding the matrices in (4.86) and rewriting their coefficients in terms of i = 0 for the Laurent series expansion using (1.110a),(1.110b), bearing in mind that Jˆ∞

i ≥ qˆr + 1, since qˆr + 1 is the index of nilpotency of Jˆ∞ we get

Q1 = (H−q−n

⎞ ⎛Bq ⎟ ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ ⎜ ⋯ H−1 ) ⎜B0 Bq ⎟ ⎟ ⎟ ⎜ ⎟ ⎜ ⋱ ⋮ ⎟ ⎜ ⎝ B0 ⎠

Q3 = (H0

Q2 = (H−q

⎞ ⎛Bq ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ Bq ⎟ ⋯ Hq ) ⎜B0 ⎟ (4.87) ⎜ ⎟ ⎜ ⋱ ⋮ ⎟ ⎜ ⎟ ⎝ B0 ⎠

⎛Bq ⎞ ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ) ⋯ Hqˆr ⎜B0 ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⋱ ⋱ ⎜ ⎟ ⎝ B0 . . . Bq ⎠

(4.88)

By replacing (4.87),(4.88) in (4.85) we obtain an equivalent reachability test:

rank (H−q−n

⎛Bq ⎞ ⎜ ⋮ ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟=r ⋯ Hqˆr ) ⎜B0 ⋱ ⎟ ⎜ ⎟ ⎜ ⎟ ⋱ ⋱ ⎜ ⎟ ⎝ B0 Bq ⎠

(4.89)


147

(4.89) indicates that the matrix Q = (Q1 Q2 Q3 ) which contains the first q + n + qˆr + 1 terms of the Laurent expansion of A(σ)−1 B(σ) does not lose rank if and only if the system (4.2) is reachable. This result is comparable to the one presented in [58] where a different approximation was used. The advantage of the approximation used here is that we find the consistent input that drives the system from the zero initial conditions to the desired output z. Remark 4.12. In the special case of state space systems β(k + 1) = Aβ(k) + Bu(k)

(4.90)

(σIr − A)β(k) = Bu(k)

(4.91)

or equivalently

the system’s matrices are A0 = −A, A1 = Ir , B0 = B, B1 = 0, with q = 1 and n = r. Now, by using the formula (σIr − A)−1 = σ −1 Ir + σ −2 A + σ −3 A2 + . . .

(4.92)

we get H−1 = Ir

H−2 = A H−3 = A2

⋯

(4.93)

and the reachability criterion takes the form rank (An−1 B . . . B) = r

(4.94)

which is the reachability criterion for discrete time state space systems provided in [6, 47]. Remark 4.13. In the special case of generalised state space or descriptor systems Eβ(k + 1) = Aβ(k) + Bu(k)

(4.95)

(σE − A)β(k) = Bu(k)

(4.96)

or equivalently

with det E = 0, the system’s matrices are A0 = −A, A1 = E, B0 = B, B1 = 0, with q = 1. The reachability criterion takes the form rank (H−n B . . . Hqˆr B) = r

(4.97)

148


This result is the same to the one presented in Chapter 2, that was established in [67] and later in [55]: rank (φ−h B . . . φ−1 B φ0 B . . . φn−1 B) = r

(4.98)

where Φi the coefficients of the Laurent expansion of (σE − A)−1 : ∞

(σE − A)−1 = σ −1 ∑ Φi σ −i = Φ−h σ h−1 + ... + Φ−1 σ 0 + Φ0 σ −1 + Φ1 σ −2 + ...

(4.99)

i=−h

Example 4.14. Let −3σ + σ 2 2+σ ⎞ ⎛ 7 − 3σ ⎛σ 2 0 0⎞ ⎜ −5 + 2σ ⎜ ⎟ 2 1 − 2σ ⎟ ⎜ ⎟ β(k) = ⎜ 0 1 1⎟ u(k) ⎝−5 + 5σ − σ 2 ⎝ σ 0 1⎠ −σ −3σ + σ 2 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ A(σ)

(4.100)

B(σ)

For the matrix A(σ) we have

C SA(σ) (σ)

0 ⎞ ⎛1 0 =⎜ 0 ⎟ ⎟, ⎜0 1 ⎝0 0 (σ − 2)2 ⎠

0 SA(σ) (σ) ˜

⎛1 0 0 ⎞ ⎟ =⎜ ⎜0 1 0 ⎟ ⎝0 0 σ 4 ⎠

(4.101)

and so q = 2, n = 2, µ = 4, with µ = q + qˆ3 = 4 ⇒ qˆ3 = 4 − q = 2. So µ ˆ = qˆ3 + 1 = 3 and a minimal realization of A(σ)−1 is given by ⎛2 1⎞ ⎟ C=⎜ ⎜1 0⎟ , ⎝0 1⎠

2 ⎛0 0⎞ 5 6 ⎝−1 − 5 − 51 ⎠

(4.102)

1 ⎛1 1 1⎞ ⎛0 1 0⎞ ⎛0 0 5 ⎞ 3 ⎜ ⎟ ⎜ ⎟ ⎜ ˆ ˆ ˆ C∞ = ⎜0 2 1⎟ , J∞ = ⎜0 0 1⎟ ,B∞ = ⎜0 5 − 52 ⎟ ⎟ ⎝1 −1 0⎠ ⎝0 0 0⎠ ⎝0 − 1 0 ⎠ 5

(4.103)

J=

⎛2 1⎞ , ⎝0 2⎠

B=

˜ are The matrices Ω, Ω

Ω = J 2 BB2 + JBB1 + BB0 =

2 2 ⎞ ⎛− 21 5 5 5 22 6 ⎝− 5 − 5 − 75 ⎠

(4.104)

with rank(Ω) = 2 and 2 1 1 ⎛− 5 − 5 − 5 ⎞ 2 ˆ ⎟ ˜ =B ˆ∞ B2 + Jˆ∞ B ˆ∞ B1 + Jˆ∞ Ω B∞ B0 = ⎜ ⎜0 0 0⎟ ⎝0 0 0⎠

(4.105)


149

˜ = 1. Computing Φ1 ,Φ2 and Z0 , Z1 in the same fashion, we get with rank(Ω) Φ1 = JBB2 + BB1 =

⎛ −1 0 0⎞ , ⎝− 11 0 0⎠ 5

1 ⎛0 0 5 ⎞ 1 ⎟ ˆ∞ B0 = ⎜0 3 Z0 = B , ⎜ 5 5 ⎟ ⎝0 − 1 − 1 ⎠ 5 5

⎛ 0 0 0⎞ ⎝−1 0 0⎠

3

1

(4.106)

1

⎛5 5 5 ⎞ ˆ∞ B1 + Jˆ∞ B ˆ∞ B0 = ⎜− 2 − 1 − 1 ⎟ Z1 = B ⎜ 5 5 5⎟ ⎝0 0 0⎠

⎛ −64 ⎜ 5 ⎜ Q1 = (CΩ CJΩ) = ⎜ ⎜ −21 ⎜ 5 ⎜ −22 ⎝ 5 Q2 = (CΦ1 + Cˆ∞ Z0

Φ2 = BB2 =

−2 5

−3 5

−172 5

−16 5

2 5

2 5

−64 5

−2 5

−6 5

−7 5

−44 5

−12 5

21 2 1 ⎛− 5 5 5 1 ⎜ ˆ ) = CΦ2 + C∞ Z1 ⎜ −1 1 5 ⎝− 11 − 3 0 5 5

2 1 1 ⎛− 5 − 5 − 5 2 Ω) ˜ =⎜ ˜ Cˆ∞ Jˆ∞ ˜ Cˆ∞ Jˆ∞ Ω Q3 = (Cˆ∞ Ω ⎜0 0 0 ⎝− 2 − 1 − 1 5 5 5

−4 ⎞ ⎟ ⎟ −3 ⎟ 5 ⎟ ⎟ ⎟ −14 5 ⎠

(4.108)

− 65 25 0⎞ 2 4 − 5 − 5 − 52 ⎟ ⎟ 2 4 2 ⎠ −5 5 5

0 0 0 0 0 0 0 0 0

(4.107)

(4.109)

0 0 0⎞ 0 0 0⎟ ⎟ (4.110) 0 0 0⎠

Overall rank (Q1 Q2 Q3 ) = 3

(4.111)

So the above system is reachable. That means that every vector in R3 can be reached from the origin within a finite number of time steps. We will now construct such an input using the methodology presented in the Proof of Theorem 4.10. First of all, we need to choose the appropriate admissible initial values for the input, so that (4.30) is satisfied. Thus, select the inputs u(0) ⎛ ⎞ ⎛u(0)⎞ ⎜ ⎟=⎜ ⋮ ⎟ ⋮ ⎜ ⎟ ⎜ ⎟ ⎝u(2q + qˆ3 − 1)⎠ ⎝u(5)⎠

(4.112)

to belong to the kernel of:

⎛A0 A1 ⎞ ⎛ H0 H1 H2 ⎝ 0 A0 ⎠ ⎝H− 1 H0 H1

⎛B0 B1 B2 0 0 0 ⎞ ⎟ 0 ⎞⎜ ⎜ 0 B0 B1 B2 0 0 ⎟ ⎜ ⎟ ⎟ H2 ⎠ ⎜ ⎜ 0 0 B0 B1 B2 0 ⎟ ⎝ 0 0 0 B0 B1 B2 ⎠

(4.113)

150


The simplest choice is ⎛u(0)⎞ ⎛0⎞ ⎜ ⋮ ⎟ = ⎜⋮⎟ ⎜ ⎟ ⎜ ⎟ ⎝u(5)⎠ ⎝0⎠

(4.114)

Now, since rankQ2 = 3, every vector in R3 can be written as a linear combination of the columns of Q2 . So we only require the input time sequences corresponding to Q2 to reach any output in R3 . Let an arbitrary vector ⎛β1 ⎞ ⎜β2 ⎟ ∈ R3 ⎜ ⎟ ⎝β3 ⎠

(4.115)

The above vector can be written as a linear combination of CΦ1 +Cˆ∞ Z0 and CΦ2 +Cˆ∞ Z1 . That is, ∃c1 , c2 ∈ R3 such that ⎛β1 ⎞ ⎜β2 ⎟ = (CΦ1 + Cˆ∞ Z0 )c1 + (CΦ2 + Cˆ∞ Z1 )c2 ⎜ ⎟ ⎝β3 ⎠

(4.116)

Solving the system we get ⎛c11 ⎞ ⎛−β1 + β2 + β3 ⎞ ⎟ ⎟ ⎜ c1 = ⎜ c12 ⎟ ⎜c12 ⎟ = ⎜ ⎠ ⎝c13 ⎠ ⎝ c13

(4.117)

c21 ⎞ ⎛c21 ⎞ ⎛ 21β 21β c13 3 2 ⎟ ⎟ ⎜ = c2 = ⎜ ⎜c22 ⎟ ⎜ −8β1 + 2 + 2 − c12 − 2 + 3c21 ⎟ 21β 31β ⎝c23 ⎠ ⎝ 1 − 2 − 13β3 + 7c12 + c13 − 5c21 ⎠ 2 2 2

(4.118)

where c12 , c13 , c21 ∈ R are arbitrary. To reach the desired output, we need to set k = 2q + qˆr = 6. Choosing as an input ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ u(k) = ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

03×1 c1 c2 03×1

k = 0, ..., 5 k=6 k=7 k = 8, ...

(4.119)

we get the output of the system (under zero initial conditions) 6

i−1 ⎛ ∑ J Ωu(6 − i)⎞ 1 2−1 6 i=1 ⎟ ˆ i+1 Φ 2 (CΦ ) β(6) = (C Cˆ∞ ) ⎜ + + C Z u(6 + i) − CJ u(i) ⇒ ∑ ∑ i+1 ∞ i ⎜ ⎟ i Ωu(8 + i) ˆ ˜ i=0 J ∑ i=0 ⎝ ∞ ⎠

i=0

β(6) = (CΦ1 + Cˆ∞ Z0 ) u(6) + (CΦ2 + Cˆ∞ Z1 ) u(7) ⇒


151 ⎛β1 ⎞ ⎟ β(6) = ⎜ ⎜β2 ⎟ ⎝β3 ⎠

(4.120)

So any arbitrary output can be reached within 6 time steps. Of course, the above input sequence is by no means unique, since it depends on c12 , c13 , c21 that can be chosen arbitrarily. One can find multiple consistent inputs that can drive the system to a desired output. The advantage of our methodology lies in the fact that we take into account the admissible initial conditions (4.30). In the above procedure, we chose the admissible initial values of the input to be zero, which is the simplest choice. Alternatively, by solving the system

⎛A0 A1 ⎞ ⎛ H0 H1 H2 ⎝ 0 A0 ⎠ ⎝H− 1 H0 H1

⎛B0 B1 B2 0 0 0 ⎞ ⎟ ⎛u(0)⎞ 0 ⎞⎜ ⎜ 0 B0 B1 B2 0 0 ⎟ ⎜ ⎟⎜ ⋮ ⎟ ⎜ ⎟=0 ⎟ H2 ⎠ ⎜ ⎜ 0 0 B0 B1 B2 0 ⎟ ⎝u(5)⎠ ⎝ 0 0 0 B0 B1 B2 ⎠

(4.121)

we find that the general form of the admissible initial values of the input is u01 ⎛ ⎞ 5u u 4u31 2u32 2u33 ⎟ 22 23 u(0) = ⎜ ⎜ u11 − 3 − 3 + 3 + 3 + 3 ⎟ ⎝−u11 + 5u22 + u23 − 4u31 − 2u32 − 2u33 ⎠ 3 3 3 3 3

(4.122)

u11 ⎛ ⎞ 3u23 2u31 u32 u33 ⎜ ⎟ u(1) = ⎜ u22 + 5 − 5 − 5 − 5 ⎟ ⎝− 13u22 − 19u23 + 46u31 + 23u32 + 23u33 ⎠

(4.123)

3

u ⎛− 322

u(2) = ⎜ ⎜ ⎝

−

15 u23 15

+

15 4u31 15

u22 u23

+

15 15 2u32 2u33 15 + 15 ⎞

⎟ ⎟ ⎠

⎛u31 ⎞ ⎛u41 ⎞ ⎛u51 ⎞ ⎟ ⎜ ⎟ ⎜ ⎟ u(3) = ⎜ ⎜u32 ⎟ , u(4) = ⎜u42 ⎟ , u(5) = ⎜u52 ⎟ ⎝u33 ⎠ ⎝u43 ⎠ ⎝u53 ⎠

(4.124)

(4.125)

where ui,j ∈ R. In the first case when we choose the admissible values to be zero, it was shown that any state in R3 can be reached in k = 2q + qˆr = 6 steps. Now, since the admissible values where chosen to be nonzero, some nonzero terms will apprear in the system’s state β(k), that need to be eliminated. So we will need extra steps to reach the desired state. So set k = 2q + n + qˆr = 8, and the output of the system becomes

152

CHAPTER 4. REACHABILITY OF ALGEBRAIC/DIFFERENCE EQUATIONS 8

1

i=1

i=0

β(8) = C ∑ J i−1 Ωu(8 − i) − CJ 8 ∑ Φi+1 u(i) + ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ A

2

1

i=0

i=0

i ˜ + Cˆ∞ ∑ Jˆ∞ Ωu(10 + i) + ∑ (CΦi+1 + Cˆ∞ Zi ) u(8 + i) (4.126)

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ B

As before, since rankQ2 = 3, we can choose ⎧ c1 k=8 ⎪ ⎪ ⎪ ⎪ u(k) = ⎨ c2 k=9 ⎪ ⎪ ⎪ ⎪ ⎩ 03×1 k = 10, ...

(4.127)

where c1 , c2 as defined in (4.117) ,(4.118), to obtain ⎛β1 ⎞ ⎟ B=⎜ ⎜β2 ⎟ ⎝β3 ⎠

(4.128)

Now for the part A, we have

8

1

i=1

i=0

A = C ∑ J i−1 Ωu(8 − i) − CJ 8 ∑ Φi+1 u(i) ⇒ 2

8

1

i=1

i=3

i=0

A = C ∑ J i−1 Ωu(8 − i) + C ∑ J i−1 Ωu(8 − i) − CJ 8 ∑ Φi+1 u(i)

(4.129) (4.130)

´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ γ

and we need to make the above term equal to zero. Since 2

i−1

γ ∈ Im W (1, 2) = ∑ CJ i−1 ΩΩT (J T )

CT

(4.131)

i=1

there exists a vector w1 ∈ R3 such that − γ = CΩΩT C T w1 + CJΩΩT J T C T w1

(4.132)

Solving the above system we obtain ⎛w11 ⎞ ⎟ w1 = ⎜ ⎜w12 ⎟ ⎝w13 ⎠

(4.133)

with w11 ∈ R and w12 = −

10(4856680u22 + 652556u23 − 5456204u31 − 2159032u32 − 2187496u33 − 483804u41 − 42u42 − 5739u43 ) + 2937579

4.3. REACHABLE SUBSPACE +

153

10(−148017u51 + 25584u52 + 27003u53 − 483804u41 − 42u42 − 5739u43 − 148017u51 + 25584u52 + 27003u53 ) − 2w11 2937579 (4.134)

w13 =

49482400u22 + 7004720u23 − 55428080u31 − 21490240u32 − 21739600u33 − 4733760u41 + 2937579 247320u42 + 222180u43 − 1271940u51 + 422280u52 + 459480u53 − 2937579w11 + 2937579

(4.135)

So, by choosing as input u(6) = ΩT J T C T w1

(4.136)

u(7) = ΩT C T w1

(4.137)

2

we obtain A = C ∑ J i−1 Ωu(8 − i) + γ = −γ + γ = 0 and overall i=1

⎛β1 ⎞ ⎟ β(8) = A + B = ⎜ ⎜β2 ⎟ ⎝β3 ⎠

(4.138)

For example, by choosing u22 = v31 = 0, u01 = 1, u11 = 1, u31 = 1, u33 = 0, u32 = 3, u41 = u42 = u43 = u51 = u52 = u53 = 0 and w11 = 0, we obtain the input sequence 2 ⎛1⎞ ⎛ 1 ⎞ ⎛1⎞ ⎛1⎞ ⎛3⎞ 13 ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ u(0) = ⎜ ⎜ 3 ⎟ , u(1) = ⎜−1⎟ , u(2) = ⎜ 0 ⎟ , u(3) = ⎜3⎟ , u(4) = ⎜3⎟ , ⎝0⎠ ⎝− 13 ⎠ ⎝0⎠ ⎝ 23 ⎠ ⎝0⎠ 3

⎛1⎞ ⎟ u(5) = ⎜ ⎜3⎟ u(6) = ⎝0⎠

3 472352960 ⎛− 2937579 ⎞ ⎜ 240023920 ⎟ , ⎜ 2937579 ⎟ ⎝ 264116840 ⎠ 2937579

(4.139)

26356120

⎛ 2937579 ⎞ 191611760 ⎟ u(7) = ⎜ ⎜ 2937579 ⎟ , u(8) = c1 , u(9) = c2 (4.140) ⎝ 71863840 ⎠ 979193

and u(i) = 0, i = 10, .... Another approach for solving this problem might be to use the formula proposed by [40, 56]. Let the general solution of the system, under zero initial conditions β(0) = β(1) = 03 : β(k) = (H−k

⎛B0 B1 B2 ⋯ 0 ⎞ ⎛ u(0) ⎞ ⎜ ⎟ ⋯ H2 ) ⎜ ⋱ ⋱ ⋮ ⎟ ⋮ ⎜ ⋮ ⎟⎜ ⎟ ⎝ 0 ⋯ B0 B1 B2 ⎠ ⎝u(k + 4)⎠

(4.141)

It is easy to check that for k = 2 ⎛ ⎛B0 B1 B2 0 0 0 0 ⎞⎞ ⎜ ⎜0 B B B ⎟ 0 0 0⎟ ⎜ ⎜ ⎟⎟ 0 1 2 ⎜ ⎜ ⎟⎟ ⎜ ⎟⎟ rank ⎜ ⎜(H−2 H−1 H0 H1 H2 ) ⎜ 0 0 B0 B1 B2 0 0 ⎟⎟ = 3 ⎜ ⎜ ⎟⎟ ⎜ ⎜ 0 0 0 B0 B1 B2 0 ⎟⎟ ⎜ ⎜ ⎟⎟ ⎝ ⎝ 0 0 0 0 B0 B1 B2 ⎠⎠

(4.142)

154


So for an arbitrary vector in R3 , ∃c1 , c2 , ..., c7 ∈ R3 such that: ⎛B0 B1 B2 ⋯ 0 ⎞ ⎛c1 ⎞ ⎛β1 ⎞ ⎜β2 ⎟ = (H−2 ⋯ H2 ) ⎜ ⋮ ⎜ ⎟ ⋱ ⋱ ⋮ ⎟ ⎜ ⎟⎜ ⋮ ⎟ ⎜ ⎟ ⎝ 0 ⋯ B0 B1 B2 ⎠ ⎝c7 ⎠ ⎝β3 ⎠

(4.143)

Solving the above system, we can find a plethora of possible choices for the input. However we need to be careful, since the admissible conditions (4.30) need to be taken into account. For example, by choosing ⎛0⎞ ⎟ u(0) = ⎜ ⎜1⎟ ⎝0⎠

⎛5⎞ ⎟ u(1) = ⎜ ⎜−5⎟ ⎝0⎠ ⎛1⎞ ⎟ u(3) = ⎜ ⎜0⎟ ⎝1⎠

17 ⎛ − 5 − β1 + β2 + β3 ⎞ 227 11 11 16 ⎟ u(2) = ⎜ ⎜ 15 + 3 β1 − 3 β2 − 3 β3 ⎟ ⎝− 209 − 70β1 + 85β2 + 95β3 ⎠ 3

⎛0⎞ ⎟ u(k) = ⎜ ⎜0⎟ ⎝0⎠

3

3

3

k = 4, ...

as an input, we get that ⎛β1 ⎞ ⎟ β(2) = ⎜ ⎜β2 ⎟ ⎝β3 ⎠

(4.144)

Yet, although we managed to reach the desired output in 2 time steps, the choice of initial conditions is wrong, since they do not satisfy (4.30). So the above input sequence is not feasible. Moreover, if we try to solve the system of equations (4.113), (4.143), we find that they have no solutions in common, so for k = 2 we cannot find an input sequence that will drive the system to any arbitrary output.

4.4

Conclusions

A formula for the general solution of discrete time ARMA representations was provided in terms of the minimal realization matrices of A(σ)−1 . Furthermore, the reachability subspace of discrete time ARMA systems has been defined and criteria regarding the reachability of a system have been proposed. The results presented in this work constitute an extension of the results presented in [55] for descriptor systems. The advantage of the method used in this work is that it provides us with the form of the consistent input that drives the system from the zero vector to the desired output. Although the input and output of the system are defined as real vector valued functions, these

4.4. CONCLUSIONS

155

results can be generalized to an arbitrary field F. Further research still remains on the controllability to the origin and the extension of these results to positive systems or fractional order systems.

156


Chapter 5 Observability of Linear Systems of Algebraic and Difference Equations 5.1

Introduction

In this Chapter, we will consider nonhomogeneous systems of higher order discrete time algebraic and difference equations that are described by the matrix equation Aq β(k + q) + ... + A1 β(k + 1) + A0 β(k) = Bq u(k + q) + ... + B1 u(k + 1) + B0 u(k) (5.1a) ξ(k) = Cq β(k + q) + ... + C1 β(k + 1) + C0 β(k)

(5.1b)

where k ∈ N, Ai ∈ Rr×r , Bi ∈ Rr×m , Ci ∈ Rp×r , with at least one of Aq , Cq and at least one of A0 , C0 is a non zero matrix. The discrete time functions u(k) ∶ N → Rm , β(k) ∶ N → Rr and ξ(k) ∶ N → Rp define the input, state and output vectors of the system respectively. Using the forward shift operator σ with σ i β(k) = β(k + i), the system (5.1) can be rewritten as A(σ)β(k) = B(σ)u(k)

(5.2a)

ξ(k) = C(σ)β(k)

(5.2b)

with A(σ) = ∑qi=0 Ai σ i ∈ R[σ]r×r , B(σ) = ∑qi=0 Bi σ i ∈ R[σ]r×m , C(σ) = ∑qi=0 Ci σ i ∈ R[σ]p×r and det A(σ) ≠ 0. Systems described by (5.2) are also called Polynomial Matrix Descriptions (PMDs). As mentioned in the previous chapters, (5.2a) is also known as an ARMA (Auto-Regressive Moving Average) representation. 157

158 CHAPTER 5.

OBSERVABILITY OF ALGEBRAIC/DIFFERENCE EQUATIONS

As mentioned in previous chapters, higher order systems (5.2) and their continuous time analogues have been a subject of extensive research and often appear in various applications. Various methods of approach have been established to compute their solution and study their properties, as was commented in the Introduction of Chapter 4. The concept of observability was introduced by Kalman in [48–50] and has been previously studied by various authors, initially for state space systems in [6, 35, 47, 97, 104]. These results have been extended to descriptor systems in [7, 9, 26, 36, 60, 67, 98, 108, 120, 129], for second order descriptor systems in [64, 74], for rectangular descriptor systems in [38] and for positive systems in [41, 42]. In general, observability refers to the determination of the initial value of the system’s state, by knowledge of its input and output values over a finite interval. In contrast to state space systems though where the state and output of the system can be determined by knowledge of x(0), in descriptor systems, knowledge of x(0) may not be necessary, since the state and output are determined by Ex(0). Thus, observability of descriptor systems indicates if different Ex(0) may yield distinct time responses for a fixed input u(k), as commented for the continuous time case in [36]. So the ability to determine Ex(0) or x(0) are considered two different properties of the system which are defined as weak and complete observability or simply as observability. These terms are used because knowledge of x(0) also gives us knowledge of the vector Ex(0), while the opposite does not hold since E is singular. This is why the ability to determine x(0) is called complete observability. A similar distinction between x(0) and Ex(0) for continuous time systems is made in [98]. As was seen in Chapter 2 though, for discrete time descriptor and higher order systems systems, which satisfy certain admissibility conditions, the two notions coincide. This is also the case for higher order systems of the form (5.2). For such systems, we make the distinction between the initial values β(0), ..., β(q − 1) and the vector ⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟⎜ ⎟ βin = ⎜ ⋮ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠

(5.3)

as it was defined in [102]. For continuous time higher order descriptor systems of the form (5.2), conditions

5.1. INTRODUCTION

159

were given in [101] for (5.2b) to be a formal mapping, that is, to uniquely specify an image ξ(t) for each smooth or impulsive solution β(t) of (3.2). It was concluded that T

δM (A(σ)T C(σ)T ) = δM (A(σ)) must hold, where δM denotes the Mc-Millan degree of a polynomial matrix (see (1.56)). In the discrete time case though, the notion of impulsive solutions does not exist and thus this condition is not required. If we were to study the solution of (5.2) over a finite time horizon though, then the state β(k) would depend not only on past, but also on future values of β(k) and u(k), as was shown in [19] for descriptor systems and in [56] for higher order systems. Thus, the impulsive behavior in continuous time is exhibited as noncausality in discrete time systems over a finite horizon [0, N ], as commented in [69]. Contrary to continuous time descriptor systems though, discrete time descriptor systems do not have a solution for every set of initial values, as was commented in Chapter 2, in [19] and later in [55, 56]. That is, in order for the descriptor systems (5.2), (2.1) to have a solution, the initial values need to satisfy certain admissibility constraints. The concept of admissible initial values also appears in continuous time and gives the conditions under which the solution of the system will be impulse free [129]. So while for discrete time the non satisfaction of the admissibility constraints means that the system has no solution, for continuous time it translates to the system exhibiting impulses. In the present chapter we present conditions for the complete observability of (5.2). This concept of complete observability appears in [26, 98, 129] in continuous time (where the determination of x(0) or x(0−) is of interest), and in [17, 19] for discrete time systems. To do so, we first transform the system (5.2) into a descriptor system, using the method proposed in [58]. The resulting system is a special form of a descriptor system with output matrix C(σ) = C0 +C1 σ, that is, the output will depend on x(k) and x(k + 1). This is the discrete time analog of continuous time descriptor systems with a derivative in the output, that have been studied in [108]. This descriptor system is then further decomposed into two subsystems, the so called causal and noncausal. In this way, we first derive a criterion for observability in terms of the matrices of the causal and noncausal subsystems. We then extend these results to give criteria that connect observability to the matrices of the descriptor system (σE − A), C, the matrices of the higher order system A(σ), C(σ), and the Laurent expansion at infinity of (σE − A)−1

160 CHAPTER 5.


and A(σ)−1 . Thus, the observability requirements for the resulting descriptor and the original polynomial system are shown to coincide in the analysis that follows. Possible applications of the results presented in this chapter can be the problem of observer design, or the problem of pole placement through output feedback of the form u(k) = Kξ(k), which for the continuous time case has been studied in [27, 128, 130, 131, 136].

5.2

Transformation of higher order systems to descriptor form

In this section we will show how a higher order descriptor system can be transformed to a first order descriptor system with a state lead in the output, as studied in Section 2.5 of Chapter 2. It will be shown that the two systems have the same initial values and thus we can study the observability of the higher order system by studying the first order system. As in Section 2.5, we will focus our attention to the homogeneous case of (5.2), that is A(σ)β(k) = 0

(5.4a)

ξ(k) = C(σ)β(k)

(5.4b)

First, we will give a formula for the output of (5.4) in terms of the Laurent expansion of A(σ)−1 . Recall from [56] and Chapter 4 that the state of (5.4) is given by

where

β(k) = (H−k−q ⋯ H−k−1 ) βin

(5.5)

⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟⎜ ⎟ βin = ⎜ ⋮ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠

(5.6)

The set of consistent initial values for the homogeneous system is given by ad HAR = {β(i) ∈ Rr , i = 0, ..., q − 1 ∣

⎫ ⎛ β(0) ⎞ ⎛ H−q ⋯ H−1 ⎞ ⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞⎪ ⎪ ⎪ ⎜ ⎟=⎜ ⋮ ⎟⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟⎪ (5.7) ⋮ ⋮ ⋮ ⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟⎬ ⎪ ⎪ ⎪ ⎝β(q − 1)⎠ ⎝H−2q+1 ⋯ H−q ⎠ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠⎪ ⎭

5.2. TRANSFORMATION OF HIGHER ORDER SYSTEMS

161

Theorem 5.1. The output ξ(k) of (5.4) is ξ(k) = (C0

⎛ H−(k+q) ⋯ H−(k+1) ⎞ ⎟ βin ⋯ Cq ) ⎜ ⋮ ⋮ ⎜ ⎟ ⎝H−(k+2q) ⋯ H−(k+q+1) ⎠

(5.8)

Proof. The above formula can be easily derived by directly substituting (5.5) into ξ(k) = C(σ)β(k). Alternatively, the formula for the output can be derived analytically by taking the Z-transform on (5.4) and making use of (5.7). Let B(z) = Z[β(k)] = ∞ ∞ ∑k=0 β(k)z −k and Ξ(z) = Z[ξ(k)] = ∑k=0 ξ(k)z −k be the Z-transforms of β(k) and ξ(k).

Applying the Z-transform on (5.4) gives A A(z)B(z) = Bin (0)

(5.9a)

C Ξ(z) = C(z)B(z) − Bin (0)

(5.9b)

where A Bin (0) = (z q Ir

⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟ ⎟⎜ ⋮ ⋯ zIr ) ⎜ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠

(5.10)

C Bin (0)

⎛Cq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟ ⎟⎜ ⋮ ⋯ zIp ) ⎜ ⎟ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎝C1 ⋯ Cq ⎠ ⎝β(q − 1)⎠

(5.11)

= (z q Ip

From (5.9) we get A B(z) = A(z)−1 Bin (0)

(5.12a)

A C Ξ(z) = C(z)A(z)−1 Bin (0) − Bin (0) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¸ ¹ ¹ ¹ ¶

(5.12b)

Y1 (z)

Y2 (z)

A we will examine Y1 (z) and Y2 (z) separately. For Y1 (z), using the formula for A(z)−1 Bin (0)

from [39, Ch. 5], we get ⎛ H−q ⋯ H−1 ⎞⎛Aq ⋯ 0 ⎞⎛ β(0) ⎞ ⎟⎜ ⎟⎜ ⎟ Y1 (z)= (Cq z + ... + C0 )(Ir z −1 Ir ⋯)⎜ ⋮ ⎜H−q−1 ⋯ H−2 ⎟⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟ ⎝ ⋮ ⎠ ⎝ ⎠ ⎝ ⋮ ⋮ A1 ⋯ Aq β(q − 1)⎠ q

(1.110b)

=

(Cq z q + ... + C0 )Cz(zIn − J)−1 (J q−1 B ⋯ B) βin ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(5.13)

βf (0)

Now, consider the function f (k) = J k , with Z-transform F (z) = Z[f (k)] = z(zIn −J)−1 . The following properties hold Z[f (k)] = F (z) = z(zIn − J)−1

162 CHAPTER 5.


Z[f (k + 1)] = zF (z) − zf (0) = z 2 (zIn − J)−1 − zIn Z[f (k + 2)] = z 2 F (z) − z 2 f (0) − zf (1) = z 3 (zIn − J)−1 − z 2 In − zJ

(5.14)

⋮ Z[f (k + q)] = z q F (z)− z q f (0)− ... −zf (q − 1)=z q+1 (zIn − J)−1 −z q In − ...−zJ q−1 Using the above properties, Y1 (z) is equal to ⎛ z(zIn − J)−1 ⎞ ⎟ βf (0) Y1 (z)= (C0 C ⋯ Cq C) ⎜ ⋮ ⎜ ⎟ q+1 −1 ⎝z (zIn − J) ⎠ F (z) ⎛ ⎞ ⎜ ⎟ zF (z) − zIn ⎜ ⎟ ⎟ βf (0) + Y1 (z)= (C0 C ⋯ Cq C) ⎜ ⎜ ⎟ ⋮ ⎜ ⎟ ⎝z q F (z) − z q In − ...−zJ q−1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(5.15)

(5.16)

Y1a (z)

0 ⎞ ⎛ ⎟ ⎜ zIn ⎟ ⎜ ⎟ βf (0) + (C0 C ⋯ Cq C) ⎜ ⎟ ⎜ ⋮ ⎟ ⎜ ⎝z q In + ...+zJ q−1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶

(5.17)

Y1b (z)

The term Y1b (z) is equal to (z q Ip

0 ⎞⎛ In ⎞ ⎛ Cq C ⋯ ⎟ q−1 ⎜ ⎜ ⋱ ⋮ ⎟ ⋯ zIp )⎜ ⋮ ⎟⎜ ⋮ ⎟(J B ⋯ B) βin ⎝C1 C ⋯ Cq C ⎠⎝J q−1 ⎠

(5.18)

= (z q Ip

CB ⎞ ⎛Cq ⋯ 0 ⎞ ⎛ CJ q−1 B ⋯ ⎟ βin ⎟⎜ ⋮ ⋮ ⋯ zIp ) ⎜ ⎟ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎝C1 ⋯ Cq ⎠ ⎝CJ 2q−2 B ⋯ CJ q−1 B ⎠

(5.19)

Y1b (z)=

Now, considering Y1b (z) − Y2 (z), we get Y1b (z) − Y2 (z) =

(z q Ip

⎛ Cq ⋯ 0 ⎞ ⎟ ⋯ zIp ) ⎜ ⎜ ⋮ ⋱ ⋮ ⎟× ⎝C1 ⋯ Cq ⎠

CB ⎞⎛Aq ⋯ 0 ⎞⎛ β(0) ⎞ ⎛ β(0) ⎞⎞ ⎛⎛ CJ q−1 B ⋯ ⎜ ⎜ ⎟⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎟−⎜ ⎟⎟ (5.7) = 0 (5.20) × ⎜⎜ ⋮ ⋮ ⋮ ⋮ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎟ (1.110b) 2q−2 q−1 ⎝⎝CJ B ⋯ CJ B ⎠⎝A1 ⋯ Aq ⎠⎝β(q − 1)⎠ ⎝β(q − 1)⎠⎠ So overall Ξ(z)= Y1 (z) − Y2 (z) = Y1a (z) ⇒

(5.21)

5.2. TRANSFORMATION OF HIGHER ORDER SYSTEMS ⎛ Jk ⎞ ⎟ q−1 ξ(k) = Z −1 [Ξ(z)] = (C0 C ⋯ Cq C) ⎜ ⎜ ⋮ ⎟ (J B ⋯ B) βin = ⎝J k+q ⎠

163

(5.22)

= (C0

⎛ CJ k+q−1 B ⋯ CJ k B ⎞ ⎟ βin (1.110b) = ⋯ Cq ) ⎜ ⋮ ⋮ ⎜ ⎟ k+2q−1 k+q ⎝CJ B ⋯ CJ B ⎠

(5.23)

= (C0

⎛ H−(k+q) ⋯ H−(k+1) ⎞ ⎟ βin ⋯ Cq ) ⎜ ⋮ ⋮ ⎜ ⎟ ⎝H−(k+2q) ⋯ H−(k+q+1) ⎠

(5.24)

Now, following the procedure of [58], by defining the new state and output vectors ⎛ β(kq + 0) ⎞ ⎟, x(k) = ⎜ ⋮ ⎟ ⎜ ⎝β(kq + q − 1)⎠

⎛ ξ(kq + 0) ⎞ ⎟ y(k) = ⎜ ⋮ ⎟ ⎜ ⎝ξ(kq + q − 1)⎠

(5.25)

the system (5.4) can be rewritten as ⎛A0 ⋯ Aq−1 ⎞ ⎛Aq ⋯ 0 ⎞ ⎜ ⋮ ⋱ ⋮ ⎟ x(k + 1) = − ⎜ ⋮ ⋱ ⋮ ⎟ ⎟ x(k) ⎜ ⎟ ⎜ ⎝ 0 ⋯ A0 ⎠ ⎝A1 ⋯ Aq ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(5.26a)

˜ A

˜ E

⎛ Cq ⋯ 0 ⎞ ⎛C0 ⋯ Cq−1 ⎞ ⎟ ⎟ ⎜ y(k) = ⎜ ⋮ ⋱ ⋮ ⎟ x(k) + ⎜ ⎜ ⋮ ⋱ ⋮ ⎟ x(k + 1) ⎝C1 ⋯ Cq ⎠ ⎝ 0 ⋯ C0 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ˜1 C

(5.26b)

˜2 C

Thus, the higher order system (5.4) has been rewritten as a descriptor system of the form ˜ ˜ Ex(k + 1) = −Ax(k)

(5.27a)

y(k) = C˜1 x(k) + C˜2 x(k + 1)

(5.27b)

˜ A˜ ∈ Rrq×rq , C˜1 , C˜2 ∈ Rpq×rq . with E, Before we continue, it should be noted that the state vector defined in [58] was (β(kq+q−1)T , ⋯, β(kq+0)T )T , so the resulting system had the block transpose matrices ˜ A. ˜ of E, For the descriptor system (5.27) we have the following important results.

164 CHAPTER 5.


Theorem 5.2. [58] The fundamental matrix sequences Hi of A(σ)−1 and Φi of (σ E˜ + ˜ −1 are connected by A) ⎛ H−q−qi ⋯ H−qi−1 ⎞ rq×rq Φi = ⎜ ⋮ ⋮ ⎟ ⎜ ⎟∈R ⎝H−2q−qi+1 ⋯ H−q−qi ⎠

(5.28)

Proof. This can be derived following the procedure in [58, Theorem 2]. ˜ and A(σ) are equal, Theorem 5.3. [58] The degrees of the determinants of (σ E˜ + A) that is ˜ = deg det A(σ) = n deg det(σ E˜ + A)

(5.29)

Theorem 5.4. Let λi ∈ C, i = 1, ..., ` be the distinct zeros of A(σ). Then the zeros of (σE + A) are given by λqi . Proof. Let λi be a zero of A(σ). It thus holds that det A(λi ) = 0. Define the following lower block triangular matrix 0 ⎛ Ir ⎜ λi Ir Ir ⎜ G=⎜ ⎜ ⋮ ⋱ ⎜ q−1 ⎝λi Ir 0

⋯ 0⎞ ⋯ 0⎟ ⎟ ⎟ , with det G = 1 ⋯ ⋮⎟ ⎟ ⋯ Ir ⎠

(5.30)

The determinant of (σE + A) for σ = λqi is equal to RRR λq A + A A1 0 RRR⎛ i q q det(λi E + A)= RRRR⎜ ⋮ ⋱ RRR⎜ q q ⎝ RRR λi A1 λi A2 RRR RRR λq A + A A1 0 RRR⎛ i q R = RRR⎜ ⋮ ⋱ RRR⎜ q q ⎝ RRR λi A1 λi A2 RRR RRR A(λ ) A1 i RRR⎛ R = RRR⎜ ⋮ RRR⎜ q−1 ⎝ RRR λi A(λi ) λqi A2 RRR I A1 ⋯ RRR⎛ r R ⎜ = RRR⎜ ⋮ ⋮ RRR q−1 q RRR⎝λi Ir λi A2 ⋯ RRR I A1 ⋯ RRR⎛ r R ⎜ = RRR⎜ ⋮ ⋮ RRR q−1 q RRR⎝λi Ir λi A2 ⋯

R ⎞RRRR ⎟RRRR = ⎟RR R q ⋯ λi Aq + A0 ⎠RRRR ⋯

Aq−1

(5.31)

0 ⋯ 0 ⎞RRRR ⎛ Ir R ⎞⎜ ⎟RRRR λ I I ⋯ 0 i r r ⎜ ⎟ ⎟⎜ ⎟RRRR = (5.32) ⎟⎜ ⋮ ⋱ ⋯ ⋮⎟ ⎟RRRR ⋯ λqi Aq + A0 ⎠ ⎜ q−1 ⎝λi Ir 0 ⋯ Ir ⎠RRRR R R ⋯ Aq−1 ⎞RRR R ⎟RRRR = (5.33) ⋮ ⎟RR RRR q ⎠ ⋯ λi Aq + A0 RR R Aq−1 ⎞ ⎛A(λi ) ⎞RRRR ⎟⎜ ⎟RRRR = (5.34) ⋱ ⎟⎜ ⎟RR R q λi Aq + A0 ⎠ ⎝ Ir ⎠RRRR Aq−1 ⎞RRRR R ⎟RRRR ∣A(λi )∣ = 0 (5.35) ⎟RR R q R λi Aq + A0 ⎠RRR ⋯

Aq−1

5.3. OBSERVABILITY

165

and so λqi are zeros of the matrix (σE + A). Since from Theorem 5.3 the determinants of the two matrices have the same degrees, the complete set of zeros of (σE + A) are given by λqi .

5.3

Observability

As described in Chapter 2, there exist nonsingular matrices P, Q ∈ Rrq×rq such that the system (5.27) is decomposed as in (2.168)-(2.170), with A1 ∈ Rn×n , N ∈ Rµ×µ , where µ = rq − n is the sum of the orders of the infinite elementary divisors of A(σ) as in (1.57), F1 , D1 ∈ Rpq×n , F2 , D2 ∈ Rpq×µ and N is nilpotent with index of nilpotency h. Now we can proceed with studying the observability of the higher order system (5.4). Definition 5.5. The higher order system (5.4) is completely observable if the initial values β(0), ..., β(q − 1), and consequently the vector βin can be uniquely determined from knowledge of the output ξ(k) over a finite time interval. Arranging the initial values β(0), ..., β(q − 1) of (5.4) in a column it can be easily observed that ⎛ β(0) ⎞ ⎟, x(0) = ⎜ ⋮ ⎟ ⎜ ⎝β(q − 1)⎠

⎛Aq ⋯ 0 ⎞ ⎛ β(0) ⎞ ⎟ = βin ⎟⎜ ˜ Ex(0) =⎜ ⋮ ⎟ ⎜ ⋮ ⋱ ⋮ ⎟⎜ ⎝A1 ⋯ Aq ⎠ ⎝β(q − 1)⎠

(5.36)

So the two systems (5.4) and (5.27) have the same initial values. In addition, the admissible initial values of (5.27) are given by ˜ x(0) = Φ0 Ex(0)

(5.37)

and considering (5.28) and (5.26a), it is clear that (5.37) is equivalent to (5.7). Thus, by also taking into account Theorem 2.23 we conclude that the higher order system (5.4), the descriptor system (5.27) and the causal/noncausal subsystems (2.168)-(2.170) have the same set of admissible initial values. Now, although the higher order system (5.4) and its corresponding descriptor system (5.27) may not be connected in a strict system equivalence sense [see 104], there is a

166 CHAPTER 5.


1-1 correspondence between their input vectors β(k), x(k) and their output vectors ξ(k), y(k), given by ⎛ β(kq + 0) ⎞ ⎟ x(k) = ⎜ ⋮ ⎜ ⎟ ⎝β(kq + q − 1)⎠ β(k) = (δ(k

mod q) Ir

δ(k

mod q)−1 Ir

⋯ δ(k

(5.38) mod q)−(q−1) Ir ) x ([

k ]) q

⎛ ξ(kq + 0) ⎞ ⎟ y(k) = ⎜ ⋮ ⎜ ⎟ ⎝ξ(kq + q − 1)⎠ ξ(k) = (δ(k

mod q) Ip

δ(k

mod q)−1 Ip

⋯ δ(k

(5.39)

(5.40) mod q)−(q−1) Ip ) y ([

k ]) q

(5.41)

where k ∈ N, δi = 0 for i ≠ 0 and δ0 = 1 and [⋅] denotes the integer part of a number. In addition, since the higher order system (5.4), the descriptor system (5.27) and its equivalent causal/noncausal subsysems (2.168)-(2.170) all have the same initial values (β(0)T , ..., β(q − 1)T )T = x(0) = Q(˜ x(0)), it should be clear that in order to study the observability of the higher order system (5.4), one can instead study the observability of the causal/noncausal subsystems (2.168)-(2.170), since the determination of the initial value x˜(0) will naturally give us the initial values of the higher order system. This leads to the following result. Remark 5.6. The higher order system (5.4) is completely observable if and only if the descriptor system (5.27) is completely observable. Corollary 5.7. Using the connection between Φi and Hi in (5.28) and the form of C˜1 , C˜2 in (5.26) the observability criterion (2.173) can be rewritten as

rankOP M D

⋯ H−1 ⎞ ⎛C0 ⋯ Cq ⋯ 0 ⎞ ⎛ H−q ⎜ ⎟ ⎜ = rank ⎜ ⋮ ⋱ ⋱ ⋱ ⋮ ⎟ ⎜ ⋮ ⋮ ⎟ ⎟=n ⎝ 0 ⋯ C0 ⋯ Cq ⎠ ⎝H−2q−qn+1 ⋯ H−q−qn ⎠

(5.42)

So overall, the rank conditions in (2.172), (2.173) and (5.42) coincide. In Theorem 2.33 we managed to connect the observability of (5.27) with the co˜ and (C˜1 + C˜2 σ). The next step is to connect primeness of the matrices (σ E˜ + A) observability with the coprimeness of the original system matrices A(σ), C(σ). To do so we need the following Theorem.

5.3. OBSERVABILITY

167

˜ A) ˜ and (C˜1 + C˜2 σ) are right coprime, or equivalently Theorem 5.8. The matrices (σ E+ rank

⎛ σ E˜ + A˜ ⎞ = rq ⎝C˜1 + C˜2 σ ⎠

(5.43)

if and only if the matrices A(σ) and C(σ) are right coprime. Proof. Before we proceed to the main part of this proof, we need to consider the division ˜ (C˜1 + C˜2 σ) separately. of the matrices A(σ), C(σ) and (σ E˜ + A), Considering the division of A(σ), C(σ), there exist Q(σ), R(σ) such that C(σ) = Q(σ)A(σ) + R(σ)

(5.44)

with R(σ)A(σ)−1 strictly proper. From (5.44), we obtain ⎛A(σ)⎞ ⎛ Ir 0 ⎞ ⎛A(σ)⎞ = ⎝R(σ)⎠ ⎝−Q(σ) Ip ⎠ ⎝C(σ)⎠

(5.45)

From the abve equation, we can easily conclude that A(σ), C(σ) are right coprime if and only if A(σ), R(σ) are right coprime. Multiplying (5.44) from the right by A(σ)−1 we get C(σ)A(σ)−1 = Q(σ) + R(σ)A(σ)−1

(5.46)

Using the Laurent expansion of A(σ)−1 from (1.109), C(σ)A(σ)−1 is equal to

C(σ)A(σ)

−1

= (C0

Hqˆr ⋯ H0 ⎛ ⎜ ⋯ Cq ) ⎜ ⋰ ⋮ ⎝Hqˆr ⋯ ⋯ H−q

H−1 ⋮ H−q−1

⎛σ q+ˆqr Ir ⎞ ⎟ ⋯⎞ ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎜ ⎟ ⋯⎟ ⎟ ⎜ Ir ⎟ (5.47) ⎜ ⎟ −1 I ⎟ ⋯⎠ ⎜ σ r ⎜ ⎟ ⎝ ⋮ ⎠

Comparing (5.46) and (5.47) we get that the strictly proper matrix T (σ) = R(σ)A(σ)−1 is equal to

R(σ)A(σ)−1 = (C0

= (C0

H−2 ⋯⎞ ⎛ σ −1 I ⎞ ⎛ H−1 ⎜ −2 ⎟ (1.110b) ⋯ Cq ) ⎜ ⋮ ⋯⎟ ⎜ ⋮ ⎟ ⎜σ Ir ⎟ = ⎝H−q−1 H−q−2 ⋯⎠ ⎝ ⋮ ⎠

(5.48)

CJB ⋯⎞ ⎛σ −1 Ir ⎞ ⎛ CB ⎜ −2 ⎟ ⋯ Cq ) ⎜ ⋮ ⋯⎟ ⎜ ⋮ ⎟ ⎜σ Ir ⎟ = ⎝CJ q B CJ q+1 B ⋯⎠ ⎝ ⋮ ⎠

(5.49)

168 CHAPTER 5.


⎛σ −1 In ⎞ −2 ⎟ = (C0 C + C1 CJ + ... + Cq CJ q ) (I J J 2 ⋯) ⎜ ⎜σ In ⎟ B = (5.50) ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ⎝ ⋮ ⎠ K ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ (σIn −J)−1

= K(σIn − J)−1 B

(5.51)

where n = deg det A(σ). So T (σ) = R(σ)A(σ)−1 = K(σIn − J)−1 B with K ∈ Rp×n , J ∈ Rn×n , B ∈ Rn×r and (K, J, B, 0m×r ) is a realization of T (σ). Now, since (C, J, B, 0m×r ) is a minimal realization of the strictly proper part of A(σ)−1 , from Theorem 1.64 it holds that rank (B JB ... J n−1 B) = n

(5.52)

As stated above, if A(σ), C(σ) are right coprime, then the matrices A(σ) and R(σ) are coprime as well and T (σ) = R(σ)A(σ)−1 is a right coprime matrix fraction description of T (σ) and since J ∈ Rn×n with n = deg det A(σ), from Theorem 1.64 it holds that (K, J, B, 0m×r ) is a minimal realization of T (σ). So it holds that ⎛ K ⎞ ⎜ KJ ⎟ ⎟ ⎜ ⎟=n rank ⎜ ⎜ ⋮ ⎟ ⎟ ⎜ ⎝KJ n−1 ⎠

(5.53)

˜ and (C˜1 + C˜2 σ) we have In the same fashion, if we consider the division of (σ E˜ + A) ¯ ˜ + R(σ) ¯ (C˜1 + C˜2 σ) = Q(σ)(σ E˜ + A)

(5.54)

¯ ˜ −1 is strictly proper. From (5.54) we obtain where R(σ)(σ E˜ + A) ⎛σ E˜ + A˜⎞ ⎛ Irq 0 ⎞ ⎛ σ E˜ + A˜ ⎞ = ¯ ¯ ⎝ R(σ) ⎠ ⎝−Q(σ) Ipq ⎠ ⎝C˜1 + C˜2 σ ⎠

(5.55)

˜ (C˜1 + C˜2 σ) are right corime if and only if (σ E˜ + A), ˜ R(σ) ¯ so (σ E˜ + A), are right coprime. ˜ −1 we get Multiplying (5.54) from the right by (σ E˜ + A) ˜ −1 = Q(σ) ¯ ¯ ˜ −1 (C˜1 + C˜2 σ)(σ E˜ + A) + R(σ)(σ E˜ + A)

(5.56)

˜ −1 , (C˜1 + C˜2 σ)(σ E˜ + A) ˜ −1 is equal to Using the Laurent expansion of (σ E˜ + A)

Φ−µ ⋯ Φ−1 ˜ −1 = (C˜1 C˜2 ) ⎛ 0 (C˜1 + C˜2 σ)(σ E˜ + A) ⎝Φ−µ Φ−µ+1 ⋯ Φ0

Φ0 Φ1

⎛ σ µ Irq ⎞ ⎜ ⋮ ⎟ ⎜ ⎟ ⎟ ⎞ ⋯ ⎜ ⎜ Irq ⎟(5.57) ⎜ ⎟ ⋯⎠ ⎜ −1 ⎟ ⎜σ Irq ⎟ ⎜ ⎟ ⎝ ⋮ ⎠

5.3. OBSERVABILITY

169

¯ Comparing (5.56) and (5.57) we get that the strictly proper matrix T¯(σ) = R(σ)(σ E˜ + ˜ −1 is equal to A) ¯ ˜ −1 T¯(σ)= R(σ)(σ E˜ + A)

(5.58) ⋯⎞ ⎛σ −1 Irq ⎞ (1.113)

⎛Φ0 Φ1 = (C˜1 C˜2 ) ⎝Φ1 Φ2 ⋯⎠ ⎝

⋮

⎠

=

(5.59)

= (C˜1

⎞ ⎛ ⎛In 0 ⎞ ⎛A1 0 ⎞ P Q P ⋯⎟ ⎜Q ⎜ ⎝ 0 0µ ⎠ ⎟ ⎛σ −1 Irq ⎞ ⎝ 0 0µ ⎠ ⎟ = C˜2 ) ⎜ ⎟⎝ ⎜ ⎛ ⎛A21 0 ⎞ ⎞ ⋮ ⎠ ⎟ ⎜ A 0 1 ⎟ ⎜Q P ⋯ P Q ⎝ 0 0µ ⎠ ⎝ ⎝ 0 0µ ⎠ ⎠

(5.60)

= (C˜1

⎛ ⎛In 0 ⎞ ⎛A1 0 ⎞ ⎞ ⋯⎟ ⎜ ⎟ ⎛σ −1 P ⎞ (2.14) ⎛Q 0 ⎞ ⎜ ⎜ ⎝ 0 0µ ⎠ ⎝ 0 0 µ ⎠ ⎟ = C˜2 ) ⎜ ⎟⎝ ⎝ 0 Q⎠ ⎜⎛A1 0 ⎞ ⎛A21 0 ⎞ ⋮ ⎠ (2.162) ⎟ ⎟ ⎜ ⋯ ⎝⎝ 0 0 µ ⎠ ⎝ 0 0 µ ⎠ ⎠

(5.61)

= (F1 F2 D1

= (F1 D1 )

⎛ In 0n×µ A1 0n×µ ⎜0µ×n 0µ 0µ×n 0µ ⎜ D2 ) ⎜ ⎜ A1 0n×µ A2 0n×µ ⎜ 1 ⎝0µ×n 0µ 0µ×n 0µ

⋯⎞ ⋯⎟ ⎟ ⎛σ −1 P ⎞ ⎟ = ⋯⎟ ⎟⎝ ⋮ ⎠ ⋯⎠

⎛ In 0n×µ A1 0n×µ ⋯⎞ ⎛σ −1 P ⎞ ⎝A1 0n×µ A21 0n×µ ⋯⎠ ⎝ ⋮ ⎠

(5.62)

(5.63)

and by decomposing the matrix P as P=

⎛P1 ⎞ ⎝P2 ⎠

(5.64)

with P1 ∈ Rn×rq , P2 ∈ Rµ×rq the above is equal to ⎛ In A1 ⋯⎞ ⎛σ −1 P1 ⎞ T¯(σ)= (F1 D1 ) = ⎝A1 A21 ⋯⎠ ⎝ ⋮ ⎠ = (F1 + D1 A1 ) (In A1 A21 ⋯) = (F1 + D1 A1 )(σIn − A1 )−1 P1

⎛σ −1 P1 ⎞ = ⎝ ⋮ ⎠

(5.65) (5.66) (5.67)

˜ = deg det A(σ), from Theorem 5.3. So T¯(σ) = R(σ)(σ ¯ where n = deg det(σ E˜ + A) E˜ + ˜ −1 = (F1 + D1 A1 )(σIn − A1 )−1 P1 and ((F1 + D1 A1 ), A1 , P1 , 0pq×rq ) is a realization of A) T¯(σ). Now we can proceed to the main part of our proof regarding the coprimeness of the ˜ matrices A(σ), C(σ) and (C˜1 + C˜2 σ), (σ E˜ + A).

170 CHAPTER 5.


˜ are right coprime. Then the matrices First, assume that (C˜1 + C˜2 σ) and (σ E˜ + A) ˜ R(σ) ¯ (σ E˜ + A), are also right coprime and ((F1 + D1 A1 ), A1 , P1 , 0pq×rq ) is a minimal realization of T¯(σ). It thus holds that ⎛ F1 + D1 A1 ⎞ ⎛ Φ0 ⎞ ⎛C˜1 C˜2 ⋯ 0 ⎞⎜ ⎟ ⎜ (F1 + D1 A1 )A1 ⎟ (5.26) Φ1 ⎟ Proof of ⎜ ⎟ ⎟⎜ ⎟=n ⎜ ⎟ = n ⇔ (5.68) rank ⎜ ⇔ rank ⎜ ⋮ ⋱ ⋱ ⋮ ⎜ ⎟ ⎜ ⎟ ⎜ ⋮ ⎟ Theorem 2.32 (5.28) ⋮ ⎜ ⎟ ⎝ 0 ⋯ C˜1 C˜2 ⎠⎜ ⎟ ⎝(F1 + D1 A1 )An−1 ⎠ ⎝Φn ⎠ 1 ⋯ H−1 ⎞ ⎛C0 ⋯ Cq ⋯ 0 ⎞ ⎛ H−q ⎟⎜ rank ⎜ ⋮ ⋮ ⎟ ⎜ ⋮ ⋱ ⋱ ⋱ ⋮ ⎟⎜ ⎟=n⇔ ⎝ 0 ⋯ C0 ⋯ Cq ⎠ ⎝H−2q−qn+1 ⋯ H−q−qn ⎠

(5.69)

KB ⎞ ⎛ KJ q−1 B ⋯ ⎜ ⎟=n⇔ rank ⎜ ⋮ ⋮ ⎟ ⎝KJ q+qn−2 B ⋯ KJ qn−1 B ⎠

(5.70)

⎛ K ⎞ ⎜ KJ ⎟ ⎟ q−1 ⎜ ⎟ (J B ⋯ B) = n rank ⎜ ⎜ ⋮ ⎟ ⎟ ⎜ ⎝KJ qn−1 ⎠

(5.71)

from the above relation we conclude that ⎛ K ⎞ ⎜ KJ ⎟ ⎜ ⎟ ⎟=n rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝KJ qn−1 ⎠

(5.72)

From the Cayley-Hamilton theorem we know that J, which is in Jordan form, n−1

satisfies its characteristic polynomial p(λ) = det (λIn − J) = λn + ∑ pi λi . Hence, J i=0

satisfies p(J) = J n + pn−1 J n−1 + ... + p0 In = 0

(5.73)

This means that the matrix J n = −pn−1 J n−1 − ... − p0 In is linearly dependent to the matrices J n−1 , J n−2 ,..., J, In . Multiplying the above equation from the right by J and replacing J n , we get that the same holds for J n+1 . Following this procedure, we get that all J k , k ≥ n are linearly dependent to J n−1 ,J n−2 ,. . . , J, In . As a result

J i ∈ Im (J n−1 ⋯ J In ) ⇒ KJ i ∈ Im (KJ n−1 ⋯ KJ K) , i ≥ n

(5.74)

5.3. OBSERVABILITY

171

Thus we conclude that the last pqn − pn = pn(q − 1) rows are linearly dependent to its first pn rows K, KJ, ..., KJ n−1 . It thus holds that ⎛ K ⎞ ⎜ KJ ⎟ ⎜ ⎟ ⎟=n rank ⎜ ⎜ ⋮ ⎟ ⎜ ⎟ ⎝KJ n−1 ⎠

(5.75)

and (K, J, B, 0p×r ) is a minimal realization of T (σ) and so A(σ), R(σ) and therefore A(σ), C(σ) are right coprime. For the reverse, assume that A(σ) and C(σ) are right coprime. The pairs (K, J) and (J, B) are observable and controllable respectively. Considering again the rank of the following matrix ⎛ K ⎞ ⎛ F1 + D1 A1 ⎞ ⎜ KJ ⎟ ⎜ (F1 + D1 A1 )A1 ⎟ ⎟ q−1 ⎜ ⎟ ⎜ ⎟ (J B ⋯ B) = rank(M L) ⎟ = rank ⎜ rank ⎜ ⎟ ⎜ ⎟ ⎜ ⋮ ⎜ ⋮ ⎟ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ⎟ ⎜ L ⎝KJ qn−1 ⎠ ⎠ ⎝(F1 + D1 A1 )An−1 1 ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(5.76)

M

with M ∈ Rpqn×n , L ∈ Rn×rq . Since (K, J) is an observable pair, we can easily conclude that rankM = n. For the product M L it holds that (see Theorem 1.8): ⎫ rank(M L) ≤ rankM = n ⎪ ⎪ ⎪ ⎪ ⎬ ⇒ rank(M L) = rankL rank(M L) ≤ rankL ⎪ ⎪ ⎪ rank(M L) ≥ rankM + rankL − n = rankL⎪ ⎭

(5.77)

Thus, it suffices to show that rankL = n. Considering the minimal realization of A(σ)−1 sp from Theorem 1.66: −1 T A(σ)−1 sp = C(σIn − J) B ⇒ A(σ)sp

−1

T

= (C(σIn − J)−1 B) = B T (σIn − J T )−1 C T (5.78)

so (B T , J T , C T ) is a minimal realization of A(σ)Tsp , thus from Theorem 1.67 it holds −1

that ⎛ BT ⎞ ⎟ = n ⇒ rankL = n rank ⎜ ⋮ ⎜ ⎟ q−1 ⎝B T J T ⎠

(5.79)

and the pair (F1 + D1 A1 , A1 ) is observable. So rank

⎛ σIn − A1 ⎞ ⎛ σ E˜ + A˜⎞ proof of =n ⇔ rank = rq Theorem (2.33) ⎝C˜1 + C˜2 ⎠ ⎝F1 + D1 A1 ⎠

˜ are right coprime. and the matrices (C˜1 + C˜2 σ), (σ E˜ + A)

(5.80)

172 CHAPTER 5.


Summarising the results from Theorems 2.31, 2.32 2.33, 5.8, Remark 5.6 and Corollary 5.7 we conclude to the following result. Theorem 5.9. The following statements are equivalent. 1. The higher order system (5.4) is completely observable. 2. rankOP M D = n. 3. The matrices A(σ) and C(σ) are right coprime. 4. The descriptor system (5.27) is completely observable. 5. rankOdescriptor = n. 6. The matrices (σ E˜ + A) and (C˜1 + C˜2 σ) are right coprime. 7. The causal subsystem (2.168) is observable. 8. rankOcausal = n. 9. The matrices (σIn − A1 ) and (F1 + D1 A1 ) are right coprime. It should be noted that the coprimeness condition for the matrices A(σ) and C(σ) was taken as an initial assumption in [128, 130, 131] where the problem of eigenstructure assignment and observer design via output feedback was considered, for continuous time descriptor systems and higher order systems of algebraic and difference equations. Example 5.10. Consider the system ⎛1 + 2σ + σ 2 1 ⎞ β(k) = 0 ⎝ 0 σ − 1⎠

(5.81a)

ξ(k) = (σ 2 2 + σ) β(k)

(5.81b)

For the matrix A(σ) we have det A(σ) = (s + 1)2 (s − 1), so q = 2, r = 2, n = 3 and n + µ = rq ⇒ µ = 1. The triple (C, J, B) is given by ⎛ 1 1 0⎞ C= , ⎝−4 0 0⎠

⎛1 0 0 ⎞ ⎟ J =⎜ ⎜0 −1 1 ⎟ , ⎝0 0 −1⎠

1 ⎛0 − 4 ⎞ 1 ⎟ B=⎜ ⎜0 4 ⎟ ⎝1 1 ⎠ 2

(5.82)

5.3. OBSERVABILITY

173

The corresponding descriptor system is ⎛A0 A1 ⎞ ⎛A2 0 ⎞ x(k) x(k + 1) = − ⎝ 0 A0 ⎠ ⎝A1 A2 ⎠ y(k) =

(5.83a)

⎛C2 0 ⎞ ⎛C0 C1 ⎞ x(k + 1) x(k) + ⎝C1 C2 ⎠ ⎝ 0 C0 ⎠

(5.83b)

or equivalently 0 0⎞ 2 0⎞ ⎛1 0 ⎛1 1 ⎜0 0 ⎜ ⎟ 0 0⎟ 0 1⎟ ⎜ ⎜0 −1 ⎟ ⎜ ⎟ x(k + 1) = − ⎜ ⎟ x(k) ⎜2 0 ⎟ ⎜ 1 0⎟ 1 1⎟ ⎜ ⎜0 0 ⎟ ⎝0 1 ⎝0 0 0 0⎠ 0 −1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ˜ E

y(k) =

˜ A

⎛0 2 ⎛1 0 0 1⎞ 0 0⎞ x(k) + x(k + 1) ⎝0 0 ⎝0 1 0 2⎠ 1 0⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ˜1 C

with x(k) =

(5.84a)

(5.84b)

˜2 C

⎛ β(2k) ⎞ , ⎝β(2k + 1)⎠

y(k) =

⎛ ξ(2k) ⎞ ⎝ξ(2k + 1)⎠

(5.85)


⎛ 0 21 0 21 ⎞ ⎛−1 −1 21 0⎞ ⎜−2 −1 1 0 ⎟ ⎜ 2 0 0 0⎟ ⎜ ⎟ ⎟ ⎜ ⎟ ⎟, Q = ⎜ P =⎜ ⎜−2 −1 2 1 ⎟ ⎜ 0 1 0 0⎟ ⎜ ⎟ ⎜ ⎟ ⎝ 0 −1 0 0 ⎠ ⎝ 2 0 0 1⎠

(5.86)

with ⎛1 ⎜0 ˜ =⎜ ⎜ P EQ ⎜0 ⎜ ⎝0

0 1 0 0

0 0 1 0

⎛6 0 0 C˜1 Q = ⎝4 0 0

0⎞ ⎛1 ⎟ ⎜0 0⎟ ˜ =⎜ ⎟ , P (−A)Q ⎜ ⎜0 0⎟ ⎟ ⎜ ⎠ ⎝0 0

0 1 1 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠

(5.87)

⎛−1 −1 1/2 1⎞ ˜ , C2 Q = ⎝2 1 2⎠ 0

0⎞ 0⎠

(5.88)

0 1 0 0

and the resulting subsystems are ⎛1 0 0⎞ ⎟ x˜1 (k + 1) = ⎜ ⎜0 1 1⎟ x˜1 (k), ⎝0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ A1

y1 (k) =

⎛5 −1 − 12 ⎞ x˜1 (k) ⎝6 1 1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ (F1 +D1 A1 )

(5.89a)

174 CHAPTER 5.


and 0 = x˜2 (k), ®

y2 (k) =

N

⎛1⎞ ⎛1⎞ x˜2 (k) = 02×1 x˜2 (k) + 02×1 x˜2 (k + 1) = ⎝2⎠ ⎝2⎠ ± D2 ±

(5.90a)

F2


⎛x˜1 (k)⎞ ⎝x˜2 (k)⎠

(5.91)


0}. The observability matrix of the causal subsystem is ⎛5 −1 − 21 ⎞ ⎜6 1 1 ⎟ ⎜ ⎟ ⎟ ⎛ F1 + D1 A1 ⎞ ⎜ ⎜5 −1 − 3 ⎟ ⎜ 2⎟ ⎟ ⎜ ⎟ Ocausal = ⎜(F1 + D1 A1 )A1 ⎟ = ⎜ ⎜6 1 2 ⎟ ⎜ ⎟ ⎝(F1 + D1 A1 )A21 ⎠ ⎜ ⎟ ⎜5 −1 − 5 ⎟ ⎜ 2⎟ ⎝6 1 3 ⎠

(5.92)

and has rankOcausal = 3, so the causal subsystem is observable and thus the complete system is observable. The same holds for the matrices Odescriptor and OP M D , where

OP M D

⎛ ⎜ ⎜ ⎜ ⎜ ⎜ =⎜ ⎜ ⎜ ⎜ ⎜ ⎜ ⎝

C0 C1 C2 0 0 0 0 0 0 C1 C2 C3 0 0 0 0 0 0 C1 C2 C3 0 0 0 0 0 0 C1 C2 C3 0 0 0 0 0 0 C1 C2 C3 0 0 0 0 0 0 C1 C2 C3

= Odescriptor

⎛C˜1 C˜2 0 =⎜ ⎜ 0 C˜1 C˜2 ⎝ 0 0 C˜1

⎛H−2 ⎜H ⎞⎜ ⎜ −3 ⎟ ⎜H ⎟ ⎜ −4 ⎟⎜ ⎟ ⎜H−5 ⎟⎜ ⎟⎜ ⎟ ⎜H−6 ⎟⎜ ⎟⎜ ⎟ ⎜H−7 ⎟⎜ ⎠⎜ ⎜H−8 ⎜ ⎝H−9

H−1 ⎞ H−2 ⎟ ⎟ ⎟ H−3 ⎟ ⎟ ⎟ H−4 ⎟ ⎟ ⎟= H−5 ⎟ ⎟ ⎟ H−6 ⎟ ⎟ ⎟ H−7 ⎟ ⎟ H−8 ⎠

⎛ 3 4 −2 2⎞ ⎜ ⎟ ⎛Φ0 ⎞ ⎜−4 1 3 4⎟ ⎟ 0 ⎞⎜ ⎟ ⎜ ⎜ 5 5 −4 1⎟ Φ 1 ⎜ ⎟ ⎜ ⎟ ⎟=⎜ ⎟ 0⎟ ⎟⎜ ⎜Φ2 ⎟ ⎜−6 0 5 5⎟ ⎜ ⎟ ⎜ ⎟ ˜ ⎠ C2 ⎟ ⎝Φ3 ⎠ ⎜ ⎜ 7 6 −6 0⎟ ⎜ ⎟ ⎝−8 −1 7 6⎠

(5.93)

(5.94)

˜ (C˜1 + C˜2 σ) are right coprime. In addition, the matrix pairs A(σ), C(σ) and (σ E˜ + A), Example 5.11. Consider the system ⎛1 + 2σ + σ 2 1 ⎞ β(k) = 0 ⎝ 0 σ − 1⎠

(5.95a)

5.3. OBSERVABILITY

175 ξ(k) = (−1 + σ 2 1 + σ) β(k)

(5.95b)

For the matrix A(σ) we have det A(σ) = (s + 1)2 (s − 1), so q = 2, r = 2, n = 3 and n + µ = rq ⇒ µ = 1. The triple (C, J, B) is given by ⎛ 1 1 0⎞ , C= ⎝−4 0 0⎠

1 ⎛0 − 4 ⎞ 1 ⎟ B=⎜ ⎜0 4 ⎟ ⎝1 1 ⎠ 2

⎛1 0 0 ⎞ ⎟ J =⎜ ⎜0 −1 1 ⎟ , ⎝0 0 −1⎠

(5.96)

The corresponding descriptor system is ⎛A0 A1 ⎞ ⎛A2 0 ⎞ x(k + 1) = − x(k) ⎝ 0 A0 ⎠ ⎝A1 A2 ⎠ y(k) =

(5.97a)

⎛C0 C1 ⎞ ⎛C2 0 ⎞ x(k) + x(k + 1) ⎝ 0 C0 ⎠ ⎝C1 C2 ⎠

(5.97b)

or equivalently

0 0⎞ 2 0⎞ ⎛1 0 ⎛1 1 ⎜0 0 ⎜ ⎟ 0 0⎟ 0 1⎟ ⎜ ⎜0 −1 ⎟ ⎜ ⎟ x(k + 1) = − ⎜ ⎟ x(k) ⎜2 0 ⎜0 0 1 0⎟ 1 1⎟ ⎜ ⎜ ⎟ ⎟ ⎝0 1 ⎝ ⎠ 0 0 0 0 0 −1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ˜ E

y(k) =

(5.98a)

˜ A

⎛−1 1 ⎛1 0 0 1⎞ 0 0⎞ x(k) + x(k + 1) ⎝0 0 ⎝0 1 −1 1⎠ 1 0⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¶ ˜1 C

(5.98b)

˜2 C

with x(k) =

⎛ β(2k) ⎞ , ⎝β(2k + 1)⎠

y(k) =

⎛ ξ(2k) ⎞ ⎝ξ(2k + 1)⎠

(5.99)

The matrices that give the decomposition into the causal and noncausal subsystems are the same as in the previous example and the resulting decomposition is the following ⎛1 ⎜0 ˜ =⎜ ⎜ P EQ ⎜0 ⎜ ⎝0

0 1 0 0

0 0 1 0

⎛5 1 − 21 C˜1 Q = ⎝2 −1 0

0⎞ ⎛1 ⎟ ⎜0 0⎟ ˜ =⎜ ⎟ , P (−A)Q ⎜ ⎜0 0⎟ ⎟ ⎜ ⎠ ⎝0 0

0 1 0 0

0 1 1 0

⎛−1 −1 1/2 1⎞ ˜ , C2 Q = ⎝2 1 1⎠ 0

0⎞ 0⎟ ⎟ ⎟ 0⎟ ⎟ 1⎠ 0⎞ 0⎠

(5.100)

(5.101)

176 CHAPTER 5.



⎛1 0 0⎞ ⎟ x˜1 (k + 1) = ⎜ ⎜0 1 1⎟ x˜1 (k), ⎝0 0 1⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹¸¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

y1 (k) =

⎛4 0 −1⎞ x˜1 (k) ⎝4 0 1 ⎠ ´¹¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¸ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¹ ¶

(5.102a)

(F1 +D1 A1 )

A1

and 0 = x˜2 (k), ®

y2 (k) =

N

⎛1⎞ ⎛1⎞ x˜2 (k) = 02×1 (5.103a) x˜2 (k) + 02×1 x˜2 (k + 1) = ⎝1⎠ ⎝1⎠ ± D2 ± F2


⎛x˜1 (k)⎞ ⎝x˜2 (k)⎠

(5.104)

ad = {˜ with x˜1 (k) ∈ R3 , x˜2 (k) ∈ R and admissibility set Hc−nc x1 (0) ∈ R3 , x˜2 (0) ∈ R ∣ x˜2 (0) =

0}. The observability matrix of the causal subsystem is ⎛4 ⎜4 ⎜ ⎛ F1 + D1 A1 ⎞ ⎜ ⎜4 ⎟=⎜ Ocausal = ⎜ (F + D A )A 1 1 1 1 ⎜ ⎟ ⎜ ⎜4 2 ⎝(F1 + D1 A1 )A1 ⎠ ⎜ ⎜ ⎜4 ⎜ ⎝4

0 −1⎞ ⎛2 ⎟ ⎜−2 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜2 0 −1⎟ ⎜ ⎟ , OP M D = ⎜ ⎟ ⎜−2 0 1⎟ ⎜ ⎟ ⎜ ⎟ ⎜2 0 −1⎟ ⎜ ⎝−2 0 1⎠

3 −2 1⎞ 1 2 3⎟ ⎟ ⎟ 3 −2 1⎟ ⎟ ⎟ 1 2 3⎟ ⎟ ⎟ 3 −2 1⎟ ⎟ 1 2 3⎠

(5.105)

and has rankOcausal = rankOP M D = 2, so the system is not observable. It also holds ˜ (C˜1 + C˜2 σ) are not right comprime, that the matrix pairs A(σ), C(σ) and (σ E˜ + A), which can be seen from the Smith forms of the compound matrices. ⎛1 ⎜0 ⎜ ⎜ ⎜0 ⎜ C S (σ) = ⎜ ⎜0 ˜ + A) ˜ ⎞ ⎛ (σ E ⎜ ⎜ ⎟ ⎜ ⎜ ⎟ ˜1 + σ C˜2 )⎠ ⎜0 ⎝(C ⎜ ⎝0

5.4

0 1 0 0 0 0

0 0 ⎞ 0 0 ⎟ ⎟ ⎟ 0 ⎞ ⎛1 1 0 ⎟ ⎟ C ⎟ ⎟, S (σ) = ⎜ ⎜0 σ + 1⎟ ⎛A(σ)⎞ 0 σ − 1⎟ ⎟ ⎜ ⎟ ⎝0 0 ⎠ ⎟ ⎜ ⎟ 0 0 ⎟ ⎝ ⎠ C(σ) ⎟ 0 0 ⎠

(5.106)

Conclusions

The observability of higher order systems of algebraic and difference equations was studied. By first transforming the system into a first order descriptor system with a state

5.4. CONCLUSIONS

177

lead in the output and then further transforming it into an equivalent causal/noncausal subsystem decomposition, observability criteria have been derived for the higher order system in all its forms. In its original form (5.4), observability criteria are given in terms of the Laurent expansion of A(σ)−1 and the coprimeness of A(σ), C(σ). In its descriptor form (5.27) observability criteria are given in terms of the Laurent expansion ˜ −1 and the coprimeness of (σ E˜ + A), ˜ (C˜1 + C˜2 σ). In its causal/noncausal of (σ E˜ + A) subsystem form, observability criteria are given in terms of the state space observability matrix of the causal subsystem and the coprimeness of (σIn − A1 ), (F1 + D1 A1 ). These results can be extended to higher order continuous time systems following the work of [108], or to higher order positive systems. The problem of observer design or pole placement via output feedback can also be studied, as well as the property of constructibility.

178 CHAPTER 5.


Conclusions This dissertation studied higher order linear systems of algebraic and difference equations. The problem of constructing homogeneous systems with prescribed forward and backward solutions over a finite time interval was addressed and two different methods were proposed for it. The first method is an extension of [33] and is the discrete time analog of the work by [54]. In the second method, the modeling problem was reduced to solving a linear system of equations, and can be used to also construct non-regular systems. For both methods presented, it was shown that when the number of vector valued functions provided (counting multiplicities) does not satisfy a certain relation, the constructed system will include additional solutions, linearly independent from the ones that are given. Further research on this subject would be the extension of the second method for continuous time systems, the conditions under which the construction of the desired system is possible, and the study of the conditions under which the undesired solutions satisfy certain properties, like stability or positivity. The reachability and observability of higher order systems was also studied. Regarding reachability, the reachable subspace was described in terms of the coefficients of the Laurent expansion of A(σ)−1 and reachability criteria have also been derived. A method for constructing an admissible input that can drive the state from the origin to any desired position in the reachable subspace has also been given. These results can be expanded for positive systems, as well as for the controllability of such systems. The observability of higher order systems was studied by transforming the system into a first order descriptor system with a state lead in the output and then further transforming it into an equivalent causal/noncausal subsystem decomposition. Observability criteria have been derived for the higher order system in all its forms. In 179

180 CHAPTER 5.


its original form, observability criteria are given in terms of the Laurent expansion of A(σ)−1 and the coprimeness of A(σ), C(σ). In its descriptor form, observability ˜ −1 and the coprimeness criteria are given in terms of the Laurent expansion of (σ E˜ + A) ˜ (C˜1 + C˜2 σ). In its causal/noncausal subsystem form, observability criteria of (σ E˜ + A), are given in terms of the state space observability matrix of the causal subsystem and the coprimeness of (σIn − A1 ), (F1 + D1 A1 ). These results can be extended to higher order continuous time systems following the work of [108], or to higher order positive systems. The problem of observer design or pole placement via output feedback can also be studied, as well as the property of constructibility. The results of this dissertation have been published or submitted for publication in [87–92].

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Index Auto-Regressive (AR) representation, 81 Laurent expansion forward/backward solutions, 82 of a rational function, 44 backward solutions, 88 of a rational matrix about zero, 47 dual system, 83 of a rational matrix at infinity, 45, 51 forward solutions, 85 matrix forward/backward solutions, 82 characteristic polynomial, 26 power, 83, 123 column space, 24 solution space, 82 column span, 24 Auto-Regressive Moving Average (ARMA) defect, 25 representation, 133 eigenvalue, 26 admissible initial values, 138 algebraic multiplicity, 26 reachability, 139 geometric multiplicity, 26 reachability criteria, 146 eigenvector, 26 solution, 137 elementary, 24 Cayley-Hamilton theorem, 26 elementary divisor, 30 Cayley-Hamilton theorem (generalized), 47 equivalence, 24 image, 24 descriptor system, 49 index of nilpotenxy, 26 causal/noncausal (forward/backward) kernel, 25 subsystems, 54, 67, 73 left kernel, 25 consistent initial values, 56, 67, 74 left nullspace, 25 observability, 68, 70, 76 nilpotent, 26 output, 52 nullspace, 25 reachability, 57, 58, 61, 64 range, 24 state, 52 rank, 24 with a state lead in the output, 72 row space, 24 elementary operations non-regular system, 83 on matrices, 23 on polynomial matrices, 28 Observability, 68 observability criteria, 68, 70, 76, 165, fundamental matrix sequence, 45, 47 166, 172 Infinite Jordan Pair, 88 observability, 50, 66, 165 observable system, 68 Jordan chain, 86, 89 Jordan Pair, 85 PMD, 157 Jordan Pairs, 41 observability, 165, 166, 172 Kronecker delta, 46 lag, 81

output, 161 polynomial matrix, 27 (greatest) common left divisor, 28 191

192

INDEX elementary divisor at ∞, 34 left equivalent, 29, 93, 106, 113, 118 right equivalent, 29 common right divisor, 29 degree, 27 equivalent, 29 finite elementary divisors, 30 finite zeros, 29 algebraic multiplicity, 30 geometric multiplicity, 30 partial multiplicity sequence, 30 invariant polynomials, 29 lag, 27 left coprime, 35 left divisor, 28 Mc-Millan degree, 34 monic, 27 multiple, 28 order, 27 poles at ∞, 33 regular, 28, 36 regular pair (E,A), 36 right coprime, 35 right divisor, 28 simple, 32 singular, 28, 36 spectral structure, 35 spectrum, 35 unimodular, 28 unimodular equivalence, 93 zeros at ∞, 33

rational function, 27 finite poles, 37 finite zeros, 37 pole at infinity, 40 strictly proper, 27 zero at infinity, 40 biproper, 38 proper, 27 standard form, 27 rational matrix, 27 (coprime) left/right matrix fraction description (MFD), 39 (minimal)realization, 41–43, 136 biproper, 38 equivalence, 37 equivalence at infinity, 38

invariant rational functions, 38 irreducible, 37 Mc-Millan degree, 40 pole, 38 pole at infinity, 40 reducible, 37 standard form, 37 zero, 38 zero at infinity, 40 reachability, 50, 139 reachability criteria, 58, 64, 146 reachability Grammian, 57 reachable state, 57 reachable subspace, 58, 61, 142 reachable system, 57 singular system, 49 Smith form, 29, 84 at ∞, 33 at local point, 30 Smith-McMillan form, 38 Smith-McMillan form of a rational matrix at infinity, 40 state space system, 49 Weierstrass Decomposition, 36, 52

Modeling, Reachability and Observability of Linear

Modeling, Reachability and Observability of Linear

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