On λ-Matrices and Their Applications in MIMO Control Systems Design ...

Int. J. Modelling, Identication and Control, Vol. 01, No. 01, 2017

On λ-Matrices and Their Applications in MIMO Control Systems Design Belkacem Bekhiti Electronics and Electrotechnics Institute, ex:INELEC University of Boumerdes, Algeria. E-mail: [email protected] E-mail: [email protected]

Abdelhakim Dahimene Signal and System Laboratory, Electronics and Electrotechnics Institute University of Boumerdes, Algeria, IGEE Ex:(INELEC). E-mail: [email protected]

Bachir Nail The applied Automation and Industrial Diagnostics Laboratory, LAADI Science and Technology Department, University of Ziane Achour Moudjbara Street BP 3117 Djelfa, Algeria. E-mail: [email protected]

Kamel Hariche Electronics and Electrotechnics Institute, ex:INELEC University of Boumerdes, Algeria. E-mail: [email protected] Abstract: in the present paper we have introduced a new control design algorithms based on the theory of matrix polynomials. The first procedure is called Block decoupling control which is based on the spectral factors of the denominator of the right matrix fraction description (RMFD), the advantages of this control are the non-interacted behavior, simplicity in control design and low order controller is obtained due to the cancellation property of the proposed algorithm. The second control algorithm is the whole set of latent-structure assignment via the approaches of block root placement, of course the procedure have been developed even if the system is not block transformable. A process done with the aid of conversion between state space and matrix fraction description. The last method defined as a MIMO PID controller design via the placement of Block roots with the help of Diophantine equation resolution, the later systematic procedure retains both regulation and tracking objectives with small gains and minimum error. Keywords: Block roots; spectral factors; (RMFD) right matrix fraction description; MIMO PID; Diophantine equation. Reference Belkacem Bekhiti, Abdelhakim Dahimene, Bachir Nail and Kamel Hariche. (2017) ’On λ-Matrices and Their Applications in MIMO Control Systems Design’, International Journal of Modelling, Identification and Control, Vol. 01, No. 01, pp.1–13. Biographical notes: Belkacem Bekhiti received his Electrical Engineering Degree in 2011 from Mohammed Bougara University (UMBB-INELEC, Algeria) and his Magister Degree in 2014 from ENP-Oran university (Algeria). He is currently a Ph.D student at the Institute of Electronics and Electro-techniques of the University of Boumerdes (ex:INELEC). His current research interests include MIMO system control,reduction,identification and algebraic theory approaches to linear and nonlinear multivariable automatic control. Abdelhakim Dahimene received a Master degree in Control from the University of Boumerdes, INELEC (1992). And a PhD degree in electrical engineering from the IGEE Boumerdes in telecommunication. Currently, he is a professor in electrical engineering at the Institute of Electronics and Electro-techniques of the University of Boumerdes, Algeria. His major fields includes Advanced matrix theory, numerical methods, control theory, communication systems. He wrote numerous papers in control systems, telecommunications and applied mathematics.

DOI:

Bachir Nail received his licence and master degrees in Electrical Engineering in 2015 from University of Ziane Achour Djelfa. He is currently a Ph.D student at the Ziane Achour University of Djelfa Algeria. His current research interests include Fault Tolerance control and Detection, MIMO linear and nonlinear control system , industrial processes estimation, identification and robotics. 10.1504/IJMIC.2017.10008337

Int. J. Modelling, Identification and Control, Vol. 01, No. 01, 2017

1 INTRODUCTION Dynamic system with inherently more than one variable at the output to be controlled are frequently encountered in industries and are known as multi-input multioutput (MIMO) or multivariable processes. Interactions usually exist between control loops, which account for the renowned difficulty in their control compared with single-input single-output (SISO) processes. The goal of controller design to achieve satisfactory loop performance has hence posed a great challenge in the area of control design Yu Zhang et al.(2002). Mathematical model of a given complex physical plant plays a role of central heart for the analysis and design in control system engineering. In the light of this Mathematical presentation a dynamical system can be described either by internal description (state space description) or external description (input-output description), instated of matching all modes of the system we should derive meaningful model of the plant, i.e. a model that captures the key dynamics of the plant in the operational range of interest Madan G.Singh et al.(1980). The Matrix Fractional Description (MFD’s) can be regarded as extensions of the classical single-input singleoutput (SISO) transfer functions to the multivariable case with coprime numerator and denominator polynomials. Several methods are available for obtaining MFD’s, to mention W. A. Wolovich et al.(1973), Patel et al.(1981). It is shown that the polynomial matrix fraction description is a straightforward generalization of transfer functions to multivariable systems. Similarly to the transfer function of a SISO system, the polynomial matrix fraction description displays two important properties of a multivariable system in a transparent manner, namely the poles and the zeros see Peter Hippe et al.(2009). Representations of linear timeinvariant systems based on polynomial matrices, called Polynomial Matrix Description (PMD) or Differential (Difference) Operator Representation (DOR) are introduced. Such representations arise naturally when differential (or difference) equations of order higher than one are used to describe the behavior of systems, and the differential (or difference) operator is introduced to represent the operation of differentiation (or of timeshift). Polynomial matrices in place of polynomials are involved since this approach is typically used to describe MIMO systems. Note that state-space system descriptions involve only first-order differential (difference) equations, and as such, PMDs include the state-space descriptions as special cases Panos J. Antsaklis (1997). The dynamic modeling of physical linear time invariant multi input multi output systems, results in high degree coupled vector differential equations with matrix constant coefficients or a matrix transfer function where in this case the relationship between the input and output is a ratio of two matrix polynomials, expressed c 2017 Inderscience Enterprises Ltd. Copyright

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as a right (or left) matrix fraction description (RMFD or LMFD):  H(s) = NR (s)DR −1 (s) (1) = DL −1 (s)NL (s) Where:NR , DR ,NL and DL are matrix polynomials and d operator. This fact has led to an ”s” stands for dt active research effort in matrix polynomials theory see also Malika Yaici et al.(2014a),(2014b) and (2014c). The dynamic properties of the system under study are determined by the Block roots of a matrix polynomial. This is why we find quit a lot of publications at the present time about those matrices polynomials theory in system and control journals K. Hariche et al. (1988), E. Periera (2003a), J. E. Dennis et al.(1976),M. K. Solak et al.(1987)and J. S. H. Tsai et al.(1992). One of the most popular and well known techniques used to assign the eigenvalues of the closed loop system to desired location is state feedback W.M. Wonham et al.(1967), T. Kailath (1980), A.N. Andry et al.(1983) and Malika Yaici et al.(2014a). In the case of multivariable systems, the feedback gain matrix permitting the assignment of the desired set of poles is not unique; this is due, in the case of solvent placement method (Block pole assignment), to the fact that different Block poles can be constructed from the same set of eigenvalues E. Periera et al.(2003b), L. S. Shieh et al.(1982a),(1982b) and (1986). Then the degree of freedom offered by the choice of the feedback gain matrix could be exploited to satisfy some desired closed loop performances (the system response characteristics, robustness, tracking, decoupling, regulation, etc...). This can be done by choosing the structure of the solvent to be placed which gives the best feedback gain matrix that verifies the desired objectives Malika Yaici et al.(2014a). For a system described in state space description (where the system order is not an integer multiple of the input and/or the output i.e. the system neither block controllable nor block observable), the first step consists in converting it to MFD via a new decoupling similarity block transformation the resulting system will be block decoupled controllable and/or block decoupled observable forms, then convert the desired eigen-structure into a latent structure, and thereafter construct its corresponding block roots. The state feedback problem assumes that all states are measurable. Unfortunately, this is impractical for most systems. To solve this, it is required to use either static output feedback or dynamic compensator. Hence for the first case we will do the state feedback control which will decouple completely the system and then trajectory tracking regulators are designed out, but for the solution of the second problem a MIMO PID controller based on Block root assignment is developed via the solution of Diophantine equation. In this study, firstly we start the work by introducing some theoretical preliminaries and a survey on matrix polynomials, after that as a main results we propose a

On λ-Matrices and Their Applications in MIMO Control Systems Design new decoupling control of linear multivariable system based on concept of Block roots of λ-matrices assignment via the state feedback, and thereafter a specific structure MIMO output compensator design is illustrated with some simulation examples are detailed, and finally the paper is finished by a conclusion.

2 THEORETICAL PRELIMINARIES

In this section, we attempt to present some of important results obtained in the theory of matrix polynomials. A more emphasis will be given to the latent structure of these matrix polynomials, which consists mainly of the latent roots and latent vectors as well as solvents. The algebraic theory of matrix polynomials has been investigated by Dennis et al. J. E. Dennis et al.(1976) and (1978), Gohberg et al.(1978) and (1982). Spectral factors of a lambda matrix and right (left) solvents, for a right (left) characteristic matrix polynomial have been defined. The different transformations between right (left) solvents and spectral factors are mainly proposed by Shieh and Tsay (1981c). Definition1: given the set of m × m complex matrices A0 , A1 , ..., Al , the following matrix valued function of the complex variable λ is called a matrix polynomial of degree l and order m: (2)

Definition2: The complex number λi is called a latent root of the matrix polynomial A(λ) if it is a solution of the scalar polynomial equation det(A(λ)) = 0 The nontrivial vector p , solution of A(λi )p = 0m is called a primary right latent vector associated with λi . Similarly the nontrivial vector q solution of q T A(λi ) = 0m is called a primary left latent vector associated with λi . Remark1: If A(λ) has a singular leading coefficient (Al ) then A(λ) has latent roots at infinity. From the definition we can see that the latent problem of a matrix polynomial is a generalization of the concept of eigenproblem for square matrices. Indeed, we can consider the classical eigenvalues/vector problem as finding the latent root/vector of a linear matrix polynomial (λI − A) . We can also define the spectrum of a matrix polynomial A(λ) as being the set of all its latent roots (notation σ(λ) ). It is essentially the same definition as the one of the spectrum of a square matrix. Definition3: A right Block root also called solvent of monic λ-matrix A(λ) (i.e. A0 = Im ) and is an m × m real matrix R such that: Rl + A1 Rl−1 + ... + Al−1 R + Al = Om l P ⇔ AR (R) = Ai Rl−i = Om i=0

While a left solvent is an m × m real matrix L such that: Ll + Ll−1 A1 + ... + LAl−1 + Al = Om l P ⇔ AL (L) = Ll−i Ai = Om

(4)

i=0

The following are important facts on solvents L. S. Shieh et al.(1982a): • Solvents of a matrix polynomial do not always exist.

2.1 Matrix polynomials

A(λ) = A0 λl + A1 λl−1 + ... + Al−1 λ + Al

3

(3)

• Generalized right (left) eigenvectors of a right (left) solvent are the generalized latent vectors of the corresponding matrix polynomial Definition4: A matrix R (respectively: L) is called a right (respectively: left) solvent of the matrix polynomial if and only if the binomial (λI − R)(respectively:(λI − L))divides exactly A(λ) on the right (respectively: left). Theorem1: (K. Hariche et al.1987) given a matrix polynomial A(λ) = A0 λl + A1 λl−1 + ... + Al−1 λ + Al

(5)

a) The remainder of the division of A(λ) on the right by the binomial (λI − X) is AR (X) b) The remainder of the division of A(λ) on the left by the binomial (λI − X) is AL (X) Means that there exist matrix polynomials Q(λ) and S(λ) such that: A(λ) = Q(λ)(λI − X) + AR (X) = (λI − X)S(λ) + AL (X)

(6)

Corollary1: (K. Hariche et al.1987) also gives the fundamental relation that exist between right solvent (respectively: left solvent) and right (respectively: left) linear factor: AR (X) = 0 AL (X) = 0

iff iff

A(λ) = Q(λ)(λI − X) A(λ) = (λI − X)S(λ)

(7)

Theorem2: (Malika Yaici et al. 2014b) Consider the set of solvents {R1 , R2 , ..., Rl } constructed from the eigenvalues (λ1 , λ2 , ..., λl ) of a matrix Ac . {R1 , R2 , ..., Rl }is a complete set of solvents if and only if:   ∪σ(Ri ) = σ(Ac ) σ(Ri ) ∩ σ(Rj ) = ∅ (8)  det(VR (R1 , R2 , ..., Rl )) 6= 0 Where: σ denotes the spectrum of the matrix. VR Vandermonde matrix corresponding to {R1 , R2 , ..., Rl } given as   Im Im ... Im  R1 R2 ... Rl    VR (R1 , R2 , ..., Rl ) =  .. (9) .. ..   . . ... .  R1l−1 R2l−1 ... Rll−1

4

B. Bekhiti et al.

Remark2: we can define a set of left solvents in the same way as in the previous theorem. The relationship between latent roots, latent vectors, and the solvents can be stated as follows: Theorem3:(L. S. Shieh et al. 1981c) If A(λ) has n = ml linearly independent right latent vectors pij (i = 1, · · · , l) and (j = 1, · · · , m) (left latent vectors qij ) corresponding to latent roots λij then Pi Λi Pi−1 , (Qi Λi Qi−1 ) is a right (left) solvent. Where: Pi = [pi1 , pi2 , ..., pim ], (Qi = [qi1 , qi2 , ..., qim ]T ) and Λi = diag(λi1 , λi2 , · · · , λim ). Proof: see (L. S. Shieh et al.(1981))

Algorithm1:

Theorem4:(L. S. Shieh et al. 1981c) If A(λ) has n = ml latent roots λi1 , λi2 , · · · , λim and the corresponding right latent vectors pi1 , pi2 , ..., pim has as well as the left latent vectors qi1 , qi2 , ..., qim are both linearly independent, then the associated right solvent Ri and left solvent Li are related by: Ri = Wi Li Wi−1

On the other hand, without prior knowledge of the eigenvalues and eigenvectors of the matrix, the NewtonRaphson method L. S. Shieh et al.(1981b) has been successfully utilized for finding the solvents. Also, the Block-power method has been developed by Tsai et al. (1988) for finding the solvents and spectral factors of a general nonsingular polynomial matrix. Moreover, there are numerous numerical methods for computing the Block roots of matrix polynomials without any prior knowledge of the eigenvalues and eigenvectors of the matrix polynomial. In this paper we will use one of the very well-known methods (Newton-like generalized method) in constructing complete set of spectral factors.

(10)

Enter the degree and the order m, l Enter the matrix polynomial coefficients Ai ∈ Rm×m X0 ∈ Rm×m =initial guess; Give some small η and (δ=initial start)> η k=0 While δ ≥ η Xk+1 = Xk [Al − AR (Xk )]−1 Al ; −Xk k ; δ = 100. kXk+1 kXk k Xk ← Xk+1 ; k = k + 1;

Where: Wi = Pi Qi and Pi = [pi1 , pi2 , ..., pim ], (Qi = [qi1 , qi2 , ..., qim ]T ). and ”T ” stands for transpose Theorem5: (L. S. Shieh et al. 1981c) if the elementary divisors of A(λ) are linear, then A(λ) can be factored into the product of l-linear monic λ-matrices called a complete set of spectral factors. A(λ) = (λIm − Ql )(λIm − Ql−1 )...(λIm − Q1 )

(11)

Where: (λIm − Qi ), i = 1...l are referred to as a complete set of linear spectral factors. The m × m complex matrices Qi , i = 1...l are called the spectral factors of the λ-matrix A(λ). The most right spectral factor Q1 is a right solvent of A(λ) and the most left spectral factor Ql is a left solvent of A(λ), whereas the spectral factors may or may not be solvents of A(λ) . The relationship between solvents and spectral factors are explored by Shieh and Tsay in (1981c), and various transformations have been developed.

2.2 Spectral Factors Construction The algebraic theory of matrix polynomials has been investigated by Dennis et al. (1978), Denman et al.(1977) and(1976), Gohberg et al. (1978), Shieh et al. (1981.b),(1986) and (1984), and Tsai et al. (1992),(1988). Various computational algorithms J. E. Dennis et al.(1978), Shieh et al. (1981.a),E. D. Denman et al.(1977) are available for finding the solvents and spectral factors of a matrix polynomial. A very wellknown method and approach Malika Yaici et al.(2014a), I. Gohberg et al.(1982) is the use of the eigenvalues and eigenvectors of the Block companion form matrix. However, it is often inefficient to explicitly determine the eigenvalues and eigenvectors of a matrix, which can be ill conditioned and either non-defective or defective.

End Remark3: Firstly, at each time factorize the linear term using the synthetic long division algorithm and repeat the process to get the complete set of spectral factors. Secondly, the proposed method was chosen among a set of algorithms, and this comparison study is a property of some authors of the paper. However, is not yet published.

2.3 Block companion form Let a system be described by standard state equations supposed controllable and/or observable. In this case the system can be transformed into companion forms (observer, controller, etc.) through similarity transformations. Two transformations are needed in our case: block controller form, block observer form. To convert a system described in SSD (A, B, C, D), the system must satisfy the following conditions:  

The number n/m = l or n/p = l is an integer. The matrix Ωc or Ωo is of full rank.

Where: Ωc = [B, AB, ..., Al−1 B] Ωo = [C T , AT C T , ..., (Al−1 )T C T ]T Then it is block controllable (block observable) with controllability index l. Remark4: If the dimension of a system matrix is not an integer multiple of the number of inputs or outputs some non-dominant eigenvalues can be added and placed at

On λ-Matrices and Their Applications in MIMO Control Systems Design the diagonal entries of the system matrix to enlarge the dimension Shieh L.S. et al.(1982). Block controller canonical form: If a system described by general state space equation (A, B, C, D) is block controllable then it can be transformed into a block controller form. So if n/m = l is an integer, and if the block controllability matrix Ωc is full rank, then we can convert the state equation into block controller form using the following similarity transformation xc = Tc x Where:  X˙ c (t) = Ac Xc (t) + Bc u(t) (12) Y (t) = Cc Xc (t) + Dc u(t)   Tc1  Tc1 A      Tc =   , Tc1 = Om , Om , ..., Im Ωc −1 (13) ..   . 

Tc1 Al−1

   Om Im · · · Om     Om Om · · · Om         ..  −1 ..  A = T AT =    c c c . . . . . O  m      Om Om . . . Im  −Al −Al−1 · · · −A1 
T    O B = T B = m Om · · · Im c c    
   Cc = CTc −1 = Cl Cl−1 · · · C1 

Consider the square matrix transfer function:  k  l −1 P P −1 i i H(s) = N (s)D (s) = Ni s Di s i=0

i=0

H(s) = (Nk s + · · · + N0 )(Dl s + · · · + D0 )−1 k

l

(18)

with: Dl = I is an m × m identity matrix and Ni ∈ Rm×m , (i = 0, 1, ..., k) Di ∈ Rm×m , (i = 0, 1, ..., l), l > k Assume that N (s) can be factorized into k Block zeros and D(s)can be factorized into l Block roots Where: (19)

and (14)

D(s) = (sI − Q1 )...(sI − Ql ).

(20) −1

Also we know that: H(s) = C(sI − A) B Now via the use of state feedback the control law becomes a state dependent and be rewritten as u(t) = −K.X(t) + F.r(t). Hance we obtain the following closed loop system:

Block Observer canonical form: If the system is block observable then it can be transformed in a block observer form. So if n/p = l is an integer, and if the block observability matrix Ωo is full rank, then we can convert the state equation into block observer form using the following similarity transformation xo = To x Where:  X˙ o (t) = Ao Xo (t) + Bo u(t) (15) Y (t) = Co Xo (t) + Do u(t) 


T To = Tol ATol ... Al−1 Tol , Tol = Ω−1 o Co    Op · · · Op −Al    Ip · · · Op −Al−1        −1 . . . ..    Ao = To ATo =   .. .. ..  .        Op · · · Op −A2   Op · · · Ip −A1    T   B Op  l    Bl−1   Op         Bo = To −1 B =   .. , Co = CTo =  ..     .   .   B1

polynomial N (s) into a complete set of spectral factors using one of the very well-known algorithms, then we place those found Block zeros by forcing the denominator to have exactly those ones via state feedback. Hence the decoupling objectives are achieved.

N (s) = Nk (sI − Z1 )...(sI − Zk )

With: Xc ∈ Rn , Ai ∈ Rm×m , Ci ∈ Rp×m , i = 1, ..., l, Im and Om are m × m identity and null matrices respectively, and the superscript T denotes the transpose.



5

(H(s))closed = C(sI − A + BK)−1 BF = N (s)Dd−1 (s)F Where: Dd (s) = (sI − Qd1 )...(sI − Qdl ) −1 Dd (s) = (sI − Qdl )−1 · · · (sI − Qd1 )−1 Qdi : are the desired spectral factors to be placed H(s)closed = N (s)Dd−1 (s)F = Nk (sI − Z1 ) · · · (sI − Zk )(sI − Qdl )−1 · · · (sI − Qd1 )−1 F (21)

Choose: Qd1 = Nk−1 J1 Nk , ..., Qd(l−k) = Nk−1 J(l−k) Nk Qd(l−k+1) = Z1 , ..., Qdl = Zk Ji = diag(λi1 , ..., λim ), F = (Nk )−1

(16) Now by assigning those Block roots the system is decoupled and the closed loop matrix transfer function become: (17)

Ip

H(s)closed = (sI − J1 )−1 ...(sI − Jl−k )−1 1 H(s)closed =

0   ∆1   ..   .    1  0 ∆m

(22)

Where: ∆i = (s + λ1i ) · · · (s + λ(l−k)i ) i = 1, ..., m

3 THE MAIN DECOUPLING RESULTS Idea: our objectives here are to decouple MIMO dynamic systems. Let first factorize the numerator matrix

Let we summarize the preceding procedure in the next algorithmic version to be more understandable and efficient for the use in linear multi-variable control system.

6

B. Bekhiti et al. Algorithm2: • Assume that measurable.

all

states

are

available

and

• Check the Block Observability and Block Controllability of a given state space model of square dynamic system. • Construct the right numerator and right denominator matrix polynomials using algorithms found in Malika Yaici et al. (2014a).

and (2014b)), Ωc and Ωo both are of full rank and then the dynamic system is Block controllable and Block Observable. Now we should construct the numerator and denominator matrix polynomials from the state space data as follow:     −18.5628 −28.2436 D0       = − B, AB −1 A2 B =  20.9730 28.1476   −3.2657 −9.5842  D1 7.4534

 • Decompose or factorize the numerator matrix polynomial into a complete set of Block spectral factors see algorithm1. • Choose the k spectral factors of numerator as Block roots to denominator and design the rest ones in diagonal form. • Construct the desired matrix polynomial form those obtained Block spectral data see Malika Yaici et al. (2014b). • Design the state feedback gain matrix in controller form and then transform it to the original base see L. S. Shieh et al.(1982a). Here at this point we are ready to design SISO tracking regulators for each input-output pairs, because the system is perfectly decoupled. Remark5: The proposed method depends on the existence of complete set of spectral factors and belong to only square systems. Therefor this is a departure point which gives one a potential to think about some extension to rectangular systems. From another point of view one may ask the question, how if the system is highly coupled? the answer of this question is illustrated in section.5 Example1: Consider the next state space of a given dynamic system   A=

−11.3730 −11.7945 2.3156 47.6250  −3.7170 −6.1683 1.4713 15.3387     −3.5962 −4.6114 −1.8504 19.3509  −2.7258 −2.5712   1.2053 9.3916 1.5582 1.3732

 1.3060 1.4047   B=  0.6631 0.6238  , D =



00 00



0.6014 0.5664

 C=

1.8816 0.8315 0.9551 −4.4856 −0.4368 0.2407 0.5181 3.0342



Let we check firstly for the Block controllability and Block Observability of the system n = 4 : the state number, m = 2 : the input number p = 2 : the output number n The controllability index is defined by:l = m n The observability index is defined by:q = p The Block Controllability and Block Observability matrices are defined by (see Malika Yaici et al.(2014a)



13.2658



   D1 D2 N0 , N1 = CB, CAB D2 O2   2.5426 −1.0809 1.9536 −1.8071 = 3.6476 2.2550 1.8019 1.7799

Let we decompose the numerator matrix polynomial and reconstruct its Block root:(use algorithm:1) N (s) = N1 s + N0 = N1 (sI − Z1 ) The desired denominator is of second order written in the form: Dd (s) = Dd2 s2 + Dd1 s + Dd0 In Block decomposition form one may consider: Dd (s) = (sI − Qd1 )(sI − Qd2 ) = Is2 − (Qd1 + Qd2 )s + Qd1 Qd2 Using our proposition (see algorithm:2) the desired set of spectral factors are formed by:   9.3509 27.1521 −1 Qd2 = Z1 = −N1 N0 = −11.5158 −28.7546

Qd1 = Z2 =

N1−1



−1 0 0 −2



 N1 =

13.7381 14.5581 −15.9326 −16.7381



Now the desired matrix coifficients of the denominator matrix polynomial of the decoupled plant are:     −23.0890 −41.7102 −304.1349 302.4902 Dd1 = , Dd0 = 27.4484

−318.3432 313.6492

45.4927

The state feedback gain matrix of the Block controller form is obtained by:  Kc0 = Dd0 − D0 ⇔ Kc = [Kc0 , Kc1 ] Kc1 = Dd1 − D1 Now let we go back to original base by the next similarity transformation:    −1   D1 D2 B, AB K = Kc Tc and Tc = D2 O2   −54.5835 −106.5499 −30.3095 373.2731 K= 51.3624

104.4725

32.4130 −362.4491

The new state space presentation of the decoupled the system after the state feedback are:   3.1500

10.7735 10.7735 −36.3061

 −4.5821 −13.7714 −4.4763 36.9931   Ad = (A − BK) =   0.5565 0.8685 −1.9724 −2.0588  1.0099

2.3365

1.0753 −9.8100

On λ-Matrices and Their Applications in MIMO Control Systems Design 



1.3539 −0.6031  −0.9354 1.7389     0.2542 0.0924  and 0.2257 0.0890

Bd = BF =

H(s)closed = C(sI − A + BK)

−1

BF =

Cd = C

1 s+1 0 0 s +1 2

!

Now we can design a SISO PID controller for each input output pair, using the known tuning methods for example the Nichol and Ziegler method or any other one.

3.1 Internal and Zero Dynamics: The dynamics of the non-observable stats are called the internal dynamics. The stability of this dynamic is required for the creation of the control law. For a MIMO linear system, the internal dynamics is stable if the Block zeros of the matrix transfer function have latent roots lie in the left half-plane of the complex field. We introduce the notion of zero dynamics to study the stability of the internal dynamics of a MIMO linear system. Unfortunately if at least one Block zero of the numerator is unstable then we have a hidden instability when we do a decoupling state feedback, then we may thing how to move or to relocate Block zeros at desired stable locations. But this can be done only in systems with input-output matrix D. st

1

end

and 2

Trajectory tracking Control

3

The output signal The input signal

2.5

From the obtained simulation results as shown in the last figures, we see that the controlled plant tracks its reference trajectory with very small error, no overshooting, no static error is obtained at both transient and steady state regimes then both tracking and regulation objectives are verified by the procedure. Finally from the error signal the BIBO stability but not internal is guaranteed and this is not surprising or new to us due to the cancellation behavior of designed controller.

4 STATE FEEDBACK DESIGN GENERAL MIMO SYSTEMS

FOR

Polynomial Matrix Descriptions (PMDs) and Matrix Fractional Descriptions (MFDs) are used to study properties such as controllability, observability, and stability, primarily of interconnected systems, and to conveniently characterize all stabilizing feedback controllers. These system descriptions are important in feedback control system analysis and design and are the key to developing control design theories Panos J. Antsaklis et al(1997). The block pole placement requires that the MIMO system is block controllable of index l i.e., the controllability indices of the system are all equal to l and n = lm. When the dimension n of the system matrix described by general state equation is not equal to lm , where l is an integer and m is the number of inputs, the proposed method cannot be directly applied. In order to avoid enlarging the dimension of the system matrix A , one may think about similarity transformation that will decompose the original system into two subsystems of dimension n0 = lm and k respectively such that n = n0 + k and k < m.

2

Amplitude

7

1.5

1

0.5

0

−0.5 0

0.5

1

1.5

2

2.5

3

3.5

In this case, here we have a detailed design procedure that will achieve the desired block pole placement for the system of dimension n0 , and a pole placement for the remaining k eigenvalues through state feedback.

Time (sec)

4.1 The Block-Decoupled Form

1st and 2end Output error 0.04

Consider a MIMO system described by general state equation where n/m is not an integer. Since m does not divide exactly n , we can write: n = lm + k with k < m. The desired block-decoupled form is chosen as,  X˙ c (t) = Ac Xc (t) + Bc u(t) (23) Y (t) = Cc Xc (t) + Dc u(t)

0.03 0.02

Amplitude

0.01 0 −0.01 −0.02 −0.03

Where the matrices Ac , Bc and Cc can be written in the following form:

−0.04 −0.05



−0.06 0

0.5

1

1.5

2

2.5

3

3.5

Time (sec)

Figure 1: The trajectory tracking control of the decoupled system

   Im · · · Om Om,k Om   Om · · · Om Om,k    Om     .. ..    .  . . . . Om . , Bc =  Ac =   ..  (24)  Om Om . . . Im Om,k       Im   −Al −Al−1 · · · −A1 Om,k  Bkm Ok,m Ok,m · · · Ok,m Λk Om Om .. .

8

B. Bekhiti et al. 

p1 0 · · ·  0 p2 · · ·  Λk =  .. ..  . . ...

0 0 .. .

 
 , Cc = Cc1 Cc2 , Dc = Op×m (25) 

0 0 . . . pk

K = Kc Tc

Where: Om,k , Ok,m are m × k and k × m null matrices respectively and Bkm is an k × m matrix satisfying Bkm = Tc(l+1) B. The desirable similarity transformation which transforms the coordinates x in general state equation into xc in (24) and (25) is defined as xc = Tc x where     −1 E[Ωc , V1 , ..., Vk ] Tc1 −1   Tc2     E[Ωc , V1 , ..., Vk ] A     ..   .  .. Tc =  .  =  (26)      −1 l−1  Tcl   E[Ωc , V1 , ..., Vk ] A    T T T Tc(l+1) T1 , T2 , ..., TkT h i E = Om · · · Im ... Om,k With: Tci are m × n matrices for i = 1, ..., l and  T Tc(l+1) = T1T , T2T , ..., TkT being a k × n matrix with Ti being a left eigenvector of A corresponding to the eigenvalues pi for i = 1, ..., k.. The system is Block if and only decoupled form  if the n × n matrix φ = B, AB, ..., Al−1 B, V1 , ..., Vk is nonsingular, with Vi being a right eigenvector of A corresponding to the eigenvalues pi for i = 1, ..., k.

4.2 Finding The State Feedback Gain Matrix In this section, we present an alternative method for constructing the linear state-feedback control law from the desired left or right latent structure using some algebraic approaches based on Lyapunov-like equation. Theorem6:(H. Loubar 1998) Given a linear timeinvariant multivariable system described by general state equation, and given a desired complete set of l block poles: {L1 , L2 , ..., Ll } and k poles {p1 , p2 , ..., pk }. If the system described by general state equation can be transformed by the similarity transformation xc = Tc x into the block-decoupled form     Ac1 Olm,k Bc1 ,  Ac =  Bc =  (27) Ok,lm Λk Bc2 Then the state feedback gain matrix that achieves the desired set of block poles and poles for the closed-loop system is given by: h i Kc = (Kc1 + Kc2 L) ... Kc2 (28) Where Kc1 is the feedback gain matrix which places the block poles of the matrix (Ac1 − Bc1 Kc1 ) at desired left solvents {L1 , L2 , ..., Ll }, and L is a solution of the following Lyapunov-like equation: L(Ac1 − Bc1 Kc1 ) − Λk L = Bc2 Kc1

And Kc2 is the feedback gain matrix which places the remaining k poles of the matrix Λk − (Bc2 + LBc1 )Kc2 at the k desired locations.

(29)

(30)

Proof: The decoupled system described previously in equation (27) rewritten in the next form:  x˙ 1 = Ac1 x1 + Bc1 u x˙ 2 = Λk x2 + Bc2 u Set the control input as state feedback defined by:     x1 u = − Kα Kβ +r x2 The closed loop system will be:  x˙ 1 = (Ac1 − Bc1 Kα ) x1 + (−Bc1 Kβ ) x2 + Bc1 r x˙ 2 = (−Bc2 Kα ) x1 + (Λk − Bc2 Kβ ) x2 + Bc2 r For making simplicity of computations let we put: A1 A2 A3 A4

= (Ac1 − Bc1 Kα ) = (−Bc1 Kβ ) = (−Bc2 Kα ) = (Λk − Bc2 Kβ )

In order to stabilize this last coupled system we use the Mc transformation which will completely  T decouples the original system.η = Mc x ⇔ η1 T η2 T =  T T T Mc x 1 x 2 with:     In1 − HL −H In1 H −1 , Mc = Mc = L

 Mc

A1 A2 A3 A4



Mc−1

−L

In2

 =

As O O Af



 ,

Bs Bf

In2 − LH



 = Mc

Bc1 Bc2



So that in the new coordinates we have  η˙ 1 = As η1 + Bs r η˙ 2 = Af η2 + Bf r Where: As Af Bs Bf

= A1 − A2 L = A4 + LA2 = Bc1 − HBc2 − HLBc1 = Bc2 + LBc1

The matrix transformation Mc exist and nonsingular satisfying the decoupling condition(see Kokotovic et al., 1986, Khalil, H. et al., 1989 and Chang, K. et al., 1972) if and only if the matrices L, H satisfy the following matrix equality: LA1 − A4 L − LA2 L + A3 = 0 (A1 − A2 L) H − H (A4 + LA2 ) + A2 = 0 Now we make the substitution of A1 , A2 , A3 , A4 , Bc1 and Bc2 into those last two equations with the following change of variables Kα = Kc1 + Kc2 L and Kβ = Kc2 we

On λ-Matrices and Their Applications in MIMO Control Systems Design

Step7: Compute a k × m matrix L satisfying the Lyapunov-like equation:

get two matrix equations First equation: LΓ1 − Γ2 L + LBc1 Kβ L = Bc2 Kα

L(Ac1 − Bc1 Kc1 ) − Λk L = Bc2 Kc1

(?)

Step8: Compute a feedback gain matrix Kc2 that places the k poles of Λk − (Bc2 + LBc1 )Kc2 ) at k the remaining desired locations. Step9: Compute the state feedback gain matrix using h i Kc = (Kc1 + Kc2 L) ... Kc2

With: Γ1 = (Ac1 − Bc1 Kα ) Γ2 = (Λk − Bc2 Kβ ) (?) ⇔ L (Ac1 − Bc1 Kc1 ) − Λk L = Bc2 Kc1 Second equation: Γ3 H − HΓ4 = Bc1 Kβ

9

(??)

With: Γ3 = (Ac1 − Bc1 Kα + Bc1 Kβ L) Γ4 = (Λk − Bc2 Kβ − LBc1 Kβ )

and compute the state feedback gain matrix in original coordinates using K = Kc Tc Example2:Consider the following 2-input, 2-output system of order 5 given by its matrices [Chia-Chi Tsui 2004]:  −0.1094 0.0628 0 0 0   1.3060 −2.1320 0.9807 0 0    0 1.5950 −3.1490 1.5470 0 A=    0 0.0355 2.6320 −4.2570 1.8550  0 0.0023 0 0.1636 −0.1625 

(??) ⇔ (Ac1 − Bc1 Kc1 ) H − HAd2 = Bc1 Kc2 With: Ad2 = (Λk − (Bc2 + LBc1 ) Kc2 ) Finally to ensure the stability of the global decoupled system we must follow the next steps: First, we design the gain matrix Kc1 that assigns complete set of block roots then we obtain the matrix L from equation (29) after that we must get the gain matrix Kc2 which will assign the remain eigenvalues to the matrix Ad2 . Algorithm3: Let n : Order of the state equation m: Number of inputs l, k are integers satisfying n = lm + k with k < m. Step1: Input the system matrices A, B, C and the complete set of l left solvents {L1 , L2 , ..., Ll } or right solvents {R1 , R2 , ..., Rl }, and the k set of poles to be assigned. Step2: Form the desired matrix polynomial Df (λ) from the given set of desired solvents using either:     Ddl , Dd(l−1) , ..., Dd1 = − R1l , R2l , ..., Rll VR−1 Step3: Compute k eigenvalues of A, respectively {p1 , p2 , ..., pk }, and find their corresponding left Ti and right Vi eigenvectors for (i = 1, ..., k). Step4: Check that the matrix φ is nonsingular Where   φ = B, AB, ..., Al−1 B, V1 , ..., Vk If φ is singular matrix then the system cannot be transformed into the block-decoupled form; hence, select a new set of k eigenvalues and go back to step3. Step5: Compute the similarity transformation xc = Tc x shown in (26) and transform the system into the blockdecoupled form (block controllable form if k) Step6: Compute a state feedback gain matrix that places the block poles of (Ac1 − Bc1 Kc1 ) at the desired l block poles using Kc1 = [Kl , Kl−1 , ..., K1 ] Where Ki = Di − Ai for (i = 1, ..., k) and Ai (i = 1, ..., l) are m × m matrices obtained from Ac1 in the block controllable form.



 0 0     0  0.0638  15 −2 −1 0 −2  0.0838 −0.1496 C= , B=   10 −1 2 1 −1  0.1004 −0.2060  0.0063 −0.0128

We have in this example n = 5 and m = 2 =⇒ l = and k=1 It follows that we can assign two block poles of dimension 2 × 2 and one remaining pole. So we can transform a given system into the block- decoupled form; we need to compute arbitrary eigenvalues of matrix A with their corresponding left and right eigenvectors. The eigenvalues of are: −5.9822, −2.8408, −0.8953, −0.0143, −0.0773. This leads to p1 = −5.9822 with the corresponding right eigenvector V1

and left eigenvector T1 given by:  −0.0015  0.1362     V1   −0.5326  , T1 = [−0.0663 0.2980 − 0.7314 0.6763 − 0.2156]  0.8350  −0.0235 

We form the matrix φ as follows φ = [B AB V1 ] Since φ is nonsingular, the given system can be transformed into the following block decoupled form:     0 0 1 0 0 0 0 0 1 0       −0.6849 −2.6039 0 Ac =  −5.9822 −0.3916   0.0472 −0.2997  0.1869 −3.1428 0 0 0 0 0 −5.9822



 Cc =



−0.0074 0.1316 0.0553 −0.0395



−0.2309 0.1829 0.2066 −0.5021

  0 0 0   0   0 Bc =  1 ,  0  1 0.0228 0.0255



0.2855 −0.3573

0 0

0 0

 



Dc =





10

B. Bekhiti et al.

With

5 DIOPHANTINE EQUATION BASED MIMO PID CONTROLLER DESIGN

  −1  Tc1 E[Ωc , V1 ] Tc =  Tc2  =  E[Ωc , V1 ]−1 A  Tc3 T1 



⇒ Tc =

 256.4048 0.0650 −0.3866 0.2182 1.0060  94.2132 −0.1792 1.1612 0.3796 −19.6806     −27.9658 15.3571 1.8555 −1.3625 0.2413     −10.5409 8.1189 −2.8332 −3.0393 3.9022  −0.0663 0.2980 −0.7314 0.6763 −0.2156

Let construct the desired block poles with a following desired eigenvalues: 0.2, 0.5, 1 ± i, 1 The desired block poles constructed in diagonal form: 

−0.2 0 0 −0.5

R1 =



 ,

R2 =

−1 1 −1 −1

 λ = −1

,

The corresponding 2 × 2 desired right denominator matrix polynomial of degree 2 is 2

H(s) = Nr (s)Dr−1 (s) = Dl−1 (s)Nl (s)

(31)

The matrix transfer function of the controller is

The control input signal is given by: u(s) = (Ks)−1 (KI + KP s + KD s2 )e(s)

This gives 

[Df 2 Df 1 ] =

0.2429 −0.5857 0.1786 0.3929



1.4143 −1.1714 0.8929 1.2857



The remaining closed-loop pole is to be assigned at 1. Now we compute 2 × 4 state feedback gain matrix Kc1 that places the block poles of (Ac1 − Bc1 Kc1 ) at R1 and R2 , taking (Ac1 − Bc1 Kc1 ) = (Ac )desired =⇒ Ki = Df i − Ai 

Kc1 =

0.2952 −0.9773 0.7294 −3.7753 0.2257 0.0931 1.0797 −1.8571



Then we compute the 1 × 4 matrix L by solving the Lyapunov-like equation L (Ac1 − Bc1 Kc1 ) − Λk L = Bc2 Kc1 This yields the next solution L = [−0.0002 − 0.0045 − 0.0038 − 0.0069] Next we compute a 2 × 1 state feedback gain matrix Kc2 that places the eigenvalue of [Λk − (Bc2 + LBc1 ) Kc2 ] at the desired closed-loop pole −1. 

Kc2 =

−64.6696 107.6896

25.9524 −31.1584 57.6345 −35.1371 15.0427 17.8122 21.3214 −70.7031 79.7112 −25.3564



The static pre-compensator (steady state tracking) is designed as follow:  + −1 Kp = H(0)+ = C (sI − A) B |s=0 = −(CA−1 B)+ 

Kp =

−1.0270 4.4713 0.3997 2.9206

(33)

Where: e(s) = r(s) − y(s) is the error between the input and the output. The closed loop transfer matrix is obtained as:  −1 Hclosed (s) = N (s) Dc (s)D(s) + Nc (s)N (s) Nc (s) Hclosed (s) = N (s)Df (s)−1 Nc (s)

(34)

The matrix equation Df (s) is called Diophantine equation where: Df (s) = Dc (s)D(s) + Nc (s)N (s)

(35)

Expanding this last equation we get:  T  T  Om Om  K T   DT Dl Nl−1 f (l+1)      T    T T    Dl−1 Nl−2 T  Nl−1 Om  T K     D  Df l   .. .. .. ..     =   (36)  .  .. . . .  T       . K   DT Om N T Om   P  0  0        T T D KI f0 Om Om Om N0T Remark6: the existence of MIMO PID controller using this procedure depends on the solvability of the last rectangular matrix equation.



  .. Using K = Kc1 + Kc2 L . Kc2 Tc its yields K=

Given a system described by right or left MFD as

C(s) = (Ks)−1 (KI + KP s + KD s2 ) = Dc−1 (s)Nc (s)(32)

Df (s) = Is + Df 1 s + Df 2  −1   I I I      Df (s) = Is2 − R12 R22  R1 R2 Is



Objectives: the main idea of this section here is the dynamic compensator design which relates inputs to outputs when the states are not measurable, the proposed compensator is of special type called three actions or MIMO PID controller based on the solution of Diophantine equation to relocate some desired Block roots of matrix polynomial achieving needed control performances.

Example3: Consider the matrix transfer function of dynamic system given in the RMFD form: H(s) = N (s)D−1 (s) Where:  D0 =



 N0 =

D2 = I , N2 = O and    3.6640 −6.2621 4.2082 −2.1862 , D1 =

−0.2896 7.0461

452.5373 78.3892 463.9935 314.0497

−0.0377 5.7919



 , N1 =

195.3567 180.7099 180.1895 177.9893



On λ-Matrices and Their Applications in MIMO Control Systems Design Design MIMO PID controller which will achieve the desired set of latent structure with tracking conditions. The desired set of solvent (Latent structure) is given below:     −3.9914 −0.1381 −2.9657 0.4224 , R2 = R1 = 0.0627 −5.0086 −0.3279 −7.0343   −7.9647 −0.3234 R3 = 0.1129 −9.0353

Construction of the desired matrix polynomial coefficients form those Block spectral data see Malika Yaici et al.(2014b): −1 I I I [Df 3 , Df 2 , Df 1 , Df 0 ] = −[R13 , R23 , R33 ]  R1 R2 R3  R12 R22 R32 

Set:



D2  N1 M =  O2 O2

D1 N0 N1 O2

 O2 O2   O2  N0

D0 O2 N0 N1

[K, KD , KP , KI ] = [Df 3 , Df 2 , Df 1 , Df 0 ]M −1 Finally the PID parameters are:     1.2548 0.7286 0.0389 −0.0436 , KD = K=  KP =

0.1882 −0.1352 −0.4025 0.5964

−0.0966 0.2183



 , KI =

0.1945 0.0210 −1.2949 1.3305



The tracking is shown in figure below: The output signals 3

2

magnitude Y1 Y2

The following case study illustrates best tracking , regulation and robustness with no oscillation and the ability of the proposed MIMO PID controller to robustly maintaining best dynamic performance and matching some desired latent structures or in other word Block pole placement preserving the output feedback compensator behavior. From the results obtained, in the above figures we see that the plant outputs coincides with its reference, no excess is recorded in both transient and permanent regimes which are well shown by the error signals. Also another discussion point can be considered and taken as an advantage which is the small controller gains that leads to smaller control signals, and thus to less energy consumption. Finally the global stability is guaranteed because all desired Block roots are stable matrices having specific latent roots latent vectors which implies large design degree of freedom and/or much more flexibility in syntheses.

6 CONCLUSION

Here in this example M is full rank square matrix then:

−20.4771 −20.4111

11

1

0

On Block roots of λ-matrices Decoupling method has been proposed to eliminate interactions between control loops in MIMO systems, concerning to this method the simulation results show that a high performance is obtained for both regulation and tracking problems with low order controllers, only we have gotten a problem of internal dynamics due to the proposed procedure nature (i.e. Block Zeros Block Poles cancellation). When the system is not block transformable (means that n is not integer multiple of m) a second algorithm based on Lyapunov-like equation is detailed here to achieve both poles and block poles placement. Lastly and more importantly a new systematic procedure based on the assignment of latent structure via MIMO output compensator design is illustrated and shown to be efficient, this dynamic compensator is of special type called MIMO PID controller, which have more degree of freedom in design and low energy consumption due to the small gains that are obtained.

−1

The output signal The input signal

−2

References

−3 0

5

10

15 time

20

25

30

Malika Yaici, Kamel Hariche, (2014a)On eigenstructure assignment using Block poles placement, European Journal of Control.

The error signals 0.2

magnitude E1 and E2

0.15 0.1 0.05

Malika Yaici, Kamel Hariche, (2014b) On Solvents of Matrix Polynomials, International Journal of Modeling and Optimization, Vol. 4, No. 4.

0 −0.05 −0.1 −0.15 −0.2 0

5

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15 time

20

25

30

Figure 2: The trajectory tracking control of MIMO PID controller

Malika Yaici, Kamel Hariche,Clark Tim, (2014c) A Contribution to the Polynomial Eigen Problem, International Journal of Mathematics, Computational, Natural and Physical Engineering, Vol:8 No 10.

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