Linear systems toolkit in Matlab: structural decompositions ... - CiteSeerX

Journal of COnJrol Theory and Applications

3 (2005)

287 - 294

Linear systems toolkit in Matlab: structural decompositions and their applications Xinmin LIUI, Ben M. CHENz,

Zongli UNI

( 1 . Charles L. Brown Department of Electrical and Computer Engineering, University of Virginia, Charlottesville, USA; 2. Department of Electrical and Computer Engineering, National University of Singapore, Singapore 117576, Singapore)

Abstract:

This paper presents a brief description of the software toolbox, linear systems toolkit, developed in Matlab

enviromnent. The

toolkit contains 66 m-functions, including structUral decompositions of linear autonomous

systems,

unforced! unsensed systems, proper systems, and singular systems, along with their applications to system factorizations, sensor/actuator selectiou, H-two and H-infinity control, and disturbance decoupling problems. Keywords:

Linear systems; StructUraldecompositions; Linear control; Software development

each other lead us to a deeper insight into how feedback

1 Introduction

control would take effect on the system, and thus to the The state space representation of linear multivariable systems is fundamental

to the analysis and design of

explicit construction of feedback laws that meet our design specifications. The search for structural canonical forms and

dynamical systems. Modem control theory relies heavily on

their applications in feedback design for various perfonnance

the state space representation of dynamical systems, which

specifications has been an active area of research for a long

facilitates characterization

time. The

of the inherent

properties

of

effectiveness of the structural decomposition

dynamical systems. Since the introduction of the concept of

approach has also been extensively explored in nonlinear

the state, the study of linear systems in the state space

systems and control theory in the recent past.

representation itself has emerged as an ever active research

In this article, we present a MATLAB toolkit, linear

area, covering a wide range of topics £fom the basic notions

systems toolkit, for realizing various structural decomposi-

of stability, controllability, observability, redundancy

minimaIity to more intricate properties of finite and infinite

tion techniques recently reported in a monograph by Chen, Lin and Sharnash [1]. The toolkit [ 2] , which has been

zero structures, invertibility, and geometric subspaces. A

built upon the earlier versions [3,4],

deeper

compute

understanding

of linear

development of modem

and

systems facilitates the

control theory. The demanding

the structural

is able to efficiently

decompositions

of autonomous

systems, unforced! unsensed systems, proper

systems, and

expectations £fom modem control theory impose an ever

singular systems, along with their properties, such as finite

increasing demand for the understanding and utilization of

and infinite

subtler properties of linear systems.

geometric

Structural properties play an important

role in our

zero structures, invertibility subspaces .

The

structures

applications

of

and these

decomposition techniques to system factorizations, structural

space

assignments via sensor or actuator selection, and Hz and

representation. The structural canonical fonn representation

Hx> control are also included. It is now used extensively in

of linear systems not only reveals the structural properties

the education

understanding

of

linear

systems

in

the

state

and research of control

theory. Its rich

but also facilitates the design of feedback laws that meet

collection of linear algebra functions are immediately useful

various control objectives. In particular, it decomposes the

to the control engineer and system analyst. Its easy-to-use

system into various subsystems. These subsystems, along

environment

with the interconnections

expressed in familiar mathematical notation.

that exist among them, clearly

show the structural properties of the system. The simplicity of the subsystems and their explicit interconnections with Received

8 June 2005;

revised 30 July 2005 .

allows the problems

and solutions to be

The detailed algorithms and proofS of the functions reported in the toolkit can be found in the monograph of

288

X. UU

et al. / Joumal of ConJrol Theory and Application:; 3 (2005) 287- 294

Chen, Lin and Shamash [1 ] , and the beta version of this

5) morseidx: Morse indices;

toolkit

6) blkz: blocking zeros; 7) invz: invariant zero structure;

is

currently

available

on

the

website

at

.

http://linearsystemskit.netorhttp://hdd.ece.nus.edu

sgl

-

bmchen.

Readers who

have our earlier versions

[ 3 , 4] of the software realization of the special coordinate

8) in£Z:infinite zero structure; 9) L invt: left invertibility structure;

basis of Sannuti and Saberi [6] are strongly encouraged to

10) r - invt: right invertibility structure;

update to the new toolkit. The special coordinate basis,

11) nonnrank: nonnal rank;

implemented in the new toolkit, is based on a numerically

12) v - star: veakly unobservable subspace;

stable algorithm recently reported in Chu, Liu and Tan

13) v - minus: stable weakly unobservable subspace;

[ 5 J, together with an enhanced procedure [1] .

15) L star: strongly controllable subspace;

reported in

The paper is organized as follows. Section 2 provides the detailed list of m-functions in the toolkit. Section 3

14) v - plus: unstabl weakly unobservable subspace; 16) s - minus: stable strongly controllable subspace;

17) s - plus: unstableweakly unobservablesubspace;

describes some key functions of the toolkit, while Section 4

18)

demonstrates some m - functions with numerical examples.

subspace ;

L star: strongly controllable weakly unobservable

Finally, Section 5 draws a brief conclusion to the paper.

19) n - star: distributionally weakly unobservable subspace;

2

20) s - lambda: geometric subspace SA;

Contents of tookit

21) v - lambda: geometric subspace VA; We list in this section the detailed contents of the toolkit. We note that some m-functions in the toolkit are interactive, which require users to enter desired parameters during execution. Others are implemented in a way that can return results either in a symbolic or numerical form. The current version of the toolkit consists of the following m-functions: A) Decompositions

1) ea- ds: decomposition of a matrix pair ( E , A ) ; of autonomous

1) ssd: continuous-time tion;

systems.

2) sd - ds: decomposition for descriptor systems;

stability structural decomposi-

tion;

3) bdosd: block diagonal observable structural decomposition; 4) csd: controllability structural decomposition; 5) ctridx: controllability index; 6) bdcsd: block diagonal controllable structural decomproperties

of

system.

1) scbraw: raw decomposition without integration chains; 2) scb: decomposition of a continuous-time system; 3) dscb: decomposition of a discrete-time system; 4) kcf: Kronecker canonical form for system matrices;

2) iofact: continuous- time inner-outer factorization; 3) gcfact: continuou~ generalized cascade factorization; 4) dmpfact: discrete minimum-phasel all-pass factorization;

position. proper

struc-

F) System factorizations. 1) mpfact:cop.tinuousminlmum-phaselall-passfactoriza-

2) obvidx: observability index;

& structural

system left invertibility

ture.

unsensed

.

Decompositions

L invt - ds: descriptor

6) L invt - ds: descriptor system right invertibility strucand

I) osd: observability structural decomposition;

C)

5)

ture;

canonical form;

4) Ijd: real Jordan decomposition. B) Decompositions of unforced

systems

3) invz- ds: descriptorsystem invariantzero structure; 4) in£Z- ds: descriptor system infinite zero structure;

2) dssd: discrete-time stability structural decomposition; 3) jcf:Jordan

D) Operatioll$ of vector subspaces. 1) ssorder:ordering of vector subspaces; 2) ssintsec:intersection of vector subspaces; 3) ssadd:addition of vector subspaces. E) Decompositions and properties of descriptor systems.

5) diofact: discrete-time inner-outer factorization. G) Structural selection.

assignment

via

sensor/actuator

1) sa- sen: structural assignment via sensor selection; 2) sa - act: structural assignment via actuator selection.

H) Asymptotic time-scale and eigenstructure assignment. 1) atea:continuous-time ATEA;

289

X. UU et 01. / Joumnl of Control Theory and Applicatiorn 3 (2005) 287 - 294

whereeachblockJi, i = 1,2, ... , k, has the following

2) gm2star: infimum for continuous-time H2control; 3) h2care: solution to continuous-time H2ARE;

form:

1

4) h2state: solution to continuous-time H2control; 5) gm8star: infimum for continuous-time Hoc control; 6) h8care: solution to continuous-time Hoc ARE;

.

Ji

='

)...

8) addps: solution to continuous disturbance decoupling; 9) datea: discrete-time ATEA;

,

or

I

10) dare: solution to general discrete-time ARE;

Ji =

11) dgm2star: infimum for discrete-time H2 control; 12) h2dare: solution to discrete-time H2 ARE; 13) dh2state: solution to discrete-time H2 control;

contains

a pair of complex

OSD

decoupling

with

static

= [.!:iWi

Ai

17) daddps: solution to discrete-time disturbance decou-

.

Observability

;

2) rosys4ddp: irreducible reduced-order system that can be used to solve DDPCM.

3 Descriptions of key m-functions In parricular, we will present the well-known decomposition

At = Ts- I ATs

=I

for

autonomous

real Jordan

systems,

a

systems, the indices

special coordinate

and weakly

basis, Morse

unobservable

geometric

subspace for proper systems, a structural decomposition for singular systems, the inner-outer

actuator

structural

selection,

assignments through the

asymptotic

system

sensor

time-scale

or and

eigenstructure assignment, the computation of the best achievable disturbance attenuation level in Hoc control, and the solution to the problem of disturbance decoupling

*

0

0

*

0

Ct

...

*

...

h-Ip

*

0

0

0

0

...

1

= To-'CT, = [ ! p

nom:

=n-

n

:Z=ki, Oidx: = Ikl,k2,"',kpt. i= 1

We note that nom is the number of unobservable modes and the set Oidx is the observability index of ( C , A ) . BDCSD Block diagonal controllable structural decomposition. [ At , Bt , Ts , Ti , ks] = BDCSD( A , B) transforms a controllable pair (A , B) into the block diago-

At = T,-t AT., =

generates a transformation that transforms a square matrix into its real Jordan canonical form, i. e. ,

[A'

JI

h [

0

0 0 ...

0

[], T] = RJD(A)

=J =

*

* 1

Real Jordan decomposition.

T-I AT

= OSD(A,C)

nal controllable structureal decomposition form, i . e. ,

through static output feedba~k. RJD

fJ.i :t jWi'

decomposition.

0

0

so-called

a block diagonal controllable structural decomposition for

factorization,

eigenvalues

structural

observability structural decomposition for unforced systems,

technique

]

returns an observability structural decomposition for the matrix pair( A, C), i. e. , * 0 0 Ao * * * 0 0 hl-I

We briefly describe some key m-functions of the toolkit.

invariance

Wi fJ.i

[At,Ct,Ts,To,uom,Oidx]

with static output feedback (DDPCM)

I .1 A-

output

1) ddpcrn: solution to disturbance decoupling problem

unsensed

:

and ).i is a real eigenvalue and

16) dh8state: solution to discrete-time Hoc control;

feedback

.

.

[A

14) dgm8star: infimum for discrete-time Hoc control; 15) h8dare: solution to discrete-time Hoc ARE;

pling. I) Disturbance

J

["

7) h8state: solution to continuous-time Hoc control;

J

Bt

= Ts-IBTi = 0 [B'

...

A2 ... : 0 ... * ...

n

Ak

*

B2 ...

*

0

Bk

...

il

290

X. UU et at. / ]ourruLlof COnlrolTheory and Applications 3 (2005) 287

where Ai and Bi, i = 1,2,"',

k, are in the fonn of

:

0*

[

]

[1] .

' Bi =

Special coordinate basis of contionuous-time

sys-

tern.

[As, Bt, Ct, Dt, Gms, Gmo, Gmi, dimJ = SCB(A, B , C, D) decomposes a continuous-time

system into the standard

MORSEIDX tems.

-

= As

Gms

LabCb

0

Lad Cd

0

Abb

0

Lbd Cd

Eca

LcbCb

Ace

LedCd

BdEdb

BdEde

Add

]

COb

Cae

BOb

0

Boe

0

.

V*

generates the structural decomposition system. The state x are decomposed as

BsJ

-

X

n =

.

0

0 0

o

].

where

computes an inner-outer factorization for a stable proper transfer function matrix G( s) with a realization(A , B , C , D), in which both [BT DTJ and [ C D J are assumedto be of full rank. The inner-outer factorizationis given as G(s) = Gi(s)Go(s), where

Add = Add + BdEdd + LddCd,

GJs)

for some constant matrices Ldd and Edd of appropriate dimensions, and Add = blkdiagiAq

Bqi

J.

=IOFACT(A,B,C,D)

. a + d1m = [ no, no, no , nb , ne, nd J ,

'

T T Xd

[Ai, Bi, Ci, Di, Ao, Bo, Co, Do J

and

(Aqi

T Xc

systems .

[1m

with

T Xb

See [2 J for detail. IOFACT Inner-outer factorization of continuous-time

Cd, CW] 0

Dt = Gmo-IDGmi= Ds =

Cd

T Xo

polynoInials or rational functions of s . In particular, Psc x + Psd Ii = Psr Xz.

0

Bd

T Xe

function as that of the original system. Ez, Psi, Psc, Psd and Psr are some matrices or vectors whose elements are either

[ Co] Cs

Cab Cae 0 0 Cb

= [ XzT

of a descriptor

The quadruple (Es, As , Bs , Cs , Ds) has the same transfer

Ct = Gmo - I C Gms

0 [C0

Weakly unobservable geometric subspace. V = V-STAR(A,B,C,D)

Gms, Gmo, iGIni, dimJ = SD - DS ( E , A , B , C, D)

CodJ,

Bad ]

[BO. Bad Bd

=

fonn;

SD - DS Structural decomposition of continuous-time descriptor system.

0 -

Morse invariance indices of proper sys-

[Es,As, Bs, Cs, Ds, Ez, Psi, Psc, Psd, Psr, Gme,

BOe [COa

=Gms-IB Gmi = [Bo

Bt

= [1,O,"',OJ.

computes a matrix whose columns span the geometric subspace

BOb

[

Cq;

14 =infinite zero structure.

V - STAR

Boa +

= [~] ,

= zero dynamics matrix in Jordan 12 = right invertibility structure; I3 = left invertibility structure;

+ BoCa

Aaa

[ BdEda

Bqi

!qi-I] 0

II

stable, marginally stable and unstable parts, and Xd being decomposed into chains of integrators .

= Gms-I A

= [~

[ II , 12, I3 , 14J = MORSEIDX (A , B , C, D) returns Morse structural invariance list:

SCB fonn with state subspaces Xo being separated into

At

,

Aqi

0

h.-I

SCB

294

= i k I , k2 , . .. , kk f ,

ks

Ai =

-

1

,Aq 2 ,"',Aq

md

= blkdiagiBq ,Bq ,"',Bq = blkdiagl Cq , Cq ,..., Cq

I,

1

2

md

I,

1

2

md

f,

' Cqi) being defined as

= Ci(s! - An-IBi + Di

is an inner, and Go(s)

= Co(s! - AO)-IBo + Do

is an outer.

SA- SEN

Structural assignment via sensor selection. C = SA- SEN(A, B)

For a given unsensed system (A, B ), returns a measurement

output

the function

matrix C such that the

X. UU

et oJ. / Journal

of Conlrol

Theory

and

Application.