An Elementary Derivation of Kronecker Canonical Form for Linear Time-Invariant Systems

Copyright © IFAC Advances in Control Education. Tokyo. Japan. 1994

AN ELEMENTARY DERIVATION OF KRONECKER CANONICAL FORM FOR LINEAR TIME-INVARIANT SYSTEMS N. SUDA Department of Systems Enginuring. Faculty of Engineeril/g Science. Osaka University. I-I Machikoneyamacho. Toyonaka . Osaka 560. Japan

Abstract. This paper demonstrates an elementary approach to derive the canonical form of a state space representation of a finite-dimensional. linear. time-invariant system under the group of transformation consisting of a state feedback. an output injection. and nonsingular coordinate transformations of state. input and output spaces. 1be prerequisites are under-graduate level of matrix algebra and a few facts from the linear systems theory. Key Words. Canonical forms: education: linear systems: matrix algebra: state space

H. under the name of special coordinate basis. via a modification of the structural algorithm of Silverman (1969).

I. INTRODUCfION

Let a quadruplet (A. B. C. D) be a state space representation of a finite-dimensional. linear. timeinvariant system x = Ax + Bu. Y =Cx + Du (I) where the dimensions of the state x. the output y. the input u are n. m. r. respectively. and a quintuplet (K. L. T. G. H) be a transformation consisting of a state feedback. K. an output injection. L. and nonsingular coordinate transformations. T. G. H. of state. input and output spaces. respectively. Thus the transformation (K. L. T. G. H) applied to (A. B. C. D) results in (A. B. D) where A= T-I(A + BK + Le + WK}T . iJ = T-I(B + WiG C= H(C + DK}T. D= HDG (2)

The purpose of this paper is to demonstrate an elementary approach to derive the canonical form . It is somewhat similar to that by Sannuti and Saberi (1987). They use mostly the row operations on matrices. while both the row and the column operations are adopted here. The procedure is quite simple to describe and the meaning of each transformation is easily understood. The prerequisites to our approach are under-graduate level of matrix algebra and a few facts from the linear systems theory. namely. the Kalman's canonical decomposition. the equivalence of the controllability and the pole-assignability. and the Brunovsky canonical form of a controllable pair. In a course of the linear systems theory. these topics are usually presented before the introduction of the Kronecker canonical form. and. therefore. assumption of the knowledge of these facts is reasonable.

t:.

The canonical form under the group of above transformation is well-known (Thorp. 1973). which corresponds to the Kronecker canonical form of the singular pencil n(s) = [ A

~SI ~]

(3)

2. CANONICAL FORM

called Rosenbrock's system matrix. This canonical form is important in that it reveals system properties such as finite and infinite zero structure (van der Weiden and Bosgra. 1979). strong controllability and observability as well as conditional controllability and observability (Suda and Mutsuyoshi. 1978). and so forth.

Three sets of nonnegative integers 0= njl = .. . = nja, < nj(aj+l) S ... :5 njp"

i = 1.2.3

(4)

are defined. (n Ij+ l)'s are the degree of infinite elementary divisors. n 2/S and n 3/s are the Kronecker row and column indices of the singular pencil n(s). respectively. Defme p,

The derivation of the canonical form is. however. not so easy. Gantmacher (1959) derives it for general pencils. but. of course. relevance to linear systems is not described. The derivation by Morse (1973) heavily relies on the geometric properties (Wonham. 1979) of system models and requires prior knowledge of this rather advanced subject. On the other hand. the derivation by van der Weiden and Bosgra (1979) is a pure algorithm and hardly gives any insight into system structure. Sannuti and Saberi (1987) derive T. G. and

nj =

I,njj. i = 1.2.3 j=a,+1

(5)

Denote the j-th column of the nxn identity matrix. In. by

eJ and define

Aj=[O e~'1

iJi} = e n'ln'l

= n ·· x 1 IJ

(6)

73

Two remarks are in order here. 1) If a1 and a3 are both zero. then one can skip this

and

Ai = block diag {A:z,+, ..... Ap.} =n, x n,. i =1.2.3 8j

= quasi diag

{8~i+I ..... 8p,}=ni x(f3,-a;).

i= 1.3

step and proceed to the next immediately. 2) If a 2 = m and/or a 3 = r. then one of the tenninating conditions. described later on. is satisfied and one can jump to the Part Two directly. Similar remarks are relevant to most of the following steps. but they will not be repeated every time.

(7)

Cj =quasi diag {C~i+I'''''Cp.} =(Pi - a;) x nj. i =1.2 Then. on application of an appropriate transfonnation (K. L. T. G. H). the system matrix is reduced to the canonical fonn shown below . ' . 0 0 0 0 0 : BI 0 AI -si 0 0 0 0 0 :0 0 A2 -si , , 0 ,, 0 83 0 0 0 0 A3 -si 0 0 0 0 0 J -si: 0 0 n{s) = --~-----------------------r------------0 0 0 0 0 0 0 Cl 0 0 0 0 0 C2 0 0 0 0 0 la J 0 0 0 0 0 0 0 0 0 0 0 0

~Let

Ho = [HOT]. Go = [G OL HOB

(16)

GOR )

(HOB : al rows. GOR: al columns)

such that

HODoGo=[~ I~J

(8)

(17)

On application of these coordinate transfonnations first.

where J is an n4xn4 matrix in the Jordan canonical fonn. corresponding to the finite elementary divisors of n(s) with n4 = n - nl - n2 - n3

(15)

rank D = rank Do = al then there exist nonsingular matrices

and then a state feedback and an output injection: Ko = [-H:BCJ

(9)

Lo = [0

(18)

-BoGo R )

one obtains the following fonn of the system matrix

There are a2 zero rows at bottom and a3 zero columns at far right.

0

In LoHO]nO{s{ In [ o Ho GOKO

AI - si

nl(s)= [ Cl

3. DERIV AnON OF CANONICAL FORM

0]

]=[nl(s)

GO

0

la,

BI]

(19)

0

where

AI = A- BoGORHOBCO'

The present derivation of the canonical fonn is divided into two parts. In Part One. the matrices B. C. Dare processed and then the subsystem with subscript '1' is extracted. Part Two takes care of the remaining part.

DJ=m-a2. rank[B

T

DTr =r-a3

(10)

then there exist nonsingular matrices H=[Z:J. G=[GL

GR)

= HOTCO (20)

HB[C

DJ=O.

(HIT: ~ rows. G IL : ~ columns)

[~fR =0

H)CIB)G) =

[H~~~I}BIGIL

By these coordinate transfonnations in the output and the input spaces the system matrix is modified as follows : 0] = [nO(s)

0

0]. 0

no{s)=

[A -

si

Co

Co

= HTC.

Do

=HTDG L

(23)

TIR)=l n

(24)

TIR 1

(25)

holds . Thus TI = [BIG IL

is nonsingular. Hence the coordinate transfonnations. T I • G I. HI. are applied. resulting in

BO] DO (13)

7i

where

= BGL .

[/~J ~]

11 is easily shown that there exist matrices S IB=(nDI)xn and TIR=nx(n-D) such that

BGL ]=(n+m-a 2 )X(r-a3 ) full column rank [ HTDG L

G

(22)

such that

HTDGd={m-a2)X(n+r-a3) full row rank (12)

I" O]n s [In [ o H () 0

=[ HIT]. GI =[G IL GIR ) HIB

HI

(11)

such that the following conditions are satisfied.

Bo

Cl

Step III This step is further divided into Step III-k (k= 1.2.... ). The Step Ill-I is as follows. Let (21) rankCIB I = 81 then there exist nonsingular matrices

Slm.! Let

[HTC

=BOGOL '

Note that BI = n x (r - al - a3) is full column rank and Cl = (m -a l -a2) x n is full row rank.

3.1. Part One

rank[C

BI

Thus a part of the canonical fonn has been extracted and the derivation continues on nl(s) part.

(14)

_I

AIT) =

[H)TCIAIBIGIL SIBAIBIG IL

TI-IBIGI=[HSITCI][BIGIL

Thus a2 zero rows at bottom and a3 zero column at far right have been extracted. Hence attention is focused on no(s) part.

IB

HITCIAITIR]

(26)

SIB A I7iR

B,G'Rl=[/l>J 0

0]

SIBBIGIR

(27)

74

;J

~:2] n 2 [G~~2

On application of a state feedback and an output [T2;1

(1)

-sI",

I",

0

: 0

0

o

-sI",

0

: I" ,

0

o 0 Ars/: 0- B, ------------------r------ · 1"2

0

0

o

0

C3

: 0

0

0

0

:

(44)

where A3 = S28A2T2R = (n -

B3 = S28B2G2R = (n -

81 - 28 2 ) x (n - 81 - 282 ) 81 - 282 ) x (r - a3 - al - 81 - 8 2 )

C3 =H28CzT2R =(m-a2 -al-81-82)X(II-~ -28 2 )

(45) where A2 = SI8AITIR = (n -~ )X(n -81) ~ =SI8BIGIR =(n-81)x(r-a3

-al-81) C2 =HI8CITIR =(m-a2 -al-81 )x(n-81)

Here it is noted that the blocks containing the identity matrix, 1c5z. after appropriate permutation of the state

(32)

variables. constitute the part of the canonical form with n lj=2 . Hence the derivation continues on the

Here it is noted that the blocks containing the identity matrix, I,,}, correspond to the part of the canonical form

rem,;,"n,

with n Ij= I . Hence the derivation continues on the remaining part, that is,

n 2 (s)= [ A2C-2 sI

B2]

(46)

It is also noted that B 3 is full column rank and C 3 is full row rank, and C3B3 = O. C3A3B3 = 0 (47) are verified in the same manner as Eq .(34).

(33)

0

,

p"'. t~:::; [A~" ~'l

It is also noted that B2 is full column rank and C 2 is full row rank, and that, in view of Eqs.(23), (27) and (28),

In a similar way, the Step III-k receives from the preceding Step III-(k- I) Bk • full column rank, and C k'

(34)

holds.

full row rank. and Ak that satisfy CkAkl Bk = O. 1 = l... .. k - 2 (48) After extracting the part of the canonical form with n Ij=k, it leaves for the succeeding Step III-(k+ I)

The Step III-2 is slightly more complicated. Let rankC2A2~ = ~

(35)

then there exist nonsingular matrices

B k .. l' full column rank, and C k.. I' full row rank, and At .. 1 that satisfy

(36)

H2=[H2T], GZ=[GZL G ZR ] HZ 8 (H zT : ~ rows, G 2L : ~ columns)

ChlAk .. 11 Bhl = O. 1 = l... .. k -1 (49) The procedure is analogous to Step 111-2 and hence the details are omitted.

such that (37) Let MZI = AzBzG 2L

3.2. Termination of Pan One

(38)

SZI = HZTCzAz(J - Mz1HzTCZ) (39) then it is easily shown that there exist matrices SZ8=(n-8 1-28 z )x(n-8 1 ) and T 1R =(n-8 1 )x(n-

When the Step III-k is concluded, the size of the matrices left are Chi

81-2~) such that

[ H~:~2l[AZB2G2L

B2G2L

T2R ]=l n

-"J

(40)

Ahl

=(m-az -al- I8

l

(

holds . Thus

Bhl = 1 n-

l

T2 =[A2~G2L B2G ZL T2R ] (41) is nonsingular. On application of the coordinate transfor-mations, T 2 • G 2 • H 2 • first. and then a state feedback and an output injection :

K, =

"

[00 00

'-2 = [

-H2TC2A/TZR]

0

-H2TC2A/B2GZL

0]

-SZC A2"B 2G2L

0

-SZ8Al B2G2L

0

J)l

n-

Ii81 J 1=1

= (n- Ii81 Jx ( n- Ii81 )1 1=1

SZ8

I

1=1

k

\

(50)

1=1

I (

•

'\

I,i8, IX I r- a 3- a l-

I,OI )

1=1

1=1

)

\

Since 8, 's are nonnegative. there exists K :5 n such that K

n-I,io,~K

(42)

(51)

1=1

holds. that is. the size of A K .. 1 is less than K. By Eq.(49) and the Cayley-Hamilton theorem. it follows that

(43)

CK"IAK .. /BK+I=O

Other possibilities are, at some k.

the system matrix becomes

75

1=1.2.....

(52)

Brunovsky canonical form . which corresponds to the (A 3 • ih) part of Eq.(8). Similarly. let the observability

k

m-u2 -UI-

IA =0

(53)

;=1

indices of the pair (A)).

holds. that is. no C -matrix is left for the next step. k

r-U3-UI-IA=0

(54)

;=1

holds. that is. no B-matrix is left for the next step, or k

n - lic5; = 0

C~)

be "U (k=a2+1. .. ..

132). then. by suitable output injection and coordinate transformations in the state and the output spaces . this pair is reduced to the dual of the Brunovsky canonical form . On application of a permutation of the state variables n2 n, (60) P = [e n2 en;_1 ... e l -

(55)

n'l

;=1

holds. that is. no A-matrix is left for the next step. The Part One terminates when any of Eqs. (52) through (55) is satisfied.

the resulting pair corresponds to the (A 2 • C2) part of Eq.(8). Finally the block A 22 • when transformed into the Jordan canonical form. corresponds to the J block in Eq.(8).

From the procedure described, it is readily noted that. if Eq .(55) holds, then Eqs.(53) and (54) are also satisfied. In this case, therefore, the Part Two is not needed and the canonical form with n2=n3=n4=0 has been derived at the conclusion of the Part One.

Thus. after an appropriate permutation of blocks. the canonical form of Eq.(8) has been derived. If Eq.(53) or Eq.(54) is satisfied. a similar procedure as above leads to the canonical form with n2=0 or n3=0. respectively.

3.3. Pan Two Equation (52) implies that the transfer function determined by the triple (A K + I. B K + I. C K + I) is identically zero, and thus. in the Kalman's canonical decomposition, there is no controllable and observable subsystem. Hence there is a coordinate transformation in the state space which brings this triple to the following form

[ A~o

4. CONCLUSION It is demonstrated that the Kronecker canonical form of linear. time-invariant systems can be derived. using only elementary matrix algebra and a few facts from the linear systems theory.

~:~ ~:~l. [~l'

[0 0 C3 1 (56) 0 A33 0 where the subscript 'I' denotes the controllable but unobservable. subscript '2' the uncontrollable and unobservable, and subscript '3' the observable but uncontrollable subsystems. respectively . I

REFERENCES Gantmacher, F. R. (1959) . The Theory of Matrices. Chelsea Publishing Company. New York. Morse, A. S. (1973) . Structural Invariants of Linear Multivariable Systems. SIAM J. Control. 11. 446465. Sannuti. P .• and A. Saberi (1987). Special Coordinate Basis for Multivariable Linear Systems - Finite and Infinite Zero Structure. Squaring Down and Decoupling. Int. J. Control. 45. 1655-1704. Silverman. L. M. (1969) . Inversion of Multivariable Linear Systems. IEEE trans. AUlOm. Control. 14. 270-276. Suda. N .. and E. Mutsuyoshi (\ 978) . Invariant Zeros and Input-Output Structure of Linear Time-invariant Systems. Int. J. Control. 28.525-535 . Thorp. J . S. (1973) . The Singular Pencil of a Linear Dynamical System . lnt. J. Control. 18.577-596. van der Weiden. A. J. J .. and O. H. Bosgra (\979). The Determination of Structural Properties of a Linear Multivariable system by Operations of System Similarity.lnt. 1. Control. 29. 835-860. Wonham. W. M. (1979). Linear Multivariahle Control:A Geometric Approach. second edition. Springer Verlag. Berlin.

The controllability of (A 11, B I) and the observability of (A 33, C 3) assure that the eigenvalues of the first and the third diagonal blocks can arbitrarily assigned. by suitable state feedback and output injection. Therefore one can assume. without loss of generality, that three diagonal blocks have no common eigenvalues. It is well-known that if square matrices A ;;=p;xp; and Ajj=PjxPj have no common eigenvalues. then. for any Fi/=p;xPj • there exists Tij=p;xPj such that A;;Tij - TijAjj + F;j = 0 (57) is satisfied . Now let T 12 and T 23 be solutions of Eq .(57) for F 12= A 12 and F 23= A 23. respectively. and T 13 be solution for F D= A 13+ A 12 T 23 . Then a coordinate transformation in the state space T=

[~ T~2 ;~:l

(58)

o

0 I brings the A-matrix into a block diagonal form block diag {A 11. A22 . A33} without affecting the B- and C -matrices.

(59)

Now let the controllability indices of the pair (A 11' B I) be n3k (k=a3+1. .... 133). then. by suitable state feedback and coordinate transformations in the state and the input spaces. this pair is reduced to the 76