Orthogonality of State Transformations to Block Diagonal Form

Copyright © IFAC 12th Triennial World Congress. Sydney. Australia, 1993

ORTHOGONALITY OF STATE TRANSFORMATIONS TO BLOCK DIAGONAL FORM M.A. Barker· and [.B Rhodes •• *Itek OpticaL Systems. 10 Maguire Road. Lexington, MA 01273. USA **University of California. Santa Barbara. Department of ELectrical and Computer Engineering. Santa Barbara, CA 93106. USA

Abstract. The computations encountered in systems control applications are most efficiently performed with the systems represented in a canonical form. Use of canonical forms is limited by the numerical stability of existing conversion algorithms and the sensitivity of the canonical matrix to small perturbations in coefficients. A block diagonal form is proposed which will. in general. be much better conditioned than the canonical forms with only a moderate increase in the number of nonzero elements. Necessary and sufficient conditions under which a system can be converted into a using an orthogonal state basis transformation are identified and an algorithm presented. Key Words. Controllability; observability; canonical forms ; decomposition; linear systems

l. INIRODUCTION

Related to this form are a set of feedback invariants (controllability indices or in the dual, observability indices) which determine the size of each diagonal block (Wonbam and Morse, 1972). Associated with each index is a controllability (observability) subspace such that the dimension of the subspace equals the corresponding controllability (observability) index. Given a controllable (observable) system, the columns of the matrix (rows of the inverse matrix) which transforms the system into Luenberger form are composed of bases for these subspaces.

Reduction of a state space model of a system to a canonical form (Luenberger, 1967) or to condensed forms (Van Dooren and Verhaegen, 1985), has numerous applications in control systems, for example, pole placement (Kailath, 1980), frequency response (Laub, 1981), determination of system controllability (Paige, 1981), and efficient Kalman ftltering (Rhodes, 1990). Use of canonical forms is desirable in order to minimize the number of nonzero elements in the system matrices, however, transformations to canonical form are commonly ill-conditioned (Wilkinson, 1965) and the coefficients of the resulting matrix highly sensluve to small perturbations (Kenney and Laub, 1988). Although much less economical in terms of the number of nonzero elements, use of condensed forms, such as the Hessenberg or staircase forms, can be accomplished with the use of numerically-stable unitary transformations (Van Dooren and Verhaegen, 1985) and are thus preferred in many applications.

Although the feedback invariants of a system are unique, the corresponding subspaces are not (Warren and Eckberg, 1975). Of special interest are conditions for which the conversion can be accomplished using an orthogonal state transformation matrix. Compared to condensed forms, a significant reduction in the number of nonzero elements can be achieved along with an associated reduction in subsequent processing. This can be accomplished while maintaining the numerical stability that is obtained by using an orthogonal state transformation. In section 3, necessary and sufficient conditions for such a state transformation to exist and an algorithm for testing existence are developed.

This paper explores a hybrid system representation, that is, a block diagonal matrix representation in which the submatrices located along the main diagonal are in a condensed form rather than companion form. The transformation is accomplished via a state basis transformation, a feedback gain, and an input or output basis transformation. The system, transformation, and gain matrices will be discussed in section 2.

2. SYSTEM DESCRIPTION AND DEFINITIONS

For the following, let block(xi) be a block diagonal matrix where Xi may be a vector or a matrix. The 799

range of the subspace spanned by the columns of a matrix, T, will be denoted by range(T) and the dimension of the subspace by dim(T). The set of integers from I to m will be denoted by m.

state space that is required to achieve block diagonalization of the system (C,A). It is not surprising that the existence of an orthogonal state transformation, T, is directly tied to the existence of a set of orthogonal observability subspaces. This will be proven in Theorem I. This theorem, by itself, is of little use unless the corresponding output transformation matrix, H, and feedback gain, K, are known. Explicit calculation of the feedback gain is possible given the system matrices (see Theorem 2). Calculation of the output transformation is somewhat more complicated, but generically involves no more than a QR decomposition (Golub and Van Loan, 1983) of CT and a Schur decomposition (Golub and Van Loan, 1983) of a single mxm matrix (see Theorem 3). Using these values it is then possible to determine if an orthogonal state transformation exists.

The system and analysis will be presented from the point of view of observability. Parallel results for controllability follow directly by duality. The system to be considered is assumed to be linear, constant and defmed by: j(t) = Ax(t)

x(t) e X = Rn, A e Rnxn

y(t) =Cx(t)

y(t) e Y = Rm, C e Rmxn

Without loss of generality assume that C has full m and the pair (C,A) is completely observable. Given these assumptions, a system (C,A) can always be decomposed into a set of m decoupled subsystems (Wonbam and Morse, 1972). Specifically, there exists a unique set of observability indexes (lCI' lC2,"" lCm) corresponding

rank

THEOREM 1: There exists a transformation triple (T,H,K) which block diagonalizes (C,A) such that T is orthogonal if and only if there exist observability subspaces, 0i' ie m such that 0i is

m

to (C,A) such that lCI :
°i=I,lCj-

To each

lCi

orthogonal to 0j for all i;t:j .

there corresponds an

Proof. For sufficiency, assume the observability

j=l

observability

subspace,

°i'

such

that

if

subspaces are orthogonal. Let 0i be an orthonormal basis for 0i and let T be the orthogonal matrix, [OI, ... ,Om]' There exists an H

(1)

and K such that 0i is A invariant and c? is the generator for 0i (Wonbam and Morse, 1972).

C=range(CT): m

dim(Oi) = lCi' dim(OinC)=I, $ 0i = X 1=1

Further, there exists a map K:X~Y with a matrix representation K (the feedback gain) and state and output basis transformations, x=T i and y=H y, such that A = rl(A - KC)T = block( Ai) Ai e RlCiXlCi

Therefore diagonal.

TT(A-KC)T

For necessity assume that

and H-ICT are block

T is orthogonal and

H-Icr and TTATT are block diagonal. If 0i=range(Oi), where 0i are the columns of T from

C = H-Icr = block( c?) DefiningA=(A-KC)T, CieRnxl as the i th column of

0i_1 to 0i. then with ci as above. the conditions of equation I are satisfied and the observability subspaces so defmed are orthogonal. •

(H-IC)T, and q= range(ci) = 0i n C, (Oi isA invariant)

Proof of the existance of an orthogonal T depends on the determination of appropriate values for H and K. The following theorem establishes that if block diagonalization is possible with an orthogonal T, then it will be possible with the feedback gain, K=ACT(CCTyl .

The triple of matrices, (T,H,K), which block diagonalize a system (C,A) will be referred to as a

transformation triple.

THEOREM 2: Given that there exists a transformation triple (t,h,k) which block diagonalizes (C,A) and such that t is orthogonal, then there exists a transformation triple (T,H,K) which block diagonalizes (C,A) such that T is orthogonal and K = ACT(CCT)"l

3. CONDmONS FOR ORTHOGONAL TRANSFORMA nON The observability subspaces as defmed by equation 1 are directly related to the basis change on the

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following lemma which characterizes all matrices,

Proof Let SE Rmx.m be a nonsingular diagonal

H-I, which can ortbogonalize the rows of C.

matrix and BE Rnxo be a block diagonal ortbogonal

LEMMA 1:

matrix. Because C is full rank and h-1Ct is block diagonal, B and s can be chosen such that s- l h- 1CtB is zero except for the (i,(Ji) elements

Given a matrix, hE Rmx.m, such that

the rows of h-IC are ortbonormal and HE Rrnxm is any other nonsingular matrix such that the rows of H-IC

which equal one, that is s- l h- 1CtB = block( ci)T

are ortbogonal, then there exists an

ortbogonal matrix, QE Rrnxm, and a nonsingular

where ci E R lCix1 = [0 0 ... 0 l]T. The diagonal

diagonal matrix, SE Rrnxm, such that H=hQs.

sub-blocks of B and the diagonal elements of s are determined from the QR decomposition of the

Proof Define s such that s-IH-ICCTH-Ts-T = I.

submatrices of h-ICt. Let T=tB and H=hs. Then (T,H,k) is a valid transformation triple, since multiplication of a block diagonal matrix with a conformably partitioned block diagonal matrix is conformably block diagonal, T is ortbogonal, and H-1cr=block( ci)T. The matrix, TT(A-kC)T, is

Since the rows of H-IC are ortbogonal, H is nonsingular and C has full rank. s can be chosen to be real, nonsingular and diagonal. Thus CCT = hbT = Hs-2HT or SWlh = s-IHTh-T (h-IHs-I)-1 = (b-IHs-I)T

block diagonal, thus TT(A-kC)T = TTAT - TTkHH-Icr = TTAT - TTkH block( ci)T

Thus b-1Hs-I is ortbogonal since its inverse equals Let b-IHs-I=Q where Q its transpose. ortbogonal and real. Thus H=hQs.

is block diagonal. Given the structure of block( ci)T, this is only possible if TTAT is in the

is •

dual of Luenberger canonical form II (Luenberger, 1967), that is, the matrix is block diagonal except for the m columns corresponding to (Ji' iE m. If

The output transformation matrix, H, used to test for the existence of an ortbogonal state transformation, T, is constructed as follows . By

these columns are made zero, the resulting matrix will be block diagonal, so that the matrix TTAT - TTA[T(:,(Jl) , ... , T(:,(Jm)] block( ci)T

Theorem I, the condition that H-IC be ortbogonal is necessary for an ortbogonal T matrix to exist. Let h- I be a matrix for which the rows of C are

is block diagonal. Since H-Icr = block( Ci)T,

ortbonormal. By Lemma I, if H=h initially, then subsequent modifications of H will be limited to post-multiplication by Qs. Again by Theorem I, it is necessary that the observability subspaces be ortbogonal. Derme a family of real matrices parameterized by the integers i and j where i,j E {O,D-l} and consisting of elements of the form

H-IC= block( Ci)T TT = [T(:,(JI), ... ,T(:,(Jm)]T and. since T and H-Icr are ortbogonal, WICTTTCTH-T = H-ICCTWT = I or (CCTr I = H-TH-I Therefore TTAT - TTA[T(:,(JI)' ... , T(:,(Jm)] block( ci)T

h-IC(A-KC)i [b-IC(A-KC~]T = (AicTh-T)T WCTh-T)

= TTAT-TT ACTWT W Icr

Given the limitation on H. the observability subspaces will be ortbogonal if and only if all elements of this family can be simultaneously diagonalized using some ortbogonal matrix, Q (if N is an element of the family, then QTNQ is diagonal). If such a Q cannot be found, then by Theorem 1 an ortbogonal T does not exist. If Q exists. then by Theorems I and 2, an ortbogonal T exists and can be found with H=hQ and

= TT(A-ACTH-T H-IC)T = TT(A-ACT(CCT)-IC)T = TT(A-KC)T

are block diagonal and K=ACT(CCTrl. (T,H.K) is a valid transformation triple.

(2)

Thus •

Application of Theorem 1 is predicated on knowing an appropriate basis change on the space of outputs. An output transformation, H. can be constructed such that if block diagonalization is possible, it will be possible with this transformation. Construction of H is based on the

K=ACT(CcT)-I. The following propoSition has thus been proved.

SOl

PROPosmON I: If h- I orthonormalizes the

identified. These conditions require knowledge of an output transformation and a feedback gain. A direct means for calculating these matrices has been developed. Given the output transformation, the state transformation and the system converted to block diagonal form with diagonal blocks in Hessenberg form can be directly calculated. In this form standard techniques for pole placement, Ka1man filtering, etc., can be applied much more efficiently than in pure condensed forms.

rows of C and K=ACT(CCl)-1 then an orthogonal state transformation matrix, T, exists if and only if there exists an orthogonal matrix, Q, which simultaneously diagonalizes the family of matrices defmed by equation 2. For Q to exist the elements of the family of matrices must all be normal and real symmetric (Horn and Johnson, 1985). Given these necessary conditions it is possible to determine the existence of an orthogonal T without actually determining Q. In addition a simple method to determine Q exists if there exists one matrix in the family with m distinct eigenvalues. This will be the case, generically.

5. REFERENCES Aplevich, J.D. (1974). Direct computation of canonical forms for linear systems by elementary matrix operations. IEEE Trans. Automat. Contr., Vol. AC-19, pp. 124-126. Golub, G.H. and Van Loan, C. F. (1983). Matrix Computations, Baltimore: Johns Hopkios University Press. Horn, R.A and Johnson, C.A (1985). Matrix Analysis, New York: Cambridge University Press. Kailath, T. (1980). Linear Systems, Englewood Cliffs, New Jersey: Prentice-Hall. Kenney, C. and Laub, A.J. (1988). Controllability and stability radii for companion form systems. Math. Control Signals Systems, Vol. I, pp. 239-256. Laub, AJ. (1981). Efficient Multivariable Frequency Response Computations. IEEE Trans. Automat. Contr., Vol. AC-26, pp. 407-408. Luenberger, D.G. (1967). Canonical forms for linear multi variable systems. IEEE Trans. Automat. Contr. , Vol. AC-12, pp. 290-293. Paige, C.C. (1981). Properties of numerical algorithms related to computing controllability. IEEE Trans. Automat. Contr., Vol. AC-26, pp 130-138. Rhodes, I.B. (1990). A parallel decomposition for Ka1man filters. IEEE Trans. Automat. Contr., Vol. AC-35, pp. 322-324. Van Dooren, P.M. and Verhaegen, M. (1985). 00 the use of unitary state-space transformations. Contemporary Mathematics, Vol. 47, pp. 447-463. Warren, M.E. and Eckberg, Jr., AE. (1975). On the dimensions of controllability subspaces: a characterization via polynomial matrices and Kronecker invariants. SIAM J. Control, Vol. 13, pp. 434-445. Wilkinson, J.H. (1965). The Algebraic Eigenvalue Problem, Oxford: Clarendon Press. Wooham, W.M. and Morse, A.S. (1972). Feedback invariants of linear multivariable systems. Auromatica, Vol. 8, pp. 93-100.

COROLLARY 1: An orthogonal state transformation matrix exists if and only if the family of matrices defined by equation 2 is a commuting family of real symmetric matrices. Proof. This is a direct consequence of the properties of a commuting family of matrices (Horn and Johnson, 1985). •

1JIEOREM 3: If I) the elements of the family of matrices defined by equation 2 are symmetric, 2) there exists an element of the family, N, which has m unique eigenvalues, 3) Q is a real orthogonal matrix which diagonalizes N, and 4) the observability subspaces generated with H=hQ and K=ACT(CCTr l are orthogonal, then an orthogonal state transformation matrix exists. Proof. A real matrix, Q, diagonalizes a real and symmetric matrix, N, if and only if the columns of Q are eigenvectors of N, since QTNQ=D where 0 is a diagonal matrix of eigenvalues of N if and only if NQ=QD which, in turn, holds if and only if the columns of Q are eigenvectors of N. Given m distinct eigenvalues, the diagooalizing matrix Q is unique modulo a permutation matrix with plus or minus one entries. Thus, by Theorem I, either this Q can diagonalize all members of the family (which implies the observability subspaces are orthogonal) and an orthogonal T exists, or a Q which diagonalizes all members of the family fails to exist and thus an orthogonal T does not exist. •

4. CONCLUSIONS Necessary and sufficient conditions for which a system, (C,A), can be block diagonalized by an orthogonal state transformation assuming C is full rank and (C,A) is completely observable have been

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