Output Maximal Dimension for the Disturbance Decoupling Problem

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OUTPUT MAXIMAL DIMENSION FOR THE DISTURBANCE DECOUPLlNG PROBLEM F. Puerta .,1,4 X. Puerta . ,2 ,4 I. Zaballa •• ,3,5

• Departament de Matemdtica Aplicada I, Universitat Politecnica de Catalunya, Av. Diagonal 647, 08028 Barcelona, Spain . •• Departamento de Matematica Aplicada y EIO. Universidad del Pais Vas co, Apartado 640, 48080 Bilbao, Spain.

Abstract: Given a linear time invariant system with a disturbance we describe a method of finding the maximal dimensional output for a generic Disturbance Decoupling Problem. Copyright ©2001 IFAC Keywords: Linear system. Brunovsky canonical form. Thorn transversality theorem. Disturbance decoupling problem.

1. INTRODUCTION

A solution of this problem can be found for example in [5J , ThA.2, where the following result is proved: the DDP has a solution if and only if

Given the control linear system x(t)

= Ax(t) + Bu(t) + Qq(t)

yet)

= Cx(t)

!D(A, B ; Ker C) :> Im Q

where !D(A , B ; Ker C) is the unique maximal (A, B) -invariant subspace contained in Ker C. We recall that a subspace S is said to be (A, B)invariant if A(S) C S + Im B.

the term q(t) represents a disturbance which is assumed not to be directly measurable by the controller. Then, the Disturbance Decoupling Problem (DDP) consists in finding (if possible) a state feedback F, such that the disturbance q(t) has no influence on the output. That is to say, F must be such that for any initial state the corresponding output yet) of the system x(t)

In this note we tackle the problem of shOwing a method of finding, for a generic Q the maximal dimension p of the output space insensitive to the disturbance q(t) . We will call briefly this problem "A maximal solution of the DDP" . In [4] is showed through an example a method of solving this problem when the pair (A, B) is controllable. In this note we solve this problem when (A, B) is a general pair. For this we make use of the bundle structure of the set of conditioned invariant subspaces given in [2J.

= (A + BF)x(t) + Bu(t) + Qq(t)

yet) = Cx(t)

does not depends on q(t).

We fix the following notation. Mp,q will denote the set of complex matrices having p rows and q columns. We write Mp for Mp ,p' If A E Mp we say that A is a p-matrix. M;,q is the set of full rank matrices in Mp ,q'

e-mail: puertaGmal . upc . es e-mail: collGmal.upc.es 3 e-mail: i~aballaOpicasso.lc.ehu . es 4 Partially supported by the Direcci6n General de lnvestigaci6n Cientifica y Tecnica, Proyecto de Investigaci6n PB97-0599-C03-03 5 Partially supported by the Direcci6n General de Investigaci6n Cientifica y Tecnica, Proyecto de Investigaci6n PB97-0599-C03-01 1

2

If A E Mp,q, A' means the conjugate transpose of A. In will be the identity matrix. If A E Mp ,q,

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but we will keep on with the same notation for these new matrices. Then,

[A] means the subspace generated by the columns of A. If E is a vector space, Grd(E) is the Grassman manifold of d-dimensional subspaces of E.

A Brunovsky (dual) matrix, B -matrix in what follows , is a matrix of the form (i) where N = diag {No , N oo }, E = diag {Eo,O}, being Noc a Jordan matrix and (Eo , No) a Brunovsky observable pair, that is to say, No = diag {NI, ... , N r } each Ni being the standard lower nilpotent k i matrix, Eo = diag {El, ' . . , Er} , each Ei being a ki-row matrix, Ei = (0, .. . ,0,1), 1::; i::; r. The integers kl ' ... , kr are called the observability indices of (i).

2. A MAXIMAL SOLUTION OF THE DDP

(2.1) Consider a linear system ±(t)

yet)

= A.x(t) + Bu(t) + Qq(t) = Cx(t)

where A E M n , B E Mn ,m, Q E Mn ,q and C E Mp,n' With the notation in the introduction, our problem can be stated as follows: Find, for a generic Q, the maximal dimension of lm C so that !D(A, B; Ker C) :::> lm Q.

c !D(A,B;Ker C) c

=

[

In-q P

1

Call this subspace F . (ii) Let (i) be the Brunovsky (dual) form of the pair (~:) (see for example [3] (6.6.3», (~) a B-matrix and Inv «B·, A·); (M, F» the set of (B· A·) -conditioned invariant subspaces W such that the restriction of (~:) to W has a fixed Brunovsky form (~) (see [2] for the definition of this restriction). We denote by k = (kl 2: k2 2: . .. 2: k r ) the observability indices of (i) and for everyeigenvalue A of N oo , 171(A) 2: 1]2(A) 2: .. . is the corresponding Segre characteristic. Analogously, !J:. = (h l 2: h2 2: ... 2: h s ) and Cl(A) 2: c2(A) 2: ... are respectively the observability indices of (~) and the Segre characteristic of A as eigenvalue of Moo. We assume that (~) is compatible with (i), that is to say that Inv «B·, A*); (M, F» is not empty. We recall that a B -matrix (~) is compatible with (i) if and only if the following conditions hold (see for example

[1]): (a) S < r, and hi::; k i for i = 0, 1,2, . . . (b) the eigenvalues of Moo are also eigenvalues of N 00, and for each one the corresponging Segre characteristics (1]1 (A), 1]2 (A), ... ) and (Cl (A), C2 (A), ... ) verify: 1]i (A) ::; c;(A), for i = 1,2, ... . In [2] (4.6) (see also [4] (2.9» the following formula is obtained:

Taking into account that Im Q

(lm Q)1.

Ker C,

finding the maximal dimension for Im C is equivalent to obtain the minimal dimension for Ker C so that Ker C is an (A, B)-invariant subspace containing Im Q. In turns, this is equivalent to finding the minimal dimension of K er C such that (Ker C) 1. is a (B·, A·) -invariant subspace of maximal dimension contained in (lm Q)1.. We recall that a subspace 5 is said to be (B·, A·)conditioned invariant (or (~:) -invariant as in [3]) if 51. is (A,B)-invariant.

dim«B*, A"); (M, F» h

=L

=

k

L mi(rHj-l - sHj-d+ + L L ei(A)(e;(A) - ei(A» + (k oo i=1 j=1

hoo)s

oX

where, mi = Si - Si+!, r. = (rl 2: ... 2: rk) is the conjugate partition of k = (kl 2: ... 2: k r ), ~ = (s 2: ... 2: S h) is the conjugate partition of !J:. = (h l 2: ... 2: h s ), A runs over the eigenvalues of Moo, Cl (A) 2: C2(A) 2: ... , el(A) 2: e2(A) 2: ... are the conjugate partitions of the Segre characteristics 1]1 (A) 2: 1]2(A) 2: ... , cl(A) 2: C2(A) 2: ... of A in Noo and Moo, respectively, and koo, hoo are the size of Noo and Moo, respectively. Set

We are going to describe a method of obtaining this dimension. We will divide our procedure in several steps: (i) Since the matrix Q is supposed to be generic, that is to say, belonging to an open and dense subset of Mn,q we can assume that Q is a full rank matrix. Then, through a convenient change of bases, Q takes the form

dim«B·, A*); (M, F»

= 8.

(iii) Now, we have to find, for the Q given in (i), the maximal dimension d so that there exists ad-dimensional (B· A·) -conditioned

where P E Mq,n-q. Matrices A, B and C modify accordingly to this change of bases,

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invariant subspace contained in F. This leads to consider the intersection Grd(F)

n Inv

in (2.1), for any set of positive integers pairwise h, i j mh-i+d such different {nij, 1 i that for i = 1,2, ... , h

s: s:

((B", A") ; (M, F»

for all possible ('~) compatibles with (~) such that ME A1 d . dim Grd(F) + <5 < dim Grd('Cn) the intersection Grd(F) n Inv ((B", A"); (M, F» will be empty for Q generic. So, we have to look for the greatest d such that

+ <5 2

s: s:

02 dq.

s: s:

Then, we have the following result Theorem 1. With the above notation, the maximal dimension of the (B", A")-conditioned invariant subspaces contained in (lm Q).L is the greatest d such that 0 2 dq and the set of such subspaces is a manifold of dimension <5 - dq. Besides of this, the Brunovsky reduced form of the restriction of (B' A*) to these subspaces is of the same type of (~) (that is to say, they have the same observability indices and for each eigenvalue of the Jordan part the Segre characteristic is the same).

s: s:

ef,

(j

e1 = (0, ... ,0,1,0, ... ,0) E (IF)ffi 4. For i

>

h

- I3+1 •

j, Yij is partitioned into blocks (L~~j+l), 1 {3 h - j + 1 in i-j+l M ' h h L suc a way t at i,B E T"mh-t!+l IS a matrix whose last Tj - Th-,B+i-j+l rows are zero and for {3 2 i - j + 1, the rows n pq , 1 p {3 - i + j, 1 q mh-p+l are also zero. (Notice that Yij follows from Yi-j+l,l)' Yij

=

s: s:

Example 2. We are going to apply this theorem (5,2), so that to the following situation: is. r. = (2 2 1 1 1), c = 3 and q = 2. We begin for the possible maximal dimension: d = 8. In this case the only possible cases are:

s:

s:

s: s:

IT Y is such a matrix, the coordinate chart is simply given by the mapping

=

and

s:

1. Yij E MT.,sj .

den - d)

(22 1 1),2;

s:

2. Yij = 0 if i < j . 3. For i = 1,2, ... , h , Yii is partitioned into blocks Yii = (Lt,B) , 1 ,B h - i + 1 in such a way that for {3 = 1,2, ... , h - i + 1, L},B E M T., ffi h_J3+1 is a matrix whose last Ti - Th-,B+l rows are zero, the rows nij, 1 i {3 - 1, 1 j mh-i+l are also zero, and the rows n,Bl, n,B2, ... ,n,B ffi h_i3+1 are unit vectors eg , ... ,e~h_i3+1 :

for some compatible (~), or equivalently

(22 1 1 1), 1;

s:

if Y (Yij) , 1 i k, 1 j h is a matrix verifying the conditions below, then the free parameters in Y define a coordinate system in Inv ((B", A."); (M, F». The conditions are the following:

We know by Thorn Transversality theorem that if

den - q - d)

s:

=

s: s:

1jJ: (Y ---+ Inv ((B*,A'); (M,F»

defined by 1jJ (free parameters in Y) = [Y].

(22),3,

Then, taking into account that F

for §. and e, respectively, and it easily checked that in any case is 0 < 15. Continuing this way we see that for d = 4 and s = (2,2), e = 0 one has that 0 = 8. Hence, in this example the maximal solution of the DDP is obtained for d = 4. Notice that if s = (2,1), e = 1, then 0 = 10. That is to say, the maximal dimension is unique but not the Brunovsky form of the restriction to the corresponding invariant subspaces.

=

~-.], it

turns out that a subspace S belongs to Grd(F) nInv (B*,A');(M,F», where we recall that d and (M, F) have been obtained in (ill), if and only if there exists X E M~_q , d such that

So, if we write Y = (~~) where Y1 E Mn-q,d, then X = Y1 and

Y2 -PY =0.

This is a linear system of q . d equations with Hence, for Q generic the solutions depend on (0 - q) . 0 free parameters, which is just the dimension of the manifold Grd(F) n Inv((B',A*);(M,F», so we have proved the following result:

(2.2) It is also possible to give a method for obtaining a parametrization of the manifold

o unknowns.

Grd(F) nInv ((B*,A*);(M,F».

However, only when the pair (A, B) is controllable, or equivalently (~:) is observable, the procedure can be given in an explicit way. So we limit ourselves to consider this case. We will use the explicit description of a coordinate system given in [4] (3.3). There it is shown that, with the notation

Proposition 3. With the above notation, the mapto each set of ping which assigns (8 -q) ·d free parameters in the linear system Y2 -

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PY1 = 0 the subspace [Y] , where Y = (~~) is the

corresponding solution of this system, defines a coordinate chart in Grd (F)nInv ((EO, .4."); (M, F)) .

References

1. Baragaiia, 1. Zaballa, Block Similarity Invariants of Restrictions to (.4., E ) -Invariants Subspaces. Linear Algebra and Appl., (220): 31- 62, 1995. [2] J. Ferrer, F. Puerta, X. Puerta, Stratification of the set of general (.4., E) -invariant subspaces. Preprint (1999). [3] 1. Gohberg, P. Lancaster, L. Rodman, Invariant Subspaces of Matrices with Applications. Wiley, New York, 1986. [4] F. Puerta, X. Puerta, 1. Zaballa, .4. coordi[1]

[5]

nate atlas of the manifold of observable conditioned invariant subspaces. Preprint (2000). W.M. Wonham, Linear Multivariable Control: A Geometrical Approach. SpringerVerlag, New York, 1984.

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