Synthesis of Nonregular Control Laws

Copyright © IFAC System Structure and Control, Nantes, France, 1998

SYNTHESIS OF NONREGULAR CONTROL LAWS Sabine Mondie· Jean J acques Loiseau··

• CINVESTAV-IPN, Sec. Control Automatico, AP 14- 740, Mexico 07000, MEXICO. Tel.: 747-7000. Fax: 747-7002. E-mail: [email protected] •• IRCyN, UMR CNRS 6597, 1 rue de la Noe, BP 92101, 44321 NANTES, Cedex 03, FRANCE. Fax: (33)240372522. E-mail.' loiseau@lan .ec-nantes·fr.

Abstract : The construction of the static state feedback that assigns to a linear system described by the state equation, a given closed loop structure, is presented. In addition , the parametrization of all control laws assigning a given structure is given. Copyright ©1998 IFAC Resume: La construction du retour d'etat statique qui perm et de placer la structure d'un systeme lineaire decrit par l'equation d'etat est donnee. De plus, on obtient une parametrisation de toutes les lois de retour d 'etat statique qui realisent le placement a une structure prescrite. Keywords: Linear sytems, Static state feedback , Structure assignment.

When G is invertible, the feedback law is said to be regular , and the structure of the system remains unchanged. When G is not invertible, the feedback is said to be nonregular and the structure of the system is modified.

1. INTRODUCTION

The study of the structure of linear systems described by

x= Ax+Bu,

(1)

Necessary and sufficient conditions that describe the possible modifications of the structure in term of the invariants of the open and closed loop systems were obtained in Mondie (1996), and Mondie and Loiseau (1997a) . This problem belongs to the family of structure assignment . It generalizes the invariant factors assignment problems tackled in the work of Sa. (1979) and Thompson (1979) and the results on controllability indices assignment of Heymann (1976) which were revisited by Loiseau and Zagalak (1994). This problem is also a generalization of the famous Rosenbrock Control Structure Theorem (1970) which determines how controllability indices limits the choice of the degrees of the pole structure of a system . Finally, notice that it is equivalent to the problem

IR nxn ,

where A E B E IRn x (m+k), is an important part of the algebraic theory of linear systems. In particular, a list of integers, that are the controllability indices of system (1) , and a list of polynomials, the invariant polynomials of the uncontrollable part of system (1), characterize the fundamental behavior of the system. When a static state feedback, described by u = Fx+Gv,

(2)

where F E IR(m+k)xn and G E IR(m+k)x m , applied, the closed loop system is given by

x= (A + BF)x + BGv .

IS

(3) 703

where

of matrix pencil completion solved by Bargaiia and Zaballa (1988).

lj = min{i I C~-i+1 < cd ,

The sufficiency proof of the result, which is presented in full details in Mondie (1996), is constructive, and relies on the so called polynomial approach . It is possible to derive from this proof an algorithmic procedure for the design of the static feedback that assigns a given closed loop structure.

.

err = and

/3{ = lcm (ai, a~_i+I<) , for i = 1, ... , q + j, and j = 0, ... , k.

In Section 2 the necessary and sufficient conditions for the existence of the desired feedback are presented and the key result concerning the equivalence of the feedback assignment problem to a polynomial matrix completion problem is stated . In Section 3, the algorithmic procedure for the design of the desired control law is presented and the parametrization of all static state feedback assigning a given structure is given. The construction of the polynomial matrix with prescribed submatrices, a key point of the construction , is intricate and , for the sake of clarity of the presentation , is presented in the appendix.

The synthesis relies mainly on the following result that establishes the equivalence of the problem of nonregular assignment with a problem of existence of polynomial matrices and submatrices with prescribed properties.

Proposition 2. (Mondie , 1996, Mondie and Loiseau, 1997a) Given a pair (A, B) E IRn x( n+m+k) with controllability indices Cl ~ C2 ~ . . . ~ Cm+k > 0 and with invariant factors 0'1 (s) ~ a2(s) ~ ... ~ aq(s) , and given a list of integers c~ ~ C2 ~ .. . ~ c~ > 0 and of monic polynomials Q~ (s) ~ a2(s) ~ ... ~ a~+I«s), there exist matrices F E IR(m+l<)xn and G E IR(m+l<)xm with Rank (G) = rn, such that c~,c2 " " ' c~ and a~ (s) , 0'2 (s) , .. . , a~+1< (s) are respectively the controllability indices and the invariant factors of the closed loop system (A+BF, BG), if and only if there exist polynomial matrices X (s) E IR m X I< [s], Y(s) E IRI
2. PREVIOUS RESULTS In this Section the necessary and sufficient conditions for the existence of a nonregular state feedback that assigns a given structure are presented . Two polynomials a(s) and /3(s) being given, lcm (a , {3) denotes their least common divisor, and a ~ /3 means that a divides /3 without remainder.

[

Theorem 1. (Mondie, 1996, Mondie and Loiseau, 1997a) Consider a pair (A, B) E IRnx (n+m+l<) with controllability indices Cl ~ C2 ~ • .. ~ Cm+k > 0 and with invariant factors of the noncontrollable part a1(s) ~ a2(s) ~ ... ~ aq(s) , and consider lists of integers c~ ~ c2 ~ .. . ~ c~ > o and of monic polynomials a~ (s) ~ 0'2 (s) ~ ... ~ a~+1< (s) . There exists a nonregular static state feedback law FE IR(m+l<)xn and G E IR(m+l<)xm with Rank (G) = rn, such that ~,C2"'" c~ and a~ (s), 0'2 (s), . .. , a~+1< (s) are respectively the controllability indices and the invariant factors of the closed loop system (A + BF, BG) if and only if I < ai+1< _ ai < _ aiI

CHI<

~ C~ ,i Ij-j

,t•

= 1, ..., q,

~

[

1

L

i=l

m

Ci

= L>~ i=l

,

,

Y(S)

o

Z(s)

]

diag {ai(s)}{=l

OF NON REGULAR CONTROL LAWS It is now possible to present an algorithmic procedure for the synthesis of nonregular control laws that assigns a prescribed structure. The parametrization of such control laws is also derived.

Step 1. Compute the structure of the open loop system. Compute the regular feedback law and transformations (Fo , Go, To) that lead to the generalized Brunowski form. Augment the system so that it is controllable and let N(s), D(s) be a Normal External Description defined in Kucera and Zagalak (1991) , for the augmented system .

(6)

I<

+ Ldegui

Y(s)

3. SYNTHESIS AND PARAMETRIZATION

for j = 1, .. . , k, and

m+1<

o

has invariant factors a~ (s), a2(s),··· , a~+q(s) .

(5)

"c· <" c' + L deg u~ , l-L..J i=l i=l<-j+1 i=l

diag {se:} X (s) ]

once column reduced , has column degrees Cl , C2 , . . " Cm+1< and, in addition , the matrix

(4)

= 1, .. . ,rn, I<

/3{/3{ .. /3~+j . . 1 j 1 j 1 ,J = 1, ... , k, /3i /32 ···/3q+j-1

(7)

i=l 704

Step 2. Chose the closed loop structure to be assigned, and make sure that it verifies the necessary and sufficient conditions of Theorem l.

Suppose that we want the closed loop system to have {cD = {4} as controllability index and {Q~(s) , Q~(s)} = {S3,Sl} as invariant factors.

Step 3. Construct the polynomial matrices W(s) and W (s) as in Lemma 4 of the appendix.

Notice that

Step 4. Solve the Diophantine equation Y N(s) + X D(s) = W(s) . Because of the properties of W(s) this equation has a constant solution (X, Y) with X invertible as shown in Kucera and Zagalak (1991) .

The necessary and sufficient conditions of Theorem 1 hold. In this case there is only one step . The path described in the appendix is given by {cn = {cD = {4} and {cL cD = {Cl, C2} = {3, 3}, and one successively obtains

Step 5. Compute a nonregular linear control law for the original system as

OL = c~ - c~ = 1 ,

[Im+k 0 1X-1YTo- l

Fl

=Fo -

Gl

= GOX- l [IQ ]

B(S)=[)l~]

Step 6. The set of all nonregular static state feedback laws that assign the same structure is parametrized as

U(s) = [

~1 ~]

, ,

F=Fl +GFi

G=G1G;

and

mxn

where F; E IR and G; E IR mxm are free parameters that can be arbitrarily choosen to meet further requirements such as prescribed pole locations or other design specifications. Remark 3. Notice that the computations involved in these steps as well as those of the construction of matrix W (s) can be performed using standard matricial and polynomial computational tools.

The solution of the Diophantine equation of step 4 leads to 4. EXAMPLE

G=

A simple example is presented, which illustrates the procedure. Consider a pair (A, B) whose controllability indices are {Cl, C2} = {3 , 3} and with a non unit invariant factor {Ql(S)} = {S2}, and assume, with no loss of generality that the system is in generalized Brunovski form. The Normal external description for the augmented system is

D(s) =

N(s) =

[00000000] 0 0 -1 0 0 0 0 -1 '

5. CONCLUSION

S3 0 s3 0

o

,F =

and one can verify that this static state feedback assigns as prescribed the structure of the closed loop system.

0 0]

[

[~1]

In this paper, the algorithmic procedure for the synthesis of nonregular control laws that assign a given structure to a system described by the state equation is presented.

0 s2

1 o0 s o0 s2 o 0 0 1 0 0 s 0 o s2 0 o01 o 0 s

The procedure presented is not straightforward, since it is based on a polynomial formulation of the problem. However, all computations can be performed with available matricial and polynomial tools. The parametrization of all feedback laws that assign such a structure is given as well .

705

This parametrization allows the designer to meet other control objectives.

once column reduced, has controllability indices Cl, C2, .. " Cm+k, are constructed. Second, a matrix Z' (s) such that

Further work include the generalization of this design procedure to the case of right invertible systems, for which necessary and sufficient conditions were reported in Mondie and Loiseau (1997) and to use the parametrization in order to fullfil additional requirements.

[

diag{o-{(s)}:=l . Z'(s) m o dlag {ai(s)}i=l

has invariant factors a~ (s), 0; (s), . . . , a~+k (s), is designed. Finally Z(s) such that

Y(S) [ o

The construction of the polynomial matrices W (s) and W(s), that is crucial for the feedback synthesis, is technical, and for this reason, is presented in the appendix. Note that the complexity of the construction is only a consequence of the difficulty of the problem : a three by three block matrix where four lists of invariants, that are either integers or polynomials, and fullfil the non evident conditions of Theorem 1, are involved. However, it is clear that the conditions are checkable, and that the construction can be readily implemented on a computer.

Step 1 Obtention of a path of indices and invariant factors. Let

j3{j3{· ·j3~+i

crt 1 -

where, for i

j3i-lj3i-l 1

diag { sc:} X (s)

0 Y(s) Z(s) 0 diag {ai(s)}

[

o

B(s) [ diago{sC:}

~~:n U(s) =

2

...

j3i- l ,j = 1, ... ,k, q+i-l

= 1 to q + j, and j = 0 to k, j3{ = lcm{ai,a:_i+k} .

Consider lists of integers Cl ~ C2 2: . . . ~ Cm+k > 0 , c~ ~ C; ~ ... ~ C;" > 0, and lists ofmonic polynomials al(s) ~ a2(s) ~ ... ~ aq(s), a~ (s) ~ 0; (s) ~ .. . 2: a~+k (s) such that the conditions of Theorem 1 hold. Then a polynomial matrix W (s) exists such that

0

]

Notice that the operations that are performed in the following construction do make sense since the necessary and sufficient conditions of Theorem 1 hold .

4.

W(s) =

Z(s) diag{ai(s)}~l

has invariant factors a~ (s), a;(s),- . " a~+k(s), is obtained, which completes the construction.

6. APPENDIX

Lemma

]

The construction starts with the initial values

{Cn~l = {Ca~l , and b?l:=l = {u~-i+l }:=1 . Hence, {cD~11, bn7~11, and, are defined as follows. Let

1 ,

l~ = min{ i I c?

diag{sCi} ,

< c;} .

If

the matrix

Y(S)

[ o

Z(s)

then let

]

diag {ai(s)}~l

1,,/ = min{i I c?

has invariant factors a~ (s), 0; (s), "', a~+k (s), and Y(s) has invariant factors uHs), ui(s), ... ,

and define

ct =c? ,i = 1, ... , 1,,/ -

u~(s).

cl

"Y

The polynomial matrix needed in the construction of Section 3 is then

W(s) = W(s)

o

Y(s)

= deg(,2)

and

, =,2 , ,f =,? ,i = 1, ... ,k-l.

Construction. The construction is done in three steps. First, the polynomial matrices X(s) and Y (s) such that the matrix [

1,

Ci=C?_I,i=I,,/+1, ... ,m+1,

[U-~(s) ~]

diag {S<}~l X(S)]

< deg(J2)},

If 1~-1

deg(J2)

,

< Cl~ ~

deg(2)

+ L (c? i=1

706

c;) ,

S8 1,I

then let

-1 s8 2 , 2

I~ -1

I C,k 2:

r = min{j

d(-Y2)

2: (c? -

+

cd} , Uj,o(s) =

i=r+l

I

o1

0

-1

0

". SIl'-I,'-1

0

and define (remarking that 1

0

< r < l~ - 1)

c; = c? ,i = I, ... , r -

o

-1

0

0 I 0

0

I,

0

=

BI ,j(s)

I~ -1

c; = c~ - CI~

2: (c? -

+

cd

+ deg(-Y2)

,

1

0

m2 ,1

1

m2 ,1

m3 ,2

i=r+l

c; = Ci , i = r + 1, ... , l~ Cik I

1 Ci

=

1,

Clk ,

o0

I

1

= c;0_ 1 , Z. = [k1 + 1 , ..., m + 1.

0

0

o0 oI

ml ,1-2

ml ,1-1

10

1

ml-l,1 ml-I ,1 ... ml-I ,I-2

The mixed path of indices and invariant factors {S~}m+j ;=1 , {_J}k-j 1i i=I' cLor J. -- 0 , ... , k , 'IS t h en d efi ne d are constructed recursively. {c{} ~ij and {i} from {c{-I}~ij-l and {i}7;/+l in the same manner.

7;/

0

0

ml , I

ml ,2

. ..

with mi ,i = s-ci+ Ci - 8i , and

-p{ (s)')'N

1

-~(S)-yN

1

Construction . Let t

eft =

1

2: et-I - <{ ,

I

k=i

-pfi-l (S)-yN 0 IN

for t 2: i , where

with IN := ILi+I(S)/sdeghLHI) .

[j

= min{i I c{-1 < c{}.

The matrices X(s), Y(s), B(s), and U(s) are finally defined as i

1 1 81 1_81 cl. _8 . Notice that sC I , sC 2- 1 , 1 , sC 3 1,2, . .. , S 11-1 1,11-2, are polynomials. Therefore, it is always possible to perform the following euclidean division

B(s) = Bds)B2(S) ... Bk-I(S)Bk(S) U(s) = Uk(S)Uk_1 (s). ·.u2(s)uds) , and

i

.

0

.

+ ri (s) sc~-8{'1 = ~(s)-y2-Hl (s) + ~(s) sC I = Qi(S)-Yk-j+l (s)

~.

S

1 " i- 2

= [ diag {sc: }~l X(S)]

o

.

-~.

'1-1

B(s) [diag {sCi}~1k] U(s)

= qfi-l (s)-Y2- j +l (s) + rfi-l (s) .

Consider the strictly proper terms

1'1 (s) = r1 (s)hLHl (s)

,i

Y(s)'

Step 2 It is now possible to build

Z'~s)

as follows

p{ (s)

= I, .. ., Ij -

1.

Define

n n ... 01

and

o

.( ) _ [Uj,o(S) 0 ] UJS 0 I k-J· ' where

where

707

o

r~+l

One can verify that this matrix, combined with the one obtained in Step 1, has the desired properties.

REFERENCES Baragaiia, I. and I. Zaballa (1990). Columncompletion of a pair of matrices. Linear and

and

multilinear algebra, 27, 243-273 .

Gantmacher, F.R. (1974). Matrix Theory, Vol. I and 11. Chelsea, New York . Heymann, M. (1976). Controllability subspaces and feedback simulation . SIAM J. Control and Optimization, 14, 769-789 . Kucera, V. and P. Zagalak (1991). Constant solutions of polynomial equations. International

for i = 1, ... ,q+j, and j = 1, ... , k. Notice that j ~ . - 1 -- 0-1 ~-i+l r ji ' r 1 -- 0-1,J - , ... , k , an d rj-i+l 1 :::; j 1, ... , k, i 1, ... , j. Hence r{ , j 2, ... , k, i 2, ... , j can be canceled through unimodular row operations and D( s) is equivalent to diag {a-{ }J=I ' . . -+1 - . With o-i :::; o-i . Then o-i ,J 1, ... , k are the invariant factors of D(s) . Let U(s) be the matrix

=

=

=

=

J. Control, 53 , 495-502 .

=

rk

rkk

- 0"1

ranean Symposium on New Directions in Control Theory and Applications, Chania, Crete,

r 32

r 33

Greece. Marques de Sa, E. (1979). Imbedding conditions for >.- matrices. Linear Algebra and its

1

r~2

Mondie, S. (1996). Contribution to the study

1 _ _2_ k-l 0"1

o U(s)

Loiseau, J.J. and P. Zagalak (1994). On feedback simulation, In Proc. 2nd IEEE Mediter-

1

1

=

- 0"2 - 0"1

o

1

- 0"1

0

1

Applications, 24, 33-50 . of structural modifications of linear systems,

1

PhD Thesis, Laboratoire d' Automatique de Nantes, Universite de Nantes, France, and CINVESTAV-IPN , Mexico. Mondie, S. and J.J . Loiseau (1997). Structure assignment of right invertible implicit systems through nonregular static state feedback . In Proc. European Control Conj., Brussels, Belgium _ Mondie, S. and J.J. Loiseau (1997a). Simultaneous zeros and controllability indices assignment through nonregular static state feedback. In Proc. 36th Conference on Decision and Control, San Diego, California. Thompson, R.e. (1979). Interlacing inequalities for invariant factors . Linear Algebra and its Applications, 24, 1-31. Zaballa, I. (1988). Interlacing inequalities and control theory. Linear Algebra and its Applications, 101 , 9-31.

then

[

U(S)

o

0]

Iq

[D(S)

Q(s)

]

diag {o;} J=1

0

= [diag{a-{(s)}~=1 o

. Z'(s) q ] . dlag{O;}j=1

Step 3 From step 1, Y (s) has invariant factors a-{ (s), j = 1, . .. , k _ Hence there exist unimodular matrices UJ(s) and U2 (s) such that UJ(s) diag {a-{ (S)}j=1 U2(S) = Y(s) , therefore

[

Ul(S) 0] [diag{a-{(s)}~=1

o

U2(S)

x [

0

Iq

0

0] _-

Iq

[Y(s)

0

. Z'(s) q dlag{o;}j=1 Z(s)

diag{o;}J=1

]

]

.

708