Decoupling and pole-zero assignment of singular ... - Springer Link

Cl~c~xrrs SYSTEmsSlON~ PROCESS Vot. 5, No. l, i986

DECOUPLING AND POLE-ZERO ASSIGNMENT OF SINGULAR SYSTEMS WITH DYNAMIC STATE FEEDBACK* B. G. M e r t z i o s 1 a n d M . A . C h r i s t o d o u l o u 2

Abstract. This paper refers to the problem of designing a linear state feedback dynamic controller for single-input; single-output decoupling of linear, time-invariant, singular systems. Sufficient conditions are established for the state-feedback decoupling problem to have a solution. In the case where the system satisfies these conditions, the class of controller matrices which decouple the system is given. Finally a method is presented for polezero placement in the decoupled singular system and a structure is described for the realization of the generalized transfer function matrices. 1. Introduction

The problem of decoupling for regular state-space systems has been widely studied. The necessary and sufficient conditions for single-input, single-output decoupling o f these systems by state feedback were first derived by Falb and Wolovich [1]. Since then, there have been many important contributions in this direction, namely by Gilbert [2], W o n h a m and Morse [3], Morse and W o n h a m [4,5], Silverman [6], Howze and Pearson [7], Mufti [8], Howze [9]. In the sequel, other extensions have been done to various classes of systems, such as timevarying systems [10], non-linear systems [11], two-dimensional systems [12], etc. In this paper we study the problem of single-input, single-output decoupling of singular systems, i.e., systems of the form MR = A x + Bu, y = Cx, where M is a singular matrix. There has recently been a growing interest in the analysis and synthesis of singular systems [13-21]. The theory of singular systems has a variety of applications in composite systems, singularly perturbed systems and electric circuits [22-25].

* ReceivedJanuary 11, 1985;revisedMay 23, 1985. i Departmentof ElectricalEngineering, Schoolof Engineering,DemocritusUniversityof Thrace, Xanthi, Greece. Departmentof ComputerEngineering,Universityof Patras and ComputerTechnologyInstitute, Patras, Greece.

50

MERTZIOSAND CHRISTODOULOU

The sufficient conditions for the problem to have a solution are established in Section 3, in terms of the nonsingularity of a matrix B*, which has an analogous form to that of the regular state-space systems. In the case of singular systems, the matrix B* as well as the controller matrices F, G depend on the Laplace variable s. Thus, the controller is of dynamic form. In Section 5 a method is proposed for controlling the poles-zeros of the decoupled system. Finally in Section 6 a controller form realization of the generalized transfer functions is presented which m a y be used for the realization of the not strictly proper d y n a m i c controllers as well as o f the resulting d e c o u p l e d system. 3. Definitions and statement of the problem Consider the linear time-invariant singular system of the form MYc = A x

+ Bu

(la) (lb)

y = Cx

where x is an n-vector of internal state variables, u and y are m-vectors representing the input and the output respectively, M is an n x n singular matrix and A, B, C are constant matrices of appropriate dimensions. It is assumed that m _< n. The controller applied to the above system is of the linear dynamic state feedback type having the form u = Fx + Gv

(2)

where F and G are operators which correspond to the dynamic controllers. The substitution of (2) in (1) yields the closed loop system MYr = ( A + B F ) x

+ BGv

(3a) (3b)

y = Cx.

The transfer function matrix of the system (3) considering zero initial conditions is H(s) = C[Ms

-

(A + BF)]-IBG

(4)

where now F, G are m x n and m x m rational matrices in s respectively. The problem of decoupling the singular system (1) by state feedback is to choose the matrices F and G so that H ( s ) can be nonsingular and diagonal. In the following section we will impose these requirements in the transfer function matrix in order to derive the sufficient conditions for the problem to have a solution. To this end we need an explicit expression of H ( s ) in terms of the matrices M, A, B, C of the given system without the inversion of the polynomial matrix. To confront this problem we write the transfer matrix as

DECOUPLINGAND POLE-ZERO ASSIGNMENT 51 C [l I z -

H(s)

[Iz

-

Ms

+ (A + BF)]

]-1B G

(5)

where z is a new variable which does n o t affect H ( s ) since it is always eliminated. Relation (5) can be written in the following f o r m H(s)

=

C[Iz

-

(6)

P]-~BG

where P = Iz -

Ms

+ A

(7)

+ BF.

Now it can be readily seen that the Leverrier algorithm can be used to compute the inverse o f the matrix I z - P a n d therefore the t r a n s f e r f u n c t i o n matrix H ( s ) . Namely, H ( s ) can be e x p a n d e d as H(s)

= q - l ( s ) C [ z " - l R o + z ~ - 2 R , + ... + z R . - 2 + R ~ - , ] B G

(8)

where q(s) = z ~ -

qlz ....

q2z ~-2 . . . . .

q~ = det [Iz -

P]

and Ro = L,

ql = tr[P]

R1 = P R o -

qlI~,

q2 = 1/2 tr [ P R d

R2 = P R 1 -

q2I~,

q3 =

1/3 tr [ P R 2 ]

Rn-1 =

-

qn =

1/n

PRn-2

qn-~In,

The matrices R i , i -- 1 . . . . . n expression R~ = P ' -

q l P '-1 -

tr

(9)

[PRn-1]

1 c a n be also c o m p u t e d b y the following

q 2 P i-2 . . . . .

q,I,

i = 1 , 2 . . . . , n - 1.

(10)

The matrices R , , i = 0 . . . . . n - 1 are n o t the coefficient matrices o f the powers o f s b u t depend o n the variable s itself. This can be seen f r o m (9) since the m a t r i x P depends o n s. As (8) is i n d e p e n d e n t o f z, in the following for the sake of simplicity, we will take z = 1. Therefore H ( s ) can be written as H ( s ) = q - l ( s ) C [ R o + R1 + ... + R n _ I ] B G

(11)

Note that we m a y have a n infinte n u m b e r o f forms of H ( s ) d e p e n d i n g o n the specific value of the pseudo-variable z. Consider n o w the q-th row of H ( s ) H ~ ( s ) = q - l ( s ) C q [ R o + R1 + ... + R n - d B G

(12)

52

MERTZIOS AND CHRISTODOULOU

Let the integers dq for each o u t p u t q = 1,2 . . . . . m be given b y d~ = rain [i: C q . 4 ' B ;~ 0, i = 0, 1. . . . . n - 1]

(13a)

or d~ = n - 1 i f C ~ A ' B = O,

for all i

(13b)

where Cq denotes the q-th r o w o f C a n d .4 = I-

(14)

Ms + A.

T h e n f r o m (7) results t h a t P=.4

+ BF=

C ~ P k B = C~(.,~ + B F ) k B : O,

I-Ms

+ A + BF

k = 0,1 . . . . . d~ -

1 (15)

= Cq.4kB = B * ,

k = dq

= C q A n ~ ( A + BF)k-aqB,

k = dq + 1, dq + 2 . . . . .

n-1

T h e r e f o r e , t a k i n g (10) a n d (15) into a c c o u n t , we have t h a t the row H q ( s ) , given b y (12), b e c o m e s H~(s) = q-X(s)C~[R~q + R~+, + ... + R . - d B G = q-~(s)[Cq.4 ~q + [Cq.4~(.4 + B F ) +

+ BF:-

c' . . . . .

q,C~.4~q + ...

(16)

q.-a(~C~A q]BG

T h e expression in s q u a r e b r a q u e t s o n the right h a n d side o f (16) is a p o l y n o m i a l r o w o f degree n - 1. N o t e t h a t the degree o f the c o r r e s p o n d i n g p o l y n o m i a l r o w in the n o n s i n g u l a r system is n - d q - 1 . I n o r d e r to write H ( s ) as a s u m o,f terms similar to (16), we define

, A*(s) =

B*(s) =

cjdmB

(17)

Cm~dm

a n d the d i a g o n a l m a t r i c e s

Sj =

~

0

, j = 1. . . . . n - d , - 1

(18)

DECOUPLING AND POLE-ZERO ASSIGNMENT

53

where

pqj =

I

1 if n - d q - j - 1

> 0

0 if n - d q - j - 1

< O, q = 1,2 ..... m

and di = min [dq~ , q = 1,2 ..... m. The matrices B* (s),A* (s), in the sequel will be noted by B* , A* for simplicity. In terms of the above notation, H ( s ) can be written as H(s) = q-'(s)[B* + SI [A*(A + BF)B - q,B*] + ... + S.-di-,[A* [ a + BF]"-di-'B . . . . .

q.-a,-1B*I]G

(19)

and by rearranging the terms in the right-hand side of (19), we obtain H(s) = q=~(s)[[I - q,S, - q2S2 . . . . .

q.-a,-,S.-d,-,lB*

+ [S, - q,S2 . . . . q.-d,-,S.-d,-~l[A*(.,~ + BF)B] + ... + [S.-a,-2-q,S.-d,-,] [A*(A+ BF)"-d'-2B] + ts.

(20)

+

3. S o l u t i o n o f the problem

In this section the main theorem which gives the sufficient condition for the decoupling problem to have a solution is established. In the sequel F is considered to always be nonzero; otherwise state feedback does not exist. In the case where F = 0, then G = Hol(s)A, where A is a nonsingular diagonal matrix provided that the giv6n transfer function matrix Ho(s) is nonsingular. T h e o r e m 1. There is a pair o f d y n a m i c controllers F , G which decouple the

open loop singular system (1) when its transfer function matrix is nonsingular,

if detB* # 0

(21)

Proof. To prove the sufficiency of (21), suppose that B* is nonsingular; we will show that there is a matrix F that diagonalizes the remaining terms of (20). To this end we choose F = - (B*)-1A *.4

(22a)

G = (B*)-'

(22b)

The above formulas are of the same form, derived by Falb and Wolovich

54

MERTZIOS AND CHRISTODOULOU

(1967), with the difference that here F , G are dynamic controllers. Application of (22) gives A * ( A + BF) = A * / t + A * B F ^ = A*A

+ B'F=

(23)

0

In view of (23) we have A * ( A + BF) k = 0,

k = 1,2 . . . . . n - d i - 1 .

(24)

At this point we want to emphasize the fact that in the case where the transfer function matrix of the open loop system is nonsingular, then Theorem 1 gives a sufficient condition for decoupling. Even in this latter case Theorem 1 is useful since it leads to a decoupling procedure without being involved in the difficult task of inverting the rational in s matrix Ho(s). The Laplace transform of the closed-loop singular system (3) gives X(s) = ( A + BF)X(s) + BGV(s)

(25a)

Y(s) = CX(s)

(25b)

where A has been defined in (14) and X ( s ) , Y(s), V(s) are the Laplace transforms of x ( t ) , y ( t ) and v(t) respectively. Now taking into account (13) and (15), and using recursively (25), we obtain

(26) where Yq(s) denotes the q-th component of Y(s). The combination of relations (26) for all q = 1,2 . . . . . m yields V(s) = ( A * A

+ e*v)X(s)

(27)

+ B*GV(s)

Clearly the substitution of the rational polynomial matrices from (23) to (27) gives Y(s) = V(s)

i.e., in this case the transfer function of the decoupled closed loop system becomes the unit matrix L,. Example 1. Let a system be described as in (1), where 2 A=

-1

0 1

1 -1

0 1

0

,B=

1~ E 1

0 1

1 1

1

1

1

0

-1

,C=

, M=

100] 0 0 0

1

o 0

1

DECOUPLING AND POLE-ZERO ASSIGNMENT

55

For this system we have , d l = l, d2 = 0 B* =

C2B

1

2

1

and A* = I

-s+2

02

s1

Hence B* is nonsingular and the system can be decoupled. The application of (22) gives

F =

1

I s2-3s-1

s+l

s+3

s

-4

s2-2s- 1

-s2+ 3s-2

G

J

I E, 2 s+2 -2 l

(s+ 1)

As a check we have s ~- s + 1 A + BF=

1

2

s

-s 2+ 3s-2

s- 3

s2 - s

s-2

2

s+l

-1 BG=

1

s2-s-1

s+2

2

-2

1

s

2(s + 1)

and H ( s ) = C [ M s - (A + B F ) ] - I B G =

[,0] 0

= 12

1

Hence, the closed-loop system is decoupled. 4. Class o f decoupling matrices In this section we establish the sufficient conditions for the matrices F and G to be a decoupling pair under the assumption that the system (1) can be decoupled, i.e. the relation (21) holds. Since the class of appropriate matrices

56

MERTZIOS AND CHRISTODOULOU

G is given b y (22b), in the sequel we will c o n s i d e r the class o f matrices F . T h e m a t r i c e s F m u s t b e such t h a t the f o l l o w i n g relations h o l d

C,.4d'(.~ + BF)k-%BG = C,(.4 + BF)kBG (28)

= ak,(s)e,, k = d,, d,+ 1. . . . . n - 1 where e, = (0,0 . . . . . 0,1,0 . . . . . 0) with the unity element in the q - t h position. I f (28) holds for q = 1,2 . . . . . m, the rows H~(s), q = 1,2 . . . . . m in (16) are also o f the f o r m a~(s)eq a n d the m a t r i x H ( s ) b e c o m e s d i a g o n a l . The rows C~(A + B F ) k B , k = dq, d~ + 1 . . . . . n - 1 m u s t be o r t h o g o n a l to t h e m - l i n d e p e n d e n t c o l u m n s g l , i = 1 . . . . . q - 1, q + 1 . . . . . m o f G. H e n c e , all these rows C , ( A + B F ) k B , k = dq, d~ + 1 . . . . . n - 1 are p r o p o r t i o n a l to each other, o r zero for k = d~ + 1 . . . . . n - 1. In this latter case where the r o w C~(A + B F ) k B is r e d u c e d to zero, the c o r r e s p o n d i n g coefficients akq(S) in (28) are also zero. In view o f the a b o v e , t h e following c o r o l l a r y is established, which is a n a l o g o u s to t h a t o f F a l b a n d W o l o v i c h [1]. C o r o l l a r y 1. L e t Qq(F) be the n x m matrix

c~(.~ + BF)"-'B c,(A +

-

BF).-2B

Qq(F, s) =

q = 1,2 . . . . . m

(29)

c~(J + BF)dqB 0

where 0 is a dq • m zero matrix. Then there is a matrix F which decouples (1) i f the rank o f Qq(F,s) is one f o r all q, p r o v i d e d that B* is nonsingular. T h e d e t e r m i n a t i o n o f the c o n s t r a i n t s a m o n g the elements o f the f e e d b a c k matrices F f r o m C o r o l l a r y 1 is generally r e d u c e d to the s o l u t i o n o f a highly n o n l i n e a r system o f e q u a t i o n s . H o w e v e r a general f o r m u l a , similar to t h a t described b y F a l b a n d W o l o v i c h [1], o f the f o r m

(30)

m a y be used where 6 = max [dq~, q = 1,2 . . . . . m a n d Ek are d i a g o n a l m a t r i c e s

57

DECOUPLING AND POLE-ZERO ASSIGNMENT

e:

0 e~ (31)

E k =

0 e~ with (32)

e~ = 0, q = 1,2, ..., m for k = dq + 1, ..., /L

E q u a t i o n (30) is b a s e d on the e q u a t i o n

A*~ + n'F = E ek c L

(33)

k=O

which ensures t h a t C o r o l l a r y 1 holds. Specifically, the q - t h r o w o f (28) m a y be written as

C.Ad~(A + BF) = C~(A + BF)~'+~BF = ea~B'~

(34)

where (32) has been used. Using (15), (31), (32) we o b t a i n

[e~CqAkl (A + BF)'-1B k=O

dq e:Cq(

(35)

A 4- B F ) k + r - 1 B ,

k=dq-r+ 1

forr

= 1,2 . . . . . n - d ~ - l .

F r o m (35) it is seen t h a t all the rows C~(A + BF)dq+'B a r e a n a l o g o u s to B*q. T h e degreesmOf f r e e d o m in the selection o f F w h i c h are a s s o c i a t e d with the r = m + q r. dq elements o f E k ,. k = 0,, ..., 6, m a y be used in o r d e r to =l . . . achieve o t h e r design r e q m r e m e n t s m t h e resulting closed l o o p system, such a s p o l e p l a c e m e n t , stability a n d time response. E x a m p l e 2. Let a system be d e s c r i b e d as in (1), where

A=

[1 01 ii 01 i 0:1 I~ 0 0

1

-1

0

2

,B=

0

-1

1

0

, C=

,Ill=

1 0

1 0-1

0

0

0

58 MERTZIOSAND CRRISTODOULOU F o r this system we have

1 E 01 =

B* =

C2AB

1

,d,

=

0 , d2 =

1

1

Thus B* is nonsingular and therefore the system can be decoupled. The class of all the feedback matrices F which decouple the considered system are those for which rank Q ~ ( F , s ) = 1,q = 1,2. The equations which F must satisfy are f12 = - 1 . 5 f,x + f13 = S 4- 0.5 --fH + f,3 + f2x -f2~ + f = = 2S + 0.5 which constitute three constraints a m o n g the six polynomial elements f,j. The above constraints are satisfied when F is given by

[_f*, = [

F.,

=(n*)-'

F= f22

f23

CAk

k=O

0 . 5 s - 1 + 0.5e~

- 1.5

L -s2-~5.5s-5+O.5e~+e](2-s)

s - 4 . 5 +e~

0.5s+ 1.5 - 0.5e~

]

- s~ + 4 . 5 s - 7.5 - O.Se~ + e~ + eff - s + 3)

J

where el = 0 according to (32) and e~, e~, e] are unspecified polynomial quantities, as was expected. If we select e~=s+3 = 3.5s - 4.5 = - s - 4.5 F becomes I s+0"5

F=

1.5

3s+ 1

0

0 1 0

and the transfer function matrix o f the closed loop system is H(s) = C[Ms - (A + BF)]-~BG

1 2.5s2 + 4 s - 2

[ -2"5s+1 0

0 -s+2

]

DECOUPLINGAND POLE-ZEROASSIGNMENT 59

5. Pole-zero assignment The pair of matrices F, G given by (30) and (24) respectively, leads to a closed loop transfer function matrix which is regular and diagonal, i.e., H ( s ) = diag[nl(s) n2(s) ... n,,(s)~

where nq(s), q = 1,2 . . . . . m are rational functions in s. As was already mentioned, the available degrees of freedom of F, which are associated with the elements of Ek, k = 0,1 . . . . . 6 in (30), as well as those of G, which are associated with the matrix A in (24), could be used for pole-zero placement. However, in the sequel we will take into account the fact that the selection of F, G from (25) makes the transfer function matrix of the closed loop system equal to the unit matrix I , , in order to easily achieve pole-zero placement o f the decoupled single-input, single-output subsystems. Then, in order to achieve a desired pole-zero configuration described by the transfer function R ( s ) = diag [rx(s) r2(s) ... r . ( s ) ]

we select the above transfer function as a compensator such that V(s) = R ( s ) W ( s )

(36)

where W ( s ) is the Laplace transform of the new input w ( t ) (Fig. 1). The new dynamic control law takes the form V(s) = F ( s ) X ( s ) + G ( s ) R ( s ) W ( s ) .

(37)

It should be mentioned that (31) does not in general represent the decoupled system ( b e f o r e the pole-zero assignment) since it may involve pole-zero cancellations. The pole-zero cancellations do not exist if the original system is c-controllable and e-observable [26]. The decoupled system is described by a realization which is presented in the next section.

u(s) . Y(s)

Figure 1. Block diagram of the overall decoupled system.

60

MERTZIOS AND CHRISTODOULOU

6. A controller form realization In this section a method will be presented for the realization of a generalized transfer function matrix which has been recently proposed in [26]. Some relevant results have also recently appeared [27, 28]. If the controllers F(s) and G(s)R(s), which appear in the control law (37), are strictly proper, then one of the many existing algorithms for the statespace realization can be used, If, however, any of these controllers is not strictly proper, the proposed structure may be used for its realization. The final decoupled system may be also realized by using the present realization structure and forming the associated augmented system where the controllers F(s), G(s)R(s) (which actually are dynamic systems) are involved. These controllers are also realized by using the same technique. Consider a right matrix fraction description (MFD) H(s) = N(s)D-I(s) where N(s) is a p x m polynomial matrix and D(s) is a square polynomial matrix m x m, with detD(s) ~ 0. Note that H(s) does not have to be strictly proper, At this point the following basic theorem proven in Kailath [29] will be stated. Theorem 2. (Division Theorem for Polynomial Matrices): Let D(s) be an m • m nonsingularpolynomial matrix. Then f or any p • m polynomial matrix N(s), there exist unique polynomial matrices Q(s), R(s) such that N(s) : Q(s)D(s) + R(s) and R(s)D-~(s) is strictly proper. Using the above theorem, H(s) can be written as H(s) = N(s)D-I(s) = Q(s) + R(s)D-I(s)

(3S)

A realization for R(s)D -l(s) which is a strictly proper right MFD was given by Wang [30] and Wolovich [31] (see also Wolovich and Falb [32]). Thus given R(s)D-l(s), one can write for D(s) D(s) = DhcS(s) + D~c~(s)

(39)

where S(s) = diag Is kl ..... sk'] and where kl are the column degrees of D(s) and Dhc is the highest-columndegree coefficient matrix of D(s) (Kailath [29]). The term D,og(s) accounts for the remaining lower-column degree terms of D(s), with D~o a matrix of coefficients and

DEr

[sk,-'...s

1

AND POLE-ZEROASSIGNMENT 61

o

".i

. . . . . . . . . . . . . . . . . . . . . . . . . . . .

'~(s)

0

s~2-~...

..... 9

.

.

.

.

.

I

1

... . ..

. l

I . . . . . . . . . . . .0

(40)

[ ..... .......... : ..... :is ..... 1~..... Following the pattern o f the scalar case we can write for R ( s ) D - l ( s ) D(s)~(s) = u(s) y~(s) = N(s)~(s)

(41a) (41b)

Then use (39) together with (41a) to write Dh,S(s)~(s) = - D , ~ I ( s ) ~ ( s ) + u(s)

(42)

Then assuming that Dh~ is invertible, (42) gives S(s)~ = - D ~ D,c'~(s)~ + D ~ u

(43)

In the scalar case, Dhc # 0 means that the equation is indeed of the n-th order. In the matrix case, the analogous assumption with n = deg det D ( s ) is that D(s) is column reduced.

As it is well known (Kailath [29]), D ( s ) can always be reduced to this form without affecting the determinantal degree. Assume now that each ~}k,~ is available and integrate each k, times to obtain all the required lower order derivatives. There exist m chains with k, integrators in each chain. The outputs of the integrators are given by the entries o f r This gives the core realization with transfer function r Now assemble these integrator outputs along the input according to the prescription on the right hand side of (43) and thus generate the left hand side of (43). Finally, closing the loop we complete the realization of (41). The output equation (41) can be written as yl(s) = N(s)~(s) = N, oCl(s)~(s)

(44)

where iV,~is an appropriate matrix of coefficients which shows clearly that y is obtained as weighted sums of the states. Since now the strictly proper part is realized, it remains to realize the polynomial part Q ( s ) . This can be written as Q(s) = Nhcp~l.(s)

(45)

62

MERTZIOS

AND

CHRISTODOULOU

where Nho~ is an appropriate matrix of coefficients and q,~(s) is the following matrix

I

1 .

,I,~(s) = a /

rill

s...s .

.

.

..........

.

.

0 0

..........

.

.

.

.

0 .

.

.

.

.

.

.

"''[ .

[1 . . .

.

.

.

.

s "~

] ................ I

l

.

.

.

...l , .

0

---'l

0

...*,

[ ............

0

*,

I

.....

9_ . . . . .

.

] ............ ]

0

(46)

0

I

I...

s "~

where n, are the degrees of the columns of Q ( s ) (highest degrees of the polynomial elements of each column). Following a procedure analogous to the previous case, we write for Q(s) y2(s) = Q(s)u(s).

(47)

Since each u, is available, differentiate each one ni times to obtain all the required higher order derivatives. There exist m chains with nl differentiators in each chain. The outputs of the differentiat0rs are given by the entries of '~p(s)u(s). This is the entire realization for the dy0amical part of Q(s). It will be shown below how it can be realized via a singular system. The output equation here can be written as

y2(s) = mh~

(48)

The final realization for Hp(s) is given as a parallel connection of the two previously studied systems. For the output it holds

y(s) = yl(s) + y2(s)

(49)

The realization is shown in Fig. 2.

A description of the above procedure in the state space will be given below. Recall that for the strictly proper part R ( s ) D - ' ( s ) , the first step is to set m chains of kl integrators each, with access to the input of the first integrator of each chain and to the outputs of every integrator of every chain. The corresponding system matrices for the realization of this core system are [29]

DECOUPLING AND POLE-ZERO ASSIGNMENT 63

I

V(s')

Llllllllllkm

-~U ~

s(s,r_

~c

D -1

III111

5

Integrators

[-J"]-~k 1

~

Yl

[",oi

^

u (s)u P

IIIIIIIII". IIIIII

>

Differentiators

" ~

n1

Figure 2. A schematic of controller form realization.

I A~ = block diag

I

(B~

fi

~

[[1

[_C ~ = /., n = d e g d e t D ( s )

o

1

0

0...0],

lxk,,

~ k,

i=l

1

ki X k , i = 1. . . . . m 1

(50a)

J i=

1, ..., m]

(SOb) (50c)

64 MERTZIOSANDCHRISTODOULOU It can be checked by direct calculation that (sI- A~

~ = C~

- A~

(51)

~ = et(s)S-~(s)

The core realization is controllable and observable. Next, the states and the input are assembled according to the right hand side of (43) and the loop is closed. This corresponds to state feedback through a gain D ~ D ~ , and to the application of an input D X ~ u . The system matrices for the realization are A 1 = A ~176 - B~

Dl,,

B1 = B ooD h-1 o

(52) Cl

~ Nle

The modification is easy to carry out. Certain rows of A ? are replaced by the rows of Dh~D~o and certain rows of B ~ by rows of D~. For the polynomial part Q ( s ) the first step is to set m chains of n, differentiators each, with access to the input of the first differentiator of each chain and to the outputs of each differentiator of every chain. The corresponding system matrices for the realization of this core system are

o f K = block diag f BC~: bl?CNio?iag[[

o

1

,(n,+l)•

0...1 "0 0...0],l•

1.....

m

t

(53a)

(53b)

i = 1..... m~

(53c) It can be checked by direct calculation that C2(sK-1)-XB2

= Q(s)

From all the above we obtain the general realization which has the form

E'o ~~ ],, = [Alo ;1 E~'lx,,E ~,]

u

(54a)

B2

,: Ec,c,1 E~1 Ix, with order n + ( ~ nl + m). i =1

(54b)

DECOUPLINOAND POLE-ZERoASSIGNMENT 65 Example 3. Let a system of the singular f o r m be described as in Example 2. Assume that we want the closed loop decoupled system to have the following transfer function:

R(s) = I 1 / s 0

01 1~(s-p)

i.e., we want to introduce a pole at 0 to the first subsystem and a pole at p to the second subsystem. The controller F(s) selected f r o m (22a) is

F(s) =

I

0.5s- 1 - s 2 + 5.5s- 5

- 1.5

0.5s+ 1.5

s-4.5

- s2 + 4 . 5 s - 7.5

J

G(s)R(s) is

while the product

G(s)R(s) = I O'5/s 0.5/s

0 l 1~(s-p)

We first construct a state-space realization for the controller F(s). Since F(s) is of a polynomial form, according to the given algorithm we have

nl = 2, n2 = 1, n3 = 2, nt+n2+n3+m = 7 Then 1 s s210 ,i,,~ =

~,~ = I-5 L

and by (53)

o - o o r ? , -s',0 --oo~ o 0,0 o; 1 i ~:

[

~176 1

0.5 5

010

5.5

0

-1.5

0

1.5

-1

-4.5

1

-7.5

0.5 4.5

01 - lJ

66

MERTZIOS

AND

CHRISTODOULOU

0

o

0

1

0

0[0

0

1

. . . . . .

0

', o',o II

0

0

o

0

1

0

0

0

0

0

0

0

0

0

0

0

0

0

0

o

0

0

1

0

. . . . . . . . . .

I I

K =

0

0

0',0

o

0

0

0

0

0

0

o

0

0

0

o

0

0

0

0

0

1

0

1

0

0

o

0

0

o

o

1

0

0

0

0

I P

0

0

. . . . . .

o', I II

,B2

=

, C2

:

-- Sh~p

. . . .

0

0

0

0

I 0',0 ,, 0',0

o

o

0',0

I I

Next we construct a state-space realization for the controller G ( s ) R ( s ) which is o f the strictly p r o p e r f o r m G(s)R(s) = N(s)D-t(s)

where

N(s) =

E~ ~ 0.5

E; ~

, D(s) =

1

s-p

A c c o r d i n g to the p r e v i o u s p r o c e d u r e it can be s h o w n t h a t kl ~ 1,

k2 --

1, k +

k2 =

2 = deg det D ( s )

T h e h i g h e s t c o l u m n degree coefficient m a t r i x i s / 2 , while

Es oI Eo ~]

S(s) =

0

D~o =

0

, Dh, =

S

, N~ =

E1 ol [o., o1

-

0

1

0.5

1

By using (50) a n d (52), we o b t a i n A1

:

, B1

0

:

/'2,

C,

:

0.5

1

It has b e e n checked t h a t the singular system in E x a m p l e 2, with the two

DECOUPLING AND POLE-ZERO ASSIGNMENT

67

dynamic compensators constructed above gives the required closed loop transfer function matrix. Conclusions In this paper the sufficient conditions for single-input, single-output decoupling of singular systems via state feedback are established. These conditions have been determined in terms of the nonsingularity of the matrix B* , as in the regular case. The main difference here is that the matrix B* , which includes, in the general case, the matrices A, B, C, M of the given system, is a function of the Laplace variable s. The controller matrices F and G are also functions of s, i.e., the matrices F and G contain dynamic elements. Therefore, there are more degrees of freedom in the controller, which m a y be used to satisfy other design specifications too. A method is presented for simultaneous decoupling and pole-zero assignment. The technique used for the solution of the problem is general and can be used for the decoupling and other classes of systems. References [1] P.L. Falb and W.A. Wolovich, "Decoupling in the design and synthesis of multivariable control systems," 1EEE Trans. Autom. Contr., Vol. AC-12, pp. 651-659, 1967. [21 E.G. Gilbert, "The decoupling of multivariable systems by state feedback," SIAM. J. Control., Vol. 7, pp. 50-63, 1969. [3] W.M. Wonham and A.S. Morse, "Decoupling and pole assignment in linear systems: A geometric approach," S I A M J. Contr., Vol. 8, pp. 1-18, 1970. [41 A.S. Morse and W.M. Wonham, "Decoupling and pole assignment by dynamic compensation," S I A M J. Control., Vol. 8, pp. 317-337, 1970. [5] A.S. Morse and W.M. Wonham, "Status of non-interactive control," IEEE Trans. on Autom. Contr., Vol. AC-16, pp. 568-581, 1971. [61 L.M. Silverman, "Decoupling with state feedback and precompensation," IEEE Trans. on Autom. Contr., Vol. AC-15, pp. 487-489, 1970. [71 J.W. Howze and J.B. Pearson, "Decoupling and arbitrary pole placement in linear systems using output feedback," IEEE Trans. on A utom. Contr., Vol. AC-15, pp. 660-663, 1970. [81 I.H. Mufti, "A note on the decoupling of multivariable systems," IEEE Trans. Autom. Contr., Vol. AC-14, p. 415, 1969. [9] J.W. Howze, "Necessary and sufficient conditions for decoupling using output feedback," 1EEE Trans. Autom. Contr., Vol. AC-18, pp. 44-46, 1973. [10l W.A. Porter, "Decoupling and inverses for time varying linear systems," IEEE Trans. Autom. Contr., Vol. AC-14, pp. 378-380, 1969. [111 A.K. Majumdar andA.K. Choudhury, "On the decoupling of nonlinear systems," lnt. J. Control, Vol. 16, pp. 705-718, 1972. [12] B.G. Mertzios, "Sensitivity-Factorization-Decoupling of linear multivariable 2-D systems," Ph.D. Thesis, Department of Electrical Engineering, Democritus U. of Thrace, Xanthi, Greece, 1982.

68 MERTZIOS AND CHRISTODOULOU [13] G.C. Verghese, P. Van Dooren and T. Kailath, "Properties of the system matrix of a generalized state space system," Int. J. Control, Vol. 30, pp. 235-243, 1979. [14] G.C. Verghese, B.C. L6vy and T. Kailath, "A generalized state space for singular systems," IEEE Trans. on Autom. Contr., Vol. AC-26, pp. 811-830, 1981. [15] R.F. Sincovec, A.M. Erisman, E.L. Yip and M.A. Epton, "Analysis of descriptor systems using numerical algorithms," IEEE Trans. on A utom. Contr., Vol. AC-26, pp. 139-147, 1981. [16] D. Cobb, "Feedback and pole placement in descriptor variable systems," Int. J. Control, Vol. 33, pp. 1135-1146, 1981. [17| E.L. Yip and R.F. Sincovec, "Solvability controllability and observability of continuous descriptor systems," IEEE Trans. on Autom. Contr., Vol. AC-26, pp. 702-707, 1981. [18] A.J.J. Van Der Weiden and O.H. Bosgra, "The determination of structural properties of a linear multivariable system by operations of system similarity, 2. Nonproper systems in generalized state-space form," Int. J. Control, Vol. 32, pp. 489-537, 1980. [19] C.E. Langenhop, "Controllability and stabilizability of singular linear systems with constant coefficients," Department of Mathematics, Southern Illinois University. [20] D.G. Luenberger, "Time invariant descriptor systems," Automatica, Vol. 14, pp. 473-480, 1978. [21] J.S. Thorp, "The singular pensil of a linear dynamical system," Int. J. Control, Vol. 18, pp. 577-596, 1973. [22] R.W. Newcomb, "The semistate description of nonlinear time variable circuits," IEEE Trans. on Circ. and Syst., Vol. CAS-28, pp. 62-71, 1981. [23] P.V. Kokotovic, R.E. O'MaIley Jr. and P. Sannuti, "Singular pertubations and order reduction in control theory-An overview,"Automatica, Vol. 12, pp. 123-132, 1976. [24] MoM. Milic, "General passive networks-Solvability, degeneracies, and order of complexity," IEEE Trans. on Circ. and Syst., Vol. CAS-21, pp. 177-183, 1974. [25] R.E. O'Malley Jr. and A. Jameson, "Singular pertubations and singular arcs, I," IEEE Trans. on Autom. Control, Vol. AC-20, pp. 218-226, 1975. [26] M.A. Christodoulou, "Analysis and synthesis of singular systems," Ph.D. Thesis, Department of Electrical Engineering, Democritus University of Thrace, Xanthi, Greece, 1984. [27] A.I.G. Vardulakis, D.N.J. Limebeer and N. Karkanias, "Structure and SmithMacMillan form of a rational matrix at infinity," Int. J. Control, Vol. 35, pp. 701-725, 1982. [28] M.A. Christodoulou and B.G. Mertzios, "Realization of singular systems via Markov parameters," Int. J. Control, Vol. 42, pp. 1433-1441, 1985. [29] T. Kailath, Linear Systems, Prentice Hall, Englewood Cliffs, N.J., 1980. [30] S.H. Wang, "Design of linear multivariable systems," Memo. No. ERL-M309, Electronics Research Lab, University of California, Berkely, 1971. [31| W.A. Wolovich, "The determination of state space representations for linear multivariable systems," Proc. Second IFAC Syrup. Multivariable Tech. Contr. Systems, Dusseldorf, 1971. [32] W.A. Wolovich and P.L. Falb, "On the structure of multivariable systems," S I A M J. Control, Vol. 7. pp. 437-451, 1969.