Model Matching with Stability for Periodic Discrete-Time Systems

Copyright © IFAC New Trends in Design of Control Systems, Smolenice, Slovak Republic, 1997

MODEL MATCHING WITH STABILITY FOR PERIODIC DISCRETE-TIME SYSTEMS Patrizio Colaneri* and Vladimir Kucera**

*

Dipartimento di Elettronica e Informazione del Politecnico di Milano

Piazza L . da Vinci 32, 20133 Milano (Italy), e-mail: colaneri@eiet .polimi.it

**

Institute of Information Theory and Automation P. O. Box 18, 182 08 Praha (Czech Republic), e-mail:kucera@utia . cas. cz

Abstract : In this paper the model matching problem with stability is studied for discrete-time periodic systems . It is shown that , under some zeros matching conditions and stabilizability/detectability assumptions, it is possible to assign a target input-output periodic system by means of a periodic controller. Most remarkably, this allows to find conditions under which a periodic system can be converted into a closed-loop time invariant system via periodic control. Keywords: Linear systems , periodic systems, model matching, periodic control.

1. INTRODUCTION

ment of stability in a geometric framework was considered in [12].

A major tool in the study of periodic systems consists in the use of suitable time-invariant representations. The most popular, among them , is the so-called lifted reformulation which dates back to an early work of Krank [6] and was later refined by Meyer and Burrus in [4] . More recently, an equivalent time-invariant reformulation, henceforth referred to as cyclic reformulation , was introduced in [10] and [2] . The two shift invariant representations are obviously in a one-to-one correspondence and one can use indifferently any of them depending on the nature of the problem at hand. Of course a time-invariant system must satisfies a structure constraint in order to be considered as a time-invariant representation of a periodic system. This is important if one wants to exploit the theory of shif invariant systems for control purposes. To this aim a complete answer to the realization problem of periodic system has been recently given in [5] . In this paper we only make use of the cyclic reformulation with the aim of solving the exact model matching problem with stability to discrete-time periodic systems . The same problem in the timeinvariant case , without the closed-loop stability requirement , was originally solved in [9], see also [7] for an updated view . More recently, the problem of model matching via open loop precompensator was shown to be equivalent to a modified disturbance rej ection problem , see [13], and the require-

The present paper develops further the conclusions reached in the previous work [14] where the state feedback model matching problem without the stability requirement was addressed. Here the state-feedback and output feedback model matching problems for periodic systems are both investigated and the stability requirement is included. Precisely, the main problem addressed herein is the existence of a periodic controller such that the closed-loop system is stable and equals a given input-output map , which can be expressed by means of the transfer function of the cyclic reformulation of the given periodic system to be matched. Necessary and sufficient conditions are given in terms of usual zeros matching conditions and stabilizability / detectability assumptions. Most interesting is the particular case in which one wants the closed-loop system to be internally stable and time-invariant. It turns out that this is always possible if the underlying periodic system is minimum phase. The paper results take advantages of the various theoretical achievements of periodc systems and control theory, in particular for what specifically concern the definition and characterizations of zeros , see [1] and [3], and the structural prop erties

[11] . 75

2. THE CYCLIC REFORMULATION

transfer function changes with r in a very simple way. Precisely, letting WT+l(Z) and WT(z) be the transfer functions of the cyclic reformulations of the periodic system at r+ 1 and r, respectively, it follows , see e.g. [11]:

The system under consideration is aT-periodic discrete-time system described by the state-space equations

x(t

+ 1) = A(t)x(t) + B(t)u(t)

(1)

= C(t)x(t) + D(t)u(t)

(2)

y(t)

WT+1(z) where ~i=

where t E Z , the input u(t) is m-dimensional and the output y(t) is p-dimensional. Matrices AC) , BC) , CC) and DC) are periodic matrices of period TE Z+.

0 [ Ii(T-l)

;. . fl~(zT)ZT-l

Vi(t)=V(t), t=kT+r+i-1 , i=1 , 2, · · ·,T In this way any subvector of v(t) equals v(t) once a period . The cyclic reformulation at time r is given by

...

where

..4=

13=

c=

.0=

0 0

0 A(r + T - 2)

0 B(r)

0 0

B(r+T-1) 0

0

0 B(r + T - 2)

0 0

[ [

0

.

0 C(r+1)

0 0

0 0

0 0

0 C(r+T-1)

D(r) 0

0 D(r + 1)

0 0

0 0

0 0

fTT(zT)

Remark 0.3 The cyclic reformulations can be associated with a time invariant system as well, provided a period T is given. Let us denote by W(z) E Cpxm the transfer function of the original time invariant system. The transfer function W(z) E cpTxmT of the cyclic reformulation does not depend on r and is given by (4) with

/;j(zT)

C(r) 0

[ [

A(TT- ll 1

0

1

(4) are p x m rational functions in zT.

+ 1) = Ax(t) + Eu(t) ii(t) = Cx(t) + .ou(t) 0

~]

Remark 0.2 The Markov parameters .0 and C..4 kT +i 13, i = 0,1"", T - 1, k 2: 0 of the transfer function of a cyclic reformulation have a very particular structure. Indeed, it turns out that

where

0 A(r)

(3)

Remark 0.1 It is apparent from (3) that the zero structure (fiinite and infinite) of W(z) does not change with r . Hence one can define the cyclic zero structure of a periodic system as the zero structure of any associated cyclic reformulation without any further specifications.

In the cyclic reformulation the dimensions of the state, input and output variables are augmented. Precisely, given a signal v(t) E RJ, t E Z and an initial time instant r, define an augmented signal v(t) E Rg T , t E Z in the following way

x(t

= ~~WT(Z)~m

. T-l

= Z;J

L

W(zel/)4>i-i , 4>

= ej27r / T

k=D

1

In particular it can be verified that if A is a zero of W(z) then AT is a zero of W(z) and all the T-th roots of >7 are zeros of W(z). Similar relations hold for the poles. 3. MODEL MATCHING WITH STABILITY

0

D(r

+T

- 1)

1

In the following we will consider the model matching problem in two situations: in the first one the state is assumed available for feedback whereas in the second only the output variable is measured . In order to render the entire exposition easier , we limit the problem to the case when the periodic system is left invertible . This means that from now on we assume that m :S p and that the cyclic tranfer function W( z ) of the system is full column rank .

1

The transfer function from u to ii will be denoted by W (z ) and will be referred to as the cyclic transfer function of system (1) ,(2) at time r. This

76

Now consider first a state feedback control law acting on system (1) of the form u(t)

= K(t)x(t) + G(t)v(t)

have the same infinite zero structure (iii) The two transfer functions

(5)

where K(t) and G(t) are T-periodic design matrices. To simplify the solution of the main problem below , we make the assumption that the matrix G(t) is nonsingular for each t. Feedback (5) can be also rewritten in terms of the cyclic reformulation. Recalling the definition of the signals in such a reformulation , it is apparent that (5) is equivalent to u(t)

and

have the same finite zero structure (iv) The two transfer functions

= Rx(t) + GiJ(t)

and W(z)

where R E nmTxnT and G E nmTxmT are block diagonal matrices with entries K(i) E nmxn and G(i) E nmxm , i = 0, 1,· · ·, T - 1 respectively. The periodic feedback control system can be redrawn in terms of the cyclic reformulation. To this purpose, let fez) = (zI - A)-1 f3 denote the transfer function from the input u to the state x. Moreover , let us denote by Wm(z) E cpTxmT the transfer function (from iJ to y) of the cyclic reformulation associated with a given closed-loop (periodic) system (with input v and output y). Notice that Wm(z) can also derive from a p x m transfer function W m (z). In this case , the aim is to work out a periodic feedback such that the closed-loop system is time-invariant . In any case , it turns out that

have the same finite unstable zero structure. Sketch of the proof. As for sufficiency, notice that conditions (1) and (ii) imply the existence of a proper m x m rational function k( z) such that k(Z)-1 is still proper and Wm(z) = W(z)k(z). For the time-invariant case see e.g. [7], [8). Moreover, Wm(z) and W(z) being the cyclic reformulations of periodic systems , the rational matrix k( z) can be considered as the cyclic r_eformulation of a periodic system too . As such, R-1(oo) is block diagonal. Moreover, let fez) = N(z)A(z)-1 be a right coprime factorization of the strictly proper function f( z). Condition (iii) ensures that k(z)-1 A(z) is indeed a polynomial_matrix. Moreover k- 1(z) - k-1(OO), as well as T(z) , have both the structure discussed in Remark 0.2 . Due to this fact the equation

Theorem 0.1 Let a T-periodic system be gzven with the left invertible full column rank W(z) E cpTxmT as input-output cyclic transfer function and assume that the periodic pair (AC), BC» is stabilizable. There exist a periodic feedback matrix KC) and a periodic matrix GC) , the latter non singular for all t , such that the closed loop system is internally stable and its cyclic transfer function is Wm(z) if and only if

k- 1(z) - k-1(OO)

= fIf(z)

(6)

admits a block-diagonal constant solution matrix fI E nmTxnT. Then, let

G = k(oo), R = -GH Matrix G is block diagonal so that R is block diagonal as well. The block entries of the matrices R and G yield the solution of the problem . We only need to show that A + f3 R is stable. Indeed this would correspond to stability of AC) + BC)KC). The stability actually fllows form condition (iv), which is in fact equivalent to the stability of the polynomial matrix k- 1(z)A(z). The necessity of conditions (i), (ii) and (iii) immediately follows from the necessity of the same conditions for the time-invariant case, see Theorem 6.4 in [7], whereas necessity of (iv) is obvious.

(i) The two transfer functions

and have the same infinite zero structure (ii) The two transfer functions

and

Remark 0.4 As already indicated the interest of Theorem 0.1 mainly consists in the case where one

W(z)

77

wants to match a closed-loop system which is actually time-invariant, with a certain transfer function Wm(z) E Cpxm . In this case, Wm(z) is nothing but the transfer function of the cyclic reformulation of Wm(z) (recall Remark 0.2 and the relation between the zeros of W m (z) and those of Wm(z)) .

contrary, illustrates the impossibility of achieving a stable closed-loop time invariant target function , when the system at hand possesses non minimum phase zeros. example 1

Now, let us turn to the problem of output feedback model matching. An easy way to build a stabilizing compensator which yields the desidered closed loop input-output operator calls for introduction of a stable periodic state observer. Actually, if the additional assumption of periodic system detectability is made, there exists a periodic matrix L( ·) such that the observer dynamic matrix A(-) + L(-)C(-) is stable . Now , consider the observer

x(t

A(O) b = B(O)

C(O) D(O)

e(t

= x(t) -

[~ ~],

= [~] , = [1 = 0,

0

T(z)

=[

z2 - 1 1 z3 _ Z

0 z3 - z 0 z2 - 1

Z

x(t) satisfies the un-

Z2

1

1 (z2 - 1)2

~ 1 ] -:-(z-=2-~---:-1)-::-2

Suppose that we want the closed loop system to be time invariant with

which is stable. Now , let K(-) be constructed as in Theorem and consider the control law

0.1

= K(t)x(t) + G(t)v(t)

= [OIl ,

C(I)

=0

+ 1) = (A(t) + L(t)C(t))e(t)

u(t)

=[~ ]

B(I)

D(I)

[~ ~] ,

A(I) =

l,

and

Then T(z) and W(z) are as follows:

+ 1) = A(t)x(t) + B(t)u(t) + (7) + L(t)(C(t)x(t) - y(t) + D(t)u(t))

The state error e(t) forced equation

= 2,

Consider system (1) ,(2) with T

(8) as target transfer function. Hence , its cyclic verslOn IS

Equations (7), (8) are those of the output feedback compensator . The closed-loop input-output map coincides with the one given by the state feedback u(t) = K(t)x(t) + G(t)v(t) since the state-error variables are completely unreachable. This map is the target one.

-

Wm(z)

= z21 [10] 0 1

Then ,

R-I(z) Theorem 0.2 Let a T-periodic system be given with left invertible and full column rank W(z) E cpTxmT as input-output cyclic transfer function. Assume that the periodic pair A(-), B(-)) is stabilizable and the periodic pair A(-), C( .)) is detectable. There exists a output periodic controller such that the closed-loop system is internally stable and its cyclic transfer function input-output is Wm(z) if and only if conditions (i)-(iv) of Theorem 0.1 are fulfilled .

= [ z2(z2 z3

1)

0

z2(z2 - 1)

]

-;-;:-_1~

(z2 - 1)2 '

o = R( (X)) = [~ ~] Conditions (i)-(iv) are fulfilled. satisfied by

H=

[~

Equation (6) is

0 0

0

1

~]

-1 0

0 0

0 -1

,

so that

R = -OH =

4. EXAMPLES

[

0 -1

]

The periodic control law is then

In this section two illustrative examples are presented for the state-feedback case . In the first , the underlying periodic system is minimum phase (there are no finite unstable zeros) , so that it is possible to work out a periodic state-feeedback control law which match a internally stable timeinvariant system . The second example , on the

u(t) = K(t)x(t)

+ G(t)v(t)

where

G(O) = 1, G(I) = 1

K(O)

78

= [-1

0

l,

K(I)

= [-1

-1

1

Notice that as a consequence of condition (iv) , the closed-loop system has two characterustic multipliers equal to zero, i.e. is stable.

Notice however that in this case the resulting closed-loop system characteristic multipliers are 0 and 9 and therefore the system is internally unstable. This is due to the fact that condition (iv) is not satisfied. It is also simple to show that in this case it is impossible to match a time-invariant system without relaxing the closed-loop stability requirement (iv) . This conclusion follows from direct investigation of condition (iv) by taking in mind the structure of the cyclic reformulation of a time-invariant system (recall Remarks 0.2 and 0.3). The reason for this negative conclusion is the fact the the periodic system is nonminimum phase.

Example 2 Consider system (1),(2) with T = 2, and

A(O)

= [~ ~],

B(O)

= [ ~ ],

A(I)

=[~ ]

B(I)

= [1 -1 J ' C(l) = [1/3 2 J,

= [~ ~] ,

C(O)

D(O)

= 0,

D(I)

=0 CONCLUDING REMARKS

Then T(z) and W(z) are as follows:

-

9

T(z) =

W(z)

l

Z3j 9z

z2

3z z3

3z 2 18

9 = [z2_ 3

1

(z4 - 18)

z3_ 9Z z2+36

2z +z

The exact model matching problem with stability for periodic systems has been solved in this paper. The main point addressed here was to discuss the conditions under which a periodic system matches a time-invariant one while asymptotic stability is ensured.

1

]

(z4-18)

Suppose we want to match a time-invariant target system and let us choose REFERENCES

Wm(z)

1

=-

[1] Bolzern P. , P.Colaneri and R.Scattolini (1986) . Zeros of discrete-time linear perIodic systems. IEEE Trans . Automatic Control, Vol. AC-31 , pp. 1057-1058.

Z

Then

Wm(z)

fl-1(z)-fl-1(oo)

=

I[OOZ] = 2" z z 1

z4 - 18

[z~ + 36 z - 9z

z3 + 36z ] -9z 2 + 180

Conditions (i)-(iii) are fulfilled. Hence equation (6) has a block diagonal solution matrix

[3] Grasselli O .M. and S.Longhi. Zeros and poles of linear periodic multi variable systems. IEEE Trans. Circuits, Systems and Signal Processing, Vol. CSSP-7 , pp . 361-380.

[1400] H = 0 0 -3 1

[4] Meyer R.A. and C.S.Burrus (1976). Design and implementation of multirate digital filters. IEEE Trans. Acoustics, Speech and Signal Processing, Vo!. ASSP-1, pp. 53-58.

Therefore

0= fl(oo) R

= [005 ~]

= -OH = [-~.5 ~2 ~ ~1]

[5] Colaneri P. and S.Longhi (1995) . The realization problem for discrete-time periodic systems. Automatica , Vol. 31 , pp . 775-779.

The periodic control law is then u(t) = K(t)x(t)

[6] Krank G.M .(1957). Input-output analysis of multirate feedback systems. IRE Trans. Automatic Contr. , Vo!. AC-32, pp. 21-28.

+ G(t)v(t)

where

G(O) K ( 0)

= 0.5 ,

=[ -

G(I) = 1 0 .5 - 2 J , If ( 1)

= [3

[2] Flamm D.S. A new shift-invariant representation for periodic system (1991). Systems and Control Letters , Vol.17, pp. 9-14.

[7] Kucera V. (1991). Analysis and Design of Discrete Linear Control Systems , Prentice Hall , London .

-1 J

79

(8) Hautus M.L.J . and M.Heymann (1978) . Linear feedback , an algebraic approach", SIAM J. Control and Optimization, Vo!. 16 , pp . 83-105. (9) Wang S.H. and C .A.Desoer (1972). The exact model matching of linear multi variable systems. IEEE Trans. Automatic Control, Vo!. AC-17, pp . 347-349 . (10) Park B. and E.I .Verriest (1989) . Canonical forms of discrete linear periodically timevarying systems and a control application . Proc. 28 th Conference on Decision and Control, Tampa (USA) , pp . 1220-1225.

[ll) Bittanti S. and P.Colaneri (1986) . Analysis of discrete-time periodic systems. In : Control and Dynamic Systems: Digital Control and Signal Processing - Systems and Techniques , C .T .Leondes Ed ., Academic Press , pp . 313339 . (12) Malabre M. and J.C .M.Garcia (1993). The modified disturbance rejection problem with stability: a structural approoach", Proc. of the European Control Conference , pp .11191124. (13) Emre E . and M.J .Hautus (1980) . A Polynomial characterization of (A ; B)-invariant and reachable subspaces. SIAM 1. Control and Optimization , Vo!. 18 , pp. 420-436 . (14) Colaneri P. and V.Kucera. The model matching problem for discrete-time periodic systems. IEEE Trans . Automatic Control, to appear. ACKNOWLEDGMENTS The paper was partially supported by the project " Periodic control and filtering techniques of dynamical systems" of the Italian National Research Council (CNR) , by MURST and by the Grant Agency of the Czech Republic under contract 102/97/086.

80