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Aulomatica. Vol. 33, No. 4, pp. 729-732, 1997 0 1997 Elsevier Science Ltd. All rights reserved

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Technical Communique

On the use of reachability Gramians for the stabilization of linear periodic systems*

GIUSEPPE DE NICOLAOt and SILVIA STRADA$

Key Words-Periodic systems; receding-horizon control; multirate sampled-data systems; time-varying systems; linear systems.

Abstmeet-This paper deals with the development of explicit formulas, based on reachability Gramians, for the stabiliza- tion of linear periodic systems. In contrast with the ‘pointwise’ strategy already known in the literature, a different ‘intervalwise’ scheme is proposed, involving reduced computational effort. Stability results are provided in both continuous and discrete time under the sole controllability assumption without requiring system reversibility. 0 1997 Elsevier Science Ltd.

1. introduction It has long been known that the reachability Gramian over a suitable finite interval can be used to construct a stabilizing gain for (continuous or discrete) linear systems; see Kleinman, (1970,1974) and Kwon and Pearson (1975) in the time-invariant case and Kwon and Pearson (1977, 1978) Anderson and Moore (1980) and De Nicolao and Strada (1996) in the time-varying one. This ‘easy way’ to stabilize a linear system, using a terminology taken from Kleinman (1970), enjoys a meaningful interpretation in terms of receding-horizon (RH) optimal control. Indeed, the control law can be associated with the solution of a finite-horizon LQ problem with a terminal constraint on the state. Besides its obvious theoretical interest, the ‘easy way’ technique was motivated by its possible application in the computation of steady-state solutions of the Riccati equation by means of the quasi-linearization method (Kleinman, 1%8), which in fact needs to be initialized with a stabilizing state feedback.

Since the ‘easy way’ technique extends to generally time-varying systems, it obviously applies to periodic systems as well. As a matter of fact, its role in the quasi-linearization algorithm for computing periodic solutions of the periodic Riccati equation (Bittanti et a/., 19881989) is even more crucial than in the time-invariant case, because of the greater difficulty in stabilizing periodic plants.

Recent studies (De Nicolao, 1994,1992), have pointed out that periodic systems, besides the traditional ‘pointwise’ RH scheme, admit an alternative ‘intervalwise’ RH stabilization strategy. In the classical ‘pointwise’ RH scheme, the fixed-length finite-horizon optimization interval is moved forward at each time instant. Conversely, in the ‘intervalwise’ scheme, the terminal point of the optimization interval is kept fixed while the initial point moves forward in time. After one period, the optimization interval is reset at its maximum length by shifting forward the terminal point by one period.

In view of the connection between the ‘easy way’ scheme and RH control, it is natural to investigate the existence of a

* Received 3 June 1996; Received in final form 31 October 1996. This paper was recommended for publication in revised form by Editor Peter Dorato. Corresponding author Guiseppe De Nicolao. Tel. + +39 382 505374; Fax + +39 382 505373; E-mail denicolao@conpro.unipv.it.

t Dipartimento di Informatica e Sistemistica, Universita di Pavia, via Ferrata 1, 27100 Pavia, Italy.

$ Dipartimento di Elettronica e Informazione, Politecnico di Milano, Piazza Leonardo da Vinci 32, 20133 Milano, Italy.

modified stabilizing gain, of ‘intervalwise’ type, based on the reachability Gramian. Hereinafter, the stabilizing properties of this alternative scheme are established and its computational advantages pointed out.

2. Stabilization of continuous-time periodic systems Consider the continuous-time linear system

i(r) = A(t)x(t) + B(t)u(t), (1)

where A(.):R+WX” and B(.):R+R”Xm are real, piecewise-continuous T-periodic matrices while x(e) and u(e) are the state and input vectors respectively. YA(t, 5) is the transition matrix of A(t) over the interval [T, t]. It is well known that A(.) is asymptotically stable iff all the eigenvalues of Y,(r + T, 7) belong to the open unit disk for some r.

The reachability Gramian over [t, tf] is defined as

50, rr) = d

(1 ‘I’,&, o)&o)BT(+‘:(t,, o) do.

Note that the controllability Gramian (Bittanti, 1986) involves the inversion of the transition matrix, and as such it may not be well defined for discrete systems.

As shown in Kwon and Pearson (1977) for generally time-varying systems, the gain K(t) = -BT(t)S-‘(t, t + L) is stabilizing, provided that (1) is controllable over [t, t + L] Vr, where L is a positive real number. However, it is not obvious that this procedure can be rendered computationally efficient, because, as observed in Kwon et al. (1983) computing such a gain requires, in principle, recomputing the Gramian S(t, t + L) for all t E [0, T]. To circumvent this problem, an alternative algorithm based on scattering theory was developed in Kwon et al. (1983), requiring the integration of a 2n x 2n matrix differential equation,

It is shown below that a different control law, still based on the reachability Gramian but involving only the integration of an n X n (linear) matrix differential equation, can be used to stabilize the system. In order to derive the result, the following technical lemma is needed.

Lemma 1. Let (A(.), f?(.)) be a controllable T-periodic pair and assume that the differential Lyapunov equation

P(t) = A(t)P(t) + P(t)AT(t) + B(t)BT(t)

admits a positive-semidefinite solution P(.) over [T, I + T] such that P(r + T)sP(r). Then the periodic matrix A(.) is asymptotically stable.

This result follows by duality from Lemma 2 in De Nicolao (1994) by observing that controllability of (A(.), B(s)) implies D-controllability of the triple (A(,), B(.), [P(r) - P(z + T)]“‘). To keep this paper self-contained, a different proof is also given in the Appendix.

Theorem 1. Assume that the periodic pair (A(.), B(.)) is controllable. Then

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Technical Communique

(i) there_ exists a positive real number L 5 nT such that if L~LthenS(t,?+L)>OVr’t[~-T,5] (nistheorder of the system);

(ii) for all L 2 L, A(.) + B(.)R(.) is asymptotically stable, where the periodic gain I?(? + T) = K(r) is defined as

Z(r) = -BT(r)S-‘(r, ? + L), t E [7 - T, 7).

Proof (i) is a direct consequence of the fact that the controllability interval of a controllable periodic pair is at most equal to nT (Bittanti, 1986).

(ii) Define s(r) = .Y(t, r + L), r s 7 + L. By the definition of the reachability Gramian S(r, rr), it is easily seen by inspection that S(t) is the solution of the differential Lyapunov equation

-i(r) = -S(r)A*(r) - A(r)S(r) + B(r)BT(r), r 5 7 + I!,, (2)

with the @al condition $( r + L) = 0. Since S(r + L - T) 2 S(r + L) =_O, in view of Lemma 5 in

De Nicolao (1994) it follows that_S(r - T) 2 S(T). Letting F(t) =A(t) - B(r)BT(r)S-‘(r), (2) can be rewritten

as

S(r) = S(r)FT(r) + F(r).?(r) + B(r)BT(r).

The controllability of (A(.), e(.)) implies the controllability of (F(.), B(a)) as well. Since S(7) 5 S(7 - T), Lemma 1 can be applied to conclude that F(.) is asymptotically stable. 0

Remark 1. As can be seen from (2) the computation of the Gramian S(c, r + L) as t varies amounts to solving a linear matrix differential equation of dimension n X n. This involves a substantial saving of computations compared with the method proposed in Kwon et al. (1983) where a 2n x 2n scattering matrix must be integrated.

Remark 2. The stabilizing control law R(.) can be seen as the result of an intervalwise RH strategy. More precisely, at a given time 7 - T, one considers the finite-horizon optimization problem

I

II minJ, J= u’(s)+) cb, (3)

4,

with r,, = T - T, tf = 7 + L, subject to (1) and the terminal constraint x(r + L) = 0. It is well known (see e.g. Kwon and Pearson, 1977) that the optimal control is given by the linear time-varying control law

u(r, 7 + L) = -B(rjw’(r, T + L)x(t).

When adopting an intervalwise RH strategy, the optimal control u(r, r + L) is applied for t E [r - r, 7). At time 7, one again solves the problem (3) with t,, = r, tr= 7 + L + T,

X(T + f. + T) = 0, and applies the corresponding optimal control over I?. r + T) (see Fie. 1). Iterating this intervalwise procedure over subsequent pehods is just equivalent to using the periodic control law u(t) = Z?(r)x(r).

Remark 3. Stabilization schemes based on intervalwise RH strategies were originally proposed by Kwon and Pearson (1982) for time-invariant systems, and subsequently extended in De Nicolao (1994) to linear periodic systems. In particular, Theorem 1 bears some resemblance to the stabilization scheme suggested in Remark 3 in De Nicolao (1994), where the gain is based on the solution of the differential Riccati equation

-S(r) = -S(r)A*(r) - A(t)S(t) + B(r)B*(r)

- S(r)Q(r)S(r), S(T + f.) = 0. (4)

Although (2) is a particular case of (4) with Q(r) = 0 Vt, the stabilizing property of the gain K(r) cannot be established using the arguments of Remark 3 in De Nicolao (1994) because they require detectability of (A(.), Q(.)).

3. Stabilization of discrete-time periodic systems Consider the discrete-time linear system

x(k + 1) = A(k)x(k) + B(k)u(k), k 2 r, (5)

where A(~):.Z+IWnX” and B(~):Z+R”xm are real T- periodic matrices (with T an integer). Y,(r, T) = A(r - l)A(t - 2) A(s) is the transition matrix associated with A(f) over the interval [T, r].

The reachability Gramian over [t, tr] is defined as ,,- 1

S(t, tr) = C YA(tf, i + l)B(i)BT(i)Y~(t,, i + 1). ,=,

In the literature it is known that the gain K(r) = -B*(t)YI(r + L, r + l)St(t, t + L)YA(r + L, t) (where t denotes the Moore-Penrose pseudoinverse) is stabilizing provided that (5) is completely controllable over [r, r + L] Vr; see e.g. Kleinman (1974), Kwon and Pearson (1978) Anderson and Moore (1980) Bemporad et al. (1994) and De Nicolao and Strada (1996) where this stability result is established under various assumptions for time-invariant and generally time-varying systems.

In analogy with the continuous-time case, the computation of the periodic gain would require the evaluation of T reachability Gramians. Again, some saving of computations may be accomplished by resorting to scattering theory (Bruckstein and Kailath, 1982).

As an alternative, one can use the following result, which extends to the discrete-time case the theorem of the previous section.

Fig. 1. The intervalwise receding-horizon scheme. The periodicsain k?(t) = -E’(f)S-‘(r) is a function of the periodic matrix e(r) = S(r, T + L), I E (7 - T, r]. Note that in order to compute x(t) as r varies, the initial point of the optimization horizon is

moved from t - T to I while the terminal point remains at r + L.

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Theorem 2. Given a positive integer L, assume that (b(a), B(.)) is controllable over [ 2, z + L] Vr. Then A( .) + B( l)K (.) is asymptotically stable, where the periodic gain K(t) = K (r + T) is such that

R(t) = -B’(r)YZ(r + L, t + l)St(r, 7 + L)Y,(t + f., I),

t E [7 - T + 1, 71.

Remark 4. The main difficulties in the following proof lie in the possible nonreversibility of A(.) and the possible singularity of S(r, 13. For these reasons, the arguments of the proof of Theorem 1 cannot be used and it is also impossible to obtain the stability result as a more or less direct consequence of existing intervalwise stability results (De Nicolao and Scattolini, 1994; De Nicolao, 1992).

Proof. Define the function

subject to (5) with initial condition x(l) = X and subject to the terminal constraint r(tf) = 0. It is apparent that V(i, t, tr) = V(Z, t + T, t, + T) VZ

Denoting by I = [u“(7 - T + 1) . . u0(7 + L - l)] the (unique) optimal solution of (6) with t = T- T + 1, tr = r + L, a simple computation based on the Lagrange- multiplier technique shows that u”(j) = K(j)x(j), j E [r - T + 1, 71.

Using the same arguments as in Bemporad er al. (1994), it is seen that, for given 7, T and L, V(.f, 7 - T + 1, 7 + L) is a quadratic function of X; i.e. there exists p = FT z 0 such that

V(Z,7-T+i,7+L)=FT~X VF.

Let Y&,_t,) denote the transition matrix of F(r) = A(r) + B(r)K(t). Then the following property holds:

V(f, 7 - T + 1, 7 + L) 2 V(YF(7 + 1, 7 - T + 1)X,

7f1,7+T+L)-tIlu0(7-T+1)...U0(7)p, (7)

where ll .I] denotes the Euclidean norm, i.e. llM]l = (Z mz)‘/2. Indeed, the control sequence iZ = [u“(T + 1) . u”(r + L - 1) 0 . . . 01, where u“(e) has been previously defined, is admissible for the problem (6) with t = 7 + 1, lf = 7 + T, and initial state Y,(r + 1, r - T + 1)X, since it results in x(r + k) = 0, k 2 L. The associated cost is II@ II’< JlU]l* = V(.f, 7 - T + 1, r+ L), so that the inequality (7) follows.

Now,letting_F=Y,(r+1,7-T+l)andQ=P-FTPF, it follows that P satisfies the algebraic Lyapunov equation

P=FTFF+Q, (8)

where, in view of (7),ZTQf? IIu“(7 - T + 1) . u“(z)(I* 20. The asymptotic stability of the periodic matrix F(.) is now

proved by showing that all the eigenvalues of F = Y,(7 + 1, r - T + 1) belong to the open unit disk. This is done by using the Lyapunov lemma, _which can be applied to (8) provided that the invariant pa” (F, Q) is reconstructible.

By contradiction, assume that (F, Q) is not reconstructible. Then there exist X#O and A #0 such that I%= A.? and QZ = 0. Letting X be the initial state of (5), it follows that

Now, two cases are possible. If & = 0 then V(X, 7 - T f 1, r + t) = 0, and the free movement of (5) with initial state X_r--T+l)=x is such that x(r+L)=O. Then U=O and K(t)x(t)=O Vt, so that &=YA(r+l,r-T+l)y=O. which contradicts A Z 0.

The second case is & #O. Then IA/ = l and V(f, 7 - T + 1, 7 + L) = V(Fy, 7 -+ 1, 7 + T + L). Now consider the optimization problem (6) subject to (5), with x(t) =X and f = 7 - T + 1. Since Qz? = 0, it follows from (7) that u”(t) = 0, 7-T+l~t57, i.e. Ila(12=Ilull’=V(x,7-T+1, 7 + L) = V(FZ 7 + 1, r + T + L), so that the sequence ii is optimal for the problem (6) with t = t + 1, tf = T + L + T and initial state F_f. Hence R(t)Y,(t, r - T + 1) Z = u”(t), t E [Z - T + 1, 7 + T]. By iterating the same rationale, it is

shown that R(r)YIIF(t, 7 - T + I)_? = u”(t) Vr < 7 + L, which implies that Y&, r - T + IF = 0 fo_r some s 5 7 + L, thus contradicting A # 0. Then the pair (F, Q) is reconstructible, and the thesis follows from the Lyapunov lemma applied to (8). D

Remark 5. Just as in the continuous-time case, the stabilization scheme of Theorem 2 is more efficient than using the standard ‘easy way’ regulator computed through scattering-theory methods (involving integration of a 2n X 2n difference matrix equation) (Bruckstein and Kailath, 1982). In particular, the matrices S(r, r + L), t = T - T + 1, . , T, needed to construct the intervalwise feedback, can be computed efficiently by means of the recursions

S(t, r + L) = S(r + 1, 7 + L)

+ Y,4(7 + L, t + 1)B(t)BT(t)YT,(7 + L, t + l),

+bA(7 + L, t + 1) = YA( 7 + L, t + 2)A(t + l),

t = r + L - 1,. . . , T - T + 1,

with the initializations

S(r+L,z+L)=O,

Y,(r+L,r+L)=I.

4. Conclusions It has been shown that a linear periodic system can be

stabilized by using its reacability Gramian in an ‘intervalwise’ fashion as opposed to the standard ‘pointwise’ method. The new scheme involves only the solution of an n X II linear matrix (difference or differential) equation, and is therefore more efficient than the standard one, which, in its best implementation, requires solving 2n X 2n matrix equations. As for memory occupation, as soon as 3 becomes available, one can compute the corresponding gain, and there is no need to store all the intermediate values of the matrix 3 (in the continuous-time case, and similarly in the discrete-time case).

To the best of the authors’ knowledge, the algorithm presented here provides the fastest way to compute a stabilizing feedback control law for a periodic system (note, however, that performance is not taken into account). At present, the available stabilization techniques for linear periodic systems include optimal control methods hinging on periodic Riccati equations (see e.g. Bittanti et al. 1988, 1989, and references therein), as well as pole-placement algorithms (Grasselli and Longhi, 1991).

The main application of the stabilization scheme seems to be in the initialization of the quasi-linearization (Newton) method for the solution of the periodic regulator problem. In this respect, it is most likely that it will prove useful in the optimal control and filtering of multirate sampled data systems. In fact, the alternative approach for solving the periodic Riccati equation is based on lifting (Berg et al., 1988), and, as observed in Bittanti et al. (1988), especially for long periods, the quasi-linearization approach (see e.g. Glasson and Dowd, 1981) may compare favourably.

Acknowledgment--This paper has been supported by the MURST project ‘Model Identification, Systems Control, Signal Processing’.

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Appendix-Proof of Lemma 1 A simple computation shows that

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P(r + T) = $A(z + T, z)P(rj$ff(z + T, 5) + S(7, 7 + T).

Hence P(7) satisfies the algebraic Lyapunov equation

P(7) = @,A(7 + T 7)P(7)#/4(7 + T 7)’

Control, Sig. Sysi., 4,437-453. Kleinman, D. L. (1968). On an iterative technique for Riccati

equations computation. IEEE Trans. Autom. Control, AC-13,114-115.

Kleinman, D. L. (1970). An easy way to stabilize a linear constant system. IEEE Trans. Autom. Control, AC-U, 692.

Kleinman, D. L. (1974). Stabilizing a discrete. constant,

+ S(7, 7 + T) + P(7) - P(7 + T).

The controllability of (A(.), B(.)) implies the controllability of the invariant pair (@A7 + T, 7), S(7,7 + 0)

(Bittanti, 1986), so that the Lyapunov lemma can be applied to conclude that all the eigenvalues of t+bA(7 + T, 7) belong to the open unit disk. This in turn implies the stability of the periodic matrix A(.). 0