Automatica, Vol. 29, No. 1, pp. 225-228, 1993 0005-1098/93 $6.00 + 0.00 Printed in Great Britain. ~) 1992 Pergamon Press Ltd

Brief Paper

Control for Discrete-time Systems Optimality and Robustness*t

P. L. D. PERES* and J. C. GEROMEL*

Key Words--Discrete-time systems; ~2-optimal control; convex programming; linear optimal control; robustness; robust control.

Airl iner--This paper proposes a new approach to determine optimal control for discrete-time linear systems, based on

convex programming. It is shown that all stabilizing state feedback control gains belong to a certain convex set, well-defined in a special parameter space. The Linear Quadratic Problem can be then formulated as the minimization of a linear objective over a convex set. The optimal solution of this convex problem furnishes, under certain conditions, the same feedback control gain which is obtained from the classical discrete-time Riccati equation solution. Furthermore, the method proposed can also handle additional constraints, for instance, the ones needed to assure asymptotical stability of discrete-time systems under actuators failure. Some examples illustrate the theory.

1. Introduction THE THEORY OF discrete-time linear systems has been developed, historically, as an extension of previous results concerning continuous-time systems. In the 1960s, the Linear Quadratic Problem--LQP has been exhaustively studied for both continuous-time and discrete-time systems, becoming a well-known technique for control design (Anderson and Moore, 1971). The discrete-time optimal state feedback solution can be calculated from the solution of an associated Riccati equation. It exhibits many good robust properties as for the continuous-time case. However, the control gain depends directly on the matrices model. This fact is one of the main reasons for the difficulties to generalize the discrete LQP to take into account additional requirements as, for instance, parameter uncertainties, output feedback, de- centralized control or sensors/actuators failure. Other difficulties stem from the fact that neither the set of stabilizing controllers nor the objective function are convex.

This paper presents a convex approach to solve the discrete-time LQP. It is formulated as a ~2 optimal state feedback control problem. First, we define a parameter space into which a convex set generates all the stabilizing state feedback control gains. An optimization problem is then formulated whose solution furnishes the state feedback gain that minimizes the ~2 norm of a closed-loop transfer function. Under certain conditions (to be defined in the sequel) this control gain equals the one provided by the classical LQP approach, that is, the associated Riccati equation solution.

Furthermore, additional requirements can be easily

*Received 17 July 1991; received in final form 18 December 1991. This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Guest Editor P. Dorato.

i" This research has been supported in part by grants from "Fundaq,~o de Amparo [1 Pesquisa do Estado de S~o PauI(>--FAPESP" and "Conselho National de Desenvol- vimento Cientlfico e Tecnol6gico----CNPq", Brazil.

:~ LAC-DT/Faculty of Electrical Engineering, UNICAMP, CP 6101, 13081, Campinas, SP, Brazil.

225

incorporated in the algorithm. To support this claim, we solve completely the discrete-time linear system stabi- lizability problem subject to actuators failure. In this case, since only one feedback gain" must be stabilizing for all prespecified contingencies, we provide an upper bound for each closed-loop transfer function ~z norm.

2. Preliminaries Let us consider a discrete-time linear system whose

dynamic behavior is given by the following difference equations

x~+ 1 = Ax k + B lw k + B2u k,

u~ = -Kxk , (1)

z k = Cxt + DUk,

where Xk ~" is the state variable, uk ~" is the control variable, Wk ~t is the external disturbance and zk ~t q is the output variable. Matrices A, B~, B 2, C and D have appropriate dimensions and are supposed to be known. Without loss of generality, the usual orthogonality hypothesis is also made, that is C'D=O and D'D>O. Defining the closed-loop matrices Aa = A - B2K and Col = C - DK and supposing that a state feedback gain K is calculated in such a way that Act is asymptotically stable, the closed-loop transfer function from w to z is given by

H(z) ~ C,[zl - AdI -1B , . (2)

The ~2 norm for a stable transfer matrix H(z) can be defined as

1 f*= Ilnll 2 =~-~ J_~ Tr {H(e-J')'H(eJ')} dto, (3)

and can also be calculated from the discrete-time associated Gramians. Let Lc be the controllability Gramian of (Ad, Bt) and Lo the observability Gramian of (Cot, Ac0. Then,

AdL~A' I - Lc + B iB ~ = 0, (4)

A'ILoA d - L o + C'lCct = 0. (5)

The ~2 norm is given by

IIHII 2 = Tr (CdL~C~O = Tr (B~LoBz). (6)

If we denote by ~ the set of all stabilizing state feedback control gains K e ~t "n, the problem of ~z optimal control can be stated as follows:

(P1) min (IIHII~:K ~}. (7)

Examined in the state feedback parameter space, this problem may have a very complicated geometry. In fact, neither the objective function nor ~r are convex with respect to the elements of the control gain K. However, it is well-known (Anderson and Moore, 1971; Dorato and Levis, 1971) that the optimal solution of (P1) is given by

K = (B~PB2 + D 'D) - tB~PA, (8)

226 Brief Paper

where, under controllability and observability assumptions, P ~" " is the unique symmetric positive definite solution of the discrete-time Riccati equation

A'PA-P -A 'PB2(B~PBz+D'D) -~B~PA+C'C=O. (9)

From (8), we note that the optimal feedback gain K depends directly on the system matrices A and B 2. Obviously (Kwakernaak and Sivan, 1972) this solution can also be regarded as the optimal one for the following discrete-time LQP:

min ~ (x't,C'Cxk + u't,D'Ouk), (P2) "k k=0 (10)

xk+ 1 = Ax k + B2Uk, x 0 given.

It is important to remark that, although the optimal control gain does not depend on the initial condition x 0, the minimal value of the quadratic criterion is such that

J* = x;,Pxo. (11)

If we rearrange equation (9) in a closed-loop form, keeping in mind that C'D = O, we have

(A - B2K)'P(A - B.,K) - P + (C - OK) ' (C - DK) = O,

(12)

and we conclude from (5) that P=Lo and IIHIbZ=J * provided we choose B~=x o. Note that the necessary optimality conditions for (P1) are given by equations (4)-(5) and

[(B~LoBz + D 'D)K - B~LoA]L~ = 0, (13)

being thus related to the solution of (P2). Indeed, from (13), if L~ is non singular then the unique solution is given by (8), which is the optimal solution for (P2) and does not depend on the initial condition x o. This fact will always occur in case (A~,B~) is observable or rank (B0=n (a sufficient condition). This is assumed throughout this paper.

3. Main results In this section, we propose a new convex programming

problem which is equivalent to (P1) in a sense to be defined in the sequel. From the above discussion, this is necessary in order to circumvent the non convexity of 9g. First of all, let us.introduce the following extended matrices FE~.~ pxp, p ~ m +n and G ~pxm

o:[0,] .4) as well as the symmetric matrices Q ~pp, R ~'~PP

o0o] .,, and the set ~:

% --~ {'W= W" ~-O:v'[F~/F' - 'W+ Q]v Lc such that

AdWA" I - W + BIB ~ ~ O. (19)

Keeping in mind that Act = A - B2K and developing (19), it gives, for any x ,~"

x'[AWA' - W - B2KWA' - AWK'B6

+ B2KWK'B~ + BtB~]x

Br ie f Paper 227

stabilizing control gain. We just have to prove that this state feedback is indeed the optimal one. First, let us prove that for K=W~W~ 1, VOW%, we have I[HII~O is a sufficiently small parameter used to approximate q~ by a closed convex set and ~2:'~ ~[ .

To illustrate the proposed method, let us consmer the following discrete-time system:

A= -0.2113 0.0087 0.4524] [0.6135 0.0824 0.8096 0.8075|, B2= 0.2749 0.7599 0.8474 0.4832_1 0.8807

C= 0 1 , D= 0 ,

0

0

0.6538] 0.4899[, 0.7741 ]

and B I=I . Since the eigenvalues of matrix A are (0.3827, -0.4919, 1.6133), it is open-loop unstable. Using the discrete-time Riccati equation, we get

KRiccati=[0.2968 0.3758 0.3114] L0.2302 0.4953 0.4997J'

and IIH112~=5.2448. With the proposed algorithm, the

AUTO 29:1-P

228 Br ie f Paper

optimal solution of (P3) is calculated to be

F 1.4582 -0.3930 -0.0926 0.2597 0.0918q

1-0.3930 1.6316 -0.1254 0.4629 0.6612 /

~t~'*= / -0 .0926 -0.1254 1.1972 0.3013 0.51511,

/ 0.2597 0.4629 0.3013 0.3489 0.4411 /

1 0.0918 0.6612 0.5151 0.4411 0.60841

providing the control gain

K .=[0.3006 0.3803 0.3148] 0.2292 0.4989 0.5002 J'

and IIHll 2 = 5.2452. Comparing both numerical solutions, we verify relative errors of about 0.8% in the control gain norm and about 0.007% in the ~g2 norm.

Now, let us suppose that one of the actuators may fail. Obviously, the optimal solution of the discrete-time LQP provides no guarantee for stability in this case. However, Theorem 3 can easily handle this possibility by defining

----&- (B21 , B22 , B23 } where

L !1 B21 = 0.2749 0.48991, B22 = 0.2749 , 0.8807 0.7741 / 0.8807 B23 = 0.4899 | .

0.7741_1

Considering the set ~2 and solving (P4), we have found /~I = 11.8127 and the control gain

[0.2966 0.6356 0.7405] Kr=L0.4823 0.6592 0.4925J'

that guarantees the closed-loop asymptotical stability under the actuators failure previously defined. Furthermore, for each matrix B2~, i = 1, 2, 3, the associated ~2 norm are given by

B21 : IIHII 2 = 7.0560,

B22 : IIHll 2 = 8.6285,

B23 : IIHII 2 = 8.4779,

making evident that /~r is a ~-norm upper bound indeed. Table 1 shows the closed-loop eigenvahies in two situations, namely B2 = B21 (the nominal one) and B2 = B22 (actuator-2 failure). It is easy to see that under the last contingency the closed-loop system with gRieeat i becomes unstable, and the same does not occur for the closed-loop system with K;. In Figs 2 and 3, we show the impulse response (and unitary impulse has been applied to the second component of w) of the closed-loop system with KRieeat i and K r, respectively, supposing that actuator 2 (corresponding to the second component of u) fails at k = 6 and remains inactive for k > 6. The unstable behavior is obvious when the Riccati control gain is used.

5. Conclusion In this paper we have proved that the optimal state

feedback solution of the discrete-time LQP can be determined by means of a convex problem. This is an important result, mainly due to two facts. First, for a given

TABLE 1. CLOSED-LOOP EIGENVALUES

Contingencies KRiccat i gf

-0.4826 0.4153 Both actuators 0.2085 -0.5326

0.4385 -0.4069

-0.4861 0.4047 Only first actuator 0.3347 0.5973

1.0959 -0.5068

LQR wi th folLure at k :6

2 I 1.0 .' o.e .- 0.6 . /

.,' I 0.4 ..' / "

0.2 " . ' " ' " " s ' J " ]i?. :'...~,~. "'. , . . . . ~:: : : : :=%- ' ' " '

-0.2

-0,4

-o . s I I I i O IO 20 30 40 50

/r

FIG. 2. Impulse response----Kruc~ti.

Robust cont ro l w i th faiLure ot k : 6 i.o

o.6

o.6 "! g 0.4 ii

"~ 0 : :' -0.2

i -0 ,4 (~ -0.6

-0.8 -i.o I I I I I

0 IO 20 30 40 50

k

FIG. 3. Impulse response--Kf.

60

60

pair (,4, B2), all stabilizing state feedback gains can be parametrized over a convex set. Second, the convexity of the YCz discrete-time control problem allows to solve it by using the most powerful mathematical programming methods. As a by-product of this fact, we claim that additional structural constraints can be easily handled. For instance, we defined and solved a problem involving actuator's failure whose solution, to our knowledge, was not available until this time in the literature. In this sense we want to emphasize that the numerical procedure proposed here is specially addressed for solving "non-classical" ~ optimal control problems, including additional convex constraints (actuators/sensors failure, robustness, uncertain systems control . . . . ) which cannot be solved by means of the algebraic Riccati equation.

References Anderson, B. D. O. and J. B. Moore (1971). Linear Optimal

Control. Prentice Hall, Englewood Cliffs, NJ. Bernussou, J., P. L. D. Peres and J. C. Geromel (1989). A

linear programming oriented procedure for quadratic stabilization of uncertain systems. Systems and Control Letters, 13, 65-72.

Dorato, P. and A. H. Levis (1971). Optimal linear regulators: the discrete time case. IEEE Trans. on Aut. Control, 16, 613-620.

Geromel, J. C., P. L. D. Peres and J. Bernussou (1991). On a convex parameter space method for linear control design of uncertain systems. SIAM J. on Control and Optimiz., 29, 381-402.

Kwakernaak, H. and R. Sivan (1972). Linear Optimal Control Systems. John Wiley, New York.

Lucnberger, D. G. (1973). Introduction to Linear Program- ruing. Addison-Wesley, Reading, MA.