Constrained MPC Algorithm for Uncertain Time-Varying Systems With ...

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 50, NO. 2, FEBRUARY 2005

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Constrained MPC Algorithm for Uncertain Time-Varying Systems With State-Delay Seung Cheol Jeong and PooGyeon Park

Abstract—In this note, we present a model predictive control (MPC) algorithm for uncertain time-varying systems with input constraints and statedelay. Uncertainty is assumed to be polytopic, and delay is assumed to be unknown but with a known upper bound. For a memoryless state-feedback MPC law, we define an optimization problem that minimizes a cost function and relaxes it to two other optimization problems by finding an upper bound of the cost function. One is solvable and the other is not. We prove equivalence and feasibilities of the two optimization problems under a certain assumption on the weighting matrix. Based on these properties and optimality, we show that feasible MPC from the optimization problems stabilizes the closed-loop system. Then, we present an improved MPC algorithm that includes relaxation procedures of the assumption on the weighting matrix and stabilizes the closed-loop system. Finally, a numerical example illustrates the performance of the proposed algorithm. Index Terms—Closed-loop stability, input constraints, model predictive control (MPC), state delay, uncertainty.

I. INTRODUCTION Model predictive control (MPC), also known as receding horizon control, has received much attention in control societies for its ability to handle both constraints and time-varying behaviors as well as its good tracking performance [1]–[10]. Moreover, it has been recognized as a successful control strategy in industrial fields, especially in chemical process control such as petrochemical, pulp, and paper control. The basic concept of MPC is to solve an open-loop constrained optimization problem at each time instant and implement only the first control move of the solution. This procedure is repeated at the next time instant. It is well known that parameter uncertainties and time-delays cannot be avoided in practice, especially, in many chemical processes where the MPC has been mainly applied. Since the parameter uncertainties

Manuscript received November 21, 2003; revised April 6, 2004, October 4, 2004, and October 5, 2004. Recommended by Associate Editor L. Magni. This work was supported in part by the Ministry of Education of Korea through its BK21 Program. The authors are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTETH), Pohang 790-784, Korea (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TAC.2004.841920

257

and time-delays are frequently the main cause of performance degradation and instability, there has been an increasing interest in the robust control of uncertain time-delay systems in the control literature [11]–[16]. However, only a few MPC algorithms [6], [17] have been published that handle time-delayed systems explicitly. In [6], after designing the novel robust constrained MPC for uncertain systems, the authors argue that the control scheme can be extended to a delayed system in a straightforward manner. This is true when the delay indices are known. However, when the delay indices are unknown, it is not straightforward and not easy to show the feasibility of the on-line optimization problem and guarantee the closed-loop stability, which is the main topic of this note. In [17], a simple receding horizon control is suggested for continuous-time systems with state-delays, where a reduction technique is used so that an optimal problem for state-delay systems can be transformed to an optimal problem for delay-free ordinary systems. They propose a set of linear matrix inequality (LMI) conditions so as to check the closed-loop stability and a numerical algorithm to compute the eigenvalues of general distributed delay systems. However, as the author admits, the guaranteed stability of the suggested control is not known. Moreover, neither model uncertainty nor constraints are considered in the note, hence the application of the controller is limited. In this note, we present an MPC algorithm for uncertain time-varying systems with input constraints and state-delay. The uncertainty is assumed polytopic and the delay is assumed unknown but with a known upper bound. After defining an optimization problem that minimizes a cost function at each time instant, we relax the problem into two other optimization problems that minimize upper bounds of the cost function. One is solvable and the other is not. We prove the equivalence and feasibility of the two optimization problems under a certain assumption on the weighting matrix. Based on these properties and optimality, we show that the MPC obtained through the solvable optimization problem stabilizes the closed-loop system. Then, we present an improved MPC algorithm that includes relaxation procedures of the assumption on the weighting matrix. The note is organized as follows. Section II states target systems, assumptions and the associated problem. Section III supplies a stabilizing MPC algorithm for uncertain delayed systems with constraints. Section IV illustrates the performance of the proposed controller through an example. Finally, in Section V, we make some concluding remarks. Notation: Notations in this note are fairly standard. n and n2m denote the n-dimensional Euclidean space and the set of all n 2 m matrices, respectively. The notation X Y and X > Y (where X and Y are symmetric matrices) means that X 0 Y is positive semidefinite and positive definite, respectively. Inequality between vectors means componentwise inequality. Finally, kxkW denotes xT W x. II. PROBLEM STATEMENTS Consider the following discrete-time uncertain time-varying systems:

x(k + 1) = A(k)x(k) + A(k)x(k 0 d) + B (k)u(k); x(k) = (k); k 2 [0d3 ; 0]

(1)

subject to input constraints

0u u(k) u; u 0; for all k 2 [0; 1) (2) where x(k) 2 n is the state, u(k) 2 m is the control and (k) 2 n

is the initial condition. d is an unknown constant integer representing the number of delay units in the state, but being assumed 0 d d3

0018-9286/$20.00 © 2005 IEEE

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with a known integer d3 . The system matrix [A(k) A(k) unknown but belongs to a polytope at all times k . That is

B (k)] is

[A(k) A(k) B (k)] 2 Cof[A1 A1 B1 ] [A2 A2 B2 ]; . . . ; [Ap Ap Bp ]g (3) where Co denotes the convex hull and [Ai Ai Bi ] are vertices of the

convex hull. It is assumed that the system (1) is stabilizable for the existence of a stabilizing feedback control, and that the state x(k) is available at each time k . The goal of this note is to find a stabilizing state-feedback control u(k) = K (k)x(k) for the system (1) by the MPC strategy. To this end, we define the following min–max problem, which is considered at each time k :

min J (k ) (kmax u(k+j j k);j 0 [A(k+j )A +j )B(k+j )]2

(4)

subject to

1

J (k )

j =0

fkx(k + j j k)kQ + ku(k + j j k)kR g

(5)

(Q1 ; X1 ; Y1 ) = arg Q;X;Y min kxkQ + s:t: 0 F1 (Q; X; Y ); . . . ; 0 Fn (Q; X; Y ) (Q2 ; X2 ; Y2 ) = arg Q;X;Y min kxkQ + s:t: 0 F1 (Q; X; Y ); . . . ; 0 Fn (Q; X; Y )

(6) (7)

M31 X (k ) Q1=2 X (k) R1=2 Y (k)

(3)

( 3)

0 0 0

0 0 0

( 3) ( 3) ( 3) ( 3) ( 3) ( 3) ( 3) ( 3) ( 3) X d (k ) ( 3) ( 3) 0 (k)I (3) 0 0 (k )I

X d (k ) ( 3) A Xd (k) X (k)

(10) where M31 = A X (k)+ B Y (k); = 1; . . . ; p, then the worst value of the cost function J (k) is bounded by (see Appendix I)

max J (k ) [A(k+j )A (k+j )B(k+j )]2 ;j 0

V (x (k j k ); P (k ); P d (k ); d )

(11)

(9)

In this section, we first derive an upper bound on the worst value of the cost to relax the original min–max problem (4) subject to (5)–(7) to another minimization problem. This can easily be done by following the approach given in [6]. Consider a quadratic function

kx(k + j j k)kP (k)

kx(k + j 0 i j k)kP (k) ; P (k) > 0; Pd (k) > 0

of the current and predicted states

(k)P 01 (k) there exist X (k)

d

i=1

kx(k 0 i j k)kP (k) :

(12)

Remark 1: If the delay index d is assumed to be known, we can replace the cost upper bound in (12) with an upper bound associated with a “full-block” matrix and show the closed-loop stability without difficulty. Therefore, the cost bound of the form (12) is necessarily conservative. If the delay index d is assumed to be unknown, however, the situation is somewhat different. Even under this assumption, the delay structure of the model does not ensure that the solution of the corresponding Riccati equation is block diagonal as would be required by (12). However, introducing a cost upper bound associated with a full-block matrix, rather than a block-diagonal matrix as in (12), makes it impossible to show the closed-loop stability with the approach which will be proposed in this note. If one might show the closed-loop stability via a cost upper bound associated with a full-block matrix, the closed-loop performance would be significantly enhanced, which belongs to future works. A. Stabilizing MPC Scheme for Uncertain Delayed Systems

III. PROPOSED MPC ALGORITHM

V (x (k + j j k ); P (k ); P d (k ); d )

V (x(k j k); P (k); Pd (k); d) = kx(k j k)kP (k)

(8)

where Q; X and Y denote optimization variables; and denote constant terms; Fi (Q; X; Y ) denote functions of Q; X and Y . The difference of the two optimization problems is only and . Then, if one of the two problems is solvable, so is the other. Moreover, minimizing arguments of the two problems are identical, that is, Q1 = Q2 ; X1 = X2 and Y1 = Y2 .

i=1

0

0

+

where Q > 0 and R > 0 are given symmetric matrices; x(k + j j k) and u(k + j j k) denote predicted variables of the state and the input, respectively, with x(k j k) = x(k) and x(k 0 i j k) = x(k 0 i) for i 1. Before ending this section, we present a fact which will play an important role in the next section. Fact 2.1: Let us consider the following two optimization problems:

+

X (k )

where

x (k + 1 + j j k ) = A(k + j )x(k + j j k) + A(k + j ) 2 x (k + j 0 d j k ) + B (k + j )u (k + j j k ) 0 u u(k + j j k)(= K (k)x(k + j j k)) u; j 2 [0; 1)

d

0; Y (k) K (k)X (k) and (k) > 0satisfying the following inequal-

ities:

x(k + j 0 i j k); i = 0; . . . d. If > 0; Xd (k) (k)Pd01 (k) >

Based on (10) and (11), the original min–max problem (4) can be redefined to the following optimization problem that minimizes an upper bound on the worst value of the original cost function J (k):

P (P (k); Pd (k); d) : K(k);Pmin V (x (k j k ); P (k ); P d (k ); d ) (k);P (k) s:t: (6), (7); and (10):

(13)

Unfortunately, we cannot directly solve the problem P (P (k); Pd (k); d) since V (x(k j k); P (k); Pd (k); d) depends on the unknown constant d. Instead, based on Fact 2.1, we can obtain the solution (or the optimizer) of P (P (k); Pd (k); d) by solving P (P (k); Pd (k); d3 ).

At this point, we make an assumption under which we will present two lemmas related to property and feasibility of P (P (k); Pd (k); d) and one theorem for the closed-loop stability. This assumption will be relaxed in the next section. A1: The matrix Pd (k) in P (P (k); Pd (k); d) is fixed to a constant matrix Pd at all times k . Remark 2: Under Assumption A1, P (P (k); Pd (k); d) and V (x(k j k); P (k); Pd (k); d) become P (P (k); Pd ; d) and V (x(k j k); P (k); Pd ; d), respectively. An easy way for choosing Pd is to solve the problem P (P (k); Pd (k); d3 ) without A1 at the initial time k = 0, and to set Pd Pd (0).


Lemma 3.1: (Equivalence of P (P (k); Pd ; d) and P (P (k); Pd ; d3 )) Suppose that A1 is satisfied. If P (P (k); Pd ; d3 ) is solvable at time k , then so is P (P (k); Pd ; d), and vice versa. Moreover, feasible

solutions of the two problems are identical. Therefore, we can obtain the solution of P (P (k); Pd ; d) by solving P (P (k); Pd ; d3 ). Proof: Since x(k 0 i j k); i = 1; . . . ; d3 are known a priori and Pd (k) is fixed to Pd at all times k; V (x(k j k); P (k); Pd (k); d) and V (x(k j k); P (k); Pd (k); d3 ) can be rewritten as V (x(k j k); P (k); Pd ; d) = kx(k j k)kP (k) + and V (x(k j k); P (k); Pd ; d3 ) = kx(k j k)kP (k) + respectively, where and are known constant terms. Since the difference of the two problems is only and , by Fact 2.1, feasibility of one of the problems P (P (k); Pd ; d) and P (P (k); Pd ; d3) at time k means feasibility of the other at time k . Feasible solutions of the two problems are identical. Lemma 3.2: (Feasibility of P (P (k); Pd ; d) and P (P (k); Pd ; d3 )) Suppose that A1 is satisfied. Any feasible solution of P (P (k); Pd ; d3 ) at time k is also a feasible solution of P (P (t); Pd ; d) and P (P (t); Pd ; d3) at all time instants t k. Thus, if the problem P (P (k); Pd ; d3) is3 feasible at time k, both problems P (P (t); Pd ; d) and P (P (t); Pd ; d ) are feasible at all time instants t k . Proof: Suppose that a feasible control sequence u(k + j j k); j 0 exists in P (P (k); Pd ; d3 ) at time k. Then, at the next time k + 1, choose the following control sequence:

u(k + 1 + j j k + 1) = u3 (k + 1 + j j k); j 0:

(14)

The control sequence satisfies the input constraint (7) at time k + 1, which implies the problem P (P (k); Pd ; d3 ) has a feasible solution at time k + 1. Hence, by induction, if the problem P (P (k); Pd ; d3 ) is feasible at time k , we observe that the problem is feasible at all time instants to be grater than k . Moreover, by Lemma 3.1, u(k + j j k); j 1 is also a feasible control sequence for the problem P (P (k); Pd ; d) at time k . The rest of the arguments for P (P (k); Pd ; d) are same as those of the problem P (P (k); Pd ; d3 ). This completes the proof. Based on Lemmas 3.1 and 3.2, robust asymptotic stability of the closed-loop system is guaranteed in the following theorem. Theorem 3.1: (Closed-loop stability) Suppose that A1 is satisfied. If the optimization problem P (P (k); Pd ; d3 ) is feasible at the initial time k = 0, then the feasible MPC from P (P (k); Pd ; d3 ) robustly asymptotically stabilizes the closed-loop system. Proof: We first show that the zero state is attractive. By Lemma 3.2, P (P (k); Pd ; d) and P (P (k); Pd ; d3 ) are always feasible when P 3(P (k); Pd ; d3) is feasible at the initial time k = 0. Let3 P 3 (k) and P (k + 1) denote the optimal values of P (P (k); Pd ; d ) at time k and k + 1, respectively. By Lemma 3.1, this implies that P 3 (k) and P 3 (k + 1) are also optimal for P (P (k); Pd ; d) at time k and k + 1, respectively. Next, let us consider a quadratic function

V (x(k j k); P 3 (k); Pd ; d) = kx(k j k)kP

+

(k)

d

i=1

kx(k 0 i j k)kP

(15)

for which we have the following inequality from optimality:

V (x(k + 1 j k + 1); P 3 (k + 1); Pd ; d) V (x(k + 1 j k + 1); P 3 (k); Pd ; d) (16) since P 3 (k + 1) is optimal while P 3 (k) is only feasible at time k + 1. Besides, it follows from (28) that

V (x(k + 1 j k); P 3 (k); Pd ; d) V (x(k j k); P 3 (k); Pd ; d) 0 (kx(k j k)kQ + ku(k j k)kR )

(17)

259

for any [A(k) A(k) B (k)] 2 . Since x(k +1 j k +1) equals [A(k)+ B (k)K (k)]x(k j k) + A(k)x(k 0 d j k) for some [A(k)A(k)B (k)] 2

, and x(k + 1 0 i j k + 1) = x(k + 1 0 i j k); i = 1; . . . ; d, (17) must hold with x(k + 1 j k + 1) in place of x(k + 1 j k). Combining this with (16), we must have

V (x(k + 1 j k + 1); P 3 (k + 1); Pd ; d) V (x(k j k); P 3 (k); Pd ; d) 0 (kx(k j k)kQ + ku(k j k)kR ): Summing inequality (18) from k

V (x(i j i); P 3 (i); Pd ; d) +

(18)

= 0 to k = i 0 1 yields

i01

fkx(k j k)kQ k=0 + ku(k j k)kRg V (x(0 j 0); P 3 (0); Pd ; d): (19)

Since V (x(i j i); P 3 (i); Pd ; d) 0 and Q > 0 and R > 0 and the left-hand side of (19) is bounded above by the constant V (x(0 0); P 3 (0); Pd ; d); x(i) = x(i j i) and u(i) = u(i j i) must go to zero as i goes to infinity. This completes the proof. Remark 3: A more precise representation of V (x(k j k); P 3 (k); Pd ; d) is V (x(k 0 i j k); i = 0; . . . ; d; P 3 (k); Pd ; d) to show that V is a quadratic function of the full state x(k 0 i j k); i = 0; . . . ; d of (1), but for notational simplicity we use V (x(k j k); P 3 (k); Pd ; d). B. MPC Algorithm for Uncertain Delayed Systems Up until now, the asymptotic stability of the closed-loop system has been proved through the monotonic decreasing property of V (x(k j k); P (k); Pd ; d) for all k 0. However, it may be conservative to use a constant matrix Pd at all times k . In this section, hence, the MPC algorithm is modified so that Pd is updated whenever a certain condition is satisfied. The following lemma plays an important role in guaranteeing the stability of the MPC algorithm that will be proposed. Lemma 3.3: Let P 3 (k) and P 3 (k + 1) denote the optimal values of P (P (k); Pd ; d3 ) at time k and P (P (k + 1); Pd ; d3 ) at time k + 1, respectively. Let Pd3 (k + 1) denote the optimal value of P (P 3 (k + 1); Pd (k +1); d3 ) at time k +1. If the following inequality is satisfied:

Pd3 (k + 1) < Pd

(20)

V (x(k j k); P 3 (k); Pd ; d) V (x(k + 1 j k + 1); P 3 (k + 1); Pd ; d) + ( kx (k j k )k Q + ku (k j k )k R ) > V (x(k + 1 j k + 1); P 3 (k + 1); Pd3 (k + 1); d) + ( kx (k j k )k Q + ku (k j k )k R ):

(21)

we have

Proof: By Lemma 3.1, P 3 (k) and P 3 (k +1) are also the optimal values of P1 (P (k); Pd ) at time k and P1 (P (k +1); Pd ) at time k +1, respectively. Hence, (21) follows from (18) and (20). Now, we summarize the proposed stabilizing MPC algorithm as follows. 1) Constrained MPC Algorithm for Uncertain Delayed Systems (CMPC-UDS): 1) (Initialization) at time k = 0, find Pd (0) by solving the optimization problem P (P (k); Pd (k); d3 ), and set Pd = Pd (0) and ag = 1.

260


2) (Generic) at time k 0, find K (k) and P (k) by solving the optimization problem P (P (k); Pd (k) = Pd ; d3 ), and set K 3 (k) = K (k) and P 3 (k) = P (k). 3) Find K (k) and Pd (k) by solving the optimization problem P (P (k) = P 3 (k); Pd (k); d3 ). If Pd (k) < Pd , set K 3 (k) = K (k); Pd = Pd (k); ag = ag + 1. Otherwise, go to step 5). 4) If ag rn , go to step 2). Otherwise, go to step 5). Here rn is a given fixed integer that represents the maximum repetition number of steps 2) and 3). 5) Apply the state-feedback control u(k) = K 3 (k)x(k) to the system. 6) At the next time, set ag = 1 and repeat steps 2)–5). Remark 4: In the previous algorithm, if Pd (k) < Pd and ag rn in steps 3) and 4), then steps 2) and 3) are repeated at most rn times at each time k . Hence, we can see that the size of rn trades off computational burden versus performance. Theorem 3.2: If the optimization problem P (P (k); Pd ; d3 ) is feasible at the initial time k = 0, then the feasible MPC from CMPC-UDS robustly asymptotically stabilizes the closed-loop system. Proof: When Pd (k) Pd and, thus, Pd is not updated in step 3), V (x(k j k); P 3 (k); Pd ; d) V (x(k + 1 j k + 1); P 3 (k + 1); Pd ; d) + (kx(k j k)kQ + ku(k j k)kR ) is satisfied by Theorem 3.1. When Pd (k) < Pd and, thus, Pd is updated in step 3), V (x(k j k); P 3 (k); Pd ; d) > V (x(k + 1 j k + 1); P 3 (k + 1); Pd ( Pd3 (k + 1)); d) + (kx(k j k)kQ + ku(k j k)kR ) holds by Lemma 3.3. Hence, by the same path as the proof of Theorem 3.1, the closed-loop system is robustly asymptotically stable. Lemma 3.4: The optimization problem P (P (k); Pd (k); d3 ) can be solved by the following semidefinite program:

min (k);X (k);Y (k);Z (k);X

(k)

(k )

(22)

subject to (10) and

1

0

0

x (k j k ) x (k 0 1 j k ) .. . x (k 0 d 3 j k ) Z (k ) Y (k ) Y T (k ) X (k )

x T (k j k ) x T (k 0 1 j k ) X (k ) 0 0 X d (k ) .. .. . .

0

0

111 111 111 ..

x T (k 0 d 3 j k )

.

111

0 0 .. .

X d (k ) (23)

Zii (k) u i ; 2

i = 1 ; 2; . . . ; m

(24)

where = 1; . . . ; p; 0 < X (k) = (k)P 01 (k); 0 < Xd (k) =

(k)Pd01 (k); Y (k) = K (k)X (k) and Zii (k) is the ith diagonal entry i is the ith element of u. of Z (k) and u Proof: See Appendix II. Remark 5: In the optimization problem of Lemma 3.4, X (k) is replaced by (k)P 01 (k) when P (k) is not an LMI variable [in step 3) of CMPC-UDS]. And Xd (k) is replaced by (k)Pd01 (k) when Pd (k) is not an LMI variable [in step 2) of CMPC-UDS]. Remark 6: If one designed an MPC by solving the optimization problem P (P (k); Pd (k; d3 ) without Assumption A1 at each time k , the closed-loop performance would be better than that of the CMPCUDS. However, its stability is not guaranteed theoretically in this note. IV. NUMERICAL EXAMPLE In this section, a numerical example is presented to illustrate the performance of the proposed MPC algorithms. The example is a backing up control of a computer simulated truck-trailer. It is well known that

backing up control of a truck-trailer is very difficult since its dynamics is nonlinear and unstable even without delay. Our goal is to design a constrained MPC such that the closed-loop system is asymptotically stable and to show that the proposed control method yields a better performance than an MPC designed without consideration on state-delay (such as [6]). The model of truck-trailer is given by [18]

v t v t v t x1 (t) 0 (1 0 a) x 1 (t 0 ) + u ( t) Lt0 Lt0 lt0 v t v t x_ 2 (t) = a x1 (t) + (1 0 a) x 1 (t 0 ) Lt0 Lt0 v t sin x2 (t) + a 2vLt x1 (t) + (1 0 a) 2vLt x1 (t 0 ) x_ 3 (t) = t0 x_ 1 (t) = 0a

(25) where the physical variables x1 ; x2 ; x3 , and u mean angle difference between the truck and the trailer, angle of the trailer, y -coordinate of the rear end of the trailer and steering angle, respectively; x1 is assumed to be perturbed by time-delay; ju(t)j . The model parameters are given as l = 2:8; L = 5:5; v = 01:0; t = 2:0; t0 = 0:5. The constant a is the retarded coefficient ranging a 2 [0; 1]. The limits 1 and 0 correspond to no delay term and to a completed delay term, respectively. In this example, we assume a = 0:7. With sampling time T = 0:1 sec., we transform the nonlinear time-delay system with the following discrete-time polytopic uncertain system:

x(k + 1) = A(k)x(k) + A(k)x(k 0 d) + B (k)u(k)

(26)

where

[A(k) A(k) B (k)] 2 Cof[A1 A1 1:0509 0 0 0 A1 = 00:0509 1:0000 0:0509 00:4000 1:0000 0:0218 0 0 A1 = 00:0218 0 0 0:0218 0 0 1:0509 0 0 A2 = 00:0509 1:0000 0 0:0810 00:6366 1:0000 0:0218 0 0 A2 = 00:0218 0 0 0:0347 0 0 00:1429 B1 = 0 0 00:1429 B2 = : 0 0

B1 ][A2

A2

B2 ]

Simulation parameters are as follows; the initial value of the state is x(0) = [0:5 0:75 0 5]T ; the state and the input weighting matrices are Q = diag(10; 10; 10) and R = 1; d = 4; d3 = 10; rn = 1. Fig. 1 shows a comparison between CMPC-UDS and CMPC-UDS without the iteration of steps 2) and 3), where we can see that the set point tracking performance as well as the cost is enhanced by the iteration of steps 2) and 3) (the first and the second figures of Fig. 1). We think that the improvement in performance due to the iteration of steps 2) and 3) is related to the delay matrix A(k). The bigger the size of the matrix A(k), the more the improvement in performance due to


261

Fig. 2. Trajectories of truck-trailer. (a) Proposed CMPC-UDS. (b) MPC designed without consideration of delay.

that the proposed CMPC-UDS outperforms the MPC designed without consideration on state-delay. V. CONCLUDING REMARKS

Fig. 1. Comparison of CMPC-UDS (dotted line) and CMPC-UDS without the iteration of steps 2) and 3) (solid line). Parts (a) and (b) are of (26) and (c) is of the artificial system with 2 3 A and 2 3 A instead of A and A as the delay matrices.

the iteration of steps 2) and 3). Therefore, the improvement shall be more evident in a system with a relatively big delay matrix. We can see that the improvement is more evident in the third figure of Fig. 1, which is a simulation result for the artificial system with 2 3 A1 and 2 3 A2 instead of A1 and A2 as the delay matrices. Fig. 2 shows the truck-trailer moves of the proposed CMPC-UDS and an MPC designed without consideration on state-delay, with d = 8. It is seen from Fig. 2

In this note, we presented an MPC algorithm for uncertain timevarying systems with input constraints and state-delay. After finding an upper bound of the cost function, we relaxed the original min–max problem into two optimization problems. The equivalence and feasibility of the two optimization problems were proved under the Assumption A1. Based on these properties and optimality, closed-loop stability of the proposed MPC scheme was proved. Then, we developed a stabilizing MPC algorithm based on the previous MPC scheme so as to reduce the conservatism due to the assumption A1. Through an example, we showed that the proposed MPC algorithm is superior to the MPC algorithm without consideration on state-delay. One of our future works is the extension of the results of this note to more general uncertain systems with both state- and input-delays or distributed delays. Design of delay-dependent MPC algorithms is another topic.

262


APPENDIX I

APPENDIX II

The difference of the quadratic function V (x(k + j j k ); P (k); Pd (k); d) with state-feedback control u(k + j j k) = K (k)x(k + j j k) is given by

We prove (22)–(24) here. Minimization of V (x(k j k); P (k); Pd (k); d3 ) is equivalent to

V (x(k + j +1 j k); P (k); Pd (k); d) 0 V (x (k + j j k ); P (k ); P d (k ); d ) = kx(k + j +1 j k)kP (k) 0 kx(k + j j k)kP (k) d

+

i=1 d

0 =

2 2

i=1

kx (k + j + 1 0 i j k )k P kx (k + j 0 i j k )k P

(k)

(k)

T 0 M(311) AkAc((kk ++ jj))kP (k)A0(kP+(kj)) d P (k)

(27)

= kAc (k + j )kP (k) 0 P (k)+ Pd (k)+Q + K T (k)RK (k),

V (x (k + j + 1 j k ); P (k ); P d (k ); d ) 0 V (x (k + j j k ); P (k ); P d (k ); d ) 0(kx(k + j j k)kQ + ku(k + j j k)kR ):

(28)

For J (k) to be finite, we must have x(1 j k) = 0 and, hence, V (x(1 j k); P (k); Pd (k); d) = 0. Summing both sides of the previous inequality from j = 0 to j = 1 yields 0V (x(k j k); P (k); Pd (k); d) 0J (k), we have (11). Now, consider the inequality condition (27). Defining P (k) =

(k)X 01 (k); X (k) > 0; (k) > 0 and Pd (k) =

(k)Xd01(k); Xd (k) > 0, pre- and postmultiplying by diag( 01=2 (k)X (k); 01=2 (k)Xd (k)), defining Y (k) = K (k)X (k) and using the Schur complement, we see that (27) is equivalent to

0

0

M21 X (k ) Q1=2 X (k) R1=2 Y (k)

x T (k j k ) x T (k 0 1 j k ) 1 1 1 x T (k 0 d 3 j k ) x (k j k ) X (k ) 0 111 0 x (k 0 1 j k ) 0 X d (k ) 111 0 .. .. .. .. .. . . . . . x (k 0 d 3 j k ) 0 0 1 1 1 X d (k )

1

where Ac (k + j ) = A(k + j ) + B (k + j )K (k). If the following inequality is satisfied for some P (k); Pd (k); K (k) and for any [A(k + j )A(k + j )B (k + j )] 2 and for all j 0;

X (k )

(31)

d subject to (k) kx(k j k)kP (k) + i=1 kx(k 0 i j k )kP (k) , from which, by defining X (k) = (k)P 01 (k) > 0; Xd (k) =

(k)Pd01 (k) > 0 and using the Schur complement, we have the LMI shown in (32)

T x (k + j j k ) x (k + j 0 d j k ) kAc (k + j )kP (k) 0 P (k)+ Pd ATc (k + j )P (k)A(k + j ) AT (k + j )P (k)Ac (k + j ) kA(k + j )kP (k) 0 Pd (k) x (k + j j k ) x (k + j 0 d j k )

where M11 we have

Minimize (k)

( 3)

( 3)

0 0 0

0 0 0

X d (k ) ( 3) A(k)Xd (k) X (k)

( 3) ( 3) ( 3) X d (k ) 0 0

( 3) ( 3) ( 3) ( 3)

(k)I 0

( 3) ( 3) ( 3) ( 3) (3)

(k )I (29)

where M21 = A(k)X (k) + B (k)Y (k). Since (29) is affine in [A(k) A(k) B (k)], it is satisfied for all [A(k) A(k) B (k)] 2 Cof[A1 A1 B1 ] [A2 A2 B2 ]; . . . ; [Ap Ap Bp ]g (30) if and only if there exist X (k) > 0; Xd (k) > 0; Y (k) and (k) satisfying (10). This completes the proof.

= K (k )X (k )

0

(32) which establishes (22) and (23). The input constraint (7) can be easily casted into the LMI (24) by using the so called invariant ellipsoid. Hence, we omit the detailed procedure. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their pertinent comments and suggestions. REFERENCES [1] W. H. Kwon and A. E. Peason, “A modified quadratic cost problem and feedback stabilization of a linear system,” IEEE Trans. Autom. Control, vol. AC-22, no. 5, pp. 838–842, May 1977. [2] J. Richalet, A. Rault, J. L. Testud, and J. Papon, “Model predictive heuristic control: Applications to industrial processes,” Automatica, vol. 14, pp. 413–428, May 1978. [3] D. W. Clarke, C. Mohtadi, and S. Tuffs, “Generalized predictive control: Part 1: The basic algorithm,” Automatica, vol. 23, pp. 137–148, Feb. 1987. [4] C. E. Garcia, D. M. Prett, and M. Morari, “Model predictive control: Theory and practice—A survey,” Automatica, vol. 25. [5] J. B. Rawlings and K. R. Muske, “The stability of constrained receding horizon control,” IEEE Trans. Autom. Control, vol. 38, no. 10, pp. 1512–1516, Oct. 1993. [6] M. V. Kothare, V. Balakrishnan, and M. Morari, “Robust constrained model predictive control using linear matrix inequalities,” Automatica, vol. 32, pp. 1361–1379, Oct. 1996. [7] J. W. Lee, W. H. Kwon, and J. H. Choi, “On stability of constrained receding horizon control with finite terminal weighting matrix,” Automatica, vol. 34, pp. 1607–1612, Dec. 1998. [8] Y. Lu and Y. Arkun, “Quasimin-max MPC algorithms for LPV systems,” Automatica, vol. 36, pp. 527–540, Apr. 2000. [9] K. B. Kim, “Implementation of stabilizing receding horizon controls for linear time-varying systems,” Automatica, vol. 38, pp. 1705–1711, Oct. 2002. [10] P. G. Park and S. C. Jeong, “Constrained MPC for LPV systems with bounded rates of parameter variations,” Automatica, vol. 40, pp. 865–872, Jul. 2004. [11] S. S. L. Chang and T. K. C. Peng, “Adaptive guaranteed cost control of systems with uncertain parameters,” IEEE Trans. Autom. Control, vol. AC-17, no. 4, pp. 474–483, Apr. 1972. [12] I. R. Petersen and D. McFarlane, “Optimal guaranteed cost control and filtering for uncertain linear systems,” IEEE Trans. Autom. Control, vol. 39, no. 9, pp. 1971–1977, Sep. 1994. [13] S. O. R. Moheimani and I. R. Petersen, “Optimal quadratic guaranteed cost control of a class uncertain time-delay systems,” Inst. Elect. Eng. Proc. Control Theory Appl., vol. 144, pp. 183–188, Feb. 1997. [14] L. Yu and J. Chu, “A LMI approach to guaranteed cost control of linear uncertain time-delay systems,” Automatica, vol. 35, pp. 1155–1159, Jun. 1999.


[15] X. Guan, Z. Lin, and G. Duan, “Robust guaranteed cost control for discrete-time uncertain systems with delay,” Inst. Elect. Eng. Proc. Control Theory Appl., vol. 146, pp. 598–602, Jun. 1999. [16] M. S. Mahmoud and L. Xie, “Guaranteed cost control of uncertain discrete systems with delays,” Int. J. Control, vol. 73, pp. 105–114, Feb. 2000. [17] W. H. Kwon, J. W. Kang, Y. S. Lee, and T. S. Moon, “A simple receding horizon control for state delayed systems and its stability criterion,” J. Process Control, vol. 13, pp. 539–551, Jun. 2003. [18] Y. Y. Cao and P. M. Frank, “Stability analysis and synthesis of nonlinear time-delay systems via linear takagi-sugeno fuzzy models,” Fuzzy Sets Syst., vol. 124, pp. 213–229, Dec. 2001.

263

A rate-based power control model is adopted in [7] and [10] for the uplink (mobile to base) power adjustment in an isolated cell, and the optimal control law involves centralized information, i.e., the control input of each user depends on the state variable of all the users. In [7], the power adjustment rate is bounded and the associated stochastic control problem is analyzed by viscosity solutions [8] and suboptimal control laws are computed using a perturbation technique. Based on [7] and [10], in this note we analyze the optimal control problem with unbounded controls and quadratic type costs and construct simplified suboptimal versions of the resulting control laws. Numerical methods for computing the suboptimal control laws are investigated under a small noise condition. Finally, a single user based suboptimal control design is applied to systems with large populations. II. PROBLEM STATEMENT

Computationally Tractable Stochastic Power Control Laws in Wireless Communications Minyi Huang, Roland P. Malhamé, and Peter E. Caines Abstract—This note considers uplink power control for lognormal fading channels by a stochastic control approach. We seek approximate solutions to the Hamilton–Jacobi–Bellman (HJB) equation in terms of a local polynomial approximation, and solve a two user system for illustration. An important issue for practical systems is to reduce implementational complexity when operating in large population conditions; we show that a single user based control design may be applied to such systems for partially decentralized optimization. Index Terms—Hamilton–Jacobi–Bellman (HJB) equations, lognormal fading channels, power control.

I. INTRODUCTION In wireless communication systems, power control is applied to regulate the transmitted power of mobile users to achieve a certain quality of service (QoS) in the presence of fading channels and user mobility. This is particularly crucial in code division multiple access (CDMA) systems for suppression of multiuser interference. Early research efforts on power control mainly dealt with static channel models subject to user signal-to-interference ratio (SIR) requirements (see, e.g., [14]). In [1], power control was studied by a noncooperative game theoretic approach for static channels. Downlink power control was analyzed in [2] for time-varying channels by averaging methods. The recent work [3] applied dynamic programming to find power control policies where each user’s channel gain is described by a finite state Markov chain. Dynamic power control for lognormal fading channels has been analyzed in a stochastic optimal control framework in [7], [10], [11]. Manuscript received January 24, 2003; revised July 8, 2003 and June 17, 2004. Recommended by Associate Editor R. S. Srikant. This work was supported by the Mathematics of Information Technology and Complex Systems (NCE-MITACS) Program 1999-2002 and by the Natural Sciences and Engineering Council of Canada under Grants 1329-00 and 1361-00. M. Huang was with the Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada. He is now with the Department of Electrical and Electronic Engineering, University of Melbourne, Parkville, 3010 VIC, Australia (e-mail: [email protected]). R. P. Malhamé is with the Department of Electrical Engineering, École Polytechnique de Montréal, Montreal, QC H3C 3A7, Canada, and is affiliated with GERAD, Montreal, QC, Canada (e-mail: [email protected]). P. E. Caines is withthe Department of Electrical and Computer Engineering, McGill University, Montreal, QC H3A 2A7, Canada, and is affiliated with GERAD, Montreal, QC, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2004.841918

Let xi (t), 1 i n, denote the power attenuation (expressed in dBs and scaled to the natural logarithm basis) at time t of the ith mobile user of a network and let i (t) = ex (t) denote the actual attenuation. Based on [4], we model the power attenuation dynamics of the n users by

dxi = 0 ai (xi + bi )dt + i dwi ; 1 i n; t 0 (1) ai ; bi ; i > 0; where fwi ; 1 i ng are n independent standard Wiener processes, and xi (0), 1 i n, are mutually independent Gaussian random variables which are also independent of the Wiener processes. Since

ai > 0, there is a long-term adjustment of the power attenuation toward the long-term mean 0bi . The pair ai and i indicates the rate of channel

variations and is related to the user mobility level and the volatility of the underlying lognormal shadowing effects [10]. We model the stepwise adjustments (see [12]) of the transmitted power pi from the ith mobile by the so-called rate adjustment model [7]

1 i n:

dpi = ui dt; T

(2)

T

Write x = [x1 ; . . . ; xn ] , p = [p1 ; . . . ; pn ] , and u = [u1 ; . . . ; un ]T . The superscript T denotes transpose of vectors or matrices. In terms of i and positive pi , 1 i n, the ith user’s SIR is given by 0i = (i pi )=( j 6=i j pj + ), where > 0 is the system background noise intensity. A standard communication QoS is given by 0i i for a set of prescribed i > 0, 1 i n, which in turn is equivalent to

00i =

n j =1

i pi j pj +

i ;

1in

(3)

where i = i =(1 + i ). There exists at least one positive power vector satisfying (3) if and only if n i=1 i < 1. In this note we n assume that i=1 i < 1 always holds. For static power optimization, it is plausible to satisfy (3) with lowest possible power consumption, and it can be verified that the equality condition H1): 0i0 = (i pi )=( nj=1 j pj + ) = i , for 1 i n, gives a power vector p minimizing n i=1 pi subject to the constraints (3). Motivated by H1) and following [10], [11], we introduce the averaged discounted integrated cost function

E

0

1 0t e

n i=1

0018-9286/$20.00 © 2005 IEEE

ex pi 0 i

n j =1

ex pj +

2

+ uT Ru dt

(4)