papers/IF04-0888.pdf, International Society of Information Fusion. [3] C. O. Savage ... T. Alamo is with the Departamento de Ingenieria de Sistemas y ...

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Control of Constrained Discrete-Time Gain Systems With Bounded Paul J. Goulart, Eric C. Kerrigan, and Teodoro Alamo

Fig. 3. Difference between all possible integer measurement schedules and the optimal, unachievable schedule with non-integer measurements as given by (47). The star indicates the rounded version of n , as given by (48) which achieves the minimal feasible cost. The x-axis is the index of the cost, on a sorted basis.

Abstract—We consider the problem of designing a control law for a constrained linear system with bounded disturbances that ensures constraint satisfaction over an infinite horizon, while also guaranteeing that the closed-loop system has bounded gain. To this end, we propose a receding horizon control strategy based on the repeated calculation of optimal finite horizon feedback policies. We parameterize these policies such that the input at each time is an affine function of current and prior states, and minimize a worst-case quadratic cost where the disturbance energy is negatively weighted as in control. We show that the resulting receding horizon controller has two advantages over previous results for this problem. First, the policy optimization problem to be solved at each time step can be rendered convex-concave, with a number of decision variables and constraints that grows polynomially with the problem size, thereby making its solution amenable to standard techniques in convex gain of the resulting closed-loop optimization. Second, the achievable system is bounded and non-increasing with increasing control horizon. A numerical example is included to demonstrate the improvement in gain relative to existing methods. achievable Index Terms—Constrained systems, receding horizon control (RHC), robust control, robust optimization, uncertain systems.

VI. CONCLUSION In this technical note, we have provided a collection of optimal measurement scheduling results for a class of linear, scalar Gauss-Markov systems using a terminal cost function. We have provided rigorous proofs for five different measurement scheduling scenarios. In each case we have shown that the resulting optimal scheduling policy is an index policy. As a consequence, this means that the optimal policy is easy to compute. Numerical examples have been given to illustrate the benefits of measurement scheduling for such systems in a number of scenarios. These result show that the gains due to optimal scheduling may be substantial. We have also shown that the optimal results can be used to derive suboptimal scheduling policies that are also easy to compute and provide acceptable performance. ACKNOWLEDGMENT The authors wish to thank Dr. B. Moran, Dr. S. Suvorova, and Dr. H. Schmitt for interesting discussions on this topic.

REFERENCES [1] R. E. Kalman, “A new approach to linear filtering and prediction problems,” Trans. ASME, vol. 82, pp. 35–45, 1960. [2] S. Howard, S. Suvorova, and B. Moran, P. Svensson and J. Schubert, Eds., “Optimal policy for scheduling of Gauss-Markov systems,” in Proc. 7th Int. Conf. Inform. Fusion, Mountain View, CA, Jun. 2004, vol. II, pp. 888–892 [Online]. Available: http://www.fusion2004.foi.se/ papers/IF04-0888.pdf, International Society of Information Fusion [3] C. O. Savage, B. F. La Scala, and B. Moran, “Optimal scheduling for state estimation using a terminal cost function,” in Proc. 9th Int. Conf. Inform. Fusion, Florence, Italy, Jul. 2006, [CD ROM]. [4] D. R. Vaughan, “A nonrecursive algebraic solution for the discrete Riccati equation,” IEEE Trans. Authomat. Control, vol. AC-15, no. 5, pp. 597–599, Oct. 1970. [5] V. Gupta, T. Chung, B. Hassibi, and R. M. Murray, “On a stochastic sensor selection algorithm with applications in sensor scheduling and dynamic sensor coverage,” Automatica, vol. 42, no. 2, pp. 251–260, Feb. 2006. [6] B. Sinopoli, L. Schenato, M. Franceschetti, K. Poolla, M. Jordan, and S. Sastry, “Kalman filtering with intermittent observations,” IEEE Trans. Automat. Control, vol. 49, no. 9, pp. 1453–1464, Sep. 2004.

I. INTRODUCTION This technical note considers the control of linear discrete-time systems subject to bounded additive disturbances on the state, and subject to mixed convex constraints on the states and inputs. Formulation of computationally tractable control laws that guarantee stability and constraint satisfaction for such systems remains an important open problem in the literature [1]. This technical note is motivated in particular by the considerable recent interest in applications of robust preredictive control to the problem of `2 gain minimization (i.e. ceding horizon control) for constrained linear [2]–[6] and nonlinear systems [7]–[9]. It is generally accepted that if disturbances are to be accounted for in the underlying finite horizon control problem, to be solved in a receding horizon scheme, then that problem must be formulated in terms of (generally nonlinear) feedback policies, rather than open-loop control sequences as in [4], [10, Chap. 9]. A standard approach to this problem is to employ methods in dynamic programming to compute a solution to the control problem over a finite horizon [11]. In general, these dynamic programming-based schemes or, alternatively, closed-loop schemes such as [12] based on finite-dimensional optimization, are computationally intractable because the size of the optimization problem required for their solution grows exponentially with the problem size. We propose an alternative scheme based on a class of finite-horizon control policies where the input at each time instant is parameterized as an affine function of current and prior system states; such policies are

H1

Manuscript received April 28, 2006; revised April 03, 2007. Current version published May 13, 2009. This work was supported by the Gates Cambridge Trust, the Royal Academy of Engineering, UK and MCYT-Spain (contract DPI2007-66718-C04-01). Recommended by Associate Editor I. Kolmanovsky. P. J. Goulart and E.C. Kerrigan are with the Department of Aeronautics and Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]; ,[email protected]). T. Alamo is with the Departamento de Ingenieria de Sistemas y Automatica, Universidad de Sevilla, Seville 41092, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TAC.2009.2013002

0018-9286/$25.00 © 2009 IEEE

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therefore more general than pre-stabilization based schemes [5]–[7], or those based on online calculation of linear control laws [2], [3]. As shown in [13], this alternative scheme is attractive since the set of feasible policy parameters can be rendered convex via a suitable reparameterization, thereby allowing a constraint admissible policy from this set to be computed using newly available techniques in robust optimization [14]. Additionally, the size of the optimization problem required to find such a policy grows only polynomially with the problem size, making it a suitable candidate for online computation. By imposing additional convex constraints on the set of policies in this class and employing a worst-case quadratic cost function, where the disturcontrol [15], [16], we show that bance is negatively weighted as in one can make additional guarantees about the `2 gain of the resulting closed-loop system when these policies are used in the synthesis of receding horizon control (RHC) laws. The proposed method is particularly attractive since the min-max optimization problem to be solved at each time is convex-concave, and thus amenable to standard computational methods in convex optimization. In cases where the system constraints are polyhedral, we show that this min-max problem can be recast as a semidefinite program (SDP)—however, unlike proposals that employ ellipsoidal invariant sets [2], [3], the proposed scheme handles asymmetric constraints and a wider variety of disturbance classes. The technical note is organized as follows. Section II discusses the problem of interest and outlines a number of standing assumptions and definitions. It also describes the class of control policies used throughout, and relates a number of results pertaining to the minimization of `2 gain over a finite horizon. Section III extends these results to the infinite horizon case; we provide conditions under which one can simultaneously guarantee constraint satisfaction for the closed-loop system for all time while ensuring bounded `2 gain, and show that the achievable bound is a decreasing function of the control horizon length. In Section IV we show how the proposed min-max optimization problem can be solved using semidefinite programming techniques for cases of practical interest. Some conclusions are drawn in Section V. Notation: For matrices A and B , A0 denotes the transpose of A, A 0(A 0) indicates that A is positive (semi)definite and A B is the Kronecker product of A and B . The standard Euclidean or 2-norm norm jjj1jjj2 . The is denoted k 1 k2 , with associated induced matrix p weighted 2-norm of a vector x is denoted kxkP = x0 Px. IN is the N 2 N identity matrix. Vertical stacking of vectors x and y is denoted (x; y ) := xy .

H1

II. FINITE HORIZON `2 GAIN MINIMIZATION Consider the discrete-time linear time-invariant system

x+ = Ax + Bu + Gw z = Cx + Du

(1a) (1b)

where x 2 n is the system state at the current time instant, x+ is the state at the next time instant, u 2 m is the control input/manipulated variable, w 2 l is a disturbance and z 2 p is the costed/controlled variable. It is assumed that at each sample instant a measurement of the state is available. The current and future values of the disturbance are unknown and may change from one time instant to the next, but are contained in a convex and compact set W , assumed known, containing the origin. The system is subject to convex constraints on the state and input, so that it must satisfy (x; u) 2 Z , where Z 2 n 2 m is a closed and convex set containing the origin in its interior. A design goal is to guarantee that the state and input of the closed-loop system remain in Z for all disturbance sequences generated from W . In addition to Z , a target/terminal constraint set Xf n is given, which is closed and

convex and contains the origin in its interior. The set Xf can be used to design an RHC law with guaranteed infinite-horizon `2 gain properties, as will be shown in Section III. Predictions of the system’s evolution over a finite control/planning horizon of length N will be used to define a suitable control policy with a bounded finite-horizon `2 gain. Define stacked versions of the predicted input, state and disturbance mN , x 2 n(N +1) and w 2 lN , respecvectors u 2 tively, as x := (x0 ; . . . ; xN ), u := (u0 ; . . . ; uN 01 ) and w := (w0 ; . . . ; wN 01 ), where x0 := x denotes the value of the initial state and xi+1 := Axi + Bui + Gwi denotes the state after i time instants. Let the set W := W N := W 2 1 1 1 2 W , so that w 2 W . Finally, define a closed and convex set Z , appropriately constructed from Z and Xf , such that the constraints to be satisfied are equivalent to requiring that (x; u) 2 Z , i.e.

Z := f(x; u)j(xi; ui ) 2 Z; i = 0; 1; . . . ; N 0 1; xN 2 Xf g : We are interested in finding a feedback policy of the form := f0 (1); 1 (1); . . . ; N 1 (1)g (2) such that the control inputs ui = i (x0 ; 1 1 1 ; xi ) will satisfy the constraints Z for all possible disturbance sequences. Given a positive scalar and an initial state x, we wish to determine whether there exists a constraint admissible feedback policy and a non-negative scalar (x) such that the following finite-horizon `2 gain property [17]–[19] holds for all allowable disturbance sequences w 2 W 0

kxN kP2 +

N 01 i=0

kzi k22 (x) + 2

N 01 i=0

kwi k22

(3)

where P 0 is given. The main aim of this section is to demonstrate that by restricting the class of policies considered, convex optimization methods can be used to solve the above finite-horizon `2 gain problem efficiently. In Section III, it will be shown that such policies can be used in the synthesis of a computationally tractable receding horizon control law with an infinite-horizon `2 gain guarantee. A. An Affine Policy Parameterization

Finding an arbitrary control policy that satisfies the constraints Z for all admissible disturbance sequences while guaranteeing satisfaction of the `2 gain property (3) is extremely difficult in general, and current proposals for defining such policies generally require solution via dynamic programming [11]. As a result, we find it convenient to restrict the class of control policies to those of the form i01 ui = vi + Mi;j Gwj ; 8i 2 f0; 1; . . . ; N 0 1g (4) j =0

where each Mi;j 2 m2n and each vi 2 m . Note that since full state feedback is assumed, the uncertain terms Gwj represent the difference between the predicted and actual states at each step, i.e. Gwj = xj +1 0 Axj 0 Buj . For notational convenience, define the variable v 2 mN and the block lower triangular matrix M 2 mN 2nN such that

0

M :=

M1;0

.. . MN 01;0

111 0 .. .

111

111 111

0 0 .. .. . . MN 01;N 02 0

; v :=

v0 v1

.. . vN 01

(5)

so that the control input and state sequences can be written as

u = v + MG w ; x = Ax + Bv + (BMG + G)w

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(6a) (6b)

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Mv

where all other matrices are defined in the Appendix. The set of admissible ( ; ), for which the constraints Z are satisfied for all allowable disturbance sequences of length N , is defined as

Mv x M v uA v BuMGwGw x u 2 Z 8w 2 W (

5N (x) :=

(

)

;

x

+

=

(

+

(7)

+

)

;

w 7! N M v w is concave. Proof: For fixed M, the inequality (12) is satisfied for any jjjHu MG Hw jjj2 . Concavity is obvious from inspection of (10). Furthermore,

N

:=

x

x

0 (Hu MG + Hw )

I

(8)

:

Remark 1: The control parameterization (4) was proposed in [14], [20] and is of particular interest because the set of feasible policy parameters 5N (x) for each state x is convex. Importantly, this class of policies is equivalent to one defined as an affine function of current and prior states, where ui = gi + ij =0 Ki;j xj [13]. B. A Finite Horizon Cost

Mv

For a given initial state x, we wish to find a control policy ( ; ) 2 Z and the `2 gain property (3). We thus define the finite-horizon quadratic cost function

5N (x) to satisfy both the system constraints

N (x; ;

J

Mvw ;

) :=

k N kP2 +

N01

kzi k22 0 2 kwi k22 i=0 N01 x and the sequence fui g i=0 is given by (4), and will ;

x

where x0 := consider a zero-sum game of the form:

(x; ; (M;v)25 (x) w2W N min

max

One can readily obtain matrices pendix) such that

N (x; ;

J

Mvw ;

;

)=

kHx

x

+

J

Mvw ;

;

):

(9)

Hx , Hu and Hw (given in the Ap-

Hu v + (Hu MG + Hw )wk22 0 2 kwk22

:

;

w2W

) := max

N (x; ;

J

Mvw ;

;

)

(11)

Mvw

w

Mv

Mv

M

N

H

:=

2

I

fM

~ N (x; ) := ( 5

the

function

0 (Hu MG + Hw )0 (Hu MG + Hw ) 0

:

M Hu MG

Hw )0 0 0

(

+

(13)

:

I

;

v

)

M

2 5N ( )j(

g

) satis es (13)

;

x

~ N ( ) := and the set of states for which a policy of this form exists as X x x; ) 6= ;g. Inclusion of these LMI constraints imposes tighter requirements on ~ N (x; ) and X ~ N ( ) than for the related admissibility for the sets 5 sets 5N (x) and XN ; i.e. for any state x and positive scalar , the set ~ N (x; ) 5N (x) hold. It is also ~ N ( ) XN and 5 inclusions X ~ ~ N ( 2 ) for ~ ~ N ( 1 ) X easily shown that 5(x; 1 ) 5(x; 2 ) and X any 1 2 .

f j5~ N (

C. Finite Horizon `2 Gain For a given initial state x 2 XN , it is of interest to compute the smallest positive value of for which one can ensure the existence of an admissible disturbance feedback policy ( ; ) such that the finitehorizon `2 gain property (3) holds. For this purpose, define the function 3

N : XN ! + as

Mv

3 N (x) := inf

j5~ N (

x; )

6= ;

(14)

:

We are interested in the following optimization problem for a given N3 (x):

Mv Mv 2 N (15) v3 , and assume We denote a minimizer of (15) as M3 that it can be partitioned in a manner identical to that in (5), so that the th element of v3 , and the th submatrix is denoted k3 3 3 is denoted i;j . of M V

3 3 N (x; ) := (min M;v) JN (x; ;

(x; )

(12)

(x; )

) s:t: (

;

(

1

k

so that the following condition, guaranteeing that the minimization problem in (9) is convex, is easily established: 3 (x; ; ; ) is convex Proposition 1: The function ( ; ) 7! JN and lower bounded by zero. 2 W , the function (x; ; ) 7! Proof: For any fixed JN (x; ; ; ; ) is convex, and the pointwise supremum of convex functions is convex. The lower bound follows from the assumption that 0 2 W . We next impose a condition on such that the maximization problem in (11) is concave, so that our eventual min-max control policy optimization problem will be convex-concave: Proposition 2: For any given , there exists a > 0 such that the following quadratic matrix inequality (QMI) holds:

Mv

then

(10)

Mv

Mv

satisfied

As a result, we will consider a policy of the form (4) to be feasible (with respect to a given ) if, given an initial state x, it is both constraint admissible and satisfies the LMI condition (13). For a given state x and positive scalar , we therefore define the set of policies satisfying these conditions as

In this section, we will provide constraints on the feedback policy ( ; ) and the gain such that this zero-sum game can be guaranteed to be convex-concave. This will ensure the existence of a saddle point solution in pure policies [15, Sect. 2.1] and that (9) is solvable via 3 (1) as convex optimization techniques. We first define the function JN

3 JN (x; ;

is

)

;

Multiplying (12) by 01= and applying the Schur complement inequality [21, Sec. A.5.5], the above QMI is equivalent to the following linear matrix inequality (LMI) in and :

;

f 2 n j5N ( ) = 6 ;g

(12)

;

and the set of initial states x for which an admissible control policy of the form (4) exists as X

if

(x; ;

J

+

) satis es (5)

;

=

1107

v

M

;

(x; );

(x; )

)

~ (x; ): 5

(x; ))

(i; j )

(x; )

The main result of this section follows immediately from the discussion thus far: 3 (x(0)), Theorem 1: For a given initial state x(0) 2 XN and N consider implementing the following time-varying control policy on the system (1): u(k )

=

v

3 k (x(0); ) k01 +

j =0

M

3 k;j (x(0); ) (x(j + 1) 0 Ax(j ) 0 Bu(j ))

1We assume that 5 ~ ( ) = for all ( ) and that a minimizer ~ ( ) and all ( ). These assumptions for (15) exists for all are guaranteed to hold if is full column rank, 0 interior( ) and there [22]. exists a bounded set such that

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for all k 2 f0; 1; . . . ; N 0 1g. For all disturbance sequences 1 fw(k)gN0 k=0 drawing values from W , we have that (x(k); u(k)) 2 Z for k 2 f0; 1; . . . ; N 0 1g, x(N ) 2 Xf and the following `2 gain property holds:

kx(N )kP2 +

N01 k=0

kz(k)k22 VN3 (x(0); ) +

N01 k=0

2 kw(k)k22 :

III. INFINITE HORIZON `2 GAIN MINIMIZATION We next consider the infinite-horizon case, and seek a control law that satisfies

1 k=0

kz(k)k22 (x(0)) + 2

1

kw(k)k22

k=0 1 kw(k)k22 < 1 k=0

(16)

and for all disturbance sequences satisfying 2 W for all k 2 f0; 1; . . .g. We construct our control law by exploiting the parameterization introduced in (4) and implementing the control in a receding horizon fashion. We therefore define the receding horizon control law N : XN 2 + ! m as

w(k)

u = N (x; ) := v03 (x; )

(17)

which is nonlinear in general. The closed-loop system dynamics become

x(k + 1) = Ax(k) + BN (x(k); ) + Gw(k) z (k) = Cx(k) + DN (x(k); ) :

(18a) (18b)

In order to show that this closed-loop system satisfies the `2 gain condition (16) while simultaneously guaranteeing that (x(k); N (x(k); )) 2 Z for all time, we require the following assumption on the terminal set Xf and cost P : Terminal Cost and Constraint: (A; B ) is stabilizable, (C; A) is detectable and (C; A; B ) has no zeros on the unit circle. The control feedback matrix K and terminal cost P satisfy the solution to an unconstrained H1 state feedback control problem [15], [16] with `2 gain f

P := C 0 C + A0 P A 0 (A0 P B + C 0 D) 2 (D0 D + B 0 P B )01 (B 0P A + D0 C ) K := 0 (D0 D + B 0 P B )01 (B 0P A + D0 C )

(19a) (19b)

is robust positively invariant for the closed-loop system (18), we first establish the following nesting condition (see the Appendix for proofs): Proposition 3: If A1 holds with f , then the following set inclusion property holds:

Xf

X~1 ( ) . . . X~ N01 ( ) X~ N ( ) X~N +1 ( ) 1 1 1

A similar nesting condition holds with respect to the minimum achievable gain: 3 +1 (x) maxf f ; N3 (x)g for Corollary 1: If A1 holds, then N all x 2 XN . The significance of Corollary 1 is that, unlike existing H1 receding horizon control schemes such as [10, Chap. 9], it will allow us to guarantee that the achievable infinite-horizon `2 gain for the closed-loop system (18) is non-increasing with increasing horizon length. Using the above results we can also demonstrate that for any f , each of ~ N ( ) is robust positively invariant for the closed-loop system the sets X (18). Lemma 1: If A1 holds with f then, for each N 2 f1; 2; . . .g, ~ N ( ) is robust positively invariant for the closed-loop system the set X ~ N ( ) then x+ = Ax + BN (x; ) + Gw 2 X ~ N ( ) (18), i.e. if x 2 X for all w 2 W . Corollary 2: If one employs instead the minimum-gain con3 (x)g), then the sequence trol law u = N (x; maxf f ; N fmaxf f ; N3 (x(k))ggk1=0 is non-increasing for the closed-loop system, provided that the disturbance sequence w(1) remains in W . B. Finite `2 Gain in Receding Horizon Control We can now state the main result of this section, which allows us to place an upper bound on the `2 gain of the closed-loop system (18) under the proposed RHC law N (1; ): 3 (x(0))g then, for the Theorem 2: If A1 holds and maxf f ; N closed-loop system (18), the `2 gain from the disturbance w(1) to the costed/controlled variable z (1) is upper bounded by . Furthermore, the ~ N ( ). constraints are satisfied for all time if the initial state x(0) 2 X Proof: For any function '(1), define

1'(x; u; w) := '(Ax + Bu + Gw) 0 '(x) and functions Vf (x) := x0 P x, `(x; u; w) := (kz k22 0 2 kwk22 ) and `f (x; u; w) := (kz k22 0 f2 kwk22 ), where z = Cx + Du. If A1 holds, then a standard result in H1 control is that

01

where P := P + P G( f2 I 0 G0 P G) G0 P and f2 I 0 G0 P G 0. We further assume that the terminal constraint set Xf is chosen to be both constraint admissible and robust positively invariant under the control u = Kx, i.e.

Xf fxj(x; Kx) 2 Z g (A + BK )x + Gw 2 Xf ; 8x 2 Xf ; 8w 2 W:

(20a) (20b)

A variety of methods are available for the computation of a set Xf satisfying the conditions (20)—the reader is referred to [23] and the references therein for details. A. Invariance Properties We first consider whether, for a given fixed f , the optimization problem (15) required to implement the receding horizon control law N (1; ) can be solved for all time for the closed-loop system ~ N ( ) (18). In order to provide conditions under which the set X

max [1Vf + `f ](x; Kx; w) = 0: w2 Since f

implies `(x; u; w) `f (x; u; w) for all (x; u; w)

[1Vf + `](x; u; w) [1Vf + `f ](x; u; w);

8(x; u; w)

so that

max [1Vf + `](x; Kx; w) 0: w2W

(21)

Given a state x+ = Ax + BN (x; ) + Gw , Lemma 1 ensures the ex~ N (x+ ; ) such that the istence of a feasible control policy ( ~ ; ~ ) 2 5 terminal control input is uN = KxN . Application of (21) guarantees that

Mv

~ N ( ); 8w 2 W [1V + `] (x; N (x; ); w) 0; 8x 2 X

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(22)

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is equivalent to (i.e. achieves the same optimal value as) the minimization problem

TABLE I (0) AND REGION OF ATTRACTION

MINIMUM GAIN

1109

y + ) subject to y K 0 min(2h (; )

y

0 and

S0y (Hx x + Hu v)

where V (1) := VN3 (1; ). For any integer q and disturbance sequence q01 fw (k )gi=0 taking values in W , it then follows from (22) that q01 k=0

` (x(k); N (x(k); ); w(k)) (23)

where x(k + 1) = Ax(k) + BN (x(k); ) + Gw(k) for all k 2 ; . . . ; q 0 1g. It follows from Proposition 1 that VN3 (x; ) 0, so that for any integer q 0, (23) implies

f0

q01

z (k)k22 VN3 (x(0); ) + 2

q01

k

k=0

w(k)k22

k

k=0

(24)

and the claim about finite gain follows. Constraint satisfaction is guaranteed by Lemma 1. Note that for any initial state x(0), Corollary 1 guarantees that the minimum achievable closed-loop gain in Theorem 2 is a non-increasing function of the horizon length N . Further, given the same assumptions as Theorem 2 and using standard results [1], one can show that the origin of the undisturbed closed-loop system x+ = Ax + BN (x; ) ~ N ( ). Finally, if is exponentially stable with region of attraction X x(0) 2 Xf and = f , then VN3 (x(0); f ) = kx(0)kP2 in (24) and N (x(k); f ) = Kx(k) for all k 2 f0; 1; . . .g, matching the well-known H1 optimal control law for unconstrained linear systems [16]. IV. COMPUTATIONAL ISSUES When the state and input constraint set Z is polyhedral, it is often possible to eliminate the universal quantifier in (7) and characterize 5N (x) using a set of convex constraints [13]. For example, if the set W is polyhedral or 2-norm bounded, then (7) can be rewritten using only linear or second-order cone constraints, respectively. We show here how the min-max optimization problem (15) can be solved numerically in such a case. We assume for the purposes of this section that the disturbance set W can be characterized using a single affine conic inequality in the form W

:=

w Sw

f

j

K

hg

with dual cone K3 , where we denote S K h , h 0 S 2 K; note that both polyhedral and norm-bounded disturbance sets can be characterized in this fashion [21]. In such a case it is possible to eliminate the maximization over implicit in (15) using duality arguments: Proposition 4 ([22]): If the QMI (12) holds, then the optimization problem for some convex cone

w

w

K

w

wmax 2W

k(Hx

x + Hu v) + (Hu MG + Hw )wk22 0 2 kwk22

I

(Hu

M

G

v

0 (Hx x + Hu )

2

(Hu

+ Hw )

M

G

I

0 + Hw )

0

:

(25)

Given this result, it is possible to solve the min-max optimization problem (12) via the solution of a single convex optimization problem, i.e.

Infeasible; result is for system (27) without constraints.

VN3 (x(q); ) VN3 (x(0); ) 0

y0 S

VN3 (x; ) =

(;

min

M;v;y)

0 2h

M; v)

s:t: (

y+

2

5N (x);

y

K

0 and (25):

(26)

In cases where the the state and input constraints are polyhedral and the disturbance set is either polytopic or 2-norm bounded, the constraint ( ; ) 2 5N (x) can be reformulated as a set of linear or secondorder-cone constraints respectively using the methods in [13]—such constraints are readily converted to LMIs using standard techniques [22], [24]. The optimization problem (26) is therefore a semidefinite program (SDP) in a tractable number of variables in these cases. Note that in posing such a problem it is not necessary to explicitly include the LMI condition (13), since its satisfaction is implied by satisfaction of (25).

Mv

A. Numerical Example Consider the system

x+ = 2x + 2u + w z = [x u]0

(27a) (27b)

subject to the constraints juj 0:7 and jwj 0:25. We calculated 3 (0) defined in (14) given terminal conditions the minimum gain N Xf = fxjjxj 0:5g and P = 2 for varying horizon lengths N . We also consider the minimum achievable gain for an open-loop (OL) = 0) defined in the sense of control policy (equivalent to setting [10, Chap. 9], as well as for a control policy based on a pre-stabilizing (PS) control sequence ui = vi + Kxi in a manner similar to [5], [6] with K = 01:25. Note that the latter case is equivalent to specifying a 01 = (I 0 ) , where := I K . fixed feedback structure 3 (0), as well as the reThe minimum guaranteed gains at the origin N gion of attraction jXN j = maxfjxjjx 2 XN g, are shown in Table I. The interesting point to note is that, for the feedback approach 3 (0) does not change with horizon adopted in this technical note, N length N . Since the aim is to minimize the closed-loop `2 gain, this is a much more favorable result than can be guaranteed by the open-loop or pre-stabilizing methods, for which the minimal gain increases with N . The feedback approach allows one to shape the matrix (Hu G + Hw )0 (Hu G + Hw ) on-line via , while still ensuring the optimization problem is convex-concave (see Proposition 2), whereas this matrix is fixed for the open-loop and pre-stabilizing methods. Furthermore, not only does the feedback approach guarantee better disturbance rejection, but it also guarantees a larger region of attraction XN than the pre-stabilized or open-loop methods, where the latter is infeasible for horizons larger than one.

M

M

M

KB KG

M

K

M

V. CONCLUSION We have shown that, by imposing additional convex constraints on the class of robust control policies proposed in [13] for control of con-

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1110

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 54, NO. 5, MAY 2009

strained linear systems with bounded additive disturbances, one can formulate a robust RHC law with guaranteed bounds on the `2 gain of the closed-loop system, while simultaneously guaranteeing feasibility and constraint satisfaction for all time. The proposed control law requires the solution, at each time instant, of a convex-concave min-max problem in a number of variables that grows polynomially in both the number of states and inputs and the length of the planning horizon, making it a suitable candidate for online implementation. APPENDIX n(N +1)2n and 2 n(N +1)2nN as

A2

Define

A

:=

G

Define N 01 B (A and

D

A A2

:=

;

.. . AN

:=

E

0 0

111 111 111

In

A

:=

.. . AN 01

.. . AN 02

..

E IN G , B N 0E IN B , G (

)

:=

1 1 1 ABB ), G A IN D . Finally, define ~ := (

CA

Hx :=

0

In

P AN

;

Hu :=

1

.

111

)

CB D +

;

~ P B

and

:=

:

.. . In

CG

:

~ P G

Proof of Proposition 3: The proof is by induction. For any x 2 ~ ( ; ) 2 5 N (x; ) by definition. If A1 holds, then it is possible to construct a pair ( ~ ; ~ ) 2 5N +1 (x) such that the final input is uN := KxN [13, Prop. 11], where ~ := 0 ~ := K (G ~ + B ~ , ~ := vv~ , M G ), v~ := K (AN x + B~ ), ~ M 0 ~ ( ), there exists a policy X N

M

Mv

v

Mv

M

M

v

and B and G are defined above. We need only show that this choice of ~

~

M also satisfies a QMI condition of the form (12). ~

After considerable, though straightforward, algebraic manipulation (see [22] for details), one can show that this is equivalent to requiring that, for horizon length N + 1 HN +1

where AK X

:=

X

0G0 P AK Y

=

A + BK ), Y

:= (

HN

+

Y

0

P

G

0Y 0 A0K P G 0

2 I 0 G0 P G B

:= ( ~ + ~

MG

)

(28)

and

0 A0K P AK 0 (C + DK )0 (C + DK )

Y:

Noting that ( 2 I 0 G0 P G) 0, which is always true if f , and computing the Schur complement, the condition (28) is satisfied iff HN

+

Y

0

P

0 A0K P AK 0 (C + DK )0 (C + DK ) 01 G0 P 2 0 + P G( I 0 G P G)

Y

0

(29)

and HN 0 by aswhere P := P sumption. Using the definitions (19) to verify the matrix identity (P 0 0 0 AK , one can show that (29) is satisfied (C + DK ) (C + DK )) = AK P if P P , which is always true for f . The proof is completed ~ ( ). by verifying, in a similar manner, that Xf X 1 3 (x)g and any Proof of Corollary 1: For any > maxf f ; N ~ ( ), the proof of Proposition 3 ensures the existence of a policy x2X N 3 ~ 5 N +1 (x; ). From the definition (14), this implies that N +1 (x) 3 3 3 (x)g. for all > maxf f ; N (x)g, so that N +1 (x) maxf f ; N Proof of Lemma 1: If A1 holds then, from Proposition 3, there ~ exists a policy pair ( ; ) satisfying ( ; ) 2 5 N +1 (x; ) with

Mv

Mv

)

0

M

via appropriate definition of

)

2 =

. We need only consider whether this pair also satisfies the

condition (12). Given this partitioning for can be partitioned as HN +1

=

U V0

M, one can show that HN

V HN

+1

0

REFERENCES

0

IN

Hw :=

0 M

M; v v and by partitioning M

KxN . From [13, Prop. 13], one can construct a pair (

+

0 0

G, B~ := G 1 1 1 AGG),{C := (IN C; 0), (

=

N (x

for appropriately defined matrices U and V , where HN is as defined as in (12). Since HN is a principal sub-matrix of HN +1 , it follows that HN 0.

E

In

uN 5

[1] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M. Scokaert, “Constrained model predictive control: Stability and optimality,” Automatica, vol. 36, no. 6, pp. 789–814, Jun. 2000, Survey paper. control with per[2] H. Chen and C. W. Scherer, “Moving horizon formance adaptation for constrained linear systems,” Automatica, vol. 42, no. 6, pp. 1033–1040, Jun. 2006. [3] H. Chen, X. Gao, and H. Wang, “An improved moving horizon control scheme through Lagrange duality,” Int. J. Control, vol. 79, no. 3, pp. 239–248, Mar. 2006. [4] K. B. Kim, “Disturbance attenuation for constrained discrete-time systems via receding horizon controls,” IEEE Trans. Automat. Control, vol. 49, no. 5, pp. 797–801, May 2004. control: Quadratic [5] Y. I. Lee, “Input constrained receding horizon programming approach,” Int. J. Control, Automat. Syst., vol. 1, no. 2, pp. 178–183, Jun. 2003. [6] Y. I. Lee and B. Kouvaritakis, “Receding horizon predictive control for systems with input saturation,” Proc. Inst. Elect. Eng., vol. 147, no. 2, pp. 153–158, Mar. 2000. [7] H. Chen, C. W. Scherer, and F. Allgöwer, “A game theoretic approach to nonlinear robust receding horizon control of constrained systems,” in Proc. Amer. Control Conf., Albuquerque, NM, Jun. 1997, pp. 3073–3077. [8] L. Magni, H. Nijmeijer, and A. J. van der Schaft, “A receding horizon control problem,” Automatica, vol. 37, approach to the nonlinear no. 3, pp. 429–435, Mar. 2001. [9] L. Magni, G. De Nicolao, R. Scattolini, and F. Allgöwer, “Robust model predictive control for nonlinear discrete-time systems,” Int. J. Robust Nonlin. Control, vol. 13, pp. 229–246, 2003. [10] C. V. Rao, “Moving Horizon Strategies for the Constrained Monitoring and Control of Nonlinear Discrete-Time Systems,” Ph.D. dissertation, University of Wisconsin, Madison, 2000. [11] D. Q. Mayne, S. V. Rakovic´, R. B. Vinter, and E. C. Kerrigan, “Characterization of the solution to a constrained optimal control problem,” Automatica, vol. 42, no. 3, pp. 371–382, Mar. 2006. [12] P. O. M. Scokaert and D. Q. Mayne, “Min-max feedback model predictive control for constrained linear systems,” IEEE Trans. Automat. Control, vol. 43, no. 8, pp. 1136–1142, Aug. 1998. [13] P. J. Goulart, E. C. Kerrigan, and J. M. Maciejowski, “Optimization over state feedback policies for robust control with constraints,” Automatica, vol. 42, no. 4, pp. 523–533, Apr. 2006. [14] A. Ben-Tal, A. Goryashko, E. Guslitzer, and A. Nemirovski, “Adjustable robust solutions of uncertain linear programs,” Mathematical Programming, vol. 99, no. 2, pp. 351–376, Mar. 2004. -Optimal Control and Related Min[15] T. Bas¸ar and P. Bernhard, imax Design Problems: A Dynamic Game Approach. Boston, MA: Birkhäuser, 1991. [16] M. Green and D. J. N. Limebeer, Linear Robust Control. Englewood Cliffs, NJ: Prentice Hall, 1995. [17] D. J. Hill and P. J. Moylan, “Connections between finite-gain and asymptotic stability,” IEEE Trans. Automat. Control, vol. AC-25, no. 5, pp. 931–936, Oct. 1980. [18] M. R. James and J. S. Baras, “Robust output feedback control for nonlinear systems,” IEEE Trans. Automat. Control, vol. 40, no. 6, pp. 1007–1017, Jun. 1995. [19] H. K. Khalil, Nonlinear Systems. Englewood Cliffs, NJ: Prentice Hall, 2002. [20] J. Löfberg, “Minimax Approaches to Robust Model Predictive Control,” Ph.D. dissertation, Linköping University, Linköping, Sweden, Apr. 2003.

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[21] S. Boyd and L. Vandenberghe, Convex Optimization. Cambridge, U.K.: Cambridge University Press, 2004. [22] P. J. Goulart, “Affine Feedback Policies for Robust Control With Constraints” Ph.D. dissertation, University of Cambridge, Cambridge, U.K., Nov. 2006 [Online]. Available: http://www-control.eng.cam.ac.uk [23] F. Blanchini, “Set invariance in control,” Automatica, vol. 35, no. 1, pp. 1747–1767, Nov. 1999. [24] M. S. Lobo, L. Vandenberghe, S. Boyd, and H. Lebret, “Applications of second-order cone programming,” Linear Algebra Appl., vol. 284, no. 1–3, pp. 193–228, Nov. 1998.

properly estimated by end-users. Although this limitation is partially resolved by the recent work [1], which proves that max-min systems with a symmetric Jacobian matrix are locally stable regardless of delay, the requirement of symmetry is very restrictive in practice and understanding whether the same result holds for a wider class of matrices remains open. In this technical note, we gain a deeper insight into stability of max-min congestion control systems under diagonal delays. Most max-min systems (e.g., MKC [1], RCP [4], and XCP [3]) can be linearized to the following shape (more on this in Section II):

x (n) =

N

i

On Delay-Independent Diagonal Stability of Max-Min Congestion Control

a

Yueping Zhang, Member, IEEE, and Dmitri Loguinov, Senior Member, IEEE

Abstract—Network feedback in a congestion-control system is subject to delay, which can significantly affect stability and performance of the entire system. While most existing stability conditions explicitly depend on delay of individual flow , a recent study [1] shows that the combination of a 1 guarantees local stability symmetric Jacobian and condition ( ) of the system regardless of . However, the requirement of symmetry is very conservative and no further results have been obtained beyond this point. In this technical note, we proceed in this direction and gain a better understanding of conditions under which congestion-control systems can achieve delay-independent stability. Towards this end, we first prove that if satisfies 1 for any monotonic induced matrix Jacobian matrix , the system is locally stable under arbitrary diagonal delay . norm We then derive a more generic result and prove that delay-independent stability is guaranteed as long as is Schur diagonally stable [2], which is also observed to be a necessary condition in simulations. Utilizing these results, we identify several classes of well-known matrices that are stable 1 and prove stability of MKC [1] with under diagonal delays if ( ) and . arbitrary parameters Index Terms—Delay-independent stability, diagonal delay, max-min congestion control.

I. INTRODUCTION Several max-min congestion control algorithms (e.g., XCP [3], RCP [4], VCP [5], MKC [1], and JetMax [6]) have been recently proposed. These protocols receive feedback from the most-congested router in their path and exhibit appealing performance from both theoretical and practical perspectives. Thus, stability of these systems, especially when delay is present in the network feedback, has recently received a fair amount of attention [1], [7]–[15]. However, most existing stability conditions (e.g., [3], [8], [10], [12]) require that parameters of the control equation be adaptively tuned according to feedback delay i of user , making them undesirable in practice due to the resulting unfairness between flows with different RTTs and oscillations when delays are not

D

i

Manuscript received February 19, 2007; revised April 22, 2008. Current version published May 13, 2009. Recommended by Associate Editor C. Abdallah. Y. Zhang is with NEC Laboratories America, Inc., Princeton, NJ 07008 USA (e-mail: [email protected]). D. Loguinov is with Texas A&M University, College Station, TX 77843 USA (e-mail: [email protected]). Color versions of one or more of the figures in this technical note are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/TAC.2009.2013005

j

=1

a x (n 0 D ) ij

j

(1)

i

N D i

n

where ij are some constants, is the number of flows, is the discrete time variable, and i ( ) and i are, respectively, the sending rate and round-trip time of user . Using this model, we first present an alternative proof of Theorem 1 in [1] under time-invariant delay i , based on which we derive a sufficient stability condition of (1) to be k k2 1, where = ( ij ) is the coefficient matrix of the system and matrix norm k k2 is induced by the 2 vector norm. Clearly, this condition is more generic than the one obtained in [1], which required to be real and symmetric. Subsequently, we prove that this result actually extends to any matrix norm induced by a monotonic vector norm (which subsumes all standard vector norms, such as k k1 , k k2 , k k , and k kw ). Moreover, we 01 k2 (where P 3 prove that a special norm k ks = inf W 2P k is the set of all positive diagonal matrices) is a monotonic induced norm and further generalize the sufficient stability condition of system (1) to k ks 1, whose necessity is also indicated by simulations. Armed with these results, we identify several classes of systems that are stable under diagonal delays if and only if they are stable under undelayed feedback. This finding allows us to prove stability of Max-min Kelly Control (MKC) [1] with arbitrary parameters i and i . We also discuss and verify obtained results using Matlab simulations. The rest of the technical note is organized as follows. In Section II, we describe modeling assumptions of this technical note and review existing work in the area of delay-independent stability. In Section III, we present our main results of the technical note and verify them via Matlab simulations. In Section IV, we conclude the technical note and suggest directions for future work.

x n

A <

A :

D

a

L

A

: A

:

:1

:1 WAW

A <

II. BACKGROUND A. Modeling Single-Link Congestion Control Assume a generic model of max-min congestion control that is comlinks and users. Each flow 1 solicits feedback i from posed of a particular link, which we call the bottleneck link of user , in its path. Assuming that bottleneck assignments of the network are fixed [11], [15], it is proven in [6] that the entire system can be decomposed into subsystems of users constrained by individual bottlenecks, which is stable independent of delays if and only if individual subsystems are. Thus, in what follows we restrict our focus to congestion control in single-link networks and note that the obtained results are applicable to all multi-link max-min systems with time-invariant bottleneck assignments. Consider a network with users accessing a single bottleneck link. Feedback delays arise from transmission, propagation, and queuing

M

N

i

i

p

N

1Throughout the technical note, we use index senders (sources).

0018-9286/$25.00 © 2009 IEEE

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to denote both flows and

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