Conditions for Sadde-Point Equiibria in Output-Feedback MPC with MHE David A. Copp and João P. Hespanha Abstract A new method for soving output-feedback mode predictive contro (MPC) and moving horizon estimation (MHE) probems simutaneousy as a singe minmax optimization probem was recenty proposed. This method aows for stabiity anaysis of the joint outputfeedback contro and estimation probem. In fact, under the main assumption that a sadde-point soution exists for the min-max optimization probem, the state is guaranteed to remain bounded. In this paper we derive sufficient conditions for the existence of a sadde-point soution to this min-max optimization probem. For the speciaized inearquadratic case, we show that a sadde-point soution exists if the system is observabe and weights in the cost function are chosen appropriatey. A numerica exampe is given to iustrate the effectiveness of this combined contro and estimation approach. I. INTRODUCTION Cassica MPC has been a prominent contro technique in academia and industria appications for decades because of its abiity to hande compex mutivariabe systems and hard constraints. MPC, which is cassicay formuated with state-feedback and invoves the repeated soution of an open-oop optima contro probem onine in order to find a sequence of future contro inputs, has a we-deveoped theory as evidenced by [1 3], and has been shown to be effective in practice [4]. However, with more efficient methods and continued theoretica work, more recent advances in MPC incude the incorporation of disturbances, uncertainties, faster dynamics, distributed systems, and output-feedback. Reated to these advances is the recent work combining noninear output-feedback MPC with MHE into a singe min-max optimization probem [5]. This combined approach simutaneousy soves an MHE probem, which invoves the repeated soution of a simiar optimization probem over a finite-horizon of past measurements in order to find an estimate of the current state [6, 7], and an MPC probem. In order to be robust to worst-case disturbances and noise, this approach invoves the soution of a min-max optimization where an objective function is minimized with respect D. A. Copp and J. P. Hespanha are with the Center for Contro, Dynamica Systems, and Computation, University of Caifornia, Santa Barbara, CA 9316 USA. dacopp@engr.ucsb.edu,hespanha@ece.ucsb.edu to contro input variabes and maximized with respect to disturbance and noise variabes, simiar to gametheoretic approaches to MPC considered in [8, 9]. The motivation for the combined MPC/MHE approach in [5] is proving joint stabiity of the combined estimation and contro probems, and, under some assumptions, the resuts guarantee boundedness of the state and bounds on the tracking error for trajectory tracking probems in the presence of noise and disturbances. The main assumption required for these resuts to hod is that there exists a sadde-point soution to the min-max optimization probem at every time step. The anaysis of the min-max probem that appears in the forward horizon of the combined MPC/MHE approach is cosey reated to the anaysis of two-payer zero-sum dynamic games as in [1] and to the dynamic game approach to H 8 optima contro as in [11]. In these anayses the contro is designed to guard against the worst-case unknown disturbances and mode uncertainties, and in both of these references, sadde-point equiibria and conditions under which they exist are anayzed. The probem proposed in [5] differs, however, in that a backwards finite horizon is aso considered in order to incorporate the simutaneous soution of an MHE probem, which aso aows the contro to be robust to worst-case estimates of the initia state. In this paper we derive conditions under which a sadde-point soution exists for the combined MPC/MHE min-max optimization probem proposed in [5] and speciaize those resuts for discrete-time inear timeinvariant systems and quadratic cost functions. We show that in the inear-quadratic case, if the system is observabe, simpy choosing appropriate weights in the cost function is enough to ensure that a sadde-point soution exists. Whie these resuts are for unconstrained probems, the weights in the cost function can be tuned to impose soft constraints on corresponding variabes, and a numerica exampe discussed at the end of the paper shows that, even for unconstrained inear-quadratic probems, faster reguation may be achieved using this MPC/MHE approach with a shorter finite horizon. The paper is organized as foows. In Section II, we formuate the contro probem under consideration
and discuss the main stabiity assumption regarding the existence of a sadde-point. In Section III, we describe a method that can be used to compute a sadde-point soution and give conditions under which this method succeeds. A numerica exampe is presented in Section IV. Finay, we provide some concusions and directions for future research in Section V. II. PROBLEM FORMULATION As in [5], we consider the contro of a time-varying discrete-time process of the form x t1 f t px t,u t,d t q, y t g t px t q n t (1) @t P Z ě, with state x t taking vaues in a set X Ă R nx. The inputs to this system are the contro input u t that must be restricted to a set U Ă R nu, the unmeasured disturbance d t that is known to beong to a set D Ă R n d, and the measurement noise n t P R nn. The signa y t P Y Ă R ny denotes the measured output that is avaiabe for feedback. The contro objective is to seect the contro signa u t P U, @t P Z ě, so as to minimize a finite-horizon criterion of the form 1 J t px t L,u t L:,d t L:,y t L:t q k t c k px k,u k q q tt px tt q t η k pn k q ρ k pd k q (2) for worst-case vaues of the unmeasured disturbanced t P D, @t P Z ě and the measurement noise n t P R nn, @t P Z ě. The functions c k p q, η k p q, and ρ k p q in (2) are a assumed to take non-negative vaues. The optimization criterion incudes T P Z ě1 terms of the running cost c t px t,u t q, which recede as the current time t advances, L 1 P Z ą1 terms of the measurement cost η t pn t q, and L T P Z ą1 terms of the cost on the disturbances ρ t pd t q. We aso incude a termina cost q tt px tt q to penaize the fina state at time t T. Just as in a two-payer zero-sum dynamic game, payer 1 (the controer) desires to minimize this criterion whie payer 2 (the noise and disturbance) woud ike to maximize it. This eads to a contro input that is designed for the worst-case disturbance input, measurement noise, and initia state. This motivates the foowing finite-dimensiona optimization 1 Given a discrete-time signa z : Z ě Ñ R n, and two times t,t P Z ě with t ď t, we denote by z t :t the sequence tz t,z t1,...,z tu. min û t:pu max ˆx t LPX,ˆd t L:tT 1PD J t pˆx t L,u t L:t 1,û t:, ˆd t L:,y t L:t q. (3) In this formuation, we use a contro aw of the form u t û t, @t ě, (4) where û t denotes the first eement of the sequence û t: computed at each time t that minimizes (3). For the impementation of the contro aw (4), the outer minimizations in (3) must ead to finite vaues for the optima that are achieved at specific sequences û t: P U, t P Z ě. However, for the stabiity resuts given in [5], we actuay ask for the existence of a sadde-point soution to the min-max optimization in (3) as foows, (as in [5, Assumption 1]): Assumption 1 (Sadde-point): The min-max optimization (3) with cost given as in (2) aways has a sadde-point soution for which the min and max commute. Specificay, for every time t P Z ě, past contro input sequence u t L:t 1 P U, and past measured output sequence y t L:t P Y, there exists a finite scaar J t pu t L:t 1,y t L:t q P R, an initia condition ˆx t L P X, and sequences û t: P U, ˆd t L: P D such that J t pu t L:t 1,y t L:t q J t pˆx t L,u t L:t 1,û t:, ˆd t L:,y t L:tq max ˆx t LPX,ˆd t L:PD J t pˆx t L,u t L:t 1,û t:, ˆd t L:,y t L:t q (5a) min û t:pu J t pˆx t L,u t L:t 1,û t:, ˆd t L:,y t L:t q (5b) ă 8. In the next section we derive conditions under which a sadde-point soution exists for the genera noninear case and then speciaize those resuts for discrete-time inear time-invariant systems and quadratic cost functions. III. MAIN RESULTS Before presenting the main resuts, for convenience we define the foowing sets of time sequences for the forward and backward horizons, respectivey, T tt,t1,...,u andl tt L,t L1,...,t 1u and use them in the seque. 2
A. Noninear systems Theorem 1: (Existence of sadde-point) Suppose there exist recursivey computed functions V k p q, for a k P T, and V j p q, for a j P L, such that for a y t L:t P Y, and u t L:t 1 P U, V tt px tt q q tt px tt q, (6a) V k px k q min max k px k,û k, ˆd k q V k1 px k1 q û k PU ˆd k PD max min k px k,û k, ˆd k q V k1 px k1 q, ˆd k PD û k PU @k P T zt, (6b) V t px t,y t q min max t px t,û t, ˆd t,y t qv t1 px t1 q û tpu ˆd tpd maxmin t px t,û t, ˆd t,y t q V t1 px t1 q, (6c) ˆd tpd û tpu V j px j,u j:t 1,y j:t q max j px j,u j, ˆd j,y j q ˆd jpd V j1 px j1,u j1:t 1,y j1:t q, @j P Lzt L, (6d) V t L pu t L:t 1,y t L:t q max ˆx t LPX,ˆd t LPD t L pˆx t L,u t L, ˆd t L,y t L q where V t L1 px t L1,u t L1:t 1,y t L1:t q, (6e) k px k,û k, ˆd k q c k px k,û k q ρ k pˆd k q,k P T zt, (7a) t px t,û t, ˆd t,y t q c t px t,û t q η t pn t q ρ t pˆd t q, (7b) j px j,u j, ˆd j,y j q η j pn j q ρ j pˆd j q, j P L. (7c) Then the soutions û k, ˆd k, ˆd j, and ˆx t L defined as foows, for a k P T and j P L, satisfy the saddepoint Assumption 1. û k arg minmax k px k,û k, ˆd k,y k q V k1 px k1 q, û k PU ˆd k PD (8a) ˆd k ˆd j ˆx t L arg max ˆd k PD min k px k,û k, ˆd k,y k q V k1 px k1 q, û k PU arg max j px j,u j, ˆd j,y j q ˆd jpd (8b) V j1 px j1,u j1:t 1,y j1:t q, (8c) arg max t L pˆx t L,u t L, ˆd t L,y t L q ˆx t LPX V t L1 px t L1,u t L1:t 1,y t L1:t q. (8d) Moreover, the sadde-point vaue is equa to J t pu t L:t 1,y t L:t q V t L pu t L:t 1,y t L:t q. Proof. We begin by proving equation (5b) in Assumption 1. Let û k be defined as in (8a), and etû k be another arbitrary contro input. To prove optimaity, we need to show that the atter trajectory cannot ead to a cost ower than the former. Since V k px k q satisfies (6b) and achieves the minimum in (6b), for every k P T zt, û k V k px k q min û k PU k px k,û k, ˆd k q V k1px k1 q k px k,û k, ˆd k q V k1px k1 q. (9) However, since û k does not necessariy achieve the minimum, we have that V k px k q min k px k,û k, ˆd k q V k1px k1 q û k PU ď k px k,û k, ˆd k q V k1px k1 q. (1) Summing both sides of (9) from k t 1 to k t T 1, we concude that V k px k q k px k,û k, ˆd k q ðñ V t1 px t1 q V k1 px k1 q k px k,û k, ˆd k q. Next, summing both sides of (1) from k t 1 to k t T 1, we concude that V k px k q ď k px k,û k, ˆd k q ðñ V t1 px t1 q ď from which we concude that V t1 px t1 q k px k,û k, ˆd k q ď V k1 px k1 q k px k,û k, ˆd k q, k px k,û k,d k q. (11) Simiary, since V t px t,y t q satisfies (6c) and û t achieves the minimum in (6c), we can concude that 3
V t px t,y t q t px t,û t, ˆd t,y t q V t1 px t1 q ď t px t,û t, ˆd t,y tq V t1 px t1 q. (12) Then from (11) and (12), we concude that tt V t px t,y t q t px t,û t, ˆd t,y 1 tq k px k,û k, ˆd k q tt ď t px t,û t, ˆd t,y 1 tq k px k,û k,d k q. (13) Next, since V j px j,u j:t 1,y j:t q satisfies (6d) and achieves the maximum in (6d), we can concude that V j px j,u j:t 1,y j:t q j px j,u j, ˆd j,y jq V j1 px j1,u j:t 1,y j:t q. (14) Summing both sides of (14) from j t L 1 to j t 1, and using (13), we concude that V j px j,u j:t 1,y j:t q V j1 px j1,u j:t 1,y j:t q ˆd j j px j,u j, ˆd j,y jq ðñ V t L1 px t L1,u t L1:t 1,y t L1:t q j px j,u j, ˆd j,y j q t px t,û t, ˆd t,y t q t 1 k px k,û k, ˆd k q ď j px j,u j, ˆd j,y jq tt t px t,û t, ˆd t,y 1 tq k px k,û k, ˆd k q. Finay, from this and the facts that V t L pu t L:t 1,y t L:t q satisfies (6e) and ˆd t L and ˆx t L achieve the maximum in (6e), we can concude that V t L pu t L:t 1,y t L:t q t L pˆx t L,u t L, ˆd t L,y t Lq V t L1 px t L1,u t L1:t 1,y t L1:t q k px k,û k, ˆd k q tpx t,û t, ˆd t,y t q j px j,u j, ˆd j,y j q t L pˆx t L,u t L, ˆd t L,y t L q ď k px k,û k, ˆd k q t px t,û t, ˆd t,y tq j px j,u j, ˆd j,y jq t L pˆx t L,u t L, ˆd t L,y t L q. Therefore û k is a minimizing poicy, for a k P T, and (5b) is satisfied with J t pu t L:t 1,y t L:t q V t L pu t L:t 1,y t L:t q. To prove (5a), et ˆd k be defined as in (8b), ˆd j be defined as in (8c), and et ˆd k and ˆd j be other arbitrary disturbance inputs. Simiary, et ˆx t L be defined as in (8d), and et ˆx t L be another arbitrary initia condition. Then, since V k px k q satisfies (6b), V t px t,y t q satisfies (6c), V j px j,u j:t 1,y j:t q satisfies (6d), and ˆd k achieves the maximum in (6b), ˆd t achieves the maximum in (6c), and ˆd j achieves the maximum in (6d), we can use a simiar argument as in the proof of (5b) to concude that V t L1 px t L1,u t L1:t 1,y t L1:t q j px j,u j, ˆd j,y jq t px t,û t, ˆd t,y tq t 1 k px k,û k, ˆd k q ě j px j,u j, ˆd j,y j q t px t,û t, ˆd t,y t q k px k,û k, ˆd k q. Finay, (5a) foows from this, (6e), (8c), (8d), and simiar arguments as used before (further detais of which can be found in [12]), which ead to V t L pu t L:t 1,y t L:t q t L pˆx t L,u t L, ˆd t L,y t L q j t L V t L1 px t L1,u t L1:t 1,y t L1:t q ě k px k,û k, ˆd k q tpx t,û t, ˆd t,y t q j px j,u j, ˆd j,y jq t L pˆx t L,u t L, ˆd t L,y t Lq k px k,û k, ˆd k q t px t,û t, ˆd t,y t q j px j,u j, ˆd j,y j q t L pˆx t L,u t L, ˆd t L,y t L q. Therefore ˆd k, for a k P T, and ˆd j, for a j P L, are maximizing poicies, ˆx t L is a maximizing poicy, and (5b) is satisfied with J t pu t L:t 1,y t L:t q V t L pu t L:t 1,y t L:t q. 4
Thus, Assumption 1 is satisfied. Next we speciaize these resuts for inear timeinvariant (LTI) systems and quadratic cost functions. B. LTI systems and quadratic costs Consider the foowing discrete inear time-invariant system, for a t P Z ě, x t1 Ax t Bu t Dd t, y t Cx t n t, (15) with x t P X R nx, u t P U R nu, d t P D R n d, n t P N R nn, and y t P Y R ny. Aso consider the quadratic cost function J t px t L,u t L:,d t L:,y t L:t q t k t x1 k Qx k λ u u 1 k u k x 1 tt Qx tt λ n py k Cx k q 1 py k Cx k q λ d d 1 kd k (16) where Q Q 1 ě is a weighting matrix, and λ u, λ d, λ n are positive constants that can be tuned to impose soft constraints on the variabes x k, u k, d k, and n k, respectivey. Again, the contro objective is to sove for a contro inputu t that minimizes the criterion (16) in the presence of the worst-case disturbance d t and initia state x t L. This motivates soving the optimization probem (3) with the cost given as in (16) subject to the dynamics given in (15). Then the contro input as defined in (4) is seected and appied to the pant. The foowing Theorem gives conditions under which a sadde-point soution exists for probem (3) with cost (16), thereby satisfying Assumption 1, as we as a description of the resuting sadde-point soution. Theorem 2: (Existence of sadde-point for inear systems with quadratic costs) LetM k andλ k, for a k P T, and P j and Z j, for a j P L, be matrices of appropriate dimensions defined by 2 M k Q A 1 M Λ 1 k1 k A; M tt Q, (17a) 1 Λ k I BB 1 1 1 DD M k1, (17b) λ u λ d P j A 1 P j1 A A 1 P DZ 1 j1 j D 1 P j1 A λ n C 1 C; P t M t λ n C 1 C, (17c) Z j λ d I D 1 P j1 D. Then, if the foowing conditions are satisfied, λ u I B 1 M k1 B ą, 2 I denotes the identity matrix with appropriate dimensions. (17d) (18a) λ d I D 1 M k1 D ą, (18b) λ d I D 1 P j1 D ą, (18c) j λn C 1 C A 1 P t L1 A A 1 P t L1 D D 1 P t L1 A λ d I D 1 ą, P t L1 D (18d) the min-max optimization (3) with quadratic costs (16) subject to the inear dynamics (15) admits a unique sadde-point soution that satisfies Assumption 1. Proof. Conditions (18) come directy from the secondorder conditions for strict-convexity/concavity of a quadratic function. If conditions (18) are satisfied, the optimizations in (6) with quadratic costs coming from (16) are stricty convex with respect to û k, for a k P T, stricty concave with respect to ˆd k and ˆd j, for a k P T and j P L, and stricty concave with respect to rˆx 1 ˆd t L 1 t L s1. Therefore, soutions as in (8) exist. Thus, a sadde point soution exists because of Theorem 1. In this inear-quadratic case, the functions (6) can be soved for expicity, and anaytica soutions exist for (8). These soutions and further detais of this proof can be found in [12]. Coroary 1: If the discrete-time inear time-invariant system given in (15) is observabe, then the scaar weights λ n and λ d can be chosen sufficienty arge such that the conditions (18a)-(18d) are satisfied. Therefore, according to Theorem 2, there exists a sadde-point soution for the optimization probem (3) with cost (16). Therefore, aso, Assumption 1 is satisfied. Proof. Condition (18a) is triviay satisfied for a k P T as ong as we choose λ u ą and weighting matrix Q ě because Q ě ùñ M k ě, @k P T. 3 Condition (18b) is satisfied if the scaar weight λ d is chosen sufficienty arge. To show this, we take the imit of the sequence of matrices M k, as given in (17a), as λ d Ñ 8 and notice that M k Ñ M k, where M k is described by M k Q A 1 p M k1 ri 1 λ u BB 1 Mk1 s 1 qa; M tt M tt, for a k P T. Then, in the imit as λ d Ñ 8, condition (18b) becomes 8I D 1 Mk1 D ą, and therefore, condition (18b) is satisfied when λ d is chosen sufficienty arge. Next we prove that conditions (18c) and (18d) are satisfied, for a j P L, when λ n and λ d are chosen sufficienty arge and the system (15) is observabe. We 3 Note that M k M 1 k due to the fact that Q Q1 and the matrix identity in [13] which says that ApI BAq 1 pi ABq 1 A. 5
first take the imit of the sequence of matrices P j, as given in (17c), as λ d Ñ 8 and notice that P j Ñ P j where P j is described by P j λ n Θ 1 jθ j A 1 t j Mt A t j, for a j P L Yt, and Θ j is defined as Θ j rc 1 A 1 C 1 A 12 C 1... A 1 t j C 1 s 1. The matrix Θ j ooks simiar to the observabiity matrix, and therefore, Θ 1 j Θ j ą if the system given in (15) is observabe. Then, the scaar weight λ n can be chosen arge enough to ensure that λ n Θ 1 j Θ j ą A 1 t j Mt A t j, for a j P L. It then foows that Pj ă for a j P L. Therefore, condition (18c) becomesλ d I D 1 Pj1 D ą in the imit as λ d Ñ 8 and is triviay satisfied if system (15) is observabe and λ n is chosen sufficienty arge. Finay, consider condition (18d). Using the Schur Compement, condition (18d) is satisfied if, and ony if, λ d I D 1 P t L1 D ą and λ n C 1 C A 1 P t L1 A A 1 P t L1 Dpλ d I D 1 P t L1 Dq 1 D 1 P t L1 A ą. We just proved that λ d I D 1 P t L1 D ą if the system (15) is observabe and the weights λ d and λ n are chosen sufficienty arge. Now we show that the second inequaity is satisfied under these same conditions. In the imit as λ d Ñ 8, the second inequaity becomes λ n C 1 C A 1 Pt L1 A ą. Then, if the system (15) is observabe,λ n can be chosen sufficienty arge such that this inequaity is satisfied, and therefore, condition (18d) is satisfied. IV. SIMULATION Intuition says that for unconstrained inear-quadratic probems, the best soution can be found by soving an infinite-horizon optimization probem. However, there are cases where a finite-horizon approach may be beneficia. Specificay, faster reguation can be achieved for the foowing unconstrained inear-quadratic exampe, where the system is subjected to impusive disturbances, using the combined MPC/MHE approach with a shorter finite-horizon. This motivates using the combined MPC/MHE finite-horizon approach over other standard infinite-horizon contro techniques for particuar types of inear-quadratic probems. Exampe 1: (Stabiizing a rideress bicyce.) Consider the foowing second order continuous-time inearized bicyce mode in state-space form: 9x t Ax t Bpu t d t q, y t Cx t n t (19) The state is given by x t φ δ φ 9 δ 9 1 where φ is the ro ange of the bicyce, δ is the steering ange of the handebars, and φ 9 and δ 9 are the corresponding anguar veocities. The contro input u t is the steering torque appied to the handebars. The matrices defining the inearized dynamics are given, as described in [14], as» fi 1 1 ffi A 13.67.225 1.319v 2.164v.552vf, 4.857 1.81 1.125v 2 3.621v 2.388v» fi ffi B.339 7.457 f, C j 1 1, where v is the bicyce s forward veocity on which the dynamics depend. Ony the ro ange and steering ange (and not their corresponding anguar veocities) are measured and avaiabe for feedback. As noted in [14], the bicyce dynamics are sefstabiizing for veocities between 3.4m{s and 4.1m{s. For our exampe, we fix the forward veocity at 2m{s; therefore, the system is unstabe. The contro objective is to stabiize the bicyce in the upright position, i.e. around a zero ro ange (φ ), by appying a steering torque to the handebars. The disturbance d t acts on the input and can be thought of as sharp bumps in the bicyce s path or simiar environmenta perturbations. Simuation resuts for two simuations of the system (19) discretized using a.1 second zero-order-hod are shown in Figures 1 and 2, respectivey. The optimization given in (3) with cost (16) was soved at each time t, and the resuting û t was appied as the contro input. The same parameters, besides the ength of the backwards horizon L, were used for both simuations. These parameters were chosen as λ u.1, λ d 1, λ n 1,T 2, andqwas chosen as a 4x4 matrix with the eement in upper eft corner equa to one and a other eements equa to zero. The measurement noise was generated as a random variabe n t N p,.1 2 q. The disturbance d t was impusive and is shown in the bottom pots of Figures 1 and 2 denoted by o s. The simuations were initiaized with zero input, and with the initia state x 1. With the same impusive disturbances, better performance is achieved when appying the contro input computed using a shorter backwards horizon ength of L 2, as shown in Figure 2, than when using a onger horizon ength of L 2, as shown in Figure 1. The contro input computed using the shorter backwards horizon ength L 2 is abe to more quicky reguate the ro ange φ back to zero without requiring 6
φ, δ [deg] 6 4 2-2 -4-6 u, d [Nm] φ, δ [deg] u, d [Nm] 3 2 1-1 -2 φ δ 5 1 15 2 u d -3 5 1 15 2 6 4 2-2 -4 Fig. 1. Longer backwards horizon (L 2). -6 5 1 15 2 3 2 1-1 -2 φ δ u d -3 5 1 15 2 Fig. 2. Shorter backwards horizon (L 2). contro inputs that are arger in magnitude than those computed using L 2. This is possibe because a short backwards finite horizon L aows the combined contro and estimation scheme to, in a sense, forget the irreguar impusive disturbances so that they do not ead to skewed estimates and more conservative contro. Therefore, it may be beneficia to use the combined MPC/MHE approach for unconstrained inear-quadratic probems when certain system characteristics, such as impusive disturbances, are present. V. CONCLUSION AND FUTURE WORK We discussed the main assumption of a new approach to soving MPC and MHE probems simutaneousy as a singe min-max optimization probem as proposed in [5]. This assumption is that a sadde-point soution exists for the resuting min-max optimization probem. First we gave conditions for the existence of a saddepoint soution when considering a genera discrete-time noninear system and a genera cost function. Next we speciaized those resuts for discrete-time inear timeinvariant systems and quadratic cost functions. For this case, we showed that observabiity of the inear system and arge weights λ d and λ n in the cost function are sufficient conditions for a sadde-point soution to exist. We showed the effectiveness of this contro approach in a numerica exampe of a inearized rideress bicyce system subjected to impusive disturbances. The simuation showed that, for this exampe, performance can be improved if a shorter backwards optimization horizon L is used. Future work may invove reaxing the requirement of a sadde-point soution to that of beingε-cose to a saddepoint soution. This coud be reated to resuts for ε-nash equiibria. REFERENCES [1] M. Morari and J. H Lee, Mode predictive contro: past, present and future, Computers & Chemica Engineering, vo. 23, no. 4, pp. 667 682, 1999. [2] D. Q. Mayne, J. B. Rawings, C. V. Rao, and P. O. Scokaert, Constrained mode predictive contro: Stabiity and optimaity, Automatica, vo. 36, no. 6, pp. 789 814, 2. [3] J. B. Rawings and D. Q. Mayne, Mode Predictive Contro: Theory and Design. Nob Hi Pubishing, 29. [4] S. J. Qin and T. A. Badgwe, A survey of industria mode predictive contro technoogy, Contro engineering practice, vo. 11, no. 7, pp. 733 764, 23. [5] D. A. Copp and J. P. Hespanha, Noninear output-feedback mode predictive contro with moving horizon estimation, in Proc. of the 53nd Conf. on Decision and Contr., Dec. 214. [6] C. V. Rao, J. B. Rawings, and D. Q. Mayne, Constrained state estimation for noninear discrete-time systems: Stabiity and moving horizon approximations, Automatic Contro, IEEE Transactions on, vo. 48, no. 2, pp. 246 258, 23. [7] A. Aessandri, M. Bagietto, and G. Battistei, Moving-horizon state estimation for noninear discrete-time systems: New stabiity resuts and approximation schemes, Automatica, vo. 44, no. 7, pp. 1753 1765, 28. [8] H. Chen, C. Scherer, and F. Agöwer, A game theoretic approach to noninear robust receding horizon contro of constrained systems, in American Contro Conference, 1997. Proceedings of the 1997, vo. 5, pp. 373 377, IEEE, 1997. [9] S. La and K. Gover, A game theoretic approach to moving horizon contro, in Advances in Mode-Based Predictive Contro, Oxford University Press, 1994. [1] T. Başar and G. J. Osder, Dynamic Noncooperative Game Theory. London: Academic Press, 1995. [11] T. Başar and P. Bernhard, H-infinity optima contro and reated minimax design probems: a dynamic game approach. Springer, 28. [12] D. A. Copp and J. P. Hespanha, Conditions for sadde-point equiibria in output-feedback MPC with MHE: Technica report, tech. rep., University of Caifornia, Santa Barbara, 215. [13] S. R. Seare, Matrix Agebra Usefu for Statistics. John Wiey and Sons, 1982. [14] V. Cerone, D. Andreo, M. Larsson, and D. Regruto, Stabiization of a rideress bicyce [appications of contro], Contro Systems, IEEE, vo. 3, no. 5, pp. 23 32, 21. 7