Move Blocking Strategies in Receding Horizon Control

Transcription

1 Move Blocking Strategies in Receding Horizon Control Raphael Cagienard, Pascal Grieder, Eric C. Kerrigan and Manfred Morari Abstract In order to deal with the comptational brden of optimal control, it is common practice to redce the degrees of freedom by fixing the inpt or its derivatives to be constant over several time-steps. This policy is referred to as move blocking. This paper will address two isses. First, a srvey of varios move blocking strategies is presented and the shortcogs of these blocking policies, sch as the lack of stability and constraint satisfaction garantees, will be illstrated. Second, a novel move blocking scheme, Moving Window Blocking MWB, will be presented. In MWB, the blocking strategy is time-dependent sch that the scheme yields stability and feasibility garantees for the closed-loop system. Finally, the reslts of a large case-stdy are presented that illstrate the advantages and drawbacks of the varios control strategies discssed in this paper. Keywords: Move Blocking, Optimal Control, Receding Horizon Control, Model Predictive Control I. INTRODUCTION In the literatre, nmeros variations of discrete-time, finite horizon constrained optimal control have been presented, e.g. [1], [7]. In most implementations, the optimal seqence of inpts is compted at each time-step and sbseqently only the first element of the seqence is applied. This policy is referred to as Receding Horizon Control RHC or Model Predictive Control MPC. One of the major open isses of MPC is the comptational complexity, which becomes prohibitive for large systems or systems with fast dynamics. In the standard problem formlation, the degrees of freedom of an RHC problem correspond to the nmber of inpts m mltiplied with the prediction horizon N. The degrees of freedom are the doating factor for complexity, regardless of whether the optimization problem is solved on-line, or the eqivalent feedback soltion is derived off-line [2], [4]. In order to deal with this isse, it is common practice to redce the degrees of freedom by fixing the inpt or its derivatives to be constant over several time-steps e.g. [8], [9]. Althogh this alleviates the comptational restrictions, the reslting control policies do not garantee stability or constraint satisfaction. In this paper, we first review varios move-blocking strategies and discss some of their disadvantages. Following this, we introdce a novel blocking strategy that provides garantees on stability as well as constraint satisfaction. The varios schemes are compared in a large casestdy that is presented in the final section before conclding. Atomatic Control Laboratory, Swiss Federal Institte of Technology ETH, CH-8092 Zrich, Switzerland Royal Academy of Engineering Post-doctoral Research Fellow, Department of Engineering, University of Cambridge, UK Corresponding Athor: cagienard@control.ee.ethz.ch II. STANDARD PROBLEM FORMULATION In this section we will give a brief overview of the optimal control problem considered in this paper. After presenting the standard finite horizon optimal control problem, a brief discssion on receding horizon control RHC, stability and feasibility properties of the feedback controller enses. We restrict orselves to the following linear, timeinvariant, discrete-time system xk + 1 = Axk + Bk, 1 with A R n n, B R n m and A, B stabilizable. We will let x or xk denote the measred state at time k and x i denote the predicted state at time k +i, given the state xk at time k and some finite inpt seqence { 0,..., i 1 }. Assme now that the states and inpts of system 1 are sbject to the constraints xk X R n, k 1 U R m, k 1 2 where X and U are compact polyhedral sets containing the origin in their interior. Before going frther, we will introdce the following definitions Definition 1: The set X I K will denote the maximm positively invariant set for the linear system 1 sbject to the stabilizing linear feedback controller K that satisfies 2: X I K := {x0 R n xk X, Kxk U, xk + 1 = A + BKxk, k 0} The set X I K is easily compted with the algorithm in [3]. Now consider the finite-time constrained LQR problem V N x := 0,..., + x N P x N, 3a sbj. to x k X, k 1 U, k {1,..., N}, 3b x N X I K 3c x k+1 = Ax k + B k, x 0 = x. 3d For the sake of simplicity, we will assme that Q = Q 0, P = P 0 and R = R 0 this can be relaxed to Q 1/2, A detectable and Q = Q 0, P = P 0. The niqe optimizer to the above optimal control problem is denoted as U x := [ 0x,..., x ]. Definition 2: We define the N-step feasible set X N R n as the set of states xk for which the optimal control problem 3 is feasible, i.e. for which problem 3

2 has a soltion satisfying the constraints: X N :={x 0 R n U = [ 0,..., ] R Nm, x k X, k 1 U, k {1,..., N}, x N X I K}. Becase problem 3 depends on the crrent state x, it can be solved either by solving the qadratic program 3 for a given x, or as shown in [2], [10], by solving problem 3 for all x within a polyhedral set of vales, i.e., by considering 3 as a mlti-parametric Qadratic Program mp-qp [2]. Theorem 1: [2] Consider the finite-time constrained reglation problem 3. The set of feasible states X N is convex, the optimizer U : X N R Nm is continos and piecewise affine PWA, i.e. U has the form: U x = F r x + G r, if x P r P r = {x R n H r x K r }, r = 1,..., R and the vale fnction V N : X N R is continos, convex and piecewise qadratic. In Receding Horizon Control RHC, the optimizer U x of problem 3 is compted at each time step bt only the first inpt 0x of the seqence is applied to the plant, i.e. k = 0xk. This control policy does not garantee closed loop stability or constraint satisfaction withot additional adjstments to the problem formlation in 3. It is standard practice [7] to obtain feasibility i.e. constraint satisfaction properties by adding the additional set constraint x N X I K to the optimal control problem 3. The teral set constraint garantees recrsive feasibility, i.e. if a feasible inpt seqence exists at time k = 0 this implies that a feasible inpt seqence will be fond for all k > 0. If the teral weight P is chosen sch that F x = x P x is a local Lyapnov fnction for the system sbject to the linear feedback law K, then stability is garanteed as well. Both stability and feasibility properties can be proven by considering the optimal inpt seqence at time k, U xk and the shifted seqence Ũ N xk = [ 1xk,..., xk, Kx N xk ] at time k+1, where x N xk is the predicted soltion of the system at time instant k + N, given the state xk at time k and the inpt seqence U xk. The shifted inpt seqence Ũ N xk is garanteed to be feasible at time k+1, becase X I K is invariant and constraint-admissible. The vale fnction VN in 3 serves as a Lyapnov fnction, de to the choice of the teral weight P. The reader is referred to [7] for frther details. III. MOVE BLOCKING SCHEMES A. Inpt Blocking IB Since the comptational complexity of solving the optimization problem 3 depends directly on the degrees of freedom i.e. nmber of free inpts, it is standard practice [8], [9] to obtain tractable optimization problem formlations by fixing the inpts to be constant over a certain nmber of time steps. For example, instead of solving for the optimal U := [ 0,..., ] R Nm, problem 3 can be restated in terms of solving for the optimal vector Û := [û 0,..., û M 1 ] sch that U = T I m Û, where denotes the Kronecker prodct and T R N M is a socalled blocking matrix with M < N. T is assmed to be a matrix of ones and zeros only, with each row containing exactly one non-zero element. For example, for a SISO system with move blocking 1 = 2 = 3 and N = 4 we might choose Û = [û 0 û 1] and 1 0 T = The N = 4 degrees-of-freedom problem has ths been redced to an M = 2 degrees-of-freedom problem. The control problem is now reformlated as V N x, T := û 0,...,û M 1 + x N P x N, 5a sbj. to x k X, k 1 U, k {1,..., N}, 5b x N X I K 5c U = [ 0,..., ] = T I m [û 0,..., û M 1], x k+1 = Ax k + B k, x 0 = x. 5d Thogh the move-blocking reformlation may make the optimization problem comptationally tractable, no stability or feasibility properties can be obtained. This is becase a simple shift in the inpt seqence is no longer possible if the inpts are blocked, i.e. the inpt seqence obtained at time k may not be sed to obtain a feasible soltion at time k + 1 since the shifted seqence as in Section II will not correspond to the initial move blocking strategy imposed by T. Frthermore, there may not exist any inpt seqence Û that satisfies the constraints in 3 for U = T I m Û. Therefore, no garantee of recrsive feasibility can be provided, i.e. obtaining a feasible inpt seqence at time k = 0 does not imply that a feasible inpt seqence will be fond for all ftre time steps. Frthermore, this move-blocking scheme may have a strong adverse effect on closed-loop performance. These drawbacks are illstrated in Section V. B. Offset Blocking OB The inpt blocking scheme described in the previos section can easily lead to very bad closed-loop performance since there is no garantee that the applied inpt will be optimal for any feasible state x X N. To some extent, this isse can be dealt with by prestabilizing the system with the nconstrained, infinitehorizon LQR control law and parameterizing the inpt

3 seqence in terms of deviations from this control law, i.e. by letting k = K LQR x k +c k and formlating the associated optimal control problem in terms of C := [c 0,..., c ], rather than U. The infinite-horizon, nconstrained LQR law K LQR is compted off-line and P is chosen to be the soltion to the associated algebraic Riccati eqation, i.e. K LQR = B T P B + R 1 B T P A, 6 where P is the soltion to: P = A T P A A T P BB T P B + R 1 B T P A + Q. 7 The optimal control problem is now restated in terms of Ĉ = [ĉ 0,..., ĉ ] and the constraints C = T I m Ĉ and k = K LQR x k + c k : V N x, T := ĉ 0,...,ĉ M 1 + x N P x N, 8a sbj. to x k X, k 1 U, k {1,..., N}, 8b x N X I K LQR, C = [c 0,..., c ] = T I m [ĉ 0,..., ĉ M 1], k = K LQR x k + c k, 8c k {0,..., N 1}, 8d x k+1 = Ax k + B k, x 0 = x. 8e The advantage of this scheme is that the obtained inpt seqence will be optimal if x X I K LQR, since Ĉ = [0,..., 0 ] will be feasible and optimal, regardless of the blocking strategy. Frthermore, only the offset to the feedback and not the actal inpt will be blocked, which may lead to greater flexibility in the controller. The comptational advantages are maintained since the nmber of decision variables is less than Nm. Althogh this scheme may improve on performance, no garantees on stability and constraint satisfaction can be given for the reasons stated in the previos section. Frthermore, in addition to the state constraints, there is the difficlty of satisfying inpt constraints, i.e. there may not exist any constant offset sch that the inpt constraints are satisfied for a fixed nmber of steps. Therefore, parameterized move blocking may not always be preferable to standard move blocking. C. Delta Blocking DIB, DOB Instead of fixing the inpt or an offset to a feedback law to be constant for a fixed nmber of steps, it is possible to impose a constraint on the difference between two consective control actions and optimize over the size of this constraint. This corresponds to a linear interpolation of inpts û for the case of delta-inpt blocking DIB. In DIB we set k k+1 to be constant over a predefined nmber of time steps. The constraint on T also has to be relaxed to allow more than one non-zero element on each row. Consider 4 for instance, where we have N = 4 and M = 2. The decision variables for the DIB problem correspond to 0, 3 and the vales for the blocked inpts 1, 2 follow directly from linear interpolation. Ths the N = 4 degrees-of-freedom problem has been redced to a M = 2 degrees-of-freedom problem. The same kind of linear interpolation can be applied to obtain offset seqences from a small nmber of offsets ĉ. This scheme will be referred to as delta-offset blocking DOB. In DOB we set c k c k+1 to be constant over a predefined nmber of time steps, bt optimize over the size of this constraint. As in the previosly presented blocking schemes, both delta-blocking strategies redce the comptational brden bt do not provide any garantees on closed-loop stability or feasibility. IV. MOVING WINDOW BLOCKING MWB In this section we will describe a novel move-blocking scheme, which provides stability and feasibility garantees in closed-loop. The scheme implements a time-dependent move blocking strategy where the blocked inpts are shifted at each time step. Specifically, the blocking matrix denoted T k is now time-dependent. Definition 3: A matrix T R N M is an admissible blocking matrix if M < N, one entry in each row of T is eqal to 1 and all other entries in the row are eqal to 0, there are no zero colmns or zero rows and the elements of the matrix are arranged in an pper staircase form, i.e. if the colmn in which a 1 occrs in the i th row is j i := {j T i,j = 1}, where T i,j denotes the element in the i th row and j th colmn of T, then j i+1 j i for all i {1,..., N 1}. Examples of admissible blocking matrices are: , 0 1 0, Examples of non-admissible blocking matrices are: , 1 0 0, The seqence of admissible blocking matrices {T 0, T 1,..., }, is generated by T k + 1 = ft k, 9 where T 0 is an admissible blocking matrix and f corresponds to the following algorithm: Algorithm 1 Blocking Matrix Generation ft : Given an admissible blocking matrix T R N M. 1 Let S := [ 0 1 I ] T be the matrix of the last N 1 rows of T, where S R N. 2 Note that the first colmn of S is, in general, not garanteed to be non-zero. If the first colmn of S is

4 zero, then we shift the colmns to the left and append the zero colmn to the right of the reslting matrix. In other words, let S [ ] if S 1,1 = 1 W := 0 1 M 1 0 S if S 1,1 = 0 I M 1 0 M Note that the first M 1 colmns of W are non-zero and in pper staircase form, bt that the last colmn may be zero. We therefore retrn the admissible blocking matrix [ ] W ft :=, 0 1 M 1 1 where ft R N M. As an example, the following seqence of admissible blocking matrices were generated by the fnction f : In order to obtain stability and feasibility garantees, we propose to parameterize the inpt as in Section III-B, i.e. P is the soltion to the ARE 7 and k = K LQR x k +c k. We also propose to replace the teral constraint 8c with a set of constraints that is dependent on the crrent blocking matrix T = T k. Specifically, the following problem is solved at each time instant k, given the crrent state x = xk and crrent blocking matrix T = T k: V N x, T := sbj. to x k X, ĉ 0,...,ĉ M 1 + x N P x N, 10a k 1 U, k {1,..., N}, 10b x l X I K LQR, l = pt, 10c [c 0,..., c l 1] = [I l, 0 l N l ]T I m [ĉ 0,..., ĉ M 1], 10d c k = 0, k {l,..., N 1}, 10e k = K LQR x k + c k, k {0,..., N 1}, x k+1 = Ax k + B k, x 0 = x. 10f where pt in 10c is the row of the first non-zero element in the last colmn of T, i.e. pt := {i T i,m = 1}. 11 i {1,...,N} The key part to garanteeing feasibility and stability is the choice of constraints 10c 10e. Clearly, if pt = N, then 10e can be removed and 10d becomes [c 0,..., c ] = T I m [ĉ 0,..., ĉ M 1]. If we denote the optimizer to the above problem as Ĉ x, T := [ĉ 0x, T,..., ĉ M 1x, T ] and C x, T := [c 0x, T,..., c x, T ] := T I m Ĉ x, T then the RHC law is given by k = K LQR xk + c 0xk, T k and the closed-loop system is xk + 1 = A + BK LQR xk + Bc 0xk, T k 12 with T k + 1 = ft k. Theorem 2: If the MWB scheme is applied as above and T 0 is an admissible blocking matrix, then the origin of the closed-loop system 12 is an asymptotically stable eqilibrim with a region of attraction eqal to the set of initial states x0 for which a soltion to the optimization problem in 10 exists. Proof: The proof is based on sing the standard method [7] of considering, at time k + 1, a shifted version of the optimal pertrbation seqence fond at time k: C xk, T k = [c 1xk, T k,..., c xk, T k, 0]. If, at time k + 1, there exists a Ĉ sch that the constraints in 10 are satisfied with C = C xk, T k, then it is easy to show that the vale fnction at time k + 1 is less than the vale fnction at time k for all xk 0, i.e. VN xk + 1, T k + 1 < V N xk, T k. This allows one to se VN as a Lyapnov fnction for the close-loop system. The key point to recognize is that, becase X I K LQR is invariant nder the control law = K LQR x, the optimal predicted teral state x N xk, T k at time k is in X I K LQR, regardless of the vale of pt k. This implies that there always exists a Ĉ sch that C = C xk, T k is feasible at time k + 1. We distingish between two distinct cases, where the proofs of feasibility differ in sbtle, bt important ways: i pt k + 1 = N implies that c = ĉ M 1 can be chosen independently of c N 2 = ĉ M 2. It also implies that the 2, 1 element of T k is zero, i.e. the first colmn of S the matrix in the definition of ft k is zero. This implies that the first row and colmn of T k was removed when compting T k + 1 and hence a shifted version of Ĉ xk, T k is feasible at time k + 1. In other words, it is easy to verify that Ĉ = [ĉ 1xk, T k,..., ĉ M 1xk, T k, 0] reslts in C = C xk, T k at time k + 1. ii If 1 pt k + 1 N 1, then we cannot se the same argments as for pt k + 1 = N; the shifted version of Ĉ xk, T k cannot be garanteed to be feasible. However, note that pt k+1 = pt k 1.

5 It is now easy to verify from 10d 10e that a nonshifted version of Ĉ xk, T k, i.e. Ĉ = [ĉ 0xk, T k,..., ĉ M 1xk, T k ] reslts in C = C xk, T k at time k + 1. Note that this corresponds to cropping the nmber of free offsets see 10e at time k + 1. Remark 1: Note that for all states x X I K LQR the inpt k = K LQR x k will be feasible, i.e. Ĉ = 0 will be a feasible and optimal soltion. Remark 2: Note that Theorem 2 does not hold if, as in Section III-A, Û is chosen as the optimizer. For example, there may not be a constant seqence of inpts which keep the state within X I K sch that recrsive feasibility can be proven. V. NUMERICAL EXAMPLES In this section, we will compare the varios move blocking schemes of Section III as well as the moving window blocking strategy in Section IV with the optimal fll degree of freedom FDOF controller. A. Problem Formlation Specifically, 10 random open-loop stable systems and 10 random open-loop nstable systems were generated. All systems have n = 2 states and m = 1 inpts. Sbseqently, we compted the mp-qp [2] soltion to 3 for N = 10 FDOF as well as the mp-qp soltions to the inpt blocking IB, offset-blocking OB, delta-inpt blocking DIB, delta-offset blocking DOB and moving window blocking MWB strategies for N = 10, M = 2. We will se the notation [i j] to denote that the first i elements and the last j elements of the optimizer are fixed to be constant k = k+1, k {0,..., i 1} and k {i,..., j 1} in IB, OB and MWB blocking schemes. Althogh we considered all possible combinations [i j], only a small sbset will be shown here de to space constraints. For delta blocking strategies 0, 9 and c 0, c 9, are the decision variables for DIB and DOB, respectively, denoted by [0 9]. MWB is initialized sch that T 0 is eqal to T of IB and OB. Note that the scheme in Section IV may ths force the initial teral set constraint to be too restrictive, i.e. pt 0 may be smaller than necessary. In order to avoid this isse, the MWB controller is initialized with a warm start as described in the following. For example, starting with T 0 = [1 9] the constraint 10c wold set l = 2, i.e. or optimization problem wold be solved with a prediction horizon of N = 2. Instead, we here solve the problem at the first time step k = 0 for the fll horizon N = 10, l = 10 with blocking matrix T = [1 9]. At the next time step k = 1, the MWB scheme is applied with T 1 = [9 1]. Feasibility and stability from k = 0 to k = 1 can be trivially garanteed by considering a shifted version of the inpt seqence k = 0 at time step k = 1. A similar warm start was applied for the other blocking matrices T 0. The control objective e.g. 3a was set to Q = I, R = 1 and the teral weight P as the soltion to the ARE 7. The partition of the fll degree of freedom controller with x N X I K LQR is invariant by design. Therefore constraint satisfaction is garanteed for all time and all states xk X N. Since this does not hold for IB, OB, DIB and DOB we compted the invariant sbset of the move blocking partitions with the method in [5]. In a second step, the invariant set was gridded to obtain at least 50 feasible initial states and the corresponding closed-loop performance of the varios controllers was compted and compared. B. Reslts Table I compares the closed-loop performance and Table II the volme of the controllable state space of the move blocking schemes verss the FDOF optimal controller. The reslts present the ratio to the FDOF case in percentages. The abbreviations, avg and max denote the imm, average and maximm vales, respectively. The abbreviation n.c. indicates the nmber of systems that were not controllable with the proposed move blocking scheme. For stable systems offset blocking strategies OB, DOB, and MWB yield slightly better performance than inpt blocking strategies IB and DIB Table I. Bt the controllable state space obtained with inpt blocking strategies is significantly larger than that obtained with offset blocking strategies Table II. This is mostly becase, in general, no constant offset ĉ exists, which satisfies the inpt constraints for those initial states that are far away from the origin. For nstable systems inpt blocking strategies are inferior verss offset blocking strategies in terms of performance. Closed-loop performance can be p to 50 times worse than that obtained with an optimal controller FDOF. In some cases it is not possible to control the system with inpt blocking strategies at all. Note the difference in the controllable systems between the three IB schemes [1 9], [2 8] and [3 7]. The reslts sggest that an neven nmber of blocked moves is more likely to yield the desired closed-loop behavior for nstable systems. This was reaffirmed in a detailed stdy of additional systems where it was observed that this behavior occrs for systems with nstable negative poles. For these types of systems, an neven nmber of blocked moves generally leads to more aggressive controller action on the first inpt, which in trn increases the invariant set. The controller is more aggressive on the first step becase the states on one end of the nstable oscillation are weighed more heavily in the cost fnction. MWB and OB give similar reslts in terms of performance and volme of the controllable state space, bt it cannot be stressed enogh that only the moving window strategy garantees closedloop stability and recrsive feasibility. Frthermore, the reslts show that for certain systems a moving-window

6 stable systems nstable systems avg % max % avg % max % n.c. IB T [1 9] IB T [2 8] IB T [3 7] DIB [0 9] OB T [1 9] OB T [2 8] OB T [3 7] DOB [0 9] MWB T [1 9] MWB T [2 8] MWB T [3 7] TABLE I COMPARISON OF MOVE BLOCKING SCHEMES VERSUS THE OPTIMAL CONTROLLER: CLOSED-LOOP PERFORMANCE DECAY E.G. 0% EQUALS OPT. CTRL.. n.c. DENOTES THE NUMBER OF SYSTEMS THAT WERE NOT CONTROLLABLE WITH THE RESPECTIVE MOVE BLOCKING SCHEME. stable systems nstable systems avg % max % avg % max % n.c. IB T [1 9] IB T [2 8] IB T [3 7] DIB [0 9] OB T [1 9] OB T [2 8] OB T [3 7] DOB [0 9] MWB T [1 9] MWB T [2 8] MWB T [3 7] X I K LQR TABLE II COMPARISON OF MOVE BLOCKING SCHEMES AND MAXIMUM INVARIANT SET VERSUS THE OPTIMAL CONTROLLER: VOLUME OF THE CONTROLLABLE STATE SPACE. n.c. DENOTES THE NUMBER OF SYSTEMS THAT WERE NOT CONTROLLABLE WITH THE RESPECTIVE MOVE BLOCKING SCHEME. strategy may yield excellent performance for a large set of states, together with stability and feasibility garantees. The examples presented in this section are only a small sbset of all the cases exaed. Different cost objectives, system sizes and blocking combinations [i j] were considered. The heristic insights gained in the case stdy are presented in the following section. All reslts were created with the MPT Toolbox [6]. VI. CONCLUSION This paper pointed ot the theoretical and practical shortcogs of varios move-blocking strategies. In addition, a novel move blocking scheme, termed Moving Window Blocking MWB, was introdced. The MWB strategy provides stability and feasibility garantees. An extensive case stdy was also presented which shows the advantages and drawbacks of the different strategies. From the large nmber of exaed systems we were able to dedce the following heristic insights: For stable systems, the IB strategy tends to give better reslts with respect to performance and size of the controllable state space than any of the other strategies which were considered. For nstable systems, the offset blocking strategies e.g. OB, DOB and MWB give better performance reslts than inpt blocking schemes e.g. IB, DIB and are always able to stabilize a sbset of the controllable state space. If the controllable state space is large in volme, then strategies sing offset blocking e.g. OB, MWB, DOB are generally not able to obtain feasible control seqences for states which are far from the origin. For stable systems the controllable state-space is therefore significantly smaller than that obtained with inpt blocking schemes. Systems with negative nstable poles are more difficlt to control with IB when choosing an even nmber of blocked moves for the first blocking seqence in the prediction horizon. MWB is the only blocking scheme that provides garantees on stability and constraint satisfaction. ACKNOWLEDGEMENT This research has been spported by the Eropean Commission Project Control and Comptation IST and the Royal Academy of Engineering, UK. REFERENCES [1] A. Bemporad and M. Morari. Robst model predictive control: A srvey. In Robstness in Identification and Control, nmber 245 in LNCS, pages [2] A. Bemporad, M. Morari, V. Da, and E.N. Pistikopolos. The explicit linear qadratic reglator for constrained systems. Atomatica, 381:3 20, Janary [3] E. G. Gilbert and K. T. Tan. Linear systems with state and control constraints: the theory and applications of maximal otpt admissible sets. IEEE Trans. Atomatic Control, 369: , [4] P. Grieder, F. Borrelli, F.D. Torrisi, and M. Morari. Comptation of the constrained infinite time linear qadratic reglator. Atomatica, 404: , April [5] P. Grieder, M. Lüthi, P. Parillo, and M. Morari. Stability & feasibility of receding horizon control. In Eropean Control Conference, Cambridge, UK, September [6] M. Kvasnica, P. Grieder, M. Baotić, and M. Morari. Mlti Parametric Toolbox MPT. In Hybrid Systems: Comptation and Control, Lectre Notes in Compter Science. Springer Verlag, http: //control.ee.ethz.ch/ mpt. [7] D. Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: Stability and optimality. Atomatica, 366: , Jne [8] S.J. Qin and T.A. Badgwell. An overview of indstrial model predictive control technology. In Chemical Process Control - V, volme 93, no. 316, pages AIChe Symposim Series - American Institte of Chemical Engineers, [9] P. Tøndel and T.A. Johansen. Complexity redction in explicit model predictive control. In IFAC World Congress, Spain, Barcelona, 2002, [10] P. Tøndel, T.A. Johansen, and A. Bemporad. An algorithm for mlti-parametric qadratic programg and explicit MPC soltions. Atomatica, 393: , March 2003.