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Receding-Horizon Control of LTI Systems with Quantized Inputs1
Copyright (;) IFAC Analysis and Design of Hybrid Systems, Brittany, France, 2003
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RECEDING-HORIZON CONTROL OF LTI SYSTEMS WITH QUANTIZED INPUTS 1 Bruno Picasso· Stefania Pancanti •• Alberto BeInporad··· Antonio Bicchi ••••
• Scuola Normale Superiore - Pisa fj Centro E. Piaggio, Universita di Pisa; e- mail: [email protected] •• Centro E. Piaggio, Universita di Pisa; e-mail: ste/[email protected] ••• Dip. di Ingegneria dell'In/ormazione, Universitd di Siena; e-mail: [email protected] •••• Centro E. Piaggio, Universitd di Pisa, e- mail: [email protected]
Abstract: This paper deals with the stabilization problem for a particular class of hybrid systems, namely discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. Results of our previous work on the subject provided a description of minimal (in a specific sense) invariant sets that could be rendered maximally attractive under any quantized feedback strategy. In this paper, we consider the design of stabilizing laws that optimize a given cost index on the state and input evolution on a finite, receding horizon. Application of Model Predictive Control techniques for the solution of similar hybrid control problems through Mixed Logical Dynamical reformulations can provide a stabilizing control law, provided that the feasibility hypotheses are met. In this paper, we discuss precisely what are the shortest horizon length and the minimal invariant terminal set for which it can be guaranteed a stabilizing MPC scheme. The simulation of the application of the control scheme to a practical quantized control problem is finally reported. Copyright, 2003, IFAC
1. INTRODUCTION
(Delchamps, 1990; Elia and Mitter, 2001; Bicchi et al., 2002)). Many papers addressed the problem of the stabilization of quantized systems (see (Brockett and Liberzon, 2000; Delchamps, 1990; Elia and Mitter, 2001; Fagnani and Zampieri , 2002; Nair and Evans, 2000; Picasso, 2002; Picasso et al., 2002; Wong and Brockett, 1999)). In (Delchamps, 1990) the author clarifies that the classical concept of stability is not applicable in this context, hence "practical" stability properties are introduced for quantized systems.
Practical applications of control theory reveal some limits of the continuous models in the description of dynamical systems: limited resources or technical constraints, which finally lead to discrete measurements or to a finite number of possible control actions, are typical situations that must be faced. This is part of a broader phenomenon which is referred to as quantization. In the past twenty years the problem of dynamic systems analysis and control synthesis in presence of quantization has developed and is currently growing in interest. It is now consolidated the idea of regarding quantization not as a phenomenon to be neglected or broadly related to approximation , but rather as an useful tool to be studied within proper models (see for instance
In the present paper we investigate the possibility of using Model Predictive Control techniques (MPC) for stabilizing the particular class of hybrid systems comprised of discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. This approach is justified by the fact that MPC is a control policy particularly suited to cope with constrained systems, which is how quantization of the control space can
1 Support from European Project Recsys-lst- 2001-37170 and from Progetto coordinato Agenzia 2000 CNR COOE714
259
2. QUANTIZED MODEL PREDICTIVE CONTROL (QMPC)
be interpreted. Moreover, we take advantage of the fact that, since a predictive controller generates the control action by solving an optimization problem, the design of the controller is reduced to a mathematical programming problem. The latter is in general more treatable a task than the computation of an explicit control law, especially when dealing with severe constraints such as input quantization and bounds on state evolution. When applicable, the model predictive approach is successful allowing to reduce the stabilization problem to the search of invariant sets and to the study of their reach ability. A detailed study of the reachability is needed to characterize the feasibility region of the optimization problem defining the model predictive controller.
Consider the following linear discrete time invariant system
x(t + 1)
(1) m where x E lRn , U E U c lR is a finite control set and (A, B) E lRnxn x lRnxm is a stabilizable pair. Consider the model predictive controller defined as
u(t) = u~(x(t)) ,
(2)
where, given a finite number of steps Hp > 0, o(x(t)) is the first element of the minimizer U* (x (t) ) E U Hp of the following optimization problem:
U
In this paper, we consider for simplicity only discrete-time systems. Indeed, control quantization of continunous time systems allows generically for results comparable to the (saturated) continuous control case, if arbitrarily fast switching is allowed (Raisch, 1994). On the other hand, if a minimum dwelling time among switches is considered, an equivalent discrete-time model can usually be derived. An interesting extension of our results would be to communication- limited control problems, whereby sampling time and quantization cardinality clearly are to be traded off. Although our results are clearly relevant to that problem, we do not pursue this topic further in this paper.
where min {J(U,x(t))} UEUHp J(U,x(t)) = tXHpPXH p+ Hp-l + (lXkQXk + tukRuk)
L
(3a)
(3b)
k=O
Xo := x(t) s.t. Xk+l:= AXk { Uk E U,
+ BUk,
k = 0, ... , Hp - 1, (3c)
where R = t R > 0, Q = tQ ~ 0, P = t P ~ 0 are matrices of suitable dimensions. Note that in (3) Xo is the current state, Xl, ... ,XHp are the predicted states for the future Hp sampling instants when the sequence of controls U := (uo, Ul , ... UHp-l) E UHp is applied. The minimizer (which exists because # U < +(0) is denoted by U* := (uo, ui , ... UHp-l) and the dependence on x(t) is omitted for simplicity. Although the cost function (3b) penalizes the trajectories getting far from the equilibrium, the model predictive controller does not guarantee stability: there are well known counter-examples showing that, even for unconstrained stable systems, the closed-loop dynamics induced by the feedback law (2) can be unstable. The reason for this undesirable behaviour is that the cost function (3b) does not properly weigh, nor it imposes a constraint on the final state. While this problem could easily be solved for the aforementioned examples, it is more difficult to face it when quantization is involved. The next Section develops stabilizing MPC strategies for quantized-input systems.
Although invariant sets are very important in control theory, in the current literature (see e.g. (Blanchini, 1999)) few results exist for quantized systems. In this paper we present some results concerning the construction of invariant sets for single-input linear systems: in this case it is possible to face the problem using direct geometric considerations allowing us to get results better suited to the practical implementation of MPC and to a quantitative analysis of the reachability problem. The paper is organized as follows: in Section 2 it is introduced the basic MPC scheme for quantized input systems, stabilizing MPC strategies fitting for quanti zed systems are developed in Section 3. The construction of invariant sets and the study of the feasibility of the optimization problem defining the model predictive controller are faced in Section 4. In the final Section 5 is shown how to render effective the theory developed and implement MPC; also an example of the application of the control scheme to a practical quantized control problem is presented.
r
= Ax(t) + Bu(t)
Notation: Qn(A) := [-~ ; ~ is the hypercube of edge length A, l x J := max {z E Z I z x} is the floor function, x+ is the standard notation for x(t+ 1), ne denotes the complementary of the set nand t x, t A are the transpose of the vector x and of the matrix A respectively.
:s
3. STABILIZING QMPC SCHEMES A suitable terminal cost P is sufficient to guarantee the closed- loop stability of system (1) when A is an asymptotically stable matrix:
Proposition 1. Assume 0 E U , let A be asymptotically stable and let P be the solution of the Lyapunov equation P = tAP A + Q . Then the origin of the closed-loop system (1), (2) is globally asymptotically stable.
260
sequent treatment. For a discrete-time system x+ = f(x,u):
Proof. Let V(x(t») = J(U·,x(t») be the minimum of (3) attained at time t, and denote by x k the corresponding optimal trajectory. At time t + 1, the sequence UI := (ui, ... UHp-I' 0) E U Hp is such that the predicted states Xk x k+ I ' Vk = 0, ... ,Hp - 1, with x(t + 1) Ax(t) + Buo . Hence,
Definition 1. The set n ~ !Rn is said to be controlled-invariant iff V x E n :3 u E U such that x+ En; Definition 2. Let n ~ Xo ~ !Rn; the set n is said to be Xo - attractive iff V x E Xo there exists a trajectory which lies within Xo and enters n in a finite number of steps. If moreover Vx E Xo such trajectory can be chosen of length Hp, then the set n is said to be Xo-attractive in Hp steps.
+ 1») :s
J(UI,x(t + 1») = Hp-I = tXHpPXH p + eXkQXk + tukRud =
V(x(t
L
k=O
= t(Ax·Hp )P(Ax·Hp ) + tx·Hp Qx·Hp + Hp-I
+
L
{txkQXk
Let n c !Rn be a bounded and controlledin its invariant set for system (1) containing interior part. Hence we can assume that a feedback law Fn : n --+ U has been defined so that V x En, Ax+BFn(x) E
°
+ tUkRuk} =
k=1
n.
We use a predictive control strategy in ne to steer the states inside n, then we switch to the feedback law Fn . To this aim a slight modification of the cost function (3b) is sufficient: let
Thus, V(x(t») is a nonnegative and non-increasing sequence, therefore it converges to a finite limit as t -+ 00 . Hence,
°:s
L(x,u)
tx(t)Qx(t)+tu(t)Ru(t) :s V (x(t») - V (x(t+ 1»)
which implies that lim u(t) = t .... oo
°
= Inc(x)·
(fxQx
+ tuRu)
(4)
(where Ine is the characteristic function of ne) and consider the MPC control law (2) based on the optimal control problem
because R is
positive definite. Since U is discrete (actually it is a finite set), there exists a finite time to such that u(t) == for all t 2: to: as A is asymptotically stable it follows that lim x(t) = 0.
°
u';"~~" {J(U,X(t)) ~ ~~' L(X"U,)}
t .... oo
Let us handle now the case of unstable systems. The quantization of the control set is a severe constraint which renders unpracticable most of the classical approaches to the stabilization problem. The approach proposed in (Chmielewski and Manousiouthakis, 1996), consisting of choosing P as the solution to the Riccati equation, can not be applied because the hypothesis that the sequence UI := (ui, ... u Hp - I ' (KXHp») is an admissible one fails when the inputs are quantized. On the other hand, an infinite horizon setup would be numerically intractable. The approach of using a terminal constraint XHp = 0, as for hybrid systems in (Bemporad and Morari, 1999), has the drawback of having a set of initial conditions which in general are limited to a zero-measure set. The idea is to use a relaxed final constraint so that the feasibility of the optimization problem is achieved for a significant set of initial conditions. The terminal equality constraint is then replaced by the requirement that the initial state can be steered in Hp steps inside a controlledinvariant neighbourhood n of the equilibrium (see next Definition 1). This policy gives rise to a quantized- input version of the so- called dualmode predictive control scheme (see Michalska and Mayne (Michalska and Mayne, 1993»: since the system is unstable and the control set is quantized, only practical stability can be achieved, on the other hand this policy has the advantage of obtaining an MPC scheme which is robust against perturbations. We introduce the definitions needed for the sub-
s.t.
{~:~: x~t) Xk+1 := AXk + BUk,
(5a)
k = 0, ... ,Hp - 1
Uk E U. (5b)
Let XCHp):= {x E!Rn I:3U EUHp s.t.
XHp En}
be the feasibility region for the optimization problem (5): note that n ~ X(Hp) . Consider the feedback law F : X(Hp) --+ U
defined by F(x) := { Fp(x) if x E ~ Uo otherwIse,
(6)
where Uo is the first element of the minimizer U· related to the optimization problem (5). Note that the optimization problem (5) is solvable (i.e. the minimum is attained) for all x(t) E XUlp) because # U < +00 .
Proposition 2. Assume n is a controlled- invariant neighbourhood of the origin, and Q > 0. Then n is XCHp) - attractive for the closed- loop system (1), (6). The proof follows arguments similar to the ones used in the classical dual mode MPC scheme (see for instance (Scokaert et al., 1999».
261
Proposition 3. Assume HI and that U = then
Remark 1. [QMPC for nonlinear systems] The quantization of the control set can be profitably exploited to state a more general result which can be proved with some modifications of the arguments leading to the proof of Proposition 2: let n be a controlled-invariant neighbourhood of the equilibrium state (which can be taken to be the origin) for the discrete time system x(t
+ 1) =
g(x(t),u(t»),
x ERn,
U
EU
c
€
Z,
i) "lA ~ €, Qn(A) is controlled- invariant. ii) VHp ~ n, V A ~ €, Qn(€) is Qn(A) attractive in Hp steps.
Moreover Qn(€) is a minimal controlled-invariant hypercube in the following sense:
Rm,
iii) For almost all x E Qn(€) 3! u E U such that x+ E Qn(€) .
where U is a discrete (not necessarily finite) control set. Consider the optimization problem (5) where in Equations (5b) Xk+1 := g(Xk,Uk) and the cost function (4) is replaced by L(x, u) = In e • C(x, u), where C(x, u) is such that there exists a norm on Rn X Rm such that C(x, u) ~ lI(x, u)112 V(x, u) E Rn X U. Then n is X(Hp) attractive for the closed- loop dynamics x(t+ 1) =
In the finite control set case we consider input sets of the type Uk := { -k€, ... ,0, ... , +h} and, for a given A ~ € , we find the condition on k ensuring the controlled- invariance of Qn(A) : Proposition 4. Assume HI and that U = Uk, then Qn(A) is controlled- invariant if and only if k ~ ko , where: 1Ann ko: = -l2~(1- ~IQil)J if ~IQil ~ 1 { o otherwise.
g( x(t), F(x(t»)) induced by the feedback law (6).
For the details of the proof see (Picasso, 2002) . It is worth noting that the only hypothesis involving the function 9 is the assumption that n is a controlled-invariant set. Of course the linear case is interesting both because there exist many efficient algorithms to implement MPC and because in this case results concerning the construction of controlled- invariant sets in the quantized- input case are available (see next Section 4).
Once we know that Qn(-r) is controlled-invariant (for , ~ €), we wish to study its attractivity properties in a fixed number of steps. The set X(Hp) from which Qn(-r) is reachable in Hp steps can be easily characterized when the matrix A is invertible: in this case, set RHp := [A H p-1 BI· . 'IABIB] , a simple calculation shows that
As a consequence of Proposition 2 and Remark 1 the model predictive approach allows to reduce the stabilization problem to the search of controlled- invariant sets and to the study of their reachability (which is indeed the study of the feasibility of the optimization problem defining the model predictive controller) .
X(Hp)
=
U
{A - Hp
(Qn(-r»)
- A - HpRHpU} .
(7) Since an increasing family of sets (Qn(-r») n€' which can be rendered invariant, is available, we have faced also another problem: to look for conditions on Uk ensuring that Qn(€) is Qn(A)attractive in a fixed number of steps (A > €) . Solving this problem allows to introduce an MPC scheme taking into account also constraints on the trajectories (see next Remark 2).
4. INVARIANT SETS In this Section we present some results concerning the construction of controlled-invariant sets for quantized-input linear systems. We restrict to the single-input case because in this framework the construction can be done using techniques that provide results which better suit the practical implementation of MPC: in fact we find invariant sets with polyhedral structure and also a quantitative analysis of the reachability problem can be done. We examine the case of a uniformly quantized control set which is fixed a priori, more precisely it is assumed that U C € Z for some € > 0 . Let us start considering systems such that the pair (A, B) is reachable : in this case a change of the coordinates allows us to work with the controller form associated to the pair (A , B). We will refer to the following hypothesis: HI) The pair (A , B) is reachable and the system (1) is in controller form. Let sn - QnSn-l - ... Q2S - Ql be the characteristic polynomial of A . The basic results about controlled- invariant sets are summarized in the following statements, for a detailed exposition we refer to (Picasso et al., 2002).
Proposition 5. Assume HI , let A > € > 0 and U = Uk, fix Hp ~ n ; Hp = n + p - 1 with p ~ 1 . A sufficient condition in order that Qn(€) is Qn(A)- attractive in Hp steps is that k ~ kp , with kp = -
l2lA~ (
~
1A-€J
1 - L..- IQi I ) - -; ~ ,
i=l
'l-'p
where the sequence {1/Jm}m EN ' is defined as follows:
1/J1 := 1 m-l
{
1/Jm := 1 + m
~
~ I,Qn-m+i+.1~ 1/Ji 2, where
Qj . -
.if .
0 If J:S O.
Ukp renders Qn(€) controlled- invariant. Moreover, if Qi ~ 0 Vi , then the condition is also necessary.
262
Remark 2. Let 0 C Xo C Rn and suppose that is controlled- invariant and Xo-attractive in Hp steps for system (1). Consider the feedback law (6) based on the optimization problem (5) where the state constraint Xk E X o, V k = 0, .. . , Hp - 1 is added to the constraints in Equation (5b): then 0 is X oattractive for the induced closed- loop system.
where the column vector p, U := t[t Uo, . " t UHp-l ]ETfJ>mH IN>
o
is the optimization vector, H = tHis positi.ve definite and H, F, Y, G, W, E are easily obtained from Q, R, and (3b) (as only the optimizer U is needed, the term involving Y is usually removed from (8)). The optimization problem (3) is an IQP. Because the problem depends on the current state x(t), the implementation of MPC requires the on-line solution of an IQP at each time step. Although efficient (mixed) integer quadratic programming solvers based on branch and bound methods are available, computing the input u(t) demands significant on-line computation effort. In (Bemporad, 2003) the author provides a method for solving nonlinear integer programming problems that depend linearly on a vector of parameters, and in particular for computing off-line the equivalent piece wise constant form of the MPC control law defined by (8) . Problem (5) can instead cast as a mixed integer quadratic program (MIQP). By defining
Let us consider now the case of a stabilizable pair (A, B) . A change of the coordinates allows us to transform the system into the following canonical form for a stabilizable pair:
A'~(U :'0 aU::) n·~m where A3 is an asymptotically stable matrix. Suppose (A, B) is a stabilizable pair in the canonical form: let Rn := ]Rnc EB ]Rna. be the underlying decomposition of the state space. As A3 is an asymptotically stable matrix, then V Q E ]Rna. xn a • such that tQ = Q > 0, ::I! P > 0 so that tA 3 PA 3 - P = -Q. V(z) := tzpz is a Lyapunov function for the system z+ = A3Z, hence Vr ~ 0, {z E Rna. I V (z) :S r} is A3 - invariant. Note that
zE{V:Sr}:::} IIA2ZII00:S,(r). With the notations introduced we can state the following: Proposition 6. Assume U := f Z, V r Or := ([ -
~
-
(ne - l)r(r); ~
~
[bk = 0] we finally obtain
0 the set
mln
{
U,Z,.6.
+ (ne - l)r(r)]
X
Hfl
0]
(10)
z( QZk + z;:' EZ;:}
k=O
subj . to zk = (Akx(t)
+ L Aj BUk - l - j )bk j=O
[Xk E 0] Hp-l Aj BUHp - l-j E 0 AHpx(t) + [bk = 0]
is controlled-invariant.
f-t
L
Remark 3. The invariant sets found in this case can be significantly larger then the ones found in the reachable case: this because of the uncontrollable part z+ = A3Z of the system which gives rise to the constant ,(r) affecting the reachable part. However lim z(t) = 0, this means that in
j=O
Uk E U, Vk
= 0, . . . , Hp -
1
(11)
t [t Zox '"
U] zHp-l t Zou ... tzHp _l' ~:= t[bo . .. bHp - d, which can be translated to MIQP according to standard techniques (see e.g. (Bemporad and Morari, 1999)).
h Z := were
t-->oo
practice the states evolve inside Or converging towards the set ([-~; ~r c ) x {O}.
Remark
4.
Xmin
It is immediate to cast problems (3) and (5) as integer quadratic programs (IQPs). Indeed , by substituting Xk = Akx(t) + L~'::~ Aj BUk - l - j , Equation (3b) can be rewritten as
t x
State constraints having the form
5. QMPC COMPUTATIONS
U
E
k- l
+ (ne - 2)r(r)] X ... x[-~-,(r); ~+,(r)]x[-~ ; ~J)x{V:sr}
x [ - ~ - (ne - 2)r(r); ~
min
[Xk
f-t
:S x(t) :S
X max
(or, more generally, Ax(t) :S fJ) can be easily taken into account by adding the linear constraints
A [AkX(t) +
{~U'HU +x'(t)FU + ~x'(t)YX(t)} 2 2
~ Aj BUk_I_j]
< fJ
J= O
subj . to GU :S W + Ex(t) UkEU, Vk=0 , .. . , H p - 1
in (11). Clearly, in this case the invaria~t set n must be contained in the polyhedron Ax :S fJ in order to guarantee constraint fulfillment of the
(8)
263
1 . . . . ....., O ft
REFERENCES
ca..c
Bemporad, A. (2003) . Multiparametric nonlinear integer programming and explicit quantized optimal control. Submitted. Bemporad, A. and M. Morari (1999). Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407- 427. Bicchi, A., A. Marigo and B. Piccoli (2002). On the reachability of quantized control systems . IEEE Transactions on Automatic ControI47(4),546- 563 . Blanchini, F . (1999). Set invariance in control. Automatica 35,1747- 1767. Brockett, R. and D. Liberzon (2000) . Quantized feedback stabilization of linear systems. IEEE Trans. Autom. Control 45(7) , 1279- 1289. Chmielewski, D. and V. Manousiouthakis (1996) . On constrained infinite-time linear quadratic optimal control. Systems and Control Letters 29(3), 121- 130. Delchamps, D. F. (1990). Stabilizing a linear system with quantized state feedback. IEEE Trans. Autom. Control 35(8), 916- 924. Elia, N. and S. Mitter (2001). Stabilization of linear systems with limited information. IEEE Trans. Autom. Control 46(9), 1384-1400. Fagnani, F. and S. Zampieri (2002). Stabilizing quantized feedback with minimal information flow : the scalar case. In: Proc. of MTNS Conference. Notre Dame, Indiana. Michalska, H. and D . Q. Mayne (1993) . Robust receding horizon of constrained non linear systems. IEEE Transactions on Automatic Control3S, 1623- 1632. Nair , G .N. and R.J. Evans (2000). Stabilization with data rate limited feedback: tightest attainable bounds. Systems and Control Letters 41,49- 56. Picasso, B. (2002). Stabilization of quantized-input systems with optimal control techniques. Degree thesis. Dipartimento di Matematica, University of Pisa, Italy. Available writing to:
':r
Fig. 1. The state trajectory from [-0.8, -0.9] with quantized control set U = {-1,0, 1} QMPC controller at all times t > 0 . Hence the algorithm can take into account state constraints also along the predicted trajectories: this is an important extension of the recent approach presented in (Quevedo et al., 2002) which is not suitable to dealing with state constraints. Example: In the following we show an application of QMPC computations to the classical problem of stabilizing an inverted pendulum. The choice of a sampling time T ~ 0.00571s and the length of the pendulum I = 0.25m gives rise to the discrete-time model in controller form with parameters o} = -0.9987 and 0:2 = 2. The system is controlled by mean of a stepper motor, hence the quantized control set is U = {- I, 0, I} . This example was previously studied in (Picasso et al., 2002), where it was shown that, while a simple quantized version of a deadbeat controller was sufficient to make Q2(1) invariant, the largest ~ for which Q2(~) could be made invariant and attracted by Q2(1) , was ~ ~ 1. By implementing a MPC scheme such as that discussed above (based on a horizon length 3) , it is possible to guarantee attractivity of Q2 (1) for all initial conditions in a substantially larger region of the state space, given by (7). An example trajectory is shown in figure (1) .
picasso~piaggio.ccii.unipi.it.
Picasso, B., F . Gouaisbaut and A. Bicchi (2002) . Construction of invariant and attractive sets for quanti zed- input linear systems. Proc. 41st IEEE Conference on Decision and Control pp. 824- 829. Quevedo, D. E ., J . A. De Dona and G. C. Goodwin (2002). Receding horizon linear quadratic control with finite input constraint sets. IFAC 15th Triennial World Congress. Raisch , J . (1994) . Simple hybrid control systems - continuous fdlti plants with symbolic measurements and quantized control inputs. In: 11th International Conference on Analysis and Optimization of Systems (G . Cohen and J .-P. Quadrat , Eds.) . Vol. 199 of LectuT'es Notes in Control and Information Sciences. Springer-VerJag. pp. 369- 376. Scokaert , P. O. M., D. Q. Mayne and J . B . Rawlings (1999). Suboptimal model predictive control (feasibility implies stability). IEEE Transactions on A utomatic Control 44(3) , 648- 654 . Wong, W . and R . Brockett (1999). Systems with finite communication bandwidth constraintspart 11: Stabilization with limited information feedback. IEEE Transactions on Automatic Control 44(5) , 1049- 1053.
CONCLUSIONS In this paper we have considered the stabilization problem for discrete-time linear systems subject to a fixed, uniform quantization of control inputs. It has been shown that the Model Predictive Control approach can be profitably used and we provided conditions for its applicability. Several open problems remain in this field , among which is the extension to (quantized) output MPC. More generally, the combination of quantization with limited communication bandwidth is a most important and challenging area to which furth er work will be devoted. Acknow ledgements Frederic Gouaisbaut is gratefully acknowledged for useful discussions and collaboration in the preliminary phase of this work.
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RECEDING-HORIZON CONTROL OF LTI SYSTEMS WITH QUANTIZED INPUTS 1 Bruno Picasso· Stefania Pancanti •• Alberto BeInporad··· Antonio Bicchi ••••
• Scuola Normale Superiore - Pisa fj Centro E. Piaggio, Universita di Pisa; e- mail: [email protected] •• Centro E. Piaggio, Universita di Pisa; e-mail: ste/[email protected] ••• Dip. di Ingegneria dell'In/ormazione, Universitd di Siena; e-mail: [email protected] •••• Centro E. Piaggio, Universitd di Pisa, e- mail: [email protected]
Abstract: This paper deals with the stabilization problem for a particular class of hybrid systems, namely discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. Results of our previous work on the subject provided a description of minimal (in a specific sense) invariant sets that could be rendered maximally attractive under any quantized feedback strategy. In this paper, we consider the design of stabilizing laws that optimize a given cost index on the state and input evolution on a finite, receding horizon. Application of Model Predictive Control techniques for the solution of similar hybrid control problems through Mixed Logical Dynamical reformulations can provide a stabilizing control law, provided that the feasibility hypotheses are met. In this paper, we discuss precisely what are the shortest horizon length and the minimal invariant terminal set for which it can be guaranteed a stabilizing MPC scheme. The simulation of the application of the control scheme to a practical quantized control problem is finally reported. Copyright, 2003, IFAC
1. INTRODUCTION
(Delchamps, 1990; Elia and Mitter, 2001; Bicchi et al., 2002)). Many papers addressed the problem of the stabilization of quantized systems (see (Brockett and Liberzon, 2000; Delchamps, 1990; Elia and Mitter, 2001; Fagnani and Zampieri , 2002; Nair and Evans, 2000; Picasso, 2002; Picasso et al., 2002; Wong and Brockett, 1999)). In (Delchamps, 1990) the author clarifies that the classical concept of stability is not applicable in this context, hence "practical" stability properties are introduced for quantized systems.
Practical applications of control theory reveal some limits of the continuous models in the description of dynamical systems: limited resources or technical constraints, which finally lead to discrete measurements or to a finite number of possible control actions, are typical situations that must be faced. This is part of a broader phenomenon which is referred to as quantization. In the past twenty years the problem of dynamic systems analysis and control synthesis in presence of quantization has developed and is currently growing in interest. It is now consolidated the idea of regarding quantization not as a phenomenon to be neglected or broadly related to approximation , but rather as an useful tool to be studied within proper models (see for instance
In the present paper we investigate the possibility of using Model Predictive Control techniques (MPC) for stabilizing the particular class of hybrid systems comprised of discrete-time linear systems subject to a uniform (a priori fixed) quantization of the control set. This approach is justified by the fact that MPC is a control policy particularly suited to cope with constrained systems, which is how quantization of the control space can
1 Support from European Project Recsys-lst- 2001-37170 and from Progetto coordinato Agenzia 2000 CNR COOE714
259
2. QUANTIZED MODEL PREDICTIVE CONTROL (QMPC)
be interpreted. Moreover, we take advantage of the fact that, since a predictive controller generates the control action by solving an optimization problem, the design of the controller is reduced to a mathematical programming problem. The latter is in general more treatable a task than the computation of an explicit control law, especially when dealing with severe constraints such as input quantization and bounds on state evolution. When applicable, the model predictive approach is successful allowing to reduce the stabilization problem to the search of invariant sets and to the study of their reach ability. A detailed study of the reachability is needed to characterize the feasibility region of the optimization problem defining the model predictive controller.
Consider the following linear discrete time invariant system
x(t + 1)
(1) m where x E lRn , U E U c lR is a finite control set and (A, B) E lRnxn x lRnxm is a stabilizable pair. Consider the model predictive controller defined as
u(t) = u~(x(t)) ,
(2)
where, given a finite number of steps Hp > 0, o(x(t)) is the first element of the minimizer U* (x (t) ) E U Hp of the following optimization problem:
U
In this paper, we consider for simplicity only discrete-time systems. Indeed, control quantization of continunous time systems allows generically for results comparable to the (saturated) continuous control case, if arbitrarily fast switching is allowed (Raisch, 1994). On the other hand, if a minimum dwelling time among switches is considered, an equivalent discrete-time model can usually be derived. An interesting extension of our results would be to communication- limited control problems, whereby sampling time and quantization cardinality clearly are to be traded off. Although our results are clearly relevant to that problem, we do not pursue this topic further in this paper.
where min {J(U,x(t))} UEUHp J(U,x(t)) = tXHpPXH p+ Hp-l + (lXkQXk + tukRuk)
L
(3a)
(3b)
k=O
Xo := x(t) s.t. Xk+l:= AXk { Uk E U,
+ BUk,
k = 0, ... , Hp - 1, (3c)
where R = t R > 0, Q = tQ ~ 0, P = t P ~ 0 are matrices of suitable dimensions. Note that in (3) Xo is the current state, Xl, ... ,XHp are the predicted states for the future Hp sampling instants when the sequence of controls U := (uo, Ul , ... UHp-l) E UHp is applied. The minimizer (which exists because # U < +(0) is denoted by U* := (uo, ui , ... UHp-l) and the dependence on x(t) is omitted for simplicity. Although the cost function (3b) penalizes the trajectories getting far from the equilibrium, the model predictive controller does not guarantee stability: there are well known counter-examples showing that, even for unconstrained stable systems, the closed-loop dynamics induced by the feedback law (2) can be unstable. The reason for this undesirable behaviour is that the cost function (3b) does not properly weigh, nor it imposes a constraint on the final state. While this problem could easily be solved for the aforementioned examples, it is more difficult to face it when quantization is involved. The next Section develops stabilizing MPC strategies for quantized-input systems.
Although invariant sets are very important in control theory, in the current literature (see e.g. (Blanchini, 1999)) few results exist for quantized systems. In this paper we present some results concerning the construction of invariant sets for single-input linear systems: in this case it is possible to face the problem using direct geometric considerations allowing us to get results better suited to the practical implementation of MPC and to a quantitative analysis of the reachability problem. The paper is organized as follows: in Section 2 it is introduced the basic MPC scheme for quantized input systems, stabilizing MPC strategies fitting for quanti zed systems are developed in Section 3. The construction of invariant sets and the study of the feasibility of the optimization problem defining the model predictive controller are faced in Section 4. In the final Section 5 is shown how to render effective the theory developed and implement MPC; also an example of the application of the control scheme to a practical quantized control problem is presented.
r
= Ax(t) + Bu(t)
Notation: Qn(A) := [-~ ; ~ is the hypercube of edge length A, l x J := max {z E Z I z x} is the floor function, x+ is the standard notation for x(t+ 1), ne denotes the complementary of the set nand t x, t A are the transpose of the vector x and of the matrix A respectively.
:s
3. STABILIZING QMPC SCHEMES A suitable terminal cost P is sufficient to guarantee the closed- loop stability of system (1) when A is an asymptotically stable matrix:
Proposition 1. Assume 0 E U , let A be asymptotically stable and let P be the solution of the Lyapunov equation P = tAP A + Q . Then the origin of the closed-loop system (1), (2) is globally asymptotically stable.
260
sequent treatment. For a discrete-time system x+ = f(x,u):
Proof. Let V(x(t») = J(U·,x(t») be the minimum of (3) attained at time t, and denote by x k the corresponding optimal trajectory. At time t + 1, the sequence UI := (ui, ... UHp-I' 0) E U Hp is such that the predicted states Xk x k+ I ' Vk = 0, ... ,Hp - 1, with x(t + 1) Ax(t) + Buo . Hence,
Definition 1. The set n ~ !Rn is said to be controlled-invariant iff V x E n :3 u E U such that x+ En; Definition 2. Let n ~ Xo ~ !Rn; the set n is said to be Xo - attractive iff V x E Xo there exists a trajectory which lies within Xo and enters n in a finite number of steps. If moreover Vx E Xo such trajectory can be chosen of length Hp, then the set n is said to be Xo-attractive in Hp steps.
+ 1») :s
J(UI,x(t + 1») = Hp-I = tXHpPXH p + eXkQXk + tukRud =
V(x(t
L
k=O
= t(Ax·Hp )P(Ax·Hp ) + tx·Hp Qx·Hp + Hp-I
+
L
{txkQXk
Let n c !Rn be a bounded and controlledin its invariant set for system (1) containing interior part. Hence we can assume that a feedback law Fn : n --+ U has been defined so that V x En, Ax+BFn(x) E
°
+ tUkRuk} =
k=1
n.
We use a predictive control strategy in ne to steer the states inside n, then we switch to the feedback law Fn . To this aim a slight modification of the cost function (3b) is sufficient: let
Thus, V(x(t») is a nonnegative and non-increasing sequence, therefore it converges to a finite limit as t -+ 00 . Hence,
°:s
L(x,u)
tx(t)Qx(t)+tu(t)Ru(t) :s V (x(t») - V (x(t+ 1»)
which implies that lim u(t) = t .... oo
°
= Inc(x)·
(fxQx
+ tuRu)
(4)
(where Ine is the characteristic function of ne) and consider the MPC control law (2) based on the optimal control problem
because R is
positive definite. Since U is discrete (actually it is a finite set), there exists a finite time to such that u(t) == for all t 2: to: as A is asymptotically stable it follows that lim x(t) = 0.
°
u';"~~" {J(U,X(t)) ~ ~~' L(X"U,)}
t .... oo
Let us handle now the case of unstable systems. The quantization of the control set is a severe constraint which renders unpracticable most of the classical approaches to the stabilization problem. The approach proposed in (Chmielewski and Manousiouthakis, 1996), consisting of choosing P as the solution to the Riccati equation, can not be applied because the hypothesis that the sequence UI := (ui, ... u Hp - I ' (KXHp») is an admissible one fails when the inputs are quantized. On the other hand, an infinite horizon setup would be numerically intractable. The approach of using a terminal constraint XHp = 0, as for hybrid systems in (Bemporad and Morari, 1999), has the drawback of having a set of initial conditions which in general are limited to a zero-measure set. The idea is to use a relaxed final constraint so that the feasibility of the optimization problem is achieved for a significant set of initial conditions. The terminal equality constraint is then replaced by the requirement that the initial state can be steered in Hp steps inside a controlledinvariant neighbourhood n of the equilibrium (see next Definition 1). This policy gives rise to a quantized- input version of the so- called dualmode predictive control scheme (see Michalska and Mayne (Michalska and Mayne, 1993»: since the system is unstable and the control set is quantized, only practical stability can be achieved, on the other hand this policy has the advantage of obtaining an MPC scheme which is robust against perturbations. We introduce the definitions needed for the sub-
s.t.
{~:~: x~t) Xk+1 := AXk + BUk,
(5a)
k = 0, ... ,Hp - 1
Uk E U. (5b)
Let XCHp):= {x E!Rn I:3U EUHp s.t.
XHp En}
be the feasibility region for the optimization problem (5): note that n ~ X(Hp) . Consider the feedback law F : X(Hp) --+ U
defined by F(x) := { Fp(x) if x E ~ Uo otherwIse,
(6)
where Uo is the first element of the minimizer U· related to the optimization problem (5). Note that the optimization problem (5) is solvable (i.e. the minimum is attained) for all x(t) E XUlp) because # U < +00 .
Proposition 2. Assume n is a controlled- invariant neighbourhood of the origin, and Q > 0. Then n is XCHp) - attractive for the closed- loop system (1), (6). The proof follows arguments similar to the ones used in the classical dual mode MPC scheme (see for instance (Scokaert et al., 1999».
261
Proposition 3. Assume HI and that U = then
Remark 1. [QMPC for nonlinear systems] The quantization of the control set can be profitably exploited to state a more general result which can be proved with some modifications of the arguments leading to the proof of Proposition 2: let n be a controlled-invariant neighbourhood of the equilibrium state (which can be taken to be the origin) for the discrete time system x(t
+ 1) =
g(x(t),u(t»),
x ERn,
U
EU
c
€
Z,
i) "lA ~ €, Qn(A) is controlled- invariant. ii) VHp ~ n, V A ~ €, Qn(€) is Qn(A) attractive in Hp steps.
Moreover Qn(€) is a minimal controlled-invariant hypercube in the following sense:
Rm,
iii) For almost all x E Qn(€) 3! u E U such that x+ E Qn(€) .
where U is a discrete (not necessarily finite) control set. Consider the optimization problem (5) where in Equations (5b) Xk+1 := g(Xk,Uk) and the cost function (4) is replaced by L(x, u) = In e • C(x, u), where C(x, u) is such that there exists a norm on Rn X Rm such that C(x, u) ~ lI(x, u)112 V(x, u) E Rn X U. Then n is X(Hp) attractive for the closed- loop dynamics x(t+ 1) =
In the finite control set case we consider input sets of the type Uk := { -k€, ... ,0, ... , +h} and, for a given A ~ € , we find the condition on k ensuring the controlled- invariance of Qn(A) : Proposition 4. Assume HI and that U = Uk, then Qn(A) is controlled- invariant if and only if k ~ ko , where: 1Ann ko: = -l2~(1- ~IQil)J if ~IQil ~ 1 { o otherwise.
g( x(t), F(x(t»)) induced by the feedback law (6).
For the details of the proof see (Picasso, 2002) . It is worth noting that the only hypothesis involving the function 9 is the assumption that n is a controlled-invariant set. Of course the linear case is interesting both because there exist many efficient algorithms to implement MPC and because in this case results concerning the construction of controlled- invariant sets in the quantized- input case are available (see next Section 4).
Once we know that Qn(-r) is controlled-invariant (for , ~ €), we wish to study its attractivity properties in a fixed number of steps. The set X(Hp) from which Qn(-r) is reachable in Hp steps can be easily characterized when the matrix A is invertible: in this case, set RHp := [A H p-1 BI· . 'IABIB] , a simple calculation shows that
As a consequence of Proposition 2 and Remark 1 the model predictive approach allows to reduce the stabilization problem to the search of controlled- invariant sets and to the study of their reachability (which is indeed the study of the feasibility of the optimization problem defining the model predictive controller) .
X(Hp)
=
U
{A - Hp
(Qn(-r»)
- A - HpRHpU} .
(7) Since an increasing family of sets (Qn(-r») n€' which can be rendered invariant, is available, we have faced also another problem: to look for conditions on Uk ensuring that Qn(€) is Qn(A)attractive in a fixed number of steps (A > €) . Solving this problem allows to introduce an MPC scheme taking into account also constraints on the trajectories (see next Remark 2).
4. INVARIANT SETS In this Section we present some results concerning the construction of controlled-invariant sets for quantized-input linear systems. We restrict to the single-input case because in this framework the construction can be done using techniques that provide results which better suit the practical implementation of MPC: in fact we find invariant sets with polyhedral structure and also a quantitative analysis of the reachability problem can be done. We examine the case of a uniformly quantized control set which is fixed a priori, more precisely it is assumed that U C € Z for some € > 0 . Let us start considering systems such that the pair (A, B) is reachable : in this case a change of the coordinates allows us to work with the controller form associated to the pair (A , B). We will refer to the following hypothesis: HI) The pair (A , B) is reachable and the system (1) is in controller form. Let sn - QnSn-l - ... Q2S - Ql be the characteristic polynomial of A . The basic results about controlled- invariant sets are summarized in the following statements, for a detailed exposition we refer to (Picasso et al., 2002).
Proposition 5. Assume HI , let A > € > 0 and U = Uk, fix Hp ~ n ; Hp = n + p - 1 with p ~ 1 . A sufficient condition in order that Qn(€) is Qn(A)- attractive in Hp steps is that k ~ kp , with kp = -
l2lA~ (
~
1A-€J
1 - L..- IQi I ) - -; ~ ,
i=l
'l-'p
where the sequence {1/Jm}m EN ' is defined as follows:
1/J1 := 1 m-l
{
1/Jm := 1 + m
~
~ I,Qn-m+i+.1~ 1/Ji 2, where
Qj . -
.if .
0 If J:S O.
Ukp renders Qn(€) controlled- invariant. Moreover, if Qi ~ 0 Vi , then the condition is also necessary.
262
Remark 2. Let 0 C Xo C Rn and suppose that is controlled- invariant and Xo-attractive in Hp steps for system (1). Consider the feedback law (6) based on the optimization problem (5) where the state constraint Xk E X o, V k = 0, .. . , Hp - 1 is added to the constraints in Equation (5b): then 0 is X oattractive for the induced closed- loop system.
where the column vector p, U := t[t Uo, . " t UHp-l ]ETfJ>mH IN>
o
is the optimization vector, H = tHis positi.ve definite and H, F, Y, G, W, E are easily obtained from Q, R, and (3b) (as only the optimizer U is needed, the term involving Y is usually removed from (8)). The optimization problem (3) is an IQP. Because the problem depends on the current state x(t), the implementation of MPC requires the on-line solution of an IQP at each time step. Although efficient (mixed) integer quadratic programming solvers based on branch and bound methods are available, computing the input u(t) demands significant on-line computation effort. In (Bemporad, 2003) the author provides a method for solving nonlinear integer programming problems that depend linearly on a vector of parameters, and in particular for computing off-line the equivalent piece wise constant form of the MPC control law defined by (8) . Problem (5) can instead cast as a mixed integer quadratic program (MIQP). By defining
Let us consider now the case of a stabilizable pair (A, B) . A change of the coordinates allows us to transform the system into the following canonical form for a stabilizable pair:
A'~(U :'0 aU::) n·~m where A3 is an asymptotically stable matrix. Suppose (A, B) is a stabilizable pair in the canonical form: let Rn := ]Rnc EB ]Rna. be the underlying decomposition of the state space. As A3 is an asymptotically stable matrix, then V Q E ]Rna. xn a • such that tQ = Q > 0, ::I! P > 0 so that tA 3 PA 3 - P = -Q. V(z) := tzpz is a Lyapunov function for the system z+ = A3Z, hence Vr ~ 0, {z E Rna. I V (z) :S r} is A3 - invariant. Note that
zE{V:Sr}:::} IIA2ZII00:S,(r). With the notations introduced we can state the following: Proposition 6. Assume U := f Z, V r Or := ([ -
~
-
(ne - l)r(r); ~
~
[bk = 0] we finally obtain
0 the set
mln
{
U,Z,.6.
+ (ne - l)r(r)]
X
Hfl
0]
(10)
z( QZk + z;:' EZ;:}
k=O
subj . to zk = (Akx(t)
+ L Aj BUk - l - j )bk j=O
[Xk E 0] Hp-l Aj BUHp - l-j E 0 AHpx(t) + [bk = 0]
is controlled-invariant.
f-t
L
Remark 3. The invariant sets found in this case can be significantly larger then the ones found in the reachable case: this because of the uncontrollable part z+ = A3Z of the system which gives rise to the constant ,(r) affecting the reachable part. However lim z(t) = 0, this means that in
j=O
Uk E U, Vk
= 0, . . . , Hp -
1
(11)
t [t Zox '"
U] zHp-l t Zou ... tzHp _l' ~:= t[bo . .. bHp - d, which can be translated to MIQP according to standard techniques (see e.g. (Bemporad and Morari, 1999)).
h Z := were
t-->oo
practice the states evolve inside Or converging towards the set ([-~; ~r c ) x {O}.
Remark
4.
Xmin
It is immediate to cast problems (3) and (5) as integer quadratic programs (IQPs). Indeed , by substituting Xk = Akx(t) + L~'::~ Aj BUk - l - j , Equation (3b) can be rewritten as
t x
State constraints having the form
5. QMPC COMPUTATIONS
U
E
k- l
+ (ne - 2)r(r)] X ... x[-~-,(r); ~+,(r)]x[-~ ; ~J)x{V:sr}
x [ - ~ - (ne - 2)r(r); ~
min
[Xk
f-t
:S x(t) :S
X max
(or, more generally, Ax(t) :S fJ) can be easily taken into account by adding the linear constraints
A [AkX(t) +
{~U'HU +x'(t)FU + ~x'(t)YX(t)} 2 2
~ Aj BUk_I_j]
< fJ
J= O
subj . to GU :S W + Ex(t) UkEU, Vk=0 , .. . , H p - 1
in (11). Clearly, in this case the invaria~t set n must be contained in the polyhedron Ax :S fJ in order to guarantee constraint fulfillment of the
(8)
263
1 . . . . ....., O ft
REFERENCES
ca..c
Bemporad, A. (2003) . Multiparametric nonlinear integer programming and explicit quantized optimal control. Submitted. Bemporad, A. and M. Morari (1999). Control of systems integrating logic, dynamics, and constraints. Automatica 35(3), 407- 427. Bicchi, A., A. Marigo and B. Piccoli (2002). On the reachability of quantized control systems . IEEE Transactions on Automatic ControI47(4),546- 563 . Blanchini, F . (1999). Set invariance in control. Automatica 35,1747- 1767. Brockett, R. and D. Liberzon (2000) . Quantized feedback stabilization of linear systems. IEEE Trans. Autom. Control 45(7) , 1279- 1289. Chmielewski, D. and V. Manousiouthakis (1996) . On constrained infinite-time linear quadratic optimal control. Systems and Control Letters 29(3), 121- 130. Delchamps, D. F. (1990). Stabilizing a linear system with quantized state feedback. IEEE Trans. Autom. Control 35(8), 916- 924. Elia, N. and S. Mitter (2001). Stabilization of linear systems with limited information. IEEE Trans. Autom. Control 46(9), 1384-1400. Fagnani, F. and S. Zampieri (2002). Stabilizing quantized feedback with minimal information flow : the scalar case. In: Proc. of MTNS Conference. Notre Dame, Indiana. Michalska, H. and D . Q. Mayne (1993) . Robust receding horizon of constrained non linear systems. IEEE Transactions on Automatic Control3S, 1623- 1632. Nair , G .N. and R.J. Evans (2000). Stabilization with data rate limited feedback: tightest attainable bounds. Systems and Control Letters 41,49- 56. Picasso, B. (2002). Stabilization of quantized-input systems with optimal control techniques. Degree thesis. Dipartimento di Matematica, University of Pisa, Italy. Available writing to:
':r
Fig. 1. The state trajectory from [-0.8, -0.9] with quantized control set U = {-1,0, 1} QMPC controller at all times t > 0 . Hence the algorithm can take into account state constraints also along the predicted trajectories: this is an important extension of the recent approach presented in (Quevedo et al., 2002) which is not suitable to dealing with state constraints. Example: In the following we show an application of QMPC computations to the classical problem of stabilizing an inverted pendulum. The choice of a sampling time T ~ 0.00571s and the length of the pendulum I = 0.25m gives rise to the discrete-time model in controller form with parameters o} = -0.9987 and 0:2 = 2. The system is controlled by mean of a stepper motor, hence the quantized control set is U = {- I, 0, I} . This example was previously studied in (Picasso et al., 2002), where it was shown that, while a simple quantized version of a deadbeat controller was sufficient to make Q2(1) invariant, the largest ~ for which Q2(~) could be made invariant and attracted by Q2(1) , was ~ ~ 1. By implementing a MPC scheme such as that discussed above (based on a horizon length 3) , it is possible to guarantee attractivity of Q2 (1) for all initial conditions in a substantially larger region of the state space, given by (7). An example trajectory is shown in figure (1) .
picasso~piaggio.ccii.unipi.it.
Picasso, B., F . Gouaisbaut and A. Bicchi (2002) . Construction of invariant and attractive sets for quanti zed- input linear systems. Proc. 41st IEEE Conference on Decision and Control pp. 824- 829. Quevedo, D. E ., J . A. De Dona and G. C. Goodwin (2002). Receding horizon linear quadratic control with finite input constraint sets. IFAC 15th Triennial World Congress. Raisch , J . (1994) . Simple hybrid control systems - continuous fdlti plants with symbolic measurements and quantized control inputs. In: 11th International Conference on Analysis and Optimization of Systems (G . Cohen and J .-P. Quadrat , Eds.) . Vol. 199 of LectuT'es Notes in Control and Information Sciences. Springer-VerJag. pp. 369- 376. Scokaert , P. O. M., D. Q. Mayne and J . B . Rawlings (1999). Suboptimal model predictive control (feasibility implies stability). IEEE Transactions on A utomatic Control 44(3) , 648- 654 . Wong, W . and R . Brockett (1999). Systems with finite communication bandwidth constraintspart 11: Stabilization with limited information feedback. IEEE Transactions on Automatic Control 44(5) , 1049- 1053.
CONCLUSIONS In this paper we have considered the stabilization problem for discrete-time linear systems subject to a fixed, uniform quantization of control inputs. It has been shown that the Model Predictive Control approach can be profitably used and we provided conditions for its applicability. Several open problems remain in this field , among which is the extension to (quantized) output MPC. More generally, the combination of quantization with limited communication bandwidth is a most important and challenging area to which furth er work will be devoted. Acknow ledgements Frederic Gouaisbaut is gratefully acknowledged for useful discussions and collaboration in the preliminary phase of this work.
264