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Automatica 40 (2004) 311–317www.elsevier.com/locate/automatica

Brief paper

Enlargement of polytopic terminal region in NMPC by interpolationand partial invariance�

Mark Cannona ;∗, Basil Kouvaritakisa, Venkatesh Deshmukhb

aDepartment of Engineering Science, University of Oxford, Oxford OX1 3PJ, UKbDepartment of Mechanical Engineering, University of Alaska Fairbanks, AK 99775, USA

Received 8 April 2003; received in revised form 2 October 2003; accepted 13 October 2003

Abstract

This paper uses the concept of partial invariance to derive a sequence of linear programs in order to maximize o5ine the volume ofan invariant polytopic set with an arbitrary prede6ned number of vertices subject to a bound on closed-loop performance. Interpolationtechniques are used to determine a nonlinear control law which is optimal with respect to a closed-loop cost bound through the on-linesolution of a linear program. The invariant polytope is also used to de6ne a receding horizon control law through an appropriate terminalconstraint and cost.? 2003 Elsevier Ltd. All rights reserved.

Keywords: Constrained control; Model predictive control; Invariant set; Optimization

1. Introduction

Controlled invariant sets and associated feedback lawsare important tools in the control of constrained systemssince they de6ne regions of state space where stability andperformance speci6cations are met. Invariant sets have beenused extensively in the design of linear and piecewise linearcontrol laws (Gutman & Cwikel, 1986; Gilbert & Tan, 1991;Wredenhagen & Belanger, 1994; Boyd, El Ghaoui, Feron,& Balakrishnan, 1994; Blanchini, 1994), as well as in theconstruction of terminal regions and terminal control lawsfor model predictive control (MPC) algorithms (Michalska& Mayne, 1993; Chen & Allgower, 1998; Magni, DeNicolao, Magnani, & Scattolini, 2001). In both applicationareas, large invariant sets imply large regions of stabilizableinitial conditions, but this has to be balanced against theneed for good closed-loop performance under the associatedcontrol law.

� This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor FrancoBlanchini under the direction of Editor Roberto Tempo.

∗ Corresponding author. Tel.: +44-1865-273189;fax: +44-1865-273906.

E-mail address: mark.cannon@eng.ox.ac.uk (M. Cannon).

The approach described in this paper improves boththe size of the stabilizable initial condition set and theperformance of control laws based on invariant target sets.Conventionally this is done by: (i) increasing the predic-tion horizon of an MPC optimization or (ii) increasing thesize of the target set. Method (i) requires an increase in thenumber of optimization variables, implying a signi6cantincrease in online computational load of the MPC law, or(using the approach of Magni et al., 2001) a greater depen-dence on the degree of optimality of the terminal controllaw. For the case of linear terminal control laws, method(ii) adversely aFects performance, since, for constrainedsystems it generally requires a more suboptimal terminalcontrol law. The current paper avoids this trade-oF by ex-panding the terminal set using reachable sets and by usingnonlinear terminal control laws.For nonlinear systems, maximal reachable sets (consist-

ing of all states that can be steered into a target set overa given horizon) are nonconvex and nonpolytopic in gen-eral (Bertsekas & Rhodes, 1971). Therefore, their o5inedetermination is extremely computationally intensive andthey can be unsuitable as terminal sets in online MPC opti-mization. The current paper employs linear diFerence inclu-sion (LDI) representations of the plant model, and this re-sults in polytopic reachable sets (Gutman & Cwikel, 1986;

0005-1098/$ - see front matter ? 2003 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2003.10.004

312 M. Cannon et al. / Automatica 40 (2004) 311–317

Blanchini, 1994). However LDI representations are neces-sarily local and no methods are available for determiningthe region of validity of the LDI simultaneously with acorresponding maximal reachable set. Moreover, the combi-natorial growth in number of vertices of maximal reachablesets with horizon length makes the approach computation-ally unwieldy, both for the oF-line determination of the setand for online MPC optimization incorporating the resultingterminal set.To retain computational tractability the paper considers

polytopic sets with 6xed numbers of vertices. Whereasapproaches based on low-complexity polytopes (Vassilaki,Hennet, & Bitsoris, 1988; Lee & Kouvaritakis, 2000;Cannon, Deshmukh, & Kouvaritakis, 2003) have beenrestricted to linear feedback laws and parallelotopes param-eterized in by n vertices in n-dimensional state-space, weconsider arbitrary numbers of vertices and nonlinear controllaws. We impose a bound on the closed-loop cost under theassociated control law by invoking a strengthened condi-tion for partial invariance (Bacic, Cannon, & Kouvaritakis,2003). As a result the maximization of the volume of thepolytopic set subject to the bound on closed-loop cost canbe performed by solving a sequence of linear program-ming (LP) problems. This procedure is used to determinea sequence of partially invariant sets corresponding to asequence of cost bounds.A nonlinear feedback law is constructed by interpolating

between controls associated with vertices. Unlike schedul-ing approaches based on gain detuning techniques (e.g.Tan & Gilbert, 1992; Wredenhagen & Belanger, 1994), orvertex interpolation approaches (Gutman & Cwikel, 1986;Blanchini, 1994) which consider stabilization alone, weshow that the resulting feedback law is optimal with respectto bounds on the closed-loop cost. Alternatively, the se-quence of partially invariant sets and interpolated feedbacklaw are incorporated into an MPC law through the derivationof suitable piecewise linear terminal cost and constraints.

2. Problem statement

We consider nonlinear discrete-time systems with model:

xk+1 = f(xk ; uk); yk = Cxk ;

xk ∈Rn; uk ∈Rl; yk ∈Rm; f(0; 0) = 0 (1)

subject to input constraints (applying elementwise):

uk ∈U; U= {u: u6 u6 Ku}; u; Ku∈Rl: (2)

Linear state constraints can also be handled by the approachdescribed below. The control objective is to minimize

J∞ =∞∑k=0

‖yk‖∞; (3)

while stabilizing the equilibrium at x = 0 and respectingconstraints. This problem formulation is easily modi6ed tohandle tracking problems and to incorporate integral action.

The dual mode prediction paradigm of MPC evaluatesconstraints and performance explicitly over a 6nite predic-tion horizon N . To ensure constraint satisfaction over thesubsequent in6nite horizon, state predictions are forced tolie within a terminal region XN on which a given feedbacklaw

uk+i|k = �(xk+i|k); i¿N; (4)

is stabilizing and feasible with respect to constraints(Michalska & Mayne, 1993; Chen & Allgower, 1998). Thisis achieved by invoking the terminal constraint xk+N |k ∈XN ,provided XN is invariant for (1) with uk = �(xk). Thereceding horizon control law is de6ned by

uk = uk|k ; (5)

where uk|k is the 6rst element of the optimal predicted inputsequence {uk+i|k} computed online by solving the NLP:

minuk+i|k

i=0;:::;N−1

N−1∑i=1

‖yk+i|k‖∞ + �(xk+N |k)

s:t:

{(1); (2)

xk+N |k ∈XN :(6)

The terminal cost �(x) is chosen so as to ensure existenceof a Lyapunov function for (1) under (5) and to account forperformance over prediction instants i¿N . The size of thestabilizable set under (5) is therefore dependent on the sizeof the terminal region XN . A modi6ed approach (Magniet al., 2001) employs the feedback law (4) for i¿N , butreplaces N in (6) with Np ¿N , thus extending the predic-tion horizon without introducing additional variables into theoptimization (6). This has the eFect of reducing the depen-dence of the stabilizable set under (5), but at the expense ofgreater reliance on the degree of optimality of the feedbacklaw (4).This paper describes a method of computing a controlled

invariant terminal regionXN of maximal size, an associatedfeasible feedback law �(x) which minimizes an upper boundon the cost (3), and the corresponding terminal cost �(x)for inclusion in the receding horizon optimization (6). Theapproach is based on a LDI representation of the plant. Wetherefore make the assumption that, for any polytopic subset� of the operating region in state-space, it is possible toconstruct a set {(Ai; Bi)

pi=1} so that the model (1) satis6es

f(x; u)∈Co {(Aix + Biu)pi=1} ∀(x; u)∈� ×U; (7)

where Co denotes the convex hull. Under this condition, alltrajectories of system (1) corresponding to input trajectories{uk ; k = 0; 1; : : :} for which (xk ; uk)∈� ×U for all k¿ 0are also trajectories of the LDI (Liu, 1968):

xk+1 ∈Co {Aixk + Biuk ; i = 1; : : : ; p}: (8)

Procedures for computing suitable sets of linear models ex-ist e.g. if f is continuously diFerentiable w.r.t. x; u (Boydet al., 1994).

M. Cannon et al. / Automatica 40 (2004) 311–317 313

3. Partially invariant polytopes and interpolation

Let� be a polytopic set for which inclusion condition (7)holds, and let � be a convex polytope containing the origin:

� = {x∈Rn: Vx6 1}; (9)

where 1 = [1 · · · 1]T. Then � is positively invariant under(1), (2) with a given feedback law u = �(x) (i.e. � is anoutput admissible set (Gilbert & Tan, 1991)) if the followingconditions hold.

V (Aix + Bi�(x))6 1 i = 1; : : : ; p ∀x∈�; (10)

� ⊆ �; (11)

�(x)∈U ∀x∈�: (12)

Here (11) and (12) ensure that (8) holds at all points in �,and that �(x) is feasible w.r.t. input constraints. Invarianceof� therefore follows from (10). More restrictive invarianceconditions can be used to impose bounds on performanceunder u= �(x), as the following lemma shows.

Lemma 3.1. If (11) and (12) hold, and for i = 1; : : : ; p

max{V (Aix + Bi�(x))}6max{Vx} − �−1‖Cx‖∞ (13)

∀x∈� is satis6ed for some �¿ 0 (max{Vx} denotes themaximum element of Vx), then for any x0 ∈�, the cost (3)along trajectories of (1) with u= �(x) has upper bound∞∑k=0

‖Cxk‖∞6 �: (14)

Proof. If x0 ∈� then conditions (11)–(13) ensure thatxk ∈� for all k ¿ 0, and the cost bound is obtained bysumming (13) over all times k = 0; 1; : : : :

Analogous invariance conditions (based on the level setsof quadratic forms rather than the piecewise linear func-tions in (13)) were used in Boyd et al. (1994), Kothare,Balakrishnan, and Morari (1996) and Chen, Ballance andO’Reilly (2001) to bound closed-loop cost in the construc-tion of ellipsoidal controlled invariant sets. However, inthe context of nonlinear systems, the use of polytopic �facilitates the simultaneous optimization of the inclusionpolytope � (Cannon et al., 2003). The invariance condi-tion (13) was used in (Cannon et al., 2003) to constructlow-complexity controlled invariant parallelotopes of theform {x : ‖Vx‖∞6 1} for V ∈Rn×n and linear feedbacklaws �(x) = Kx. The current paper extends the approachof (Cannon et al., 2003) to polytopic sets of the form (9)and uses interpolation techniques to remove the restrictionto linear feedback laws. This generalization enables greaterOexibility in shape of � and a higher degree of optimalityof �(x), but could signi6cantly increase the complexity ofthe volume maximization problem. To avoid this diPcultywe use the concept of partial invariance (Bacic et al., 2003),de6ned as follows.

De�nition 3.2 (Partial invariance). �(−1) ⊂ Rn is partiallyinvariant for xk+1 = f(xk ; �(xk)) if f(x; �(x))∈�(0) ∀x∈�(−1) where �(0) ⊂ � for some invariant set �.

Partial invariance can be used to construct invariant setssince �(−1) ∪ Q is necessarily invariant. Moreover the re-quirement that a polytopic set �(−1) be partially invariantunder the closed-loop system formed by a LDI model rep-resentation and piecewise linear feedback can be expressedas a set of linear constraints on its vertices. As shown inSection 4, this leads to a more computationally convenientmethod of constructing feasible invariant sets than the con-ventional invariance de6nitions (such as that implied by(10)), which instead result in bilinear constraints (Bitsoris,1988; Blanchini, 1999; Cannon et al., 2003).Conditions for partial invariance can be extended in an

obvious way to ensure a bound on performance. Let �(j) ={x :V (j)x6 1} for j =−1; 0, then for i = 1; : : : ; p

max{V (0)(Aix + Bi�(x))}6max{V (−1)x}−�−1‖Cx‖∞ (15)

∀x∈�(−1) ensures that �(−1) is partially invariant. Satis-faction of (15) also guarantees that the cost bound (14)holds for any initial condition x0 ∈�(−1) whenever �(0) sat-is6es inclusion, feasibility and invariance conditions of form(11)–(13). More generally, if the image of a partially in-variant set under the closed-loop dynamics is itself partiallyinvariant, then the bound (14) holds provided that (15) issatis6ed by each of a sequence of partially invariant sets�(−M); �(−M+1); : : : terminating in �(0) satisfying (13).

The procedure of Section 4 for constructing invariantpolytopes and associated feedback laws is based on optimiz-ing the vertices of �(−1), denoted x(−1)

j j=1; : : : ; r, simulta-

neously with the control u(−1)j associated with each vertex.

The feedback law u=�(x) is then obtained by interpolatingbetween the elements of {u(−1)

j ; j = 1; : : : ; r}:�(x; q) =

∑j

qju(−1)j ; (16)

where the parameters q= [q1 · · · qr]T satisfy constraints:

x =∑j

qjx(−1)j ; q¿ 0;

∑j

qj6 1: (17)

Let �=∑

j qj, then (16) and (17) imply Aix+ Bi�(x; q) =

�∑

j q̃j(Aix(−1)j + Biu

(−1)j ), where q̃j = qj=�, j = 1; : : : ; r,

and∑

j q̃j = 1. Hence (16) and (17) ensure that the imageof any x∈�(−1) under the closed-loop dynamics lies insCo {�(Aix

(−1)j + Biu

(−1)j )} for some �6 1. Consequently

conditions for partial invariance of �(−1) can be based onits vertices alone: if Aix

(−1)j + Biu

(−1)j ∈Q(0) for all i; j then

Co {�(Aix(−1)j +Biu

(−1)j )} ⊂ Q(0) since 0∈Q(0) by assump-

tion, and it follows that every x∈�(−1) maps into�(0) undera control law satisfying (16) and (17). In order to satisfy themore restrictive partial invariance condition (15) however,

314 M. Cannon et al. / Automatica 40 (2004) 311–317

we de6ne �(x) as the solution of the following LP.

�(x) =∑j

qju(−1)j

q= argminq

{∑j

qj s:t: x =∑j

qjx(−1)j ; q¿ 0

}: (18)

Before showing that (15) and the associated cost bound holdunder �(x), we 6rst derive a useful property of (18).

Lemma 3.3. For any x∈�(−1),

minq

{∑j

qj s:t: x =∑j

qjx(−1)j ; q¿ 0

}=max{V (−1)x}:

Proof. If x =∑

j qjx(−1)j for q¿ 0, then max{V (−1)x} =

max{∑j qjV (−1)x(−1)j }6∑

j qj, and furthermore thisbound is achieved for a particular choice of q, as we nowshow. Let �=max{V (−1)x} and de6ne x̃ = x=�, then x̃ liesin the boundary of �(−1) and therefore, the vector of inter-polation parameters q̃¿ 0 satisfying x̃ =

∑j q̃jx

(−1)j has

the property that∑

j q̃j = 1. Thus the choice q= �q̃ resultsin

∑j qj = �=max{V (−1)x}.

The theorem below gives a convenient reformulationof the partial invariance condition (15) under control lawof (18).

Theorem 3.4. The following condition, invoked for i =1; : : : ; p, j = 1; : : : ; r, is equivalent to (15) with (18).

max{V (0)(Aix(−1)j + Biu

(−1)j )}6 1− �−1‖Cx(−1)

j ‖∞: (19)

Proof. The necessity of (19) is obvious since (15) musthold for all x∈�(−1). To show suPciency, let q satisfy (18)for x∈�(−1). Then (19) and Lemma 3.3 imply∑

j

qj[max{V (0)(Aix(−1)j + Biu

(−1)j )}+ �−1‖Cx(−1)

j ‖∞]

6∑j

qj =max{V (−1)x}; (20)

whereas x =∑

j qjx(−1)j and �(x) =

∑j qju

(−1)j imply

max{V (0)(Aix + Bi�(x))}+ �−1‖Cx‖∞6

∑j

qj[max{V (0)(Aix(−1)j + Biu

(−1)j )}

+ �−1‖Cx(−1)j ‖∞]: (21)

Combining (20) and (21) yields (15).

Remark 3.5. If (19) is satis6ed with equality for some iat every vertex j, then it becomes necessary (as well assuPcient) to de6ne the interpolation control law �(x) in (16)

and (17) via the minimization in (18) in order to be able tosatisfy the partial invariance condition (15). This is because(20) then holds with equality for some i, whereas (21) nec-essarily holds with equality for some x∈�(−1). Condition(15) is therefore violated if

∑j qj ¿max{V (−1)x}.

The use of interpolation between controls associated withthe vertices of an invariant polytopic set in order to de-6ne a nonlinear feedback law was proposed in Gutman andCwikel (1986). However Gutman and Cwikel (1986) con-siders only stabilization, whereas the cost bound implied by(15) also enables performance to be addressed via the LP(18). Below we show that the feedback law (18) is optimalin the sense that it minimizes a bound on the cost (3) overall interpolation control laws of the form (16) and (17).

Corollary 3.6. If (19) is satis6ed at the vertices of �(−1),and �(0) satis6es (11)–(13), then the following inequalityholds for any initial condition x0 ∈�(−1):

�∑j

qj¿∞∑k=0

‖Cxk‖∞: (22)

Proof. From (13) and the assumption that �(0) is invari-ant, it is clear that

∑∞k=1 ‖Cxk‖∞6 �max{V (0)x1}. For any

initial condition x0 ∈�(−1), we therefore have∞∑k=0

‖Cxk‖∞6 �max{V (0)(Aix0 + Bi�(x0; q))}+ ‖Cx0‖∞

6 �∑j

qj[max{V (0)(Aix(−1)j + Biu

(−1)j )}

+ �−1‖Cx(−1)j ‖∞]

6 �∑j

qj;

which implies that the objective minimized in (18) is equiv-alent to an upper bound on the cost index (3).

4. Polytopic set volume maximization

To determine the volume of a polytopic set of the form(9) as a function of its vertices can be computationally de-manding, and to avoid this we propose volume maximiza-tion subject to inclusion, feasibility and partial invarianceconstraints by successively optimizing individual vertices.This requires a sequence of LP’s (solved o5ine) based on(11), (12), (19).To show that maximization of volume over a single vertex

can be cast as a linear objective, consider the problem ofoptimizing a vertex xj of �, with the remaining vertices{(xi)i �=j} 6xed. Given an initial polytope �̂ with vertices{x̂j ; (xi)i �=j}, this is equivalent to maximizing vol(� − �̂),

M. Cannon et al. / Automatica 40 (2004) 311–317 315

which can be expressed as the sum of volumes of all sim-plexes formed by xj with the facets of �̂ containing x̂j. De-noting Xjk as the matrix with columns formed from the ver-tices of the kth such facet, it follows that vol(� − �̂) =1=n!

∑k det(Xjk − xj1T) = 1=n!

∑k(det(Xjk) − 1TX−T

jk xj)(see e.g. Gritzmann & Klee, 1994), where the sum is takenover all facets containing x̂j. Since Xjk is independent of xj,maximization of vol(�−�̂) is equivalent to minimizing thelinear objective

cTj xj; cj =∑k

X−Tjk 1: (23)

Theorem 4.1. The volume of �(−1) is maximized overx(−1)j , u(−1)

j subject to inclusion (11), feasibility (12) andpartial invariance constraints (19) for 6xed � by the LP:

minx(−1)j ;u(−1)

j

cTj x(−1)j

s:t:

x(−1)j ∈�; u(−1)

j ∈U;

max{V (0)(Aix(−1)j + Biu

(−1)j )}

6 1− �−1‖Cx(−1)j ‖∞; i = 1; : : : ; p:

(24)

Proof. Constraints x(−1)j ∈�, u(−1)

j ∈U are clearly linear,whereas (19) can be invoked through linear constraints byintroducing a slack variable $: −$16Cx(−1)

j 6 $1 , $¿ 0,

V (0)(Aix(−1)j + Biu

(−1)j )6 (1− �−1$)1.

The method of optimizing individual polytope vertices inTheorem 4.1 suggests the following procedure for maximiz-ing the volume of �(−1) over all vertices {x(−1)

j ; u(−1)j ; j =

1; : : : ; r} subject to bound (14) on closed-loop cost under(18) and feasibility with respect to input constraints.

Algorithm 4.1 (Polytopic set volume maximization).DATA: {x(−1)

j ; u(−1)j ; j=1; : : : ; r} satisfying u(−1)

j ∈U and(19) for 6xed �¿ 0, a tolerance %¿ 0, and j = 1.

1. Compute the convex hull �(−1) of {x(−1)j j = 1; : : : ; r}

and determine the linear objective in (24).2. Solve the LP (24) for (x(−1)

j ; u(−1)j ).

3. If the volume increase indicated by the optimal objectivevalue is less than the tolerance %, then stop. Otherwiseincrement j (or set j := 1 if j = r) and return to 1.

Remark 4.2. Each LP solved in step 2 increases the vol-ume of �(−1). Algorithm 4.1 therefore terminates after a 6-nite number of iterations in a maximum volume set �(−1),although the sequential vertex optimization procedure doesnot guarantee convergence to the global solution of the vol-ume maximization problem subject to (11), (12), (19).

Remark 4.3. Clearly the inclusion polytope � employedin the LP (24) need not be the same as that used eitherin the computation of �(0) or in the previous optimization

of a vertex of �(−1). By de6ning � in terms of a scaledversion of �(−1) in step 1, the Algorithm can therefore beused to compute an inclusion polytope which is identical tothe maximum volume �(−1), simultaneously with �(−1).

A sequence of partially invariant sets �(−M); : : : ; �(−1),corresponding to a sequence of cost bounds: �(−M) ¿ · · ·¿�(−1) can be computed through successive application ofAlgorithm 4.1 using the following procedure. The schemecan be initialized with �(0) and associated control law �(x)computed for example using the approach of Cannon et al.(2003).

Algorithm 4.2 (Sequence of partially invariant sets).DATA: �(0) = Co {x(0)j ; j = 1; : : : ; r} and �(x) satisfying(11)–(13) for � = �(0) and �= �(0) ¡�(−1).

1. Set u(0)j := �(x(0)j ); j = 1; : : : ; r and set i := −1.

2. Optimize {(x(i)j ; u(i)j )rj=1} using Algorithm 4.1 with

{(x(−1)j ; u(−1)

j )rj=1} replaced by {(x(i)j ; u(i)j )rj=1} and ini-

tial value {(x(i)j ; u(i)j )rj=1} := {(x(i+1)j ; u(i+1)

j )rj=1}.3. If i=−M then stop; else set i := i− 1 and return to 2.

Since the initial set of vertices {x(i)j ; u(i)j } in step 2 ofAlgorithm 4.2 is de6ned in terms of �(i+1), the resultingsequence has the property vol(�(i))¿ vol(�(i+1)) for all i.

5. Receding horizon control

Given a sequence of partially invariant sets �(−M); : : : ;�(−1); �(0) computed o5ine using Algorithm 4.2, at eachsampling instant k the following control strategy determinesthe set �(m) containing xk with smallest associated costbound and then implements the interpolation control law(18) corresponding to {x(m)j ; u(m)j ; j = 1; : : : ; r}.

Algorithm 5.1 (LP-based control law). At times k¿ 0:

1. Determine m=argmini �(i) max{V (i)xk}, where the min-

imization is performed over i for which xk ∈�(i).2. Compute �(xk) using (18), with {x(−1)

j ; u(−1)j } replaced

by {x(m)j ; u(m)j }.3. Implement uk = �(xk).

The stabilizing properties of Algorithm 5.1 are obviousdue to the partial invariance and feasibility of �(i), i =−M; : : : ; 0. It is also easy to show that the cost (3) has upperbound �(−M), and that the stabilizable set contains the unionof polytopes �(−M) ∪ · · · ∪ �(0). Also on-line computationis minimal, requiring only the solution of the LP in step 2and evaluation of �(i)V (i)xk for each i in step 1.However, if more time is available for online compu-

tation, then both the stabilizable region and closed-loopperformance can be improved by incorporating the sequence

316 M. Cannon et al. / Automatica 40 (2004) 311–317

of partially invariant polytopes into a control law based onan NLP of the form (6), as we now show.

Algorithm 5.2 (NLP-based control law). At times k¿ 0:

1. Find the solution of (6) with terminal cost and constraint

�(x) = mini=−M;:::;0

�(i)(x) =

{�(i) max{V (i)x} if x∈�(i)

�(−M) otherwise

XN =0⋃

i=−M

�(i): (25)

2. Implement uk = uk|k .

Remark 5.1. In (25) �(x) is continuous and piecewise lin-ear in x∈XN . Also �(x) can equal �(i)(x) only if x∈�(i),since �(i) max{V (i)x}6 �(−M) whenever x∈�(i), thus en-suring that the terminal cost corresponds to an upper boundon the cost associated with �(i) containing xk+N |k .

Theorem 5.2. If (6), with � and XN de6ned by (25), isfeasible at k = 0, then Algorithm 5.2 ensures that theclosed-loop cost (3) is bounded by the optimal value of theobjective in (6) at k = 0, and (provided (1) is observable)renders the equilibrium at x = 0 asymptotically stable.

Proof. Let the optimal value of the objective and corre-sponding predicted input sequence at time k be Jk and{uk|k ; : : : ; uk+N−1|k}, and denote the corresponding value of�(xk+N |k) in (6) as

�(xk+N |k) = �(m) max{V (m)xk+N |k}:Then, from the de6nitions of � and XN in (25), xk+N |k nec-essarily lies in �(m), and from the partial invariance and fea-sibility properties of�(m), a feasible but suboptimal input se-quence at k+1 is given by {uk+1|k ; : : : ; uk+N−1|k ; �(xk+N |k)},where �(x) is given by (18) with {x(−1)

j ; u(−1)j } replaced

by {x(m)j ; u(m)j }. Thus initial feasibility ensures future feasi-bility. Furthermore the terminal cost corresponding to thissuboptimal predicted input sequence satis6es

mini

�(i)(f(xk+N |k ; �(xk+N |k)))

6 �(m+1) max{V (m+1)f(xk+N |k ; �(xk+N |k))}¡�(m) max{V (m+1)f(xk+N |k ; �(xk+N |k))};

where f(xk+N |k ; �(xk+N |k))∈�(m+1) and �(m+1) ¡�(m) hasbeen used. From the partial invariance condition (15), wetherefore have

mini

�(i) max{V (i)f(xk+N |k ; �(xk+N |k))}

6 �(m) max{V (m)xk+N |k} − ‖Cxk+N |k‖∞and it follows that Jk+16 Jk − ‖Cxk+1‖∞. This impliesthat the closed-loop cost is bounded by J0, and thatJk(xk) is a Lyapunov function demonstrating stability of

-5 -4 -3 -2 -1 0 1 2 3 4-3

-2

-1

0

1

2

3

4

x1

x 2

Fig. 1. Partially invariant sets �(−4); : : : ; �(−1); �(0) computed usingAlgorithm 4.2.

x = 0 under the observability assumption. Furthermore thebound

∑∞k=0 ‖yk‖∞6 J0 implies asymptotic convergence

‖yk‖∞ → 0 and hence xk → 0 if (1) is observable.

6. Numerical example

Consider the bilinear system with model:

xk+1 =

[0:28 −0:78

−0:78 −0:59

]xk

+

{[0:71

1:62

]+

[0:34 0:36

0:41 −0:65

]xk

}uk ;

yk = [− 0:69 0:2]xk ;

which is unstable, nonminimum-phase at x=0, and has inputconstraint |u|6 0:5.Fig. 1 shows the sequence of maximum area polytopic

sets �(−4); : : : ; �(−1); �(0) with 8 vertices computed usingAlgorithm 4.2 for the cost bound sequence:

{�(−5); : : : ; �(0)}= {50:0; 20:0; 10:0; 5:0; 3:0}:This sequence was chosen so as to achieve the desiredbalance between computational complexity and perfor-mance: the use of fewer elements �(i) and hence fewersets in the sequence {�(i)} would reduce the oF-line com-putation of Algorithm 4.2 and the online computationof Algorithms 5.1 or 5.2, but would result in less accu-rate bounds on closed-loop cost. The inclusion polytopewas optimized simultaneously with �(m) for each m us-ing the approach discussed in Remark 4.3. Table 1 givesthe values of vol(�(m)). Comparison with the maximumvolume low-complexity invariant polytope computed us-ing the approach of (Cannon et al., 2003) with no boundon closed-loop cost (i.e. � → ∞), which has a volumeof 11.59, shows the advantage of higher complexity

M. Cannon et al. / Automatica 40 (2004) 311–317 317

Table 1Areas of partially invariant sets in Fig. 1

m 0 −1 −2 −3 −4�(m) 3.0 5.0 10.0 20.0 50.0vol(�(m)) 3.51 5.78 8.59 13.22 21.71

Table 2Closed-loop costs J∞

Algorithm x0 = (1; 1) x0 = (2;−1)

5.1 2.07 5.945.2, XN =

⋃0i=−4 �(m) 1.65 4.97

5.3, XN = �(−4) 1.70 5.15

polytopic sets and nonlinear feedback laws: an increase involume of 14% for �= 20% and of 89% for �= 50.Table 2 compares values of the cost (3) for the LP-based

control law of Algorithm 5.1 and the NLP-based control lawof Algorithm 5.2 for two initial conditions (marked ‘+’ inFig. 1). The reduction in cost of Algorithm 5.1 for initialcondition x0 = (1; 1) over that for x0 = (2;−1) is due tox0 = (1; 1) lying close to the boundary of �(0), for whichthe cost bound is J∞6 3:0. As expected, Algorithm 5.2achieves much better performance than Algorithm 5.1, andfurthermore there is a signi6cant improvement in cost dueto the use of the union of partially invariant sets �(−4) ∪· · ·∪�(0) instead of �(−4) alone in the de6nition of terminalcost and terminal constraint.

Acknowledgements

This work was supported by the Engineering and PhysicalSciences Research Council (EPSRC).

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Mark Cannon was born in England in 1971.He received M.Eng. and D.Phil. degreesfrom Oxford University and S.M. from Mas-sachusetts Institute of Technology. He is cur-rently a University Lecturer in Engineeringand Fellow of St. John’s College, Universityof Oxford.

Basil Kouvaritakis was born in Athens,Greece in 1948. He was awarded First-Class Honours in Electrical Engineeringfrom the University of Manchester Instituteof Science and Technology, where he alsoreceived his Masters and Doctorate. He iscurrently a Professor in Engineering at theDepartment of Engineering Science and aTutorial Fellow at St Edmund Hall, OxfordUniversity.

Venkatesh Deshmukh was born in 1971 inMumbai, India. He obtained his B.E. fromVJTI, Mumbai, M.E. from the Indian In-stitute of Science, Bangalore, and Ph.D. inMechanical Engineering from Auburn Uni-versity. During 2000-2002 he worked as apost-doctoral research assistant at OxfordUniversity. At present he is a post-doctoralfellow at the University of Alaska Fairbanks.