Polytopic Computations in Constrained Optimal Control

ISSN 0005-1144ATKAFF 50(3–4), 119–134(2009)Mato BaotićPolytopic Computations in Constrained Optimal ControlUDKIFAC 681.513.525.5;2.8.1Original scientific paperIn the last decade a lot of research has focused on the explicit solution of optimal and robust control problemsfor the class of constrained discrete-time systems. Many newly developed control algorithms for such controlproblems internally use operations on polytopic sets. We review basic polytopic manipulations and analyze themin the context of the computational effort.We especially consider the so-called regiondiff problem where the set difference between a polyhedron andunion of polyhedra needs to be computed. Regiondiff problem and related polycover problem – checking if apolytope is covered by the union of other polytopes – are utilized very often in derivation of the explicit solutionsto the constrained finite time optimal control problems for piecewise affine systems. Similar observation holds forthe computation of the (positive) controlled invariant sets, infinite time optimal control solution and/or controllerswith reduced complexity for piecewise affine systems.We describe an in-place depth-first exploration algorithm that solves the regiondiff problem in an efficient manner.We derive strict upper bound for the computational complexity of the described algorithm. In extensivetesting we show that our algorithm is superior to the mixed integer linear programming approach when solving thepolycover problem.Key words: Polytope, Set difference, Set cover, Constrained optimal control, Discrete-time systemsOperacije nad politopskim skupovima kod optimalnog upravljanja sustava s ograničenjima. U posljednjihdesetak godina znatna istraživačka aktivnost usmjerena je na pronalaženje eksplicitnih rješenja optimalnogi robusnog upravljanja za klasu vremenski diskretnih sustava s ograničenjima. Brojni razvijeni algoritmi internokoriste operacije nad politopskim skupovima. U ovom radu analiziramo osnovne operacije nad politopskimskupovima sa stajališta njihove računske kompleksnosti.Naročita pozornost dana je takozvanom regiondiff problemu, odnosno problemu proračuna razlike poliedarskogskupa i unije poliedara. Isto tako je analiziran i srodni polycover problem – provjera je li poliedarski skup upotpunosti prekriven unijom poliedara. Oba ova problema često se sreću pri konstruiranju ekplicitnih rješenja optimalnogupravljanja po dijelovima afinih sustava uz konačan horizont predikcije, kao i pri proračunu pozitivnihinvarijantnih skupova, optimalnog upravljanja uz beskonačan horizont predikcije i/ili proračunu regulatora smanjenekompleksnosti za po dijelovima afine sustave.Razvijen je efikasan algoritam za rješenje regiondiff problema zasnovan na dubinskom pretraživanju stablastestrukture problema. Izvedena je teoretska gornja ograda za kompleksnost dobivenog algoritma, i pokazano je zaštoje takva ograda konzervartivna u praksi. Na nizu simulacija pokazana je računsku superiornost razvijenog algoritamza polycover problem u odnosu na pristup zasnovan na rješavanju mješovitog cjelobrojnog programa.Ključne riječi: politopski skupovi, razlika skupova, pokrivenost skupa, optimalno upravljanje sustava s ograničenjima,vremenski diskretni sustavi1 INTRODUCTIONSet-theoretic methods for analysis and design of controlsystems have a long history since the first contributionstrace back to the early 70s (cf. [1–5]). The theorywas almost abandoned once it became clear that, althoughthe techniques are appropriate to solve many theoreticaland practical problems, the complexity of the required algorithmswas not compatible with the computer technologyof the time. In recent years, the theory is receivinga renewed interest due the current computer performanceswhich are suitable to face non-trivial problems [5].In the last decade a lot of research has focused on thesolution of optimal and robust control problems for theclass of discrete-time constrained linear systems [6–11].Furthermore many algorithms have been extended to themore general class of so-called PieceWise Affine (PWA)systems [12–15].AUTOMATIKA 50(2009) 3–4, 119–134 119

Polytopic Computations in Constrained Optimal ControlM. BaotićMotivation for this paper is the computation of the explicitstate-feedback optimal controllers. The term “explicit”means that such design yields the optimal controllerin a closed form, i.e. as an explicit function of the systemstates [6]. Constrained optimal and/or robust controlmethods ensure systematic design of high-quality controlsystems for demanding applications. The optimal controlsystems satisfies prescribed constraints while achievingthe performance that is optimal with respect to the designobjective. Robustly controlled system can cope withthe measurement/modelling uncertainty i.e. it can achievedesired goal while being exposed to the adverse effects ofthe environment. The computation of the optimal or robustcontroller for a general constrained non-linear systemis a very difficult problem. Consequently, with theexisting computational capabilities, the solution to the optimalcontrol problem is tractable for a limited class of systems/constraints.Several optimal control formulations may be used in explicitcontroller design. The most common formulationsare the Constrained Finite Time Optimal Control (CFTOC)and the Constrained Time Optimal Control (CTOC). Theperformance objective of the CFTOC is the sum over afinite prediction horizon of piecewise linear or quadraticfunctions of the control inputs and (predicted values of)system states. An attractive property of the CFTOC problemis the fact that the optimizer, i.e. the optimal controlinput, is a piecewise affine function of the (current) systemstates [6,13]. In the CTOC problem formulation the goal isto find the control inputs that in a minimal number of timesteps move the (current) system states into a controlled invariantset around the origin. The computation of explicitcontroller for the CTOC problem can be solved by manipulationsof polytopes, and solution can be stored in a formof a PWA function of the current system states [14]. BothCFTOC and CTOC methodology rely on availability of avalid discrete-time model of the system. In practice thismeans that we need a systematic procedure for discretetimePWA model identification of nonlinear process [15].Almost all developed CFTOC and CTOC algorithms intheir core use some type of operations on polytopes andunions of polytopes. Furthermore, many of them rely onextensive computation of set difference between polytopeand union of polytopes [16]. In this paper we review basicpolytopic manipulations and analyze them in the contextof the computational effort. We derive efficient algorithmsfor solving two specific problems that emerge very oftenwhen deriving the explicit solution to the constrained optimalcontrol problem for PWA systems: (i) regiondiff problem– computation of the set difference between a polyhedronand union of polyhedra; (ii) polycover problem –checking if a polytope is covered by the union of otherpolytopes.2 NOTATION AND DEFINITIONSÆis the set of positive integers,Êm×n , with m, n ∈Æ,is the space of m by n real matrices,Ên is the space of n-dimensional real column vectors, ∅ denotes the empty set,andÆ1:m is the shorthand notation for the set {1, . . . , m}.The symbol ’:=’ denotes that the left-hand side is definedby the right-hand side, and ’=:’ is used for the reverse logic(e.g.,Æ1:m := {1, . . . , m}). Let r ∈Ên , R ∈Êm×n , i ∈Æ1:m, I ⊆Æ1:m, then R (i) denotes the i-th row of R; R (I)denotes the matrix formed from the rows of R indexed byI; R T denotes the matrix transpose of R; ‖ · ‖ denotesthe Euclidean vector norm, i.e., ‖r‖ := √ r T r, and card(·)denotes cardinality of the set (e.g. card(Æ1:m) = m).Throughout the paper all vector inequalities are consideredcomponent-wise (e.g., a ≤ b, with a, b ∈Êm , isequivalent to a (i) ≤ b (i) , ∀i ∈Æ1:m). As a general rulevectors and matrices are denoted with italic letters (e.g.,a, b, R, . . .) and sets are denoted with the upper case calligraphicletters (e.g., I, Q, . . .).We use the shorthand notation {Q i } NQi=1for a collectionof sets, i.e., {Q i } NQi=1 := {Q 1, Q 2 , . . . , Q NQ }. Note thatfor Q = {Q i } NQi=1 we have card(Q) = N Q.Definition 1 (Linear Program) A linear program (LP) isa convex optimization problem that can be expressed in theformminxc T x (LP)subj. to Gx ≤ g,where x ∈Ên is the optimization variable, and matricesc ∈Ên , G ∈Êm×n , g ∈Êm are given problemparameters.□Theoretically every LP (with rational parameters) issolvable in polynomial time by both the ellipsoid methodof Khachiyan [17, 18] and various interior point methods[19, 20]. A practical algorithm to solve an LP with nvariables and m constraints requires roughly O(n 3 m 0.5 +n 2 m 1.5 ) operations [21]. In this paper the computationalexpense of a single LP has similar meaning like single additionor multiplication have in algebraic computations. Inmany instances we will express complexity of algorithmsin terms of the number of LPs one needs to solve, thusestablishing the basis for relative comparison between differentalgorithms. Henceforth with lp(n, m) we denote thecomplexity of a single LP with n variables and m constraints.Most of the following definitions are standard. For additionaldetails the reader is referred to [22, 23].120 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal ControlDefinition 2 A hyperplane inÊn is a set of the form{x ∈Ên | a T x = b},where a ∈Ên , b ∈Ê, ‖a‖ > 0.Definition 3 A half-space inÊn is a set of the formH = {x ∈Ên | a T x ≤ b},where a ∈Ên , b ∈Ê, ‖a‖ > 0.Definition 4 (Polyhedron and polytope) A convex setP = {x ∈Ên | P x ≤ p}, (1)with P ∈Ên p×n , p ∈Ên p, n p < ∞, is called polyhedron.Bounded polyhedron is called polytope. The vectorinequality in (1) is considered component-wise. □We say that a polytope P = {x ∈Ên | P x ≤ p}is full-dimensional if it is possible to fit a non-empty n-dimensional ball in P, i.e.,where∃x 0 ∈Ên , ǫ > 0 : B(x 0 , ǫ) ⊂ P, (2)B(x 0 , ǫ) := {x ∈Ên | ‖x − x 0 ‖ ≤ ǫ}. (3)Equivalently, polytope P is full-dimensional if∃x 0 ∈Ên , ǫ > 0 : ‖δ‖ ≤ ǫ ⇒ P (x 0 + δ) ≤ p. (4)Otherwise, we say that polytope P is lower-dimensional.A polytope P is referred to as empty if∄x ∈Ên : P x ≤ p. (5)Furthermore, if ‖P (i) ‖ = 1, i = 1, . . . , n p , we say that thepolytope P is normalized. One of the fundamental propertiesof a polytope is that it can be described in half-spacerepresentation (1) or in vertex representation (cf. [22]), asgiven below,P = {x ∈Ên | x =v P∑i=1α i V Pi , 0 ≤ α i ≤ 1,v P∑i=1□□α i = 1},where ViP ∈Ên denotes the i-th vertex of P, and v P is thetotal number of vertices of P.We will henceforth refer to the half-space representation(1) and vertex representation (6) as H- and V-representation respectively (see Fig. 1).(6)Definition 5 (Face) A linear inequality a T x ≤ b is calledvalid for a polyhedron P if a T x ≤ b holds for all x ∈ P.A subset of a polyhedron is called a face of P if it can berepresented asF = P ∩ {x ∈Ên | a T x = b}, (7)for some valid inequality a T x ≤ b. The faces of polyhedronP of dimension 0, 1, (n − 2) and (n − 1) are calledvertices, edges, ridges and facets, respectively. □We see that a set F ⊆ P is called a face of P if it is either ∅,P itself or the intersection of P with a hyperplane derivedfrom a valid inequality. An empty set ∅ is a face of everypolyhedron and by convention it has dimension −1.According to our definition every polytope represents aconvex, compact (i.e. bounded and closed) set. We willsee later that polytopes – as simple objects as they are –play an instrumental role in the derivation of optimal controlstrategies for constrained linear and piecewise affinesystems. Very often, however, we also encounter sets thatare disconnected or non-convex but can be represented as aunion of finite number of polytopes. Therefore, it is usefulto define the following mathematical concept.Definition 6 A polytopal complex C is a finite collectionof polytopes inÊn such that• the empty set is in C• if P ∈ C, then all the faces of P are also in C,• the intersection P ∩ Q of two polytopes P, Q ∈ C isa face both of P and Q.□Example 1 Here are several polytopal complexes inÊ1 :{∅, 0, 2, [0, 2]}, {∅, 0, 2, 4, [0, 2]}. Contrary to those,the following sets are not polytopal complexes inÊ1 :{∅, 0, [0, 2]} – because of a missing polytope {2}, and{∅, −2, 0, 2, [−2, 0], [−2, 2], [0, 2]} – because intersectionof two polytopes [−2, 0] and [−2, 2] is not a face of [−2, 2].□Characterization of a polytopal complex is quite expensive.For every polytope in the complex we need to computeall the faces (i.e. lower-dimensional polytopes). Sincewe usually work only with full-dimensional polytopes (cf.Remark 1) it is reasonable to introduce a more practicalconcept.Definition 7 (P-collection) A P-collection C is a finite collectionof full-dimensional polytopes inÊn , i.e.,C = {C i } NCi=1 , (8)where N C = card(C) < ∞, C i := {x ∈Ên | C i x ≤ c i },C i full-dimensional, i = 1, . . . , N C .□AUTOMATIKA 50(2009) 3–4, 119–134 121

Polytopic Computations in Constrained Optimal ControlM. Baotić1098765x 2P432100 1 2 3 4 5 6 7 8 9 10x 11098765x 2P432100 1 2 3 4 5 6 7 8 9 10x 1(a) H-representation of a polytope inÊ2 . The hyperplanes{x | P (i) x = p (i) }, i = 1, . . . , 5, are depicted with dashed lines(b) V-representation of a polytope inÊ2 .1, . . . , 5, are depicted with circlesThe vertices V Pi , i =Fig. 1. Illustration of a polytope in H-representation and V- representationDefinition 8 (Underlying Set) The underlying set of a P-collection C = {C i } NCi=1 is the point set inÊnC :=N⋃Ci=1C i . (9)Example 2 A collection R = {[−2, −1], [0, 2], [2, 4]} is aP-collection inÊ1 with the underlying set R = [−2, −1] ∪[0, 4]. As another example, R = {[−2, 0], [−1, 1], [0, 2]}is a P-collection inÊ1 with underlying set R = [−2, 2].Clearly, polytopes that define P-collection can overlap,while the underlying sets can be disconnected and nonconvex.□Usually it is clear from the context if we are talkingabout the P-collection or we are referring to the underlyingset of a P-collection, in which case, for simplicity, weuse the same notation for both.Definition 9 (Partition) A collection of sets {R i } NRi=1 is apartition of a set P if: (i) P = ∪ NRi=1 R i, and (ii) R i ∩R j =∅, ∀i ≠ j, with i, j ∈ {1, . . . , N R }. □Definition 10 (Polyhedral Partition) A collection of sets{R i } NRi=1 is a polyhedral partition of a set P if {R i} NRi=1is a partition of P and the sets ¯R i are polyhedra, wherei = 1, . . . , N R , and ¯R i denotes the closure of R i . □3 OPERATIONS ON POLYTOPESWe will now define some basic operations and functionson polytopes. Note that although we focus on polytopes□and polytopic objects most of the operations described hereare directly (or with minor modifications) applicable topolyhedral objects. Additional details on polytope computationcan be found in [22, 23]. All operations and functionsdescribed in this section are contained in the MPTtoolbox [24].3.1 Minimal RepresentationWe say that a polytope P ⊂Ên , P = {x ∈Ên | P x ≤p} is in a minimal representation if the removal of any rowin P x ≤ p would change it (i.e., there are no redundanthalf-spaces). The computation of minimal representationof polytopes is discussed in [25] and generally requiressolution of one LP for each half-space defining the nonminimalrepresentation of P. We report one straightforwardimplementation of the minimal representation computationin Algorithm 1.Algorithm 1 Minimal representation: minrep(P)Input: P := {x ∈Ên | P x ≤ p}, p ∈Ên pOutput: Minimal representation of P1. I ← {1, . . . , n p }2. for i = 1 to n p do3. I ← I \ {i}4. f ∗ ← maxx {P (i)x | P (I) x ≤ p (I) , P (i) x ≤ p (i) +1}5. if f ∗ > p (i) then6. I ← I ∪ {i}7. end if8. end for9. return ˜P := {x | P (I) x ≤ p (I) }122 AUTOMATIKA 50(2009) 3–4, 119–134

Polytopic Computations in Constrained Optimal ControlM. Baotićwhat are we referring to, as is the case here, we will abusethe notation and use the former, more compact form.10864QP10864R 1R 2R 3R 41, . . . , N R , is a polytope(NR)⋃∑N Rco R i := {x ∈Ên | x = α i x i , x i ∈ R i ,i=1i=1∑N R0 ≤ α i ≤ 1, α i = 1}.i=1(14)200 2 4 6 8 10(a) Polytopes P and Q200 2 4 6 8 10(b) R = ⋃ i R i = P \ QAn illustration of the convex hull operation is given inFig. 5. Construction of the convex hull is an expensive op-Fig. 4. Illustration of the set difference operation10810866Remark 2 The set difference of two intersecting polytopes(or any closed sets) P and Q is, in general, not a closedset. This means that some borders of polytopes R i froma P-collection R = P \ Q are open, while other bordersare closed. Even though it is possible to keep track of theorigin of particular borders of R i , thus specifying if theyare open or closed, we are not doing it in the algorithmsdescribed in this paper nor in MPT [24], cf. Remark 1. Incomputations, we will henceforth only consider the closureof sets R i .□Related to the above operation are the followingtwo computationally very demanding problems/questions:what is the set difference between a polytope P and a P-collection Q := ∪ i Q i ; and is a polytope P fully coveredby a P-collection Q := ∪ i Q i ? For instance, the latteroperation is needed to check if two unions of polyhedracover the same non-convex set [11, 14] (e.g., step 5 of Algorithm4.1 in [14]). In Section 4 and Section 5 we analyzethese problems in more details, and derive efficientalgorithms that perform these tasks.3.5 Convex HullThe convex hull of a set of points V = {V i } NVi=1 , withV i ∈Ên , is a polytope defined as42R 2R 100 2 4 6 8 10(a) P-collection R = ⋃ i R i42co(R)00 2 4 6 8 10(b) Convex hull of RFig. 5. Illustration of the convex hull operationeration which is exponential in the number of facets of theoriginal polytope. An efficient software implementation isavailable from [32, 33].3.6 EnvelopeThe envelope of two H-polyhedra P = {x ∈Ên | P x ≤ p} and Q = {x ∈Ên | Qx ≤ q} is an H-polyhedronenv(P, Q) = {x ∈Ên | ˜P x ≤ ˜p, ˜Qx ≤ ˜q}, (15)where ˜P x ≤ ˜p is the subsystem of P x ≤ p obtained byremoving all the inequalities not valid for the polyhedronQ, and ˜Qx ≤ ˜q are defined in a similar way with respect toQx ≤ q and P [34]. In a similar fashion definition can beextended to the case of the envelope of a P-collection. Anillustration of the envelope operation is depicted in Fig. 6.The computation of the envelope is relatively cheap sinceit only requires the solution to one LP for each facet of Pand Q. Note that envelope of two (or more) polytopes isnot necessarily bounded set (e.g. when P ∪ Q is shapedlike a star).∑N V∑N V3.7 Vertex Enumerationco(V) = {x ∈Ên | x = α i V i , 0 ≤ α i ≤ 1, α i = 1}.i=1i=1(13)The convex hull operation is used to switch from a V-representation of a polytope to a H-representation. Theconvex hull of a union of polytopes R i ⊂Ên , i =The operation of extracting the vertices of a polytopeP given in H-representation is referred to as vertex enumeration.This operation is the dual to the convex hulloperation and the algorithmic implementation is identicalto a convex hull computation, i.e. given a set of extreme124 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal Control108642R 2R 100 2 4 6 8 10(a) P-collection R = ⋃ i R i108642env(R)00 2 4 6 8 10(b) Envelope of RFig. 6. Illustration of the envelope operationpoints V = {V i } NVi=1 = vert(P) of a polytope P givenin H-representation it holds that P = co(V), where theoperator vert denotes the vertex enumeration. The necessarycomputational effort is in the worst case exponentialin the number of the polytope facets. Two different approachesto vertex enumeration exist: the double descriptionmethod [35] and reverse search [36]. An efficient implementationof the double description method is availablein [32, 33].3.8 Pontryagin DifferenceThe Pontryagin difference (also known as Minkowskidifference) of two polytopes P and Q is a polytopeP ⊖ Q := {x ∈Ên | x + q ∈ P, ∀q ∈ Q}. (16)The Pontryagin difference can be efficiently computed forpolytopes by solving a sequence of LPs [37]. For specialcases (e.g. when Q is a hypercube), even more efficientcomputational methods exist [38]. An illustration of thePontryagin difference is given in Fig. 7.hull computation inÊn or a projection fromÊ2n down toÊn . The implementation of the Minkowski sum via projectionis based on the following observation. ForP = {y ∈Ên | P y ≤ p},it holds thatQ = {z ∈Ên | Qz ≤ q},R = P ⊕ Q{}= x = y + z | y, z ∈Ên , P y ≤ p, Qz ≤ q{}= x ∈Ên | ∃y ∈Ên : P y ≤ p, Q(x − y) ≤ q{[ [ ] [ }= x ∈Ên | ∃y ∈Ên 0 P x p:≤Q −Q]y q]( { [ ] [ )0 P p}= proj n w ∈Ê2n | w ≤ .Q −Q q]Both the projection and vertex enumeration based methodsare implemented in the MPT toolbox [24]. An illustrationof the Minkowski sum is given in Fig. 8.21.510.50−0.5−1−1.5QP−2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2(a) Polytopes P and Q21.510.50−0.5−1−1.5P ⊕ Q−2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2(b) Minkowski sum P ⊕ QFig. 8. Illustration of the Minkowski sum operation21.510.50QP21.510.50P ⊖ QRemark 3 The Minkowski sum is not the complement ofthe Pontryagin difference. For two polytopes P and Q, itholds that (P ⊖ Q) ⊕ Q ⊆ P (cf. Fig. 7 and Fig. 8). □−0.5−0.5−1−1.5−1−1.54 REGIONDIFF PROBLEM−2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2(a) Polytopes P and Q−2−2 −1.5 −1 −0.5 0 0.5 1 1.5 2(b) Pontryagin difference P ⊖QFig. 7. Illustration of the Pontryagin difference operation3.9 Minkowski SumThe Minkowski sum of two polytopes P and Q is apolytopeP ⊕ Q := {x + q ∈Ên | x ∈ P, q ∈ Q}. (17)The Minkowski sum is a computationally expensive operationwhich requires either vertex enumeration and convexLet P = {x | P x ≤ p} be a polyhedron inÊn givenas the intersection of n p half-spaces and Q i = {x ∈Ên | Q i x ≤ q i } be N Q polyhedra inÊn given as the intersectionof n qi half-spaces (i.e. Q i ∈Ên qi ×n ). We want todetermine the set difference P \ (∪ NQi=1 Q i). We refer to thisproblem as the regiondiff problem (to differentiate it fromthe problem of computing the set difference between twopolytopes, cf. Section 3.4).We outline the procedure of finding the exact solutionto the regiondiff problem in Algorithm 2. It computes theset R = P \ (∪ NQi=1 Q i), where R (if not empty) is given asunion of non-overlapping polyhedra.AUTOMATIKA 50(2009) 3–4, 119–134 125

Polytopic Computations in Constrained Optimal ControlM. Baotić1010981P981R 1765Q 3221765Q 32R 3R 2214141321323Q 2Q 13213R 423Q 2Q 100 1 2 3 4 5 6 7 8(a) Polytope P and P-collection Q00 1 2 3 4 5 6 7 8(b) R := {R i } 4 i=1 = regiondiff(P, Q)Fig. 9. Example for the set difference computation with Algorithm 2 inÊ2 . Polytope P is depicted with full line, activeconstraints (see (18) for definition) of Q i with dashed lines, and non-active constraints of Q i with thin lines. The indicesof corresponding active constraints of each Q i are denoted as encircled numbers.One example of solving the regiondiff problem withAlgorithm 2 for 2-dimensional polytope P and P-collection {Q i } 3 i=1 is illustrated in Fig. 9.Algorithm 2 solves the regiondiff problem by basicallyimplementing an in-place depth-first exploration ofthe tree of feasible constraint combinations, where eachbranch/node corresponds to the inversion of a facet of Q i(step 12 of Algorithm 2). The approach used to partitionthe space is based on [6].Note that the output of Algorithm 2 is a P-collection Rthat comprises only non-overlapping polyhedra, regardlessof possible overlaps in the input P-collection Q. We alsonote that Algorithm 2 would work even if steps 2-10 werenot present. However, those steps potentially have a hugeimpact on the speed of computation since they exclude unnecessarychecks at later stages (i.e. lower branches) of thealgorithm.Remark 4 Step 5 of Algorithm 2 is a simple feasibility LPof complexity lp(n + 1, n p + n qk ). Similarly, step 12 is afeasibility LP of complexity lp(n + 1, n p + 1). □Remark 5 (Preprocessing) From a practical point ofview it makes sense to preprocess the data before usingAlgorithm 2 to compute the set difference. Namely, wecan speed up the computation by ensuring that P and Q i ,i = 1, . . . , N Q , are full-dimensional polyhedra inÊn givenin the minimal H-representation: P = {x | P x ≤ p},Q i = {x | Q i x ≤ q i }, P ∈Ên p×n , p ∈Ên p, Q i ∈Ên qi ×n , q i ∈Ên qi. Note that it is always possible to obtainnormalized polyhedron in minimal representation (seeSection 3.1) and to remove those Q i that do not intersectP, e.g. by checking if Chebyshev ball (see Section 3.2) ofa joint polyhedra {x | P x ≤ p, Q i x ≤ q i } is non-empty.Furthermore, it is useful to pre-compute the set of socalledactive constraints:A i = { j ∈ {1, . . . , n qi } | ∃x : P x ≤ p, [Q i ] (j) x > [q i ] (j)},(18)for each Q i , i = 1 . . . , N Q . This can be done at the expenseof at most ∑ N Qi=1 n q iLPs:N Q∑n qi · lp(n + 1, n p + n qi ). (19)i=1We note that the removal of non-active constraints fromthe description of each Q i does not change the result ofAlgorithm 2, i.e.,whereP \ {Q i } NQi=1 = P \ { ˜Q i } NQi=1 , (20)˜Q i := {x ∈Ên | ˜Q i x ≤ ˜q i }, (21)˜Q i = [Q i ] (Ai), ˜q i = [q i ] (Ai). (22)As a matter of fact, since card(A i ) ≤ n qi , this removal ofnon-active constraints can significantly reduce dimensionof the problem parameters in Algorithm 2 and thus drasticallyreduce complexity of Algorithm 2.□Remark 6 (Postprocessing) Note that Algorithm 2 returnsP-collection R = {R i } NRi=1 , with R i that are in (possibly)non-minimal H-representation. As a postprocessingstep one might call Algorithm 1 to obtain the minimal representationfor each R i of the result.□126 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal ControlAlgorithm 2 Set difference: regiondiff(P, Q)Input: P := {x ∈Ên | P x ≤ p}, p ∈Ên p, Q :={Q i } NQi=1 , Q i := {x ∈Ên | Q i x ≤ q i }, q i ∈Ên qi,i = 1, . . . , N QOutput: P-collection representing P \ Q1. R ← ∅, k ← 12. if ∃i ∈Æ1:N Q: n qi = 0 then3. return R4. end if5. while ∄x ∈Ên : P x < p, Q k x < q k do6. k ← k + 17. if k > N Q then8. return P9. end if10. end while11. for j = 1 to n qk do12. if ∃x ∈Ên : P x ≤ p, [Q k ] (j) x > [q k ] (j) then13. ˜P = P ∩ {x | [Qk ] (j) x ≥ [q k ] (j) }14. if k < N Q then15. R ← R ∪ regiondiff( ˜P, {Q i } NQi=k+1 )16. else17. R ← R ∪ ˜P18. end if19. end if20. P ← P ∩ {x | [Q k ] (j) x ≤ [q k ] (j) }21. end for22. return RThe most time consuming part of Algorithm 2 lies inrecursive calls to itself (step 15). Therefore in the followingcomputational complexity analysis we neglect the costof removal of the non-active constraints from Q i (cf. Remark5) or computing the minimal representation of theregions R i that describe the set difference (cf. Remark 6).To establish the worst case complexity of Algorithm 2one needs to look at its tree-like exploration of the space.The depth of the tree is equal to N Q (i.e., the number ofregions Q i ). Clearly, every node on the level k has at mostn qk+1 children nodes on the level k + 1, where n qk denotesthe number of constraints of polytope Q k (if preprocessingstep from Remark 5 is done then n qk is the number of activeconstraints) . Therefore the number of nodes at level kis bounded from above bywherek∏i=1n qi≤H k :=(Hkk) k, (23)k∑n qi , (24)i=1i.e., H k is the cumulative number of constraints up to thelevel k of the tree, and k = 1, . . . , N Q .One should note that (23) gives an overly conservativeupper bound for the number of nodes at level k. For example,if (23) was a strict upper bound, it would imply thatthe number of nodes at the bottom of the tree is of the order( ( ) )NQMN QO, (25)where M denotes the total number of constraintsN Q∑M := n qi . (26)i=1To better estimate the upper bound on the number ofnodes at level k we recall the Buck’s formula [39] for thehyperplane arrangement problem. Buck’s formula gives anupper bound for the maximal number of cells created byH k hyperplanes inÊn . The upper bound is obtained by thehyperplanes in the so-called general position [22]. Therefore,we can obtain the strict upper bound on the numberof nodes at level kT k ≤n∑i=0(Hki)= O(Hknn!), (27)where T k denotes the number of nodes at level k of the treefor Algorithm 2, and k = 1, . . . , N Q .Since every region in the output of Algorithm 2 correspondsto exactly one feasible node at the bottom of theexploration tree, we clearly see that the number of nodesT NQ is an upper bound on the number of regions generatedby Algorithm 2N R ≤ T NQ . (28)Therefore, by taking into account that H NQ = M, the followingstrict upper bound on N R can be computed from(27)n∑( ) ( )M MnN R ≤= O . (29)in!i=0Feasible nodes of level k + 1, with k = 1, . . . , N Q − 1,are computed from feasible nodes at level k. The computationat every node of level k involves solution of at mostn qk+1 LPs with n+1 variables and at most n p +H k+1 constraints(cf. Remark 4). Therefore, the strict upper boundon the overall complexity of Algorithm 2 is given byN∑Q−1k=1T k · n qk+1 · lp(n + 1, n p + H k+1 ), (30)i.e., the overall complexity of Algorithm 2 is of the order( )Mn+1O · lp(n + 1, n p + M) . (31)n!AUTOMATIKA 50(2009) 3–4, 119–134 127

Polytopic Computations in Constrained Optimal ControlM. BaotićFrom (31) it is clear that for n > 1 the overall complexityof Algorithm 2 is hugely affected by M – the total numberof (active) constraints for the problem at hand. Therefore,for specific applications, where regiondiff problem issolved via Algorithm 2, one should investigate if it is possibleto split the original problem in subproblems (assumingthat the dimension n remains fixed) that would result incalls to Algorithm 2 with a reduced value of M (see [40]for one application of this idea).Remark 7 The expression (29) represents the strict upperbound on the output complexity (i.e. the set difference representation)for Algorithm 2, and therefore it cannot befurther improved. The same claim holds also for the expression(30) (and, consequently, for (31)) that defines thestrict upper bound on the computational complexity of Algorithm2. However, in practice (n ≪ N Q ≪ M) theseexpressions are found to be very conservative (cf. testingresults in Section 6).□Remark 8 Extension of the regiondiff problem to the casewhere both P and Q are P-collections is straightforward.Namely, the set difference is a P-collection R = {P i } NPi=1 \{Q j } NQj=1 that can be computed by cycling through all polytopesP i . We haveN⋃PR =i=1P i \ {Q j } NQj=1 . (32)Note that here the output P-collection R may contain overlappingpolyhedra if the input P-collection P has someoverlapping polyhedra.□5 POLYCOVER PROBLEMThe problem of checking if some polytope is coveredwith the union of other polytopes emerges very often inthe construction of explicit solution for the constrained finitetime optimal control of piecewise affine systems. Theneed for solving this type of a problem is also common inthe computation of the (positive) controlled invariant sets,infinite time solution or controllers with reduced complexityfor PWA systems. Thus, it is important to have an algorithmthat checks if a polytope P is fully covered by aP-collection Q := ∪ i Q i in an efficient manner. We referto this problem as the polycover problem.One idea of solving the polycover problem is inspiredby the following observationP ⊆ ∪ i Q i ⇔ P = ∪ i (P ∩ Q i ).Therefore, we could create R i = Q i ∩ P, for i =1, . . . , N Q and then compute the union of the collection ofpolytopes {R i } by using the polyunion algorithm for computingthe convex union of H-polyhedra reported in [34].If approach [34] succeeds (i.e., the union is a convex set)and the resulting polytope is equal to P then P is coveredby {Q i } NQi=1 , otherwise it is not. However, this approach iscomputationally very expensive. The algorithm in [34] isbased on the idea that whenever P := ∪ i Q i is a polyhedronthen P = env({Q i } NQi=1). Unfortunately the underlyingcomputation is very expensive, and one needs to solveO(Π NQi=1 n q i) LPs with n variables and M = ∑ N Qi=1 n q iconstraints. Thus, the polyunion computation from [34]becomes quickly prohibitive with the increasing numberof polytopes N Q and constraints M.In the following we propose two different approaches.5.1 Polycover: Regiondiff Based AlgorithmClearly, polycover is just a special case of regiondiff,where resulting P-collection R = ∅. In a special case whenwe only want to check if P ⊆ (∪ NQi=1 Q i) finding any feasibleR j in Algorithm 2 provides a negative answer and wecan abort further search. Implementation of this strategy isgiven in Algorithm 3.Similarly to Algorithm 2 the worst case complexity ofAlgorithm 3 is bounded by (30) (see also Remark 5 andRemark 7).Remark 9 Note that it is straightforward to extend thepolycover problem to the case where both P and Q areP-collections:• Verification of {P i } NPi=1 ⊆ {Q j} NQj=1 :{P i } NPi=1 ⊆ {Q j} NQj=1 ⇔ {P i} NPi=1 \ {Q j} NQj=1 = ∅.• Verification of {P i } NPi=1 = {Q j} NQj=1 :{P i } NPi=1 = {Q j} NQj=1 ⇔{P i } NPi=1 \{Q j} NQj=1 = ∅ & {Q j} NQj=1 \ {P i} NPi=1 = ∅.Here the set difference between two P-collections can becomputed according to (32).□5.2 Polycover: MILP formulationWhen some of the optimization variables in a linear program(see Definition 1) are constrained to integer valuesthe ensuing problem is called a mixed integer linear program(MILP).Definition 11 A mixed integer linear program (MILP) is anon-convex optimization problem that can be expressed inthe formminxf T x (MILP)subj. to Gx ≤ g,128 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal ControlAlgorithm 3 Set cover: polycover(P, Q)Input: P := {x ∈Ên | P x ≤ p}, p ∈Ên p, Q :={Q i } NQi=1 , Q i := {x ∈Ên | Q i x ≤ q i }, q i ∈Ên qi,i = 1, . . . , N QOutput: Check if P ⊆ ∪ NQi=1 Q i, {true, false}1. k ← 12. if ∃i ∈Æ1:N Q: n qi = 0 then3. return R4. end if5. while ∄x ∈Ên : P x < p, Q k x < q k do6. k ← k + 17. if k > N Q then8. return false9. end if10. end while11. for j = 1 to n qk do12. if ∃x ∈Ên : P x ≤ P c , [Q k ] (j) x > [q k ] (j) then13. if k = N Q then14. return false15. end if16. ˜P = P ∩ {x | [Qk ] (j) x ≥ [q k ] (j) }17. if polycover( ˜P, {Q i } NQi=k+1) = false then18. return false19. end if20. P ← P ∩ {x | [Q k ] (j) x ≤ [q k ] (j) }21. end if22. end for23. return truewhere x ∈Ên r× {0, 1} n bis the optimization variable, n ris the number of real valued variables, n b is the numberof binary (or, in general, integer) variables, and matricesf ∈Ên , G ∈Ên g×n , g ∈Ên g, with n = n r + n b , aregiven problem parameters.□We note that P is not fully covered by Q, i.e.P (∪ NQi=1 Q i) (33)if and only if there is a point x inside of P that violatesat least one of the constraints of each Q i , i = 1, . . . , N Q .This is equivalent to the following set of conditions∃x ∈ P : ∃j i ∈ {1, . . . , n qi }, [Q i ] (ji)x − [q i ] (ji) > 0,i = 1, . . . , N Q .(34)To express this violation of constraints we introduce slackvariables{ [Qi ] (j) x − [q i ] (j) if [Q i ] (j) x − [q i ] (j) ≥ 0,y i,j (x) =0 if [Q i ] (j) x − [q i ] (j) ≤ 0,j = 1, . . . , n qi , i = 1, . . . , N Q .(35)The expression (34) can now be posed as a feasibility questionin x and y i,jP x ≤ p,∑ nqij=1 y (36)i,j > 0, i = 1, . . . , N QChecking the condition (36) is still not possible with standardsolvers, since the relation (35) describes a non-linearfunction. However, by introducing auxiliary binary variablesone can rewrite (35) as the following equivalent setof linear inequalities (cf. [41])⎡⎢⎣⎤0 −m L0 −M U1 −M U−1 m L⎥1 −m L⎦−1 M U]δ i,j ⎢⎣[yi,j⎡δ i,j ∈ {0, 1},j = 1, . . . , n qi ,[Q i] (j) x − [q i] (j) − m L−[Q i] (j) x + [q i] (j)00[Q i] (j) x − [q i] (j) − m L−[Q i] (j) x + [q i] (j) + M Ui = 1, . . . , N Q ,(37)where δ i,j are auxiliary binary variables and m L , M U ∈Êare bounds on constraint expressions that can be precomputed(or overestimated) beforehandm L ≤M U ≥minx,i,jsubj. tominx,i,jsubj. to[Q i ] (j) x − [q i ] (j)P x ≤ pj ∈ {1, . . . , n qi }i ∈ {1, . . . , N Q }[Q i ] (j) x − [q i ] (j)P x ≤ pj ∈ {1, . . . , n qi }i ∈ {1, . . . , N Q }(38)(39)Actually, in terms of the number of inequalities that areused, (37) can be further simplified to⎡⎤−1 0⎡⎤[ ]0⎢−1 0⎥ yi,j ⎢ −[Q i] (j) x + [q i] (j) ⎥⎣ 1 −m L ⎦ δ⎣i,j [Q i] (j) x − [q i] (j) − m⎦L1 −M U 0δ i,j ∈ {0, 1},j = 1, . . . , n qi ,i = 1, . . . , N Q .(40)⎤⎥⎦AUTOMATIKA 50(2009) 3–4, 119–134 129

Polytopic Computations in Constrained Optimal ControlM. Baotić10 2 N QMILPAlg. 3MILPAlg. 310 2 N Q10 110 1time [s]10 0time [s]10 010 −110 −110 −210 20 30 40 50 60(a) Computational times for ¯n q ≈ 4.0 and dimension n = 210 −210 20 30 40 50 60(b) Computational times for ¯n q ≈ 6.8 and dimension n = 3Fig. 10. Setup 1. Comparison of MILP (41) and Algorithm 3 when solving polycover(P, {Q i } NQi=1 ), with P ⊂Ên ,n ∈ {2, 3}, and varying N Q = 11, . . . , 60. Reported computational times are average values for 100 random tests pereach scenario, i.e., value of N Q .Since (36) and (40) describe a Mixed Integer Linear Programming(MILP) feasibility problem it follows that wecan check if P (∪ NQi=1 Q i) by solving an MILP feasibilityproblem.However, instead of solving a feasibility MILP problemwith (36) it may be more useful (and numerically robust)to solve the following optimality MILP problemmaxλ,x,δ i,j,y i,jλ⎧⎪⎨P x ≤ p,∑ nqij=1subj. toy i,j ≥ λ, i = 1, . . . , N Q∑ nqij=1⎪⎩δ i,j ≥ 1, i = 1, . . . , N Qconstraints (40)(41)Effectively, the optimal value λ ∗ is related to the size ofthe largest non-covered part of P.Theorem 1 Let λ ∗ be the solution to the problem (41),then P (∪ NQi=1 Q i) if and only if λ ∗ > 0. □Proof: Follows from the construction of the MILPproblem (41).Remark 10 Strictly speaking, condition ∑ n qij=1 δ i,j ≥ 1in (41) is redundant, but it reduces the problems with theintegrality tolerances in existing MILP solvers. Also notethat when solving (41) there is no need for condition y i,j ≥0 (first row in constraints (40)). □The MILP problem (41) has n p + 2N Q + 3 ∑ N Qi=1 n q iconstraints, n + 1 + ∑ N Qi=1 n q ireal variables and ∑ N Qi=1 n q ibinary variables.6 TESTING OF ALGORITHMSIn this section we compare performances of thetwo approaches proposed in Section 5 for computingpolycover(P, {Q i } NQi=1 ), i.e., checking if P ⊆ {Q i} NQi=1 .Testing was carried out on a 2.4 GHz Pentium 4 machinewith 1.5 GB RAM. The polycover problem wassolved with MPT 2.6 [24] under MATLAB 7.01 in caseof Algorithm 3, and with CPLEX 9.0 [42] in case of theMILP formulation (41).Setup 1Series of random tests were performed in several dimensionsn, with varying number of regions N Q and averagenumber of constraints∑ NQi=1¯n q :=n q i= M . (42)N Q N QIn all testing instances P ⊂Ên was chosen as a fulldimensionalpolytope centered at the origin with (a random)Chebyshev radius between 10 and 15. P-collection Qis such that Q i ⊂Ên , i = 1, . . . , N Q , are full-dimensionalpolytopes with an average Chebyshev radius equal to 10and Chebyshev centers randomly placed (with uniform distribution)within a hypercube [−10, 10] n . Testing resultsfor various scenarios are reported in Fig. 10 and Fig. 11.We note that Algorithm 3 is always better than the MILPformulation (41). This superiority is more pronounced forlarger dimensions of the space.Setup 2To illustrate effectiveness of Algorithm 3 we considerthe following polycover problem setup that should always130 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal Control10 3 ¯n qMILPAlg. 3MILPAlg. 310 3 ¯n q10 210 2time [s]10 110 0time [s]10 110 010 −110 −110 −25 10 15 20 25 30 35(a) Computational times for dimension n = 210 −25 10 15 20 25 30 35(b) Computational times for dimension n = 3Fig. 11. Setup 1. Comparison of MILP (41) and Algorithm 3 when solving polycover(P, {Q i } NQi=1 ), with P ⊂Ên ,n ∈ {2, 3}, N Q = 20, and varying ¯n q ∈ [6.8 33]. Reported computational times are average values for 100 random testsper each scenario, i.e., value of ¯n q .1010101055550000−5−5−5−5−10−10 −5 0 5 10(a) Partition for N Q = 5−10−10 −5 0 5 10(b) Partition for N Q = 40−10−10 −5 0 5 10(c) Partition for N Q = 100−10−10 −5 0 5 10(d) Partition for N Q = 500Fig. 12. Setup 2. Illustration of a P-collection Q generation inÊ2 .give the value true. Polytope P ⊂Ên is chosen as a hypercube[−10, 10] n centered at the origin. For a fixed dimensionn ∈ {2, . . . , 12} and successive values of N Qthe P-collection Q is constructed as a polytopic partitionof P in the following (iterative) manner: for N Q = 1,Q 1 = P; for N Q > 1 a randomly generated point x NQin [−10, 10] n decides which polytope Q i , i ∈Æ1:N Q−1, issplit into two polytopes by a random hyperplane passingtrough x NQ (i.e., through the interior of Q i ) thus creatingnew polytopic partition Q of P with N Q polytopes. SeeFig. 12 for illustration of the above procedure inÊ2 .Testing results for varying N Q and different dimensionsof the space n are reported in Fig. 13. We point out thatfor most of the instances reported in Fig. 13 the MILP formulation(41) is practically intractable (i.e., it produces nosolution even after 24 hours of computation). Perhaps themost informative view of the testing results is reported inFig. 14, which indicates that the computational time of Algorithm3 (for all dimensions considered in Setup 2) growsapproximately with the square of the number of regions inP-collection Q.7 CONCLUSIONWe have reviewed standard polytopic operations that areutilized when deriving explicit form of the optimal and robustcontrol for discrete-time systems.We have analyzed in great detail two special classes ofthese polytopic operations: the so-called regiondiff problemwhere the set difference between a polyhedron andunion of polyhedra is computed, and the polycover problemwhere one is interested in checking if a polytope iscovered by the union of other polytopes.We have devised an in-place depth-first explorationalgorithm that solves in an efficient manner both theregiondiff problem and, as a special case, the polycoverproblem. Furthermore, we have derived the strict upperbound for the computational complexity of this algorithm.In large number of random test we have shown that ourpolycover algorithm is superior to the mixed integer linearprogramming solution to the same problem.AUTOMATIKA 50(2009) 3–4, 119–134 131

Polytopic Computations in Constrained Optimal ControlM. Baotić12time [s]10 210 110 010 −1n = 2, . . . , 1210 −210 3 N Q10 −30 100 200 300 400 500(a) Computational timesav. number of active constraints11109876543n = 2, . . . , 12N Q20 100 200 300 400 500(b) Average number of active constraintsFig. 13. Setup 2. Computational times and average number of active constraints in Algorithm 3 for different dimensionsof the space n = 2, . . . , 12, and varying number of polytopes N Q ∈ {5, . . . , 500}. Reported values are average valuescomputed over 100 random tests per each scenario, i.e., value of n and N Q .log(comp. time)2.521.510.50−0.5−1−1.5−2n = 2, . . . , 12log(N Q )−2.50.5 1 1.5 2 2.5 3Fig. 14. Setup 2. Logarithmic scale representation of computationaltimes reported in Fig. 13ACKNOWLEDGMENTA large portion of reported research was carried outwhile the author was working at the Automatic ControlLaboratory, ETH Zurich, Switzerland. Support ofthe Croatian Ministry of Science under grant No. 036-0361621-3012 and grant No. 036-0363078-3018 is kindlyacknowledged. The author would also like to thank reviewersfor helpful suggestions for improvement of the paper.REFERENCES[1] H. S. Witsenhausen, A Minimax Control Problem forSampled Linear Systems, IEEE Trans. on Automatic Control,vol. 13, no. 1, pp. 5–21, 1968.[2] D. P. Bertsekas and I. B. Rhodes, On the Minimax Reachabilityof Target Sets and Target Tubes, Automatica,vol. 7, no. 2, pp. 233–247, March 1971.[3] J. Glover and F. Schweppe, Control of linear dynamic systemswith set constrained disturbances, IEEE Trans. onAutomatic Control, vol. 16, no. 5, pp. 411–423, 1971.[4] D. P. Bertsekas, Infinite-time Reachability of State-SpaceRegions by Using Feedback Control, IEEE Trans. on AutomaticControl, vol. 17, no. 5, pp. 604–613, 1972.[5] F. Blanchini and S. Miani, Set-Theoretic Methods in Control.Berlin: Birkhäuser, 2007.[6] A. Bemporad, M. Morari, V. Dua, and E. Pistikopoulos,The explicit linear quadratic regulator for constrainedsystems, Automatica, vol. 38, no. 1, pp. 3–20, Jan. 2002.[7] A. Bemporad, F. Borrelli, and M. Morari, Model PredictiveControl Based on Linear Programming—The ExplicitSolution, IEEE Trans. on Automatic Control, vol. 47,no. 12, pp. 1974–1985, Dec. 2002.[8] A. Bemporad, F. Borrelli, and M. Morari, Min-Max Controlof Constrained Uncertain Discrete-Time LinearSystems, IEEE Trans. on Automatic Control, vol. 48, no. 9,pp. 1600–1606, Sept. 2003.[9] F. Borrelli, Constrained Optimal Control of Linear andHybrid Systems, ser. Lecture Notes in Control and InformationSciences. Springer, 2003, vol. 290.[10] M. Baotić, F. Borrelli, A. Bemporad, and M. Morari,Efficient On-Line Computation of Constrained OptimalControl, SIAM Journal on Control and Optimization,vol. 47, no. 5, pp. 2470–2489, Sept. 2008. [Online].Available: http://control.ee.ethz.ch[11] S. V. Raković, E. C. Kerrigan, D. Q. Mayne, and J. Lygeros,Reachability Analysis of Discrete-Time Systems With132 AUTOMATIKA 50(2009) 3–4, 119–134

M. Baotić Polytopic Computations in Constrained Optimal ControlDisturbances, IEEE Trans. on Automatic Control, vol. 51,no. 4, pp. 546–561, 2006.[12] M. Baotić, F. J. Christophersen, and M. Morari, ConstrainedOptimal Control of Hybrid Systems With aLinear Performance Index, IEEE Trans. on AutomaticControl, vol. 51, no. 12, pp. 1903–1919, Dec. 2006.[13] F. Borrelli, M. Baotić, A. Bemporad, and M. Morari, Dynamicprogramming for constrained optimal control ofdiscrete-time linear hybrid systems, Automatica, vol. 41,pp. 1709–1721, Oct. 2005.[14] P. Grieder, M. Kvasnica, M. Baotić, and M. Morari, Stabilizinglow complexity feedback control of constrainedpiecewise affine systems, Automatica, vol. 41, pp. 1683–1694, Oct. 2005.[15] M. Vašak and N. Perić, Combining identification and constrainedoptimal control of piecewise affine systems, Automatika,Journal for Control, Measurement, Electronics,Computing and Communications, vol. 48, no. 3–4, pp. 145–160, 2007.[16] M. Baotić, Optimal Control of Piecewise Affine Systems– a Multi-parametric Approach, Ph.D. dissertation,Swiss Federal Institute of Technology (ETH), Zürich,Switzerland, Mar. 2005. [Online]. Available: http://control.ee.ethz.ch[17] L. Khachiyan, A polynomial algorithm in linear programming,Soviet Mathematics Doklady, vol. 20, pp. 191–194, 1979.[18] A. Schrijver, Theory of Linear and Integer Programming.New York: John Wiley & Sons, Inc., 1986.[19] N. Karmarkar, A new polynomial-time algorithm for linearprogramming, Combinatorica, vol. 4, pp. 373–395,1984.[20] C. Roos, T. Terlaky, and J. Vial, Theory and Algorithmsfor Linear Optimization: An Interior Point Approach.New York: John Wiley & Sons, Inc., 1997.[21] D. den Hertog, Interior Point Approach to Linear,Quadratic and Convex Programming: Algorithms andComplexity, ser. Mathematics and its Applications. Dordrecht:Kluwer Academic Publishers, 1994, no. 277.[22] G. M. Ziegler, Lectures on Polytopes. Springer, 1994.[23] B. Grünbaum, Convex Polytopes, 2nd ed. Springer-Verlag, 2000.[24] M. Kvasnica, P. Grieder, and M. Baotić, Multi-ParametricToolbox (MPT), 2004. [Online]. Available: http://control.ee.ethz.ch/~mpt[25] K. L. Clarkson, More output-sensitive geometric algorithms,In Proceedings of the 35th Annual Symposium onFoundations of Computer Science, 1994, pp. 695–702.[26] S. Boyd and L. Vandenberghe, Convex Optimization.Cambridge University Press, 2004.[27] S. Cernikov, Contraction of finite systems of linear inequalities,Doklady Akademiia Nauk SSSR, vol. 152, no. 5,pp. 1075–1078, 1963, in Russian (English translation in SocietMathematics Doklady, Vol. 4, No. 5 (1963), pp.1520–1524).[28] S. S. Keerthi and K. Sridharan, Solution of parametrizedlinear inequalities by Fourier elimination and its applications,J. Opt. Theory and Applications, vol. 65, no. 1, pp.161–169, 1990.[29] E. Balas, Projection with a minimum system of inequalities,Computational Optimization and Applications, vol. 10,pp. 189–193, 1998.[30] K. Fukuda, T. M. Liebling, and C. Lütolf, Extended convexhull, in 12th Canadian Conference on ComputationalGeometry, July 2000, pp. 57–64.[31] C. Jones, E. Kerrigan, and J. Maciejowski, Equality setprojection: A new algorithm for the projection of polytopesin halfspace representation, Department of Engineering,Cambridge University, UK, Tech. Rep. CUEDTechnical Report CUED/F-INFENG/TR.463, 2004, http://www-control.eng.cam.ac.uk/~cnj22.[32] C. B. Barber, D. P. Dobkin, and H. T. Huhdanpaa, Thequickhull algorithm for convex hulls, ACM Trans. onMathematical Software, vol. 22, no. 4, pp. 469–483, Dec.1996. [Online]. Available: http://www.qhull.org[33] K. Fukuda, Cdd/cdd+ Reference Manual, Dec. 1997,www.cs.mcgill.ca/~fukuda/soft/cdd_home/cdd.html.[34] A. Bemporad, K. Fukuda, and F. Torrisi, Convexity recognitionof the union of polyhedra, Computational Geometry,vol. 18, pp. 141–154, Apr. 2001.[35] K. Fukuda and A. Prodon, Double description method revisited,Combinatorics and Computer Science, vol. 1120,pp. 91–111, 1996.[36] D. Avis, lrs: A revised implementation of the reversesearch vertex enumeration algorithm, Polytopes, Combinatoricsand Computation, pp. 177–198, 2000.[37] I. Kolmanovsky and E. G. Gilbert, Theory and computationof disturbance invariant sets for discrete-time linearsystems, Mathematical Problems in Egineering, vol. 4, pp.317–367, 1998.[38] E. C. Kerrigan and J. M. Maciejowski, On robust optimizationand the optimal control of constrained linearsystems with bounded state disturbances, in Proc. 2003European Control Conference, Cambridge, UK, Sept. 2003.[39] R. Buck, Partition of space, American MathematicalMonthly, vol. 50, pp. 541–544, 1943.[40] M. Vašak, M. Baotić, and N. Perić, Efficient Computationof the One-Step Robust Sets for Piecewise AffineSystems With Polytopic Additive Uncertainties, in Proceedingsof the European Control Conference 2007, Kos,Greece, July 2007, pp. 1430–1435.[41] A. Bemporad and M. Morari, Control of systems integratinglogic, dynamics, and contraints, Automatica, vol. 35,no. 3, pp. 407–427, Mar. 1999.[42] ILOG, Inc., CPLEX 9.0 User Manual, Gentilly Cedex,France, 2005, http://www.ilog.fr/products/cplex.AUTOMATIKA 50(2009) 3–4, 119–134 133

Polytopic Computations in Constrained Optimal ControlM. BaotićMato Baotić received the B.Sc. andM.Sc. degrees, both in Electrical Engineering,from the Faculty of ElectricalEngineering and Computing (FER Zagreb),University of Zagreb, Croatia, in 1997and 2000, respectively. As a recipientof the ESKAS scholarship of the SwissGovernment he was a visiting researcher atthe Automatic Control Lab, Swiss Federal Institute of Technology (ETH)Zurich, Switzerland, during the academic year 2000/2001. In 2005 hereceived the Ph.D. from the ETH Zurich, Switzerland. Currently he isAssistant Professor at the Department of Control and Computer Engineering,FER Zagreb, Croatia. His research interests include mathematicalprogramming, hybrid systems, optimal control and model predictivecontrol.AUTHOR’S ADDRESSAsst. Prof. Mato Baotić, Ph.D.Department of Control and Computer Engineering,Faculty of Electrical Engineering and Computing,University of Zagreb,Unska 3, HR-10000 Zagreb, Croatiaemail: mato.baotic@fer.hrReceived: 2009-11-02Accepted: 2009-11-27134 AUTOMATIKA 50(2009) 3–4, 119–134