Stabilizing Tube-Based Model Predictive Control: Terminal Set and Cost Construction for LPV Systems (extended version)
SStabilizing Tube-Based Model Predictive Control:Terminal Set and Cost Construction for LPV Systems(extended version) (cid:63)
J. Hanema a , ∗ , M. Lazar a , R. Tóth a a Department of Electrical Engineering, Eindhoven University of Technology, Eindhoven, The Netherlands.
Abstract
This paper presents a stabilizing tube-based MPC synthesis for LPV systems. We employ terminal constraint sets which arerequired to be controlled periodically contractive. Periodically (or finite-step) contractive sets are easier to compute and canbe of lower complexity than “true” contractive ones, lowering the required computational effort both off-line and on-line. Undercertain assumptions on the tube parameterization, recursive feasibility of the scheme is proven. Subsequently, asymptoticstability of the origin is guaranteed through the construction of a suitable terminal cost based on a novel Lyapunov-like metricfor compact convex sets containing the origin. A periodic variant on the well-known homothetic tube parameterization thatsatisfies the necessary assumptions and yields a tractable LPV MPC algorithm is derived. The resulting MPC algorithmrequires the on-line solution of a single linear program with linear complexity in the prediction horizon. The properties of theapproach are demonstrated by a numerical example.
Key words:
Tube model predictive control; Linear parameter-varying systems; Periodic invariance
In a linear parameter-varying (LPV) system, the statetransition map is linear, but this linear map dependson an external scheduling variable denoted by θ . Thepresent work considers systems represented in the state-space form x ( k + 1) = A ( θ ( k )) x ( k ) + Bu ( k ) . In this set-ting, the current value θ ( k ) can be measured for all time k , but the future behavior of θ is generally not knownexactly. This uncertainty complicates the application of model predictive control (MPC), as guaranteeing recur-sive feasibility and closed-loop stability necessitates theuse of an MPC which is “robust” against all possible fu-ture scheduling variations. Predictive control under un-certainty gives rise to a so-called feedback min-max op-timization problem [1, 2], which can be solved, e.g., by (cid:63) This work was supported by the Impulse 1 program ofthe Eindhoven University of Technology and ASML. Thematerial in this paper was not presented at any conference.* Corresponding author J. Hanema.c (cid:13) http://creativecommons.org/licenses/by-nc-nd/4.0/
Email addresses: [email protected] (J. Hanema), [email protected] (M. Lazar), [email protected] (R. Tóth). dynamic programming. Because of its inherent complex-ity, it is useful to search for more conservative, but effi-cient, approximations of this difficult problem which arecomputationally tractable in practice [3].
Tube MPC (TMPC) is a paradigm devised to reducecomplexity with respect to the min-max solution. Aprincipal advantage is that its computational complex-ity scales well (often, linearly) in the length of predic-tion horizon. Tube-based approaches were originally pro-posed to control constrained linear systems subject to additive disturbances [4–8]. In this type of TMPC, thepurpose of constructing a tube is to keep the perturbedsystem trajectories close to a nominal trajectory. Thework [5] uses rigid tubes, consisting of a sequence oftranslated copies of a pre-designed robustly positivelyinvariant basic shape set, and ensures recursive feasibil-ity by tightening constraints. In [4, 6], so-called homo-thetic tubes are introduced: these tubes consist of a se-quence of translated and, additionally, scaled copies ofthe basic shape set. The scalings are optimized on-line,so no constraint tightening is necessary. A much moreflexible, but more expensive, parameterization is givenin [7], which drops the restriction that the basic shapesets are designed off-line. The elastic tubes of [8] also im-prove upon the flexibility of homothetic tubes, by scaling
Preprint available on arXiv 27 June 2017 a r X i v : . [ c s . S Y ] J un ach hyperplane of the basic shape set individually. In allthe TMPC approaches mentioned so far, persistent ad-ditive disturbances are assumed to be present, and henceasymptotic stability of the origin can not be established.Instead, convergence to some limit set centered aroundthe origin is attained. In contrast, for an LPV system,it should be possible to asymptotically stabilize the ori-gin because the uncertainty enters multiplicatively in thestate transition matrix. Thus, the notion of robust sta-bility required for LPV systems is different from the oneemployed in the aforementioned TMPC approaches foradditively disturbed systems.Besides robust control, tubes can also be applied for thepurpose of stabilization. A framework for the construc-tion of “stabilizing” tubes, with application to the pre-dictive control of linear systems on assigned initial con-dition sets, was presented in [9]. All system trajectoriesemanating from the initial condition set are restrictedto be inside a tube, which terminates in a controlled λ -contractive terminal set. The specific form of tube usedin [9] is homothetic to the terminal set, and in that sensesimilar to the tubes from [4, 6]. However, the purposeof the tube (stabilizing the origin, versus a robustly in-variant set around the origin) is markedly different. Thetheoretical conditions in [9] are in principle not limitedto the homothetic case, but would then require a newterminal cost design.The authors of [4] discuss a possible adaptation of “addi-tive” TMPC to parametrically uncertain systems, how-ever without investigating closed-loop stability. Special-ized TMPC approaches for multiplicatively uncertainsystems are [10,11]: in these works, there is no schedulingvariable that can be measured on-line, classifying themas “robust” rather than “LPV” approaches. An LPV tubeMPC based on the “stabilizing tube” setting of [9] waspresented in [12]: therein, the constructed tubes are ho-mothetic to a λ -contractive terminal set, and the on-lineoptimization of predicted feedback policies is carried outover vertex controls. The cross sections in [10, 11] aremore flexible than in [12], as each hyperplane of the ba-sic shape set is scaled individually. From a different per-spective, the tubes of [10,11] are more restricted becausethe considered feedback policy consists of control ac-tions superimposed upon a pre-determined linear statefeedback, and the tube center has to satisfy a “nominal”trajectory.In this paper, we adopt the basic setting of [12], whichevolved from [9]. An important issue in these works isthat particularly in the LPV case, the λ -contractive setsrequired for stability can be hard to compute and theycan be of very high complexity as the state dimensionincreases. The main contribution of the current paper isa terminal cost computation based on periodic LPV setdynamics, which allows the construction of stabilizingtube-based predictive controllers for LPV systems usingfinite-step (or “periodic”) contractive terminal sets. Such sets can be viewed as a relaxation of λ -contractive sets,and are often easier to compute. The resulting MPCalgorithm requires the on-line solution of a single linearprogram (LP) with a number of variables and constraintswhich grows linearly in the prediction horizon.The paper is organized as follows. In Section 2, we dis-cuss notation, the problem setup, and present the mainconcepts of finite-step contractive sets. The general for-mulation of TMPC with finite-step terminal conditionsis given in Section 3. Suitable parameterizations to en-able efficient implementation are presented in Section 4.Finally, in Section 5, the method is demonstrated on anumerical example.In this extended paper, the proofs of all lemmas, propo-sitions and theorems can be found in the Appendix. The set of nonnegative real numbers is denoted by R + and N denotes the set of nonnegative integers includ-ing zero. Define the index set N [ a,b ] with ≤ a ≤ b as N [ a,b ] := { i ∈ N | a ≤ i ≤ b } . The predicted value ofa variable z at time instant k + i given the informationavailable at time k is denoted by z i | k . In this paper, thenotation (cid:107) x (cid:107) always refers to the ∞ -norm of a vector x ∈ R n , i.e., (cid:107) x (cid:107) = (cid:107) x (cid:107) ∞ = max i ∈ N [1 ,n ] | x i | . Let C n de-note the set of all compact convex subsets of R n . A set X ∈ C n which contains the origin in its non-empty in-terior is called a proper C-set, or PC-set. The convexhull of a set X ⊂ R n is denoted by convh { X } . A subsetof R n is a polyhedron if it is an intersection of finitelymany half-spaces. A polytope is a compact polyhedronand can equivalently be represented as the convex hullof finitely many points in R n . For sets Y, Z ⊂ R n anda scalar α ∈ R let αY = { αy | y ∈ Y } . Minkowski setaddition is defined as Y ⊕ Z = { y + z | y ∈ Y, z ∈ Z } and for a vector v ∈ R n let v ⊕ Y := { v } ⊕ Y . TheHausdorff distance between a nonempty set X ⊂ R n andthe origin is d H ( X ) = d H ( X, { } ) = sup x ∈ X (cid:107) x (cid:107) . Fora vector x ∈ R n , observe that d H ( { x } ) = (cid:107) x (cid:107) . A func-tion f : R + → R + is of class K ∞ when it is continuous,strictly increasing, f (0) = 0 , and lim ξ →∞ f ( ξ ) = ∞ .The gauge function ψ S : R n → R + of a given PC-set S ⊂ R n is ψ S ( x ) = inf { γ | x ∈ γS } [13]. We introducea generalized “set”-gauge function. Definition 1
The set-gauge function Ψ S : C n → R + corresponding to a PC-set S ⊂ R n is Ψ S ( X ) := sup x ∈ X ψ S ( x ) = inf { γ | X ⊆ γS } . ψ S ( · ) and Ψ S ( · ) are K ∞ -bounded. Lemma 2
Let S ⊂ R n be a PC-set. Then, the followingproperties hold:(i) ∃ s , s ∈ K ∞ such that ∀ x ∈ R n : s ( (cid:107) x (cid:107) ) ≤ ψ S ( x ) ≤ s ( (cid:107) x (cid:107) ) ,(ii) ∃ s , s ∈ K ∞ such that ∀ X ∈ C n : s (cid:0) d H ( X ) (cid:1) ≤ Ψ S ( X ) ≤ s (cid:0) d H ( X ) (cid:1) .2.2 Problem Setup We consider a constrained LPV system, represented bythe following state-space equation x ( k + 1) = A ( θ ( k )) x ( k ) + Bu ( k ) , k ∈ N , (1)with x (0) = x , and where u : N → U ⊆ R n u is theinput, x : N → X ⊆ R n x is the state variable, and θ : N → Θ ⊆ R n θ is the scheduling signal. The sets U and X are the input- and state constraint sets, respectively,while Θ is called the scheduling set. The matrix A ( θ ) in(1) is assumed to be a real affine function of θ , i.e., A ( θ ) = A + n θ (cid:88) i =1 θ i A i . (2)We consider systems with a constant B -matrix, becausethen all resulting optimization problems will turn outto be convex. It is, however, possible to transform anysystem with a parameter-varying B into the form (1) byincluding a stable input filter or a pre-integrator [14].The system (1) satisfies the following assumptions. Assumption 3 (i) The values x ( k ) and θ ( k ) can bemeasured at every time k ∈ N . (ii) The system repre-sented by (1) is stabilizable under the constraints ( X , U ) .(iii) The sets X and U are polytopic PC-sets. (iv) Theset Θ is a polytope with q vertices, i.e., Θ = convh { ¯ θ j | j ∈ N [1 ,q ] } . Our principal goal is to design a controller to achieveconstrained regulation of (1) to the origin. To this end,we propose a tube-based approach using stabilizing ter-minal conditions based on finite-step contractive sets.
Definition 4
Let M ≥ be an integer, let λ ∈ [0 , ,let S M = { S , . . . , S M − } be a sequence of PC-sets, anddefine σ ( k ) := k mod M . The PC-set S ⊆ X is calledcontrolled ( M, λ ) -contractive, if there exists a periodiccontrol law u ( k ) = κ σ ( k ) ( x ( k ) , θ ( k )) with κ i : S i × Θ → U , i ∈ N [0 ,M − such that: • ∀ i ∈ N [0 ,M − , ∀ x ∈ S i , ∀ θ ∈ Θ : (3a) A ( θ ) x + Bκ i ( x, θ ) ∈ S i +1 • ∀ x ∈ S M − , ∀ θ ∈ Θ : (3b) A ( θ ) x + Bκ M − ( x, θ ) ∈ λS , • ∀ i ∈ N [0 ,M − : { } ⊂ S i ⊆ X . (3c) It is assumed that κ σ ( · ) ( · , · ) is (i) continuous and (ii)positively homogeneous, i.e., ∀ ( i, x, θ, α ) ∈ N [0 ,M − × R n x × Θ × R + : κ i ( αx, θ ) = ακ i ( x, θ ) . Observe that (3b) means that contraction of S isachieved after M time instances. Sets satisfying theseproperties were first used in reference governor de-sign [15], and can also be interpreted as an instance ofthe positively invariant families from [16]. Finite-stepinvariant sets were used in (nominal) non-linear MPCin [17], and the framework of [18] employs periodi-cally invariant sets in MPC of linear periodic systems.Finite-step invariant ellipsoids for LPV systems wereintroduced in [19], and applied in a stabilizing open-loop min-max algorithm. A practical controller basedon these ellipsoids was given in [20], where only a singlefree control action is being optimized on-line. In the cur-rent paper, sets satisfying Definition 4 will be employedin the construction of a stabilizing tube-based MPC forLPV systems.If S is a polytope the periodic control laws in Defini-tion 4 can always be selected as gain-scheduled vertexcontrollers, because – by convexity – existence of suit-able controls on the vertices of S i × Θ implies existenceof suitable controls over the full sets S i × Θ , i ∈ N [0 ,M − (compare, e.g., [21, Corollaries 4.46 and 7.8]). Finally,the closed-loop set-valued dynamics of (1) under a givenlocal periodic controller κ σ ( · ) ( · , · ) are X ( k + 1) = G ( k, X ( k ) | κ )= (cid:8) A ( θ ) x + Bκ σ ( k ) ( x, θ ) | x ∈ X ( k ) , θ ∈ Θ (cid:9) . (4)The local uncertain closed-loop dynamics (4) are fun-damentally different from those in nominally stabilizingMPC [22]. Hence, constructing a suitable terminal costis a challenge which will be addressed in this paper. The algorithm constructs, at each time instant k ∈ N , aso-called constraint invariant tube [9, 12]. Definition 5
A constraint invariant tube for the con-straint set ( X , U ) ⊂ R n x × R n u is defined as T k := (cid:0)(cid:8) X | k , . . . , X N | k (cid:9) , (cid:8) Π | k , . . . , Π N − | k (cid:9)(cid:1) here X i | k ⊂ R n x , i ∈ N [0 ,N ] are sets and Π i | k : X i | k × Θ i | k → U , i ∈ N [0 ,N − are control laws satisfying thecondition ∀ ( x, θ ) ∈ X i | k × Θ i | k : A ( θ ) x + B Π i | k ( x, θ ) ∈ X i +1 | k ∩ X . The sequence of sets X k is called the statetube, and each set X i | k is called a cross section. Since θ ( k ) can be measured according to Assumption 3,normally we have ∀ k ∈ N : Θ | k = { θ ( k ) } . The rest ofthe sequence Θ k := (cid:8) Θ | k , . . . , Θ N − | k (cid:9) must satisfy thefollowing assumption and could, e.g., describe a knownand bounded rate-of-variation on θ or an “anticipated”approximate future scheduling trajectory, see [12]. Assumption 6 (i) At any two successive time instants,the sequences Θ k +1 and Θ k are related such that ∀ i ∈ N [0 ,N − : Θ i | k +1 ⊆ Θ i +1 | k (continuity). (ii) It holds ∀ ( k, i ) ∈ N × N [0 ,N − : Θ i | k ⊆ Θ (well-posedness). (iii)All sets Θ i | k are polytopes with q vertices, i.e., Θ i | k =convh { ¯ θ ji | k | j ∈ N [1 ,q ] } . The above Assumptions 6.(i) and 6.(ii) are critical inobtaining recursive feasibility of the MPC scheme. As-sumption 6.(iii) is invoked merely to simplify notation.To synthesize tubes satisfying Definition 5 on-line, thecross sections and control laws must be finitely parame-terized. We introduce the 2-tuple of tube parameters p i | k = (cid:16) p Xi | k , p Π i | k (cid:17) ∈ P = P X × P Π ⊆ R q Xp × R q Π p where (cid:0) q Xp , q Π p (cid:1) ∈ N . Each parameter p Xi | k uniquelycharacterizes a cross section X i | k and each p Π i | k definesthe corresponding controller Π i | k . The set P = P X × P Π iscalled the parameterization class. In the sequel, it has tobe understood that any pair (cid:0) X i | k , Π i | k (cid:1) is assumed to beparameterized by a corresponding tube parameter p i | k ∈ P . That is, we can always construct a time-dependentfunction ¯ P ( · , · ) , mapping tube parameters into corre-sponding sets and controllers, such that (cid:0) X i | k , Π i | k (cid:1) =¯ P ( k + i, p i | k ) . A suitable parameterization, which will becovered fully in Section 4, is a periodically time-varyinghomothetic tube where X i | k = z i | k ⊕ α i | k S σ ( k + i ) . Then, p Xi | k = ( α i | k , z i | k ) and S i , i ∈ N [0 ,M − are sets chosenoff-line. The control laws Π i | k are parameterized as thevertex controllers induced by the sets X i | k , so that each p Π i | k corresponds to a finite number of control actions.The tube construction can be formulated as the follow-ing optimization problem, to be solved on-line: V (cid:0) k, x | k , Θ k (cid:1) = min d k ∈ D N − (cid:88) i =0 (cid:96) ( X i | k , Π i | k ) + F k (cid:0) X N | k (cid:1) s.t. ∀ i ∈ N [0 ,N − : ∀ x ∈ X i | k , ∀ θ ∈ Θ i | k : A ( θ ) x + B Π i | k ( x, θ ) ∈ X i +1 | k ∩ X ,X | k = { x | k } , X N | k ⊆ X f | k ⊆ X , (5) where (cid:96) ( · , · ) is the stage cost chosen to meet some desiredobjective, and where the time-varying terminal set X f | k and terminal cost F k ( · ) are selected to guarantee feasibil-ity and stability. The decision variable consists of the se-quence of tube parameters and is therefore a tuple d k = (cid:16) p X | k , p Π0 | k , . . . , p XN − | k , p Π N − | k , p XN | k (cid:17) ∈ D = P N +1 X × P N Π .In (5), the sets X i | k and controllers Π i | k are functions ofthese tube parameters, but this dependency is omittedfrom the notation for brevity. The state measurement attime k is captured in the constraint X | k = { x | k } . Be-cause the value θ ( k ) is measured exactly, the first con-trol law always reduces to a single control action, i.e., Π | k ( x, θ ) = u | k . After solving (5), we set u ( k ) = u | k and repeat the optimization at the next sample. In thesequel, we use a worst-case homogeneous stage cost (cid:96) (cid:0) X i | k , Π i | k (cid:1) = max ( x,θ ) ∈ X i | k × Θ i | k (cid:0) (cid:107) Qx (cid:107) + (cid:107) R Π i | k ( x, θ ) (cid:107) (cid:1) (6)where Q ∈ R n x × n x and R ∈ R n u × n u are tuning parame-ters. Let a sequence S M of polytopic controlled ( M, λ ) -contractive sets satisfying Definition 4 be given. Then,we select the periodically time-varying terminal set as X f | k = S σ ( k + N ) . (7)To guarantee recursive feasibility the following assump-tion, an extended variant of [9, Assumption 7], on thetube parameterization is necessary. Assumption 7
The terminal set and the associated lo-cal controller are “homogeneously parameterizable” in P ,i.e., ∀ ( k, γ ) ∈ N × R + : ∃ p f | k ∈ P such that ¯ P (cid:0) k, p f | k (cid:1) = γ (cid:0) S σ ( k ) , κ σ ( k ) (cid:1) . Later, in Section 4, a concrete parameterization is givenwhich satisfies Assumption 7. Now, recursive feasibilityof (5) can be shown.
Proposition 8
Let S M be a sequence of controlled ( M, λ ) -contractive sets for (1) according to Definition 4,and let the associated closed-loop dynamics G ( · , ·| κ ) beas in (4) . Define the terminal set X f | k as in (7) . Supposethat Assumptions 6 and 7 are satisfied. Then the TMPCdefined by (5) is recursively feasible. To guarantee stability of the MPC scheme, an appropri-ate terminal cost has to be constructed. The first stepis to find a Lyapunov-type function which is monotoni-cally decreasing along the set-valued trajectories of (4).For this, we need the following finite-step decrease prop-erty of the function Ψ S i ( · ) . The abbreviated notations ψ i ( · ) := ψ S i and Ψ i ( · ) := Ψ S i ( · ) are used in the sequel. Lemma 9
Let S M be a sequence of controlled ( M, λ ) -contractive sets for (1) in the sense of Definition 4. De-fine the resulting closed-loop dynamics G ( · , ·| κ ) as in (4) . hen, Ψ σ ( k ) ( · ) satisfies ∀ k ∈ N and ∀ X ⊆ S σ ( k ) : Ψ σ ( k +1) ( G ( k, X | κ )) ≤ (cid:26) Ψ σ ( k ) ( X ) , σ ( k ) ∈ N [0 ,M − ,λ Ψ σ ( k ) ( X ) , σ ( k ) = M − . The above lemma can be exploited to construct aLyapunov-type function enabling the computation of astabilizing terminal cost for (5). The following proposi-tion applies a suitably modified version of the construc-tion of [23, Theorem 20] to sequences of sets, yieldingthe desired function.
Proposition 10
Suppose that the conditions fromLemma 9 are satisfied. Then, the function W ( k, X ) := ( M + ( λ − σ ( k )) Ψ σ ( k ) ( X ) is a Lyapunov-type function for the dynamics (4), i.e., itsatisfies the following properties:(i) ∃ s , s ∈ K ∞ such that ∀ k ∈ N : ∀ X ∈ C n : s (cid:0) d H ( X ) (cid:1) ≤ W ( k, X ) ≤ s (cid:0) d H ( X ) (cid:1) holds,(ii) ∃ (cid:37) ( k ) : N → [0 , such that ∀ k ∈ N : ∀ X ⊆ S σ ( k ) : W ( k + 1 , G ( k, X | κ )) ≤ (cid:37) ( k ) W ( k, X ) ,(iii) ∃ (cid:37) ∈ [0 , such that ∀ k ∈ N : ∀ X ⊆ S σ ( k ) : W ( k + 1 , G ( k, X | κ )) ≤ (cid:37)W ( k, X ) . The next step towards a stability proof is to constructa scaling of W ( · , · ) to obtain a terminal cost for (5). Forall i ∈ N [0 ,M − , let ¯ (cid:96) i = max ( x,u ) ∈ S i × U ( (cid:107) Qx (cid:107) + (cid:107) Ru (cid:107) ) s.t. ∀ θ ∈ Θ : (cid:26) A ( θ ) x + Bu ∈ S i +1 , i ∈ N [0 ,M − ,A ( θ ) x + Bu ∈ λS , i = M − . (8)From Proposition 10 the next Corollary follows directly. Corollary 11
Let ¯ (cid:96) i be as in (8) and define ¯ (cid:96) =max i ∈ N [0 ,M − ¯ (cid:96) i . Define W ( k, X ) and (cid:37) as in Proposi-tion 10 and X f | k as in (7) . Then the function W ( k, X ) := ¯ (cid:96) − (cid:37) W ( k, X ) (9) satisfies ∀ k ∈ N and ∀ X ⊆ S σ ( k ) : W ( k + 1 , G ( k, X | κ )) − W ( k, X ) ≤ − ¯ (cid:96)W ( k, X ) . Furthermore, ∀ k ∈ N : 1 ≤ W (cid:0) k + N, X f | k (cid:1) ≤ M . Before proving asymptotic stability of the TMPCscheme, the following assumptions on the stage cost andvalue function are required.
Assumption 12 (i) Let ( k, p ) ∈ N × P such that ¯ P ( k, p ) = ( X k , Π k ) with Π k : X × Θ → U . Thenthere exists a K ∞ -function s such that s (cid:0) d H ( X k ) (cid:1) ≤ (cid:96) ( X k , Π k ) . (ii) There exist K ∞ -functions s , s such thatfor all k ∈ N and for all x | k ∈ R n x for which (5) is feasi-ble it holds s (cid:0) (cid:107) x | k (cid:107) (cid:1) ≤ V (cid:0) k, x | k , Θ k (cid:1) ≤ s (cid:0) (cid:107) x | k (cid:107) (cid:1) . In Section 4 it is proven that, for a certain choice oftube parameterization, the stage cost (6) and the valuefunction of (5) indeed satisfy the above assumptions.Now we state the main result.
Theorem 13
Suppose that the conditions of Proposi-tion 8 and Assumption 12 are satisfied. Let F k ( · ) := W ( k + N, · ) according to (9) . Then the TMPC definedby (5) asymptotically stabilizes the origin. In this section, it is shown how the general results pre-sented previously can be used in practice by developinga specific parameterization that satisfies Assumptions 7and 12. To satisfy Assumption 7, consider a “periodic”variant on the homothetic parameterization [4, 6, 9] byparameterizing the tube cross sections as X i | k = z i | k ⊕ α i | k S σ ( k + i ) (10)where z i | k ∈ R n x and α i | k ∈ R + are optimized on-line.Thus, each cross section X i | k is considered homotheticto S σ ( k + i ) with center z i | k and scaling α i | k . The sets S i , i ∈ N [0 ,M − are the same as in (7) and they arepolytopes represented by the convex hull of t i vertices as ∀ i ∈ N [0 ,M − : S i = convh (cid:8) ¯ s i , . . . , ¯ s t i i (cid:9) . (11)The associated control laws are parameterized as gain-scheduled vertex controllers, i.e., Π i | k ( x, θ ) = t σ ( k + i ) (cid:88) j =1 ζ j q (cid:88) l =1 η l u ( j,l ) i | k (12)where u ( j,l ) i | k ∈ U are control actions and ζ ∈ R t σ ( k + i ) and η ∈ R q are convex multipliers in the state- andscheduling spaces, respectively. At each prediction timeinstant k + i , the control u ( j,l ) i | k is associated with the j -th vertex of the cross section X i | k and the l -th vertexof the relevant scheduling set (see Assumption 6). Thetube parameters p i | k = (cid:0) p Xi | k , p Π i | k (cid:1) corresponding to thegiven parameterization are p Xi | k = (cid:0) α i | k , z i | k (cid:1) , p Π i | k = (cid:16) u (1 , i | k , . . . , u ( t σ ( k + i ) ,q ) i | k (cid:17) . Because the representation (1) has a constant B -matrix,it is sufficient to verify the existence of the individual5ontrol actions u ( j,l ) i | k to establish the existence of a tubesatisfying Definition 5, i.e., computation of the convexmultipliers ( ζ, η ) is not necessary. With this parameter-ization, the stage cost (6) becomes (cid:96) ( X i | k , Π i | k )= max j ∈ N [1 ,tσ ( k + i )] ,l ∈ N [1 ,q ] (cid:16) (cid:107) Q ¯ x ji | k (cid:107) + (cid:107) Ru ( j,l ) i | k (cid:107) (cid:17) (13)where ¯ x ji | k = z i | k + α i | k ¯ s jσ ( k + i ) . The scaling ¯ (cid:96) in Corol-lary 11 can be efficiently computed as follows. For all ( i, l ) ∈ N [0 ,M − × N [1 ,q ] and all corresponding j ∈ N [1 ,t i ] ,compute the control actions u ( i,j,l ) f = arg min u ∈ U (cid:16) (cid:107) Q ¯ s ji (cid:107) + (cid:107) Ru (cid:107) (cid:17) s.t. (cid:40) A (¯ θ l )¯ s ji + Bu ∈ S i +1 , i ∈ N [0 ,M − ,A (¯ θ l )¯ s ji + Bu ∈ λS , i = M − , to obtain a local periodic vertex control law which isfeasible and asymptotically stabilizing on S M . Then, theconstants ¯ (cid:96) i are directly found by computing ∀ i ∈ N [0 ,M − :¯ (cid:96) i = max j ∈ N [1 ,ti ] ,l ∈ N [1 ,q ] (cid:16) (cid:107) Q ¯ s ji (cid:107) + (cid:107) Ru ( i,j,l ) f (cid:107) (cid:17) , (14)and the next Lemma follows. Lemma 14
Suppose that the tuning parameter Q ∈ R n x × n x is strictly positive definite and therefore of rank n x . Then, the stage cost (13) and value function of (5) satisfy Assumption 12. It has now been shown that the parameterization definedin this section satisfies all the necessary assumptionsfrom Section 3. The next conclusion follows directly.
Corollary 15
The LPV TMPC algorithm with tubeparameterization (10) - (12) is recursively feasible andasymptotically stabilizing. With the choice of stage cost (6) and under the assump-tion that all involved sets are polytopes, the optimiza-tion problem (5) is a linear program (LP). Its complex-ity, in terms of the number of decision variables and con-straints, scales linearly in the prediction horizon N . Asthey are polytopes, each set S i in (11) can be equivalentlyrepresented in a half-space form with r i hyperplanes.Half-space representations of X and U are also assumedto be available with r X and r U hyperplanes, respectively.According to the discussion below Definition 5 and toAssumption 6, let q ( i ) = 1 when i = 0 and q ( i ) = q otherwise. Then, by using the half-space representationsto verify set inclusions similarly to the implementation described in [12], an LP can be formulated which has n d ( k, N ) = 1 + ( N + 1) ( n x + 3) + N (cid:88) i =0 n u q ( i ) t σ ( k + i ) decision variables, and n ineq ( k, N ) = 1+ r σ ( k + N ) t σ ( k + N ) + N (cid:88) i =0 (cid:16)(cid:0) r X q ( i )+ r U q ( i )+ r σ ( k + i +1) q ( i ) + 2 n x + 2 n u q ( i ) (cid:1) t σ ( k + i ) (cid:17) linear inequality constraints. The initial state constraint,finally, gives rise to n x + 1 linear equality constraints.Note that alternative formulations of the LP avoidingthe computation of the hyperplane representations of S i , i ∈ N [0 ,M − , can be constructed, however their exactcomplexities were not studied in the context of this work.The construction of the sequence of finite-step contrac-tive sets S M for an LPV system can be done in sev-eral ways. One can pick an arbitrary PC-set S and findthe smallest M for which a sequence S M exists, using astraightforward extension of the algorithm for the LTIcase from [24]. Due to exponential complexity in M ,this method is only practical when contraction can beachieved for small M . Alternatively, it is possible to firstdetermine any stabilizing controller for (1). Then againwe can choose an arbitrary PC-set S and propagatethis set forwards under the resulting closed-loop dynam-ics until finite-step contraction is achieved, as proposedin [17]. The number of vertices of the sets in the resultingsequence S M grows exponentially in principle, but oftenmany vertices are redundant and can be eliminated usingstandard algorithms: a similar technique was employedin [25] for the stability analysis of switched systems. The approach is now demonstrated on an example. Weconsider a second-order LPV system defined in the state-space form of (1) with two scheduling variables where A = (cid:34) (cid:35) , A = (cid:34) . − . . . (cid:35) ,A = (cid:34) .
23 00 − . (cid:35) , B = (cid:34) (cid:35) and furthermore Θ = (cid:8) θ ∈ R | (cid:107) θ (cid:107) ≤ (cid:9) , U = { u ∈ R | | u | ≤ } , X = (cid:8) x ∈ R | | x | ≤ , | x | ≤ (cid:9) . M, λ ) n d n ineq Avg. (max.) time (1 , .
276 4034 14 (20) [ms] (5 , . The MPC tuning parameters are N = 8 , Q = I , and R = 0 . . This tuning assigns a low weight to the controlinput, leading to a fast response. For simplicity, we set Θ i | k = Θ for all ( k, i ) .A set S was chosen which leads to a sequence S M of (5 , . -contractive sets, as depicted in Figure 2. Theset S was designed with 4 vertices, and all subsequentsets also have 4 vertices except for S , which has 6. Forcomparison, the maximal controlled . -contractive setwas also calculated using the algorithm from [21] and ithas 8 vertices.The relative difference in computational load of the re-sulting TMPC algorithm, based on an LP implementa-tion where both the vertex- and hyperplane represen-tations of the sets were used, is displayed in Table 1.The simulations were carried out on a . GHz IntelCore i7-4790 with 8 GB RAM, running Arch Linux, andusing the Gurobi 7.0.2 LP solver with its default set-tings. Because the complexity of the terminal set in the (5 , . -contractive case is time-dependent, the numberof variables and constraints varies periodically betweenthe numbers shown. To illustrate their linear growth, themaximum number of variables and constraints for the (5 , . -contractive case is calculated as a function of N and shown in Figure 1.An example closed-loop output trajectory of the con-troller with finite-step terminal condition is shown inFigure 3. The scheduling trajectory was generated ran-domly and the initial state was x (0) = (cid:2) − (cid:3) , i.e.,taken at the boundary of the state constraint set. As ex-pected, the system’s state variables are steered to theorigin and input- and state constraints are satisfied. Wealso compare the achieved domains of attraction of thecontroller with the finite-step terminal condition to thatof the controller from [12], which uses the maximal . -contractive terminal set (Figure 4). The feasible set wascalculated for a fixed initial value θ (0) = (cid:2) − (cid:3) (cid:62) . Inthe present case, the reduction in computational loaddue to the lesser complexity of the sets in S M is paid forby a marginally smaller feasible set. The present work has introduced finite-step terminalconditions in tube-based MPC for LPV systems. It wasshown that, under certain assumptions on the tube pa-rameterization, the method is recursively feasible. A new
Fig. 1. Max. number of variables ¯ n d ( N ) = max k n d ( k, N ) and constraints ¯ n ineq ( N ) = max k n ineq ( k, N ) .Fig. 2. Constructed sequence of (5 , . -contractive sets(solid) compared with the maximal . -contractive set(dashed). Fig. 3. Closed-loop state- and input trajectories withfinite-step terminal condition. Fig. 4. Approximate domains of attraction with finite-stepterminal condition (filled) and with maximal contractive ter-minal set (dashed line), for θ (0) = (cid:2) − (cid:3) (cid:62) . The innermostset (solid line) is S from Figure 2, and the outer box repre-sents the state constraints. Lyapunov-like function on periodic sequences of PC-setswas constructed: it was subsequently used to derive aterminal cost, enabling a proof of closed-loop asymp-totic stability. Extension to constrained output referencetracking for LPV systems is a future direction of interest.
Appendix. ProofsProof of Lemma 2.
Denote B ∞ = { x | (cid:107) x (cid:107) ≤ } .Note that ψ B ∞ ( x ) = (cid:107) x (cid:107) . For sets S , S ⊂ R n with S ⊆ S , it holds ψ S ( x ) ≥ ψ S ( x ) for all x ∈ R n [26, Lemma 1]. Because S is a PC set, ∃ a, b ∈ R + such that a B ∞ ⊆ S ⊆ b B ∞ . Thus, ∀ x ∈ R n : b − ψ B ∞ ( x ) ≤ ψ S ( x ) ≤ a − ψ B ∞ ( x ) , i.e.,statement (i) holds with s ( ξ ) = b − ξ and s ( ξ ) = a − ξ .Next, observe that Ψ B ∞ ( X ) = sup x ∈ X (cid:107) x (cid:107) = d H ( X ) .For sets S , S ⊂ R n with S ⊆ S , it similarlyholds Ψ S ( X ) ≥ Ψ S ( X ) for all X ∈ C n . Hence, ∀ X ∈ C n : b − Ψ B ∞ ( X ) ≤ Ψ S ( X ) ≤ a − Ψ B ∞ ( X ) ,i.e., statement (ii) follows with s ( ξ ) = b − ξ and s ( ξ ) = a − ξ . (cid:3) Proof of Proposition 8.
Suppose that (5) is feasibleat time k and let T (cid:63)k = (cid:0)(cid:8) X | k , . . . , X N | k (cid:9) , (cid:8) Π | k , . . . , Π N − | k (cid:9)(cid:1) be the tube resulting from the optimal solution of (5) attime k . By construction, X | k = { x | k } and ∃ γ ∈ [0 ,
1] : X N | k ⊆ γX f | k . Note that γ = 1 would be sufficient here,but keeping it variable simplifies the subsequent stabilityproof of Theorem 13. After applying Π | k to the system,by definition of the terminal set and under Assumption 6a feasible tube at time k + 1 can be explicitly given as T ◦ k +1 = (cid:0)(cid:8) X | k +1 , X | k , . . . , X N − | k , γX f | k ,γG (cid:0) k + N, X f | k | κ (cid:1)(cid:9) , (cid:8) Π | k , . . . , Π N − | k , γκ N (cid:9)(cid:1) , where X | k +1 = { x | k +1 } ⊂ X | k , which impliesfeasibility of Π | k +1 = Π | k . Since (5) only op-timizes over finitely parameterized sets and con-trollers, there must exist parameters (cid:0) p f | k , p f | k +1 (cid:1) ∈ P such that ¯ P (cid:0) k + N, p f | k (cid:1) = γ (cid:0) X f | k , κ N (cid:1) and ¯ P (cid:0) k + N + 1 , p f | k +1 (cid:1) = γ (cid:0) G (cid:0) k + N, X f | k | κ (cid:1) , ∗ (cid:1) where ∗ denotes an irrelevant quantity. This is guaran-teed by Assumption 7, and therefore it follows that (5)is feasible at time k + 1 . (cid:3) Proof of Lemma 9.
Let ∂S denote the boundary ofa set S ⊂ R n . By Definition 4, ∀ ¯ x ∈ ∂S σ ( k ) : G ( k, { ¯ x }| κ ) ∈ (cid:26) S σ ( k +1) , σ ( k ) ∈ N [0 ,M − ,λS σ ( k +1) , σ ( k ) = M − . Now let x ∈ S σ ( k ) . By definition of the gauge function itholds x ∈ ψ σ ( k ) ( x ) ∂S σ ( k ) [13]. Thus, ∃ ¯ x ∈ ∂S σ ( k ) : x = ψ σ ( k ) ( x )¯ x . By homogeneity it follows directly that G ( k, { x }| κ ) = G (cid:0) k, { ψ σ ( k ) ( x )¯ x }| κ (cid:1) = ψ σ ( k ) ( x ) G ( k, { ¯ x }| κ ) and therefore ∀ x ∈ S σ ( k ) : G ( k, { x }| κ ) ∈ (cid:26) ψ σ ( k ) ( x ) S σ ( k +1) , σ ( k ) ∈ N [0 ,M − ,λψ σ ( k ) ( x ) S σ ( k +1) , σ ( k ) = M − . From the above we get that ∀ X ⊆ S σ ( k ) : G ( k, X | κ ) ⊆ (cid:26) sup x ∈ X ψ σ ( k ) ( x ) S σ ( k +1) , σ ( k ) ∈ N [0 ,M − ,λ sup x ∈ X ψ σ ( k ) ( x ) S σ ( k +1) , σ ( k ) = M − and by applying Definition 1 the desired property fol-lows. (cid:3) Proof of Proposition 10.
Since ( M + ( λ − σ ( k )) isa positive number for all k ∈ N , it follows from Lemma 2that ∃ s i , s i ∈ K ∞ for each i ∈ N [0 ,M − such that ∀ X ∈ C n : s σ ( k )6 (cid:0) d H ( X ) (cid:1) ≤ W ( k, X ) ≤ s σ ( k )7 (cid:0) d H ( X ) (cid:1) .As the minimum- and maximum over a finite set of K ∞ -functions is again K ∞ , statement (i) holds with s ( ξ ) =min i ∈ N [0 ,M − s i ( ξ ) and s ( ξ ) = max i ∈ N [0 ,M − s i ( ξ ) . Forthe proof of (ii), consider first that k is such that σ ( k ) ∈ N [0 ,M − . Then by Lemma 9, Ψ σ ( k +1) ( G ( k, X | κ )) ≤ σ ( k ) ( X ) , and therefore W ( k + 1 , G ( k, X | κ ))= ( M + ( λ − σ ( k + 1)) Ψ σ ( k +1) ( G ( k, X | κ )) ≤ ( M + ( λ − σ ( k + 1)) Ψ σ ( k ) ( X )= ( M + ( λ − σ ( k + 1))( M + ( λ − σ ( k )) W ( k, X ) . Next, let k be such that σ ( k ) = M − . Again byLemma 9, Ψ σ ( k +1) ( G ( k, X | κ )) ≤ λ Ψ σ ( k ) ( X ) , so W ( k + 1 , G ( k, X | κ )) = M Ψ ( G ( M − , X | κ )) ≤ λM Ψ M − ( X )= λMλ ( M −
1) + 1 W ( k, X ) . Hence, statement (ii) is satisfied with (cid:37) ( k ) = (cid:40) ( M +( λ − σ ( k +1))( M +( λ − σ ( k )) , σ ( k ) ∈ N [0 ,M − , λMλ ( M − , σ ( k ) = M − , and (iii) follows with (cid:37) = max k ∈ N (cid:37) ( k ) = (cid:37) (0) . (cid:3) Proof of Theorem 13.
Let G f | k ( · ) := G ( k + N, ·| κ ) according to (4). Consider the optimal solution T (cid:63)k andthe feasible, but not necessarily optimal, solution T ◦ k +1 constructed in the proof of Proposition 8. Further, let Θ k and Θ k +1 be two anticipated scheduling sequencessatisfying Assumption 6. By definition of F k ( · ) , it followsthat we can take γ = Ψ σ ( k + N ) (cid:0) X N | k (cid:1) . Substitute thesolutions T (cid:63)k and T ◦ k +1 in the cost function of (5) andcompute the difference between the value functions attime k and time k + 1 to obtain ∆ V k = V (cid:0) k + 1 , x | k +1 , Θ k +1 (cid:1) − V (cid:0) k, x | k , Θ k (cid:1) ≤ (cid:96) (cid:0) X | k +1 , Π | k (cid:1) + γ(cid:96) (cid:0) X f | k , Π f | k (cid:1) + γF k +1 (cid:0) G f | k (cid:0) X f | k (cid:1)(cid:1) − F k (cid:0) X N | k (cid:1) + N − (cid:88) i =2 (cid:96) (cid:0) X i | k , Π i | k (cid:1) − N − (cid:88) i =0 (cid:96) (cid:0) X i | k , Π i | k (cid:1) . Observe that X | k +1 = { x | k +1 } ⊂ X | k , so (cid:96) (cid:0) X | k +1 , Π | k (cid:1) ≤ (cid:96) (cid:0) X | k , Π | k (cid:1) and therefore ∆ V k ≤ N − (cid:88) i =1 (cid:96) (cid:0) X i | k , Π i | k (cid:1) − N − (cid:88) i =0 (cid:96) (cid:0) X i | k , Π i | k (cid:1) + γ(cid:96) (cid:0) X f | k , Π f | k (cid:1) + γF k +1 (cid:0) G f | k (cid:0) X f | k (cid:1)(cid:1) − F k (cid:0) X N | k (cid:1) = − (cid:96) (cid:0) X | k , Π | k (cid:1) + γ(cid:96) (cid:0) X f | k , Π f | k (cid:1) + γF k +1 (cid:0) G f | k (cid:0) X f | k (cid:1)(cid:1) − F k (cid:0) X N | k (cid:1) ≤ − (cid:96) (cid:0) X | k , Π | k (cid:1) + γ ¯ (cid:96) + γF k +1 (cid:0) G f | k (cid:0) X f | k (cid:1)(cid:1) − F k (cid:0) X N | k (cid:1) where the last inequality follows from the definition of ¯ (cid:96) in Corollary 11. Since X N | k ⊆ γX f | k , according to thedefinition of the terminal cost F k (cid:0) X N | k (cid:1) = ¯ (cid:96) − (cid:37) ( M + ( λ − σ ( k + N )) Ψ σ ( k ) (cid:0) X N | k (cid:1) = γ ¯ (cid:96) − (cid:37) ( M + ( λ − σ ( k + N )) Ψ σ ( k ) (cid:0) X f | k (cid:1) = γF k (cid:0) X f | k (cid:1) . Hence, ∆ V k ≤ − (cid:96) (cid:0) X | k , Π | k (cid:1) + γ (cid:0) ¯ (cid:96) + F k +1 (cid:0) G f | k (cid:0) X f | k (cid:1)(cid:1) − F k (cid:0) X f | k (cid:1)(cid:1) ≤ − (cid:96) (cid:0) X | k , Π | k (cid:1) + γ (cid:0) ¯ (cid:96) − ¯ (cid:96)W (cid:0) k + N, X f | k (cid:1)(cid:1) ≤ − (cid:96) (cid:0) X | k , Π | k (cid:1) ≤ − s (cid:0) (cid:107) x | k (cid:107) (cid:1) where the second and third inequalities follow fromCorollary 11, and the last inequality from Assump-tion 12.(i). The fact that V (cid:0) k, x | k , Θ k (cid:1) is monotoni-cally decreasing with rate s (cid:0) (cid:107) x | k (cid:107) (cid:1) is, in conjunctionwith the bounds of Assumption 12.(ii), sufficient toconclude that V ( · , · , · ) is a (time-varying) Lyapunovfunction. Hence, asymptotic stability of the controlledsystem follows [27, Theorem 2]. (cid:3) Proof of Lemma 14.
For any i ∈ N [0 ,M − let p Xi =( z i , α i ) and p Π i = (cid:16) u (1 , i , . . . , u ( t i ,q ) i (cid:17) be arbitrary butfixed tube parameters such that X i = z i ⊕ α i S i and Π i : X i × Θ → U is an associated set-induced vertex con-troller according to (12). Since rank( Q ) = n x , ∃ a i , b i > such that ∀ x ∈ X i : a i (cid:107) x (cid:107) ≤ (cid:107) Qx (cid:107) ≤ b i (cid:107) x (cid:107) [28, Corol-lary II.8]. Thus from (6), (cid:96) ( X i , Π i ) ≥ max x ∈ X i (cid:107) Qx (cid:107) ≥ a i max x ∈ X i (cid:107) x (cid:107) = a i d H ( X i ) . s ( ξ ) = (cid:0) min i ∈ N [0 ,M − a i (cid:1) ξ . It is immedi-ate that a lower bound on V ( x ) is s ( · ) = s ( · ) . Theexistence of a K ∞ -upper bound s ( x ) on V ( x ) canthen be shown proceeding as in [9, Lemma 2]. (cid:3) References [1] J. H. Lee and Z. Yu, “Worst-case Formulations of ModelPredictive Control for Systems with Bounded Parameters,”
Automatica , vol. 33, pp. 763–781, 1997.[2] A. Bemporad, F. Borrelli, and M. Morari, “Min–Max Controlof Constrained Uncertain Discrete-Time Linear Systems,”
IEEE Transactions on Automatic Control , vol. 40, pp. 1234–1236, 2003.[3] S. V. Raković, “Robust Model-Predictive Control,” in
Encyclopedia of Systems and Control . Springer, 2015, pp.1225–1233.[4] W. Langson, I. Chryssochoos, S. V. Raković, and D. Q.Mayne, “Robust model predictive control using tubes,”
Automatica , vol. 40, pp. 125–133, 2004.[5] D. Q. Mayne, M. M. Seron, and S. V. Raković, “Robust modelpredictive control of constrained linear systems with boundeddisturbances,”
Automatica , vol. 41, pp. 219–224, 2005.[6] S. V. Raković, B. Kouvaritakis, R. Findeisen, and M. Cannon,“Homothetic tube model predictive control,”
Automatica ,vol. 48, pp. 1631–1638, 2012.[7] S. V. Raković, B. Kouvaritakis, M. Cannon, C. Panos, andR. Findeisen, “Parameterized tube model predictive control,”
IEEE Transactions on Automatic Control , vol. 57, pp. 2746–2761, 2012.[8] S. V. Raković, W. S. Levine, and B. Açıkmeşe, “Elastic TubeModel Predictive Control,” in
Proc. of the 2016 AmericanControl Conference , 2016, pp. 3594–3599.[9] F. D. Brunner, M. Lazar, and F. Allgöwer, “An ExplicitSolution to Constrained Stabilization via Polytopic Tubes,”in
Proc. of the 52nd IEEE Conference on Decision andControl , 2013, pp. 7721–7727.[10] D. Muñoz-Carpintero, M. Cannon, and B. Kouvaritakis,“Robust MPC strategy with optimized polytopic dynamicsfor linear systems with additive and multiplicativeuncertainty,”
Systems & Control Letters , vol. 81, pp. 34–41,2015.[11] J. Fleming, B. Kouvaritakis, and M. Cannon, “Robust TubeMPC for Linear Systems With Multiplicative Uncertainty,”
IEEE Transactions on Automatic Control , vol. 60, pp. 1087–1092, 2015.[12] J. Hanema, R. Tóth, and M. Lazar, “Tube-based anticipativemodel predictive control for linear parameter-varyingsystems,” in
Proc. of the 55th IEEE Conference on Decisionand Control , 2016, pp. 1458–1463.[13] R. Schneider, “Basic Convexity,” in
Convex Bodies: TheBrunn-Minkowski Theory , 2nd ed. Cambridge UniversityPress, 2013, pp. 1–73.[14] F. Blanchini, S. Miani, and C. Savorgnan, “Stabilityresults for linear parameter varying and switching systems,”
Automatica , vol. 43, pp. 1817–1823, 2007.[15] E. Gilbert and I. Kolmanovsky, “Nonlinear tracking controlin the presence of state and control constraints: A generalizedreference governor,”
Automatica , vol. 38, pp. 2063–2073,2002. [16] Z. Artstein and S. Raković, “Set invariance under outputfeedback: A set-dynamics approach,”
Int. J. of SystemsScience , vol. 42, pp. 539–555, 2011.[17] M. Lazar and V. Spinu, “Finite–step Terminal Ingredientsfor Stabilizing Model Predictive Control,” in
Proc. of the 5thIFAC Conference on Nonlinear Model Predictive Control ,2015, pp. 9–15.[18] R. Gondhalekar and C. N. Jones, “MPC of constraineddiscrete-time linear periodic systems - A framework forasynchronous control: Strong feasibility, stability andoptimality via periodic invariance,”
Automatica , vol. 47, pp.326–333, 2011.[19] Y. I. Lee, M. Cannon, and B. Kouvaritakis, “Extendedinvariance and its use in model predictive control,”
Automatica , vol. 41, pp. 2163–2169, 2005.[20] Y. I. Lee and B. Kouvaritakis, “Constrained robust modelpredictive control based on periodic invariance,”
Automatica ,vol. 42, pp. 2175–2181, 2006.[21] F. Blanchini and S. Miani,
Set-Theoretic Methods in Control ,2nd ed. Birkhäuser, 2015.[22] D. Q. Mayne, J. B. Rawlings, C. V. Rao, and P. O. M.Scokaert, “Constrained model predictive control: Stabilityand optimality,”
Automatica , vol. 36, pp. 789–814, 2000.[23] R. Geiselhart, R. H. Gielen, M. Lazar, and F. R. Wirth,“An alternative converse Lyapunov theorem for discrete-timesystems,”
Systems & Control Letters , vol. 70, pp. 49–59, 2014.[24] N. Athanasopoulos and M. Lazar, “Scalable Stabilization ofLarge Scale Discrete-Time Linear Systems via the 1-Norm,”in
Proc. of the 4th IFAC Workshop on Distributed Estimationand Control in Networked Systems , 2013, pp. 277–284.[25] ——, “Alternative Stability Conditions for Switched DiscreteTime Linear Systems,” in
Proc. of the 19th IFAC WorldCongress , 2014, pp. 6007–6012.[26] S. V. Raković and M. Lazar, “Minkowski terminal costfunctions for MPC,”
Automatica , vol. 48, pp. 2721–2725,2012.[27] D. Aeyels and J. Peuteman, “A New Asymptotic StabilityCriterion for Nonlinear Time-Variant Differential Equations,”
IEEE Transactions on Automatic Control , vol. 43, pp. 968–971, 1998.[28] M. Lazar, “On Infinity norms as Lyapunov Functions:Alternative Necessary and Sufficient Conditions,” in
Proc. ofthe 49th IEEE Conference on Decision and Control , 2010,pp. 5936–5942., 2010,pp. 5936–5942.