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Finite-step Terminal Ingredients for Stabilizing Model Predictive Control
Preprints, 5th IFAC Conference on Nonlinear Model Predictive Preprints, 5th IFAC Conference on Nonlinear Model Predictive Control Preprints, 5th 5th IFAC IFAC Conference Conference on on Nonlinear Nonlinear Model Model Predictive Predictive Preprints, Control September 17-20, 2015. Seville, SpainAvailable online at www.sciencedirect.com Control Control September 17-20, 2015. Seville, Spain September September 17-20, 17-20, 2015. 2015. Seville, Seville, Spain Spain
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Finite–step Terminal Ingredients Finite–step Terminal Ingredients Finite–step Terminal Ingredients Finite–stepModel Terminal Ingredients Stabilizing Predictive Control Stabilizing Model Predictive Control Stabilizing Model Predictive Control Stabilizing Model Predictive Control ∗ ∗∗
Mircea Lazar ∗ Veaceslav Spinu ∗∗ Mircea Lazar ∗∗ Veaceslav Spinu ∗∗ ∗∗ Mircea Mircea Lazar Lazar Veaceslav Veaceslav Spinu Spinu ∗ Eindhoven University of Technology, Eindhoven, Netherlands ∗ of Technology, Eindhoven, Netherlands ∗ Eindhoven University (e-mail: [email protected]) ∗ Eindhoven University of Technology, Eindhoven, of Technology, Eindhoven, Netherlands Netherlands (e-mail: [email protected]) ∗∗Eindhoven University of Technology, Eindhoven, Netherlands (e-mail: [email protected]) ∗∗ Eindhoven University (e-mail: [email protected]) of Technology, Eindhoven, Netherlands ∗∗ Eindhoven University (e-mail: [email protected]) ∗∗ Eindhoven University of Eindhoven, Eindhoven University of Technology, Technology, Eindhoven, Netherlands Netherlands (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper proposes a novel construction of the terminal cost and terminal set in Abstract: This paper proposes a novel of the terminal cost and and terminal set in stabilizing model predictive control thatconstruction uses finite–step Lyapunov finite–step Abstract: This proposes a construction of terminal cost terminal set Abstract: This paper paper proposes a novel novel construction of the the terminalfunctions cost and and and terminal set in in stabilizing model predictive control that uses finite–step Lyapunov functions finite–step invariant sets. Thispredictive construction results inuses a periodic terminal set constraint associated with a stabilizing model control that finite–step Lyapunov functions and finite–step stabilizing model predictive control that uses finite–step Lyapunov functions and finite–step invariant sets. This construction results in a periodic constraint associated with finite sequence of terminal sets, out of which none is terminal required set to be invariant. We argue thataaa invariant sets. construction results in terminal set constraint associated with invariant sets. This This construction results in aa periodic periodic terminal set constraint associated with finite sequence of terminal sets, out of which none is required to be invariant. We argue constructing such sets is easier and more scalable compared to invariant sets, while a comparable finite sequence sequence of of terminal terminal sets, sets, out out of of which which none none is is required required to to be be invariant. invariant. We We argue argue that that finite that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same prediction horizon. In the one stepwhile caseaathe proposed constructing such sets is easier and more scalable compared to invariant sets, comparable constructing such sets is easier and more scalable compared to invariant sets, while comparable region of attraction isreduce obtained for theto same horizon. In the one step case the stability naturally the prediction standard terminal cost setproposed stability region of attraction obtained for same prediction horizon. the one step proposed region of conditions attraction is isreduce obtained for the theto same prediction horizon. In In theand oneconstraint step case case the the proposed stability conditions naturally the standard terminal cost and constraint set stability conditions. stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions reduce naturally to the standard terminal cost and constraint set stability stability conditions. conditions. conditions. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Predictive control, Stabilization methods, Lyapunov function, Invariants. Keywords: Predictive Predictive control, Stabilization Stabilization methods, Lyapunov Lyapunov function, Invariants. Invariants. Keywords: Keywords: Predictive control, control, Stabilization methods, methods, Lyapunov function, function, Invariants. 1. INTRODUCTION structing terminal equality constraints and enlarging the 1. INTRODUCTION INTRODUCTION structingofterminal terminal equality constraints and enlarging the domain attraction in nonlinear MPCand wasenlarging proposedthe in 1. structing equality constraints 1. INTRODUCTION structing terminal equality constraints and enlarging the domain of attraction in nonlinear MPC was proposed in and Teel, 2013). Usage of predicted trajectories domain of attraction in nonlinear MPC was proposed This paper considers the problem of designing model pre- (Fagiano domain ofand attraction in nonlinear MPC wastrajectories proposed in in (Fagiano Teel, 2013). Usage of predicted in This paper considers the problem of designing model prewith scaled and translated invariant sets for (Fagiano and Teel, 2013). Usage of predicted trajectories in dictive controllers (MPC) with anof adesigning priori guarantee of combination This paper considers the problem model pre(Fagiano and Teel, 2013). Usage of predicted trajectories in This paper considers the problem of designing model precombination with scaled and translated invariant sets for dictive controllers (MPC) with an a priori guarantee of constructing a larger invariant terminal set and enlarging combination with scaled and translated invariant sets for recursive feasibility(MPC) and asymptotic in the Lyadictive controllers with an aastability priori guarantee of combination with scaled and translated invariant sets for dictive controllers (MPC) with an priori guarantee of constructing a larger invariant terminal set and enlarging recursive feasibility and asymptotic asymptotic stability inproblem the LyaLyadomain of attraction in linear MPC proposed constructing aa larger invariant terminal set and punov sense. The established solution to thisin is the recursive feasibility and stability the constructing larger invariant terminal set was and enlarging enlarging recursive feasibility and asymptotic stability inproblem the Lyathe(Brunner domain of of attraction in linear MPC was proposed punov sense. The and established solution to this thiswhich is in et al., 2013). While the above collection of the domain attraction in linear MPC was proposed the terminal cost constraint set design, origipunov sense. The established solution to problem is domain of attraction in linear MPC was proposed punov sense. cost The and established solution to thiswhich problem is the in (Brunner et al., 2013). While the above collection of the terminal constraint set design, origiapproaches is not meant to be exhaustive, it is enough in (Brunner et al., 2013). While the above collection of nated in (Michalska and Mayne, 1993). Other stabilizing the terminal cost and constraint set design, which origi(Brunnerisetnot al.,meant 2013).toWhile the above it collection of the terminal cost and constraint set design, which origi- in approaches be exhaustive, is enough nated in (Michalska and Mayne, 1993). Other stabilizing concluding thatmeant to a to large extent, the conservatism approaches is not be exhaustive, it is enough design methods for model predictive control (MPC) in- for nated in (Michalska and Mayne, 1993). Other stabilizing approaches is not meant to be exhaustive, it is enough nated in (Michalska and Mayne, 1993). Other stabilizing for concluding that to a large extent, the conservatism designstabilization methods for for model model predictive predictive control constraints (MPC) inin- forthe terminal that cost and set design stems from concluding to large the clude without terminal design methods control (MPC) for concluding to aaconstraint large extent, extent, the conservatism conservatism designstabilization methods forconditions model predictive control constraints (MPC) in- of of the the terminal that cost and constraint set design design stems from from clude conditions without terminal the shortcomings of the available methods for constructing of terminal cost and constraint set stems (Limon et al., 2006; Grune, 2012), sub-optimal stabilizclude stabilization conditions without terminal constraints of the terminal cost andavailable constraint set design stems from clude stabilization conditions without terminal constraints the shortcomings of the methods for constructing (Limon et al., 2006; Grune, 2012), sub-optimal stabilizLyapunov functions and (controlled) invariant the shortcomings of available methods for ing MPCet (Scokaert al., 1999), MPC (control) (Limon al., Grune, 2012), sub-optimal the shortcomings of the the available methods for constructing constructing (Limon al., 2006; 2006; et 2012),Lyapunov-based sub-optimal stabilizstabiliz(control) Lyapunov functions and (controlled) invariant ing MPC MPCet et (Scokaert etGrune, al., 1999), 1999), Lyapunov-based MPC sets. (control) Lyapunov functions and (controlled) invariant (Mhaskar al., 2006; Lazar, 2009). For a recent survey ing (Scokaert et al., Lyapunov-based MPC (control) Lyapunov functions and (controlled) invariant ing MPC et (Scokaert et Lazar, al., 1999), Lyapunov-based MPC sets. (Mhaskar al., 2006; 2009). For a recent survey sets. on stabilizing MPC the reader is referred to (Mayne, 2013) (Mhaskar et al., 2006; Lazar, 2009). For aa recent survey sets. (Mhaskar et al., 2006; Lazar, 2009). For recent survey Recently, a finite-step (also called finite-time) relaxation of on stabilizing stabilizing MPC the reader reader is referred referred to topics (Mayne, 2013) and for a recent survey on MPC research we2013) refer Recently, a finite-step (also called finite-time) relaxation of on MPC the is to (Mayne, on stabilizing MPC the reader is referred to topics (Mayne, 2013) these concepts was investigated (Lazar et al., 2013) for Recently, aa finite-step (also finite-time) relaxation of and for a recent survey on MPC research we refer Recently, finite-step (also called calledin finite-time) relaxation of to (Mayne, 2014).survey It is worth to mention that apart from and for aa recent on MPC research topics we refer these concepts concepts was investigated investigated in (Lazar et al., al., 2013) for and for recent survey on MPC research topics we refer homogeneous nonlinear systems and in (Geiselhart et al., these was in (Lazar et 2013) for to (Mayne, 2014). It is worth to mention that apart from these concepts was investigated in (Lazar et al., 2013) for these theoretical in practice one commonly to 2014). It to that from homogeneous nonlinear systems andgeneral in (Geiselhart (Geiselhart et sysal., to (Mayne, (Mayne, 2014).methods, It is is worth worth to mention mention that apart apart relies from 2014; Gielen and Lazar,systems 2015) for nonlinearet homogeneous nonlinear and in al., these theoretical methods, in practice practice one commonly relies homogeneous nonlinear systems andgeneral in (Geiselhart et sysal., on thetheoretical conjecturemethods, that a sufficiently long prediction horizon these in one commonly relies 2014; Gielen and Lazar, 2015) for nonlinear these theoretical methods, in practice one commonly relies tems. finite-step Lyapunov function is a candidate 2014; Gielen and 2015) for nonlinear syson the the stability conjecture that a sufficiently sufficiently long prediction prediction horizon 2014; A Gielen and Lazar, Lazar, 2015) for general general nonlinear Lyasysyields and makes use of so-called rate models for on conjecture that a long horizon tems. A finite-step Lyapunov function is a candidate Lyaon the stability conjecture that a sufficiently long prediction horizon i.e.,Lyapunov satisfyingfunction class K∞ and lower tems. Afunction, finite-step is candidate Lyayields and makes use of so-called so-called rate models for punov tems. finite-step function is aaupper candidate Lyaintroducing integral actionuse in MPC (Wang,rate 2004). yields stability and makes of models for punovAfunction, function, i.e.,Lyapunov satisfying class K K and lower ∞ upper yields stability and makes use of so-called rate models for bounds, that is required to decrease after a finite number punov i.e., satisfying class upper lower ∞ upper and introducing integral action in MPC (Wang, 2004). punov function, i.e., satisfying class K and lower ∞ a finite number introducing integral action in (Wang, 2004). that is required decrease after introducing integral action in MPC MPC to (Wang, 2004). of discrete time instants, to instead of at every time instant. bounds, that is to decrease after aa finite number Efforts within the MPC community improve the design bounds, bounds, that is required required to decrease after finite number of discrete time instants, instead of at every time instant. Efforts within the MPC community to improve the design One of the benefits of these relaxed concepts is that, under of discrete time instants, instead of at every time instant. of the terminal cost and constraint set include several apEfforts within the MPC community to improve the design of discrete time instants, instead of at every time instant. Efforts within the MPC community to improve the design One of the benefits of these relaxed concepts is that, under of the the terminal terminal cost and andcontractive constraint condition set include includeonseveral several ap- the assumption of exponential stability of the equilibrium, One of the benefits of these relaxed concepts is that, under proaches. A finite–step the MPC of cost constraint set apOne of the benefits of these relaxed concepts is that, under of the terminal cost andcontractive constraint condition set includeonseveral ap- the assumption of exponential stability of the equilibrium, proaches. A finite–step the MPC any candidate Lyapunov function and of any proper C-set the assumption of exponential stability the equilibrium, cost was proposed already in (Alamir, 2006)on tothe achieve proaches. A finite–step contractive condition MPC the assumption of exponential stability of the equilibrium, proaches. A finite–step contractive condition on the MPC candidate Lyapunov function and any proper C-set cost was proposed proposed already inscheme (Alamir, 2006) to achieve achieve any (Lazar et al., 2013) are a finite-step Lyapunov function any candidate Lyapunov function and any proper C-set acost stabilizing nonlinear MPC for continuous–time was already in (Alamir, 2006) to candidate Lyapunov function andLyapunov any proper C-set cost was proposed already inscheme (Alamir, 2006) to achieve any (Lazar et al., 2013) are a finite-step function a stabilizing nonlinear MPC for continuous–time and a finite-step (contractive) invariant set, respectively, (Lazar et al., 2013) are a finite-step Lyapunov function systems without explicitly using stabilizing constraints in aa stabilizing nonlinear MPC scheme for continuous–time (Lazar et al., 2013) are a finite-step Lyapunov function stabilizing nonlinear MPC scheme for continuous–time and a finite-step (contractive) invariant set, respectively, systems without explicitly using The stabilizing constraints in and some finite number of steps.invariant An explicit, lower aa finite-step (contractive) set, respectively, the MPCwithout optimization problem. notion constraints of a invariant systems explicitly using stabilizing in and finite-step (contractive) invariant set,feasible respectively, systems without explicitly using The stabilizing constraints in for for some finite number of steps. An explicit, feasible lower the MPC optimization problem. notion of a 2010) invariant bound on this number was established from exponential for some finite number of steps. An explicit, feasible lower family of sets was proposed in (Rakovic et al., for the MPC optimization problem. The notion of a invariant for some finite number ofwas steps. An explicit, feasible lower the MPC optimization problem. The notion of a 2010) invariant bound on this number established from exponential family of sets was proposed in (Rakovic et al., for stability constants in (Gielen and Lazar, 2015) and an bound on this number was established from exponential reducing conservatism of invariant set construction methfamily of sets was proposed in (Rakovic et al., 2010) for bound on this number was established from exponential family of sets was proposed in (Rakovic et al., 2010) for constants in (Gielen and Lazar, 2015) and an reducing conservatism systems. of invariant invariant set construction construction meth- stability explicit construction of a Lyapunov function from a finite– stability constants in (Gielen and Lazar, 2015) and an ods for decentralized Methods for constructing reducing conservatism of set methstabilityconstruction constants inof (Gielen and function Lazar, 2015) and an reducing conservatism systems. of invariant set construction meth- explicit a Lyapunov from a finite– ods for decentralized Methods for constructing step Lyapunov function was established in (Geiselhart explicit construction of a Lyapunov function from a finite– periodically invariant terminal sets were introduced in ods for decentralized systems. Methods for constructing explicit construction of a Lyapunov function from a finite– ods for decentralized systems. Methods forintroduced constructing step Lyapunov function was established in (Geiselhart periodically invariant terminal sets were in al.,Lyapunov 2014). step function (Gondhalekar and Jones, 2011) sets (set were iterates based conperiodically invariant terminal introduced in step Lyapunov function was was established established in in (Geiselhart (Geiselhart periodically invariant terminal introduced in et et al., 2014). (Gondhalekar and Jones, Jones, 2011) sets (set were iterates based conet al., 2014). struction of polyhedral periodically invariant sets) and (Gondhalekar and 2011) (set iterates based conet al., 2014). (Gondhalekar and Jones, 2011) (set iterates based conThis paper proposes a novel construction of the terminal struction of polyhedral periodically invariant sets) sets and (Bohm et of al.,polyhedral 2012) (ellipsoidal periodically invariant struction periodically invariant sets) and This paper proposes aainnovel construction of the terminal struction of polyhedral periodically invariant sets) and cost and terminal set stabilizing model predictive conThis paper proposes novel construction of the (Bohmon et quadratic al., 2012) 2012) (ellipsoidal (ellipsoidal periodically invariant sets This paper proposes a novel construction of the terminal terminal based periodic Lyapunov functions). These (Bohm et al., periodically invariant sets cost and terminal set in stabilizing model predictive con(Bohm et al., 2012) (ellipsoidal periodically invariant sets trol that uses finite–step Lyapunov functions and finite– cost and terminal set in stabilizing model predictive conbased on quadratic periodic Lyapunov functions). These cost and terminal set in stabilizing model predictive conworks are quadratic motivatedperiodic by stabilizing MPC for periodically based on Lyapunov functions). These trol that uses finite–step Lyapunov functions and finite– based on quadratic periodic Lyapunov functions). These step invariant sets. This construction results in a periodic trol that uses finite–step Lyapunov functions and finite– works are motivated by stabilizing MPC for periodically trol that uses sets. finite–step Lyapunov functions and finite– time-varying systems. Usage of multiple equilibria for conworks are motivated by stabilizing MPC for periodically step invariant This construction results in a periodic works are motivated stabilizing MPC for periodically time-varying systems.by Usage of multiple multiple equilibria for concon- step step invariant invariant sets. sets. This This construction construction results results in in a a periodic periodic time-varying time-varying systems. systems. Usage Usage of of multiple equilibria equilibria for for con-
Copyright © 2015, 2015 IFAC 9 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 9 Copyright © 2015 IFAC 9 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 9 Control. 10.1016/j.ifacol.2015.11.256
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If M = 1 then the above theorem yields the original global and regional (with S ⊆ X a invariant set) Lyapunov theorem with V a Lyapunov function. In this case the K– boundedness property of G is not used in the proof. As such, Theorem 4 can be invoked to establish KL–stability when both classical Lyapunov functions and finite–step Lyapunov functions are used, in combination with a corresponding (finite–step) invariant set S.
terminal set constraint associated with a finite sequence of terminal sets, out of which none is required to be invariant. We argue that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same prediction horizon. In the one step case the proposed stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions. 2. PRELIMINARIES
3. MAIN RESULTS
Consider a discrete–time dynamical system x(k + 1) = G(x(k)), k ∈ N, (1) n n where G : R → R is the system dynamics. We assume that G(0) = 0 and we are interested in characterizing stability of the zero equilibrium. Let G0 (x) := x and Gi (x) := G ◦ Gi−1 (x) for all i ∈ N ∩ [1, ∞). Definition 1. The origin of the dynamical system (1) is KL–stable in X ⊆ Rn if there exists a KL–function β : Rn × Rn → Rn such that �x(k)� ≤ β(�x(0)�, k) for all x(0) ∈ X and for all k ∈ N. Definition 2. A set S ⊂ X is called a finite–step contractive set for the dynamical system (1) with respect to the safe set X if there exists a pair (M, λ) ∈ {N∩[1, ∞)}×[0, 1] such that for all x(0) ∈ S it holds that (i) x(i) ∈ X for all i ∈ N ∩ [1, M − 1] and (ii) x(M ) ∈ S. A finite–step contractive set will also be called a (M, λ)–contractive set.
To set-up the MPC problem, consider a discrete–time dynamical system with inputs, i.e., x(k + 1) = g(x(k), u(k)), n
where g : R × R
m
k ∈ N,
(3)
n
→ R is the system dynamics.
Sometimes we consider u(k) to have an explicit, state– feedback form u(k) = h(x(k)). In this case let G(x(k)) := g(x(k), h(x(k))) denote the corresponding closed–loop dynamics. We will use the notation xi|k to denote the predicted state at time i ∈ {0, . . . , N }, given measured state x0|k := x(k), and similarly ui|k to denote the predicted input at time i, while setting u(k) := u0|k . Here N denotes the prediction horizon. The prediction model is a copy of (3), i.e., xi+1|k = g(xi|k , ui|k ),
Observe that (M, 1)–contractive sets are periodically invariant sets and (1, 1)–contractive sets are standard invariant sets. To analyze stability of the dynamical system (1) we will make use of finite–step Lyapunov functions. Definition 3. A function V : Rn → R+ that satisfies the following properties: ∃α1 , α2 ∈ K∞ : (2a) α1 (�x�) ≤ V (x) ≤ α2 (�x�), ∀x ∈ Rn ; ∃M ∈ N ∩ [1, ∞) and ρ ∈ K, ρ < id :
i ∈ {0, . . . , N − 1}, k ∈ N.
Defining uk := {u0|k , . . . , uN −1|k } yields the MPC optimization problem: N −1 l(xi|k , ui|k ) min J(x(k), uk ) := min L(xN |k ) + uk
uk
i=0
(4a) subject to constraints: xi+1|k = g(xi|k , ui|k ), i ∈ {0, . . . , N − 1}, (4b) (4c) xi|k ∈ X, i ∈ {1, . . . , N − 1}, ui|k ∈ U, i ∈ {0, . . . , N − 1}, (4d) xN |k ∈ X (k). (4e)
(2b) V (GM (x)) ≤ ρ(V (x)), ∀x ∈ S ⊆ Rn , is called a finite–step Lyapunov function in S for the dynamical system (1). A finite–step LF will also be called a M –step LF.
In equation (4) L : Rn → R+ denotes the terminal cost, l : Rn × Rm → R+ denotes the stage cost, X denotes the set where the predicted states are constrained, U denotes the set where the predicted inputs are constrained, and X (k) denotes the set where the terminal predicted state xN |k is constrained to lie.As usually done in terminal cost and constraint set MPC, we will employ an explicit, known state–feedback control law h : Rn → Rm and the closed– loop dynamics G(x) = g(x, h(x)). We will also make an abuse of notation for simplicity and define:
The following result was proven in (Geiselhart et al., 2014), the global version (S = X = Rn ), while the regional version (S ⊆ X ⊂ Rn ) was proven in (Lazar et al., 2013) and (Bobiti and Lazar, 2014). Theorem 4. Finite–step Lyapunov Theorem Suppose that the set {0} ⊂ S ⊆ X is a (M, 1)–contractive set for the dynamical system (1) with respect to the safe set X ⊆ Rn . Additionally, suppose that 1 the dynamics G is K–bounded, i.e., �G(x)� ≤ ω(�x�) for some ω ∈ K and all x ∈ X ⊆ Rn . Then if the dynamical system (1) admits a M –step Lyapunov function V in S, its origin is KL–stable in S.
h(S) := {y ∈ Rm : y = h(x), x ∈ S}, for any subset S of Rn and similarly for any other maps with arguments in a vector space, such as G.
Observe that the corresponding admissible domain of i attraction will be given by ∪M−1 {∪ G (x)} ⊆ X, x∈S i=0 which is a standard invariant set.
For any M ∈ N, M ≥ 1, consider a finite sequence of subsets of Rn , i.e., {S0 , . . . , SM−1 } ,
1
This property is a direct consequence of the KL–stability definition (Gielen and Lazar, 2015).
with the following properties: 10
(5)
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function V (x) is a Lyapunov function in Q(N, ∪M−1 i=0 Si ) and hence, the closed–loop MPC dynamical system is KL– stable in Q(N, ∪M−1 i=0 Si ) (for any N ∈ N ∩ [1, ∞)).
G(Si ) ⊆ Si+1 , i ∈ {0, . . . , M − 2}, (6a) (6b) G(SM−1 ) ⊆ S0 , (6c) h(Si ) ⊆ U, i ∈ {0, . . . , M − 1}, (6d) {0} ⊂ Si ⊆ X. i ∈ {0, . . . , M − 1}. Observe that for any i ∈ {0, . . . , M − 1}, the set Si is M –step invariant for the dynamics G(x) = g(x, h(x)) and constraints admissible with respect to input and state constraints. Notice also that if M = 1 we obtain that S0 is a standard invariant and constraints admissible set.
Proof. (i) For any N ∈ N ∩ [1, ∞) and k ∈ N there exists an index p ∈ {0, . . . , M − 1} such that x∗N |k ∈ X (k) = S(j+k)modM = Sp , where the ∗ denotes optimality. Then at time k + 1 it holds that X (k + 1) = S(j+k+1)modM = Sp+1
The terminal set is defined as follows: initial condition X (0) can be chosen arbitrarily within the sequence (5), as long as the corresponding optimization problem (4) is feasible for the initial state x(0). Suppose that X (0) = Sj , for some j ∈ {0, . . . , M − 1}. Then X (1) = Sj+1 , X (2) = Sj+2 and so on, until X (M − 1 − j) = SM−1 and then X (M − j) = S0 and so on, until X (M − 1) = Sj . Based on this assignment we can obtain the generic terminal set definition, for all k ∈ N: ∀j ∈ {0, . . . , M − 1} : X (0) = Sj ⇒ X (k) := S(j+k)modM , (7) where mod stands for the standard modulo operation. Next, we define the terminal cost: M−1 L(x) := F (Gi (x)),
if p < M −1 or X (k+1) = S(j+k+1)modM = S0 if p = M −1. In either case consider the shifted sequence of inputs uk+1 = {u∗1|k , . . . , u∗N −1|k , h(x∗N |k )}
(ii) For any k ∈ N and x(k) ∈ Q(N, ∪M−1 i=0 Si ) following the usual steps (Mayne, 2013) in proving stability of MPC and keeping in mind that uk+1 defined as in (10) is suboptimal but feasible, yields: V (x(k + 1)) − V (x(k)) = J(x(k + 1), u∗k+1 ) − J(x(k), u∗k ) ≤ J(x(k + 1), uk+1 ) − J(x(k), u∗k )
i=0
where F : Rn → R+ satisfies the property: F (GM (x)) − F (x) + l(x, h(x)) ≤ 0,
∀x ∈
= L(xN |k+1 ) + Si . (9)
i=0
− L(x∗N |k ) −
Under the standard assumption that l(x, h(x)) ≥ α3 (�x�) for all x ∈ Rn and some α3 ∈ K∞ , F becomes a finite–step Lyapunov function for dynamics G(x) with respect to the M−1 invariant set i=0 Si . Indeed,
N −1
i=0 N −1
l(xi|k+1 , ui|k+1 ) l(x∗i|k , u∗i|k )
i=0
≤ L(G(x∗N |k )) − L(x∗N |k ) + l(x∗N |k , h(x∗N |k )) − l(x∗0|k , u∗0|k ).
F (GM (x)) − F (x) ≤ −α3 (�x�) is equivalent with property (2b), as proven in (Geiselhart et al., 2014). Observe also that by taking M = 1 we obtain that F is a standard Lyapunov function that satisfies the standard terminal cost condition in stabilizing MPC. Theorem 5. Consider the closed–loop MPC dynamical system (3) with the control input u(k) = u∗0|k , with u∗k denoting the optimum of Problem 4, and with X (k) as defined in (7), for all k ∈ N.
Then directly from (8) and property (9) we obtain: V (x(k + 1)) − V (x(k)) ≤ F (GM (x∗N |k )) − F (x∗N |k ) + l(x∗N |k , h(x∗N |k )) − l(x∗0|k , u∗0|k ) ≤ −α3 (�x(k)�). Above we have used that ui|k+1 = u∗i+1|k for i ∈ {0, . . . , N − 2} and xi|k+1 = x∗i+1|k for i ∈ {0, . . . , N − 1}. The statement (ii) then follows from Theorem 4.
(i) Suppose that Problem 4 is feasible at time k ∈ N for x0|k = x(k). Then Problem 4 is feasible at time k + 1 for x0|k+1 = x(k + 1) = g(x(k), u∗0|k ) and thus, the N –step Q(N, ∪M−1 i=0 Si )
(10)
and observe that by property (6) it holds that h(x∗N |k ) ∈ U and G(x∗N |k ) ∈ X (k+1) and thus, xN |k+1 ∈ X (k+1) holds, which completes the proof of statement (i).
(8)
M−1
11
In what follows we will suggest a simple design method for obtaining a terminal cost and sequence of terminal sets that satisfy (9) and (6) for linear systems. Consider F (x) = x⊤ P x, l(x, u) = x⊤ Qx+u⊤ Ru, g(x, u) = Ax+Bu and h(x) = Kx for P, Q, R ≻ 0 and (A, B, K) of suitable dimensions. Lemma 6. Suppose that the following inequality holds:
N
:= {x ∈ X : ∃u ∈ U : controllable set S , {x(0) = x, x(1), . . . , x(N − 1)} ∈ XN } is x(N ) ∈ ∪M−1 i i=0 an invariant set. (ii) Suppose that 2 the MPC value function V (x) := J(x, u∗ ) satisfies property (2a) for some α1 , α2 ∈ K∞ for all x ∈ X. Furthermore, suppose that the terminal cost as defined in (8) satisfies property (9) and the sequence of terminal sets {S0 , . . . , SM−1 } used to generate X (k) at every k ∈ N satisfies properties (6). Then the MPC value
(A + BK)⊤ P (A + BK) − P + (Q + K ⊤ RK) ≤ 0. Then (9) holds. Proof. Pre and post multiplying the above inequality with (A + BK)⊤ and (A + BK) successively up to M − 1 times and summing up yields
2
This is typically insured by lower bounding the stage cost and lower and upper bounding the terminal cost with class K∞ functions (Mayne, 2013).
11
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((A + BK)M )⊤ P (A + BK)M − P +
xi+1|k xi|k ui|k xN −j|k
M−1
((A + BK)i )⊤ (Q + K ⊤ RK)(A + BK)i
i=0
≤0 and hence ((A + BK)M )⊤ P (A + BK)M − P + (Q + K ⊤ RK) ≤ 0, which completes the proof.
= g(xi|k , ui|k ), i ∈ {0, . . . , N − 1}, ∈ X, i ∈ {1, . . . , N } \ {N − j} ∈ U, i ∈ {0, . . . , N − 1} ∈ S0 .
(11a) (11b) (11c) (11d)
This modified problem still retains the recursive feasibility property under the same assumptions, as briefly outline next. By applying an appropriately shifted suboptimal input sequence based on repeatedly applying the local control law h(x) from prediction time N − j onwards, it can be easily seen that if the problem is feasible at time k := M i, i ∈ N, it is also feasible at times k + j, where j ∈ {1, . . . , M − 1}. Recall that at time k + M − 1, x∗N −M+1|k+M−1 ∈ S0 . By shifting the sequence and applying uN −M+j|k+M = h(xN −M+j|k+M ) one obtains that xN −M+j+1|k+M ∈ Sj+1 mod M for all j ∈ {0, . . . , M − 1}. Consequently, xN |k+M ∈ S0 .
Therefore, the P matrix corresponding to the LQR control law can readily be used to obtain the terminal cost (8). However, other designs are possible such as F (x) := Fj (x) = x⊤ Pj x and h(x) = hj (x) = Kj x if x ∈ Sj , which makes Fj a finite–step (control) Lyapunov function in Sj for each j ∈ {0, . . . , M − 1}. This leads to a periodically time varying terminal cost and control law which can still be synthesized (for periodically time-varying linear systems with period M ) via an appropriate set of LMIs and reduces conservatism considerable compared to the periodic control Lyapunov function of (Bohm et al., 2012).
Observe that the predicted states xN −j|k in (11a) do not contribute to the recursive feasibility guarantee and can be dropped. Therefore a simplified optimization problem can be constructed N −j−1 min J(x(k), uk ) := min L(xN −j|k ) + l(xi|k , ui|k )
The proposed MPC design can still be solved explicitly for linear (and PWA) systems, only that an explicit solution will be obtained for each terminal set Sj and corresponding terminal cost. However, the complexity of each problem will be less in general, as shown in the illustrative example, because each terminal set Sj will have a much lower complexity than a standard invariant set.
uk
uk
i=0
(12a)
subject to constraints: xi+1|k = g(xi|k , ui|k ), i ∈ {0, . . . , N − j − 1}, (12b) (12c) xi|k ∈ X, i ∈ {1, . . . , N − j − 1} ui|k ∈ U, i ∈ {0, . . . , N − j − 1} (12d) xN −j|k ∈ S0 , (12e)
Next, a simple design of the sequence of sets (5) is proposed for the linear case: take S0 as a subset of X such that it contains the origin and KS0 ⊆ U. Generate the set sequence Sj = (A + BK)Sj−1 iteratively starting from j = 1 and scale S0 such that Sj ⊆ X and KSj ⊆ U for all j. Stop when Sj ⊆ S0 . The iterations will convergence after a finite number of steps if ρ(A + BK) < 1 − ε for any ε > 0 (Gielen and Lazar, 2015). If polytopic sets are used, each iteration requires only basic polytopic operations in vertex representation. A similar algorithm can be derived using backward propagation and hyperplane representations.
where j = k mod M . Note that in this setting N must be greater than M . The corresponding stability theorem is stated next. Theorem 7. Consider the closed–loop MPC dynamical system (3) with the control input u(k) = u∗0|k , with u∗k denoting the optimum of Problem 12 for all k ∈ N.
The finite–step terminal ingredients proposed in this paper, which allow for a simple computation of the sequence of terminal sets, open up the application of stabilizing and thus reliable, terminal cost and constraint MPC to systems of sensible state and input space dimensions. A sequence of terminal sets and a control law that satisfy properties (6) for a linear system with 1000 states and 100 inputs was computed in (Athanasopoulos and Lazar, 2013), where even more complex iterative algorithms and control laws were used.
(i) Suppose that Problem 12 is feasible at time k ∈ N for x0|k = x(k). Then Problem 12 is feasible at time k + 1 for x0|k+1 = x(k + 1) = g(x(k), u∗0|k ) and thus, the N –step controllable set Q(N, S0 ) := {x ∈ X : ∃u ∈ UN : x(N ) ∈ S0 , {x(0) = x, x(1), . . . , x(N − 1)} ∈ XN } is an invariant set. (ii) Suppose that the MPC value function V (x) := J(x, u∗ ) satisfies property (2a) for some α1 , α2 ∈ K∞ for all x ∈ X. Furthermore, suppose that the terminal cost as defined in (8) satisfies property (9) and the sequence of terminal sets {S0 , . . . , SM−1 } satisfies properties (6). Then the MPC value function V (x) is a finite–step Lyapunov function in Q(N, S0 ) and hence, the closed–loop MPC dynamical system is KL–stable in Q(N, S0 ) (for any N ∈ N ∩ [2M, ∞)).
Next, we will discuss a variant of the MPC problem (4) which requires using a single terminal set, but a timevarying terminal constraint time instant. The advantage of this method is that it will not require storage of more terminal sets, but it may require a longer prediction horizon to reach the same domain of attraction. 3.1 Time-varying terminal constraint time instant
Proof. (i) The recursive feasibility proof follows from recursive feasibility of Problem 11.
Consider a MPC optimization problem with the same optimization criterion as in (4), however, the constraints (4c) and (4e) change, i.e., for a given j := k mod M
(ii) Let V (x) = J(x(k), u∗k ), where u∗k is the optimal input sequence at time k. Then it holds that 12
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feasible, as given x∗N −j|k = xN −j−M|k+M ∈ S0 , it follows that Gi (x∗N −j|k ) ∈ Si mod M , ∀i ∈ N.
V (x(k)) − V (x(k + M )) = L(x∗N −j|k )
+
N −j−1
l(x∗i|k , u∗i|k )−
i=0 N −j−1
L(x∗N −j|k+M ) −
(13)
Establishing stability for the case when N ≥ M + 1 is the subject of ongoing work.
l(x∗i|k+M , u∗i|k+M ).
4. ILLUSTRATIVE EXAMPLE
i=0
Since N ≥ 2M , N − j > M for all j ∈ {0, . . . , M − 1}. Then it holds that: V (x(k)) − V (x(k + M )) = L(x∗N −j|k ) +
M−1
The main results of this paper are illustrated on the simple example of the discrete-time double integrator: 2 Ts 1 Ts x(k + 1) = Ax(k) + Bu(k), A = , B= 2 , 0 1 Ts where Ts = 0.4. For simplicity, the local controller h(x) := Kx, where K is the LQR state-feedback, which is designed by taking Q = I2 and R = 100. The terminal cost is defined with F (x) := x⊤ P x, with P the solution of the discrete time algebraic Riccati equation, and clearly satisfies (9).
l(x∗i|k , u∗i|k )+
i=0
N −j−1
l(x∗i|k , u∗i|k )−
(14)
i=M
L(x∗N −j|k+M ) −
N −j−M−1
l(x∗i|k+M , u∗i|k+M )−
i=0
N −j−1
Three controllers are compared in what follows, i.e., standard MPC with invariant terminal set constraint and F (x) as the terminal cost, the construction from (4), and the controller from (12). The prediction horizon of 6 steps was taken for the first two formulations. The set S0 was taken as an unit-box centered in origin and scaled such that Sj = (A + BK)Sj−1 , j ∈ {1, . . . , M − 1}, are control admissible and stay within the state constraints. It was found that in this particular case M = 19. The prediction horizon for the controller in (12) is taken M + 1.
l(x∗i|k+M , u∗i|k+M ).
i=N −j−M
By applying the shifted control sequence [u∗M|k , . . . , u∗N −j−1|k , h(x∗N −j|k ), . . . , h(GM−1 (x∗N −j|k ))] one obtains V (x(k)) − V (x(k + M )) ≥ L(x∗N −j|k ) − L(xN −j|k+M ) + N −j−1
l(x∗i|k , u∗i|k ) −
i=M M−1
M−1
l(x∗i|k , u∗i|k )+
i=0 N −j−M−1
The explicit solution for each of the methods has been computed employing the MPT3 (Herceg et al., 2013) and YALMIP (Lofberg, 2004) toolboxes. Note that, for each of the methods proposed in this paper M controllers are computed. As far as the complexity goes, the standard solution results in 57 regions, and the largest controller partition for (4) and (12) contain 37 and 49 regions, and the smallest 13 and 11, respectively. The larger complexity of the standard solution is mainly driven by the complexity of the terminal set which has 26 hyperplanes, compared to 4 hyperplanes of Sj , for all j. Moreover, the relatively large number of regions in (12) can be explained by the long prediction horizon, i.e., 20 compared to the prediction horizon of 6 used in the other two control approaches. Nevertheless, the solution in (12) can be build in the Dynamic Programming fashion from j = M − 1 to j = 0, which ultimately can save memory space compared to the approach in (4).
l(x∗i+M|k , u∗i+M|k )−
i=0
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k ))) =
(15)
i=0
L(x∗N −j|k ) − L(xN −j|k+M ) +
M−1
l(x∗i|k , u∗i|k )−
i=0
M−1
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k ))).
i=0
By substituting (9) into (15) it follows that V (x(k)) − V (x(k + M )) ≥ M−1
i=0 M−1
13
l(x∗i|k , u∗i|k )−
The domain of attraction (DoA), and the terminal set for the standard MPC are shown in Fig. 1. For comparison, the same sets are shown in Fig. 2 and Fig. 3, for the methods in (4) and (12). Note that the DoA and the terminal sets are shown for all M controllers in the same figure for the latter two methods. It can be observed how the DoA changes for each of the M controllers. In Fig. 2 this change is driven by the change of the terminal set Sj , whereas in Fig. 3 the terminal set remains the same, i.e., S0 , but the prediction horizon is reduced, leading to a smaller DoA for controllers corresponding to j close to M.
(16) (F (Gi (x∗N −j|k )) − F (Gi+M (x∗N −j|k ))−
i=0
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k )))),
where last two rows are positive, and consequently M−1 l(xi|k , u∗i|k ) ≥ α3 (�x(k)�). V (x(k)) − V (x(k + M )) ≥ i=0
(17)
Hence V is a finite-step Lyapunov function for the closedloop system and the statement then follows directly from Theorem 4. Note that the shifted control sequence is
The DoA for all controllers is the same, because of the linear MPC setup and conservative choice of the local 13
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Fig. 1. Domain of attraction (cyan) and the terminal set (blue) of the standard MPC formulation.
Fig. 3. Domain of attraction (cyan) and the terminal set (yellow ) of the method in (12). The terminal set of the standard MPC is depicted in blue.
[x]
1
1 0 −1 0
2
4
6
8
10
12
14
16
18
0
2
4
6
8
10
12
14
16
18
[x]
2
1 0 −1
u
1
Fig. 2. Domain of attraction (cyan) and the terminal set (yellow ) of the method in (4).
Standard Sequenced T. sets Single T. set
0 −1 0
controller. Moreover, the prediction horizon of 5 appears to be sufficient to cover the maximal possible DoA for this system. Nevertheless, by varying the local controller throughout the sequence of the method in (4) can potentially yield a larger DoA than the standard MPC formulation and reduce the controller design time and complexity. The solution in (12) is somewhat more conservative when it comes to the DoA size, but can reduce the complexity of the design and implementation of the controller even further.
2
4
6
8
10
12
14
16
18
Fig. 4. Time response of the system in closed–loop with each of the MPC controllers for all vertexes of the feasible set. prediction horizon. In the one step case the proposed stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions. 5.1 Remarks for future developments
To conclude the time evolution of the state and input trajectories from the vertexes of the DoA are shown in Fig. 4. Clearly, the approach from (4) forces faster convergence via the sequence of the terminal sets, whereas (12) behaves fairly similarly to the standard MPC setup.
The finite–step terminal ingredients for stabilizing MPC proposed in this paper have an appeal towards the design of robust MPC schemes and tracking MPC schemes. In the robust MPC case, a sequence of terminal sets which are finite–step robustly invariant is much more easy to construct and it will help in avoiding the complexity of working with approximations of the minimal RPI set.
5. CONCLUSIONS This paper proposed a novel construction of the terminal cost and terminal set in stabilizing model predictive control that used finite–step Lyapunov functions and finite– step invariant sets. This construction resulted in a periodic terminal set constraint associated with a finite sequence of terminal sets, out of which none was required to be invariant. We argue that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same
In the tracking MPC case, for periodic reference signals, which are common in motion control systems, electromechanical, power electronics and robotics applications, a sequence of terminal sets out of which none is invariant can be constructed so that convergence to the periodic reference is attained. In fact, the proposed sequence of terminal sets is a natural solution when the desired operating set–point is given by a limit cycle. 14
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ity and optimality via periodic invariance. Automatica, 47(2), 326 – 333. Grune, L. (2012). NMPC without Terminal Constraints. In Proceedings of 4th IFAC Conference on Nonlinear Model Predictive Control, 1–13. Leeuwenhorst, Netherlands. Herceg, M., Kvasnica, M., Jones, C., and Morari, M. (2013). Multi-Parametric Toolbox 3.0. In Proc. of the European Control Conference, 502–510. Z¨ urich, Switzerland. http://control.ee.ethz.ch/ mpt. Lazar, M. (2009). Flexible control Lyapunov functions. In American Control Conference, 2009. ACC ’09., 102– 107. Lazar, M., Doban, A.I., and Athanasopoulos, N. (2013). On stability analysis of discrete–time homogeneous dynamics. In Proceedings of 17th International Conference on Systems Theory, Control and Computing, 297–305. Sinaia, Romania. Limon, D., Alamo, T., Salas, F., and Camacho, E.F. (2006). On the stability of constrained MPC without terminal constraint. Automatic Control, IEEE Transactions on, 51(5), 832–836. Lofberg, J. (2004). Yalmip : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan. URL http://users.isy.liu.se/johanl/yalmip. Mayne, D.Q. (2013). An apologia for stabilising terminal conditions in model predictive control. International Journal of Control, 86(11), 2090 – 2095. Mayne, D.Q. (2014). Model predictive control: Recent developments and future promise. Automatica, 50(12), 2967 – 2986. Mhaskar, P., El-Farra, N.H., and Christofides, P.D. (2006). Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Systems & Control Letters, 55(8), 650 – 659. Michalska, H. and Mayne, D.Q. (1993). Robust receding horizon control of constrained nonlinear systems. Automatic Control, IEEE Transactions on, 38(11), 1623– 1633. Rakovic, S.V., Kern, B., and Findeisen, R. (2010). Practical Set Invariance for Decentralized Discrete Time Systems. In 49th IEEE Conf. on Decision and Control. Scokaert, P., Mayne, D., and Rawlings, J. (1999). Suboptimal model predictive control (feasibility implies stability). Automatic Control, IEEE Transactions on, 44(3), 648–654. Wang, L. (2004). A tutorial on model predictive control using velocity form model. Developments of Chemical Engineering and Mineral Processing, 12, 573–614.
In terms of industrial applications, the proposed method for designing the terminal cost L can be used to determine the length of the prediction horizon required for achieving stability in the unconstrained case, when a specific terminal weight matrix is not used. As preferred in industry, the terminal weight can be kept equal to the stage weight matrix, i.e., F (x) = x⊤ Qx, and then it suffices to find the number M for which the finite–step decrease property (9) holds. Such an M always exists for any Q ≻ 0 chosen by the practitioner, under the assumption of exponential stability of the local dynamics (A + BK), according to the results in (Gielen and Lazar, 2015). The resulting cost will be: M−1 ⊤ J(x(k), uk ) = Gi (xN |k ) Q Gi (xN |k ) +
i=0 N −1
15
⊤ x⊤ i|k Qxi|k + ui|k Rui|k ,
i=0
and it can be regarded as an extension of the stage cost to a prediction horizon of N + M , where the last M control actions are not free, but they are set equal to the local controller h(x), which can represent a classical frequency– domain controller. Such a cost is expected to preserve the performance of the local controller compared with using an aggressive terminal weight matrix P and it is more likely to be accepted by practitioners. REFERENCES Alamir, M. (ed.) (2006). Stabilization of Nonlinear Systems Using Receding-horizon Control Schemes. Lecture Notes in Control and Information Sciences. SpringerVerlag, London. Athanasopoulos, N. and Lazar, M. (2013). Scalable Stabilization of Large–Scale Discrete–Time Linear Systems via the 1–norm. In 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems, 277–284. Koblenz, Germany. Bobiti, R.V. and Lazar, M. (2014). On the computation of Lyapunov functions for discrete–time nonlinear systems. In 18th International Conference on System Theory, Control and Computing, 93 – 98. Sinaia, Romania. Bohm, C., Lazar, M., and Allgower, F. (2012). Stability of periodically time-varying systems: Periodic lyapunov functions. Automatica, 48(10), 2663 – 2669. Brunner, F.D., Lazar, M., and Allgower, F. (2013). Stabilizing linear model predictive control: On the enlargement of the terminal set. In Control Conference (ECC), 2013 European, 511–517. Zurich, Zwitzerland. Fagiano, L. and Teel, A.R. (2013). Generalized terminal state constraint for model predictive control. Automatica, 49(9), 2622 – 2631. Geiselhart, R., Gielen, R.H., Lazar, M., and Wirth, F.R. (2014). An alternative converse Lyapunov theorem for discrete-time systems. Systems and Control Letters, 70(0), 49 – 59. Gielen, R.H. and Lazar, M. (2015). On stability analysis methods for large-scale discrete-time systems. Automatica, 55(0), 66 – 72. Gondhalekar, R. and Jones, C.N. (2011). MPC of constrained discrete-time linear periodic systems a framework for asynchronous control: Strong feasibility, stabil15
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for for for for
Finite–step Terminal Ingredients Finite–step Terminal Ingredients Finite–step Terminal Ingredients Finite–stepModel Terminal Ingredients Stabilizing Predictive Control Stabilizing Model Predictive Control Stabilizing Model Predictive Control Stabilizing Model Predictive Control ∗ ∗∗
Mircea Lazar ∗ Veaceslav Spinu ∗∗ Mircea Lazar ∗∗ Veaceslav Spinu ∗∗ ∗∗ Mircea Mircea Lazar Lazar Veaceslav Veaceslav Spinu Spinu ∗ Eindhoven University of Technology, Eindhoven, Netherlands ∗ of Technology, Eindhoven, Netherlands ∗ Eindhoven University (e-mail: [email protected]) ∗ Eindhoven University of Technology, Eindhoven, of Technology, Eindhoven, Netherlands Netherlands (e-mail: [email protected]) ∗∗Eindhoven University of Technology, Eindhoven, Netherlands (e-mail: [email protected]) ∗∗ Eindhoven University (e-mail: [email protected]) of Technology, Eindhoven, Netherlands ∗∗ Eindhoven University (e-mail: [email protected]) ∗∗ Eindhoven University of Eindhoven, Eindhoven University of Technology, Technology, Eindhoven, Netherlands Netherlands (e-mail: [email protected]) (e-mail: [email protected]) (e-mail: [email protected]) Abstract: This paper proposes a novel construction of the terminal cost and terminal set in Abstract: This paper proposes a novel of the terminal cost and and terminal set in stabilizing model predictive control thatconstruction uses finite–step Lyapunov finite–step Abstract: This proposes a construction of terminal cost terminal set Abstract: This paper paper proposes a novel novel construction of the the terminalfunctions cost and and and terminal set in in stabilizing model predictive control that uses finite–step Lyapunov functions finite–step invariant sets. Thispredictive construction results inuses a periodic terminal set constraint associated with a stabilizing model control that finite–step Lyapunov functions and finite–step stabilizing model predictive control that uses finite–step Lyapunov functions and finite–step invariant sets. This construction results in a periodic constraint associated with finite sequence of terminal sets, out of which none is terminal required set to be invariant. We argue thataaa invariant sets. construction results in terminal set constraint associated with invariant sets. This This construction results in aa periodic periodic terminal set constraint associated with finite sequence of terminal sets, out of which none is required to be invariant. We argue constructing such sets is easier and more scalable compared to invariant sets, while a comparable finite sequence sequence of of terminal terminal sets, sets, out out of of which which none none is is required required to to be be invariant. invariant. We We argue argue that that finite that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same prediction horizon. In the one stepwhile caseaathe proposed constructing such sets is easier and more scalable compared to invariant sets, comparable constructing such sets is easier and more scalable compared to invariant sets, while comparable region of attraction isreduce obtained for theto same horizon. In the one step case the stability naturally the prediction standard terminal cost setproposed stability region of attraction obtained for same prediction horizon. the one step proposed region of conditions attraction is isreduce obtained for the theto same prediction horizon. In In theand oneconstraint step case case the the proposed stability conditions naturally the standard terminal cost and constraint set stability conditions. stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions reduce naturally to the standard terminal cost and constraint set stability stability conditions. conditions. conditions. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: Predictive control, Stabilization methods, Lyapunov function, Invariants. Keywords: Predictive Predictive control, Stabilization Stabilization methods, Lyapunov Lyapunov function, Invariants. Invariants. Keywords: Keywords: Predictive control, control, Stabilization methods, methods, Lyapunov function, function, Invariants. 1. INTRODUCTION structing terminal equality constraints and enlarging the 1. INTRODUCTION INTRODUCTION structingofterminal terminal equality constraints and enlarging the domain attraction in nonlinear MPCand wasenlarging proposedthe in 1. structing equality constraints 1. INTRODUCTION structing terminal equality constraints and enlarging the domain of attraction in nonlinear MPC was proposed in and Teel, 2013). Usage of predicted trajectories domain of attraction in nonlinear MPC was proposed This paper considers the problem of designing model pre- (Fagiano domain ofand attraction in nonlinear MPC wastrajectories proposed in in (Fagiano Teel, 2013). Usage of predicted in This paper considers the problem of designing model prewith scaled and translated invariant sets for (Fagiano and Teel, 2013). Usage of predicted trajectories in dictive controllers (MPC) with anof adesigning priori guarantee of combination This paper considers the problem model pre(Fagiano and Teel, 2013). Usage of predicted trajectories in This paper considers the problem of designing model precombination with scaled and translated invariant sets for dictive controllers (MPC) with an a priori guarantee of constructing a larger invariant terminal set and enlarging combination with scaled and translated invariant sets for recursive feasibility(MPC) and asymptotic in the Lyadictive controllers with an aastability priori guarantee of combination with scaled and translated invariant sets for dictive controllers (MPC) with an priori guarantee of constructing a larger invariant terminal set and enlarging recursive feasibility and asymptotic asymptotic stability inproblem the LyaLyadomain of attraction in linear MPC proposed constructing aa larger invariant terminal set and punov sense. The established solution to thisin is the recursive feasibility and stability the constructing larger invariant terminal set was and enlarging enlarging recursive feasibility and asymptotic stability inproblem the Lyathe(Brunner domain of of attraction in linear MPC was proposed punov sense. The and established solution to this thiswhich is in et al., 2013). While the above collection of the domain attraction in linear MPC was proposed the terminal cost constraint set design, origipunov sense. The established solution to problem is domain of attraction in linear MPC was proposed punov sense. cost The and established solution to thiswhich problem is the in (Brunner et al., 2013). While the above collection of the terminal constraint set design, origiapproaches is not meant to be exhaustive, it is enough in (Brunner et al., 2013). While the above collection of nated in (Michalska and Mayne, 1993). Other stabilizing the terminal cost and constraint set design, which origi(Brunnerisetnot al.,meant 2013).toWhile the above it collection of the terminal cost and constraint set design, which origi- in approaches be exhaustive, is enough nated in (Michalska and Mayne, 1993). Other stabilizing concluding thatmeant to a to large extent, the conservatism approaches is not be exhaustive, it is enough design methods for model predictive control (MPC) in- for nated in (Michalska and Mayne, 1993). Other stabilizing approaches is not meant to be exhaustive, it is enough nated in (Michalska and Mayne, 1993). Other stabilizing for concluding that to a large extent, the conservatism designstabilization methods for for model model predictive predictive control constraints (MPC) inin- forthe terminal that cost and set design stems from concluding to large the clude without terminal design methods control (MPC) for concluding to aaconstraint large extent, extent, the conservatism conservatism designstabilization methods forconditions model predictive control constraints (MPC) in- of of the the terminal that cost and constraint set design design stems from from clude conditions without terminal the shortcomings of the available methods for constructing of terminal cost and constraint set stems (Limon et al., 2006; Grune, 2012), sub-optimal stabilizclude stabilization conditions without terminal constraints of the terminal cost andavailable constraint set design stems from clude stabilization conditions without terminal constraints the shortcomings of the methods for constructing (Limon et al., 2006; Grune, 2012), sub-optimal stabilizLyapunov functions and (controlled) invariant the shortcomings of available methods for ing MPCet (Scokaert al., 1999), MPC (control) (Limon al., Grune, 2012), sub-optimal the shortcomings of the the available methods for constructing constructing (Limon al., 2006; 2006; et 2012),Lyapunov-based sub-optimal stabilizstabiliz(control) Lyapunov functions and (controlled) invariant ing MPC MPCet et (Scokaert etGrune, al., 1999), 1999), Lyapunov-based MPC sets. (control) Lyapunov functions and (controlled) invariant (Mhaskar al., 2006; Lazar, 2009). For a recent survey ing (Scokaert et al., Lyapunov-based MPC (control) Lyapunov functions and (controlled) invariant ing MPC et (Scokaert et Lazar, al., 1999), Lyapunov-based MPC sets. (Mhaskar al., 2006; 2009). For a recent survey sets. on stabilizing MPC the reader is referred to (Mayne, 2013) (Mhaskar et al., 2006; Lazar, 2009). For aa recent survey sets. (Mhaskar et al., 2006; Lazar, 2009). For recent survey Recently, a finite-step (also called finite-time) relaxation of on stabilizing stabilizing MPC the reader reader is referred referred to topics (Mayne, 2013) and for a recent survey on MPC research we2013) refer Recently, a finite-step (also called finite-time) relaxation of on MPC the is to (Mayne, on stabilizing MPC the reader is referred to topics (Mayne, 2013) these concepts was investigated (Lazar et al., 2013) for Recently, aa finite-step (also finite-time) relaxation of and for a recent survey on MPC research we refer Recently, finite-step (also called calledin finite-time) relaxation of to (Mayne, 2014).survey It is worth to mention that apart from and for aa recent on MPC research topics we refer these concepts concepts was investigated investigated in (Lazar et al., al., 2013) for and for recent survey on MPC research topics we refer homogeneous nonlinear systems and in (Geiselhart et al., these was in (Lazar et 2013) for to (Mayne, 2014). It is worth to mention that apart from these concepts was investigated in (Lazar et al., 2013) for these theoretical in practice one commonly to 2014). It to that from homogeneous nonlinear systems andgeneral in (Geiselhart (Geiselhart et sysal., to (Mayne, (Mayne, 2014).methods, It is is worth worth to mention mention that apart apart relies from 2014; Gielen and Lazar,systems 2015) for nonlinearet homogeneous nonlinear and in al., these theoretical methods, in practice practice one commonly relies homogeneous nonlinear systems andgeneral in (Geiselhart et sysal., on thetheoretical conjecturemethods, that a sufficiently long prediction horizon these in one commonly relies 2014; Gielen and Lazar, 2015) for nonlinear these theoretical methods, in practice one commonly relies tems. finite-step Lyapunov function is a candidate 2014; Gielen and 2015) for nonlinear syson the the stability conjecture that a sufficiently sufficiently long prediction prediction horizon 2014; A Gielen and Lazar, Lazar, 2015) for general general nonlinear Lyasysyields and makes use of so-called rate models for on conjecture that a long horizon tems. A finite-step Lyapunov function is a candidate Lyaon the stability conjecture that a sufficiently long prediction horizon i.e.,Lyapunov satisfyingfunction class K∞ and lower tems. Afunction, finite-step is candidate Lyayields and makes use of so-called so-called rate models for punov tems. finite-step function is aaupper candidate Lyaintroducing integral actionuse in MPC (Wang,rate 2004). yields stability and makes of models for punovAfunction, function, i.e.,Lyapunov satisfying class K K and lower ∞ upper yields stability and makes use of so-called rate models for bounds, that is required to decrease after a finite number punov i.e., satisfying class upper lower ∞ upper and introducing integral action in MPC (Wang, 2004). punov function, i.e., satisfying class K and lower ∞ a finite number introducing integral action in (Wang, 2004). that is required decrease after introducing integral action in MPC MPC to (Wang, 2004). of discrete time instants, to instead of at every time instant. bounds, that is to decrease after aa finite number Efforts within the MPC community improve the design bounds, bounds, that is required required to decrease after finite number of discrete time instants, instead of at every time instant. Efforts within the MPC community to improve the design One of the benefits of these relaxed concepts is that, under of discrete time instants, instead of at every time instant. of the terminal cost and constraint set include several apEfforts within the MPC community to improve the design of discrete time instants, instead of at every time instant. Efforts within the MPC community to improve the design One of the benefits of these relaxed concepts is that, under of the the terminal terminal cost and andcontractive constraint condition set include includeonseveral several ap- the assumption of exponential stability of the equilibrium, One of the benefits of these relaxed concepts is that, under proaches. A finite–step the MPC of cost constraint set apOne of the benefits of these relaxed concepts is that, under of the terminal cost andcontractive constraint condition set includeonseveral ap- the assumption of exponential stability of the equilibrium, proaches. A finite–step the MPC any candidate Lyapunov function and of any proper C-set the assumption of exponential stability the equilibrium, cost was proposed already in (Alamir, 2006)on tothe achieve proaches. A finite–step contractive condition MPC the assumption of exponential stability of the equilibrium, proaches. A finite–step contractive condition on the MPC candidate Lyapunov function and any proper C-set cost was proposed proposed already inscheme (Alamir, 2006) to achieve achieve any (Lazar et al., 2013) are a finite-step Lyapunov function any candidate Lyapunov function and any proper C-set acost stabilizing nonlinear MPC for continuous–time was already in (Alamir, 2006) to candidate Lyapunov function andLyapunov any proper C-set cost was proposed already inscheme (Alamir, 2006) to achieve any (Lazar et al., 2013) are a finite-step function a stabilizing nonlinear MPC for continuous–time and a finite-step (contractive) invariant set, respectively, (Lazar et al., 2013) are a finite-step Lyapunov function systems without explicitly using stabilizing constraints in aa stabilizing nonlinear MPC scheme for continuous–time (Lazar et al., 2013) are a finite-step Lyapunov function stabilizing nonlinear MPC scheme for continuous–time and a finite-step (contractive) invariant set, respectively, systems without explicitly using The stabilizing constraints in and some finite number of steps.invariant An explicit, lower aa finite-step (contractive) set, respectively, the MPCwithout optimization problem. notion constraints of a invariant systems explicitly using stabilizing in and finite-step (contractive) invariant set,feasible respectively, systems without explicitly using The stabilizing constraints in for for some finite number of steps. An explicit, feasible lower the MPC optimization problem. notion of a 2010) invariant bound on this number was established from exponential for some finite number of steps. An explicit, feasible lower family of sets was proposed in (Rakovic et al., for the MPC optimization problem. The notion of a invariant for some finite number ofwas steps. An explicit, feasible lower the MPC optimization problem. The notion of a 2010) invariant bound on this number established from exponential family of sets was proposed in (Rakovic et al., for stability constants in (Gielen and Lazar, 2015) and an bound on this number was established from exponential reducing conservatism of invariant set construction methfamily of sets was proposed in (Rakovic et al., 2010) for bound on this number was established from exponential family of sets was proposed in (Rakovic et al., 2010) for constants in (Gielen and Lazar, 2015) and an reducing conservatism systems. of invariant invariant set construction construction meth- stability explicit construction of a Lyapunov function from a finite– stability constants in (Gielen and Lazar, 2015) and an ods for decentralized Methods for constructing reducing conservatism of set methstabilityconstruction constants inof (Gielen and function Lazar, 2015) and an reducing conservatism systems. of invariant set construction meth- explicit a Lyapunov from a finite– ods for decentralized Methods for constructing step Lyapunov function was established in (Geiselhart explicit construction of a Lyapunov function from a finite– periodically invariant terminal sets were introduced in ods for decentralized systems. Methods for constructing explicit construction of a Lyapunov function from a finite– ods for decentralized systems. Methods forintroduced constructing step Lyapunov function was established in (Geiselhart periodically invariant terminal sets were in al.,Lyapunov 2014). step function (Gondhalekar and Jones, 2011) sets (set were iterates based conperiodically invariant terminal introduced in step Lyapunov function was was established established in in (Geiselhart (Geiselhart periodically invariant terminal introduced in et et al., 2014). (Gondhalekar and Jones, Jones, 2011) sets (set were iterates based conet al., 2014). struction of polyhedral periodically invariant sets) and (Gondhalekar and 2011) (set iterates based conet al., 2014). (Gondhalekar and Jones, 2011) (set iterates based conThis paper proposes a novel construction of the terminal struction of polyhedral periodically invariant sets) sets and (Bohm et of al.,polyhedral 2012) (ellipsoidal periodically invariant struction periodically invariant sets) and This paper proposes aainnovel construction of the terminal struction of polyhedral periodically invariant sets) and cost and terminal set stabilizing model predictive conThis paper proposes novel construction of the (Bohmon et quadratic al., 2012) 2012) (ellipsoidal (ellipsoidal periodically invariant sets This paper proposes a novel construction of the terminal terminal based periodic Lyapunov functions). These (Bohm et al., periodically invariant sets cost and terminal set in stabilizing model predictive con(Bohm et al., 2012) (ellipsoidal periodically invariant sets trol that uses finite–step Lyapunov functions and finite– cost and terminal set in stabilizing model predictive conbased on quadratic periodic Lyapunov functions). These cost and terminal set in stabilizing model predictive conworks are quadratic motivatedperiodic by stabilizing MPC for periodically based on Lyapunov functions). These trol that uses finite–step Lyapunov functions and finite– based on quadratic periodic Lyapunov functions). These step invariant sets. This construction results in a periodic trol that uses finite–step Lyapunov functions and finite– works are motivated by stabilizing MPC for periodically trol that uses sets. finite–step Lyapunov functions and finite– time-varying systems. Usage of multiple equilibria for conworks are motivated by stabilizing MPC for periodically step invariant This construction results in a periodic works are motivated stabilizing MPC for periodically time-varying systems.by Usage of multiple multiple equilibria for concon- step step invariant invariant sets. sets. This This construction construction results results in in a a periodic periodic time-varying time-varying systems. systems. Usage Usage of of multiple equilibria equilibria for for con-
Copyright © 2015, 2015 IFAC 9 Hosting by Elsevier Ltd. All rights reserved. 2405-8963 © IFAC (International Federation of Automatic Control) Copyright © 2015 IFAC 9 Copyright © 2015 IFAC 9 Peer review under responsibility of International Federation of Automatic Copyright © 2015 IFAC 9 Control. 10.1016/j.ifacol.2015.11.256
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If M = 1 then the above theorem yields the original global and regional (with S ⊆ X a invariant set) Lyapunov theorem with V a Lyapunov function. In this case the K– boundedness property of G is not used in the proof. As such, Theorem 4 can be invoked to establish KL–stability when both classical Lyapunov functions and finite–step Lyapunov functions are used, in combination with a corresponding (finite–step) invariant set S.
terminal set constraint associated with a finite sequence of terminal sets, out of which none is required to be invariant. We argue that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same prediction horizon. In the one step case the proposed stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions. 2. PRELIMINARIES
3. MAIN RESULTS
Consider a discrete–time dynamical system x(k + 1) = G(x(k)), k ∈ N, (1) n n where G : R → R is the system dynamics. We assume that G(0) = 0 and we are interested in characterizing stability of the zero equilibrium. Let G0 (x) := x and Gi (x) := G ◦ Gi−1 (x) for all i ∈ N ∩ [1, ∞). Definition 1. The origin of the dynamical system (1) is KL–stable in X ⊆ Rn if there exists a KL–function β : Rn × Rn → Rn such that �x(k)� ≤ β(�x(0)�, k) for all x(0) ∈ X and for all k ∈ N. Definition 2. A set S ⊂ X is called a finite–step contractive set for the dynamical system (1) with respect to the safe set X if there exists a pair (M, λ) ∈ {N∩[1, ∞)}×[0, 1] such that for all x(0) ∈ S it holds that (i) x(i) ∈ X for all i ∈ N ∩ [1, M − 1] and (ii) x(M ) ∈ S. A finite–step contractive set will also be called a (M, λ)–contractive set.
To set-up the MPC problem, consider a discrete–time dynamical system with inputs, i.e., x(k + 1) = g(x(k), u(k)), n
where g : R × R
m
k ∈ N,
(3)
n
→ R is the system dynamics.
Sometimes we consider u(k) to have an explicit, state– feedback form u(k) = h(x(k)). In this case let G(x(k)) := g(x(k), h(x(k))) denote the corresponding closed–loop dynamics. We will use the notation xi|k to denote the predicted state at time i ∈ {0, . . . , N }, given measured state x0|k := x(k), and similarly ui|k to denote the predicted input at time i, while setting u(k) := u0|k . Here N denotes the prediction horizon. The prediction model is a copy of (3), i.e., xi+1|k = g(xi|k , ui|k ),
Observe that (M, 1)–contractive sets are periodically invariant sets and (1, 1)–contractive sets are standard invariant sets. To analyze stability of the dynamical system (1) we will make use of finite–step Lyapunov functions. Definition 3. A function V : Rn → R+ that satisfies the following properties: ∃α1 , α2 ∈ K∞ : (2a) α1 (�x�) ≤ V (x) ≤ α2 (�x�), ∀x ∈ Rn ; ∃M ∈ N ∩ [1, ∞) and ρ ∈ K, ρ < id :
i ∈ {0, . . . , N − 1}, k ∈ N.
Defining uk := {u0|k , . . . , uN −1|k } yields the MPC optimization problem: N −1 l(xi|k , ui|k ) min J(x(k), uk ) := min L(xN |k ) + uk
uk
i=0
(4a) subject to constraints: xi+1|k = g(xi|k , ui|k ), i ∈ {0, . . . , N − 1}, (4b) (4c) xi|k ∈ X, i ∈ {1, . . . , N − 1}, ui|k ∈ U, i ∈ {0, . . . , N − 1}, (4d) xN |k ∈ X (k). (4e)
(2b) V (GM (x)) ≤ ρ(V (x)), ∀x ∈ S ⊆ Rn , is called a finite–step Lyapunov function in S for the dynamical system (1). A finite–step LF will also be called a M –step LF.
In equation (4) L : Rn → R+ denotes the terminal cost, l : Rn × Rm → R+ denotes the stage cost, X denotes the set where the predicted states are constrained, U denotes the set where the predicted inputs are constrained, and X (k) denotes the set where the terminal predicted state xN |k is constrained to lie.As usually done in terminal cost and constraint set MPC, we will employ an explicit, known state–feedback control law h : Rn → Rm and the closed– loop dynamics G(x) = g(x, h(x)). We will also make an abuse of notation for simplicity and define:
The following result was proven in (Geiselhart et al., 2014), the global version (S = X = Rn ), while the regional version (S ⊆ X ⊂ Rn ) was proven in (Lazar et al., 2013) and (Bobiti and Lazar, 2014). Theorem 4. Finite–step Lyapunov Theorem Suppose that the set {0} ⊂ S ⊆ X is a (M, 1)–contractive set for the dynamical system (1) with respect to the safe set X ⊆ Rn . Additionally, suppose that 1 the dynamics G is K–bounded, i.e., �G(x)� ≤ ω(�x�) for some ω ∈ K and all x ∈ X ⊆ Rn . Then if the dynamical system (1) admits a M –step Lyapunov function V in S, its origin is KL–stable in S.
h(S) := {y ∈ Rm : y = h(x), x ∈ S}, for any subset S of Rn and similarly for any other maps with arguments in a vector space, such as G.
Observe that the corresponding admissible domain of i attraction will be given by ∪M−1 {∪ G (x)} ⊆ X, x∈S i=0 which is a standard invariant set.
For any M ∈ N, M ≥ 1, consider a finite sequence of subsets of Rn , i.e., {S0 , . . . , SM−1 } ,
1
This property is a direct consequence of the KL–stability definition (Gielen and Lazar, 2015).
with the following properties: 10
(5)
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function V (x) is a Lyapunov function in Q(N, ∪M−1 i=0 Si ) and hence, the closed–loop MPC dynamical system is KL– stable in Q(N, ∪M−1 i=0 Si ) (for any N ∈ N ∩ [1, ∞)).
G(Si ) ⊆ Si+1 , i ∈ {0, . . . , M − 2}, (6a) (6b) G(SM−1 ) ⊆ S0 , (6c) h(Si ) ⊆ U, i ∈ {0, . . . , M − 1}, (6d) {0} ⊂ Si ⊆ X. i ∈ {0, . . . , M − 1}. Observe that for any i ∈ {0, . . . , M − 1}, the set Si is M –step invariant for the dynamics G(x) = g(x, h(x)) and constraints admissible with respect to input and state constraints. Notice also that if M = 1 we obtain that S0 is a standard invariant and constraints admissible set.
Proof. (i) For any N ∈ N ∩ [1, ∞) and k ∈ N there exists an index p ∈ {0, . . . , M − 1} such that x∗N |k ∈ X (k) = S(j+k)modM = Sp , where the ∗ denotes optimality. Then at time k + 1 it holds that X (k + 1) = S(j+k+1)modM = Sp+1
The terminal set is defined as follows: initial condition X (0) can be chosen arbitrarily within the sequence (5), as long as the corresponding optimization problem (4) is feasible for the initial state x(0). Suppose that X (0) = Sj , for some j ∈ {0, . . . , M − 1}. Then X (1) = Sj+1 , X (2) = Sj+2 and so on, until X (M − 1 − j) = SM−1 and then X (M − j) = S0 and so on, until X (M − 1) = Sj . Based on this assignment we can obtain the generic terminal set definition, for all k ∈ N: ∀j ∈ {0, . . . , M − 1} : X (0) = Sj ⇒ X (k) := S(j+k)modM , (7) where mod stands for the standard modulo operation. Next, we define the terminal cost: M−1 L(x) := F (Gi (x)),
if p < M −1 or X (k+1) = S(j+k+1)modM = S0 if p = M −1. In either case consider the shifted sequence of inputs uk+1 = {u∗1|k , . . . , u∗N −1|k , h(x∗N |k )}
(ii) For any k ∈ N and x(k) ∈ Q(N, ∪M−1 i=0 Si ) following the usual steps (Mayne, 2013) in proving stability of MPC and keeping in mind that uk+1 defined as in (10) is suboptimal but feasible, yields: V (x(k + 1)) − V (x(k)) = J(x(k + 1), u∗k+1 ) − J(x(k), u∗k ) ≤ J(x(k + 1), uk+1 ) − J(x(k), u∗k )
i=0
where F : Rn → R+ satisfies the property: F (GM (x)) − F (x) + l(x, h(x)) ≤ 0,
∀x ∈
= L(xN |k+1 ) + Si . (9)
i=0
− L(x∗N |k ) −
Under the standard assumption that l(x, h(x)) ≥ α3 (�x�) for all x ∈ Rn and some α3 ∈ K∞ , F becomes a finite–step Lyapunov function for dynamics G(x) with respect to the M−1 invariant set i=0 Si . Indeed,
N −1
i=0 N −1
l(xi|k+1 , ui|k+1 ) l(x∗i|k , u∗i|k )
i=0
≤ L(G(x∗N |k )) − L(x∗N |k ) + l(x∗N |k , h(x∗N |k )) − l(x∗0|k , u∗0|k ).
F (GM (x)) − F (x) ≤ −α3 (�x�) is equivalent with property (2b), as proven in (Geiselhart et al., 2014). Observe also that by taking M = 1 we obtain that F is a standard Lyapunov function that satisfies the standard terminal cost condition in stabilizing MPC. Theorem 5. Consider the closed–loop MPC dynamical system (3) with the control input u(k) = u∗0|k , with u∗k denoting the optimum of Problem 4, and with X (k) as defined in (7), for all k ∈ N.
Then directly from (8) and property (9) we obtain: V (x(k + 1)) − V (x(k)) ≤ F (GM (x∗N |k )) − F (x∗N |k ) + l(x∗N |k , h(x∗N |k )) − l(x∗0|k , u∗0|k ) ≤ −α3 (�x(k)�). Above we have used that ui|k+1 = u∗i+1|k for i ∈ {0, . . . , N − 2} and xi|k+1 = x∗i+1|k for i ∈ {0, . . . , N − 1}. The statement (ii) then follows from Theorem 4.
(i) Suppose that Problem 4 is feasible at time k ∈ N for x0|k = x(k). Then Problem 4 is feasible at time k + 1 for x0|k+1 = x(k + 1) = g(x(k), u∗0|k ) and thus, the N –step Q(N, ∪M−1 i=0 Si )
(10)
and observe that by property (6) it holds that h(x∗N |k ) ∈ U and G(x∗N |k ) ∈ X (k+1) and thus, xN |k+1 ∈ X (k+1) holds, which completes the proof of statement (i).
(8)
M−1
11
In what follows we will suggest a simple design method for obtaining a terminal cost and sequence of terminal sets that satisfy (9) and (6) for linear systems. Consider F (x) = x⊤ P x, l(x, u) = x⊤ Qx+u⊤ Ru, g(x, u) = Ax+Bu and h(x) = Kx for P, Q, R ≻ 0 and (A, B, K) of suitable dimensions. Lemma 6. Suppose that the following inequality holds:
N
:= {x ∈ X : ∃u ∈ U : controllable set S , {x(0) = x, x(1), . . . , x(N − 1)} ∈ XN } is x(N ) ∈ ∪M−1 i i=0 an invariant set. (ii) Suppose that 2 the MPC value function V (x) := J(x, u∗ ) satisfies property (2a) for some α1 , α2 ∈ K∞ for all x ∈ X. Furthermore, suppose that the terminal cost as defined in (8) satisfies property (9) and the sequence of terminal sets {S0 , . . . , SM−1 } used to generate X (k) at every k ∈ N satisfies properties (6). Then the MPC value
(A + BK)⊤ P (A + BK) − P + (Q + K ⊤ RK) ≤ 0. Then (9) holds. Proof. Pre and post multiplying the above inequality with (A + BK)⊤ and (A + BK) successively up to M − 1 times and summing up yields
2
This is typically insured by lower bounding the stage cost and lower and upper bounding the terminal cost with class K∞ functions (Mayne, 2013).
11
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((A + BK)M )⊤ P (A + BK)M − P +
xi+1|k xi|k ui|k xN −j|k
M−1
((A + BK)i )⊤ (Q + K ⊤ RK)(A + BK)i
i=0
≤0 and hence ((A + BK)M )⊤ P (A + BK)M − P + (Q + K ⊤ RK) ≤ 0, which completes the proof.
= g(xi|k , ui|k ), i ∈ {0, . . . , N − 1}, ∈ X, i ∈ {1, . . . , N } \ {N − j} ∈ U, i ∈ {0, . . . , N − 1} ∈ S0 .
(11a) (11b) (11c) (11d)
This modified problem still retains the recursive feasibility property under the same assumptions, as briefly outline next. By applying an appropriately shifted suboptimal input sequence based on repeatedly applying the local control law h(x) from prediction time N − j onwards, it can be easily seen that if the problem is feasible at time k := M i, i ∈ N, it is also feasible at times k + j, where j ∈ {1, . . . , M − 1}. Recall that at time k + M − 1, x∗N −M+1|k+M−1 ∈ S0 . By shifting the sequence and applying uN −M+j|k+M = h(xN −M+j|k+M ) one obtains that xN −M+j+1|k+M ∈ Sj+1 mod M for all j ∈ {0, . . . , M − 1}. Consequently, xN |k+M ∈ S0 .
Therefore, the P matrix corresponding to the LQR control law can readily be used to obtain the terminal cost (8). However, other designs are possible such as F (x) := Fj (x) = x⊤ Pj x and h(x) = hj (x) = Kj x if x ∈ Sj , which makes Fj a finite–step (control) Lyapunov function in Sj for each j ∈ {0, . . . , M − 1}. This leads to a periodically time varying terminal cost and control law which can still be synthesized (for periodically time-varying linear systems with period M ) via an appropriate set of LMIs and reduces conservatism considerable compared to the periodic control Lyapunov function of (Bohm et al., 2012).
Observe that the predicted states xN −j|k in (11a) do not contribute to the recursive feasibility guarantee and can be dropped. Therefore a simplified optimization problem can be constructed N −j−1 min J(x(k), uk ) := min L(xN −j|k ) + l(xi|k , ui|k )
The proposed MPC design can still be solved explicitly for linear (and PWA) systems, only that an explicit solution will be obtained for each terminal set Sj and corresponding terminal cost. However, the complexity of each problem will be less in general, as shown in the illustrative example, because each terminal set Sj will have a much lower complexity than a standard invariant set.
uk
uk
i=0
(12a)
subject to constraints: xi+1|k = g(xi|k , ui|k ), i ∈ {0, . . . , N − j − 1}, (12b) (12c) xi|k ∈ X, i ∈ {1, . . . , N − j − 1} ui|k ∈ U, i ∈ {0, . . . , N − j − 1} (12d) xN −j|k ∈ S0 , (12e)
Next, a simple design of the sequence of sets (5) is proposed for the linear case: take S0 as a subset of X such that it contains the origin and KS0 ⊆ U. Generate the set sequence Sj = (A + BK)Sj−1 iteratively starting from j = 1 and scale S0 such that Sj ⊆ X and KSj ⊆ U for all j. Stop when Sj ⊆ S0 . The iterations will convergence after a finite number of steps if ρ(A + BK) < 1 − ε for any ε > 0 (Gielen and Lazar, 2015). If polytopic sets are used, each iteration requires only basic polytopic operations in vertex representation. A similar algorithm can be derived using backward propagation and hyperplane representations.
where j = k mod M . Note that in this setting N must be greater than M . The corresponding stability theorem is stated next. Theorem 7. Consider the closed–loop MPC dynamical system (3) with the control input u(k) = u∗0|k , with u∗k denoting the optimum of Problem 12 for all k ∈ N.
The finite–step terminal ingredients proposed in this paper, which allow for a simple computation of the sequence of terminal sets, open up the application of stabilizing and thus reliable, terminal cost and constraint MPC to systems of sensible state and input space dimensions. A sequence of terminal sets and a control law that satisfy properties (6) for a linear system with 1000 states and 100 inputs was computed in (Athanasopoulos and Lazar, 2013), where even more complex iterative algorithms and control laws were used.
(i) Suppose that Problem 12 is feasible at time k ∈ N for x0|k = x(k). Then Problem 12 is feasible at time k + 1 for x0|k+1 = x(k + 1) = g(x(k), u∗0|k ) and thus, the N –step controllable set Q(N, S0 ) := {x ∈ X : ∃u ∈ UN : x(N ) ∈ S0 , {x(0) = x, x(1), . . . , x(N − 1)} ∈ XN } is an invariant set. (ii) Suppose that the MPC value function V (x) := J(x, u∗ ) satisfies property (2a) for some α1 , α2 ∈ K∞ for all x ∈ X. Furthermore, suppose that the terminal cost as defined in (8) satisfies property (9) and the sequence of terminal sets {S0 , . . . , SM−1 } satisfies properties (6). Then the MPC value function V (x) is a finite–step Lyapunov function in Q(N, S0 ) and hence, the closed–loop MPC dynamical system is KL–stable in Q(N, S0 ) (for any N ∈ N ∩ [2M, ∞)).
Next, we will discuss a variant of the MPC problem (4) which requires using a single terminal set, but a timevarying terminal constraint time instant. The advantage of this method is that it will not require storage of more terminal sets, but it may require a longer prediction horizon to reach the same domain of attraction. 3.1 Time-varying terminal constraint time instant
Proof. (i) The recursive feasibility proof follows from recursive feasibility of Problem 11.
Consider a MPC optimization problem with the same optimization criterion as in (4), however, the constraints (4c) and (4e) change, i.e., for a given j := k mod M
(ii) Let V (x) = J(x(k), u∗k ), where u∗k is the optimal input sequence at time k. Then it holds that 12
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feasible, as given x∗N −j|k = xN −j−M|k+M ∈ S0 , it follows that Gi (x∗N −j|k ) ∈ Si mod M , ∀i ∈ N.
V (x(k)) − V (x(k + M )) = L(x∗N −j|k )
+
N −j−1
l(x∗i|k , u∗i|k )−
i=0 N −j−1
L(x∗N −j|k+M ) −
(13)
Establishing stability for the case when N ≥ M + 1 is the subject of ongoing work.
l(x∗i|k+M , u∗i|k+M ).
4. ILLUSTRATIVE EXAMPLE
i=0
Since N ≥ 2M , N − j > M for all j ∈ {0, . . . , M − 1}. Then it holds that: V (x(k)) − V (x(k + M )) = L(x∗N −j|k ) +
M−1
The main results of this paper are illustrated on the simple example of the discrete-time double integrator: 2 Ts 1 Ts x(k + 1) = Ax(k) + Bu(k), A = , B= 2 , 0 1 Ts where Ts = 0.4. For simplicity, the local controller h(x) := Kx, where K is the LQR state-feedback, which is designed by taking Q = I2 and R = 100. The terminal cost is defined with F (x) := x⊤ P x, with P the solution of the discrete time algebraic Riccati equation, and clearly satisfies (9).
l(x∗i|k , u∗i|k )+
i=0
N −j−1
l(x∗i|k , u∗i|k )−
(14)
i=M
L(x∗N −j|k+M ) −
N −j−M−1
l(x∗i|k+M , u∗i|k+M )−
i=0
N −j−1
Three controllers are compared in what follows, i.e., standard MPC with invariant terminal set constraint and F (x) as the terminal cost, the construction from (4), and the controller from (12). The prediction horizon of 6 steps was taken for the first two formulations. The set S0 was taken as an unit-box centered in origin and scaled such that Sj = (A + BK)Sj−1 , j ∈ {1, . . . , M − 1}, are control admissible and stay within the state constraints. It was found that in this particular case M = 19. The prediction horizon for the controller in (12) is taken M + 1.
l(x∗i|k+M , u∗i|k+M ).
i=N −j−M
By applying the shifted control sequence [u∗M|k , . . . , u∗N −j−1|k , h(x∗N −j|k ), . . . , h(GM−1 (x∗N −j|k ))] one obtains V (x(k)) − V (x(k + M )) ≥ L(x∗N −j|k ) − L(xN −j|k+M ) + N −j−1
l(x∗i|k , u∗i|k ) −
i=M M−1
M−1
l(x∗i|k , u∗i|k )+
i=0 N −j−M−1
The explicit solution for each of the methods has been computed employing the MPT3 (Herceg et al., 2013) and YALMIP (Lofberg, 2004) toolboxes. Note that, for each of the methods proposed in this paper M controllers are computed. As far as the complexity goes, the standard solution results in 57 regions, and the largest controller partition for (4) and (12) contain 37 and 49 regions, and the smallest 13 and 11, respectively. The larger complexity of the standard solution is mainly driven by the complexity of the terminal set which has 26 hyperplanes, compared to 4 hyperplanes of Sj , for all j. Moreover, the relatively large number of regions in (12) can be explained by the long prediction horizon, i.e., 20 compared to the prediction horizon of 6 used in the other two control approaches. Nevertheless, the solution in (12) can be build in the Dynamic Programming fashion from j = M − 1 to j = 0, which ultimately can save memory space compared to the approach in (4).
l(x∗i+M|k , u∗i+M|k )−
i=0
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k ))) =
(15)
i=0
L(x∗N −j|k ) − L(xN −j|k+M ) +
M−1
l(x∗i|k , u∗i|k )−
i=0
M−1
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k ))).
i=0
By substituting (9) into (15) it follows that V (x(k)) − V (x(k + M )) ≥ M−1
i=0 M−1
13
l(x∗i|k , u∗i|k )−
The domain of attraction (DoA), and the terminal set for the standard MPC are shown in Fig. 1. For comparison, the same sets are shown in Fig. 2 and Fig. 3, for the methods in (4) and (12). Note that the DoA and the terminal sets are shown for all M controllers in the same figure for the latter two methods. It can be observed how the DoA changes for each of the M controllers. In Fig. 2 this change is driven by the change of the terminal set Sj , whereas in Fig. 3 the terminal set remains the same, i.e., S0 , but the prediction horizon is reduced, leading to a smaller DoA for controllers corresponding to j close to M.
(16) (F (Gi (x∗N −j|k )) − F (Gi+M (x∗N −j|k ))−
i=0
l(Gi (x∗N −j|k ), h(Gi (x∗N −j|k )))),
where last two rows are positive, and consequently M−1 l(xi|k , u∗i|k ) ≥ α3 (�x(k)�). V (x(k)) − V (x(k + M )) ≥ i=0
(17)
Hence V is a finite-step Lyapunov function for the closedloop system and the statement then follows directly from Theorem 4. Note that the shifted control sequence is
The DoA for all controllers is the same, because of the linear MPC setup and conservative choice of the local 13
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Fig. 1. Domain of attraction (cyan) and the terminal set (blue) of the standard MPC formulation.
Fig. 3. Domain of attraction (cyan) and the terminal set (yellow ) of the method in (12). The terminal set of the standard MPC is depicted in blue.
[x]
1
1 0 −1 0
2
4
6
8
10
12
14
16
18
0
2
4
6
8
10
12
14
16
18
[x]
2
1 0 −1
u
1
Fig. 2. Domain of attraction (cyan) and the terminal set (yellow ) of the method in (4).
Standard Sequenced T. sets Single T. set
0 −1 0
controller. Moreover, the prediction horizon of 5 appears to be sufficient to cover the maximal possible DoA for this system. Nevertheless, by varying the local controller throughout the sequence of the method in (4) can potentially yield a larger DoA than the standard MPC formulation and reduce the controller design time and complexity. The solution in (12) is somewhat more conservative when it comes to the DoA size, but can reduce the complexity of the design and implementation of the controller even further.
2
4
6
8
10
12
14
16
18
Fig. 4. Time response of the system in closed–loop with each of the MPC controllers for all vertexes of the feasible set. prediction horizon. In the one step case the proposed stability conditions reduce naturally to the standard terminal cost and constraint set stability conditions. 5.1 Remarks for future developments
To conclude the time evolution of the state and input trajectories from the vertexes of the DoA are shown in Fig. 4. Clearly, the approach from (4) forces faster convergence via the sequence of the terminal sets, whereas (12) behaves fairly similarly to the standard MPC setup.
The finite–step terminal ingredients for stabilizing MPC proposed in this paper have an appeal towards the design of robust MPC schemes and tracking MPC schemes. In the robust MPC case, a sequence of terminal sets which are finite–step robustly invariant is much more easy to construct and it will help in avoiding the complexity of working with approximations of the minimal RPI set.
5. CONCLUSIONS This paper proposed a novel construction of the terminal cost and terminal set in stabilizing model predictive control that used finite–step Lyapunov functions and finite– step invariant sets. This construction resulted in a periodic terminal set constraint associated with a finite sequence of terminal sets, out of which none was required to be invariant. We argue that constructing such sets is easier and more scalable compared to invariant sets, while a comparable region of attraction is obtained for the same
In the tracking MPC case, for periodic reference signals, which are common in motion control systems, electromechanical, power electronics and robotics applications, a sequence of terminal sets out of which none is invariant can be constructed so that convergence to the periodic reference is attained. In fact, the proposed sequence of terminal sets is a natural solution when the desired operating set–point is given by a limit cycle. 14
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ity and optimality via periodic invariance. Automatica, 47(2), 326 – 333. Grune, L. (2012). NMPC without Terminal Constraints. In Proceedings of 4th IFAC Conference on Nonlinear Model Predictive Control, 1–13. Leeuwenhorst, Netherlands. Herceg, M., Kvasnica, M., Jones, C., and Morari, M. (2013). Multi-Parametric Toolbox 3.0. In Proc. of the European Control Conference, 502–510. Z¨ urich, Switzerland. http://control.ee.ethz.ch/ mpt. Lazar, M. (2009). Flexible control Lyapunov functions. In American Control Conference, 2009. ACC ’09., 102– 107. Lazar, M., Doban, A.I., and Athanasopoulos, N. (2013). On stability analysis of discrete–time homogeneous dynamics. In Proceedings of 17th International Conference on Systems Theory, Control and Computing, 297–305. Sinaia, Romania. Limon, D., Alamo, T., Salas, F., and Camacho, E.F. (2006). On the stability of constrained MPC without terminal constraint. Automatic Control, IEEE Transactions on, 51(5), 832–836. Lofberg, J. (2004). Yalmip : A toolbox for modeling and optimization in MATLAB. In Proceedings of the CACSD Conference. Taipei, Taiwan. URL http://users.isy.liu.se/johanl/yalmip. Mayne, D.Q. (2013). An apologia for stabilising terminal conditions in model predictive control. International Journal of Control, 86(11), 2090 – 2095. Mayne, D.Q. (2014). Model predictive control: Recent developments and future promise. Automatica, 50(12), 2967 – 2986. Mhaskar, P., El-Farra, N.H., and Christofides, P.D. (2006). Stabilization of nonlinear systems with state and control constraints using Lyapunov-based predictive control. Systems & Control Letters, 55(8), 650 – 659. Michalska, H. and Mayne, D.Q. (1993). Robust receding horizon control of constrained nonlinear systems. Automatic Control, IEEE Transactions on, 38(11), 1623– 1633. Rakovic, S.V., Kern, B., and Findeisen, R. (2010). Practical Set Invariance for Decentralized Discrete Time Systems. In 49th IEEE Conf. on Decision and Control. Scokaert, P., Mayne, D., and Rawlings, J. (1999). Suboptimal model predictive control (feasibility implies stability). Automatic Control, IEEE Transactions on, 44(3), 648–654. Wang, L. (2004). A tutorial on model predictive control using velocity form model. Developments of Chemical Engineering and Mineral Processing, 12, 573–614.
In terms of industrial applications, the proposed method for designing the terminal cost L can be used to determine the length of the prediction horizon required for achieving stability in the unconstrained case, when a specific terminal weight matrix is not used. As preferred in industry, the terminal weight can be kept equal to the stage weight matrix, i.e., F (x) = x⊤ Qx, and then it suffices to find the number M for which the finite–step decrease property (9) holds. Such an M always exists for any Q ≻ 0 chosen by the practitioner, under the assumption of exponential stability of the local dynamics (A + BK), according to the results in (Gielen and Lazar, 2015). The resulting cost will be: M−1 ⊤ J(x(k), uk ) = Gi (xN |k ) Q Gi (xN |k ) +
i=0 N −1
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⊤ x⊤ i|k Qxi|k + ui|k Rui|k ,
i=0
and it can be regarded as an extension of the stage cost to a prediction horizon of N + M , where the last M control actions are not free, but they are set equal to the local controller h(x), which can represent a classical frequency– domain controller. Such a cost is expected to preserve the performance of the local controller compared with using an aggressive terminal weight matrix P and it is more likely to be accepted by practitioners. REFERENCES Alamir, M. (ed.) (2006). Stabilization of Nonlinear Systems Using Receding-horizon Control Schemes. Lecture Notes in Control and Information Sciences. SpringerVerlag, London. Athanasopoulos, N. and Lazar, M. (2013). Scalable Stabilization of Large–Scale Discrete–Time Linear Systems via the 1–norm. In 4th IFAC Workshop on Distributed Estimation and Control in Networked Systems, 277–284. Koblenz, Germany. Bobiti, R.V. and Lazar, M. (2014). On the computation of Lyapunov functions for discrete–time nonlinear systems. In 18th International Conference on System Theory, Control and Computing, 93 – 98. Sinaia, Romania. Bohm, C., Lazar, M., and Allgower, F. (2012). Stability of periodically time-varying systems: Periodic lyapunov functions. Automatica, 48(10), 2663 – 2669. Brunner, F.D., Lazar, M., and Allgower, F. (2013). Stabilizing linear model predictive control: On the enlargement of the terminal set. In Control Conference (ECC), 2013 European, 511–517. Zurich, Zwitzerland. Fagiano, L. and Teel, A.R. (2013). Generalized terminal state constraint for model predictive control. Automatica, 49(9), 2622 – 2631. Geiselhart, R., Gielen, R.H., Lazar, M., and Wirth, F.R. (2014). An alternative converse Lyapunov theorem for discrete-time systems. Systems and Control Letters, 70(0), 49 – 59. Gielen, R.H. and Lazar, M. (2015). On stability analysis methods for large-scale discrete-time systems. Automatica, 55(0), 66 – 72. Gondhalekar, R. and Jones, C.N. (2011). MPC of constrained discrete-time linear periodic systems a framework for asynchronous control: Strong feasibility, stabil15