Economic model predictive control without terminal constraintsforoptimalperiodicbehavior

Transcription

1 Economic mode predictive contro without termina constraintsforoptimaperiodicbehavior MatthiasA.Müer a LarsGrüne b a Institute for Systems Theory and Automatic Contro, University of Stuttgart, Stuttgart, Germany. b Mathematica Institute, University of Bayreuth, Bayreuth, Germany. Abstract In this paper, we anayze economic mode predictive contro schemes without termina constraints, where the optima operating behavior is not steady-state operation, but periodic behavior. We first show by means of a counterexampe, that a cassica receding horizon contro scheme does not necessariy resut in an optima cosed-oop behavior. Instead, a muti-step MPC scheme may be needed in order to estabish near optima performance of the cosed-oop system. This behavior is anayzed in detai, and we show that under suitabe dissipativity and controabiity conditions, desired cosed-oop performance guarantees as we as convergence to the optima periodic orbit can be estabished. Key words: Economic mode predictive contro, Optima periodic operation, Noninear systems 1 Introduction In recent years, the study of economic mode predictive contro (MPC) schemes has received a significant amount of attention. In contrast to standard stabiizing MPC, the contro objective is the minimization of some genera performance criterion, which needs not be reated to any specific steady-state to be stabiized. These type of contro probems arise in many different fieds of appication, ranging, e.g., from the process industry over buiding cimate contro or the contro of wind turbines to the deveopment of sustainabe cimate poicies (see, e.g., [2,4,11,17,18,21]). In the iterature, various different economic MPC schemes have been deveoped for which desired cosed-oop properties such as performance estimates or convergence can be guaranteed. These incude schemes with additiona termina equaity or termina region constraints[1, 3], with generaized termina constraints[8, 22], without termina constraints [12], and Lyapunov-based schemes [17] (see aso the recent survey artice [7]). A distinctive feature of economic MPC is the fact that A preiminary version of parts of this paper has been presented at the 54th IEEE Conference on Decision and Contro (CDC) 2015, see [24]. The work of Matthias A. Müer was supported by DFG Grant MU3929/1-1. The work of Lars Grüne was supported by DFG Grant GR1569/13-1. Emai addresses: matthias.mueer@ist.uni-stuttgart.de (Matthias A. Müer), ars.gruene@uni-bayreuth.de (Lars Grüne). the cosed-oop trajectories are not necessariy convergent to a steady-state, but can exhibit more compex, e.g., periodic, behavior. In particuar, the optima operating behavior for a given system depends on its dynamics, the considered performance criterion and the constraints which need to be satisfied. The case where steady-state operation is optima is by now fairy we understood, and various cosed-oop guarantees have been estabished in this case. For exampe, a certain dissipativity property is both sufficient [3] and (under a mid controabiity condition) necessary [23] for a system to be optimay operated at steady-state. The same dissipativity condition (strengthened to strict dissipativity) wasusedin[1,3]toproveasymptoticstabiityofthe optima steady-state for the resuting cosed-oop system with the hep of suitabe termina constraints. Simiar (practica) stabiity resuts were estabished in [12, 15] without such termina constraints. On the other hand, the picture is sti much ess compete in case that the optima operating behavior is nonstationary. This situation occurs in many cases of practica interest, such as in certain chemica reactors (see, e.g.[2, 3, 7]) or in appications with time varying(energy) prices or demand (see, e.g., [20,21]). For such cases, in [3] it was shown that when using some periodic orbit as (periodic) termina constraint within the economic MPC probem formuation, then the resuting cosed-oop system wi have an asymptotic average performance which is at east as good as the average cost of the periodic orbit. Convergence to the optima periodic orbit was es- Preprint submitted to Automatica 21 January 2016

2 tabishedin 1 [19,27]usingsimiarterminaconstraints, and in [20] for inear systems and convex cost functions using ess restrictive generaized periodic termina constraints. Furthermore, dissipativity conditions which are suited as sufficient conditions such that the optima operating behavior of a system is some periodic orbit were recenty proposed in [16]. In this paper, we consider economic MPC without termina constraints for the case where periodic operation is optima. Using no termina constraints is in particuar desirabe in this case as the optima periodic orbit then needs not be known a priori (i.e., for impementing the economic MPC scheme). Furthermore, the onine computationa burden might be ower and a arger feasibe region is in genera obtained. We first show by means of a counterexampe (see Section 3), that the cassica receding horizon contro scheme, consisting of appying the first step of the optima predicted input sequence to the system at each time, does not necessariy resut in an optima cosed-oop performance. We then prove in Section 4 that this undesirabe behavior can be resoved by possiby using a muti-step MPC scheme instead. In particuar, we show that the resuting cosed-oop system has an asymptotic average performance which is equa to the average cost of the optima periodic orbit (up to an error term which vanishes as the prediction horizon increases). This recovers the resuts of [3], where periodic termina constraints were used as discussed above. In Section 5 we derive checkabe sufficient conditions based on dissipativity and controabiity in order to appy the resuts of Section 4. Furthermore, in Section 6 we show that under the same conditions, aso (practica) convergence of the cosed-oop system to the optima periodic orbit can be estabished. We cose this section by noting that our anaysis party buids on the one in [12], where cosed-oop performance guarantees and convergence resuts for economic MPC without termina constraints were estabished for the case where the optima operating behavior is steadystate operation. However, whie some of the empoyed concepts and ideas are simiar to those in [12], various properties of predicted and cosed-oop sequences are different in the periodic case considered in this paper, and hence aso different anaysis methods are required. Finay, we note that a preiminary version of some of the resuts of this paper have appeared in the conference paper [24]. Compared to [24], the main noveties of the present paper are a more comprehensive exposition of the subject incuding various additiona remarks and exampes as we as a proofs (which were party missing in [24]), the deveopment of our resuts using a more genera dissipativity condition, and the estabishment of cosed-oop (practica) convergence to the optima periodic orbit. 1 In [19], however, again the standard assumption as in stabiizing MPC, i.e., positive definiteness of the cost function, was imposed, which means that no genera performance criterion as in economic MPC coud be considered. 2 Preiminaries and setup LetI [a,b] denotetheset ofintegersintheinterva[a,b] R, and I a the set ofintegers greaterthan orequa to a. For a set A I 0, #A denotes its cardinaity (i.e., the number of eements). For a R, a is defined as the argest integer smaer than or equa to a. The distance of a point x R n to a set A R n is defined as x A := inf a A x a. For a set A R n and ε > 0, denote by B ε (A) := {x R n : x A ε}.bylwedenotethesetof functions ϕ : R 0 R 0 which are continuous, nonincreasing and satisfy im k ϕ(k) = 0. Furthermore, by KL wedenotethe set offunctions γ : R 0 R 0 R 0 suchthatforeachϕ L,thefunctionγ(k) := γ(ϕ(k),k) satisfiesγ L.NotethatthedefinitionofaKL-function requires weaker properties than those for cassica KLfunctions, i.e., each KL-function is aso a KL-function (but the converse does not hod). We consider noninear discrete-time systems of the form x(k +1) = f(x(k),u(k)), x(0) = x (1) with k I 0 and f : R n R m R n. System (1) is subject to pointwise-in-time state and input constraints x(k) X R n and u(k) U R m for a k I 0. For a given contro sequence u = (u(0),...,u(k)) U K+1 (or u = (u(0),...) U ), denote by x u (k,x) the corresponding soution of system (1) with initia condition x u (0,x) = x. For a given x X, the set of a feasibe contro sequences of ength N is denoted by U N (x), where a feasibe contro sequence is such that u(k) U for a k I [0,N 1] and x u (k,x) X for a k I [0,N]. Simiary, the set of a feasibe contro sequences of infinite ength is denoted by U (x). In the foowing, we assumeforsimpicitythatu (x) forax X.This means that for a initia conditions x X, there exists a trajectory which stays in X for a times, i.e., the set X is contro invariant under contros in U. This assumptionmightberestrictiveingenera,anditcanbereaxed if desired, using, e.g., methods simiar to those in [14, Chapter 8] or [9]. However, the technica detais of such an extension are beyond the scope of this paper. Remark 1 For ease of presentation, we use decouped state and input constraint sets X and U in the statement of our resuts. Nevertheess, a resuts in this paper are aso vaid for possiby couped state and input constraints, i.e., (x(k),u(k)) Z for a k I 0 and some Z R n R m, which wi aso be used in the exampes. System (1) is equipped with a stage cost function : X U R,whichisassumedtobe boundedfrombeow on X U, i.e., min := inf x X,u U (x,u) is finite. Note that this is, e.g., the case if X U is compact and is continuous. Without oss of generaity, in the foowing we assume that min 0. We then define the foowing finite horizon cost functiona J N (x,u) := N 1 (x u (k,x),u(k)) (2) 2

3 and the corresponding optima vaue function V N (x) := inf J N (x,u). (3) u U N (x) In the foowing, we assume that for each x X, a contro sequence u UN (x) exists such that the infimum in (3) is attained, i.e., u satisfies V N (x) = J N (x,u ). Since we assume that U (x) for a x X, this is, e.g., satisfied if f and are continuous and U is compact. A standard MPC scheme without additiona termina cost and termina constraints then consists of minimizing, at each time instant k I 0 with current system state x = x(k), the cost functiona (2) with respect to u U N (x) and appying the first eement of the resuting optima input sequence u to the system. This means that the resuting receding horizon contro input to system (1) is given by u MPC (k) := u umpc (k,x) (0), where x umpc (,x) denotes the corresponding cosedoop state sequence. For T I 0, the cost aong T steps of these cosed-oop sequences is denoted by JT c(x,u MPC) = T 1 (x u MPC (k,x),u MPC (k)), and the resuting asymptotic average cost is given by J,av c (x,u MPC) := imsup T (1/T)JT c(x,u MPC). In [12], it was shown that if system (1) is optimay operated at some steady-state (x,u ) with cost 0 := (x,u ), then under suitabe conditionsthe asymptotic average performance of the cosed-oop system, J,av c, equas 0 (up to an error term which vanishes as N ), and the cosed-oop system converges to x (again up to an error term which vanishes as N ). In this paper, we consider the more genera case where the optima operating behavior of system(1) is not necessariy stationary but given by some periodic orbit with period P I 1. To this end, consider the foowing definitions. Definition 2 A set of state/input pairs Π = {(x p 0,up 0 ),...,(x p,up )} with P I 1 is caed a feasibe P- periodicorbitofsystem (1),ifx p k X andup k U for a k I [0,], x p k+1 = f(xp k,up k ) for a k I [0,P 2], and x p 0 = f(xp,up ). It is caed a minima P-periodic orbitifx p k 1 x p k 2 fora k 1,k 2 I [0,] withk 1 k 2. In the foowing, denote by Π X the projection ofπon X, i.e., Π X := {x p 0,...,xp }. Definition 3 System (1) is optimay operated at a periodic orbit Π if for each x X and each u U (x) the foowing inequaity hods: im inf T T 1 (x u(k,x),u(k)) 1 (x p k T P,up k ) Definition 3 means that each feasibe soution wi resut in an asymptotic average performance which is at most as good as the average performance of the periodic u= 1 (x,u)=1 u=0 (x,u)=1 u=0 (x,u)=1+ε -1 0 u=1 1 (x,u)=1 2ε Fig. 1. Iustration of the states x (nodes) and feasibe transitions (edges) with corresponding input u and cost in Exampe 4. orbit Π. Furthermore, for P = 1 the notion of optima steady-state operation [3, 23] is recovered. Note that if system (1) is optimay operated at some periodic orbit Π = {( x p 0,ūp 0 ),...,( xp,ūp )}, then Π is necessariy an optima periodic orbit for system (1), i.e. we have 1 P ( x p k,ūp k ) = inf P I 1,Π S P Π 1 P (x p k,up k ), (4) where SΠ P denotes the set of a feasibe P-periodic orbits. Note that in genera, there can exist mutipe(minima) optima periodic orbits for system(1). On the other hand, if the assumption of strict dissipativity as required in Section 5 is satisfied for some minima periodic orbit Π (cf. Assumption 9), then Π is necessariy the unique minima optima periodic orbit. Incasethat asystemisoptimayoperatedataperiodic orbit Π, the cosed-oop system resuting from appication of the economic MPC scheme exhibits optima performance if J,av(x,u c MPC ) = (1/P) (xp k,up k ). AsdiscussedintheIntroduction,in[3]itwasshownthat this can be achieved in case that Π X is used as a periodic termina constraint. When using no termina constraints, this equaity may in genera not be achieved, as weshowinthefoowingsectionbymeansofacounterexampe. Nevertheess, optima performance and (practica) convergence to the optima periodic orbit can sti be guaranteed aso without termina constraints in case a muti-step MPC scheme is used, as wi be shown in Sections Motivating exampe Exampe 4 Consider the one-dimensiona system x(k + 1) = u(k) with state and input constraint set Z = {( 1, 1),( 1,0),(0,1),(1,0)} consisting of four eements ony and cost (x,u) defined as ( 1, 1) = 1, ( 1,0) = 1, (0,1) = 1 2ε, (1,0) = 1+ε for some constant ε > 0, see aso Figure 1. The system is optimay operated at the two-periodic orbit given by Π = {(0,1),(1,0)}, and with average cost 0 := (1/2) 1 (xp k,up k ) = 1 ε/2. For initia condition x 0 = 1, it foows that for any even prediction horizon N I 2, the optima open-oop input sequence u 0 issuchthatx u 0 (1,x 0 ) = 0andthenx u 0 (,x 0 ) stays on Π X. This means that aso the cosed-oop system converges to the set Π X and J,av( 1,u c MPC ) = 0. 3

4 On the other hand, for any odd prediction horizon N I 3, the optima open-oop input sequence u 0 is such that x u 0 (1,x 0 ) = 1, x u 0 (2,x 0 ) = 0, and then x u (,x 0 ) stays on Π X. But this means that 0 the cosed-oop system stays at x = 1 for a times, i.e., x umpc (k,x 0 ) = 1 for a k I 0, and hence J c,av ( 1,u MPC) = 1 > 1 ε/2 = 0. The above exampe shows that the phase on the periodic orbit is decisive, i.e., what is the optima time to converge to the periodic orbit. Simiar exampes can aso be constructed where the cost aong the optima periodic orbit is constant (see [24, Exampe 5]). The above shows that in genera, one cannot guarantee that for a sufficienty arge prediction horizons N, the cosed-oop asymptotic average performance satisfies ) (pus some error term which vanishes as N ), as coud be estabished in [12] for the case of optima steady-state operation, i.e., P = 1. On the other hand, one observes in the above exampe that if the MPC scheme is modified in such a way that not ony the first vaue of the optima contro sequence is appied to the system, but the first two vaues, then the cosed-oop system converges to the optima periodic orbit and hence J c,av (x,u MPC) = (1/P) (xp k,up k (xp k,up k ), for a prediction horizons N I 2. In the foowing, we wi see that this behavior can be rigorousy estabished under suitabe assumptions on the probem. J c,av (x,u MPC) = (1/P) Remark 5 We note that the above observations do not depend on the fact that the constraint set Z ony consists of a finite number of points. Whie for carity of presentation, we chose the above two simpe motivating exampes, it is not difficut to find exampes where the same effects occur and Z has non-empty interior. 4 Cosed-oop performance guarantees As mentioned above, in the foowing we consider a muti-step MPC scheme where for some P I 1, an optima input sequence u is ony cacuated every P time instants, and then the first P eements of this sequence are appied to system (1). This means that the contro input to system (1) at time k is given by u MPC (k) = u ([k]), (5) where x = x umpc (P k/p,x) and [k] := k mod P. Remark 6 The fact that the P-step MPC scheme operates in open oop for P time steps wi in genera reduce robustness with respect to perturbations compared to a cassica MPC scheme in which the oop is cosed in each time step. However, this probem can be resoved by the foowing variant, for which the subsequent resuts sti appy. In this variant, an optima input sequence is computedateach timeandonythe firsteement is appied to the system as in standard MPC, but the prediction horizon is periodicay time-varying, i.e., N in (2) is repaced by N [k]. By the dynamic programming principe, in absence of perturbations the cosed-oop sequences resuting from appication of these two schemes are the same. However, the second wi in genera exhibit better robustness properties in case of uncertainties and disturbances (see [13]). In the foowing, we estabish cosed-oop performance guarantees for the P-step MPC scheme as defined via (5). We first derive a preparatory resut (Proposition 7), which is a generaization of Proposition 4.1 in [12]. This resut wi ater be appied to ensure the desired cosed-oop average performance guarantees. Proposition 7 Assume there exist N,P I 1 and δ 1,δ 2 L such that for each x X and each N I N there exists a contro sequence ū U N+P (x) and time instants k 1,...,kP I [0,...,N+] satisfying the foowing conditions. (i) The inequaity J N (x) V N(x)+δ 1 (N) hods for J N(x) := N+ (xū(k,x),ū (k)). k/ {k 1,...,kP } (ii) There exists 0 R such that for a x X the foowing inequaity is satisfied: 1 (xū (k,x),ū (k)) 0 +δ 2 (N) P k {k 1,...,kP } Then the inequaities J c KP(x,u MPC ) V N (x) V N (x umpc (KP,x))+KP 0 +Kδ 1 (N P)+KPδ 2 (N P) (6) and J c,av(x,u MPC ) 0 +δ 1 (N P)/P +δ 2 (N P)(7) hod for a x X, a N I N+P and a K I 0. Proof: Fix x X and N I N+P. Using the abbreviation x(k) = x umpc (k,x), from the dynamic programming principe and the definition of the muti-step MPC contro input in (5), we obtain that for a i I 0 (x(ip +k),u MPC (ip +k)) = V N (x(ip)) V N P (x((i+1)p)). Summing up for i = 0,...,K 1 then yieds J c KP(x,u MPC ) = K 1 i=0 (x(ip +k),u MPC (ip +k)) = V N (x(0)) V N P (x(kp)) + K 1 i=1 ( VN (x(ip)) V N P (x(ip)) ).(8) 4

5 Now consider the summands in (8). Condition (i) of the proposition with N P in pace of N and x = x(ip) impiesthatv N P (x(ip)) J N P (x(ip)) δ 1(N P). Furthermore, by optimaity of V N we get V N (x(ip)) J N (x(ip),ū N P,x(iP) ). Combining the above and defining I := {kn P,x(iP) 1,...,kP N P,x(iP) }, from condition (ii) of the proposition and the definitions of J N and J N we obtain V N (x(ip)) V N P (x(ip)) J N (x(ip),ū N P,x(iP) ) J N P (x(ip))+δ 1(N P) = (xūn P,x(iP) (k,x(ip)),ū N P,x(iP) (k)) k I +δ 1 (N P) P 0 +Pδ 2 (N P)+δ 1 (N P). (9) Recaing that x(0) = x and inserting (9) into (8) for i = 1,...,K 1 yieds JKP c (x,u MPC) V N (x) V N P (x(kp))+(k 1)(P 0 +Pδ 2 (N P)+δ 1 (N P)). Moreover,using (9) for i = K yieds V N P (x(kp)) V N (x(kp))+p 0 +Pδ 2 (N P)+δ 1 (N P).Together with the above, this resuts in (6). Finay, (7) foows from (6) by dividing by KP and etting K, due to the fact that (x,u) min 0 for a (x,u) X U and V N (x(kp)) N min 0. As is the case in most MPC proofs, Proposition 7 compares the cost aong different input sequences. In particuar, existence of an input sequence ū of ength N + P is required such that when summing up the stage cost aong this sequence for a but P time instantsk 1,...,kP,theresutingcostJ N (x)iscoseto the optima cost V N (x) of probem (3) with horizon N (see condition (i)). If the sum of the stage cost aong is cose to some these P time instants k 1,...,kP vaue 0 (see condition (ii)), then (7) guarantees that the cosed-oop average performance is upper-bounded by this constant 0 (up to an error term which vanishes as N ). In the foowing, we construct contro sequences ū such that Proposition 7 can be appied with 0 = (1/P) (xp k,up k ) forsomep-periodicorbit Π. Then, inequaity (7) yieds the desired property that the asymptotic average performance of the cosedoop system resuting from appication of the P-step MPC scheme is ess than or equa to the average performance of the periodic orbit Π (up to an error term which vanishes as N ). As discussed above, this approximatey recovers asymptotic average performance resuts obtained in MPC schemes with (periodic) termina constraints [3]. The foowing theorem gives sufficient conditions under which the controsequences ū required in Proposition 7 can be constructed. As wi be seen in the proof, for these sequences the time instants k 1,...,kP are consecutive time instants. However, we note that this is not necessariy needed in order to appy Proposition 7. Theorem 8 Assume that there exist constants 0 0, δ > 0, and P I 1 and a set Y X such that the foowing properties hod. (a) There exists γ K such that for a δ (0, δ] and a x B δ (Y) X there exists a contro sequence u x U P (x) such that the inequaity (1/P) (x u x (k,x),u x (k)) 0 +γ (δ) hods. (b) There exist N 0 I 1 and a function γ V KL such thatforaδ (0, δ],an I N0,ax B δ (Y) X and the contro sequence u x U P from (a) the inequaity V N (x) V N (x ux (P,x)) γ V (δ,n) hods. (c) There exist σ L and N 1 I N0 with N 0 from (b) such that for a x X and a N I N1, each optima trajectory x u (,x) satisfies x u (k x,x) Y σ(n) for some k x I [0,N N0]. Then the conditions of Proposition 7 are satisfied. Proof: See Appendix A. IntheproofofTheorem8,themainideafordefiningthe contro sequence ū U N+P (x) required in Proposition 7 is the foowing: given an optima input sequence u UN (x), we insert P additiona contro vaues (thosefrom(a))startingattimek x (from(c)),wherethe optima trajectory x u (,x) is cose to the set Y. The corresponding P (consecutive) time instants are then the time instants k 1,...,kP in Proposition 7 (see Appendix A for detais). Theorem 8 uses simiar conditions astheorem4.2in[12],whichwereshowntohodincase of optima steady-state operation. However, there are some crucia differences. Namey, [12, Theorem 4.2] requiresthat 2 V N (x) V N (y) γ V (δ)hastohodfora y Y and a x B δ (Y) with γ V K, which in particuar impies that V N (x) = V N (y) for a x,y Y, i.e., theoptima vauefunction is constantony.in casethat Y = Π X forsomeperiodicorbitπ,thiscaningeneranot be satisfied, as is the case in our motivating exampes in Section 3. In Theorem 8, condition (b) instead ony requires that V N (x) V N (x ux (P,x)) γ V (δ,n) hods for a x B δ (Y) X, where u x is the contro sequence from condition (a). Furthermore, γ V may depend on N, and in particuar for fixed N, V N (x) V N (x ux (P,x)) needs not go to zero as δ 0, but we ony require that γ V (δ,n) 0 if both N and δ 0. These reaxations are crucia such that Theorem 8 can be appied with Y = Π X for some periodic orbit Π, as shown in the foowing. 5 Checkabe sufficient conditions based on dissipativity and controabiity It is easy to verify that the motivating exampe satisfies the conditions of Theorem 8 with Y = Π X, which ex- 2 We note that in [12], a resuts were stated in terms of averaged cost functionas, i.e., the right hand side of (2) mutipied by 1/N; this is why the condition in [12, Theorem 4.2] in fact reads V N(x) V N(y) γ V(δ)/N. 5

6 painsthefactthata2-stepmpcschemeresutsinoptima cosed-oop performance, as observed in Section 3. In genera, however, the conditions of Theorem 8 might be difficut to check since they invove properties of optima trajectories and the optima vaue function. The goa of this section is to provide checkabe sufficient conditions for conditions (a) (c) of Theorem 8 for the case where Y = Π X for some periodic orbit Π of system (1). First, we briefy discuss that condition(a) foows in a straightforwardwayfromcontinuityoff and.second,weshow that a certain dissipativity condition resuts in a turnpike behavior of the system with respect to the optima periodic orbit, from which together with suitabe controabiity assumptions condition (c) foows (see Section 5.1). Third, we discuss in Section 5.2 how condition (b) can be estabished under the same dissipativity and controabiity assumptions. Finay, in Section 5.3 we combine a these aspects to obtain the main resut concerning cosed-oop performance under dissipativity and controabiity. Before turning our attention to conditions(b) and(c) of Theorem 8, we briefy discuss how for the case that Y = Π X for some P-periodic orbit Π int(x U), condition (a) with 0 = (1/P) (xp k,up k ) foows if f and arecontinuous.inthiscase,foreachx B δ (Y) forsome δ (0, δ], by definition of Y it hods that x B δ (x p j ) for some j I [0,]. Then, if f and are continuous, the controsequenceu x U P incondition(a)canbechosen as u x = (u p j,...,up,up 0,...,up j 1 ), (10) and the function γ can be computed as foows. As f and are continuous, for each compact set W X U there exist η f,η K such that f(x,u) f(x,u ) η f ( (x,u) (x,u ) )and (x,u) (x,u ) η ( (x,u) (x,u ) ) for a (x,u),(x,u ) W. Now choose W X U arge enough and δ > 0 sma enough such that B max{ δ,η P f ( δ)}(π) W; note that this is possibe since Π int(x U). Considering the above, we obtain x ux (1,x) x p [j+1] = f(x,up j ) f(xp j,up j ) η f ( (x,u p j ) (xp j,up j ) ) = η f( x x p j ) η f(δ). Using this argument recursivey resuts in x ux (k,x) x p [j+k] ηk f(δ) (11) for a k I [1,P]. Furthermore, ( 1 (x ux (k,x),u x (k)) 1 (x p P P [j+k],up [j+k] ) ) +η ( (x ux (k,x),u p [j+k] ) (xp [j+k],up [j+k] ) ) P η (η k f(δ)) =: 0 +γ (δ). (12) Since B max{ δ,η P f ( δ)}(π) W X U, it foows that x ux (k,x) X for a k I [1,P] and a x B δ(y), i.e., u x U P (x). Hence condition (a) of Theorem 8 is satisfied with γ given by (12) and 0 = (1/P) (xp k,up k ). 5.1 Turnpike behavior with respect to periodic orbits We now turn our attention to condition (c) of Theorem 8, which requires that each optima soution is cose to the set Y for at east one time instant in the interva [0,N N 0 ]. To this end, for system (1) we define the corresponding P-step system with state x = (x 0,...,x ) X P, input ũ = (u 0,...,u ) U P, dynamics x(k + 1) = f P ( x(k),ũ(k)) and initia condition 3 x (0) = x X, where f P ( x,ũ) := xũ(1,x )... xũ(p,x ) = f(x,u 0 ) f(f(x,u 0 ),u 1 ).... (13) For a given contro sequence ũ U KP with K I 1, the corresponding soution of system (13) is denoted by xũ(k,x) for k I [1,K]. This means that for a given contro sequence u U KP with K I 1, partitioned into ũ(k) = (u(kp),...,u((k+1)p 1)) for a k I [0,K 1], we have that xũ(k,x) = (x u ((k 1)P +1,x),...,x u (kp,x)) for a k I [1,K]. For the P-step system (13), define the cost function ( x,ũ) := j=0 (x ũ(j,x ),u j ). Furthermore, for ( x,ũ) X P U P and a P-periodic orbit Π, define ( x,ũ) Π := j=0 (x ũ(j,x ),u j ) Π and x ΠX := j=0 x ũ(j,x ) ΠX. We can now make the foowing assumption of strict dissipativity for the P- step system (13). Assumption 9 (Strict dissipativity) The P-step system (13) is stricty dissipative with respect to a periodic orbit Π and the suppy rate s( x,ũ) = ( x,ũ) (xp k,up k ), i.e., there exists a storage function λ : X P R and a function α K such that λ(f P ( x,ũ)) λ( x) ( x,ũ) (x p k,up k ) α ( ( x,ũ) Π ) (14) for a x X P and a ũ U P. Furthermore, the storage function λ is bounded on X P. As was discussed in [16], Assumption 9 is a sufficient condition for system (1) to be optimay operated at 3 Initia conditions for the first P 1 components of x, i.e., x 0(0),...,x P 2(0), can be arbitrary. 6

7 the periodic orbit Π. In fact, as shown in [25], Assumption 9 is sufficient and (under an additiona controabiity condition) aso necessary for a sighty stronger property than optima periodic operation, namey suboptima operation off the periodic orbit Π, meaning that the unique optima operating behavior is the periodic orbit Π. Furthermore, note that for P = 1, we recover the standard dissipativity condition which is typicay used in economic MPC in case that steady-state operation is optima (see, e.g., [3, 23]). Finay, we remark that our dissipativity condition here is weaker than the one used in the preiminary conference version [24]. Besides the above dissipativity assumption, we impose two controabiity conditions on system (1), namey oca controabiity in a neighborhood of the periodic orbit Π and finite time controabiity into this neighborhood. Assumption 10 (Loca controabiity on B κ (Π X )) There exists κ > 0, M I 0 and ρ K such that for a z Π X and a x,y B κ (z) X there exists a contro sequence u U M (x) such that x u (M,x) = y and (x u (k,x),u(k)) Π ρ(max{ x ΠX, y ΠX }) hods for a k I [0,M 1]. contro sequence u U N (x) as specified in the theorem we obtain that J N (x,u) = J N (x,u) N/P (x p k,up k ) + λ( xũ(0,x)) λ( xũ( N/P,x)) J N (x,u) ( N P 1) = J N (x,u) N P (x p k,up k )+C (x p k,up k )+C ν +C. (15) Now assume for contradiction that Q ε < N P +1 P(ν + C)/α (ε). Then there exists a set N I [0,N 1] of N Q ε > P 1 + P(ν + C)/α (ε) time instants such that (x u (k,x),u(k)) Π > ε for a k N. Hence there exists a set Ñ I [0, N/P 1] of at east (N Q ε P +1)/P > (ν+c)/α (ε) time instants such that ( xũ(k,x),ũ(k)) Π > ε for a k Ñ. By the assumption of strict dissipativity, this impies that Assumption 11 (Finite time controabiity into B κ (Π X )) For κ > 0 from Assumption 10 there exists M I 0 such that for each x X there exists k I [0,M ] and u U k (x) such that x u (k,x) B κ (Π X ). We are now in a position to state the foowing theorem which estabishes a turnpike property [6, 26, 28] for system (1) with respect to a periodic orbit Π. Turnpike properties with respect to an optima steady-state have recenty been studied in the context of economic MPC both in discrete-time [5, 12] and continuous-time [10]. The foowing resut can be seen as a generaization to the case of time-varying periodic turnpikes. Theorem 12 Suppose that Assumption 9 is satisfied. Then there exists C > 0 such that for each x X, each ν > 0, each contro sequence u U N (x) satisfying J N (x,u) (N/P) (xp k,up k ) + ν, and each ε > 0 thevaueq ε := #{k I [0,N 1] : (x u (k,x),u(k)) Π ε} satisfiestheinequaityq ε N P+1 P(ν+C)/α (ε). Proof: Let C := 2sup x X P λ( x), where x is the state of the P-step system (13), and C := C + (xp k,up k ). Define the rotated cost function L( x,ũ) := ( x,ũ) (xp k,up k )+ λ( x) λ(f P ( x,ũ)), and note that from Assumption 9, it foows that L( x,ũ) α ( ( x,ũ) Π ). Now for a given x X and u U N (x), consider the modified cost functiona JN (x,u) := N/P 1 L( xũ(k,x),ũ(k)) + N 1 k= N/P P (x u(k,x),u(k)),whereũ(k) = (u(kp),..., u((k+1)))forak I [0, N/P 1].Fromthedefinition of L and the fact that N/P N/P 1, for each J N (x,u) N/P 1 α ( ( xũ(k,x),ũ(k)) Π ) > ν +C α (ε) α (ε) = ν +C, which contradicts(15) and hence proves the theorem. Theorem 12 gives a ower bound Q ε for the number of time instants where the considered trajectory is cose to the periodic orbit Π. This turnpike resut can now be used together with the controabiity conditions specifiedby Assumptions10and11toconcudecondition (c) of Theorem 8, as shown in the foowing. Theorem 13 Suppose that Assumptions 9, 10, and 11 hod and is bounded on X U. Then condition (c) of Theorem 8 hods for Y = Π X. Proof: From Assumptions 10 and 11, it foows that for each x X there exists a contro sequence u such that the system is steered to a point on Π X in at most M +M steps and then stays on the periodic orbit Π for an arbitrary number of time steps. Hence for each N I 1 we have for some j I [0,] V N (x) J N (x,u) = min{n,m +M } 1 + N 1 (x p [k+j],up [k+j] ) (x u (k,x),u(k)) (x p [k+j],up [k+j]). (16) 7

8 Now define C := (P 1)max (x,u) Π (x,u) and consider the first sum in (16). We obtain N 1 N/P (x p [k],up [k] ) = N 1 (x p k,up k ) + N P (x p k,up k )+C. k= N/P P (x p [k],up [k] ) Furthermore, each summand in the second sum of inequaity (16) can be upper bounded by Ĉ := sup x X,u U (x,u) min (x,u) Π (x,u) <. Hence (16) yieds V N (x) (N/P) (xp k,up k ) + ν with ν := C + (M + M )Ĉ. Now choose N 1 := N 0 + P and define σ(n) arbitrary for N I [0,N1 1] and σ(n) := α 1 (P(ν+C)/(N N 0 P+1))forN I N1, with C as defined in the proof of Theorem 12. From the above considerations, it foows that for each x X and each N I N1, Theorem 12 can be appied with contro sequence u N and ε = σ(n), resuting in Q σ(n) N P + 1 P(ν + C)/α (σ(n)) = N 0. This means that there are at east N 0 time instants k I [0,N 1] such that (x u (k,x),u N N (k)) Π σ(n), and hence aso x u N (k,x) ΠX σ(n). As there are at east N 0 such time instants k, at east one of these k must satisfy k I [0,N N0], i.e., condition (c) of Theorem 8 hods with k x equa to this k and Y = Π X. Remark 14 Assumption 9 is sighty stronger than the usua definition of strict dissipativity. Namey, in Assumption 9 strictness both with respect to x and u is considered (via the function α in (14)), whie typicay this is ony required with respect to x. In Theorem 12, this resuts in a turnpike property where both the states and inputs are cose to the periodic orbit Π. In fact, the preceding resuts woud sti hod in a simiar fashion if α ( ( x,ũ) Π ) in (14) was repaced by α ( x ΠX ). In Theorem12,thedefinitionofQ ε woudthenneedtobesighty changed to Q ε := #{k I [0,N 1] : x u (k,x) ΠX ε}, which woud constitute a more cassica turnpike property invoving ony the states (but not the inputs). Whie this woud sti be sufficient for estabishing Theorem 13, strict dissipativity as in Assumption 9 (i.e.,using α ( ( x,ũ) Π ) in (14)) wi be needed for the resuts in Section Loca optima vaue function properties Next, we turn our attention to condition (b) of Theorem 8 and derive checkabe sufficient conditions for it for the case where Π is a minima periodic orbit of system (1). In this case,a state and controsequencessatisfying (x u (k,x),u(k)) Π for k I [a,b] with a,b I 0 must necessariy foow the unique P-periodic orbit specifiedbyπduringthistimeinterva 4,i.e.,thereexistsj I [0,] such that x u (k,x) = x p [k+j] and u(k) = up [k+j] 4 If Π is not minima, this is not necessariy the case, but different soutions staying inside Π for a times might exist. for a k I [a,b]. The foowing auxiiary resut shows thatasoastateandcontrosequencesstayinginasufficienty sma neighborhood of Π during some time interva must necessariy approximatey foow the unique P-periodic orbit specified by Π during this time interva. Lemma 15 Let Π be a minima P-periodic orbit for system (1), and assume that the function f in (1) is continuous. Then there exists ε > 0 such that for a 0 ε < ε and each state and contro sequence satisfying (x u (k,x),u(k)) B ε (Π) for a k I [a,b] with a,b I 0, there exists j I [0,] such that (x u (k,x),u(k)) B ε ((x p [k+j],up [k+j] )) for a k I [a,b]. Proof: See Appendix B. With the hep of the above, we can now prove the foowing resut. Theorem 16 Suppose that Assumptions 9 and 10 are satisfied for some minima P-periodic orbit Π int(x U) of system (1) and with M = ip for some i I 1. Furthermore, assume that f and are continuous and that the contro sequence u x in condition (a) of Theorem 8 is chosen according to (10). Then condition (b) of Theorem 8 is satisfied for Y = Π X. Proof: See Appendix C. Remark 17 It is worth noting the foowing interesting and fundamenta difference in the behavior of the optima statesequencesx u (,x) incase ofoptima periodic operation (as is considered in this paper) compared to the case of optima steady-state operation as in[12]. Namey, for the atter case it was estabished in [12] that each optima state sequence, which starts cose to the optima steady-state, wi stay cose to the optima steady-state for at east the first N/2time instants.this is not necessariy the case anymore when considering periodic orbits as in this paper, i.e., an optima state sequence starting cose to the optima periodic orbit does not necessariy stay cose to the periodic orbit at the beginning (see aso Exampe 18 beow for an iustration of this fact). However, as shown in the proof of Theorem 16, each optima state sequence wi be sufficienty cose to the optima periodic orbitforateast2m +1consecutivetimeinstants at some point in the interva [0,N 1] (but not necessariy at the beginning), which can be expoited to ensure condition(b) of Theorem 8. This observation aso impies that in genera ony convergence of the cosed oop to the optima periodic orbit can be expected, but not asymptotic stabiity of the set Π X.This wi be shown in Section 6. Exampe 18 With this exampe, we iustrate that even if the initia condition x is cose to Π X, this does not necessariy impy that (x u (k,x),u (k)) Π is sma for the first 2M + 1 time instants, but ony for 2M + 1 consecutive time instants at some point in the interva [0, N 1]. Namey, consider the system x(k + 1) = u(k) with state and input constraint set 8

9 u=ε (x,u)=2+ ε ε u=ε (x,u)=1+ε ε u=1 (x,u)=1+ε 3 1 u=3 (x,u)=1 u=2 (x,u)=1 u=2 (x,u)= u=4 (x,u)= 10 Fig. 2. Iustration of the states x (nodes) and feasibe transitions (edges) with corresponding input u and cost in Exampe 18. Z = {(1,2),(2,3),(2,4),(4,2),(3,ε),(ε,1),(ε,ε) : 0 ε ε} for some 0 < ε < 1 and cost function defined by (1,2) = (2,3) = 1, (2,4) = 10, (4,2) = 30, (3,ε) = (ε,1) = 1+ε and (ε,ε) = 2+ ε ε, see aso Figure 2. The system is optimay operated at the minima 4-periodic orbit Π = {(0,1),(1,2),(2,3),(3,0)}with average cost 0 := 3 (xp k,up k )/4 = 1. Furthermore, it is straightforward to verify that the Assumptions of Theorem16aresatisfied; 5 inparticuar,thecorresponding P-step system is stricty dissipative with respect to Π and the suppy rate s( x,ũ) = ( x,ũ) (xp k,up k ), i.e., (14) is satisfied with α (r) = r and storage function λ( x) = { 0 0 x 3 20(x 3) 3 < x 4 (17) Now consider an initia condition x 0 = δ with 0 δ ε and prediction horizon N = 4i for arbitrary i I 0. The (unique) optima input sequence u 0 is such that the system stays at the initia state x 0 for one step, then foows the optima periodic orbit unti it moves to x = 4 in the ast step, i.e., u 0 = (u 0 (0),...,u 0 (N 1)) = (δ,1,2,3,0,1,...,2,4) and x u = (x u (0,x 0 0 ),...,x u ( 0 0 )) = 0 (δ,δ,1,2,3,0,1,...,2,4). However, this means that (x u (0,x),u (0)) Π = ( δ,1 δ) 1/2 independent δ and N, i.e., the optima soution is not cose to Π for the first time instant. On the other hand, (x u (k,x),u (k)) Π = 0 for a k I [1,N 2]. 5.3 Cosed-oop performance under dissipativity and controabiity Combining a the above, under the assumptions of strict dissipativity of the P-step system with respect to a periodic orbit Π, oca controabiity on a neighborhood of Π and finite time controabiity into this neighborhood 5 Note that can easiy be extended such that it is continuous on R 2. Furthermore, Π is not contained in the interior of the constraint set as required in Theorem 16. However, this assumption is ony needed in the proof to ensure that the contro sequence u x defined by (10) is feasibe. Here, this is sti the case and hence the concusions of Theorem 16 are vaid. of Π, it foows that the cosed-oop asymptotic average performance is near optima, i.e., equas the average cost of the periodic orbit Π up to an error term which vanishes as N. This is summarized in the foowing coroary. Coroary 19 Consider the P-step MPC scheme as definedvia (5)andsuppose thatassumptions9, 10,and11 are satisfied for some minima P-periodic orbit Π int(x U) of system (1) and with M = ip for some i I 1. Furthermore, assume that f and are continuous and is bounded on X U. Then system (1) is optimay operated at the periodic orbit Π and there exist δ 1,δ 2 LandN I 1 suchthatfor theresutingcosedoop system, the performance estimates (6) and (7) with 0 = (1/P) (xp k,up k ) are satisfied for a x X, a N I N+P and a K I 0. Remark 20 Coroary 19 gives cosed-oop performance guarantees for the P-step MPC scheme as defined via (5), where P is the period of the corresponding minima periodic orbit Π. Nevertheess, we note that in certain cases, Theorem 8 can possiby aso be appied with 0 equa to the average cost of the optima periodic orbit, but for some vaue of P which is ess than the period of the optima periodic orbit. This is iustrated with the foowing exampe. Exampe 21 Consider the one-dimensiona system x(k + 1) = u(k) with state and input constraint set Z = {(0,1),(1,2),(2,3),(3,0)} consisting of four eements ony and cost (x,u) defined as (0,1) = (2,3) = 1, (1,2) = (3,0) = 1+ε, forsomeconstantε > 0.Thismeansthattheonyfeasibe state and input sequence foows the periodic orbit Π = Z for a times. It is easy to see that the conditions of Theorem 8 can be satisfied with 0 = 3 (xp k,up k ) = 1 + ε/2 and P = 2k for a k I 1, and hence the performance estimates (6) and (7) aso hod if, e.g., a 2-step MPC scheme is appied. Remark 22 Some comments on impications of the presented resuts on practica impementation aspects of the proposed MPC scheme are in order. First, note that a priori knowedge of the period ength P of the optima periodic orbit is needed for impementing the corresponding P-stepMPC scheme. On the other hand, the optima periodic orbit itsef needs not be known for impementing the proposed MPC method, in contrast to the case when using additiona termina constraints as in[3, 27]. In our setting, the optima periodic orbit ony needs to be known for a theoretica vaidation of the proposed method, in particuar for verifying Assumptions In practica appications, the period ength P of the optima periodic orbit coud, e.g., be determined via (offine) simuations of the system. 9

10 6 Convergence to the optima periodic orbit In the foowing, we show that under the same conditions as needed to estabish near optima performance, the cosed-oop system aso asymptoticay converges into a neighborhood of the optima periodic orbit Π, i.e., the cosed-oop system finds the optima operating behavior. This neighborhood can be made arbitrariy sma by choosing N arge enough. Theorem 23 Suppose that the conditions of Coroary 19 are satisfied and that the storage function λ is continuous on X P. Then there exists ˆN I 1 such that for a N I ˆN, the cosed-oop system resuting from appication of the P-step MPC controer (5) practicay asymptoticay converges to the optima periodic orbit Π, i.e., there exists ν L and for each x X some ˆk I 0 and j I [0,] such that (x umpc (k,x),u MPC (k)) B ν(n) (x p [k+j],up [k+j]) for a k I ˆk. Proof: As shown above, the conditions of Coroary 19 impy that the conditions of Theorem 8 and hence aso those of Proposition 7 are satisfied. Now et ˆδ(N) := δ 1 (N P) + Pδ 2 (N P) with δ 1 and δ 2 from Proposition 7, fix β > 0 and define a := α 1 ((1+β)ˆδ(N)) with α from (14). Since f and λ are assumed to be continuous, there exist η f,η λ K such that f(x,u) f(x,u ) η f ( (x,u) (x,u ) ) for a(x,u),(x,u ) B a (Π)and λ( x) λ(ỹ) η λ ( x ỹ ) for a x Π X := {(x p k,...,xp,xp 0,...,xp k 1 ) : k I [0,] } and a ỹ such that x ỹ P max{a,η f (a)}. Next,eta := ˆδ(N)+2γ V (a,n)+2η λ (()a+η f (a)) with γ V from Theorem 8 and a := α 1 (a + ˆδ(N)). Then, aong the cosed-oop sequence, for each i I 0 define the vaues ˆV N (x umpc (ip,x)) := V N (x umpc (ip,x))+ λ( x(i)) with 6 x(i) := (x umpc ((i 1)P + 1,x),...,x umpc (ip,x)). Furthermore, for each j I [0,], et ˆV N (x p j ) := V N(x p j ) + λ( x p j ) with x p j := (x p j+1,...,xp,xp 0,...,xp j ). Finay, choose ˆN I 1 argeenoughsuchthatthefoowingfourconditions are satisfied 7 for a N I ˆN: (i) max{a,a } ε with ε from Lemma 15, (ii) η f (a) max{ δ,η f P( δ)} with δ chosen as in the proof of Theorem 16, (iii) a < min ˆV(x p i,j I[0,],ˆV(x p i ) ˆV(x p j )>0 i ) ˆV(x p j ), and (iv) ˆN N + P with N from Coroary 20. In the foowing, consider an arbitrary N I ˆN. 6 Fori = 0,thevauesx umpc ( P+1,x),...,x umpc ( 1,x) can be arbitrary. 7 Note that this is possibe since a, a, and a are decreasing to zero for increasing N. Furthermore, condition (iii) does not appy if ˆV(x p i ) = ˆV(x p j ) for a i,j I [0,]. Appying (6) with K = 1, we obtain that for a i I 0 V N (x umpc ((i+1)p,x)) V N (x umpc (ip,x)) (i+1)p 1 (x umpc (k,x),u MPC (k))+p 0 + ˆδ(N). k=ip Using Assumption 9, this resuts in ˆV N (x umpc ((i+1)p,x)) ˆV N (x umpc (ip,x)) = V N (x umpc ((i+1)p,x)) V N (x umpc (ip,x)) + λ( x(i+1)) λ(x(i)) (18) (i+1)p 1 (18),(14) α ( (x umpc (k,x),u MPC (k)) Π )+ ˆδ(N). k=ip (19) Since ˆV is bounded from beow (which foows from the fact that both and λ are bounded from beow), from (19)itfoowsthatforeachinitiaconditionx X,there exists some i I 0 such that (i+1)p 1 k=ip α ( (x umpc (k,x),u MPC (k)) Π ) (1+β)ˆδ(N); (20) denotethe smaestsuch iby i 1. In particuar,from(20) it foows that (x umpc (k,x),u MPC (k)) Π α 1 ((1 + β)ˆδ(n)) = a for a k I [i1p,(i 1+1)]. Now define ī 1 := inf{i I i1+1 : (i+1)p 1 k=ip α ( (x umpc (k,x),u MPC (k)) Π ) > (1+β)ˆδ(N)}, (21) and consider the foowing. If ī 1 =, we have (x umpc (k,x),u MPC (k)) Π a for a k I i1p, and hence the statement of the theorem with ν(n) = a and ˆk = i 1 P foowsfromappicationoflemma15(considering the fact that a ε). On the other hand, in case that ī 1 <, we have (x umpc (k,x),u MPC (k)) Π a for a k I [i1p,ī 1]. Hence we can again appy Lemma 15 to concude that there exists j i1 I [0,] such that (x umpc (k,x),u MPC (k)) B a (x p [k+j i1 ],up [k+j i1 ] ) for a k I [i1p,ī 1]. By continuity of f, it then foows that x umpc (ī 1 P,x) x p [ī 1P+j i1 ] η f(a). Since [i 1 P +j i1 ] = [ī 1 P +j i1 ] = j i1 and due to the fact that η f (a) max{ δ,η P f ( δ)}, the proof of Theorem 16 shows that V(x umpc (ī 1 P,x)) V(x p j i1 ) γ V (a,n), (22) 10

11 and hence, by definition of ˆV, ˆV(x umpc (ī 1 P,x)) ˆV(x p j i1 ) γ V (a,n)+ λ( x(ī 1 )) λ( x p j i1 ) γ V (a,n)+η λ ((P 1)a+η f (a)). (23) Furthermore, by the same argument as above, it foows from (19) that there exists some i I ī1 such that (20) is satisfied; denote the smaest such i by i 2. This means that (x umpc (k,x),u MPC (k)) Π a for a k I [i2p,(i 2+1)]. Using again Lemma 15, this impies that there exists j i2 I [0,] such that (x umpc (k,x),u MPC (k)) B a (x p [k+j i2 ],up [k+j i2 ]) for a k I [i2p,(i 2+1)] and x umpc ((i 2 +1)P,x) x p [j i2 ] η f (a). Using the same reasoning as in the derivation of (23), we can concude that ˆV(x umpc ((i 2 +1)P,x)) ˆV(x p j i2 ) γ V (a,n)+η λ ((P 1)a+η f (a)). (24) Combining (23) (24) with the fact that ˆV N (x umpc ((i+ 1)P,x)) ˆV N (x umpc (ip,x)) 0 for a i I [ī1,i 2 1] by definition of ī 1 and i 2 and ˆV N (x umpc ((i 2 + 1)P,x)) ˆV N (x umpc (i 2 P,x)) ˆδ(N) by (19), it foows that ˆV(x p j i2 ) ˆV(x p j i1 )+ ˆδ(N)+2γ V (a,n) +2η λ ((P 1)a+η f (a)) = ˆV(x p j i1 )+a. But then, since N is chosen such that a < min ˆV(x p i,j I[0,],ˆV(x p i ) ˆV(x p j )>0 i ) ˆV(x p j ), it foows that ˆV(x p j i2 ) ˆV(x p j i1 ). Nowincasethat ˆV(x p j i2 ) = ˆV(x p j i1 ),considerthefoowing. Combining (23) (24) with the fact that ˆV(x p j i2 ) = ˆV(x p j i1 ), we obtain ˆV(x umpc ((i 2 +1)P,x)) ˆV(x umpc (ī 1 P,x)) 2γ V (a,n) 2η λ ((P 1)a+η f (a)) = a + ˆδ(N). (25) Since ˆV N (x umpc ((i + 1)P,x)) ˆV N (x umpc (ip,x)) 0 for a i I [ī1,i 2 1] by definition of ī 1 and i 2 and ˆV N (x umpc ((i 2 +1)P,x)) ˆV N (x umpc (i 2 P,x)) ˆδ(N) by (19), from (19) and (25) it foows that (i+1)p 1 k=ip α ( (x umpc (k,x),u MPC (k)) Π ) a + ˆδ(N) for a i I [ī1,i 2 1], and hence in particuar (x umpc (k,x),u MPC (k)) Π α 1 (a + ˆδ(N)) = a for a k I [ī1p,i 2]. Now for both the cases where (a) ˆV(x p j i2 ) = ˆV(x p j i1 ) and (b) ˆV(x p j i2 ) < ˆV(x p j i1 ), we can go back to (21) and repeat a the steps from there (repacing i 1 by i 2 etc.). Doing this recursivey, it foows that case (b) can occur at most P 1 times (since Π ony contains P eements). Denote by i r the ast vaue of the sequence i 1,i 2,... obtained in this way such that case (b) occurs, i.e., such that ˆV(x p j ir ) < ˆV(x p j ir 1 ). Combining the above, for a k I ir we have (x umpc (k,x),u MPC (k)) Π max{a,a }. The concusion of the theorem with ˆk := i r P and ν(n) := max{a,a } then foows from appication of Lemma 15. Noting that ν L as required then concudes the proof of Theorem 23. Remark 24 A cose inspection of the proof reveas that itcanbemodified suchthatrequiring continuityof λony on Π X := {(x p k,...,xp,xp 0,...,xp k 1 ) : k I [0,]} (instead of X P ) is enough. Remark 25 One might wonder why from (19), one cannot use standard Lyapunov and invariance arguments which are typicay empoyed when estabishing practica asymptotic stabiity (or convergence). The reason for this is that whie both the inequaities (19) and ˆV N (x umpc (ip,x)) α ( x umpc (ip,x) ΠX ) hod, in genera there is no upper bound on ˆV of the form ˆV N (x umpc (ip,x)) α 2 ( x umpc (ip,x) ΠX ) for some α 2 K. This resuts in the fact that the cosed-oop stateandinputneednotstayinaneighborhood oftheoptima periodic orbit once the cosed-oop state is cose to Π X.This was,e.g.,thecaseinexampe18,where wehad (x umpc (0,x),u MPC (0)) Π 1/2 for arbitrariy sma x ΠX. In the proof of Theorem 23, practica asymptotic convergence of the cosed-oop system coud nevertheess be estabished by showing that such a situation can ony occur a finite number of times. Remark 26 As aready mentioned above, in case of optima steady-state operation, not ony (practica) asymptotic convergence but (practica) asymptotic stabiity of the optima steady-state can be estabished. In both cases with and without additiona termina constraints, this is typicay done by using as a Lyapunov function the optima vaue function of a modified optimization probem, where J N (x,u) as defined in the proof of Theorem 12 is used as objective function. This is possibe since the optima input sequences resuting from this modified optimization probem are the same (when using termina constraints) or, at east initiay, sufficienty cose (when using no termina constraints) to the ones of the originaoptimizationprobem(see,e.g.,[1,3,12,15]).onthe other hand, in case of optima periodic operation, this is not necessariy the case anymore (such as, e.g., in Exampe 18), and hence the optima vaue function of the modified probem cannot be used as a Lyapunov function. Instead, in the proof of Theorem 23 we empoy ˆV N as a Lyapunov-ike function in order to estabish practica asymptotic convergence of the cosed-oop system, using the more invoved argument as discussed in Remark