Asymptotic Stability of Economic NMPC: The Importance of Adjoints

6th IFAC Conference on Nonlinear Model Predictive Control 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, USA, August 19-22, 2018 6th IFAC IFAC Conference on Nonlinear Nonlinear Model Predictive Predictive Control Control 6th on Model Available online at www.sciencedirect.com Madison,Conference WI, USA, August 19-22, 2018 6th IFAC Conference on Nonlinear Model Predictive Control Madison, WI, 19-22, Madison, WI, USA, USA, August August 19-22, 2018 2018 Madison, WI, USA, August 19-22, 2018

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IFAC PapersOnLine 51-20 (2018) 157–168

Asymptotic Asymptotic Stability Stability of of Economic Economic NMPC: NMPC: Asymptotic Stability of Economic NMPC: The Importance Adjoints Asymptotic Stability of of Economic NMPC: The Importance of Adjoints The Importance of Adjoints The Importance of Adjoints Timm Faulwasser ∗∗ Mario Zanon ∗∗ ∗∗

Timm Faulwasser ∗∗ Mario Zanon ∗∗ Timm Faulwasser Faulwasser Mario Mario Zanon Zanon ∗∗ Timm ∗ Timm Faulwasser Mario Zanon ∗∗ ∗ ∗ Institute for Automation and Applied Informatics, Karlsruhe Institute for Automation and Applied Informatics, Karlsruhe ∗ ∗ Institute for and Applied Informatics, Karlsruhe Institute of Technology Germany. (e-mail: for Automation Automation and(KIT), Applied Informatics, Karlsruhe Institute of Technology (KIT), Germany. (e-mail: ∗ Institute Institute for Automation and Applied Informatics, Karlsruhe Institute of Technology (KIT), Germany. (e-mail: [email protected]). Institute of [email protected]). Technology (KIT), Germany. (e-mail: ∗∗ Institute of [email protected]). Technology (KIT), Germany. (e-mail: Advanced Studies Lucca, Italy. (e-mail: ∗∗ IMT School for [email protected]). Advanced Studies Lucca, Italy. (e-mail: ∗∗ IMT School for [email protected]). ∗∗ IMT School for Advanced [email protected]) Advanced Studies Studies Lucca, Lucca, Italy. Italy. (e-mail: (e-mail: [email protected]) ∗∗ IMT School for IMT School for Advanced Studies Lucca, [email protected]) [email protected]) Italy. (e-mail: [email protected]) Abstract: Recently, it has been shown Abstract: Recently, it has been shown in in aa sampled-data sampled-data continuous-time continuous-time setting setting that that under under Abstract: Recently, it shown in aa sampled-data continuous-time setting that certain regularity assumptions simple end penalty enforces exponential of Abstract: Recently, it has has been beena shown inlinear sampled-data continuous-time setting stability that under under certain regularity assumptions a simple linear end penalty enforces exponential stability of Abstract: Recently, it has been shown in a sampled-data continuous-time setting that under certain regularity regularity assumptions a simple simple Control linear end end penalty enforces exponential stability of Economic (nonlinear) Model Predictive (EMPC) without terminal constraints. This certain assumptions a linear penalty enforces exponential stability of Economic (nonlinear) Model Predictive Control (EMPC) without terminal constraints. This certain regularity assumptions a simple Control linear end penalty enforces exponential stability of Economic (nonlinear) Model Predictive Control (EMPC) without terminal constraints. This paper investigates the same framework in the discrete-time case, i.e. we establish sufficient Economic (nonlinear) Model Predictive (EMPC) without terminal constraints. This paper investigates the Model same framework in the discrete-time case, terminal i.e. we establish sufficient Economic (nonlinear) Predictive Control (EMPC) without constraints. This paper investigates the same framework in the discrete-time case, i.e. we establish sufficient conditions for stability of steady an scheme without paper investigates the same framework in the discrete-time case, i.e. we establish conditions for asymptotic asymptotic stability of the the optimal optimal steady state state under under an EMPC EMPC schemesufficient without paper investigates theThe same framework in discrete-time case, i.e. we be establish conditions for stability of steady state under an EMPC scheme without terminal constraints. key ingredient is aathelinear end penalty that understood as conditions for asymptotic asymptotic stability of the the optimal optimal steady state under ancan EMPC schemesufficient without terminal constraints. The key ingredient is linear end penalty that can be understood as aa conditions for asymptotic stability of the optimal steady state under an EMPC scheme without terminal correction constraints.ofThe The key ingredient is aa linear linear end penalty that that can be beofunderstood understood as aa gradient the stage cost by means of the adjoint/dual variable the underlying terminal constraints. key ingredient is end penalty can as gradient correction ofThe the key stageingredient cost by means of the adjoint/dual variable the underlying terminal constraints. is a linear end penalty that can for beof understood ason a gradient correction correction of the the problem. stage cost costAlthough by means means of the the adjoint/dual variable ofEMPC the underlying underlying steady-state optimization almost all stability proofs focus gradient of stage by of adjoint/dual variable of the steady-state optimization problem. Although almost all stability proofs for EMPC focus on gradient correction of the stage cost by means of the adjoint/dual variable of the underlying steady-state optimization problem. Although almost all stability proofs for EMPC focus on primal variables, our elucidate the of the for steady-state optimization problem. Although all stability for EMPC focus on primal variables, our developments developments elucidate almost the importance importance of proofs the adjoints adjoints for achieving achieving steady-state optimization problem. Although almost all stability proofs foradaptive EMPC focus on primal our developments elucidate the importance of the for asymptotic stability terminal constraints. we propose an gradient primal variables, variables, ourwithout developments elucidate the Moreover, importance of the adjoints adjoints for achieving achieving asymptotic stability without terminal constraints. Moreover, we propose an adaptive gradient primal variables, our developments elucidate the Moreover, importance the adjoints for achieving asymptotic stability without terminal constraints. Moreover, we of propose an adaptive adaptive gradient correction strategy which alleviates the need for solving explicitly the steady-state optimization. asymptotic stability without terminal constraints. we propose an gradient correction strategy which alleviates the constraints. need for solving explicitly the steady-state optimization. asymptotic stability without terminal we results. propose an adaptive gradient correction strategy which alleviates theexamples need for for solving solving explicitly the steady-state optimization. Finally, we draw upon two simulation to Moreover, illustrate our correction strategy which alleviates the need explicitly the steady-state optimization. Finally, we draw upon two simulation examples to illustrate our results. correction strategy which alleviates the need for solving explicitly the steady-state optimization. Finally, we draw upon two simulation examples to illustrate our results. Finally, we draw upon two simulation examples to illustrate our results. © 2018, IFAC (International of Automatic Hosting Elsevier Ltd. All rights reserved. Finally, we draw upon twoFederation simulation examples Control) to illustrate ourbyresults. Keywords: Keywords: Economic Economic MPC, MPC, dissipativity, dissipativity, asymptotic asymptotic stability stability Keywords: Keywords: Economic Economic MPC, MPC, dissipativity, dissipativity, asymptotic asymptotic stability stability Keywords: Economic MPC, dissipativity, asymptotic stability 1. INTRODUCTION property property in in the the open-loop open-loop predictions, predictions, whereby whereby the the turnturn1. INTRODUCTION 1. property in the the open-loop predictions, whereby the turnpike happens to be the optimal steady state (Gr¨ u ne, 2013; 1. INTRODUCTION INTRODUCTION property in open-loop predictions, whereby the turnpike happens to be the optimal steady state (Gr¨ u ne, 2013; 1 predictions, 1. INTRODUCTION property in et the open-loop whereby turnpike happens happens toal., be the the optimal steady state state (Gr¨ uthe ne, 2013; 2013; Faulwasser 2014). again under mild 1 Moreover, pike to be optimal steady (Gr¨ u ne, The notion of Economic Model Predictive Control (EMPC) Faulwasser ettoal., 2014). Moreover, again(Gr¨ under mild 1 The notion of Economic Model Predictive Control (EMPC) pike happens be the optimal steady state u ne, 2013; 1 Faulwasser et al., 2014). Moreover, again under mild assumptions, the existence of a turnpike implies recursive The notion of Economic Model Predictive Control (EMPC) Faulwasser et al., 2014). Moreover, again under mild has become part of standard MPC terminology. It refers The notion of Economic Model Predictive Control (EMPC) assumptions, the existence 1 of a turnpike implies recursive has become part of standard MPC terminology. It refers Faulwasser et al., 2014). Moreover, again under mild The notion of Economic Model Predictive Control (EMPC) assumptions, the existence of a turnpike implies recursive feasibility of the NMPC loop (Faulwasser and Bonvin, has become part of standard MPC terminology. It refers assumptions, the existence of a turnpike implies recursive to linear and nonlinear MPC schemes with economic (i.e. has become part of standard MPC terminology. It refers feasibility of the NMPC loop (Faulwasser and Bonvin, to linear and nonlinear MPC schemes with economic (i.e. assumptions, the existence of a turnpike implies recursive has become part of standard MPC terminology. It refers feasibility of the NMPC loop (Faulwasser and Bonvin, 2015, 2017). to linear and nonlinear MPC schemes with economic (i.e. feasibility of the NMPC loop (Faulwasser and Bonvin, almost generic) objectives, see e.g. (Rawlings and Amto lineargeneric) and nonlinear MPCsee schemes with economic (i.e. 2015, 2017). almost objectives, e.g. (Rawlings and Amfeasibility of the NMPC loop (Faulwasser and Bonvin, to linear and nonlinear schemes with economic (i.e. 2015, 2017). almost generic) objectives, see e.g. and 2015, 2017). rit, 2009; Diehl et al., al., MPC 2011; Angeli et al., al., 2012). The almost generic) objectives, see e.g. (Rawlings (Rawlings and AmAmDissipativity-based approaches rit, 2009; Diehl et 2011; Angeli et 2012). The 2015, 2017). almost generic) objectives, see e.g. (Rawlings and AmDissipativity-based approaches with with and and without without termitermirit, et 2011; Angeli et 2012). promise ofDiehl EMPC is to to avoid the quite quite cumbersome rit, 2009; 2009;of Diehl et al., al., 2011; Angeli et al., al., cumbersome 2012). The The Dissipativity-based Dissipativity-based approaches with and without terminal constraints differ crucially with respect to the closedpromise EMPC is avoid the approaches with and without termirit, 2009; Diehl et al., 2011; Angeli et al., 2012). The nal constraints differ crucially with respect to the closedpromise of EMPC is to avoid the quite cumbersome translation ofEMPC economic objectives into quite corresponding con- nal promise of of is objectives to avoid the cumbersome Dissipativity-based approaches withrespect and without termiconstraints differ crucially with to the closedloop stability properties: Using appropriate terminal contranslation economic into corresponding connal constraints differ crucially with respect to the closedpromise of(Morari is al., to 1980; avoid Kadam the cumbersome stability properties: Usingwith appropriate terminal contranslation of economic objectives into corresponding control tasks et and Marquardt, translation ofEMPC economic objectives into quite corresponding con- loop nal constraints differ crucially respect to the closedloop stability properties: Using appropriate terminal constraints and penalties one establishes Lyapunov stability trol tasks (Morari et al., 1980; Kadam and Marquardt, loop stability properties: Using appropriate terminal contranslation of economic objectives into corresponding constraints and penalties one establishes Lyapunov stability trol tasks (Morari et al., 1980; Kadam and Marquardt, 2007). This is achieved via the explicit consideration of trol tasks (Morari et al., 1980; Kadam and Marquardt, loop stability properties: Using appropriate terminal constraints and penalties one establishes Lyapunov stability of the optimal steady state (Diehl et al., 2011; Angeli et al., 2007). This is achieved via the explicit consideration of straints and penalties one establishes Lyapunov stability trol tasks (Morari et al., 1980; Kadam and Marquardt, of the optimal steady state (Diehl et al., 2011; Angeli et al., 2007). This is achieved via the explicit consideration of user-provided economic objectives (instead of steady-state 2007). This is achieved via the explicit consideration of straints and penalties one establishes Lyapunov stability of the optimal steady state (Diehl et al., 2011; Angeli et al., 2012) or periodic orbit. However, without them one shows user-provided economic objectives (instead of steady-state of the optimal steady state (Diehl et al., 2011; Angeli et al., 2007). This is achieved via the explicit consideration of 2012) or periodic orbit. However, without them one shows user-provided economic objectives (instead of steady-state tracking ones).economic objectives (instead of steady-state 2012) user-provided of the optimal steady state (Diehl et al., 2011; Angeli et al., or periodic orbit. However, without them one shows practical stability (Gr¨ u ne, 2013; Gr¨ u ne and Stieler, 2014; tracking ones). 2012) or periodic orbit. However, without them one shows user-provided economic objectives (instead of steady-state practical stabilityorbit. (Gr¨ une, 2013; Gr¨ une and Stieler, 2014; tracking ones). tracking ones). 2012) or periodic However, without them one shows practical stability (Gr¨ u ne, 2013; Gr¨ u ne and Stieler, 2014; Faulwasser and Bonvin, 2015); i.e. convergence to a neighpractical stability (Gr¨ une, 2013; une and Stieler, 2014; The recent progress on on EMPC EMPC comprises comprises Lyapunov-based Lyapunov-based Faulwasser tracking ones). and Bonvin, 2015); i.e.Gr¨ convergence to a neighThe recent progress practical stability (Gr¨ une, 2013; Gr¨ une and Stieler, 2014;a Faulwasser and Bonvin, 2015); i.e. convergence to a a neighneighborhood of the optimal steady state without requiring The recent progress on EMPC comprises Lyapunov-based Faulwasser and Bonvin, 2015); i.e. convergence to stability results (Ellis et al., 2014); approaches using dissiThe recent progress on EMPC comprises Lyapunov-based borhood of the optimal steady state without requiring a stability results (Ellis et al., 2014); approaches using dissiFaulwasser and Bonvin, 2015); i.e. convergence to a neighThe recent progress onet comprises Lyapunov-based borhood of the the optimal optimal steady state state without without requiring requiring a a priori knowledge of this target. stability results (Ellis al., approaches using dissiborhood of steady pativity conditions and terminal constraints (Diehl al., stability results (Ellis etEMPC al., 2014); 2014); approaches usinget dissipriori knowledge of this target. pativity conditions and terminal constraints (Diehl et al., borhood of the optimal steady state without requiring a stability results (Ellis et terminal al., 2014); approaches usingM¨ dissipriori knowledge of this target. pativity conditions and constraints (Diehl et al., priori knowledge of this target. 2011; Angeli et al., 2012; Zanon et al., 2013, 2017b; u ller pativity conditions and terminal constraints (Diehl et al., we in 2011; Angeli et al., 2012; Zanon etconstraints al., 2013, 2017b; M¨ ller priori knowledge this target. pativity conditions and terminal (Diehl etu al., Recently, Recently, we have haveofshown shown in (Zanon (Zanon and and Faulwasser, Faulwasser, 2018) 2018) 2011; et al., Zanon al., M¨ u ller et al.,Angeli 2015); and dissipativity approaches without end 2011; Angeli etand al., 2012; 2012; Zanon et etapproaches al., 2013, 2013, 2017b; 2017b; M¨ uend ller Recently, we have shown in (Zanon and Faulwasser, 2018) in a sampled-data continuous-time NMPC setting that et al., 2015); dissipativity without Recently, we have shown in (Zanon and Faulwasser, 2018) 2011; Angeli et al., 2012; Zanon et al., 2013, 2017b; M¨ u ller in a sampled-data continuous-time NMPC setting that et al., 2015); and dissipativity approaches without end penalties and and terminal constraints (Gr¨ une, ne, 2013; 2013; Gr¨ u ne in et al., 2015); dissipativity approaches without end Recently, we have shown in assumptions) (Zanon and Faulwasser, 2018) a sampled-data continuous-time NMPC setting that (under certain regularity a simple linear penalties and terminal constraints (Gr¨ u Gr¨ u ne in a sampled-data continuous-time NMPC setting that et al., 2015); and dissipativity approaches without end (under certain regularity assumptions) a simple linear penalties and terminal constraints (Gr¨ u ne, 2013; Gr¨ u ne and Stieler, 2014; Faulwasser and Bonvin, 2015). Appenalties and2014; terminal constraints (Gr¨ une, 2013; Gr¨ une in a sampled-data continuous-time NMPC setting that (under certain regularity assumptions) a simple linear end penalty closes this evident stability gap between and Stieler, Faulwasser and Bonvin, 2015). Ap(under certain regularity assumptions) a simple linear penalties and terminal constraints (Gr¨ u ne, 2013; Gr¨ u ne end penalty closes this evident stability gap between and 2014; and Bonvin, Approximate schemes with stability stability guarantees have been been and Stieler, Stieler,schemes 2014; Faulwasser Faulwasser andguarantees Bonvin, 2015). 2015). Ap- (under certaincloses regularity assumptions) a simple linear end penalty closes this evident stability gap between dissipativity-based EMPC with and without terminal conproximate with have end penalty this evident stability gap between and Stieler, 2014; Faulwasser Bonvin, 2015). Ap- dissipativity-based EMPC with andstability without terminal conproximate with guarantees have proposed inschemes (Zanon et al., al.,stability 2014,and 2016, 2017a). We been refer proximate schemes with stability guarantees have been end penalty closes this aevident gap between dissipativity-based EMPC with and without terminal constraints. More precisely, linear end penalty can restore proposed in (Zanon et 2014, 2016, 2017a). We refer dissipativity-based EMPC with and without terminal conproximate schemes with stability guarantees have been straints. More precisely, a linear end penalty can restore proposed in (Zanon et al., 2014, 2016, 2017a). We refer to (Faulwasser et al., 2018; Ellis et al., 2014) for recent proposed in (Zanon et2018; al., 2014, 2016, 2017a). Werecent refer straints. dissipativity-based EMPC with and without terminal conMore precisely, a linear end penalty can restore asymptotic stability of the optimal steady state without to (Faulwasser et al., Ellis et al., 2014) for straints. More precisely, a linear end penalty can restore proposed in (Zanon al., 2014, 2016, Werecent refer asymptotic stability of the optimal steady state without to et Ellis al., 2014) for literature overviews. In2018; dissipativity-based EMPC withto (Faulwasser (Faulwasser et al., al.,et 2018; Ellis et et al., 2017a). 2014) for recent straints. More precisely, a linear end penalty can restore asymptotic stability of the optimal steady state without any need for a terminal constraint. The crucial key insight literature overviews. In dissipativity-based EMPC withasymptotic of constraint. the optimalThe steady state to (Faulwasser et al., Ellis et al., 2014) for recent need forstability a terminal crucial keywithout insight literature overviews. dissipativity-based without end penalties, penalties, theIn crucial observation isEMPC that, under literature overviews. In2018; dissipativity-based EMPC with- any asymptotic of constraint. the optimalThe steady state any need need for forstability terminal constraint. The crucial keywithout insight out end the crucial observation is that, under any aa terminal crucial key insight literature overviews. In dissipativity-based EMPC without end penalties, the crucial observation is that, under mild reachability assumptions, dissipativity of the Optimal out end penalties, the crucial dissipativity observation of is the that, under 11any need for a terminal constraint. The crucial insight Turnpike properties are wellknown in optimal controlkey approaches mild reachability assumptions, Optimal out end penalties, the crucial observation is of that, under 11 Turnpike properties are wellknown in optimal control approaches mild reachability assumptions, dissipativity of the Optimal Control Problem (OCP) implies the existence a turnpike mild reachability assumptions, dissipativity of the Optimal to control problems in economics, see McKenzie (1976). Early obserTurnpike properties are wellknown in optimal control approaches Turnpike properties are wellknown in optimal(1976). controlEarly approaches Control Problem (OCP) implies the existence a control problems in economics, see McKenzie obsermild reachability assumptions, ofof the Optimal to 1 Turnpike Control Problem implies the of aa turnpike turnpike properties are wellknown in optimal control approaches Control Problem (OCP) (OCP) impliesdissipativity the existence existence of turnpike vations of problems turnpike properties cansee be McKenzie traced back to von Neumann to control problems in economics, economics, see McKenzie (1976). Early obserto control in (1976). Early observations of turnpike properties can be traced back to von Neumann Control Problem (OCP) implies the existence of a turnpike  Both authors to control in economics, see (1976). obser(1945) while the term has been by Dorfman etEarly al. (1958). are supported by the Deutsche Forschungsgemeinvations of problems turnpike properties cancoined be McKenzie traced back to von Neumann 

 Both authors are supported by the Deutsche Forschungsgemeinvations of turnpike properties cancoined be traced back to von Neumann (1945) while the term has been by Dorfman et al. (1958).   vations of there turnpike properties cancoined be traced back to von Neumann Recently, has been re-newed interest the context of optimal schaft, 2056/1 and 2056/4-1.ForschungsgemeinFurthermore, TF (1945) while while the term has been coined byinDorfman Dorfman et al. al. (1958). Both Grants authors WO are supported supported by WO the Deutsche Deutsche Forschungsgemein(1945) the term been by et (1958). Both authors are by the Recently, there has beenhas re-newed interest in the context of optimal schaft, Grants WO 2056/1 and WO 2056/4-1. Furthermore, TF  (1945) while the term has been coined et al. (1958). Both Grants authors are supported by the Deutsche Forschungsgemeinand predictive control (Angeli et al., 2009;byTr´ ethe lat context and Zuazua, 2015; acknowledges support from the Baden-W¨ urttemberg Stiftung under Recently, there has been re-newed interest in of schaft, WO 2056/1 and WO 2056/4-1. Furthermore, TF Recently, there has been re-newed interest inDorfman of optimal optimal schaft, Grants WO 2056/1 and WO 2056/4-1. Furthermore, TF and predictive control (Angeli et al., 2009; Tr´ ethe lat context and Zuazua, 2015; acknowledges support from the Baden-W¨ urttemberg Stiftung under Recently, has been re-newed interest inGugat context of optimal schaft, WO 2056/1 and WO 2056/4-1. Furthermore, TF Damm et there al., 2014; Faulwasser etal., al., 2017;Tr´ et al., 2016). the EliteGrants Programme for Postdocs. and predictive control (Angeli et 2009; eethe lat and Zuazua, 2015; acknowledges support from the Baden-W¨ u rttemberg Stiftung under and predictive control (Angeli et al., 2009; Tr´ lat and Zuazua, 2015; acknowledges support from the Baden-W¨ u rttemberg Stiftung under Damm et al., 2014; Faulwasser et al., 2017; Gugat et al., 2016). the Elite Programme for Postdocs. and predictive control (Angeli etet elat and Zuazua, acknowledges supportfor from the Baden-W¨ urttemberg Stiftung under Damm et al., al., 2014; 2014; Faulwasser etal., al.,2009; 2017;Tr´ Gugat et al., al., 2016).2015; the Elite Elite Programme Programme for Postdocs. Damm et Faulwasser al., 2017; Gugat et 2016). the Postdocs. Damm et al., 2014; Faulwasser et al., 2017; Gugat et al., 2016). the Elite Programme for (International Postdocs. 2405-8963 © 2018 2018, IFAC IFAC Federation of Automatic Control) Copyright © 182 Hosting by Elsevier Ltd. All rights reserved. Copyright © under 2018 IFAC 182 Control. Peer review responsibility of International Federation of Automatic Copyright © 182 Copyright © 2018 2018 IFAC IFAC 182 10.1016/j.ifacol.2018.11.009 Copyright © 2018 IFAC 182

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is that a linear end penalty corresponds to a linear rotation of the stage cost. If this rotation is done such that it corrects for the gradient of the steady-state Lagrangian (modulo technical assumptions) asymptotic stability of the optimal steady state is recovered. In other words, in (Zanon and Faulwasser, 2018) we propose to correct by means of the adjoint/dual variable of the underlying steady-state optimization problem which highlights the importance of the adjoints for achieving asymptotic stability without terminal constraints. The main contribution of the present paper is twofold. We show that the main sampled-data results of (Zanon and Faulwasser, 2018) carry over to the discrete-time setting. However, the proposed gradient correcting end penalty still requires knowledge of the adjoint/dual solution of the steady-state optimization problem. To overcome this issue, we propose a novel EMPC scheme without terminal constraints that employs an adaptive gradient correction strategy. The remainder of the paper is structured as follows: Section 2 discusses the problem statement, while in Section 3 we present the main stability results. Section 4 proposes a novel scheme for adaptive gradient correction EMPC; Section 5 illustrates our finding via simulations; and Section 6 concludes the paper. Notation We denote the state and control of a system, respectively, as x ∈ Rnx and u ∈ Rnu . We denote partial derivatives of functions by a subscript: e.g. for  : Rnx × Rnu → R   ∂  we use x (v, w) = ∂x  x=v . Moreover, we define z := u=w

, and use the shorthand notation (x, u) ∈ R = R (z) = (x, u). We omit the dependence of the function on its variables whenever it is clear from context, especially in the case z (z). We define sequences of a variable by using a bold-face notation, i.e. x = (x0 , . . . , xN , . . .). The length of the sequence is defined by context. For any function f we write f (x) = (f (x0 ), . . . , f (xN ), . . .). Finally, we define I[a,b] := {a, . . . , b}, a, b ∈ Z, i.e. integers. nz

nx +nu

2. PROBLEM STATEMENT AND PRELIMINARIES In this paper, we consider discrete-time economic NMPC formulations. In the following, we introduce the problem, recall important results and definitions from the literature.

In this paper, we consider economic NMPC based on the following OCP N −1  x0 ) := min (xk , uk ) + E(xN ) (2a) VN (ˆ x,u

k=0

ˆ0 , (2b) s.t. x0 = x (2c) xk+1 = f (xk , uk ), k ∈ I[0,N −1] , k ∈ I[0,N −1] . (2d) g(xk , uk ) ≤ 0, To avoid cumbersome technicalities, we assume that the problem data of (2) is sufficiently smooth, i.e. at least twice differentiable, and that the minimum exists. Here  denotes the stage cost, E is usually called terminal cost. If it exists, we denote the optimal pair of OCP as z  := (x , u ). Moreover, we denote the corresponding optimal adjoint (i.e. the multiplier of the equality constraint induced by the dynamics) as λ . 2.2 Necessary Optimality Conditions In our later developments, we rely on the Necessary Conditions of Optimality (NCO) of OCP (2). For the sake of compact notation, we define for k ∈ I[0,N −1]

 Lk := (xk , uk ) + λ k+1 (f (xk , uk ) − xk+1 ) + µk g(xk , uk ) and for k = N LN := E(xN ). Thus, the Lagrangian of Problem (2) reads N  L(z, λ, µ) = λ ˆ0 ) + Lk . (3) 0 (x0 − x k=0

The NCO of OCP (2) then read as 0 = L uk , k ∈ I[0,N −1] , (4a) 0 = Lxk , (4b) 0 = Ex (xN ) − λN , (2b), (2c), (2d), (4c) 0 = µk,j gj , µk,j ≥ 0, j ∈ I[1,ng ] , k ∈ I[0,N −1] , (4d) where Lxk is the gradient of Lk with respect to xk and Luk is the gradient of Lk with respect to uk . We remark that the evaluation of the first part of (4a) for k < N and k = N yields the adjoint dynamics λk = −fx λk+1 − x − gx µk , λN = Ex (xN ). We define the Steady-state Optimization Problem (SOP ) corresponding to OCP (2) as min (¯ x, u ¯) s.t. x ¯ = f (¯ x, u ¯), g(¯ x, u ¯) ≤ 0. (5) x ¯,¯ u

2.1 Nonlinear Model Predictive Control Consider the dynamic system given by xk+1 = f (xk , uk ), x0 = x ˆ0 ,

(2) Solve OCP (2); (3) Apply the optimal control law u0 and go to Step 1.

The optimal solution of SOP (5) is denoted as z¯ = (¯ x , u ¯ ). (1)

and subject to the constraints z := (x, u) ∈ Z ⊂ Rnx +nu , Z = {z ∈ Rnx +nu | gj (z) ≤ 0, j ∈ I[1,ng ] }, where Z is assumed to be compact. We assume w.l.o.g. that f (0, 0) = 0 and, occasionally, we use the shorthand notation g(x, u) = [g1 (x, u), . . . , gng (x, u)] . NMPC is based on repeatedly solving a given OCP according to the following strategy: (1) Get the current state x ˆ0 ; 183

Note that the differentiability assumption on the problem data of OCP (2) implies similar smoothness in SOP (5). We define the Lagrangian of Problem (5) as ¯ z , λ, ¯ µ ¯  (f (¯ L(¯ ¯) = (¯ z) + λ x, u ¯) − x ¯) + µ ¯ g(¯ x, u ¯). The NCO of SOP then read as follows ¯ x, ¯ u, 0=L 0=L (6a) x ¯ = f (¯ x, u ¯), g(¯ x, u ¯) ≤ 0, (6b) 0=µ ¯ j gj , µj ≥ 0, j ∈ I[1,ng ] . (6c) Furthermore, we require regularity in the following sense:

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159

Assumption 1. (Regularity of SOP (5)). SOP (5) has an optimal solution in the interior of the inequality constraints, i.e. z¯ ∈ int Z, and the corresponding ¯  are unique. dual variables λ 

Proof. The proof is given in (Zanon et al., 2016, Theorem 4) for the context at hand by exploiting the results developed in (Diehl, 2001) and (Guddat et al., 1990, Theorem 3.3.4 and Corollary 3.3.1). 

Note that w.l.o.g. we shall assume that z¯ = 0. Moreover, we remark that the last assumption is equivalent to requiring the Linear Independence Constraint Qualification (LICQ), see (Wachsmuth, 2013).

2.4 Economic and Tracking NMPC

2.3 Local LQ Approximation In this paper, we will occasionally approximate OCP (2) locally around optimal solutions z¯ ∈ int Z to SOP (5) by the following linear-quadratic problem OCPLQ : min x,u

N −1 

1  2 zk W zk



+ w zk +

1  2 x N PN x N

+

x N pN

(7a)

k=0

ˆ0 , (7b) s.t. x0 = x (7c) xk+1 = Axk + Buk , k ∈ I[0,N −1] ,  z ) ≤ 0, k ∈ I[0,N −1] . (7d) Cxk + Duk − g(¯ where the linear dynamics and path constraints are defined via the Jacobians A = fx , B = fu , C = gx , D = gu , and the quadratic objective is given by     Q S q W = , w = , r S R ¯ xu , R = L ¯ uu , q = x , r = u , ¯ xx , S = L with Q = L ¯  and PN = Exx , pN = Ex , all evaluated at z¯ = 0, λ  µ ¯ = 0. For the purpose of this paper, we assume that OCPLQ satisfies a certain stabilizability condition. Assumption 2. (OCPLQ is receding-horizon stabilizing). (i) The (possibly indefinite) finite-horizon OCPLQ (7) satisfies the NCOs stated in Appendix A. ˜ ∈ N such that for (ii) There exists a horizon length N ˜ all N ≥ N the receding-horizon (1-step) application of OCPLQ (7) asymptotically stabilizes the linearized system (A, B) to some point z¯LQ that in general may differ from z¯ .  The fact that the system might be stabilized to a point different from z¯ is discussed in detail in Lemma 2 and Lemma 5, while the NCO of OCPLQ are given explicitly in Appendix A. Note that in (i) the satisfaction of the NCO Appendix A requires that det(R + B  Pk,N B) = 0, where Pk,N is the solution of a corresponding Riccati difference equation. It can be shown that this condition does not pose any problem whenever R  0. We comment on item (ii) in Remark 2 and in the proof of Lemma 5. The next result states the local approximation properties of OCPLQ . Proposition 1. (Properties of LQ approximation). Let Assumption 2 hold and suppose strict complementarity holds in OCP (2) with x0 = x ¯ . Then the opti mal solution z LQ (ˆ x0 ) of OCPLQ and the optimal solution x0 ) of OCP satisfy z LQ (ˆ x0 ) − z  (ˆ x0 ) = O(ˆ x0 2 ). z  (ˆ Moreover, also the adjoint, respectively, the dual variables satisfy λLQ (ˆ x0 ) − λ (ˆ x0 ) = O(ˆ x0 2 ) and µLQ (ˆ x0 ) −  2 x0 ) = O(ˆ x0  ).  µ (ˆ 184

We recall the definitions of economic and tracking MPC used throughout the paper. Definition 1. (Tracking and economic MPC). Let z¯ ∈ int Z be the optimal steady-state from SOP (5). We say a predictive control scheme is a tracking MPC (TMPC) if (¯ z ) < (z),

∀ z ∈ Z \ z¯.

(8)

We label as economic those MPC schemes for which (8) does not hold; i.e. the cost function does not have a strict (global) minimum at the optimal steady state.  Note that (8) implies that (z) is a positive-definite function, thus it holds that ∃ α ∈ K : α(z − z¯) ≤ (z). ¯  = 0. In the followMoreover, z¯ = argminz∈Z (z) and λ ing, whenever it will be necessary to clearly distinguish tracking and economic MPC formulations, we will denote the tracking stage cost as t , while will refer to economic stage costs. In the view of Definition 1, tracking and economic MPC differ in how stability can be proven. In classical tracking MPC one may enforce stability in different ways: (a) Choose E(x) as a local control Lyapunov function and enforce additional terminal constraints (Rawlings and Mayne, 2009); (b) if no terminal constraints are imposed, the terminal cost E(x) is either chosen as a global control Lyapunov function (Jadbabaie et al., 2001) or it can be constructed from a local control Lyapunov function (Lim´on et al., 2006). In applications a common choice is E(x) = 0 and no terminal constraint is enforced. Note that asymptotic stability can still be proven for TMPC provided that the prediction horizon is long enough and certain controllability assumptions hold (Gr¨ une and Pannek, 2017). In contrast to TMPC, stability results for EMPC often require that there exists a so-called storage function S : Rnx → R satisfying the following Strict Dissipation Inequality (SDI) S(f (z)) − S(x) ≤ −α(z − z¯) + (z) − (¯ z ),

(9)

where α is of class K, see (Diehl et al., 2011; Angeli et al., 2012; Gr¨ une, 2013; Faulwasser et al., 2018). As we will prove in Theorem 3, this implies that the storage function must have a specific slope. Lemma 1. (Rotation invariance of primal solutions). The rotated OCP, i.e. (2) formulated with the cost N −1 

ˆ k , uk ) + E(x ˆ N ), (x

k=0

ˆ u) = (x, u) − (¯ (x, x , u ¯ ) + S(x) − S(f (x, u)) ˆ E(x) = E(x) + S(x), provides the same primal solution as the original OCP (2). 

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The proof is similar to the one given in (Diehl et al., 2011; Amrit et al., 2011) and thus omitted. This result will be very useful throughout the paper. Relying on S being a storage function, using E(x) = 0, and considering no terminal constraint, one can show that the closed-loop system converges to a neighborhood of the optimal steady-state x ¯ , cf. (Gr¨ une, 2013; Gr¨ une and Stieler, 2014; Faulwasser et al., 2018). We denote by uEM P C the feedback policy generated by EMPC and x(k; x0 , uEM P C ) is the value of the corresponding closedloop trajectory at time k originating at x0 .

3. MAIN RESULTS Similar to the sampled-data continuous-time case discussed in (Zanon and Faulwasser, 2018) the main question we answer in this paper is: Under which conditions does the linear end penalty E(x) = x pN , in OCP (2) enforce asymptotic stability of the optimal steady state x ¯ ? 3.1 Stability of EMPC at Optimal Steady States

(A1) for all x0 ∈ X0 , the SDI (9) holds along all optimal pairs z  (x0 ); (A2) the optimal steady state x ¯ is exponentially reachable by admissible controls from any x0 ∈ X0 ; (A3) the Jacobian linearization of (1) at z¯ ∈ int Z, (A, B), is nx -step controllable.

Theorem 2. (EMPC is not stabilizing z¯ ). Consider an EMPC controller based on OCP (2) with N < ∞, E(x) = 0 and no terminal constraint applied to system (1). Let Assumptions 1 and 2 hold. Furthermore, suppose ¯  = 0, i.e. the cost has a non-zero gradient at (i) that λ the optimal steady state z¯ , which implies that the scheme is not of tracking type; and (ii) that OCP (2) satisfies LICQ and strict complementarity for x0 = x ¯ .

Then, there exists a finite horizon N < ∞, and a constant ρ(N ) > 0 such that

Then, the EMPC controller cannot stabilize the system to  x ¯ .

(i) OCP (2) is recursively feasible (ii) and

Before proving Theorem 2, we turn to the easier linearquadratic case. Lemma 2. Consider OCPLQ with the problem data from (7). Suppose that Assumptions 1 and 2 hold and that the optimal steady state is z¯ = 0 ∈ int Z and OCPLQ satisfies LICQ for x ˆ0 = x ¯ . Assume, moreover, that the system is locally controllable at z¯ . ¯  = 0, any EMPC scheme based on Then, whenever λ OCPLQ with E(x) = 0 and without additional terminal constraints does not stabilize the system at the optimal  steady state z¯ .

Theorem 1. (Practical stability of EMPC). Consider an EMPC controller based on OCP (2) with N < ∞, and E(x) = 0. Suppose that

lim

k→∞



 max{x(k; x0 , uEM P C ) − x ¯ , ρ(N )} = ρ(N ),

holds for all x0 ∈ X0 , and lim ρ(N ) = 0. N →∞

(10) 

The above result is a minor modification to (Gr¨ une and Pannek, 2017, Thm. 8.33). In essence the result from Gr¨ une and Pannek (2017) is extended by the recursive feasibility statement, cf. (Faulwasser et al., 2018, Prop. 4.2 and Thm. 4.1). The detailed proof relies on the existence of a turnpike property at z¯ , which is implied by the dissipativity and reachability assumptions. The turnpike property allows concluding that, for a sufficiently long horizon N , the open-loop predictions stay near z¯ during large parts of the horizon. However, they may leave the neighborhood of z¯ towards the end of the horizon. Furthermore, it is important to note that the size of the neighborhood to which the closed-loop EMPC solutions eventually converge, ρ(N ), shrinks with increasing horizon length. It is worth to be remarked that the existence of a turnpike in the open-loop predictions is not affected by end penalties, cf. for example the proof of (Faulwasser et al., 2018, Prop. 4.1). Yet, we note that the shape of the end piece of open-loop optimal solutions—i.e., the shape of the socalled leaving arc of the turnpike—is usually affected by end penalties. Hence, we have the following corollary to Theorem 1. Corollary 1. (Terminal penalties preserve prac. stability). Let the conditions of Theorem 1 hold and consider a differentiable end penalty E(x) = 0 in OCP (2). Then, there exists a finite horizon N > ∞, and a constant ρ(N ) > 0 such that (10) holds for the closed EMPC loop.  185

Proof. First we remark that satisfaction of LICQ by OCPLQ can be obtained as a consequence of satisfaction of LICQ by SOPLQ , provided that the path constraints are adequately formulated. However, for the sake of brevity, we omit the proof and instead assume that OCPLQ satisfies LICQ. The optimality conditions of SOPLQ , which is the LQ pendant to OCPLQ (7), entail ¯, ¯. 0 = r + Bλ (11) 0 = q + (A − I)λ The optimality conditions of OCPLQ entail λk = Qxk + Suk + q + A λk+1 , 



(12a)

0 = S xk + Ruk + r + B λk+1 , (12b) and the transversality condition λN = 0. For the sake of contradiction, assume that for x ˆ0 = x ¯ = 0, the optimal pair zk remains at the steady state (¯ x , u ¯ ) until time ¯ k > 1. In order for this to hold, we must have that, for ¯ k ≤ k: λk = q + A λk+1 , 0 = r + B  λk+1 . (13) We now regard the second equation as the linear output of the first one and observe that controllability of (A, B) implies observability of (A , B  ). This implies that the solution of (13) must remain constant. Since we assume

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¯ is unique. Since by assumpLICQ for SOPLQ (Ass. 1), λ tion LICQ also holds for OCPLQ then (13) must have ¯ a unique solution. Moreover, since λk = λk+1 = λ ¯ solves (13), then for k ≤ k this is also the only admissible dual solution. ¯ Using (12b), one obtains that xk = 0 and λk = λk+1 = λ imply uk = 0. This, in turn, implies that xk+1 = 0 ¯  . Therefore, the solution and, consequently λk+2 = λ must remain constant until the end of the horizon. This contradicts the boundary (transversality) condition λN = 0. Hence, the system immediately leaves the optimal steady state.  We are now ready to prove Theorem 2. Proof. Assume by contradiction that EMPC stabilizes the system to z¯ = 0. We exploit the fact that the LQ approximation yields a control law which is a first-order approximation of the control law yielded by nonlinear MPC, cf. Proposition 1. Since the LQ MPC feedback immediately steers the system away from x ¯ , we conclude  that x ¯ = 0 cannot be a steady-state for the closed-loop system.  This result has several consequences that are elaborated next. The first important consequence regards closedloop performance, evaluated via the asymptotic average, defined as    N   1 vk = v¯ . (14) Av[v] := v¯  lim inf N →∞ N k=0

Henceforth, we consider the closed-loop performance as measured by the asymptotic average cost, i.e. Av[(z)]. Note that we will use the economic cost  also when referring to closed-loop trajectories obtained by tracking MPC schemes. Corollary 2. (TMPC performing better than EMPC). Consider a stabilizing TMPC (in the sense of Definition 1, with z¯ the economic optimum given by (5)) and a stabilizing EMPC based on OCP (2) with N < ∞, E(x) = cl ¯  = 0. Denote byz cl 0 and λ TMPC and z EMPC the closedloop state and control trajectories obtained with TMPC and EMPC respectively. cl Then Av[(z cl TMPC )] ≤ Av[(z EMPC )], i.e. the TMPC controller yields a better average closed-loop performance than the EMPC controller. Moreover, if z¯ is a strict global optimum, then the inequality is strict. 

The proof follows from the same arguments as the ones of the sampled-data result (Zanon and Faulwasser, 2018, Cor. 2) and is thus omitted. Lemma 3. (Linear cost rotation in TMPC). Let there be a stabilizing TMPC with t (x, u) satisfying (8) and E(x) = 0. Consider a linear rotation of the stage cost, i.e. consider ˆ u) = t (x, u) + a f (x, u), using the cost defined by (x, a = 0 and E(xN ) = 0.

Then, the obtained MPC scheme is economic in the sense of Definition 1 and does not stabilize the system to the origin.  The proof follows along the lines of the sampled-data result (Zanon and Faulwasser, 2018, Lem. 4) and is thus omitted. 186

161

3.2 Storage Function Geometry and Steady-State Multiplier We turn towards the investigation of the relation between the Lagrange multipliers of SOP (5) and the local geometry of the storage function. Theorem 3. (Storage function slope at x ¯ ). Let S satisfy the SDI (9) along any optimal pair z. Suppose that S is continuously differentiable on some open neighborhood B(¯ x ) of the optimal steady state x ¯ . Then the slope of S at x ¯ is given by the Lagrange ¯. multiplier of SOP (5), i.e. Sx (¯ x ) = −λ  x ) × B(¯ u ), consider Proof. On the open set B(¯ z  ) := B(¯ the rotated cost ˆ := (z) − (¯ (z) z  ) + S(x) − S(f (z)). As  and f are assumed to be continuously differentiable on Z, ˆ is so on B(¯ z  ). Strict dissipativity implies ˆ ≥ α(z − z¯ ), ∀z ∈ B(¯ z  ). (15) (z) Hence, z¯ is a strict local minimizer of ˆ on B(¯ z  ). Differentiability of ˆ on B(¯ z  ) implies that ˆx (¯ z  ) = 0.

Consider now two variants of (5) formulated using (a) the original cost , i.e. and (b) the rotated cost ˆ with S as specified above. For (b) let the corresponding multiplier ˆ ¯ Because of the steady state constraint be denoted as λ. ˆ  ¯ ˆ z ) = 0, we have that λ = 0. Comparison of the NCOs x (¯ ˆ¯ = λ ¯  + Sx (¯ x ). Hence the for both SOPs shows that λ assertion.  The above theorem connects the findings of (Diehl et al., 2011), which make use of a linear storage function, with the results in (Amrit et al., 2011) using a nonlinear storage function. While in the former paper, the connection to the Lagrange multiplier of the SOP is explicitly stated. In the latter one, this connection has not been investigated. For the continuous-time counterpart see (Zanon and Faulwasser, 2018, Thm. 4). 3.3 Recovering Stability of the Optimal Steady State In the following, we show how closed-loop stability of the optimal steady state can be recovered and, consequently, optimal average performance can be achieved also in the absence of terminal constraints. Lemma 4. (Nonlinear rotation of cost functions). Any EMPC scheme based on OCP with N < ∞, and E(x) = −S(x) is equivalent to a TMPC scheme with zero terminal cost in the sense of Definition 1, provided that S(x) is a storage function satisfying the SDI (9).  Proof. Lemma 1 implies that OCP (2) gives the same ˆ primal solutions as swapping  for ˆ and E(x) for E(x). Moreover, using E(x) = −S(x) yields a zero rotated ˆ terminal cost, i.e. E(x) = 0. The SDI (9) yields ˆ k ) = (zk ) − (¯ z  ) + S(x) − S(f (x, u)) ≥ α(z − z¯ ). (z Since w.l.o.g. we can set (¯ z  ) = 0 and z¯ = 0, the rotated stage cost is a positive definite function. Consequently, setting E(x) = −S(x) in (2) is equivalent to a tracking MPC formulation with zero terminal cost. 

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As a consequence, whenever a storage function S is known, EMPC can be formulated as a TMPC penalizing the deviation from the optimal steady state z¯ . Therefore, by an appropriate choice of the prediction horizon and under suitable controllability assumptions, stability of x ¯ can be concluded (Gr¨ une and Pannek, 2017; Jadbabaie et al., 2001). We refer to (Amrit et al., 2011) for the counterpart for EMPC with terminal constraints. Unfortunately this approach is impractical, as it requires knowledge of a storage function, the computation of which is in general as difficult as the computation of Lyapunov functions (Ebenbauer and Allg¨ ower, 2006; Faulwasser et al., 2017). Next, we analyse how to tackle this issue by means of end penalties in the linear-quadratic setting. Lemma 5. (Properties of stabilizing LQ EMPC). ¯  = 0. Let (A, B) be Consider OCPLQ (7) whereby λ stabilizable, consider an EMPC formulation with E(x) = 1   2 x PN x, with PN = PN . Suppose that, with the chosen N and PN , the EMPC stabilizes the system to some z¯LQ = z¯ = 0; i.e. Assumption 2 holds. Then the following statements hold: (i) For increasing prediction horizons N , the closed-loop steady-state z¯LQ tends to z¯ with an exponential decay in N . (ii) The EMPC formulation with E(x) = 12 x PN x + x pN stabilizes the closed-loop system to a steadystate z¯LQ , which tends to the optimal steady-state ¯.  z¯ = 0 linearly as pN tends to λ Proof. Recall that the fact that the EMPC formulation with E(x) = 0 does not stabilize the system to z¯ = 0 is a consequence of Theorem 2. Recall that the optimality conditions of SOPLQ entail ¯, ¯. 0 = q + (A − I)λ 0 = r + Bλ (16) Because we assume that EMPC stabilizes the system, in the limit for N → ∞, one obtains from the NCOs given in Appendix A 0 = A P∞ A + Q − (A P∞ B + S)K∞ , K∞ = (R + B  P∞ B)−1 (B  P∞ A + S  ),        A −I q  0 = I −K∞ p + . ∞ r B    Then, because I −K∞ is full row rank, we have  ¯ p∞ = λ , B  p∞ + r = 0,

(17)

(18) u∞ = −K∞ x. Using (18), the finite-horizon feedback control u0,N can be written as   u0,N = −K0,N x − (R + B  P0,N B)−1 r + B  p0,N . (19)

Then, defining T := B(R + B  P0,N B)−1 B  , the steadystate x ¯LQ satisfies 0 =(A − BK0,N − I)¯ xLQ − T (p0,N − p∞ ). Since T = 0, p0,N = p∞ implies x ¯LQ = 0.    ¯, Pre-multiplying (16) by I −Kk,N and using p∞ = λ we obtain        A −I q  I −Kk,N p∞ + = 0. r B 187

 Therefore, q − Kk,N r = p∞ − (A − BKk,N ) p∞ . Hence, we use pN,N = pN to rewrite (A.1c) as

pk,N − p∞ = (A − BKk,N ) (pk+1,N − p∞ ) =

N 

j=k



(A − BKj,N ) (pN − p∞ ).

(20)

Rewriting (20) we obtain pk,N − p∞ = ·

˜ N −N j=k

(A − BKj,N )

N 

˜ −1) j=N −(N





(A − BKj,N ) (pN − p∞ ).

(21)

Observe that item (ii) of Assumption 2 implies σ (A − BK0,N ) < 1 ˜ . Moreover, for N and N + 1, it for all fixed N ≥ N follows from the NCO (A.1) that K0,N = K1,N +1 . In ˜ ) = turn, this gives that the time-varying system A(N ˜ is uniformly exponentially (A − BK0,N ) with N ≥ N stable, cf. (Hinrichsen and Pritchard, 2005, Rem 3.3.11 and Thm. 3.3.15). Hence we have from Assumption 2 (ii) that ˜ , the first product of matrices for sufficiently large N ≥ N in (21) enforces exponential stability. Thus p0,N − p∞ decays exponentially with increasing N . Moreover, for a fixed N , p0,N depends linearly on pN − p∞ . Hence, also x ¯LQ depends linearly on pN −p∞ and decays exponentially for increasing N .  ¯  , i.e. using The above result suggests considering pN = λ ¯. the terminal penalty E(x) = x λ ¯  ). Remark 1. (Primal and adjoint view on E(x) = x λ As already noted in the continuous-time companion paper (Zanon and Faulwasser, 2018), the end penalty ¯ E(x) = x λ allows for interesting interpretations. Either one can view ¯  from a primal variables point of view. Then E(x) = x λ it is a correction of the gradient of the stage cost at steady state; i.e. the linearly rotated stage cost becomes ¯, (z) + (x − f (z)) λ cf. (Zanon and Faulwasser, 2018, Rem. 3). Alternatively, ¯  from the dual/adjoint one may interpret E(x) = x λ point of view. Then it is a means to enforce a suitable transversality condition for the adjoint λ, i.e. it enforces ¯, λN = λ cf. (Zanon and Faulwasser, 2018, Rem. 4). However, it is ¯  in OCP (2) does crucial to note that using E(x) = x λ not necessarily define a TMPC scheme, since the cost  is in general indefinite. Finally, it is worth to be remarked ¯  indicate the that these interpretations of E(x) = x λ importance of considering adjoint/dual variables in the analysis of economic NMPC schemes.  Remark 2. (Stabilization via indefinite LQR feedback). Recall that a sufficient condition for the infinite-horizon LQR feedback to be stabilizing is R  0, S = 0, Q = Q  0 with (A, C) (C  C = Q) detectable, cf. Anderson and Moore (1990). However, in many relevant finite-horizon EMPC applications, this is not the case; i.e. the local

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LQ approximation is often indefinite (Zanon et al., 2014, 2016). Moreover, we remark that if the set Z is not compact, then strict dissipativity does not automatically imply stability of infinite horizon optimal solutions. A simple example is given by xk+1 = 2xk + uk , (x, u) = u2 . Strict dissipativity holds with e.g. S(x) = −x2 , but the optimal solution is u = 0 and the system is unstable. If, on the other hand, Z is compact and the problem is feasible, the optimal solution stabilizes the system to the origin. For more insight on this problem see (Gr¨ une, 2012; Gr¨ une and Guglielmi, 2018). However, we remark that to the best of the authors’ knowledge the question for necessary and sufficient stability conditions for infinite and finite horizon indefinite LQR feedback is still open.  Now, we are ready to establish sufficient conditions for local exponential stability of the closed EMPC loop without terminal constraints. ¯  ). Theorem 4. (Exp. stable EMPC with E(x) = x λ Consider an EMPC controller based on OCP (2) with ¯  . Suppose conditions A1-A3 of N < ∞ and E(x) = x λ Theorem 1 are satisfied for all x0 ∈ X0 , let Assumptions 1–2 hold, and assume that strict complementarity holds in OCP (2) with x0 = x ¯ . Then, there exists N ∈ N such that for all x0 ∈ X0 the closed-loop EMPC state trajectory x(k; x0 , uEM P C (x)) is locally exponentially stable at x ¯ .  Proof. We begin by observing that the above Theorem satisfies the conditions of Corollary 1. Thus we note that ¯  does not jeopardize the terminal penalty E(x) = x λ recursive feasibility of OCP (2) nor convergence to a neighborhood of x ¯ . Due to z¯ = (¯ x , u ¯ ) ∈ int Z (cf. A3), we have that for ¯  ) is a steady-state solution ¯ the triple (¯ x , u ¯ , λ x0 = x to the NCO (4) of OCP (2). W.l.o.g. we assume z¯ = 0. Linearizing the closed-loop EMPC dynamics xk+1 = f (xk , uEM P C (x)) yields xk+1 = Axk + BuLQ (x) + O(x2 ). Here, uLQ (x) denotes the static feedback generated by OCPLQ (7) and the error term O(x2 ) follows from Proposition 1. By part (ii) of Assumption 2 the linearized dynamics are an asymptotically stable LTI system; thus the nonlinear closed-loop system is locally exponentially stable at x ¯ , cf. (Hinrichsen and Pritchard, 2005, Thm. 3.3.41). Finally, using Corollary 1, we can pick a sufficiently large horizon N ∈ N such that the closed EMPC loop converges to the region of attraction of x ¯ induced by the feedback induced by OCPLQ (7).  While the sampled-data counterpart of this result shows ¯  = 0 there might exist small that with E(x) = 0 and λ  limit cycles around z¯ —see (Zanon and Faulwasser, 2018, Thm. 5)—the same is not true in the discrete-time setting. ¯  = 0 then due to (ii) of Indeed, if E(x) = 0 and λ Assumption 2 the closed EMPC loop will exhibit a locally exponentially stable equilibrium z¯EM P C = z¯ . 188

163

We conclude the discussion with a direct consequence of Theorem 4. Corollary 3. (Recovering Avg. Performance for EMPC). The average performance of the EMPC scheme from Theorem 4 is no worse than that of any TMPC scheme.  4. ADAPTIVE GRADIENT CORRECTION IN EMPC One of the advantages of economic MPC without terminal constraints and no terminal penalty is the fact that the optimal steady-state does not need to be pre-computed. This means that if the cost function changes occasionally— e.g. due to rare parametric updates (prices, etc.)—one can expect (under suitable technical conditions) that EMPC without terminal constraints will approach the optimal steady state (or a neighborhood of it) without any explicit computation of this steady state. Moreover, OCPs without terminal constraints are often easier to solve numerically. Unfortunately, as we have proven in the previous section, a terminal cost based on the optimal steady-state adjoint is necessary in order to guarantee that the system is asymptotically stabilized to the optimal steady-state. This raises the question of whether or not one can avoid explicitly computing the optimal steady state. Subsequently, we propose an adaptation strategy of the proposed linear terminal constraints. While ideas on selftuning terminal costs in EMPC have been discussed in (M¨ uller et al., 2013), they do not apply to the setting without some form of terminal constraints. Our proposed strategy relies on an intuitive idea using the proposed linear terminal penalty. The key observation is that, in the presence of a turnpike in OCP (2), the adjoint λ will approach the steady-state optimal Lagrange multiplier at least in some part of the prediction horizon. 2 The main of adaptive gradient correction is summarized in Algorithm 1. Note that in Algorithm 1 the user has to choose kˆ ∈ I[0,N −1] which defines how the steady-state adjoint is approximated. Algorithm 1. EMPC with Adaptive Gradient Correction

i ← 0 Set p0 = 0, choose kˆ ∈ I[0,N −1] while controller enabled do Get the current state and set x0 = x ˆi Solve OCP (2) with E(x) = x pˆi Apply the optimal control law ui,EM P C = u0   ˆ x ˆi Set pˆi+1 = λ k; i←i+1 end while

We will demonstrate the performance of EMPC with Adaptive Gradient Correction (AGC) by means of simulation examples in Section 5. However, prior to that we analyze a variant of Algorithm 1 tailored to the LQ setting. Observe that in the LQ setting (assuming no active input or state constraints) the adjoint λ satisfies λk = Pk,N xk + pk,N 2

The formal proof of this so-called extremal turnpike property follows via LICQ from Proposition 1 and the standard dissipativity assumption. Alternatively, one may derive it using an approach similar to (Tr´ elat and Zuazua, 2015). Due to space limitations, we do not discuss the details here.

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with Pk,N and pk,N as defined in Appendix A. In other words, pk,N is the part of the adjoint which does not depend on the state xk . Proposition 2. (LQ-EMPC with AGC). Consider Algorithm 1 with kˆ = 0 and the adaptation pi ) = p0,N (ˆ pi ) pˆi+1 := pk,N ˆ (ˆ with pk,N from (A.1c) and the transversality condition pN,N (ˆ pi ) = pˆi . Suppose that OCPLQ (7) satisfies Assumption 2, i.e. the conditions of Lemma 5. ˜, Then, for sufficiently large N ≥ N ¯ , lim x(k; x0 , uEM P C (x)) = x i→∞

i.e, the closed EMPC loop converges asymptotically.



Proof. The proof follows from Lemma 5 with minor modifications. Under the proposed adaptive gradient correction we obtain from (A.1c) that pi ) − p ∞ = pˆi+1 − p∞ = p0,N (ˆ

N 

j=0



(A − BKj,N ) (ˆ pi − p∞ ).

(22) Recall that in Lemma 5 we have shown that the only steady state of (22) is 0; respectively of (A.1c) it is ¯  , cf. (18). Moreovoer, note that Assumption 2 p∞ = λ ˜ , the time-varying entails that, for sufficiently large N ≥ N system (22) is exponentially stable, cf. proof of Lemma 5. ¯  . In turn this gives Hence, we have that lim pˆi+1 = λ lim x ˆi = x ¯ .

i→∞

i→∞



This result does not directly yield asymptotic stability. As a consequence of Lemma 2: if the system starts from x ¯ , the closed-loop trajectory first leaves x ¯ (since pˆ0 = 0) and converges back to it only after the gradient correction ¯  sufficiently well. Moreover, one should obapproximates λ serve that the proposed AGC turns the EMPC scheme into a dynamic feedback strategy since pˆi can be regarded as an internal state of the controller. We conjecture that with appropriate modifications Algorithm 1 yields to closed-loop stability also in the nonlinear case. Indeed the simulation results of Section 5 indicate this as well. However, at this point we leave a formal proof of the stability properties of EMPC with AGC for future research.

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In this section, we provide two simple numerical examples illustrating the theoretical developments of the paper. We remark that the dynamic systems are stabilizable but not locally controllable. By simple modifications to the dynamics of both considered systems, local controllability can be obtained and the numerical results qualitatively match the ones provided in the previous sections. We conjecture that Theorem 4, Theorem 1 and Lemma 2 will hold in this case (modulo some adaptation). Consider the stage cost and the discrete-time system (23a) (x, u) = −2uxB + 0.5u + 0.1(u − 12)2 ,   xA + 0.01 u (1 − xA ) − 0.12 xA f (x, u) = . (23b) xB + 0.01 u xB + 0.12 xA 189

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Fig. 3. Closed-loop simulations starting from three different initial conditions using the gradient-correction estimation from Algorithm 1. Top figure: LQ system, bottom figure: nonlinear system. ¯ = 12 The optimal steady-state is x ¯ = [0.5, 0.5] , u   ¯ with λ = [−100, −200] . We use a prediction horizon N = 4 and consider the three initial conditions [0.5, 0.8] , [0.2, 0.4] , [0.5, 0.5] . The closed-loop trajectories obtained by the original formulation and by the formulation with the linearly rotated cost are displayed in Figure 1 as a phase plot and in Figure 5 as time series. It can be seen that the absence of gradient correction/rotation is pushing

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Fig. 4. Time series of the LQ closed-loop simulations starting from three different initial conditions. Left plot: no gradient correction, right plot: gradient correction.

Fig. 5. Time series of the nonlinear closed-loop simulations starting from three different initial conditions. Left plot: no gradient correction, right plot: gradient correction.

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Fig. 6. Time series of the closed-loop simulations with the gradient-correction estimation from Algorithm 1 starting from three different initial conditions. Left plot: LQ system, right plot: nonlinear system. ¯ . the system away from x ¯ even when the initial state is x On the contrary, the formulation with the linear rotation is stabilizing the system to x ¯ . Consider now the quadratic stage cost and linear discretetime system (x, u) = xA u + 0.1u2 + 24xB − 0.5u,      0.76 0 xA 0.005 f (x, u) = + u. 0.12 0.88 −0.005 xB

(24a) (24b)

¯ = 0 with The optimal steady-state is x ¯ = [0, 0] , u ¯  = [−100, −200] . The closed-loop trajectories obtained λ by the original formulation and by the formulation with the linearly rotated cost are displayed in Figure 2 as a phase plot and in Figure 4 as time-series. Since system (24) is the LQ approximation of the nonlinear system (23), the behaviour of the two systems is similar. As outlined before, system (24b) is stabilizable but not controllable. As a consequence of the assumptions of Lemma 2, the terminal cost E(x) = x pˆ, with pˆ =  ¯  is such that the system does not leave [ 0 100 ] = λ the optimal steady-state. Also, the closed-loop simulations obtained with this alternative choice yields the behavior of Figure 2, right plot. In this example,  we observe that ¯  + L , L ∈ R stabilizes any terminal cost E(x) = x λ the system to x ¯ . Such a relationship is not easy to find in the general case and, though there might exist multiple terminal costs yielding closed-loop stability, the choice ¯  is guaranteed to do so by Theorem 4. E(x) = x λ By applying the gradient-correction estimation Algorithm 1 to the LQ system, we obtain the results displayed 191

in Figure 3 as a phase plot and in Figure 6 as time series. In particular, it can be seen that the correct value is estimated rather quickly and this reflects in the open-loop predictions quickly pointing towards the optimal steadystate. When the system is initialized at the optimal steadystate it first leaves it but, as the estimated gradient correction improves, the system is progressively brought back to the optimal steady state. This constitutes a proof by counterexample that EMPC with AGC does not yield asymptotic stability, but only attractivity. We conjecture that, after a first phase of gradient adaptation, asymptotic stability is recovered. The simulations for the adaptive gradient-correction Algorithm 1 in the nonlinear case are displayed in Figure 3 as a phase plot and in Figure 6 as time series. The results are inline with the observations made for the linearized problem. 6. CONCLUSIONS In this paper we have investigated conditions under which economic MPC without terminal constraints nor cost does not yield asymptotic stability to the economically optimal steady state x ¯ . This situation implies a worse average economic performance than for a corresponding tracking MPC scheme. We have proven that (under suitable assumptions) the introduction of a gradient-correcting linear terminal cost resolves this issue. The key idea is to use the adjoint/dual variable of the underlying steady-state optimization problem to rotate the objective. While the majority of stability proofs for economic MPC does not consider adjoint/dual variables, our analysis elucidates the

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importance of adjoints for achieving asymptotic stability. Furthermore, we have proposed an algorithm for adapting the gradient correcting end penalty. Finally, we have illustrated the theoretical developments with two numerical examples. Future work will aim at further investigating the connections of the proposed framework with existing results on economic MPC and at thoroughly analyzing the stability properties of EMPC when deploying the adaptive gradient correction strategy. Appendix A. NCOS OF OCPLQ (7) Let Vk−1,N : Rnx → R denote the optimal value function of OCPLQ (7) with truncated horizon I[k−1,N ] . Using Bellman’s optimality principle, we write the LQR one-step propagation as Vk−1,N (x) =      Q + A Pk,N A S + A Pk,N B x x min u u  R + B  Pk,N B u     x q + A pk,N + u r + B  pk,N The optimal control is then uk,N =   −(R + B  Pk,N B)−1 (S  + B  Pk,N A)x + r + B  pk,N . By replacing uk,N in the cost, the value function at time k − 1 reads as Vk−1,N (x) = x Pk−1,N x + p k−1,N x, with Kk−1,N = (R + B  Pk,N B)−1 (S  + B  Pk,N A), 

(A.1a)



Pk−1,N = Q + A Pk,N A − (S + A Pk B)Kk−1,N , (A.1b)

 pk−1,N = q + A pk,N − Kk−1,N (r + B  pk,N ), (A.1c) PN,N = PN , pN,N = pN . (A.1d) Alternatively, the NCOs can be derived following the one approach for LQ tracking problems in (Anderson and Moore, 1990).

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