oss Institute for Systems Theory in Engineering Uni ersity of Stuttgart 70550 Stuttgart German 64257ndeiseallgo wer istunistuttgartde Department of Engineering Cybernetics Norwe gian Uni ersity of Science and echnology 7491 rondheim Norw ay LarsImsl ID: 23944 Download Pdf

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oss Institute for Systems Theory in Engineering Uni ersity of Stuttgart 70550 Stuttgart German 64257ndeiseallgo wer istunistuttgartde Department of Engineering Cybernetics Norwe gian Uni ersity of Science and echnology 7491 rondheim Norw ay LarsImsl

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State and Output Feedback Nonlinear Model Predicti Control: An Ov ervie Rolf Findeisen Lars Imsland Frank Allg ower Bjarne A. oss Institute for Systems Theory in Engineering, Uni ersity of Stuttgart, 70550 Stuttgart, German ﬁndeise,allgo wer @ist.uni-stuttgart.de Department of Engineering Cybernetics, Norwe gian Uni ersity of Science and echnology 7491 rondheim, Norw ay Lars.Imsland,Bjarne.F oss @itk.ntnu. no Abstract The purpose of this paper is tw ofold. In the ﬁrst part we gi re vie on the current state of nonlinear model predicti control (NMPC). After brief

presentation of the basic principle of predicti control we outline some of the theoretical, computational, and implementational aspects of this control strate gy Most of the theoretical de elopments in the area of NMPC are based on the assumption that the full state is ailable for measurement, an assumption that does not hold in the typical practical case. Thus, in the second part of this paper we focus on the output feedback problem in NMPC. After brief ervie on xisting output feedback NMPC approaches we deri conditions that guarantee stability of the closed-loop if an NMPC state feedback

controller is used together with full state observ er for the reco ery of the system state. eyw ords: nonlinear model predicti control, output feedback, stability performance Intr oduction In man control problems it is desired to design stabilizing feedback such that performance criterion is minimized while satisfying constraints on the controls and the states. Ideally one ould look for closed solution for the feedback la satisfying the constraints while optimizing the performance. Ho we er typically the optimal feedback la cannot be found analytically en in the unconstrained case, since it in

olv es the solution of the corresponding Hamilton- Jacobi-Bellman partial dif ferential equations. One approach to circumv ent this problem is the repeated solution of an open-loop optimal control problem for gi en state. The ﬁrst part of the resulting open-loop input signal is implemented and the whole process is repeated (see Section 2). Control approaches using this strate gy are referred to as model predicti control (MPC), mo ving horizon control or receding horizon control. In general one distinguishes between linear and nonlinear model predicti control (NMPC). Linear MPC refers to

amily of MPC schemes in which linear models are used to predict the system dynamics and considers linear constraints on the states and inputs and quadratic cost function. Ev en if the system is linear the closed-loop dynamics are in general nonlinear due to the presence of constraints. NMPC refers to MPC schemes that are based on nonlinear models and/or consider non-quadratic cost-functionals and general nonlinear constraints on the states and inputs. Since its ﬁrst in ention in the 70s of the last century linear MPC has crystallized as one of the adv anced control strate gies. By no

linear model predicti control, is widely used in industrial applications especially in the process industry see for xample [38, 39, 73, 77, 78]. The practical success is mainly based on the possibility to tak constraints on the states and inputs systematically into account while operating the process optimally Corresponding author

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Ov ervie on industrial linear MPC techniques can be found in [77] and[78]. In [78] more than 4500 applications spanning wide range from chemicals to aerospace industries are reported. By no linear MPC theory can be considered as quite mature.

Important issues such as online computation, the interplay between modeling/identiﬁcation and control and system theoretic issues lik stability are well addressed [53, 73]. Man systems are, ho we er inherently nonlinear Higher product quality speciﬁcations and increasing producti vity demands, tighter en vironmental re gulations and demanding economical considerations require to operate systems er wide range of operating conditions and often near the boundary of the admissible re gion. Under these conditions linear models are often not suf ﬁcient to describe the process

dynamics adequately and nonlinear models must be used. This inadequac of linear models is one of the moti ations for the increasing interest in nonlinear model predicti control. The purpose of this paper is tw ofold. In the ﬁrst part we pro vide re vie on the current state of NMPC. After presentation of the basic principle of predicti control we present some of the theoretical, computational and implementational aspects of this control strate gy furthermore discuss the inherent adv antages and disadv antages of NMPC. Note that this part is not intended to pro vided complete re vie of

xisting NMPC approaches. or xample we mainly focus on NMPC for continuous time systems using sampled measurement information and do not go into details on discrete time NMPC strate gies. or more self contained re vie ws on NMPC we refer to [3, 17, 22, 69, 80]. In the second part of the paper the output feedback problem for NMPC is considered. One of the obstacles of NMPC is that it is inherently state feedback control scheme using the current state and system model for prediction. Thus, for an application of predicti control in general the full state information is necessary and must be recon-

structed from the ailable state information. Ho we er en if the state feedback NMPC controller and the observ er used for the state reconstruction are both stable, there is no guarantee that the erall closed-loop is stable with reasonable re gion of attraction, since no general separation principle for nonlinear systems xists. After re vie of xisting solutions of the output feedback NMPC problem in Section 3.1 we present in Sections 3.2- 3.4 an unifying approach for output feedback NMPC that is based on separation ideas. In the follo wing, denotes the Euclidean ector norm in (where the

dimension follo ws from conte xt) or the associated induced matrix norm. ectors are denoted by boldf ace symbols. Whene er semicolon “; occurs in function ar gument, the follo wing symbols should be vie wed as additional parameters, i.e. means the alue of the function at with the parameter State eedback Nonlinear Model Pr edicti Contr ol In this section we pro vide an up to date ervie on the area of state feedback NMPC. Note, ho we er that we limit the discussion to NMPC for continuous time systems using sampled measurement information. brieﬂy refer to this as sampled-data NMPC. do not

go into details on NMPC for discrete time systems. Ho we er most of the outlined approaches ha dual discrete time counterparts, see for xample [3, 22, 69, 80, 81]. 2.1 The Principle of Pr edicti Contr ol Model predicti control is formulated as the repeated solution of (ﬁnite) horizon open-loop optimal control problem subject to system dynamics and input and state constraints. Figure depicts the basic idea behind model predicti control. Based on measurements obtained at time the controller predicts the dynamic beha vior of the system er prediction horizon in the future and determines (o

er control horizon the input such that predetermined open-loop performance objecti is minimized. If there were no disturbances and no model-plant mismatch, and if the optimization problem could be solv ed er an inﬁnite horizon, then the input signal found at could be open-loop applied to the system for all 0. Ho we er due to disturbances, model-plant mismatch, and the ﬁnite prediction horizon the actual system beha vior is dif ferent from the predicted one. incorporate feedback, the optimal open-loop input is implemented only until the ne xt sampling instant. In principle the

time between each ne optimization, the sampling time, can ary assume for simplicity of presentation, that it is ﬁx ed, i.e the optimal control problem is re-e aluated after the constant sampling time Using the ne system state at time the whole procedure prediction and optimization is repeated, mo ving the control and prediction horizon forw ard.

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closed-loop closed-loop control hor iz on prediction hor iz on state input set-point future/prediction past open loop input predicted state Figure 1: Principle of model predicti control. In Fig. the open-loop optimal input is

depicted as continuous function of time. allo numerical solution of the open-loop optimal control problem the input is often parametrized by ﬁnite number of “basis functions, leading to ﬁnite dimensional optimization problem. In practice often piece wise constant input is used, leading to decisions for the input er the control horizon. Summarizing, standard NMPC scheme orks as follo ws: 1. Obtain estimates of the states of the system 2. Calculate constraint-conforming optimal input minimizing the desired cost function er the pr ediction hori- zon using the system model and the

current state estimate for prediction 3. Implement the ﬁrst part of the optimal input until the ne xt sampling instant 4. Continue with 1. While it can be desirable for computational and performance reasons to choose unequal lengths of the prediction and control horizon (see e.g. [60]), we assume in the follo wing that for our presentation. 2.2 Mathematical ormulation of State eedback NMPC Consider the stabilization of time-in ariant nonlinear systems of the form )) (1) subject to the input and state constraints: (2) (3) ith respect to the ector ﬁeld we assume that it is locally

Lipschitz continuous in the re gion of interest (typically the re gion of attraction) and satisﬁes 0. Furthermore, the set is compact, is connected, and ypically and are (con x) box constraints of the form: min max (4) min max (5) This property is needed for the output feedback considerations in Section 3.

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with the constant ectors min max and min max In sampled-data NMPC an open-loop optimal control problem is solv ed at discrete sampling instants based on the current state information Since we consider constant sampling time the sampling instants are gi en by When the

time and occurs in the same setting, should be tak en as the closest pre vious sampling instant The open-loop input signal applied in between the sampling instants is gi en by the solution of the follo wing optimal control problem: Pr oblem 1: min )) subject to: )) (6a) (6b) (6c) The bar denotes predicted ariables (internal to the controller), i.e. is the solution of (6a) dri en by the input with the initial condition The distinction between the real system ariables and the ariables in the controller is necessary since en in the nominal case the predicted alues will not be the same as the

actual closed-loop alues. The cost functional minimized er the control horizon is typically gi en by )) )) )) (7) where the stage cost is often assumed to be continuous, satisﬁes 0, and lo wer bounded by class function The terminal penalty term and the so called terminal re gion constraint might or might not be present. These are often used to enforce nominal stability (see Section 2.3). The stage cost can for xample arise from economical and ecological considerations. Often, quadratic form for is used: Qx Ru (8) with and 0. The state measurement enters the system via the initial

condition in (6a) at the sampling instant, i.e. the system model used to predict the future system beha vior is initialized by the actual system state. Since all state information is necessary for the prediction, the full state must be either measured or estimated. The solution of the optimal control problem (6) is denoted by )) It deﬁnes the open-loop input that is applied to the system until the ne xt sampling instant )) )) (9) The control )) is feedback, since it is recalculated at each sampling instant using the ne state measurement. In comparison to sampled-data NMPC for continuous

time systems, in instantaneous NMPC (also often referred to as receding horizon control) the input is deﬁned by the solution of Problem at all times: )) )) i.e. no open-loop input is applied, see e.g. [68, 69]. The solution of (1) starting at time from an initial state applying an input is denoted by )) will refer to an admissible input as: Deﬁnition 2.1 (Admissible input) An input for state is called admissible if it is: a) piece wise continuous, b) c) d) Furthermore, one refers to the so-called alue function as: Deﬁnition 2.2 (V alue function) The value function of the

open-loop optimal contr ol Pr oblem is deﬁned as the minimal value of the cost for the state x: )) continuous function is class function, if it is strictly increasing and 0.

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The alue function plays central role in the stability analysis of NMPC, since it often serv es as yapuno function candidate [3, 69]. ypically no xplicit controllability assumption on the system is considered in NMPC. Instead, the stability results are based on the assumption of initial feasibility of the optimal control problem, i.e. the xistence of input trajectory s.t. all constraints are

satisﬁed. From theoretical and practical point of vie one ould lik to use an inﬁnite prediction and control horizon, i.e. and in Problem are set to This ould lead to minimization of the cost up to inﬁnity Ho we er normally the solution of nonlinear inﬁnite horizon optimal control problem can not be calculated (suf ﬁciently ast). or this reason ﬁnite prediction and control horizons are considered. In this case the actual closed-loop input and states will dif fer from the predicted open-loop ones, en if no model plant mismatch and no disturbances are

present. At the sampling instants the future is only predicted er the prediction horizon. At the ne xt sampling instant the prediction horizon mo es forw ard, allo wing to obtain more information thus leading to mismatch of the trajectories. The unequalness of the predicted and the closed-loop trajectories has tw immediate consequences. Firstly the actual goal to compute feedback such that the performance objecti er the inﬁnite horizon of the closed-loop is minimized is not achie ed. In general it is by no means true that the repeated minimization er mo ving ﬁnite horizon leads

to an optimal solution for the inﬁnite horizon problem. The solutions will often dif fer signiﬁcantly if short ﬁnite horizon is chosen. Secondly there is in general no guarantee that the closed-loop system will be stable. It is indeed easy to construct xamples for which the closed-loop becomes unstable if short ﬁnite horizon is chosen. Hence, when using ﬁnite prediction horizons special attention is required to guarantee stability (see Section 2.3). The summarize the characteristics and properties of NMPC are: NMPC allo ws the direct use of nonlinear models

for prediction. NMPC allo ws the xplicit consideration of state and input constraints. In NMPC speciﬁed time domain performance criteria is minimized on-line. In NMPC the predicted beha vior is in general dif ferent from the closed loop beha vior or the application of NMPC typically real-time solution of an open-loop optimal control problem is necessary perform the prediction the system states must be measured or estimated. Basing the applied input on the solution of an optimal control problem that must be solv ed on-line is adv antageous and disadv antageous at the same time. First,

and most important, this allo ws to directly consider constraints on states and inputs which are often dif ﬁcult to handle otherwise. Furthermore, the desired cost objecti e, the constraints and en the system model can in principle be adjusted on-line without making complete redesign of the controller necessary Ho we er solving the open-loop optimal control problem, if attack ed blindly can be dif ﬁcult or en impossible for lar ge systems. 2.3 State eedback NMPC and Nominal Stability One of the most important questions in NMPC is whether ﬁnite horizon NMPC strate gy does

guarantee stability of the closed-loop or not. The problem with ﬁnite prediction and control horizon is due to the dif ference between the predicted open-loop and the resulting closed-loop beha vior Ideally one ould seek for an NMPC strate gy which achie es closed-loop stability independently of the choice of the parameters and, if possible, approximates the inﬁnite horizon NMPC scheme as well as possible. An NMPC strate gy that achie es closed-loop stability independently of the choice of the performance parameters is often referred to as an NMPC approach with guar anteed

stability Dif ferent approaches to achie closed-loop stability using ﬁnite horizon lengths xist. Here only some central ideas are re vie wed and no detailed proofs are gi en. Moreo er no attempt is made to co er all xisting methods. Most of the technical details are left out for reasons of simple presentation. ithout loss of generality it is assumed that the origin and 0) is the steady state to be stabilized. Inﬁnite Horizon NMPC: Probably the most intuiti ay to achie stability is to use an inﬁnite horizon cost, i.e. in Problem is set to In this case the open-loop input

and state trajectories computed as the solution of the

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NMPC optimization Problem at speciﬁc sampling instant are in act equal to the closed-loop trajectories of the nonlinear system due to Bellman principle of optimality[7]. Thus, the remaining parts of the trajectories at the ne xt sampling instant are still optimal (end pieces of optimal trajectories are optimal). This also implies con er gence of the closed-loop. Detailed deri ations can be found in [46, 47, 68, 69]. Finite Horizon NMPC Schemes with Guaranteed Stability: Dif ferent possibilities to achie closed-loop

stability using ﬁnite horizon length xist. Most of these approaches modify the standard NMPC setup such that stability of the closed-loop can be guaranteed independently of the plant and performance speciﬁcations. This is usually achie ed by adding suitable equality or inequality constraints and suitable additional penalty terms to the standard setup. The additional terms are generally not moti ated by physical restrictions or performance requirements ut ha the sole purpose to enforce stability Therefore, the are usually called stability constr aints One possibility to enforce

stability with ﬁnite prediction horizon is to add the so called zer terminal equality constr aint at the end of the prediction horizon, i.e. (10) is added to Problem [15, 47, 68, 70]. This leads to stability of the closed-loop, if the optimal control problem has solution at 0. Similar to the inﬁnite horizon case the feasibility at one sampling instant does imply feasibility at the follo wing sampling instants and decrease in the alue function. One disadv antage of zero terminal constraint is that the predicted system state is forced to reach the origin in ﬁnite time. This

leads to feasibility problems for short prediction/control horizon lengths, i.e. to small re gions of attraction. From computational point of vie an xact satisf action of zero terminal equality constraint does require in general an inﬁnite number of iterations in the optimization and is thus not desirable. The main adv antages of zero terminal constraint are the straightforw ard application and the conceptual simplicity Man schemes xist that try to ercome the use of zero terminal constraint of the form (10) Most of them use the terminal gion constr aint and/or terminal penalty term ))

to enforce stability and feasibility ypically the terminal penalty and the terminal re gion are determined of f-line such that the cost function )) )) )) (11) gi es an upper bound on the inﬁnite horizon cost and guarantees decr ease in the value function as the horizon recedes in time. do not go into details about the dif ferent approaches. Instead we state the follo wing theorem, which gi es conditions for the con er gence of the closed-loop states to the origin. It is slight modiﬁcation of the results gi en in [36] and [16, 17]. Theor em 2.1 (Stability of sampled-data NMPC)

Suppose that a) the terminal gion is closed with and that the terminal penalty is positive semi-deﬁnite (b) the terminal gion and terminal penalty term ar hosen suc that ther xists an (admissible) input suc that and )) )) (12) (c) the NMPC open-loop optimal contr ol pr oblem is feasible for Then in the closed-loop system (1) with (9) con ver es to the origin for and the gion of attr action consists of the states for whic an admissible input xists. Pr oof The proof is gi en here for sak of completeness. It bases on using the alue function as decreasing yapuno v- lik function. As usual in

predicti control the proof consists of tw parts: in the ﬁrst part it is established that initial feasibility implies feasibility afterw ards. Based on this result it is then sho wn that the state con er ges to the origin.

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easibility: Consider an sampling instant for which solution xists (e.g. ). In between and the op- timal input )) is implemented. Since no model plant mismatch nor disturbances are present, )) )) The remaining piece of the optimal input )) satisﬁes the state and input constraints. Furthermore, ))) and we kno from Assumption (b) of the theorem

that for all there xists at least one input that renders in ariant er Picking an such input we obtain as admissible input for an time )) )) (13) Speciﬁcally we ha for the ne xt sampling time that )) is feasible input, hence feasibility at time implies feasibility at Thus, if (6) is feasible for 0, it is feasible for all 0. Furthermore, if the states for which an admissible input xists con er ge to the origin, it is clear that the re gion that consists of those points belongs to the re gion or attraction. Con ver ence: ﬁrst sho that the alue function is decreasing starting from

sampling instant. Remember that the alue of at is gi en by: )) )) )) )) )) ))) (14) and the cost resulting from (13) starting from an )) )) using the input )) is gi en by: ))) )) )) ))) )) ))) (15) Reformulation yields ))) )) )) )) ))) )) ))) )) )) ))) )) ))) (16) Inte grating inequality (12) from to starting from we obtain zero as an upper bound for the last three terms on the right side. Thus, ))) )) ))) ))) (17) Since is only feasible ut not necessarily the optimal input for it follo ws that )) )) )) )) ))) (18) i.e. the alue function is decreasing along solution trajectories starting at

sampling instant Especially we ha that: )) )) )) )) ))) (19) By assumption, this decrease in the alue function is strictly positi for 0. Since this holds for all sampling instants, con er gence can be established as in [18, 36] by an induction ar gument and the application of Barbalat lemma.

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Loosely speaking, is an -conform local control yapuno function in The terminal re gion constraint en- forces feasibility at the ne xt sampling instant and allo ws, similarly to the inﬁnite horizon case, to sho that the alue function is strictly decreasing. Thus stability can be

established. Note that this result is nonlocal in nature, i.e. there xists re gion of attraction which is of at least the size of ypically the re gion of attraction resulting from this scheme is much lar ger than and contains the set of all states for which the open-loop optimal control problem has feasible solution. arious ays to determine suitable terminal penalty term and terminal re gion xist. Examples are the use of control yapuno function as terminal penalty [45, 75, 84] or the use of local nonlinear or linear control la to determine suitable terminal penalty and terminal re gion [17,

18, 20, 63, 71]. Quasi-inﬁnite horizon NMPC: emplify the choice of suitable terminal re gion constraint and terminal penalty term considering the so called quasi-inﬁnite horizon NMPC (QIH-NMPC) approach [16, 18, 31]. In QIH- NMPC and are obtained on the basis of locally stabilizing linear control la considering quadratic stage cost of the form (8) The terminal penalty term in this case is quadratic and of the form Px The follo wing procedure gi es systematic approach to compute the terminal re gion and terminal penalty matrix of f-line [18], assuming that the Jacobian

linearization of (1) is stabilizable, where and Step Solv the linear control problem based on the Jacobian linearization of (1) to obtain locally stabi- lizing linear state feedback Step Deﬁne BK and choose constant satisfying max and solv the yapuno equation RK (20) to get positi deﬁnite and symmetric Step Find the lar gest possible deﬁning re gion Px (21) such that for all Step Find the lar gest possible specifying terminal re gion Px (22) such that the optimal alue of the follo wing optimization problem is non-positi e: max Px Px (23) where This procedure allo ws to

calculate and if the linearization of the system at the origin is stabilizable. If the terminal penalty term and the terminal re gion are determined accordingly the open-loop optimal trajectories found at each time instant approximate the optimal solution for the inﬁnite horizon problem. Remark 2.1 Theor em 2.1 allows to consider the stabilization of systems that can only be stabilized by feedbac that is discontinuous in the state [36, 37], .g nonholonomic mec hanical systems. This is possible since the optimal input signal is applied ver the sampling time open-loop to the system. or

suc systems, howe ver it is in ener al ather dif ﬁcult to determine suitable terminal gion and terminal penalty term for instance the QIH-NMPC method cannot be used.

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Remark 2.2 The use of terminal inequality constr aint leads to computational and feasibility advanta es compar ed to the inﬁnite horizon and zer terminal constr aint appr oac h. No zer terminal constr aint must be met in ﬁnite time The solution time necessary for solving the open-loop optimal contr ol pr oblem is decr eased, since no “boundary- value pr oblem stemming fr om the zer terminal

constr aint must be solved. Furthermor note that in NMPC it is in ener al not necessary to ﬁnd always an optimal solutions of Pr oblem in or der to guar antee stability [18, 45, 82]. Only feasible solution leading to decr ease in the value function is necessary This can be utilized to decr ease the necessary on-line solution time and mak es the pr actical application mor ob ust. Summarizing, the nominal stability question of NMPC in the state feedback case is well understood. arious NMPC approaches that guarantee stability xist. 2.4 Rob ustness and NMPC The NMPC schemes presented are

based on the assumption that the actual system is identical to the model used for prediction, i.e. that no model/plant mismatch or unkno wn disturbances are present. Clearly this is ery unrealistic for practical applications and the de elopment of an NMPC frame ork to address rob ustness issues is of paramount importance. In general one distinguishes between the inherent rob ustness properties of NMPC and NMPC designs taking the uncertainty/disturbances directly into account. The inherent rob ustness of NMPC is related to the act that nominal NMPC can cope with uncertainties and disturbances

without taking them directly into account. This act stems from the close relation of NMPC to optimal control and in erse optimality considerations [15, 64]. ithout going into details, we refer to [64, 69, 71, 83]. NMPC schemes that tak the uncertainty/disturbance directly into account are often based on min-max considera- tions. series of dif ferent approaches can be distinguished. or details we refer for xample to [13, 19, 49, 52, 58, 61, 62]. 2.5 Ef˛cient Real-time Solution of NMPC One important precondition for the application of NMPC, is the ailability of reliable and ef ﬁcient

numerical dynamic optimization algorithms, since at ery sampling time nonlinear dynamic optimization problem must be solv ed in real-time. Solving such an optimization problem ef ﬁciently and ast is, ho we er not tri vial task and has attracted strong research interest in recent years (see e.g. [6, 10, 11, 23, 25–27, 55, 65–67, 86, 90]). ypically so called direct solution methods [11, 12, 76] are used, i.e. the original inﬁnite dimensional problem is turned into ﬁnite dimensional one by discretizing the input (and also possibly the state). Basically this is done by

parameterizing the input (and possibly the states) by ﬁnite number of parameters and to solv e/approximate the dif ferential equations during the optimization. do not go into further details here and instead refer to [12, 25, 67]. Recent studies ha sho wn, that using special dynamic optimizers and tailored NMPC schemes allo ws to emplo NMPC to practically rele ant problems (see e.g. [6, 27, 32, 66, 86],[35]) en with todays computational po wer NMPC Output eedback Pr oblem So ar we assumed that the full state information can be measured and is ailable as initial condition for predicting

the future system beha vior In man applications, ho we er the system state can not be fully measured, i.e. only an output is directly ailable for feedback: (24) where are the measured outputs and where maps the state and input to the output. Thus, to apply predicti control methods the state must be estimated from the measured outputs using suitable state observ ers or ﬁlters. Ho we er en if the state feedback NMPC controller and the observ er used are both stable, there is no guarantee that the erall closed loop is stable with reasonable re gion of attraction, since no general alid

separation principle for nonlinear systems xists.

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achie non-local stability results of the observ er based output feedback NMPC controller tw possibilities (and mixtures thereof) seem to be attracti e: Separated Designs/Certainty Equi alence ppr oach: In this approach the observ er state estimate is used as the real system state follo wing the “certainty equi alence principle. sho stability of the closed-loop, one tries to separate the observ er error from the state feedback for xample by time scale separation. Utilizing observ ers for which the speed of con er gence of the

observ er error can be made suf ﬁciently ast and the absolute achie ed observ er error can be made suf ﬁciently small, semi-re gional stability results for the closed-loop can be established. Consideration of the obser er err or in the NMPC contr oller In this approach the observ er error is ackno wledged in the controller typically by using some bounds on the observ er error This solution is closely related to the design of rob ustly stabilizing NMPC schemes and typically requires observ ers that deli er an estimate of the observ er error ollo wing the ﬁrst approach we

deri in this section, for broad class of state feedback nonlinear model predicti controllers, conditions on the observ er that guarantee that the closed-loop is semi-globally practically stable. The result is based on the results presented in [34, 43], where high-gain observ ers are used for state reco ery Basically we xploit that sampled-data predicti controllers that possess continuous alue function are inherently rob ust to small disturbances, i.e. we will consider the estimation error as disturbance acting on the closed-loop. Before we deri the approach, we gi brief re vie of the xisting

output feedback NMPC approaches. 3.1 Existing Output-F eedback Results arious researchers ha addressed the question of output feedback NMPC using observ ers for state reco ery restrict the discussion to output feedback model predicti control schemes relying on state space models for prediction and dif ferentiate between the tw output feedback design approaches as outlined abo e. The “certainty equi alence”- method is often used in some what ad-hoc manner in industry [78], e.g. based on the (e xtended) Kalman ﬁlter as state observ er In the presence of separation principle, this ould be

theoretically sound ay to achie stabilizing output feedback scheme. Unfortunately general separation principle does not xist for MPC en in the case of linear models, the separation principle for linear systems is oid due to the presence of constraints. Thus, at the outset, nothing can be said about closed loop stability in this case, and it seems natural that one has to restrict the class of systems one considers to obtain results. As an xample, [91] sho ws global asymptotic stability for the special case of discrete-time linear open-loop stable systems. or more general class of nonlinear

systems, it can be sho wn that the properties of the alue function as yapuno function gi es some rob ustness of NMPC to “small estimation errors. or “weakly detectable discrete- time systems, this as ﬁrst pointed out in [83] (see also [57, 59], and an early ersion in [74]). Ho we er these results must be interpreted as “local”, in the sense that en though that an approximated re gion of attraction can be calculated in principle, it is not clear ho parameters in the controller or observ er must be tuned to inﬂuence the size of the re gion of attraction. In [24], local uniform

asymptotic stability of contracti NMPC in combination with “sampled EKF state estimator is established. Non-local results are obtained in [72], where an optimization based mo ving horizon observ er combined with the NMPC scheme proposed in [71] is sho wn to lead to (semi-global) closed-loop stability or the results to hold, ho we er global optimization problem for the mo ving horizon observ er with an imposed contraction constraint must be solv ed. More recently “re gional separation principle-based approaches ha appeared for wide class of NMPC schemes. In [43, 44] it as sho wn that based on

the results of [5, 85], semi-global practical stability results could be obtained for instantaneous NMPC based on special class of continuous-time models, using high gain observ ers for state estimation. In this conte xt, semi-global practical stability means that for an compact re gion inside the state feedback NMPC re gion of attraction, there xists sampling time and an observ er gain such that for system states starting in this re gion, the closed loop tak the state into an small re gion containing the origin. The result of [43] are de eloped further to the more realistic sampled-data case

in [33, 34], still considering class (albeit lar ger one) of continuous- time systems. In [30] it is pointed out ho these results can be seen as consequence of NMPC state feedback rob ustness. In [42], conditions are gi en on the system and the observ er for the state to actually con er ge to the origin. 10

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Related results appeared recently in [1], where for the same system class as considered in [43], semi-global prac- tical stability results are gi en for sampled-data systems using sampled high-gain observ ers. In [89] scheduled state feedback NMPC scheme is combined with

an xponential con er gent observ er and re gional stability results are established. On related note, the same authors sho in [88] ho an NMPC controller can be combined with con er gent observ er to obtain stability where stability is tak en care of of f-line. In the rob ust design approach the errors in the state estimate are directly accounted for in the state feedback predicti controller or linear systems, [8] introduces set membership estimator to obtain quantiﬁable bounds on the estimation error which are used in rob ust constraint-handling predicti controller The setup of [8] is

tak en further in [21], using more general observ er and considering more ef fecti computational methods. or the same class of systems, [56] does joint estimation and control calculation based on minimax formulation, ho we er without obtaining stability guarantees. or linear systems with input constraints, the method in [54] obtains stability guarantees through computation of in ariant sets for the state ector augmented with the estimation error In similar ashion, by constructing in ariant sets for the observ er error [50] adapts the NMPC controller in [14] such that the total closed loop is

asymptotically stable. 3.2 Output eedback NMPC with Stability–Setup In the follo wing we present one speciﬁc approach to output feedback NMPC. It is based on the act that sampled-data predicti controllers that possess continuous alue function are inherently rob ust to small disturbances, i.e. we will consider the estimation error as disturbance acting on the closed-loop. This inherent rob ustness property of NMPC is closely connected to recent results on the rob ustness properties of discontinuous feedback via sample and hold [48]. Ho we er here we consider the speciﬁc case of

sampled-data NMPC controller and we do not assume that the applied input is realized via hold element. Setup: Instead of the real system state at ery sampling instant only state estimate is ailable. Thus, instead of the optimal feedback (9) the follo wing “disturbed feedback is applied: )) )) (25) Note that the estimated state can be outside the re gion of attraction of the state feedback NMPC controller oid feasibility problems we assume that in this case the input is ﬁx ed to an arbitrary ut bounded alue. The NMPC scheme used for feedback is assumed to ﬁt the setup of Theorem

2.1. Additionally we mak the follo wing assumptions: Assumption In the nominal gion of attr action the following holds: 1. Along solution tr ajectories starting at sampling instant at the value function satisﬁes for all positive )) )) ))) (26) 2. The value function is uniformly continuous. 3. or all compact subsets ther is at least one le vel set s.t. ollo wing Theorem 2.1, Assumption 1.1 imply stability of the state feedback NMPC scheme (compare equation (18) in the proof of Theorem 2.1), and is typically satisﬁed for stabilizing NMPC schemes. Ho we er in general there is no

guarantee that stabilizing NMPC schemes satisﬁes Assumption 1.2 and 1.3, especially if state constraints are present. As is well kno wn [36, 40, 70], NMPC can also stabilize systems that cannot be stabilized by feedback that is continuous in the state. This in general also implies discontinuous alue function. Man NMPC schemes, ho we er satisfy this assumption at least locally around the origin [18, 20, 69]. Furthermore, for xample NMPC schemes that are based on control yapuno functions [45] and that are not subject to constraints on the states and inputs satisfy Assumption 1. 11

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Remark 3.1 Note that the uniform continuity assumption on implies that for any compact subset ther xists -function suc that for any do not state an xplicit observ ability assumptions, since the depend on the observ er used for state reco ery As outlined later se eral dif ferent observ ers satisfy the conditions required for the semi-global practical stability results to be deri ed. Concerning the observ er used, ho we er we assume that after an initial phase, the observ er error at the sampling instants can be made suf ﬁciently small, i.e. we assume that Assumption

Obser er err or con er gence) or any desir ed maximum estimation err or max ther xist ob- server par ameter s, suc that max con (27) Here con is freely chosen, ut ﬁx ed number of sampling instants after which the observ er error has to satisfy (27) Remark 3.2 Depending on the observer used, further conditions on the system (e .g observability assumptions) might be necessary Note that the observer does not have to oper ate continuously since the state information is only necessary at the sampling times. Note that there xist series of observ ers which satisfy Assumption 2, see Section

3.4. Examples are high-gain observ ers and mo ving horizon observ ers with contraction constraint. Since we do not assume that the observ er error con er ges to zero, we can certainly not achie asymptotic stability of the origin, nor can we render the complete re gion of attraction of the state feedback controller in ariant. Thus, we consider in the follo wing the question if the system state in the closed loop can be rendered semi-globally practically stable, under the assumption that for an maximum error max there xist observ er parameters such that (27) holds. In this conte xt semi-globally

practically stable means, that for arbitrary sets there xist observ er parameters and maximum sampling time such that 1. 0, 2. s.t. or clariﬁcation see Fig. 2. Figure 2: Set of initial conditions desired maximum attainable set and desired re gion of con er gence Note, that in the follo wing we only consider le el sets for the desired set of initial conditions ), the maximum attainable set and the set of desired con er gence ). do this for pure simpliﬁcation of the presentation. In principle one can consider arbitrary compact sets which contain the origin, and subsets of each

other and of since due to Assumption 1.3 it is al ays possible to ﬁnd suitable co ering le el sets. Basic Idea: The deri ed results are based on the observ ation that small state estimation errors lead to (small) dif ference between the predicted state trajectory based on the estimated state and the real state trajectory (as long as both of them are contained in the set ). As will be sho wn, the inﬂuence of the estimation error (after the con er gence time con of the observ er can in principle be bounded by )) )) )) )) ))) (28) 12

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where corresponds to the state

estimation error contrib ution. Note that the inte gral contrib ution is strictly ne gati e. Thus, if “scales with the size of the observ er error (it certainly also scales with the sampling time one can achie contraction of the alue function. Ho we er considering Assumption 2, the bounding is possible after certain time. oid that the system state lea es the set during this time we ha to decrease the sampling time (or use smaller sampling times initially). In order to bound the inte gral contrib ution on the right side of (28) we state the follo wing act: act or any with the lower bound min on

the value function xists and is non-trivial for all min min )) 3.3 Output eedback NMPC with Stability–Main Result Under the gi en setup the follo wing theorem holds Theor em 3.1 Assume that Assumption and hold. Then given arbitr ary le vel sets ther xists maximum allowable observer err or max and maximum sampling time max suc that for all initial conditions the state stays in the gion and con ver es in ﬁnite time to the set Pr oof The proof is di vided in three parts. In the ﬁrst part it is ensured that the system state does not lea the maximum admissible set during the con er

gence time con of the observ er This is achie ed by decreasing the sampling time suf ﬁciently In the second part is sho wn, that by requiring suf ﬁciently small max the system state con er ges into the set In the third part it is sho wn that the state will not lea the set once it has entered at sampling time. ir st part (x con ): ﬁrst deri conditions guaranteeing that states starting in stay in for certain time for all possible inputs. Note that is strictly contained in and thus also in with 2. Thus, there xists time such that The xistence of such time is guaranteed, since

as long as )) where is constant depending on the Lipschitz constants of and on the bounds on tak as the smallest (w orst case) time to reach the boundary of from an point allo wing to tak an alue in By similar ar guments there xists also time such that for all where 2. pick no the maximum sampling time max as max min con (29) The sampling time is assumed to be less or equal to this maximum alue for the remainder of the proof. This ensures that the state will be at least during the con er gence of the observ er completely contained in Furthermore we need for the second part of the proof that

for an after the observ er has con er ged. Note that due to Assumption there al ays xist observ er parameters such that after max con the observ er error is smaller than an desirable max If we require that max (30) it is guaranteed that if Second part (decr ease of the value function after observer con ver ence and ﬁnite time con ver ence to ): assume that or simplicity of notation, denotes the optimal input resulting from and denotes the input that correspond to the real state Furthermore, and Note that since we kno by the deri ations in the ﬁrst part of the proof that and that

Under these conditions the follo wing equality is alid: )) )) )) )) (31) 13

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can bound the last tw terms since is uniformly continuous in compact subsets of Also note that the third and forth term start from the same and that the ﬁrst term can be bound via )) (32) Here we used an upper bound for based on the Gronw all-Bellman lemma. If we no assume that and that max (33) then we kno that Thus we we obtain from (31) using act that )) min (34) guarantee that is decreasing from sampling instant to sampling instant along the le el sets, and to achie con er gence to the set

in ﬁnite time we need that the right hand side is strictly less than zero. One possibility to obtain this is to require, that the observ er parameters are chosen such that: min min min (35) Thus, if we choose the observ er parameters such that max max min and max (36) we achie ﬁnite time con er gence from an point to the set Thir part (x ): If equation (37) is still alid. Skipping the inte gral contrib ution on the right we obtain: )) (37) Thus if we assume that max max (38) then Combining all three steps, we obtain the theorem if max min con (39) and if we choose the observ er

error max such that max max min min (40) Remark 3.3 Explicitly designing an observer based on (40) and (39) is in ener al not possible Howe ver the theor em underpins that if the observer err or can be suf ﬁciently fast decr eased that the closed-loop system state will be semi- globally pr actically stable Theorem 3.1 lays the basis for the design of observ er based output feedback NMPC controllers that achie semi- global practical stability While in principle Assumption is dif ﬁcult to satisfy quite number of observ ers designs achie the desired properties as sho wn in the ne xt

Section. The alues and are chosen for simplicity 14

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3.4 Output eedback NMPC with Stability–P ossible Obser er Designs arious observ ers satisfy Assumption and thus allo the design of semi-globally stable output feedback controllers follo wing Theorem 3.1. will go into some detail for standard high-gain observ ers [87] and optimization based mo ving horizon observ ers with contraction constraint [72]. Note, that further observ er designs that satisfy the assump- tions are for xample observ ers that possess linear error dynamics where the poles can be chosen arbitrarily (e.g.

based on normal form considerations and output injection [9, 51]), or observ ers that achie ﬁnite con er gence time such as sliding mode observ ers [28] or the approach presented in [29]. High Gain Obser ers: One possible observ er design approach that satisﬁes Assumption are high-gain observ ers. Basically high-gain observ ers obtain state estimate based on approximated deri ati es of the output signals. The are in general based on the assumption that the system is uniformly completely observ able. Uniform complete ob- serv ability is deﬁned in terms of the observ ability

map which is gi en by successi dif ferentiation of the output (41) Here is the ector of output deri ati es. Note that we assume for simplicity of presentation that does not depend on the input and its deri ati es. More general results allo wing that depends on the input and its deri ati es can be found in [34]. assume that the system is uniformly completely observ able, i.e. Assumption The system (1) is uniformly completely observable in the sense that ther xists set of indices suc that the mapping depends only on x, is smooth with espect to and its in ver se fr om to is smooth and onto.

Application of the coordinate transformation where is the observ ability mapping, to the system (1) leads to the system in observ ability normal form in coordinates (42) The matrices and ha the follo wing structure blockdiag (43a) blockdiag (43b) blockdiag (43c) and is the “system nonlinearity in observ ability normal form. The high-gain observ er (44) allo ws reco ery of the states [4, 87] from information of assuming that Assumption in (44) is globally bounded. The function is the approximation of that is used in the observ er and which may be deduced from The observ er gain matrix is gi en

by blockdiag with where is the so-called high-gain parameter since goes to inﬁnity for 0. The are design parameters and must be chosen such that the polynomials :; are Hurwitz. use hatted ariables for the observ er states and ariables. 15

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Note that estimates obtained in coordinates can be transformed back to the coordinates by As sho wn in [4] and utilized in [34], under the assumption that the initial observ er error is out of compact set and that the system state stays in bounded re gion, for an desired max and an con er gence time con there xists maximum such that

for an the observ er error stays bounded and satisﬁes: con Thus, the high-gain observ er satisﬁes Assumption 2. Further details can be found in [33, 34]. Mo ving Horizon Obser ers: Mo ving horizon estimators (MHE) are optimization based observ ers, i.e. the state estimate is obtained by the solution of dynamic optimization problem in which the de viation between the measured output and the simulated output starting from estimated initial state is minimized. arious approaches to mo ving hori- zon state estimation xist [2, 72, 79, 92]. focus here on the MHE scheme with contraction

constraint as introduced in [72], since it satisﬁes the assumptions needed. In the approach proposed in [72] basically at all sampling instants dynamic optimization problem is solv ed, considering the output measurements spanning er certain estimation windo in the past. Assuming that certain reconstructability assumptions hold and that no disturbances are present, one could in principle estimate the system state by solving one single dynamic optimization problem. Ho we er since this ould in olv the solution of global optimization problem in real-time, it is proposed in [72] to only

impro the estimate at ery sampling time by requiring that the inte grated error between the measured output and the simulated output is decreased from sampling instant to sampling instant. Since the contraction rate directly corresponds to the con er gence of the state estimation error and since it can in principle be chosen freely this MHE scheme satisﬁes the assumptions on the state estimator Thus, it can be emplo yed together with state feedback NMPC controller to achie semi-global practical stability as xplained in Section 3.3. Conclusions Model predicti control for linear

constrained systems has been pro en as useful control solution for man practical applications. It is xpected that the use of nonlinear models in the predicti control frame ork, leading to non- linear model predicti control, results in impro ed control performance and allo ws the direct use of ﬁrst principle based models. Ho we er the consideration of nonlinear models also poses challenging theoretical, computational and implementational problems. In the ﬁrst part of this paper an ervie of arious aspects of NMPC has been gi en. mainly focused on sampled-data NMPC, since the plant

model is usually deri ed based on ﬁrst principles. As outlined, some of the challenges occurring in NMPC such as stability ef ﬁcient solution of the dynamic optimization problem etc. are already (at least partially) solv ed. Ne ertheless man unsolv ed questions remain. In the second part of the paper we addressed the important question of output feedback NMPC. Speciﬁcally we xpanded the sampled-data output feedback NMPC approach for continuous time systems as presented in [33, 34] to wider class of observ ers stating conditions the observ er error must satisfy such that

the closed-loop is semi-globally practically stable. As sho wn, se eral observ er designs satisfy the required conditions. Ho we er the results should be seen as conceptual rather then as practically applicable. or xample it is in general dif ﬁcult to establish priori that the alue function is continuous. In general there is no guarantee that nominally stable NMPC schemes satisfy this assumption, especially if constraints on the states are present, see [40]. Thus, future research has to focus on either relaxing this condition, or to deri conditions under which an NMPC scheme does

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