Sampled-Data Model Predictive Control for Constrained ... - CiteSeerX

Sampled-Data Model Predictive Control for Constrained Continuous Time Systems Rolf Findeisen, Tobias Raff, and Frank Allg¨ ower Institute for Systems Theory and Automatic Control, University of Stuttgart, Germany {findeise,raff,allgower}@ist.uni-stuttgart.de

Summary. Typically one desires to control a nonlinear dynamical system in an optimal way taking constraints on the states and inputs directly into account. Classically this problem falls into the field of optimal control. Often, however, it is difficult, if not impossible, to find a closed solution of the corresponding Hamilton-Jacobi-Bellmann equation. One possible control strategy that overcomes this problem is model predictive control. In model predictive control the solution of the Hamilton-Jacobi-Bellman equation is avoided by repeatedly solving an open-loop optimal control problem for the current state, which is a considerably simpler task, and applying the resulting control open-loop for a short time. The purpose of this paper is to provide an introduction and overview to the field of model predictive control for continuous time systems. Specifically we consider the so called sampled-data nonlinear model predictive control approach. After a short review of the main principles of model predictive control some of the theoretical, computational and implementation aspects of this control strategy are discussed and underlined considering two example systems. Key words. Model predictive control, constrained systems, sampled-data

1 Introduction Many methods for the control of dynamical systems exist. Besides the question of stability often the achieved performance as well as the satisfaction of constraints on the states and inputs are of paramount importance. One classical approach to take these points into account is the design of an optimal feedback controller. As is well known, however, it is often very hard, if not impossible, to derive a closed solution for the corresponding feedback controller. One possible approach to overcome this problem is the application of model predictive control (MPC), often also referred to as receding horizon control or moving horizon control. Basically in model predictive control the

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optimal control problem is solved repeatedly at specific sampling instants for the current, fixed system state. The first part of the resulting open-loop input is applied to the system until the next sampling instant, at which the optimal control problem for the new system state is solved again. Since the optimal control problem is solved at every sampling instant only for one fixed initial condition, the solution is much easier to obtain than to obtain a closed solution of the Hamilton-Jacobi-Bellmann partial differential equation (for all possible initial conditions) of the original optimal control problem. In general one distinguishes between linear and nonlinear model predictive control (NMPC). Linear MPC refers to MPC schemes that are based on linear dynamical models of the system and in which linear constraints on the states and inputs and a quadratic cost function are employed. NMPC refers to MPC schemes that use for the prediction of the system behavior nonlinear models and that allow to consider non-quadratic cost functions and nonlinear constraints on the states and inputs. By now linear MPC is widely used in industrial applications [40, 41, 75, 77, 78]. For example [78] reports more than 4500 applications spanning a wide range from chemicals to aerospace industries. Also many theoretical and implementation issues of linear MPC theory have been studied so far [55, 68, 75]. Many systems are, however, inherently nonlinear and the application of linear MPC schemes leads to poor performance of the closed-loop. Driven by this shortcoming and the desire to directly use first principles based nonlinear models there is a steadily increasing interrest in the theory and application of NMPC. Over the recent years many progress in the area of NMPC (see for example [1, 17, 68, 78]) has been made. However, there remain a series of open questions and hurdles that must be overcome in order that theoretically well founded practical application of NMPC is possible. In this paper we focus on an introduction and overview of NMPC for continuous time systems with sampled state information, i.e. we consider the stabilization of continuous time systems by repeatedly applying input trajectories that are obtained from the solution of an open-loop optimal control problem at discrete sampling instants. In the following we shortly refer to this as sampled-data NMPC. In comparison to NMPC for discrete time systems (see e.g. [1, 17, 68]) or instantaneous NMPC [68], where the optimal input is recalculated at all times (no open-loop input signal is applied to the system), the inter sampling behavior of the system while the open-loop input is applied must be taken into account, see e.g. [25, 27, 44, 45, 62]. In Section 2 we review the basic principle of NMPC. Before we focus on the theoretical questions, we shortly outline in Section 2.3 how the resulting openloop optimal control problem can be solved. Section 3 contains a discussion on how stability in sampled-data NMPC can be achieved. Section 4 discusses robustness issues in NMPC and Section 5 considers the output feedback problem for NMPC. Before concluding in Section 8 we consider in Section 6 the sampled-data NMPC control of a simple nonlinear example system and in Section 7 the pendulum benchmark example considered throughout this book.

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2 Principles of Sampled-Data Model Predictive Control In model predictive control the input applied to the system (1) is given by the repeated solution of a (finite) horizon open-loop optimal control problem subject to the system dynamics, state and input constraints: Based on measurements obtained at a sampling time (in the following denoted by ti ), the controller predicts the dynamic behavior of the system over the so called control/prediction horizon Tp and determines the input such that an open-loop performance objective is minimized. Under the assumption that the prediction horizon spans to infinity and that there are no disturbances and no model plant mismatch, one could apply the resulting input open-loop to the system and achieve (under certain assumptions) convergence to the origin. However, due to external disturbances, model plant mismatch and the use of finite prediction horizons the actual predicted state and the true system state differ. Thus, to counteract this deviation and to suppress the disturbances it is necessary to in cooperate feedback. In model predictive control this is achieved by applying the obtained optimal open-loop input only until the next sampling instant at which the whole process – prediction and optimization – is repeated (compare Figure 1), thus moving the prediction horizon forward.

closed-loop input u

control/prediction horizon Tp

open loop input u ¯

closed-loop

control/prediction horizon Tp

closed-loop input u

open loop input u ¯

closed-loop

state x

state x

predicted state x ¯

predicted state x ¯ ti

ti+1 sampling time ti

ti + T p

ti

ti+1

ti+2

ti+1 + Tp

sampling time ti+1

Fig. 1. Principle of model predictive control.

The whole procedure can be summarized by the following steps: 1. Obtain estimates of the current state of the system 2. Obtain an admissible optimal input by minimizing the desired cost function over the prediction horizon using the system model and the current state estimate for prediction 3. Implement the obtained optimal input until the next sampling instant 4. Continue with 1. Considering this control strategy various questions such as closed-loop stability, robustness to disturbances/model uncertainties and the efficient solution of the resulting open-loop optimal control problem arise.

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2.1 Mathematical Formulation of Sampled-Data NMPC Throughout the paper we consider the stabilization of time-invariant nonlinear systems of the form x(t) ˙ = f (x(t), u(t))

a.e. t ≥ 0,

x(0) = x0 ,

(1)

where x ∈ Rn denotes the system state and u ∈ Rm is the control or input to the system. We assume that the vector field f : Rn ×Rm → Rn is locally Lipschitz continuous with f (0, 0) = 0. The objective is to (optimally) stabilize the system subject to the input and state constraints: u(t) ∈ U ⊂ Rm , x(t) ∈ X ⊆ Rn , ∀t ≥ 0, where U ⊂ Rm is assumed to be compact and X ⊆ Rn is assumed to be simply connected with (0, 0) ∈ X ×U. Remark 1. (Rate constraints on the inputs) If rate constraints u(t) ˙ ∈ U˙ , ∀t ≥ 0

(2)

on the inputs must be considered, they can be transformed to the given form by adding integrators in the system before the inputs, see for example Section 7. Note, however, that this transforms the input constraint u ∈ U to constraints on the integrator states. We denote the solution of (1) (if it exists) starting at a time t1 from a state x(t1 ), applying a (piecewise continuous) input u : [t1 , t2 ] → Rm by x(τ ; u(·), x(t1 )), τ ∈ [t1 , t2 ]. In sampled-data NMPC an open-loop optimal control problem is solved at the discrete sampling instants ti . We assume that these sampling instants are given by a partition π of the time axis: Definition 1. (Partition) A partition is a series π = (ti ), i ∈ N of (finite) positive real numbers such that t0 = 0, ti < ti+1 and ti → ∞ for i → ∞. Furthermore, π ¯ := supi∈N (ti+1 −ti ) denotes the upper diameter of π and π := inf i∈N (ti+1 −ti ) denotes the lower diameter of π. Whenever t and ti occur together, ti should be taken as the closest previous sampling instant with ti < t. The input applied in between the sampling instants, i.e. in the interval [ti , ti+1 ), in NMPC is given by the solution of the open-loop optimal control problem min

[0,Tp ]

J(x(ti ), u ¯(·))

(3a)

u ¯(·)∈L∞

subject to: x ¯˙ (τ ) = f (¯ x(τ ), u¯(τ )),

x¯(ti ) = x(ti )

(3b)

u ¯(τ ) ∈ U, x ¯(τ ) ∈ X τ ∈ [ti , ti + Tp ]

(3c)

x ¯(ti + Tp ) ∈ E.

(3d)

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Here the bar denotes predicted variables, i.e. x¯(·) is the solution of (3b) driven by the input u ¯(·) : [ti , ti + Tp ] → U with the initial condition x(ti ). The distinction between the real system state x of (1) and the predicted state x ¯ in the controller is necessary since due to the moving horizon nature even in the nominal case the predicted states will differ from the real states at least after one sampling instant. As cost functional J minimized over the control horizon Tp ≥ π ¯ > 0 we consider Z J(x(ti ), u ¯(·)) :=

ti +Tp

F (¯ x(τ ), u ¯(τ ))dτ + E(¯ x(ti + Tp )),

(4)

ti

where the stage cost F : X × U → X is assumed to be continuous, satisfies F (0, 0) = 0, and is lower bounded by positive semidefinite function αF : R → R+ 0 , i.e. αF (x) ≤ F (x, u) ∀(x, u) ∈ X × U. We furthermore assume that the autonomous system f (x, 0) is zero state detectable via α(x), i.e. ∀(x0 ) ∈ X , αF (x(τ ; x0 )) = 0 ⇒ x(τ ; x0 ) as t → ∞, where x(τ ; x0 ) denotes the solution of the system x˙ = f (x, 0) starting from x(0) = x0 . The so called terminal region constraint E and the so called terminal penalty term E are typically used to enforce stability or to increase the performance of the closed-loop, see Section 3. The solution of the optimal control problem (3) is denoted by u ¯ ? (·; x(ti )). It defines the open-loop input that is applied to the system until the next sampling instant ti+1 : u(t; x(ti )) = u ¯? (t; x(ti )),

t ∈ [ti , ti+1 ) .

(5)

As noted above, the control u(t; x(ti )) is a feedback, since it is recalculated at each sampling instant using the new state measurement. We limit the presentation to input signals that are piecewise continuous and refer to an admissible input as: Definition 2. (Admissible Input) An input u : [0, Tp ] → Rm for a state x0 is called admissible, if it is: a) piecewise continuous, b) u(τ ) ∈ U ∀τ ∈ [0, Tp ], c) x(τ ; u(·), x0 ) ∈ X ∀τ ∈ [0, Tp ], d) x(Tp ; u(·), x0 ) ∈ E. We furthermore consider an admissible set of problem (3) as: Definition 3. (Admissible Set) A set X ⊆ X is called admissible, if for all x0 ∈ X there exists a piecewise continuous input u ˜ : [0, Tp ] → U such that a) x(τ ; x0 , u ˜(·)) ∈ X, τ ∈ [0, Tp ] and b) x(Tp ; x0 , u ˜(·)) ∈ E. Without further (possibly very strong) restrictions it is often not clear if for a given x an admissible input nor if the minimum of (3) exists. While the existence of an admissible input is related to constrained controllability, the existence of an optimal solution of (3) is in general non trivial to answer. For simplicity of presentation we assume in the following, that the set R denotes an admissible set that admits an optimal solution of (3), i.e. one obtains the following assumption:

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Assumption 1 (Set R) There exists an admissible set R such that (3) admits for all x0 ∈ R an optimal (not necessarily unique) solution. It is possible to derive existence results for (3) considering measurable inputs and imposing certain convexity and compactness see for example [36, 37, 73] and [4, 35, 82]. However, often it is not possible to check the necessary conditions a priory. The main reason for imposing Assumption 1 is the requirement that an optimal/feasible solution at one sampling instant should guarantee (under certain assumptions) the existence of an optimal/feasible solution at the next sampling instant (see Section 3). The optimal value of the cost functional (4) plays an important role in many considerations. It is typically denoted as value function: Definition 4. (Value function) The value function V (x) is defined as the minimal value of the cost for the state x: V (x) = J(¯ u? (·; x); x). The value function is for example used in the proof of convergence and stability. It often serves as a “Lyapunov function”/decreasing function candidate, see Section 3 and [1, 68]. In comparison to sampled-data NMPC for continuous time systems, in instantaneous NMPC the input is defined by the solution of the optimal control problem (3) at all times: u(x(t)) = u ¯? (t; x(t)), i.e. no open-loop input is applied, see e.g. [67, 68]. Considering that the solution of the open-loop optimal control problem requires an often non negligible time, this approach can not be applied in practice. Besides the continuous time considerations results for NMPC of discrete time systems are also available (see e.g. [1, 17, 68]). We do not go into further details here. Remark 2. (Hybrid nature of sampled-data predictive control) Note, that in sampled-data NMPC the input applied in between the recalculation instants ti and ti+1 is given by the solution of the open-loop optimal control problem (3) at time ti , i.e. the closed-loop is given by x(t) ˙ = f (x(t), u(t; x(ti ))) .

(6)

Thus, strictly speaking, the behavior of the system is not only defined by the current state. Rigorously one has to consider a hybrid system [43, 46, 74, 84] consisting of the “discrete” state x(ti ), the continuous state x(t). This is especially important for the stability considerations in Section 3, since the the “discrete memory” x(ti ) must be taken into account. 2.2 Inherent Characteristics and Problems of NMPC One of the key problems in predictive control schemes is that the actual closedloop input and states differ from the predicted open-loop ones, even if no model plant mismatch and no disturbances are present. This stems from the fact, that at the next sampling instant the (finite) prediction horizon moves

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forward, allowing to consider more information thus leading to a mismatch of the trajectories. The difference between the predicted and the closed-loop trajectories has two immediate consequences. Firstly, the actual goal to compute a feedback such that the performance objective over an often desired infinite horizon of the closed-loop is minimized is not achieved. Secondly there is in general no guarantee that the closed-loop system will be stable at all. It is indeed easy to construct examples for which the closed-loop becomes unstable if a short finite horizon is chosen. Hence, when using finite prediction horizons special attention is required to guarantee stability (see Section 3). Summarizing, the key characteristics and properties of NMPC are: • • • •

NMPC allows the direct use of nonlinear models for prediction. NMPC allows the explicit consideration of state and input constraints. In NMPC a time domain performance criteria is minimized on-line. In NMPC the predicted behavior is in general different from the closedloop behavior. • For the application of NMPC an open-loop optimal control problem must be solved on-line. • To perform the prediction the system states must be measured or estimated. Remark 3. In this paper we mainly focus on NMPC for the stabilization of time-invariant continuous time nonlinear systems. However, note that NMPC is also applicable to a large class of other systems, i.e. discrete time systems, delay systems, time-varying systems, and distributed parameter systems, for more details see for example [1, 17, 68]. Furthermore, NMPC is also well suited for tracking problems or problems where one has to perform transfer between different steady states optimally, see e.g. [28, 58, 70]. Before we summarize the available stability results for sampled-data NMPC, we comment in the next section on the numerical solution of the open-loop optimal control problem. 2.3 Numerical Aspects of Sampled-Data NMPC Predictive control circumvents the solution of the Hamilton-Jacobi-Bellman equation by solving the open-loop optimal control problem at every sampling instant only for the currently measured system state. An often untraceable problem is replaced by a traceable one. In linear MPC the solution of the optimal control problem (3) can often be cast as a convex quadratic program, which can be solved efficiently. This is one of the main reasons for the practical success of linear MPC. In NMPC, however, at every sampling instant a general nonlinear open-loop optimal control problem (3) must be solved on-line. Thus one important precondition for the application of NMPC, is the availability of reliable and efficient numerical dynamic optimization algorithms for the optimal control problem (3). Solving (3) numerically efficient and fast is,

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however, not a trivial task and has attracted many research interest in recent years (see e.g. [2, 5, 6, 18, 22–24, 56, 64–66, 81, 83]). Typically so called direct solution methods [6, 7, 76] are used, i.e. the original infinite dimensional problem is turned into a finite dimensional one by discretizing the input (and also possibly the state). Basically this is done by parameterizing the input (and possibly the states) finitely and to solve/approximate the differential equations during the optimization. We do not go into further details and instead refer to [7, 22, 66]. However, we note that recent studies have shown the usage of special dynamic optimizers and tailored NMPC schemes allows to employ NMPC to practically relevant problems (see e.g. [2, 24, 29, 34, 65, 81]), even with todays computational power. Remark 4. (Sub optimality and NMPC) Since the optimal control problem (3) is typically non convex, it is questionable if the globally minimizing input can be found at all. While the usage of a non optimal admissible input might lead to an increase in the cost, it is not crucial to find the global minima for stability of the closed-loop, as outlined in the next Section.

3 Nominal Stability of Sampled-Data NMPC As outlined one elementary question in NMPC is whether a finite horizon NMPC strategy does guarantee stability of the closed-loop. While a finite prediction and control horizon is desirable from an implementation point of view, the difference between the predicted state trajectory and the resulting closed-loop behavior can lead to instability. Here we review some central ideas how stability can be achieved. No attempt is made to cover all existing approaches and methods, especially those which consider instantaneous or discrete time NMPC. We do also only consider the nominal case, i.e. it is assumed that no external disturbances act on the system and that there is no model mismatch between the system model used for prediction and the real system. Stability by an infinite prediction horizon: The most intuitive way to achieve stability/convergence to the origin is to use an infinite horizon cost, i.e. Tp in the optimal control problem (3) is set to ∞. In this case the open-loop input and state trajectories resulting from (3) at a specific sampling instant are coincide with the closed-loop trajectories of the nonlinear system due to Bellman’s principle of optimality [3]. Thus, the remaining parts of the trajectories at the next sampling instant are still optimal (end pieces of optimal trajectories are optimal). Since the first part of the optimal trajectory has been already implemented and the cost for the remaining part and thus the value function is decreasing, which implies under mild conditions convergence of the states. Detailed derivations can for example be found in [51, 52, 67, 68].

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Stability for finite prediction horizons: In the case of finite horizons the stability of the closed-loop is not guaranteed a priori if no precautions are taken. By now a series of approaches exist, that achieve closed-loop stability. In most of these approaches the terminal penalty E and the terminal region constraint E are chosen suitable to guarantee stability or the standard NMPC is modified to achieve stability. The additional terms are not motivated by physical restrictions or performance requirements, they have the sole purpose to enforce stability. Therefore, they are usually called stability constraints. Stability via a zero terminal constraint: One possibility to enforce stability with a finite prediction horizon is to add the so called zero terminal equality constraint at the end of the prediction horizon, i.e. x ¯(t + Tp ) = 0

(7)

is added to the optimal control problem (3) [9, 52, 67, 69]. This leads to stability of the closed-loop, if the optimal control problem has a solution at t = 0. Similar to the infinite horizon case the feasibility at one sampling instant does imply feasibility at the following sampling instants and a decrease in the value function. One disadvantage of a 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 regions of attraction. Furthermore, from a computational point of view, an exact satisfaction of a zero terminal equality constraint does require in general an infinite number of iterations in the optimization and is thus not desirable. The main advantages of a zero terminal constraint are the straightforward application and the conceptual simplicity. Dual mode control: One of the first sampled-data NMPC approaches avoiding an infinite horizon or a zero terminal constraint is the so called dual-mode NMPC approach [71]. Dual-mode is based on the assumption that a local (linear) controller is available for the nonlinear system. Based on this local linear controller a terminal region and a quadratic terminal penalty term which are added to the open-loop optimal control problem similar to E and E such that: 1.) the terminal region is invariant under the local control law, 2.) the terminal penalty term E enforces a decrease in the value function. Furthermore the prediction horizon is considered as additional degree of freedom in the optimization. The terminal penalty term E can be seen as an approximation of the infinite horizon cost inside of the terminal region E under the local linear control law. Note, that the dual-mode control is not strictly a pure NMPC controller, since the open-loop optimal control problem is only repeatedly solved until the system state enters the terminal set E, which is achieved in finite time. Once the system state is inside E the control is switched to the local control law u = Kx, thus the name dual-mode NMPC. Thus the local control is utilized to establish asymptotic stability while the NMPC feedback is used to increase the region of attraction of the local control law.

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Based on the results in [71] it is shown in [12] that switching to the local control law is not necessary to establish stability. Control Lyapunov function approaches: In the case that E is a global control Lyapunov function for the system, the terminal region constraint x ¯(t + Tp ) ∈ E is actual not necessary. Even if the control Lyapunov is not globally valid, convergence to the origin can be achieved [50] and it can be established that for increasing prediction horizon length the region of attraction of the infinite horizon NMPC controller is recovered [48, 50]. Approaches using a control Lyapunov functions as terminal penalty term and no terminal region constraint are typically referred to as control Lyapunov function based NMPC approaches. Unified conditions for convergence: Besides the outlined approaches there exist a series of approaches [11, 12, 14, 61, 71] that are based on the consideration of an (virtual) local control law that is able to stabilize the system inside of the terminal region and where the terminal penalty E provides an upper bound on the optimal infinite horizon cost. The following theorem covers most of the existing stability results. It establishes conditions for the convergence of the closed-loop states under sampleddata NMPC. It is a slight modification of the results given in [10, 11, 36]. The proof is outlined here since it gives a basic idea on the general approach how convergence and stability is achieved in NMPC. Theorem 1. (Convergence of sampled-data NMPC) Suppose that (a) the terminal region E ⊆ X is closed with 0 ∈ E and that the terminal penalty E(x) ∈ C 1 is positive semi-definite (b) ∀x ∈ E there exists an (admissible) input uE : [0, π ¯ ] → U such that x(τ ) ∈ E and ∂E f (x(τ ), uE (τ )) + F (x(τ ), uE (τ )) ≤ 0 ∂x

∀τ ∈ [0, π ¯]

(8)

(c) x(0) ∈ R Then for the closed-loop system (1), (5) x(t) → 0 for t → ∞. Proof. See [26]. Loosely speaking, E is a F -conform local control Lyapunov function in the terminal set E. The terminal region constraint enforces feasibility at the next sampling instant and allows, similarly to the infinite horizon case, to show that the value function is strictly decreasing. Thus stability can be established. Note that this result is nonlocal in nature, i.e. there exists a region of attraction R which is of at least the size of E. Various ways to determine a suitable terminal penalty term and terminal region exist. Examples are the use of a control Lyapunov function as terminal penalty E [49, 50] or the use of a local nonlinear or linear control law to determine a suitable terminal penalty E and a terminal region E [11, 12, 14, 61, 71].

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Remark 5. (Sub optimality) Note that we need the rather strict Assumption 1 on the set R to ensure the existence of a new optimal solution at ti+1 based on the existence of an optimal solution at ti . The existence of an admissible input at ti+1 , i.e. u ˜ is already guaranteed due to existence of local controller, i.e. condition (b). In principle the existence of an optimal solution at the next time instance is not really required for the convergence result. The admissible input, which is a concatenation of the remaining old input and the local control already leads to a decrease in the cost function and thus convergence. To increase performance from time instance to time instance one could require that the cost decreases from time instance to time instance more than the decrease resulting from an application of the “old” admissible control, i.e. feasibility implies convergence [12, 79]. Remark 6. (Stabilization of systems that require discontinuous inputs) In principle Theorem 1 allows to consider the stabilization of systems that can only be stabilized by feedback that is discontinuous in the state [36], e.g. nonholonomic mechanical systems. However, for such systems it is in general rather difficult to determine a suitable terminal region and a terminal penalty term. To weaken the assumptions in this case, it is possible to drop the continuous differentiability requirement on E, requiring merely that E is only Lipschitz continuous in E. From Rademacker’s theorem [16] it then follows that E is continuously differentiable almost everywhere and that (8) holds for almost all τ and the proof remains nearly unchanged. More details can be found in [37]. Remark 7. (Special input signals) Basically it is also possible to consider only special classes of input signals, e.g. one could require that the input is piecewise continuous in between sampling instants or that the input is parameterized as polynomial in time or as a spline. Modifying Assumption 1, namely that the optimal control problem posses a solution for the considered input class, and that condition (8) holds for the considered inputs, the proof of Theorem 1 remains unchanged. The consideration of such inputs can for example be of interest, if only piecewise constant inputs can be implemented on the real system or if the numerical on-line of the optimal control problem allows only the consideration of such inputs. One example of such an expansion are the consideration of piecewise constant inputs as in [61, 62]. So far only conditions for the convergence of the states to the origin where outlined. In many control applications also the question of asymptotic stability in the sense of Lyapunov is of interest. Even so that this is possible for the sampled-data setup considered here, we do not go into further details, see e.g. [26, 37]. Concluding, the nominal stability question of NMPC is by now well understood and a series of NMPC schemes exist, that guarantee the closedloop stability.

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4 Robustness of Sampled-Data NMPC The results reviewed so far base on the assumption that the real system coincides with the model used for prediction, i.e. no model/plant mismatch or external disturbances are present. Clearly, this is very unrealistic and the development of a NMPC framework to address robustness issues is of paramount importance. In general one distinguishes between the inherent robustness properties of NMPC and the design of NMPC controllers that take the uncertainty/disturbances directly into account. Typically NMPC schemes that take uncertainty that acts on the system directly into account are based on game-theoretic considerations. Practically they often require the on-line solution of a min-max problem. A series of different approaches can be distinguished. We do not go into details here and instead refer to [8, 13, 38, 53, 54, 57, 59, 60]. Instead we are interested in the so called inherent robustness properties of sampled-data NMPC. By inherent robustness we mean the robustness of NMPC to uncertainties/disturbances without taking them directly into account. As shown sampled-data NMPC posses under certain conditions inherent robustness properties. This property stems from the close relation of NMPC to optimal control. Results on the inherent robustness of instantaneous NMPC can for example be found in [9, 63, 68]. Discrete time results are given in [42, 80] and results for sampled-data NMPC are given in [33, 71]. Typically these results consider additive disturbances of the following form: x˙ = f (x, u) + p(x, u, w)

(9)

where p : Rn × Rm × Rl → Rn describes the model uncertainty/disturbance, and where w ∈ W ∈ Rl might be an exogenous disturbance acting on the system. However, assuming that f locally Lipschitz in u these results can be simply expanded to the case of input disturbances. This type of disturbances is of special interrest, since it allows to capture the influence of numerical solution of the open-loop optimal control problem. Further examples of input disturbances are neglected fast actuator dynamics, computational delays, or numerical errors in the solution of the underlying optimal control problem. For example, inherent robustness was used in [20, 21] to establish stability of a NMPC scheme that employs approximated solutions of the optimal control problem. Summarizing, some preliminary results for the inherent robustness and the robust design of NMPC controller exist. However, these result are either not implementable since they require a high computational load or they are not directly applicable due to their restrictive assumptions.

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5 Output Feedback Sampled-Data NMPC One of the key obstacles for the application of NMPC is that at every sampling instant ti the system state is required for prediction. However, often not all system states are directly accessible, i.e. only an output y = h(x, u)

(10)

is directly available for feedback, where y ∈ Rp are the measured outputs and where h : Rn ×Rm → Rp maps the state and input to the output. To overcome this problem one typically employs a state observer for the reconstruction of the states. In principle, instead of the optimal feedback (5) the “disturbed” feedback u(t; x ˆ(ti )) = u ¯? (t; xˆ(ti )), t ∈ [ti , ti+1 ) (11) is applied. Yet, due to the lack of a general nonlinear separation principle, stability is not guaranteed, even if the state observer and the NMPC controller are both stable. Several researchers have addressed this problem (see for example for a review [32]). The approach in [19] derives local uniform asymptotic stability of contractive NMPC in combination with a “sampled” state estimator. In [58], see also [80], asymptotic stability results for observer based discrete-time NMPC for “weakly detectable” systems are given. The results allow, in principle, to estimate a (local) region of attraction of the output feedback controller from Lipschitz constants. In [72] an optimization based moving horizon observer combined with a certain NMPC scheme is shown to lead to (semi-global) closed-loop stability. In [30, 31, 47], where semi-global stability results for output feedback NMPC using high-gain observers are derived. Furthermore, in [32], based on the inherent robustness properties of NMPC as outlined in Section 4 for a broad class of state feedback nonlinear model predictive controllers, conditions, on the observer that guarantee that the closed-loop is semi-global practically stable. Even so that a series of output feedback results for NMPC using observers for state recovery exist, most of these approaches are fare away from being implementable. Thus, further research has to address this important question to allow for a practical application of NMPC.

6 A Simple Nonlinear Example The following example is thought to show some of the inherent properties of sampled-data NMPC and to show how Theorem 1 can be used to design a stabilizing NMPC controller that takes constraints into account. We consider the following second order system [39] x˙ 1 (t) = x2 (t) x˙ 2 (t) = −x1 (t) + x2 (t) sinh(x21 (t) + x22 (t)) + u(t),

(12a) (12b)

14

Findeisen, Raff, Allg¨ ower

which should be stabilized with the bounded control u(t) ∈ U := {u ∈ R| |u| ≤ 1} ∀t ≥ 0 where the stage cost is given by F (x, u) = x22 + u2 .

(13)

According to Theorem 1 we achieve stability if we can find a terminal region E and a C 1 terminal penalty E(x) such that (8) is satisfied. For this we consider the unconstrained infinite horizon optimal control problem for (12). One can verify that the control law 2

2

u∞ (x) = −x2 ex1 +x2

(14)

minimizes the corresponding cost J∞ (x, u(·)) =

Z

∞ 0


x22 (τ ) + u2 (τ ) dτ,

(15)

and that the associated value function, which will be used as terminal penalty term, is given by 2 2 E(x) := V∞ (x) = ex1 +x2 − 1. (16) It remains to find a suitable terminal region. According to Theorem 1 (b) for all x ∈ E there must exist an open-loop input u which satisfies the constraints such that (8) is satisfied. If we define E as E := {x ∈ R2 |E(x) ≤ α}

(17)

we know that along solution trajectories of the closed-loop system controlled by u∞ (x), i.e. x˙ = f (x, u∞ ), the following holds ∂E f (x, u∞ (x)) + F (x(τ ), u∞ (x)) = 0, ∂x

(18)

however, α must be chosen such that u∞ (x) ∈ U. It can be verified, that for 2 α = β1 − 1, where β satisfies 1 − βeβ = 0, u∞ (x) ∈ U ∀x ∈ E. The derived terminal penalty term E(x) and the terminal region E are designed to satisfy the conditions of Theorem 1, thus the resulting NMPC controller should be able to stabilize the closed-loop. The resulting NMPC controller with the prediction horizon set to Tp = 2 is compared to a feedback linearizing controller and the optimal controller (14) (where the input of both is limited to the set U by saturation). The feedback linearizing controller used is given by:
uFl (x) := −x2 1 + sinh(x21 + x22 ) , (19) which stabilizes the system globally, if the input is unconstrained. The really implemented input for the feedback linearizing controller (and the unconstrained optimal controller (14)) is given by

Sampled-Data Model Predictive Control for Constrained Systems

15

u(x) = sign(uFl (x)) min{1, |uFl(x)|},

(20) n

x