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 over-comes this problem is model predictive control. In model predictive control the solution of the Hamilton-Jacobi-Bellman equation is avoided by repeat-edly 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 predic-tive 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 ques-tion of stability often the achieved performance as well as the satisfacques-tion 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 op-timal 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
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 so-lution 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 lin-ear dynamical models of the system and in which linlin-ear 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 indus-tries. 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 perfor-mance of the closed-loop. Driven by this shortcoming and the desire to di-rectly 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 exam-ple [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 in-stants. 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 instan-taneous 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 open-loop 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 prob-lem 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 Sec-tion 7 the pendulum benchmark example considered throughout this book.
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 mea-surements obtained at a sampling time (in the following denoted by ti), the
controller predicts the dynamic behavior of the system over the so called con-trol/prediction horizon Tp and determines the input such that an open-loop
performance objective is minimized. Under the assumption that the predic-tion 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 pre-diction horizons the actual predicted state and the true system state differ. Thus, to counteract this deviation and to suppress the disturbances it is nec-essary 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 state x closed-loop input u closed-loop state x closed-loop input u
control/prediction horizon Tp control/prediction horizon Tp
ti+1 ti
predicted state ¯x
ti+ Tp ti ti+1 ti+2 ti+1+ Tp predicted state ¯x open loop input ¯u
sampling time ti sampling time ti+1
open loop input ¯u
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 func-tion over the predicfunc-tion 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 stabil-ity, robustness to disturbances/model uncertainties and the efficient solution of the resulting open-loop optimal control problem arise.
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 Sec-tion 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 :=
infi∈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 ¯ u(·)∈L[0,Tp]∞ J(x(ti), ¯u(·)) (3a) subject to: ˙¯x(τ)=f(¯x(τ), ¯u(τ )), x(t¯ i) = x(ti) (3b) ¯ u(τ ) ∈ U, ¯x(τ ) ∈ X τ ∈ [ti, ti+ Tp] (3c) ¯ x(ti+ Tp) ∈ E. (3d)
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
J(x(ti), ¯u(·)) :=
Z ti+Tp
ti
F (¯x(τ ), ¯u(τ ))dτ + E(¯x(ti+ Tp)), (4)
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(t i)).
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:
Assumption 1 (Set R) There exists an admissible set R such that (3) ad-mits 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 condi-tions 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 stabil-ity. 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 in-stantaneous 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
ap-plied, 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+1is 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 closed-loop input and states differ from the predicted open-closed-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
forward, allowing to consider more information thus leading to a mismatch of the trajectories. The difference between the predicted and the closed-loop tra-jectories 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 unsta-ble 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
closed-loop 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 esti-mated.
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 gen-eral nonlinear open-loop optimal control problem (3) must be solved on-line. Thus one important precondition for the application of NMPC, is the avail-ability of reliable and efficient numerical dynamic optimization algorithms for the optimal control problem (3). Solving (3) numerically efficient and fast is,
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 prob-lem 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 equa-tions 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 result-ing 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 exist-ing 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. Tpin 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 tra-jectories 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].
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 sta-bility 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 stabil-ity 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 satis-faction 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 avoid-ing 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 (lin-ear) 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 ter-minal 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 repeat-edly 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.
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 attrac-tion 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 con-sideration 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 estab-lishes conditions for the convergence of the closed-loop states under sampled-data 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) ∈ C1 is positive semi-definite
(b) ∀x ∈ E there exists an (admissible) input uE : [0, ¯π] → U such that x(τ ) ∈ E
and
∂E
∂xf (x(τ ), uE(τ )) + F (x(τ ), uE(τ )) ≤ 0 ∀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].
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 prin-ciple 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. nonholo-nomic 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 closed-loop stability.
4 Robustness of Sampled-Data NMPC
The results reviewed so far base on the assumption that the real system co-incides 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 ro-bustness 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 di-rectly into account are based on game-theoretic considerations. Practically they often require the on-line solution of a min-max problem. A series of dif-ferent 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 ac-count. As shown sampled-data NMPC posses under certain conditions in-herent 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.
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→ Rpmaps 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; ˆx(ti)) = ¯u?(t; ˆx(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 con-troller 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 out-put 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 de-rived. 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]
˙x1(t) = x2(t) (12a)
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 C1terminal 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
u∞(x) = −x2ex 2
1+x22 (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
E(x) := V∞(x) = ex 2
1+x22− 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 uwhich 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
∂xf (x, u∞(x)) + F (x(τ ), u∞(x)) = 0, (18) however, α must be chosen such that u∞(x) ∈ U. It can be verified, that for
α = 1β− 1, where β satisfies 1 − βeβ2 = 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 uncon-strained optimal controller (14)) is given by
u(x) = sign(uFl(x)) min{1, |uFl(x)|}, (20)
where the sign operator is defined as usual, i.e. sign(x) :=n−1,1, x<0x≥0. For the NMPC controller the sampling instants where given by an equidistant parti-tion of the time axis, i.e. π = (ti) with ti+1= ti+δ and t0= 0, where the
sam-pling time δ is δ = 0.1. The open-loop optimal control problem (3) is solved by a direct solution method. Specifically the input signal is parametrized as piece-wise constant with a time discretization of 0.05 over the prediction horizon, i.e. at every sampling instant an optimization problem with 40 free variables is solved. Figure 2 shows the simulation results in the phase plan for the initial conditions x1(0) = −1.115 and x2(0) = −0.2 of all three controllers. Note that
−1.5 −1 −0.5 0 0.5 1 1.5 −0.6 −0.4 −0.2 0 0.2 0.4 0.6 0.8 1 1.2 x 1 x2
Fig. 2. Phase plot x1 over x2 starting from the initial condition x(0) = [−1.115, −0.2] for the NMPC controller (black solid), the saturated feedback linearizing con-troller (dark gray solid) and the saturated optimal concon-troller (gray solid). The inner ellipsoid (gray dashed) is the border of the terminal region E of the NMPC con-troller, while the outer curve (black dashed) are the points for which the optimal controller u∞(x) just satisfies the input constraint (saturation not active).
the initial conditions are such that for all controllers the input constraints are not active at the beginning. However, after some time the maximum applica-ble input is reached, i.e. the saturation in (20) is active. As can be seen, both, the optimal controller and the feedback linearizing controller are not able to stabilize the system for the considered initial conditions. In comparison the NMPC controller is able to stabilize the system while meeting the input
con-−1 −0.5 0 x1 −1 0 1 x2 0 2 4 6 8 10 12 −1 0 1 Time u
Fig. 3.Simulation results starting from the initial condition x(0) = [−1.115, −0.2] for the NMPC controller (black solid), the saturated feedback linearizing controller (dark gray solid) and the saturated optimal controller (gray solid).
straints (see Figure 3). Note, that inside of the terminal region the NMPC controller and the optimal control law u∞(x) coincide, since the constraints
are not active and since (18) is satisfied with equality. Thus, the terminal penalty term E(x) can be seen as an approximation of the cost that appears up to infinity. As can be seen from this example, if a value function/Lyapunov function and a local controller as well as the corresponding region of attraction is know, NMPC can be utilized to increase the overall region of attraction of the closed-loop while satisfying the input and state constraints.
7 Inverted Pendulum Benchmark Example
As a second example, underling the achievable performance in the case of input and state constraints, we consider the benchmark inverted pendulum on a cart system ˙x(t) = 0 −1 0 −1 0 1 0 1 0 x(t) + 0 0 1 u(t) + 1 0 0 z(t) (21)
around its upright position. The variable x1 denotes the horizontal speed
the horizontal speed of the cart. The load z represents a horizontal force on the pendulum which is persistent with unknown but bounded magnitude. Furthermore, u is the force by the actuator on the cart which is constrained in the magnitude by |u| ≤ 1.25 and in the slew rate by |du(t)/dt| ≤ 2s−1. In order to take the slew rate constraint on the control input into account, the system (21) is augmented by an integrator at the control input. Thus, the input constraint |u| ≤ 1.25 is transformed to a state constraint on the new state, e.g. |ξ4| ≤ 1.25. With the state ξ = [ξ1, ξ2, ξ3, ξ4]T = [x1, x2, x3, u]T and
the new control input v(t) = du(t)/dt one obtains the augmented system
˙ξ(t) = 0 −1 0 0 −1 0 1 0 0 1 0 1 0 0 0 0 ξ(t) + 0 0 0 1 v(t) + 1 0 0 0 z(t). (22)
Therefore, the input constraints of the system (21) are |ξ4| < 1.25 and
|dv(t)/dt| < 2s−1 for the system (22). Note, that the input constraints of
the system (22) can be casted in the optimal control problem (3). In the
ϕ
u
x3
x1 z
Fig. 4.Inverted pendulum on a cart.
following, two control problems are considered. The control objective of the first problem is to track a reference signal r while the control objective of the second problem is stabilize the system under the influence of a disturbance z. For both control problems the stage cost is chosen as
F (ξ, v) = (ξ − ξs)T 10 0 0 0 0 10 0 0 0 0 10 0 0 0 0 1 (ξ − ξs) + v2,
where is a small positive parameter, e.g. = 0.00001, and ξs the set point
in the stage cost is chosen so small in order to recover the classical quadratic stage cost on the state x and on the input u. To guarantee closed-loop stability, the terminal cost E and the terminal region E are are calculated off-line by a procedure as in the quasi infinite horizon model predictive control scheme described in [12, 15]. The resulting terminal cost E is given by
E(ξ) = (ξ − ξs)T 142.6 −148.8 −67.0 −17.3 −148.8 169.0 79.2 21.1 −67.0 79.2 44.5 12.1 −17.3 21.1 12.1 5.3 (ξ − ξs)
and the terminal region E is given by
E = {ξ ∈ R4|E(ξ) ≤ 3.2}. (23)
Note, that the design of the terminal penalty term and terminal region con-straint is rather easy, since the system itself is linear. Furthermore, the control and prediction horizon is chosen to TP = 6 and the sampling time is chosen
to δ = 0.1. 7.1 Tracking
In the following the tracking problem is studied. The control objective is that the state variable x1 tracks asymptotically the reference signal r. However,
the tracking problem cannot directly solved via the NMPC controller with the optimal control problem (3). Therefore, the tracking problem was considered as a sequence of set point changes. The set points of the system (22) depend on the reference signal r, i.e. ξs = [r 0 r 0]T. Figure 5 shows the
closed-loop system states x, the control input u and the reference signal r. Figure 5 shows that the reference signal r is asymptotically tracked while satisfying the constraints.
7.2 Disturbance Attenuation
In the following the task is to stabilize the state x1 under a persistent
dis-turbance z with unknown but bounded magnitude. It is assumed that the full state ξ can be measured but not the disturbance z. Also in this control problem the NMPC controller with the optimal control problem (3) cannot directly be applied to stabilize the state x1under the disturbance z. A typical
approach to solve such kind of disturbance attenuation problems in model pre-dictive control is to estimate the disturbance z via an observer and to use the estimated disturbance ˆz in the prediction of the model predictive controller. The disturbance z can be estimated via the observer
0 5 10 15 20 25 30 35 40 45 0 1 2 2.5 x1 , r 0 5 10 15 20 25 30 35 40 45 −1 0 1 2 2.5 x2 , x 3 0 5 10 15 20 25 30 35 40 45 −2 −1.25 0 1.25 2 Time u
Fig. 5. Simulation results of r (gray solid), x1 (black solid), x2 (gray solid), x3 (black solid), and u (black solid) for the tracking problem.
˙ˆ φ(t) = 0 −1 0 0 1 −1 0 1 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 ˆ φ(t) + 0 0 0 1 0 v(t) + L(y(t) − ˆy(t)) ˆ y(t) = 1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 1 0 ˆ φ(t), (24)
where ˆφ = [ ˆξ ˆz]T is the augmented state. The observer gain L was chosen
such that the disturbance z is estimated sufficiently fast in order to obtain a good performance. Figure 6 shows the closed-loop system states x, the control input u and the disturbance z. As can be seen the state x1 is asymptotically
stabilized under the disturbance z while satisfying the constraints.
In summary, in all considered cases NMPC shows good performance while satisfying the constraints.
0 5 10 15 20 25 30 35 40 −1 0 1 2 x1 , z 0 5 10 15 20 25 30 35 40 −1 0 1 2 x2 , x 3 0 5 10 15 20 25 30 35 40 −2 −1.25 0 1.25 2 Time u
Fig. 6. Simulation results of z (gray solid), x1 (black solid), x2 (gray solid), x3 (black solid), and u (black solid) for the disturbance attenuation problem.
8 Conclusions
Model predictive control, especially linear model predictive control, is by now widely applied in practice. However, increasing productivity demands, tighter environmental regulations, higher quality specifications and demanding eco-nomical considerations require to operate process over a wide region of op-erating conditions, for which linear models are often not adequate. This in-adequacy has lead in recent years to an increased theoretical and practical interest in NMPC.
In this paper reviewed the main principles and the existing results of sampled-data NMPC for continuous time systems subject to constraints. As outlined, in NMPC an open-loop optimal control problem is solved repeatedly at fixed sampling instant considering the current system state and the resulting control is applied loop for a short time. Since NMPC is based on an open-loop optimal control problem, it allows the direct consideration of a nonlinear system model and the inclusion of constraints on states and inputs. As outlined a series of questions for NMPC, such as the stability of the closed-loop, are by now well understood. Nevertheless, many open questions remain, before NMPC can be applied successfully in practice.
References
1. F. Allg¨ower, T.A. Badgwell, J.S. Qin, J.B. Rawlings, and S.J. Wright. Nonlinear predictive control and moving horizon estimation – An introductory overview. In P.M. Frank, editor, Advances in Control, Highlights of ECC’99, pages 391–449. Springer, London, 1999.
2. R.A. Bartlett, A. W¨achter, and L.T. Biegler. Active set vs. interior point strate-gies for model predictive control. In Proc. Amer. Contr. Conf., pages 4229–4233, Chicago, Il, 2000.
3. R. Bellman. Dynamic Programming. Princeton University Press, Princeton, New Jersey, 1957.
4. L.D. Berkovitz. Optimal Control Theory. Springer-Verlag, New York, 1974. 5. L. Biegler. Efficient solution of dynamic optimization and NMPC problems. In
F. Allg¨ower and A. Zheng, editors, Nonlinear Predictive Control, pages 219–244. Birkh¨auser, Basel, 2000.
6. L.T. Biegler and J.B Rawlings. Optimization approaches to nonlinear model predictive control. In W.H. Ray and Y. Arkun, editors, Proc. 4th International Conference on Chemical Process Control - CPC IV, pages 543–571. AIChE, CACHE, 1991.
7. T. Binder, L. Blank, H.G. Bock, R. Burlisch, W. Dahmen, M. Diehl, T. Kro-nseder, W. Marquardt, J.P. Schl¨oder, and O. von Stryk. Introduction to model based optimization of chemical processes on moving horizons. In M. Groetschel, S.O. Krumke, and J. Rambau, editors, Online Optimization of Large Scale Sys-tems: State of the Art, pages 295–339. Springer, Berlin, 2001.
8. R. Blauwkamp and T. Basar. A receding-horizon approach to robust output feedback control for nonlinear systems. In Proc. 38th IEEE Conf. Decision Contr., pages 4879–4884, San Diego, 1999.
9. C.C. Chen and L. Shaw. On receding horizon feedback control. Automatica, 18(3):349–352, 1982.
10. H. Chen. Stability and Robustness Considerations in Nonlinear Model Predictive Control. Fortschr.-Ber. VDI Reihe 8 Nr. 674. VDI Verlag, D¨usseldorf, 1997. 11. H. Chen and F. Allg¨ower. Nonlinear model predictive control schemes with
guaranteed stability. In R. Berber and C. Kravaris, editors, Nonlinear Model Based Process Control, pages 465–494. Kluwer Academic Publishers, Dodrecht, 1998.
12. H. Chen and F. Allg¨ower. A quasi-infinite horizon nonlinear model predictive control scheme with guaranteed stability. Automatica, 34(10):1205–1218, 1998. 13. H. Chen, C.W. Scherer, and F. Allg¨ower. A game theoretic approach to non-linear robust receding horizon control of constrained systems. In Proc. Amer. Contr. Conf., pages 3073–3077, Albuquerque, 1997.
14. W. Chen, D.J. Ballance, and J. O’Reilly. Model predictive control of nonlin-ear systems: Computational burden and stability. IEE Proceedings, Part D, 147(4):387–392, 2000.
15. W. Chen, D.J. Ballance, and J. O’Reilly. Optimisation of attraction domains of nonlinear mpc via lmi methods. In Proc. Amer. Contr. Conf., pages 3067–3072, Arlington, 2002.
16. F.H. Clark, Y.S. Leydaev, R.J. Stern, and P.R. Wolenski. Nonsmooth Analysis and Control Theory. Number 178 in Graduate Texts in Mathematics. Springer Verlag, New York, 1998.
17. G. De Nicolao, L. Magni, and R. Scattolini. Stability and robustness of nonlin-ear receding horizon control. In F. Allg¨ower and A. Zheng, editors, Nonlinear Predictive Control, pages 3–23. Birkh¨auser, Basel, 2000.
18. N.M.C. de Oliveira and L.T. Biegler. An extension of Newton-type algorithms for nonlinear process control. Automatica, 31(2):281–286, 1995.
19. S. de Oliveira Kothare and M. Morari. Contractive model predictive control for constrained nonlinear systems. IEEE Trans. Aut. Control, 45(6):1053–1071, 2000.
20. M. Diehl, R. Findeisen, F. Allg¨ower, J.P. Schl¨oder, and H.G. Bock. Stability of nonlinear model predictive control in the presence of errors due to numerical online optimization. In Proc. 43th IEEE Conf. Decision Contr., pages 1419– 1424, Maui, 2003.
21. M. Diehl, R. Findeisen, H.G. Bock, J.P. Schl¨oder, and F. Allg¨ower. Nominal stability of the real-time iteration scheme for nonlinear model predictive control. IEE Control Theory Appl., 152(3):296–308, 2005.
22. M. Diehl, R. Findeisen, Z. Nagy, H.G. Bock, J.P. Schl¨oder, and F. Allg¨ower. Real-time optimization and nonlinear model predictive control of processes gov-erned by differential-algebraic equations. J. Proc. Contr., 4(12):577–585, 2002. 23. M. Diehl, R. Findeisen, S. Schwarzkopf, I. Uslu, F. Allg¨ower, H.G. Bock, and J.P. Schl¨oder. An efficient approach for nonlinear model predictive control of large-scale systems. Part I: Description of the methodology. Automatisierungstechnik, 12:557–567, 2002.
24. M. Diehl, R. Findeisen, S. Schwarzkopf, I. Uslu, F. Allg¨ower, H.G. Bock, and J.P. Schl¨oder. An efficient approach for nonlinear model predictive control of large-scale systems. Part II: Experimental evaluation considering the control of a distillation column. Automatisierungstechnik, 1:22–29, 2003.
25. A.M. Elaiw and Gyurkovics ´E. Multirate sampling and delays in receding hori-zon stabilization of nonlinear systems. In Proc. 16th IFAC World Congress, Prague, Czech Republic, 2005.
26. R. Findeisen. Nonlinear Model Predictive Control: A Sampled-Data Feedback Perspective. Fortschr.-Ber. VDI Reihe 8 Nr. 1087, VDI Verlag, D¨usseldorf, 2005. 27. R. Findeisen and F. Allg¨ower. Stabilization using sampled-data open-loop feed-back – a nonlinear model predictive control perspective. In Proc. Symposium on Nonlinear Control Systems, NOLCOS’2004, Stuttgart, Germany, 2004. 28. R. Findeisen, H. Chen, and F. Allg¨ower. Nonlinear predictive control for setpoint
families. In Proc. Amer. Contr. Conf., pages 260–265, Chicago, 2000.
29. R. Findeisen, M. Diehl, I. Uslu, S. Schwarzkopf, F. Allg¨ower, H.G. Bock, J.P. Schl¨oder, and E.D. Gilles. Computation and performance assessment of nonlin-ear model predictive control. In Proc. 42th IEEE Conf. Decision Contr., pages 4613–4618, Las Vegas, 2002.
30. R. Findeisen, L. Imsland, F. Allg¨ower, and B.A. Foss. Output feedback nonlinear predictive control - a separation principle approach. In Proc. of 15th IFAC World Congress, Barcelona, Spain, 2002. Paper ID 2204 on CD-ROM.
31. R. Findeisen, L. Imsland, F. Allg¨ower, and B.A. Foss. Output feedback stabi-lization for constrained systems with nonlinear model predictive control. Int. J. of Robust and Nonlinear Control, 13(3-4):211–227, 2003.
32. R. Findeisen, L. Imsland, F. Allg¨ower, and B.A. Foss. State and output feedback nonlinear model predictive control: An overview. Europ. J. Contr., 9(2-3):190– 207, 2003.
33. R. Findeisen, L. Imsland, F. Allg¨ower, and B.A. Foss. Towards a sampled-data theory for nonlinear model predictive control. In W. Kang, C. Borges, and M. Xiao, editors, New Trends in Nonlinear Dynamics and Control, volume 295 of Lecture Notes in Control and Information Sciences, pages 295–313, New York, 2003. Springer-Verlag.
34. R. Findeisen, Z. Nagy, M. Diehl, F. Allg¨ower, H.G. Bock, and J.P. Schl¨oder. Computational feasibility and performance of nonlinear model predicitve con-trol. In Proc. 6th European Control Conference ECC’01, pages 957–961, Porto, Portugal, 2001.
35. W. H. Fleming and R. W. Rishel. Deterministic and stochastic optimal control. Springer, Berlin, 1982.
36. F.A. Fontes. A general framework to design stabilizing nonlinear model predic-tive controllers. Syst. Contr. Lett., 42(2):127–143, 2000.
37. F.A. Fontes. Discontinuous feedbacks, discontinuous optimal controls, and continuous-time model predictive control. Int. J. of Robust and Nonlinear Con-trol, 13(3-4):191–209, 2003.
38. F.A. Fontes and L. Magni. Min-max predictive control of nonlinear systems using discontinuous feedback. IEEE Trans. Aut. Control, 48(10):1750–1755, 2003.
39. R. Freeman and J. Primbs. Control Lyapunov functions: New ideas from an old source. In Proc. 35th IEEE Conf. Decision Contr., pages 3926–3931, Kobe, Japan, December 1996.
40. J. B. Froisy. Model predictive control: Past, present and future. ISA Transac-tions, 33:235–243, 1994.
41. C.E. Garc´ıa, D.M. Prett, and M. Morari. Model Predictive Control: Theory and practice – A survey. Automatica, 25(3):335–347, 1989.
42. G. Grimm, M.J. Messina, S. Tuna, and A.R. Teel. Model predictive control: for want of a local control lyapunov function, all is not lost. IEEE Trans. Aut. Control, 50(5):546–558, 2005.
43. R. Grossman, A. Nerode, A. Ravn, and H. Rischel, editors. Hybrid Dynamical Systems. Springer-Verlag, New York, 1993.
44. L. Gr¨une and D. Ne˘si´c. Optimization based stabilization of sampled-data non-linear systems via their approximate discrete-time models. SIAM J. Contr. Optim., 42:98–122, 2003.
45. L. Gr¨une, D. Ne˘si´c, and J. Pannek. Model predictive control for nonlinear sampled-data systems. In R. Findeisen, L.B. Biegler, and F. Allg¨ower, editors, Assessment and Future Directions of Nonlinear Model Predictive Control, Lec-ture Notes in Control and Information Sciences, Berlin, 2006. Springer-Verlag. to appear.
46. L. Hou, A.N. Michel, and H. Ye. Some qualitative properties of sampled-data control systems. IEEE Trans. Aut. Control, 42(42):1721–1725, 1997.
47. L. Imsland, R. Findeisen, E. Bullinger, F. Allg¨ower, and B.A. Foss. A note on stability, robustness and performance of output feedback nonlinear model predictive control. J. Proc. Contr., 13(7):633–644, 2003.
48. K. Ito and K. Kunisch. Asymptotic properties of receding horizon optimal control problems. SIAM J. Contr. Optim., 40(5):1585–1610, 2002.
49. A. Jadbabaie and J. Hauser. On the stability of receding horizon control with a general cost. IEEE Trans. Aut. Control, 50(5):674–678, 2005.
50. A. Jadbabaie, J. Yu, and J. Hauser. Unconstrained receding horizon control of nonlinear systems. IEEE Trans. Aut. Control, 46(5):776 –783, 2001.
51. S.S. Keerthi and E.G. Gilbert. An existence theorem for discrete-time infinite-horizon optimal control problems. IEEE Trans. Aut. Control, 30(9):907–909, 1985.
52. S.S. Keerthi and E.G. Gilbert. Optimal infinite-horizon feedback laws for a general class of constrained discrete-time systems: Stability and moving-horizon approximations. J. Opt. Theory and Appl., 57(2):265–293, 1988.
53. M.V. Kothare, V. Balakrishnan, and M. Morari. Robust constrained model predictive control using linear matrix inequalities. Automatica, 32(10):1361– 1379, 1996.
54. S. Lall and K. Glover. A game theoretic approach to moving horizon control. In D. Clarke, editor, Advances in Model-Based Predictive Control. Oxford Uni-versity Press, 1994.
55. J.H. Lee and B. Cooley. Recent advances in model predictive control and other related areas. In J.C. Kantor, C.E. Garcia, and B. Carnahan, editors, Fifth International Conference on Chemical Process Control – CPC V, pages 201– 216. American Institute of Chemical Engineers, 1996.
56. W.C. Li and L.T. Biegler. Multistep, Newton-type control strategies for con-strained nonlinear processes. Chem. Eng. Res. Des., 67:562–577, 1989.
57. L. Magni, G. De Nicolao, R. Scatollini, and F. Allg¨ower. Robust model predictive control for nonlinear discrete-time systems. Int. J. of Robust and Nonlinear Control, 13(3-4):229–246, 2003.
58. L. Magni, G. De Nicolao, and R. Scattolini. Output feedback and tracking of nonlinear systems with model predictive control. Automatica, 37(10):1601–1607, 2001.
59. L. Magni, G. De Nicolao, R. Scattolini, and F. Allg¨ower. Robust receding horizon control for nonlinear discrete-time systems. In Proc. of 15th IFAC World Congress, Barcelona, Spain, 2001. Paper ID 759 on CD-ROM.
60. L. Magni, H. Nijmeijer, and A.J. van der Schaft. A receding-horizon approach to the nonlinear H∞control problem. Automatica, 37(5):429–435, 2001. 61. L. Magni and R. Scattolini. State-feedback MPC with piecewise constant control
for continuous-time systems. In Proc. 42th IEEE Conf. Decision Contr., pages 4625 – 4630, Las Vegas, 2002.
62. L. Magni and R. Scattolini. Model predictive control of continuous-time non-linear systems with piecewise constant control. IEEE Trans. Aut. Control, 49(5):900–906, 2004.
63. L. Magni and R. Sepulchre. Stability margins of nonlinear receding–horizon control via inverse optimality. Syst. Contr. Lett., 32(4):241–245, 1997.
64. R. Mahadevan and F.J. Doyle III. Efficient optimization approaches to nonlinear model predictive control. Int. J. of Robust and Nonlinear Control, 13(3-4):309– 329, 2003.
65. F. Martinsen, L.T. Biegler, and B.A Foss. Application of optimization algo-rithms to nonlinear MPC. In Proc. of 15th IFAC World Congress, Barcelona, Spain, 2002. Paper ID 1245 on CD-ROM.
66. D.Q. Mayne. Optimization in model based control. In Proc. IFAC Symposium Dynamics and Control of Chemical Reactors, Distillation Columns and Batch Processes, pages 229–242, Helsingor, 1995.
67. D.Q. Mayne and H. Michalska. Receding horizon control of nonlinear systems. IEEE Trans. Aut. Control, 35(7):814–824, 1990.
68. D.Q. Mayne, J.B. Rawlings, C.V. Rao, and P.O.M. Scokaert. Constrained model predictive control: stability and optimality. Automatica, 26(6):789–814, 2000.
69. E.S. Meadows, M.A. Henson, J.W. Eaton, and J.B. Rawlings. Receding hori-zon control and discontinuous state feedback stabilization. Int. J. Contr., 62(5):1217–1229, 1995.
70. H. Michalska. Trajectory tracking control using the receding horizon stratrgy. In Symposium on Control, Optimization and Supervision, CESA’96 IMACS Mul-ticonference, pages 298–303, Lille, 1996.
71. H. Michalska and D.Q. Mayne. Robust receding horizon control of constrained nonlinear systems. IEEE Trans. Aut. Control, 38(11):1623–1633, 1993. 72. H. Michalska and D.Q. Mayne. Moving horizon observers and observer-based
control. IEEE Trans. Aut. Control, 40(6):995–1006, 1995.
73. H. Michalska and R.B. Vinter. Nonlinear stabilization using discontinuous moving-horizon control. IMA Journal of Mathematical Control & Information, 11:321–340, 1994.
74. A.N. Michel. Recent trends in the stability analysis of hybride dynamical sys-tems. IEEE Trans. on Circuits and System, 45(1):120–133, 1999.
75. M. Morari and J.H. Lee. Model predicitve control: Past, present and future. Comp. & Chem. Eng., 23(4/5):667–682, 1999.
76. R. Pytlak. Numerical Methods for Optimal Control Problems with State Con-straints. Lecture Notes in Mathematics. Springer, Berlin, 1999.
77. S.J. Qin and T.A. Badgwell. An overview of nonlinear model predictive control applications. In F. Allg¨ower and A. Zheng, editors, Nonlinear Predictive Control, pages 369–393. Birkh¨auser, 2000.
78. S.J. Qin and T.A. Badgwell. A survey of industrial model predictive control technology. Control Engineering Practice, 11(7):733–764, July 2003.
79. P.O.M. Scokaert, D.Q. Mayne, and J.B. Rawlings. Suboptimal model predictive control (feasibility implies stability). IEEE Trans. Aut. Control, 44(3):648–654, 1999.
80. P.O.M. Scokaert, J.B. Rawlings, and E.S. Meadows. Discrete-time stability with perturbations: Application to model predictive control. Automatica, 33(3):463– 470, 1997.
81. M.J. Tenny and J.B. Rawlings. Feasible real-time nonlinear model predictive control. In 6th International Conference on Chemical Process Control – CPC VI, AIChE Symposium Series, 98(326), pages 187–193, 2001.
82. R. Vinter. Optimal Control. Systems & Control: Foundations & Applications. Birkh¨auser Verlag, Boston, 2000.
83. S. J. Wright. Applying new optimization algorithms to model predictive con-trol. In J.C. Kantor, C.E. Garcia, and B. Carnahan, editors, Fifth International Conference on Chemical Process Control – CPC V, pages 147–155. AIChE Sym-posium Series, 93(316), 1996.
84. H. Ye, A.N. Michel, and L. Hou. Stability theory for hybrid dynamical systems. IEEE Trans. Aut. Control, 43(4):461–474, 1998.
