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This chapter argues that optimization can be used to solve the design problem completely. The case for optimization becomes stronger when one moves outside the class of linear systems. Design of controllers for nonlinear systems is at a less advanced state despite recent spectacular progress due to the complexity and variety of the systems encountered in practice. In practice, of course, controllers are nearly always nonlinear, even if designed using linear theory. The work involved in constructing a Lyapunov function for a given autonomous system is comparable to that of determining a nonlinear feedback law. The first requirement for a design environment is a facility for simulating the system and its controller, together with a suitable friendly interface for inputting the system descriptions and modifying them. Because the need to assess stability may recur frequently in the design process, it appears necessary to avoid repeated modification of the Lyapunov function.

Two competing versions for robust and stochastic model predictive control of constrained linear systems are compared and contrasted. Implications for robust and stochastic model predictive control are discussed.

An important reason for the success of model predictive control is the fact that, for deterministic systems, feedback and open-loop control are equivalent so that ‘optimal’ (but implicit) feedback may be obtained by solving an open-loop optimal control problem at each state encountered. This equivalence is not maintained in the presence of uncertainty complicating the development of robust model predictive control. The advantages and limitations of tube-based model predictive control for dealing with uncertainty of various forms are discussed, firstly in the context of constrained linear systems, and an extension to deal with robustness against unstructured uncertainty is briefly described. Tube-based control requires the determination both of a nominal or reference trajectory and an ancillary controller that constrains deviations of the state of the uncertain systems from the nominal trajectory and is difficult to extend to nonlinear systems. A novel ancillary controller for tube-based control of constrained nonlinear systems with additive disturbances that overcomes some disadvantages of a previous version and is relatively simple to implement is described and assessed; the addition of a terminal equality constraint to the nominal optimal control problem and its removal from the ancillary optimal control problem simplifies the analysis, removes some restrictive assumptions, and enables stronger results to be obtained.

The problem of output feedback model predictive control of discrete time systems in the presence of additive but bounded state and output disturbances is considered. The overall controller consists of two components, a stable state estimator and a tube based, robustly stabilizing model predictive controller. Earlier results are extended by allowing the estimator to be time varying. The proposed robust output feedback controller requires the online solution of a standard quadratic program. The closed loop system renders a specified invariant set robustly exponentially stable.

This paper introduces the concept of optimized robust control invariance for discrete-time linear time-invariant systems subject to additive and bounded state disturbances. A novel characterization of two families of robust control invariant sets is given. The existence of a constraint admissible member of these families can be checked by solving a single and tractable convex programming problem in the generic linear-convex case and a standard linear/quadratic program when the constraints are polyhedral or polytopic. The solution of the same optimization problem yields the corresponding feedback control law that is, in general, set-valued. A procedure for selection of a point-valued, nonlinear control law is provided.

A successful method for model predictive control of constrained linear systems uses a local linear control law that, in the presence of disturbances, maintains the disturbed trajectory of the controlled system in a tube whose centre is the nominal trajectory (generated ignoring the disturbance) and whose ‘cross-section’ is a robust, positively invariant set; robust exponential stability of an invariant set centred on the origin may be established. The purpose of this paper is to show how this successful procedure may be extended to provide robust model predictive control of constrained nonlinear systems. An ancillary problem is proposed, the solution of which provides a local nonlinear control law. The disturbed trajectories lie in a tube and this provides the means for constructing a tube-based robust nonlinear model predictive controller.

This paper provides a solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of bounded state and output disturbances. The proposed output feedback controller consists of a simple, stable Luenberger state estimator and a recently developed, robustly stabilizing, tube-based, model predictive controller. The state estimation error is bounded by an invariant set. The tube-based controller ensures that all possible realizations of the state trajectory lie in a simple uncertainty tube the 'center' of which is the solution of a nominal (disturbance-free) system and the 'cross-section' of which is also invariant. Satisfaction of the state and input constraints for the original system is guaranteed by employing tighter constraint sets for the nominal system. The complexity of the resultant controller is similar to that required for nominal model predictive control.

This note provides results on approximating the minimal robust positively invariant (mRPI) set (also known as the 0-reachable set) of an asymptotically stable discrete-time linear time-invariant system. It is assumed that the disturbance is bounded, persistent and acts additively on the state and that the constraints on the disturbance are polyhedral. Results are given that allow for the computation of a robust positively invariant, outer approximation of the mRPI set. Conditions are also given that allow one to a priori specify the accuracy of this approximation.

This paper provides a novel solution to the problem of robust model predictive control of constrained, linear, discrete-time systems in the presence of bounded disturbances. The optimal control problem that is solved online includes, uniquely, the initial state of the model employed in the problem as a decision variable. The associated value function is zero in a disturbance invariant set that serves as the 'origin' when bounded disturbances are present, and permits a strong stability result, namely robust exponential stability of the disturbance invariant set for the controlled system with bounded disturbances, to be obtained. The resultant online algorithm is a quadratic program of similar complexity to that required in conventional model predictive control.

In this paper we introduce the concept of optimized robust control invariance for a discrete-time, linear, time-invariant system subject to additive state disturbances. A novel characterization of a family of the robust control invariant sets is given. The existence of a constraint admissible member of this family can be checked by solving a single linear programming problem. The solution of the same linear programming problem yields the corresponding feedback controller.

A variety of control problems require the control action and/or state to be positive. Typical applications include situations where the operating point maximizes (steady state) efficiency so that the steady state control and/or the steady state itself lie on the boundaries of their respective constraint sets. Any deviation of the control and/or state from its steady state value must therefore be directed to the interior of its constraint set. To address these problems, we characterize a novel family of the robust control invariant sets for linear systems under positivity constraints. The existence of a constraint admissible member of this family can be checked by solving a single linear or quadratic programming problem. The solution of this optimization problem yields the corresponding controller. These results are then used to devise a robust time-optimal control scheme for regulation of uncertain linear systems under positivity constraints. Robust finite-time attractivity of an appropriately chosen member of this family is also established.

Min-max feedback formulations of model predictive control are discussed, both in the fixed and variable horizon contexts. The control schemes the authors discuss introduce, in the control optimization, the notion that feedback is present in the receding-horizon implementation of the control. This leads to improved performance, compared to standard model predictive control, and resolves the feasibility difficulties that arise with the min-max techniques that are documented in the literature. The stabilizing properties of the methods are discussed as well as some practical implementation details.

A version of dynamic programming, which computes level sets of the value function rather than the value function set itself, is used to design robust non-linear controllers for linear, discrete-time, dynamical systems subject to hard constraints on controls and states. The controller stabilizes the system and steers all trajectories emanating in a prescribed set to a control invariant set in minimum time. For the robust regulator problem, the control invariant terminal set is a neighborhood, preferably small, of the origin; for the robust tracking problem, the control invariant terminal set is a neighborhood of the invariant set in which the tracking error is zero. Two non-linear controllers which utilize the level sets of the value function, are described. The first requires the controller to solve, on-line, a modest linear program whose dimension is approximately the same as that of the control variable. The second decomposes each level set into a set of simplices; a piecewise linear control law, affine in each simplex, is then constructed.

This paper proposes a simple model predictive control scheme for linear systems, tracking a random reference and analysis its performance. In such situations it is usual to assume that the reference eventually converges to a constant in which case convergence to zero of the tracking error can be established. In this note we characterize the set to which the tracking error converges and the associated region of convergence when the reference does not converge to a constant.

This note shows that the problems of optimal control of linear, continuous-time, deterministic systems with piecewise constant control and optimal filtering of linear, continuous-time stochastic systems using sampled filtered observations are dual to each other.

An optimization-based procedure is used to design nonlinear controllers for linear dynamical systems subject to hard constraints on controls and states. The controller stabilizes the system and regulates to the origin or tracks the output of an exosystem which generates a reference signal. Earlier results are briefly reviewed and extended to achieve two objectives: firstly, the online computation of the nonlinear control is appreciably simplified, thus increasing the applicability of the approach; secondly, the approach is extended to computer control where the control is necessarily discrete-time (sampled-data). The nonlinear feedback control law is constructed using the solutions to a finite set of open-loop optimal control problems. This procedure, which relies on linearity of the system being controlled and convexity of the constraint sets, permits incremental improvement of the control law in those regions of state space where simulation reveals improvement is desirable.

This paper addresses the problem of obtaining nonlinear state-feedback controllers for controlling constrained dynamic systems, a problem which is, perhaps, the most important outstanding problem in control design. A novel off-line, optimization-based design procedure is employed; the control law is constructed from the solution to a finite number of optimal open-loop control problems. The design procedure permits relatively simple modification and improvement to the control law to be made in those regions of the state-space in which simulation reveals deficiencies. This potential for incremental improvement constitutes an important advantage over alternative methods. The results are relevant when high performance is required of systems subject lo constraints on controls and states.

The importance of the obstacle avoidance problem is stressed in [4]. Computation of reachability sets for the obstacle avoidance problem is addressed, for continuous-time systems in [4, 5] and for discrete-time systems in [12]; further results appear in, for instance [2, 17, 18]. The obstacle avoidance problem is inherently non-convex. Most existing results are developed for the deterministic case when external disturbances are not present. The main purpose of this paper is to demonstrate that the obstacle avoidance problem in the discrete time setup has considerable structure even when disturbances are present. We extend the robust model predictive schemes using tubes (sequences of sets of states) [9, 11, 14] to address the robust obstacle avoidance problem and provide a mixed integer programming algorithm for robust control of constrained linear systems that are required to avoid specified obstacles. The resultant robust optimal control problem that is solved on-line has marginally increased complexity compared with that required for model predictive control for obstacle avoidance in the deterministic case.

This paper introduces set robust control invariance, a concept that generalizes robust control invariance for systems described by difference equations to systems described by difference inclusions of special structure. The concept is useful for the analysis and synthesis of uncertain systems where a given control policy results in, for each initial state, a tube of trajectories rather than a sing trajectory; it also reveals the properties required for the terminal constraint set in receding horizon control of constrained linear systems with bounded disturbances and shows how improved terminal sets may be constructed. The family of set robust control invariant sets is characterized and the most important members of this family, the minimal and the maximal, are identified.

This paper provides a new and efficient method for the computation of an arbitrarily close outer robust positively invariant (RPI) approximation to the minimal robust positively invariant (mRPI) set for linear difference inclusions. It is assumed that the linear difference inclusion is absolutely asymptotically stable (AAS) in the absence of an additive state disturbance, which is the case for parametrically uncertain or switching linear discrete-time systems controlled by a stabilizing linear state feedback controller.

This paper provides a novel solution to the problem of robust output feedback model predictive control of constrained, linear, discrete-time systems in the presence of additive but bounded state and output disturbances. The overall controller consists of two components, a stable state estimator and a tube based, robustly stabilizing model predictive controller. The proposed robust output-feedback controller requires the on-line solution of a quadratic program, which is only marginally more complex then a standard model predictive control formulation. The closed loop system renders a specified invariant set robustly exponentially stable.

This paper provides a solution to the problem of computing a robustly positively invariant outer approximation of the minimal robustly positively invariant set for a discrete-time, linear, time-invariant system. It is assumed that the disturbance is additive and persistent, but bounded.

This paper presents new results that permit the computation of the set of states that can be robustly steered, using state feedback, to a given target set in a finite number of steps. It is assumed that the system is discrete-time, nonlinear, time-invariant and subject to mixed constraints on the state and input. A persistent disturbance, dependent on the current state and input, acts on the system. The results in this paper generalize previously published results that are not able to address state-input dependent disturbances. The application of the results to the computation of the maximal robustly controlled invariant set is briefly discussed. It is shown how polyhedral algebra, linear programming and computational geometry may be employed for set computations relevant to the analysis of linear and piecewise affine systems with additive state disturbances. Some simple examples are given to demonstrate that convexity of the robustly controllable sets cannot be guaranteed even if all relevant sets are convex and the system is linear.

The receding horizon control strategy provides a relatively simple method for determining feedback control for linear or nonlinear systems. The method is especially useful for the control of slow nonlinear systems, such as chemical batch processes, where it is possible to solve, sequentially, open-loop fixed-horizon, optimal control problems online. The method has been shown to yield a stable closed-loop system when applied to time-invariant or time-varying linear systems. It is shown that the method also yields a stable closed-loop system when applied to nonlinear systems. >

The problem of tracking an exogenous signal in the presence of bounded disturbances is considered. A version of dynamic programming, which computes level sets of the value function rather than the value function itself, is used to design robust nonlinear controllers for linear, discrete-time, dynamical systems with hard constraints on controls and states, an external bounded disturbance, and an exogenous input to be tracked. The controller stabilizes the system and regulates to a control invariant set in minimum time, and thereafter maintains the state in this set despite the disturbance; the tracking error is bounded in this set and is small if the disturbance is small. Two nonlinear controllers which utilize the level sets of the value function, are described. The first requires the controller to solve, online, a modest linear program whose dimension is approximately the same as that of the control variable. The second decomposes each level set into a set of simplices; a piecewise linear control law, affine in each simplex, is then constructed.

An optimization-based procedure is used to design nonlinear controllers for linear dynamical systems subject to hard constraints on controls and states. The controller stabilizes the system and regulates to the origin or tracks the output of an exosystem which generates a reference signal. Firstly, a linear controller which maintains zero tracking error is determined. Secondly, a finite set of initial states is selected. For each initial state a control, which steers the combined state to the zero error manifold in a finite time is determined using numerical optimization. A nonlinear feedback control law is then constructed using the solutions to this finite set of open-loop optimal control problems. The solution method permits incremental improvement to the control law in those regions of state space where simulation reveals improvement is desirable. >

The hysteresis switching algorithm of R.H. Middleton et al. (ibid., vol.33, no.1, p.50-8, Jan. 1988) is reexamined in a broader context. To demonstrate its utility, the algorithm is applied to various families of identifier-based parameterized controllers of both the direct and indirect control types. Application to the direct control type results in a model reference adaptive controller capable of stabilizing, without excitation, any SISO process which can be modeled by a minimum phase linear system whose transfer function has relative and McMillan degrees not exceeding prescribing integers m and n, respectively. It is shown that such processes can also be adaptively stabilized with indirect adaptive controllers and hysteresis switching. A simple numerical example involving a non-minimum-phase process model is used to illustrate how hysteresis switching might be applied to implicitly tuned parameterized controllers to realize an adaptive controller with capabilities which might prove very difficult, if not impossible, to achieve without hysteresis switching or some other form of discontinuous control. >

Saturating actuators are present in all real control systems. With saturation, the system becomes nonlinear and a nonlinear controller design is called for. This paper presents a hybrid scheme composed of several linear controllers and a switching strategy using the concept of "allowed over-saturation".

It is known that stability of a model predictive control system is ensured if the terminal conditions of the optimal control problem solved online satisfy certain criteria. The usual requirement is that the terminal cost function is a control Lyapunov function defined on the terminal constraint set. Conventionally the terminal cost function is chosen, when the system being controlled is linear, to be the value function for the infinite horizon unconstrained optimal control problem and the terminal constraint set is chosen to be the output admissible set for the closed-loop system using the optimal unconstrained controller u=-Kx. The purpose of the paper is to relax these terminal conditions thereby facilitating online solution of the optimal control problem. Using some earlier results, we present alternative conditions that employ, as the terminal cost, the infinite horizon cost resulting from a nonlinear controller u=-sat(Kx) and, as the terminal constraint set, the set in which this controller is optimal for the infinite horizon constrained optimal control problem. It is shown that this solution provides a considerably larger terminal constraint set.

Conventional model predictive control implements a version of receding horizon control where uncertainty (in the form of model error, estimation error, or disturbances) is absent. Robustness against uncertainty usually requires minimisation over control policies rather than control sequences; this may be prohibitively difficult in the presence of hard constraints on controls and states. It is shown how robustness may be achieved by a relatively simple modification to any conventional model predictive controller that is guaranteed to be stabilising in the absence of uncertainty.

This paper presents an algorithm for adaptive control of linear systems having unknown time delay. A rational approximation is used for the delay and it is argued that within this framework, the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics. A global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms. It is argued in the paper that a rational approximation gives a highly accurate and effective way of dealing with variable time delays. The analysis is supported by simulation results.

This paper presents an algorithm for adaptive control of linear systems having unknown time delay. A rational approximation is used for the delay and it is argued that within this framework, the analysis of the properties of the corresponding adaptive control algorithm essentially reduces to the issue of robustness to unmodelled dynamics. A global convergence theory is presented using recent methods for analyzing robustness properties of adaptive control algorithms. The analysis is supported by simulation results.

This paper presents a survey of the basic aspects involved in the design of linear multivariable control systems via semi-infinite optimization. Specific topics treated are a) database and simulation requirements, b) techniques for the transcription of design specifications into semi-infinite inequalities, and c) semi-infinite optimization algorithms for control system design.

A method for obtaining optimal estimates of the states of a linear system given a record of observations is derived. The method uses Kalman's filtering theory to obtain an estimate using 'past' observations, and optimal control theory to obtain an estimate using 'future' observations, and shows how the two estimates can be combined to give a solution to the smoothing or interpolation problem. The method also yields the variance of the estimate. The method is illustrated by means of a numerical example.

A linear system with gnusian inputs can be specified by its state transition equations. Assuming that the coefficients of the state transition matrix are initially unknown, a procedure, which gives a continuously up-dated estimate of these coefficients, is outlined. The procedure is extended to give an optimal estimate when the coefficients are random processes. Full knowledge of the state of the system as time proceeds is required. The puper shows how the result can be used to obtain an optimal estimate of the coefficients of the state transition and control matrices of a linear control system with gaussian inputs, provided that full knowledge of the value of the control signal is also available.

A method of determining the control action of a piecewise linear system to minimise a quadratic cost function is presented. The piecewise linear system considered is described by a different set of linear differential equations for each of two regions of the state space ; the two regions are separated by a hyperplane. It is shown that, with certain restrictions, conventional linear control theory can be used, provided the time taken by an optimal trajectory to reach the separating hyperplane can be determined. A mothod for determining this time is given, thus completing the solution.

A second-order method of successively improving a control sequence for a non-linear discrete-time system is derived. One step convergence is obtained for linear systems with quadratic performance functions. Although the results are of interest in their own right, a second-order method for continuous-time systems is obtained by formally allowing the sampling interval to approach zero. The equations so obtained differ slightly, because of the method of derivation, from results already obtained using the calculus of variations approach. The difference, which is an advantage of the method described in this paper, is that one vector differential equation less has to be integrated. The approach used in the derivation is motivated by dynamic programming and facilitates the application of gradient methods to stochastic problems which will be the subject of a future paper.

Most minimal realization procedures utilize the Hankel matrix or properties of the controllability matrix. Recently a useful algorithm for obtaining minimal realizations has been developed by Rosenbrock using the system theory developed by him. This note shows that the algorithm can be simply developed using the properties of similarity transformation and matrix reduction procedures.

Powerful algorithms exist for identifying linear systems given input/output data. In many cases, however, it is convenient to obtain the data in the form of correlation functions. The data in this form are often more condensed. An extension of Astrom's identification algorithm for making it able to use correlation functions in the place of an input/output record is described.

Luenberger's procedure for the design of controllers of linear systems with inaccessible state vectors is extended to deal with time-delay systems.