In previous chapters we have shown how to compute the solution to constrained finite time optimal control problems. Two approaches can be used. The first approach consists of solving a mathematical program for a given initial state. The second approach employs multiparametric programming to compute the solution as an explicit piecewise affine function of the initial state. This implies that for constrained linear systems, a Receding Horizon Control (RHC) policy requires either the on-line solution of a quadratic or linear program, or the evaluation of a piecewise affine on polyhedra function.

In this chapter we focus on efficient on-line methods for the computation of RHC control laws. The main drawback of explicit optimal control laws is that the number of polyhedral regions can grow exponentially with the number of constraints in the optimal control problem. Section 14.1 presents efficient on-line methods for the evaluation of explicit piecewise affine control laws.

If the on-line solution of a quadratic or linear program is preferred, Sections 14.2 and 14.3 briefly discuss how to improve the efficiency of a mathematical programming solver by exploiting the structure of the RHC control problem.

Storage and On-Line Evaluation of the PWA Control Law

In Chapter 11 we have shown how to compute the solution to the constrained finite time optimal control (CFTOC) problem as an explicit piecewise affine function of the initial state. Such a function is computed off-line by using a multiparametric program solver, which divides the state space into polyhedral regions, and for each region determines the linear gain and offset which produces the optimal control action.

This method reveals its effectiveness when applied to Receding Horizon Control (RHC). Having a precomputed solution as an explicit piecewise affine on polyhedra (PPWA) function of the state vector reduces the on-line computation of the RHC control law to a function evaluation, therefore avoiding the on-line solution of a quadratic or linear program.

The main drawback of such an explicit optimal control law is that the number of polyhedral regions can grow dramatically with the number of constraints in the optimal control problem. In this chapter we focus on efficient on-line methods for the evaluation of such a piecewise affine control law.

The operations research community has addressed parameter variations in mathematical programs at two levels: sensitivity analysis, which characterizes the change of the solution with respect to small perturbations of the parameters, and parametric programming, where the characterization of the solution for a full range of parameter values is studied. In this chapter we introduce the concept of multiparametric programming and recall the main results of nonlinear multiparametric programming. The main goal is to make the reader aware of the complexities of general multiparametric nonlinear programming. Later in this book we will use multiparametric programming to characterize and compute the state feedback solution of optimal control problems. There we will only make use of Corollary 5.1 for multiparametric linear programs and Corollary 5.2 for multiparametric quadratic programs, which show that these specific programs are “well behaved.”

Introduction to Multiparametric Programs

Consider the mathematical program

where z is the optimization vector and x is a vector of parameters.We are interested in studying the behavior of the value function J*(x) and the optimizer z*(x) as we vary the parameter x. Mathematical programs where x is a scalar are referred to as parametric programs, while programs where x is a vector are referred to as multiparametric programs.

There are several reasons to look for efficient solvers of multiparametric programs. Typically, mathematical programs are affected by uncertainties due to factors that are either unknown or that will be decided later. Parametric programming systematically subdivides the space of parameters into characteristic regions, which depict the feasibility and corresponding performance as a function of the uncertain parameters, and hence provide the decision maker with a complete map of various outcomes.

Our interest in multiparametric programming arises from the field of system theory and optimal control. For example, for discrete-time dynamical systems, finite time constrained optimal control problems can be formulated as mathematical programs where the cost function and the constraints are functions of the initial state of the dynamical system. In particular, Zadeh and Whalen [293] appear to have been the first ones to express the optimal control problem for constrained discrete-time linear systems as a linear program. We can interpret the initial state as a parameter. By using multiparametric programming we can characterize and compute the solution of the optimal control problem explicitly as a function of the initial state.

In this chapter we review the basics of Receding Horizon Control (RHC). In the first part we discuss the stability and the feasibility of RHC and we provide guidelines for choosing the terminal weight so that closed-loop stability is achieved.

The second part of the chapter focuses on the RHC implementation. Since RHC requires at each sampling time to solve an open-loop constrained finite time optimal control problem as a function of the current state, the results of the previous chapters imply two possible approaches for RHC implementation.

In the first approach a mathematical program is solved at each time step for the current initial state. In the second approach the explicit piecewise affine feedback policy (that provides the optimal control for all states) is precomputed off-line. This reduces the on-line computation of the RHC law to a function evaluation, thus avoiding the on-line solution of a quadratic or linear program. This technique is attractive for a wide range of practical problems where the computational complexity of on-line optimization is prohibitive. It also provides insight into the structure underlying optimization-based controllers, describing the behavior of the RHC controller in different regions of the state space. Moreover, for applications where safety is crucial, the correctness of a piecewise affine control law is easier to verify than that of a mathematical program solver.

RHC Idea

In the previous chapter we discussed the solution of constrained finite time and infinite time optimal control problems for linear systems. An infinite horizon suboptimal controller can be designed by repeatedly solving finite time optimal control problems in a receding horizon fashion as described next. At each sampling time, starting at the current state, an open-loop optimal control problem is solved over a finite horizon (top diagram in Figure 12.1). The computed optimal manipulated input signal is applied to the process only during the following sampling interval [t, t+1]. At the next time step t+1 a new optimal control problem based on new measurements of the state is solved over a shifted horizon (bottom diagram in Figure 12.1). The resulting controller is referred to as a Receding Horizon Controller (RHC). A receding horizon controller where the finite time optimal control law is computed by solving an optimization problem on-line is usually referred to as Model Predictive Control (MPC).

This book is designed to be used in a quarter- or semester-long senior-level undergraduate linear control systems class. Readers are assumed to have had some exposure to differential equations and complex numbers (good references are [BD12] and [BC14]), and to have some familiarity with the engineering notion of signals and systems (a standard reference is [Lat04]). It is also assumed that the reader has access to a high-level software program, such as MATLAB, to perform calculations in many of the homework problems. In order to keep the focus on the content, examples in the book do not discuss MATLAB syntax or features. Instead, we provide supplementary MATLAB files which can produce all calculations and figures appearing in the book. These files can be downloaded from http://www.cambridge.org/deOliveira.

Chapters 1 and 2 provide a quick overview of the basic concepts in control, such as feedback, tracking, dynamics, disturbance rejection, integral action, etc. Math is kept at a very basic level and the topics are introduced with the help of familiar examples, such as a simplistic model of a car and a toilet bowl.

Chapter 3 formalizes the concept of a transfer-function for dynamic linear system models. Its first part is a review of the Laplace transform and its application to linear ordinary differential equations. The second part introduces systems concepts such as stability, transient and steady-state response, and the frequency response method. Some topics, e.g. complex integration, the calculus of residues, and norms of signals and systems, are covered in more depth than is usually found in typical introductory courses, and can be safely skipped at first read.

Equipped with the concept of a transfer-function, Chapter 4 formalizes fundamental concepts in feedback analysis, such as tracking, sensitivity, asymptotic and internal stability, disturbance rejection, measurement noise, etc. Homework problems in this chapter expose readers to these concepts and anticipate the more sophisticated analytic methods to be introduced in the following chapters.

Chapter 5 takes a slight detour from classic methods to introduce the reader to statespace models. The focus is on practical questions, such as realization of dynamic systems and controllers, linearization of nonlinear systems, and basic issues that arise when using linear controllers with nonlinear systems.

In controls we make use of the abstract concept of a system: we identify a phenomenon or a process, the system, and two classes of signals, which we label as inputs and outputs. A signal is something that can be measured or quantified. In this book we use real numbers to quantify signals. The classification of a particular signal as an input means that it can be identified as the cause of a particular system behavior, whereas an output signal is seen as the product or consequence of the behavior. Of course the classification of a phenomenon as a system and the labeling of input and output signals is an abstract construction. A mathematical description of a system and its signals is what constitutes a model. The entire abstract construction, and not only the equations that we will later associate with particular signals and systems, is the model.

We often represent the relationship between a system and its input and output signals in the form of a block-diagram, such as the ones in Fig. 1.1 through Fig. 1.3. The diagram in Fig. 1.1 indicates that a system, G, produces an output signal, y, in the presence of the input signal, u. Block-diagrams will be used to represent the interconnection of systems and even algorithms. For example, Fig. 1.2 depicts the components and signals in a familiar controlled system, a water heater; the block-diagram in Fig. 1.3 depicts an algorithm for converting temperature in degrees Fahrenheit to degrees Celsius, in which the output of the circle in Fig. 1.3 is the algebraic sum of the incoming signals with signs as indicated near the incoming arrows.

Models and Experiments

Systems, signals, and models are often associated with concrete or abstract experiments. A model reflects a particular setup in which the outputs appear correlated with a prescribed set of inputs. For example, we might attempt to model a car by performing the following experiment: on an unobstructed and level road, we depress the accelerator pedal and let the car travel in a straight line. We keep the pedal excursion constant and let the car reach constant velocity. We record the amount the pedal has been depressed and the car's terminal velocity.