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.

In Chapter 1 we contemplated solutions to our first control problem, a much simplified cruise controller for a car, without taking into account possible effects of time. System and controller models were static relations between the signals: the output signal, y, the input signal, u, the reference input, y, and then the disturbance, w. Signals in block-diagrams flow instantaneously, and closed-loop solutions derived from such block-diagrams were deemed reasonable if they could be implemented fast enough. We drew encouraging conclusions from simple analysis but no rationale was given to support the conclusions if time were to be taken into consideration.

Of course, it is perfectly fine to construct a static mathematical model relating a car's pedal excursion with its terminal velocity, as long as we understand the model setup. Clearly a car does not reach its terminal velocity instantaneously! If we expect to implement the feedback cruise controller in a real car, we have to be prepared to say what happens between the time at which a terminal target velocity is set and the time at which the car reaches its terminal velocity. Controllers have to understand that it takes time for the car to reach its terminal velocity. That is, we will have to incorporate time not only into models and tools but also into controllers. For this reason we need to learn how to work with dynamic systems.

In the present book, mathematical models for dynamic systems take the form of ordinary differential equations where signals evolve continuously in time. Bear in mind that this is not a course on differential equations, and previous exposure to the mathematical theory of differential equations helps. Familiarity with material covered in standard text books, e.g. [BD12], is enough. We make extensive use of the Laplace transform and provide a somewhat self-contained review of relevant facts in Chapter 3.

These days, when virtually all control systems are implemented in some form of digital computer, one is compelled to justify why not to discuss control systems directly from the point of view of discrete-time signals and systems. One reason is that continuous-time signals and systems have a long tradition in mathematics and physics that has established a language that most scientists and engineers are accustomed to.

The book you have in your hands grew out of a set of lecture notes scribbled down for MAE 143B, the senior-level undergraduate Linear Control class offered by the Department of Mechanical and Aerospace Engineering at the University of California, San Diego.

The focus of the book is on classical methods for analysis and design of feedback systems that take advantage of the powerful and insightful representation of dynamic linear systems in the frequency domain. The required mathematics is introduced or revisited as needed. In this way the text is made mostly self-contained, with accessory work shifted occasionally to homework problems.

Key concepts such as tracking, disturbance rejection, stability, and robustness are introduced early on and revisited throughout the text as the mathematical tools become more sophisticated. Examples illustrate graphical design methods based on the rootlocus, Bode, and Nyquist diagrams. Whenever possible, without straying too much from the classical narrative, the reader is made aware of contemporary tools and techniques such as state-space methods, robust control, and nonlinear systems theory.

With so much to cover in the way of insightful engineering and relevant mathematics, I tried to steer clear of the curse of the engineering systems and control textbook: becoming a treatise with 1000 pages. The depth of the content exposed in fewer than 300 pages is the result of a compromise between my utopian goal of at most 100 pages on the one hand and the usefulness of the work as a reference and, I hope, inspirational textbook on the other. Let me know if you think I failed to deliver on this promise.

I shall be forever indebted to the many students, teaching assistants, and colleagues whose exposure to earlier versions of this work helped shape what I am finally not afraid of calling the first edition. Special thanks are due to Professor Reinaldo Palhares, who diligently read the original text and delighted me with an abundance of helpful comments.

I would like to thank Sara Torenson from the UCSD Bookstore, who patiently worked with me to make sure earlier versions were available as readers for UCSD students, and Steven Elliot from Cambridge University Press for his support in getting this work to a larger audience.