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.

The system concept was the very first one introduced in Chapter 1. We argued that using the word “design” instead of “system” allows us to make use of the extensive work done on studying systems. In this chapter, we take up the system discussion again and assert that every design is a system.What is the exact “system” is a matter of subjective perspective and focus of the design effort. Following the discussion in Section 1.2, our working concept of a system is that the system comprises elements, probably interacting with each other, that function together. A system can be partitioned into its elements (system partitioning), and its overall function is achieved through properly tracking how the elements function together (system coordination). The elements can be physical parts (object partitioning) or function disciplines (aspect partitioning).

System optimization acknowledges this partitioning and coordination character and follows the basic principle of decomposition: break the problem into simpler problem pieces (subproblems), solve each subproblem separately, and coordinate the subproblem solutions so that you obtain the solution to the original problem. This decomposition-based optimization strategy is useful if: (i) the effort for partitioning, subproblem solution, and coordination is less than the effort required to solve the original problem; or (ii) the original problem is simply not solvable without decomposition; and (iii) the decomposition-based solution is indeed the same as the solution to the original nonpartitioned problem. The first two cases are a practical matter for when to use decomposition strategies. The third one is both practical and theoretical; it requires a mathematical convergence proof that the decomposition strategy will yield solutions within the solution set of the original problem. This mathematical requirement turns out to be quite demanding, as many strategies developed for solving practical system design problems tend to be heuristic with no convergence proofs. Still, getting a good answer for a complex problem is better than no answer.

Designing is a complex human process that has resisted comprehensive description and understanding. All artifacts surrounding us are the results of designing. Creating these artifacts involves making a great many decisions, which suggests that designing can be viewed as a decision-making process. In the decision-making paradigm of the design process we examine the intended artifact in order to identify possible alternatives and select the most suitable one. An abstract description of the artifact using mathematical expressions of relevant natural laws, experience, collected data, and geometry is the mathematical model of the artifact. This mathematical model may contain many alternative designs, and so criteria for comparing these alternatives must be introduced in the model. Within the limitations of such a model, the best, or optimum, design can be identified with the aid of mathematical methods.

In this first chapter we define the design optimization problem and describe most of the properties and issues that occupy the rest of the book.We outline the limitations of our approach and caution that an “optimum” design should be perceived as such only within the scope of the mathematical model describing it and the inevitable subjective judgment of the modeler.

Mathematical Modeling

Although this book is concerned with design, almost all the concepts and results described can be generalized by replacing theword design by theword system. We will then start by discussing mathematical models for general systems.

The System Concept

A system may be defined as a collection of entities that perform a specified set of tasks. For example, an automobile is a system that transports passengers. It follows that a system performs a function, or process, which results in an output. It is implicit that a system operates under causality; that is, the specified set of tasks is performed because of some stimulation, or input.

It is almost three decades since this book was first written. Much has changed since then. Perhaps the change most relevant to our readers is the central role that design has taken in society's interests, and in education and research, but also in how it impacts our lives. Design of products and systems is recognized as an important element of a vibrant economy and an innovative society. More importantly, there is increased awareness that the many big problemswe face today, such as environmental sustainability, can be addressed through thoughtful design and up-front assessment of the trade-offs involved, rather than as remedial efforts made after the fact.

Understanding and quantifying such trade-offs to support our collective decision making means that design optimization is now more important than ever. Optimal design is the goal not only of engineering, but also of every other social effort to shape our world. Many of our problems usually grow from our inability to agree on what is “optimal.”

The book was born out of our own desire to address explicitly what we mean by “optimal” and to put the concept of optimal design on a firm, rigorous foundation. There is an intimate relationship between the mathematical model that describes a design and the solution methods that optimize it. A basic premise from the start was that a good model can make optimization almost trivial, whereas a bad one can make correct optimization difficult or impossible. Software tools today provide capabilities for intricate analysis of many difficult performance aspects of a system. These analysis models, often referred to also as simulations, can be coupled with numerical optimization software to generate better designs iteratively. This virtual prototyping ability has grown dramatically and is an important contributor to reducing product development time and increasing robustness of systems.

In designing, as in other endeavors, one learns by doing. In this sense, the present chapter, although at the end of the book, is the beginning of the action. The principles and techniques of the previous chapters will be summarized and organized into a problem-solving strategy that can provide guidance in practical design applications. Students in a design optimization course should fix these ideas by applying them to a term project. For the practicing designer, actual problems at the workplace can serve as first trials for this new knowledge, particularly if sufficient experience exists for verifying the first results.

The chapter begins with a review in Section 9.1 of some modeling implications derived from the discussion in previous chapters about how numerical algorithms work. Although the subject is quite extensive, our goal here is to highlight again the intimacy between modeling and computation that was explored first in Chapters 1 and 2. The reader should be convinced by now of the validity of this approach and experience a sense of closure on the subject.

Sections 9.2 and 9.3 deal with two extremely important practical issues: the computation of derivatives and model scaling. Local computation requires knowledge of derivatives. The accuracy by which derivatives are computed can have a profound influence on the performance of the algorithm. A closed-form computation would be best, and this has become dramatically easier with the advent of symbolic computation programs. When this is not possible, numerical approximation of the derivatives can be achieved using finite differences. The newer methods of automatic differentiation are a promising way to compute derivatives of functions defined implicitly by computer programs.

Scaling the model's functions and variables is probably the single most effective “trick” that one can do to tame an initially misbehaving problem. Although there are strong mathematical reasons why scaling works (and sometimes even why it should not work), effective scaling in practice requires a judicious trial and error approach.