Introduction and definition of controllability

The concepts of controllability and observability, which were first introduced by Kalman (1960), play an important role in modern system theory. We shall see later in this chapter how they are used to obtain minimal realizations for systems described by sets of differential equations, once any realization is obtained by the methods discussed in Chapter 2. These concepts also appear as necessary conditions for the existence of solutions to many optimal control problems and when studying stability properties of systems. We begin with the concept of controllability.

Definition 8.1A system is called (completely) controllable if for any initial time to, any initial state x(t0) can be transferred to any final state x* (i.e., x(t)=x*) using some input [to, t] over a finite time interval (i.e., t is finite).

Note: In the literature, “completely” is used to denote a system that is controllable for any initial time t0 (as in our definition) as opposed to a system controllable over a specified time interval. There are numerous refinements of this concept, as, for example:

Total controllability if the system is completely controllable over every (or almost every) finite interval.

Strong controllability if the system is controllable from each input terminal.

Output controllability if the system output (rather than state) can be set arbitrarily at some finite time t, by using an appropriate input.

This textbook is designed to help the student learn the basic techniques necessary to begin the practice of automatic control engineering. The subject matter of the book, classical control theory, is a study of the dynamics of feedback control systems. Particular emphasis is placed upon the way in which the physical properties of the individual interconnected devices in a system influence the dynamic performance of the entire control system. Although classical control theory is normally taught to engineering students during their third or fourth year of study, it may also be learned by interested persons having a comparable background in mathematics, physics, and engineering.

Classical control theory was developed from the feedback amplifier technology of the 1920s and 1930s. It was first applied to automatic control systems for machine tools, industrial processes, and military equipment of many types. Although these applications bore little outward resemblance to their electronic amplifier antecedents, they all relied on the same principle of feedback for their remarkable performance. Increased demand for control systems in the 1940s gave rise to the branch of engineering known as automatic control. Research groups were formed in industry to develop and apply classical control theory. Engineering departments in universities offered seminars and courses in automatic control, and textbooks on the subject began to appear, including my earlier book which was published in 1962 (Introduction to Automatic Control Systems, Wiley & Sons, New York).

Introduction

We now derive state-variable models for several types of actuators that are used in automatic control systems to drive massive loads that must be precisely positioned. These models are used in subsequent chapters, where they appear as vital components in automatic systems.

One of the most commonly used actuators is the DC motor, an ideal subject for our study because of its immediate practical application in many engineering design problems. It is the simplest device that requires the simultaneous application of Newton's laws of mechanics, Kirchhoff's laws of electric circuits (subjects fundamental to Chapters 2 and 3), and the first principles of electromechanical energy conversion manifested in Faraday's law and Ampere's rule (which are reviewed in this chapter).

We then continue our study with the analysis of another commonly used system, the electromagnet driven by an electronic amplifier. This combination of two basically nonlinear devices is used to drive loads which require substantial forces but which are intended to travel only small distances from an equilibrium position. Because only limited displacements are required, our linear approximations to the nonlinear force–displacement–current relationships of the amplifier–magnet combination yield a useful linear state-variable model for this system.

The chapter concludes with the analysis of an electrohydraulic actuating system that combines the amplifier–magnet system with an hydraulic servovalve–ram system. This type of system is used in applications where high power (greater than several horsepower) is required in a small space and where the actuator is commanded by an electrical driving signal, in many cases delivered from a remote site.

Introduction

We first study a very simple class of mechanical systems – those consisting of a single rigid body or of two rigid bodies simply connected. The bodies will be restricted to planar motion in most cases, but they may rotate as well as translate in the plane. We recognize that most practical mechanical systems feature parts that move in three dimensions, but our restrictions are necessary here because the dynamics of massive bodies in three-dimensional motion is, in most cases, beyond the scope of this introductory book on automatic control. However, our use of simple systems has some advantages. We can use the freebody diagram to formulate directly from Newton's and Euler's laws the differential equations that describe the dynamics of the motion without resorting to the more abstract approach of using variational principles, which is usually necessary in the three-dimensional case. It is also possible in this simple setting to illustrate an important principle of mechanics that one must observe when expressing Euler's law for a body undergoing angular acceleration with respect to inertial space. Furthermore, since the motions of bodies in many practical applications are approximately planar, our simple approach in such cases will yield useful results.

The mathematical model we seek for each of the systems studied here is a set of differential equations that describe the physical system and its environment, plus certain auxiliary information that permits us to use these equations to determine the dynamic behavior of our system.

Introduction

In the type of dynamic system analysis that concerns us here, we normally identify the input quantity (or quantities) and the system state variables as our first step. We then write the differential equation (or equations) that describe the relationships between the input variables, the state variables, and their derivatives. The process of establishing these equations requires an understanding of the physical principles that govern the dynamics of our system. We have seen in the first four chapters that the principles of mechanics (the laws of Newton and Euler), those of electromechanics (the laws of Faraday, Ampere, Ohm, and others), and those of fluid mechanics, including aerodynamics, are all basic to the systems of interest here. The equations that result are generally nonlinear, ordinary, differential equations. In our work the differential equations are also restricted to those having constant physical parameters. In much of our work we also concentrate on the study of the dynamics of systems in a restricted regime of operation, usually for motions of the system near an equilibrium state (called a bias point in electronic circuits, or a trim condition in aircraft flight-control systems). With this further restriction on our analysis, the nonlinear differential equations may usually be approximated by linear differential equations having constant coefficients. We have seen several examples of this form of approximation in the first four chapters, and in the ensuing chapters our attention will be focused almost exclusively on linear systems of differential equations.

Background

Dynamics plays the central role in automatic control engineering. The analytical techniques and design principles examined in this book are simply methods of dealing with dynamics problems from the specialized point of view of the automatic feedback control system. This collection of methods and procedures – known as servomechanism theory, basic control theory, and, in recent years, as classical control theory – constitutes the basic subject to be mastered by a beginning control system engineer.

A typical automatic control system consists of several interconnected devices designed to perform a prescribed task. For example, the task may be to move a massive object such as the table of a machine tool in response to a command. The interconnected devices of the system are typically electromechanical actuators, sensors that measure the position and velocity of the controlled object and the currents or voltages at the actuator, and a control computer that processes the sensor signals along with the command. These interconnected dynamic elements work simultaneously, and they also embody a feedback connection. Frequently the engineer must determine the dynamic response of the entire system to a given command when only the physical properties of the individual component elements of the system are known. This formidable task requires quantitative dynamic analysis even in relatively simple systems.

Modern Control Theory and the Digital Computer

Classical control theory is directly applicable to systems which have only single input variables and single response variables.