Chapter 1
Introduction

One of the main obstacles related to key assumptions in many appealing feedback control theories lies in the need to perfectly know the system to be controlled. Even though mathematical models can be derived precisely for many areas of physical systems, using well-established physical laws and principles, the problems remain of gathering the precise values of the relevant system parameters (or, obtaining the information stored in the inaccessible-for-measurement states of the system) and, very importantly, dealing with unknown (i.e., non-structured) perturbations affecting the system evolution through time. These issues have been a constant concern in the feedback control systems literature and a wealth of approaches have been developed over the years to separately, or simultaneously, face some, or all, of these challenging realities involved in physical systems operation. To name but a few, system identification, adaptive control, energy methods, neural networks, and fuzzy systems have all been developed and have tried out disciplines that propose related approaches, from different viewpoints, to deal with, or circumvent, the three fundamental obstacles to make a clean control design work: parameter identification, state estimation, and robustness with respect to external perturbations.

This book deals with a new approach to the three fundamental problems associated with the final implementation of a nicely justified feedback control law. We concentrate on the ways to handle these obstacles from an algebraic viewpoint, that is one resulting from an algebraic vision of systems dynamics and control. As for the preferred theory to deal with the ideal control problem, we emphasize the fact that the methods to be presented are equally applicable to any of the existing theories. We propose examples where sliding-mode control is used, others where passivity-based control methods are preferred, and yet others where flatness and generalized proportional integral controllers are implemented. The algebraic approach is equally suitable when dynamic observers are used. The book therefore does not concentrate on, or favor, any particular feedback control theory. We choose the controller as we please. Naturally, since the theoretical basis of the proposed algorithms and techniques stems from the differential algebraic approach to systems analysis and control, we often present the background material in detail at the end of chapters, so that the mathematically inclined reader has a source for the basics being illustrated in that chapter through numerous physically oriented examples.

1.1 Feedback Control of Dynamic Systems

The control systems presented here are designed using an algebraic estimation methodology in combination with well-known control design techniques, which are applicable to linear and nonlinear systems. Since the algebraic estimation methodology is independent of the particular controller design method being used and, furthermore, it is quite easy to understand, it will be profitable for the reader to combine this tool with his preferred controllers or with conventional control techniques. This book introduces a wide variety of application examples and detailed explanations to illustrate the use of the algebraic methodology in identification, state estimation, and disturbance estimation. However, this work is not an introductory control textbook. A basic control course is a prerequisite for a deeper understanding of these examples.

1.1.1 Feedback

A control system is one whose objective is to positively influence the performance of a given system in accordance with some specific objectives. A control law or controller is a set of rules that allows us to determine the commands to be sent to the governed plant (via an actuator) to achieve the desired evolution. These rules can be described as either open-loop control or closed-loop (feedback) control. Figure 1.1 shows both control strategies, where and are, respectively, the output and input of the plant, and represents a finite collection of internal variables (called state variables) that completely characterize, from the present, the system behavior in the future provided the future control inputs are available. The first scheme, an open-loop control system, does not measure (or feed back) the output to determine the control action; its accuracy depends on its calibration, and thus it is only used when there are no perturbations and the plant is perfectly known. A perturbation (or disturbance) is an unknown signal that tends to adversely affect the output value of the plant. This undesired signal may be internal (endogenous) or external (exogenous) to the system. Figure 1.1(b) shows a closed-loop control system, where a sensor is used to obtain an error signal. This signal is processed by the controller to determine the action necessary to reduce the difference between the output and its desired value.

Figure 1.1 Typical control scheme

Feedback is a mechanism to command a system to evolve in a desired fashion so that the states, and outputs, exhibit a desired evolution (e.g., to track a reference trajectory) or stay at a prescribed equilibrium. Feedback enables the current state or the current outputs (feedback signal) to be measured, determining how far the behavior is from the desired state or desired outputs (i.e., to assess the error signal and then automatically generate a suitable control signal to bring the system closer and closer to the desired state). Feedback can be used to stabilize the state of a system, while also improving its performance.

Feedback control has ancient origins, as noted by Mayr (1970). Throughout history, many examples of ingenious devices, based on feedback, can be found. In ancient Greece, China, during the Middle Ages, and in the Renaissance, many examples have been recovered and explained in modern terms. These artifacts were improved and specialized to become pressure regulators, float valves, and temperature regulators. One of these devices was the fly-ball or centrifugal governor, used to control the speed of windmills; it was later adapted by James Watt, in 1788, to control steam engines.

It was only in the 1930s that a theory of feedback control was fully developed by Black and Nyquist at Bell Laboratories. They studied feedback as a means to vastly improve the amplifier performance in telephone lines. They had to face the (well-known) closed-loop instability problem when the feedback gain was set too high, transforming the amplifier into an oscillator.

1.1.2 Why Do We Need Feedback?

Fundamentally, feedback is necessary for the following reasons.

• Counteracting disturbance signals affecting the plant: A controller must reject the effects of unknown undesirable inputs and maintain the output of the plant within desired values.
• Improving system performance in the presence of model uncertainty: Uncertainty arises from two sources – unknown or unpredictable inputs (disturbance, noise, etc.) and uncertain or unmodeled dynamics.
• Stabilizing an unstable plant.

This book addresses only additive perturbations, which will be divided into structured and unstructured perturbations. Structured perturbations are generated by the initial conditions of the system and unknown exogenous inputs that can be modeled as families of time polynomials. The unstructured perturbations are considered as highly fluctuating, or oscillating, phenomena affecting the behavior of the system. A common unstructured perturbation is the case of a zero-mean noisy signal. Notwithstanding this, it is still possible to build an algebraic estimator that takes into account these disturbances and mitigates their effects.

The main objectives of feedback control are to ensure that the variables of interest in the system either track prescribed reference trajectories (tracking problem) or are maintained close to their constant set-points (regulation problem).

1.2 The Parameter Identification Problem

A model is a mathematical representation of the essential characteristics of an existing system. When a system model can be defined by a finite number of variables and parameters, it is called a parametric model. Examples of parametric models are the transfer function of an electrical circuit, the equations of motion of a mechanical suspension system, and so on. The whole family of functions and equations that integrate a model is called the model structure. In general, this structure can be linear or nonlinear. The models used in modern control theory are, with a few exceptions, parametric models in terms of linear and nonlinear state equations. To implement a model-based controller, it is necessary to know precisely the structure of the model of the system and its associated parameters. Therefore, if parameters are initially unknown, the process of parameter identification is quite important for the design of the control system.

1.2.1 Identifying a System

According to Eykhoff (1974), model building begins with the application of basic physical laws (Newton's laws, Maxwell's laws, Kirchhoff's laws, etc.). From these laws, a number of relations are established between variables describing the system (e.g., ordinary differential or difference equations or, sometimes, partial differential equations). If all external and internal conditions of the system are known, and if our physical...