M. Sami Fadali

Introduction to Digital Control

There is a need to control the evolution with time of one or more of the system variables in many engineering systems. Controllers are required to ensure satisfactory transient and steady-state behavior for these engineering systems. To guarantee satisfactory performance in the presence of disturbances and model uncertainty, most controllers in use today employ some form of negative feedback. A sensor is needed to measure the controlled variable and compare its behavior to a reference signal. Control action is based on an error signal defined as the difference between the reference and the actual values. This chapter explains the reasons for the popularity of digital control systems. Following this, it illustrates a block diagram for digital control of a given analog control system. Furthermore, it explains the structure and components of a typical digital control system. Digital control offers distinct advantages over analog control that explain its popularity. The advantages are—accuracy, implementation errors, flexibility, speed, and cost. Finally, the chapter discusses examples of control systems where digital implementation is now the norm.

State Feedback Control

State variable feedback allows the flexible selection of linear system dynamics. Often, not all state variables are available for feedback, and the remainder of the state vector must be estimated. This chapter includes an analysis of state feedback and its limitations. It also includes the design of state estimators for use when some state variables are not available and the use of state estimates in feedback control.

Chapter 10 – Optimal Control

Publisher Summary Many problems in engineering can be solved by minimizing a measure of cost or maximizing a measure of performance. The designer must select a suitable performance measure based on his or her understanding of the problem to include the most important performance criteria and reflect their relative importance. The designer must also select a mathematical form of the function that makes solving the optimization problem tractable. This chapter introduces optimal control theory for discrete-time systems. It begins with unconstrained optimization of a cost function and then generalizes to optimization with equality constraints. It also considers the problem of minimizing a cost function or performance measure; then extends the solution to problems with equality constraints. Following this, it covers the optimization or optimal control of discrete-time systems. It then specializes to the linear quadratic regulator and obtains the optimality conditions for a finite and for an infinite planning horizon. In addition, this chapter addresses the regulator problem where the system is required to track a nonzero constant signal.

Digital Control System Design

To design a digital control system, a z -domain transfer function or difference equation model of the controller that meets given design specifications, is seeked. The controller model can be obtained from the model of an analog controller that meets the same design specifications. Alternatively, the digital controller can be designed in the z -domain using procedures that are almost identical to s -domain analog controller design. This chapter elucidates both these approaches. It begins with an explanation of the z -domain root locus and describes the method of sketching the z -domain root locus for a digital control system or obtaining it using MATLAB. This approach is based on the relation between any time function and its s -domain poles and zeros. If the time function is sampled and the resulting sequence is z- transformed, the z -transform contains information about the transformed time sequence and the original time function. The poles of the z -domain function can therefore be used to characterize the sequence, and possibly the sampled continuous time function, without inverse z -transformation. However, this latter characterization is generally more difficult than characterization based on the s -domain functions.

Modeling of Digital Control Systems

This chapter explores the modeling concept of digital control systems. The configuration includes a digital-to-analog converter (DAC), an analog subsystem, and an analog-to-digital converter (ADC). The DAC converts numbers calculated by a microprocessor or computer into analog electrical signals that can be amplified and used to control an analog plant. The analog subsystem includes the plant as well as the amplifiers and actuators necessary to drive it. The output of the plant is periodically measured and converted to a number that can be fed back to the computer using an ADC. It begins by developing models for the ADC and DAC, then for the combination of DAC, analog subsystem, and ADC. Following this, it develops models for the various components of this digital control configuration. Many other configurations that include the same components can be similarly analyzed. It then describes the procedure of obtaining the transfer function of an analog system with analog-to-digital and digital-to-analog converters including systems with a time delay. Finally, it describes the steady-state tracking error for a closed-loop control system and the steady-state error caused by a disturbance input for a closed-loop control system.

Properties of State–Space Models

This chapter examines the properties of state–space models. It examines controllability, which determines the effectiveness of state feedback control; observability, which determines the possibility of state estimation from the output measurements; and stability. These three properties are independent, so that a system can be unstable but controllable, uncontrollable but stable, and so on. However, systems whose uncontrollable dynamics are stable are stabilizable, and systems whose unobservable dynamics are stable are called detectable. Stabilizability and detectability are more likely to be encountered in practice than controllability and observability. This chapter begins by discussing the asymptotic stability of state–space realizations. The natural response of a linear system because of its initial conditions may converge to the origin asymptotically, remain in a bounded region in the vicinity of the origin, or grow unbounded. Following this, the chapter discusses the BIBO stability of their input-output responses. Finally, it demonstrates how state–space representations of a system in several canonical forms can be obtained from its input-output representation.

Stability of Digital Control Systems

This chapter examines different definitions and tests of the stability of linear time-invariant (LTI) digital systems based on transfer function models. Stability is a basic requirement for digital and analog control systems. Digital control is based on samples and is updated every sampling period, and there is a possibility that the system will become unstable between updates. This chapter also considers input-output stability and internal stability. Thereafter it provides several tests for stability—the Routh-Hurwitz criterion, the Jury criterion, and the Nyquist criterion and defines the gain margin and phase margin for digital systems. The most commonly used definitions of stability are based on the magnitude of the system response in the steady state. If the steady-state response is unbounded, the system is said to be unstable. Finally, the chapter discusses two stability definitions that concern the boundedness or exponential decay of the system output.

Analog Control System Design

This chapter deals with the analog control system design. It begins with a review of a classical control design and then explains the design of analog controllers in the s-domain. Analog controllers can be implemented using analog components or approximated with digital controllers using standard analog-to-digital transformations. In addition, direct digital control system design in the z-domain is very similar to the s-domain design of analog systems. Following this, the chapter obtains root locus plots for analog systems. The root locus method provides a quick means of predicting the closed-loop behavior of a system based on its open-loop poles and zeros. It characterizes a systems step response based on its root locus plot and discusses design proportional (P), proportional-derivative (PD), proportional-integral (PI), and proportional-integral-derivative (PID) controllers in the s-domain. The objective of control system design is to construct a system that has a desirable response to standard inputs. A desirable transient response is one that is sufficiently fast without excessive oscillations. A desirable steady-state response is one that follows the desired output with sufficient accuracy. Finally, the chapter describes the process of empirical tuning PID controllers using the Ziegler-Nichols approach.

Chapter 9 – State Feedback Control

Publisher Summary This chapter presents an analysis of state feedback and its limitations. It also includes the design of state estimators for use when some state variables are not available and the use of state estimates in feedback control. State feedback involves the use of the state vector to compute the control action for specified system dynamics. State variable feedback allows the flexible selection of linear system dynamics. This chapter shows a linear system (A, B, C) with constant state feedback gain matrix K. Using the rules for matrix multiplication, it deduces that the matrix K is m X n so that for a single-input system K is a row vector. The dynamics of the closed-loop system depend on the eigenstructure(eigenvalues and eigenvectors) of the matrix Acl. Thus, the desired system dynamics can be chosen with appropriate choice of the gain matrix K. Using output or state feedback, the poles or eigenvalues of the system can be assigned subject to system-dependent limitations. This is known as pole placement, pole assignment, or poleallocation. This is elucidated in the chapter.

Chapter 12 – Practical Issues

Publisher Summary The designer of a digital control system must be mindful of the fact that the control algorithm is implemented as a software program that forms part of the control loop. Successful practical implementation of digital controllers requires careful attention to several hardware and software requirements. During the design phase, designers make several simplifying assumptions that affect the implemented controller. They usually assume uniform sampling with negligible delay due to the computation of the control variable. Thus, they assume no delay between the sampling instant and the instant at which the computed control value is applied to the actuator. This chapter discusses the most important of these requirements and their influence on controller design. It then analyzes the choice of the sampling frequency in more detail in the presence of antialiasing filters and the effects of quantization, rounding, and truncation errors. In particular, it examines the effective implementation of a proportional–integral–derivative (PID) controller. Finally, it examines changing the sampling rate during control operation as well as output sampling at a slower rate than that of the controller.