Analysis and Control of Linear Dynamic Systems

Abstract

In this chapter, we will develop the basic relationships to solve the motion control problem for dynamic systems using analog and digital proportional-integralderivative (PID) controllers as well as state-space control algorithms It should be emphasized that the PID control laws use the tracking errore(t)whereas the state-space controllers use the tracking errore(t)and the state variablesx (t).Proportional-integral-derivative and state-space control algorithms are widely used in dynamic systems to stabilize systems, attain tracking and disturbance attenuation, guarantee robustness and accuracy, and so forth. This chapter provides the introduction to nonlinear control and feedback tracking, and single-input/singleoutput as well as multi-input/multi-output systems are studied. We model the dynamic systems in thes-and z-domains using transfer functionsG _sys (s)and Gs_ys(z) to synthesize the PID-type controllers. The state-space models are used to design control algorithms applying the Hamilton—Jacobi and Lyapunov theories for multi-input/multi-output systems. The theoretical foundations in analysis and design of dynamic systems modeled using linear differential equations are needed to be covered to fully understand the basic concepts in control of nonlinear systems. It should be emphasized that analysis of linear continuous-and discrete-time closed-loop systems with control constraints will be accomplished. As mathematical models are found in the form of differential or difference equations, system parameters (coefficients of differential or difference equations) are defined, and the analysis can be performed to study stability and stability margins, time response, accuracy, and so on. The system characteristics and performance can be improved and “shaped” using PID-type and state-space stabilizing and tracking controllers studied in this chapter. The general problem approached is the design of PID-type and state-space controllers to ensure the specifications imposed on the desired performance of closed-loop dynamic systems. The synthesis of analog and digital control laws involve the design of controller structures as well as adjusting feedback coefficients to attain certain desired criteria and characteristics. These performance specifications relate stability, robustness, dynamics, accuracy, tracking, disturbance attenuation, as well as other criteria needed to be achieved through the use of control algorithms. The tradeoff between stability and accuracy, robustness and system response, and complexity and implementability is well known. The design procedures are reported to find the structures and feedback coefficients of control laws.