STATE-SPACE ANALYSIS AND FEEDBACK THEORY

In this chapter, we formulate the network equations in the time-domain as a system of first-order differential equations that govern the dynamic behavior of a network. The advantages of representing the network equations in this form are numerous. First of all, such a system has been widely studied in mathematics and its solution, both analytical and numerical, is known and readily available. Secondly, the representation can easily and naturally be extended to time-varying and nonlinear networks. In fact, nearly all time-varying and nonlinear networks are characterized by this approach. Finally, the first-order differential equations are easily programmed for a digital computer or simulated on an analog computer.

We begin the chapter by first presenting procedures for the systematic formulation of network equations in the form of a system of first-order differential equations known as the state equations, and then discuss the number of dynamically independent state variables required in the formulation of these state equations. We shall demonstrate how this number or its upper bounds can be determined from network topology alone. Finally, we express the familiar feedback matrices in terms of the coefficient parameter matrices of the state equations, and discuss their physical significance. In this chapter, we consider only linear lumped networks that may be passive or active, reciprocal or nonreciprocal, and time-invariant or time-varying.