Introduction

The description of the dynamics of large systems, such as power systems, by their transfer functions is unsatisfactory for a number of reasons. For example, for a system of order n, say 100, the characteristic polynomial has degree 100 and 101 coefficients of s. Moreover, such systems typically have more than one output variable and more than one input signal. modelling based on the multi-input multi-output state equations of the system is simpler and problems of loss of accuracy are reduced. Moreover, such modelling has a number of advantages and features some of which are described in the following sections. To illustrate the formation of the state equations of a plant or an electro-mechanical system, let us consider two examples.

Much of the material on linear systems analysis in the later sections is covered by [1].

Example 3.1.

Find a set of state and output equations for the simple RLC circuit shown in Figure 3.1. The voltage supplied by an ideal source is vs(t), and the required outputs are the capacitor voltage are vC(t) and iL(t) inductor current.

Loop voltages:

where p is the differential operator d/dt.

Current flow:

Note that each of the right-hand equations is a first-order differential equation with the derivative specifically sited on the left-hand side of the equation.

There are two independent energy storage elements, C and L. Because the instantaneous energy stored in C and L is and, respectively, the variables x1 = vC and x2 = iL are ‘natural’ selections for states. Hence, the state equations are formed as follows:

The two output equations required are y1= vC = x1 and y2= iL = x2.

The state and output equations can thus be written in matrix form as follows:

Where

Example 3.2

A drive system shown in Figure 3.2 consists of a DC motor driving an inertial load through a speed-reducing gearbox. The controlled DC supply voltage to the armature is supplied by a power amplifier. The motor field current is maintained constant (i.e. the flux/pole is constant). Write down the equations of motion for this system.