]> 5.4.2 Example 2: RC–transmission line

5.4.2 Example 2: RC–transmission line

Consider the problem of parametric optimization of the system consisting of a proportional controller (operational amplifier) and a distributed parameter plant (the RC–transmission line) discussed in Chapter 2. It relies on finding minimum of the function

J(K) = fL2(0,)2

in the interval ( cosh π, 1) being the stability region in the space of the parameter K and the function f is given by its Laplace transform

f̂(s) = E(cosh s 1) s(1 K)(cosh s K) (5.12)

In Section 2.7, to solve the problem we employed spectral properties of the operator describing the closed–loop system in the state space L2(0, 1). It was shown that the formula expressing the performance index as a function of K has the form (2.53). Recall that the formula (2.53) admits a simple, highly stable computer implementation which is due to the fast convergence of the series in (2.53).

The efforts to implement a reasonable approximation of Pnf L2(0,)2 with the aid of both (5.4) and (5.10) failed down on Matlab 6 out of ill–conditioning. Recall that the standard Matlab calculations are carried on 32 positions. Therefore to control the accuracy of floating–point calculations the formula (5.10) was implemented on Maple 6 rather than on Matlab 6. This was achieved by Digits:=90; command. The comparison of the exact values of the performance index (2.53) with their approximations are depicted in Figure 5.3.


Figure 5.3: Plots of the performance index for the system examined in Example 2. The upper curve is exact, the lower ones are the approximations for n = 20, n = 30, n = 40 and n = 50