Determination of quadratic performance indices

The integral performance indices and from Table 7.2 have the disadvantage that they have to be evaluated in the time domain, either by laborious computation or by simulation. All the other squared error type of indices are more pleasant because of calculating its value in the s-domain rather than in the time domain. Therefore the analysis is simpler. This will be shown in the following for .

The performance index is given by

Applying the convolution theorem in the frequency domain from Eq. (2.12) for and , one obtains Parseval's theorem

When is expressed as a ratio of polynomials

different methods are available to evaluate the integral. For calculation of the integral a recursion formula is available. Its solution has also been tabulated up to quite high values of in terms of the coefficients of the polynomials. Table 7.3 below gives a short list. For a detailed analysis, e.g. when the integral depends on some parameters, a general algebraic approach using determinants is more suitable as shown in section A.7.

Table 7.3: Values for the integral

The more general form

of a squared performance index can be easily evaluated. Since the Laplace transform of according to the complex differentiation theorem Eq. (2.8) is equal to , one obtains

Since it is easy to compute the derivatives of a polynomial the above integral can be computed using simple algebra and the aforementioned recursive formula.

Example 7.3.1   Determining the best damping ratio for a second-order system is a simple example of the use of performance indices. Let us assume that the command transfer function of a control system is described by Eq. (4.54) with as

The control error for a unit step input is

From Table 7.3 we have

As this function is a parabola in , with minimum given by

the minimum square error to a step input occurs for .

Demonstration Example 7.1   A virtual experiment using manual control