Design by specifying the closed-loop transfer function

The desired transfer function of the closed loop is given by

where and are polynomials in . In the following design methods, the distribution of poles and zeros of will be chosen, such that the performance indices for the step response are fulfilled. Often a detailed investigation of the distribution of poles and zeros of the desired transfer function is not necessary, especially when the transfer function does not contain zeros and when - because of the requirement - just yields and in the simplest case

For a closed loop with the transfer function according to Eq. (10.2) different possibilities exist, the so called standard forms, which can be used by a table lookup for the step response , distribution of poles of and the coefficients of the denominator polynomial .

A first possibility is a distribution of poles with a real multiple pole at . Here and in the following sections, the term is a relative frequency, not the natural frequency. Thus one obtains for the step response of the desired behaviour

This is a series connection of elements with the same time constant . This representation is also called a binomial form. The standard polynomials of different order are given in Table A.2. As this table further shows, the normalised step response will become slower with increasing order . A design using this binomial form is only considered when the step response is required to have no overshoot.

A further possibility of a standard form for of Eq. (10.2) is the Butterworth form. In this form, all poles of are equally distributed on a semicircle with radius in the left-half  plane and centred at the origin. Table A.2 contains the standard polynomials and the associated normalised step responses .

Numerous further possibilities for the development of standard forms of Eq. (10.2) can be derived from the integral criteria given in Table 7.2. For example, the minimum performance index is the basis of a standard form that is also shown in Table A.2. Further, often the minimum settling time is used as the criterion. This table contains for the corresponding standard form.

Furthermore, for pole assignment the Weber method can be used. This specifies the desired closed-loop transfer function

by a real pole with multiplicity and a pair of complex poles. Table A.1 contains for different values of and the normalised step responses . By a proper choice of , and a closed-loop transfer function can be found that fulfils in many instances the desired performance.