Optimal transient response shaping of the servomechanism problem

The problem of constructing a controller which gives a desirable transient response, e.g., a time response which is smooth, ripple-free, fast, and with negligible interaction effects, is considered in this paper for the servomechanism problem for continuous linear time invariant (LTI) systems. It is shown that this may be accomplished by constructing a controller which minimizes the following cheap control performance index: J = integral(0)(infinity) {(theta,(e) over dot + e)' (theta(e) over dot + e) + epsilonphi(u)'phi(u)} dtau, where e is the error in the system and phi(u) is a polynomial function of u and its derivatives; here, epsilon > 0 and theta is a diagonal matrix with non-negative elements. A number of examples are included to illustrate the type of results which may be obtained.