 # Determination of the Optimum Conditions of Response of Systems

Babister, Arthur William (1955) Determination of the Optimum Conditions of Response of Systems. PhD thesis, University of Glasgow.

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## Abstract

In this thesis we consider the transient response of systems satisfying linear differential equations with constant coefficients. Simple mathematical criteria for optimising the response are given in terms of [mathematical equations] where e is the error at time t. Expressions are obtained for both L and L1 in terms of (i) the roots of the characteristic equation, (ii) the coefficients of the characteristic equation, and (iii) the frequency response spectrum of the system. It is shown how the response due to (i) a step function disturbance, (ii) an initial impulse, (iii) a constant velocity input, and (iv) a, constant acceleration input can be simply related to the response in the free motion. The response following an arbitrary disturbance is also considered. The response of a linear system having one degree of freedom is considered for (1i) azero-displacemnent-error system, (ii) a zero-veloclty-error system and (ii) a zero-acceleration-error system. By considering the response to a step-function disturbance it is found that systems making 1 a minimum have a lightly damped oscillatory response. The smaller L1 is,the "smoother" is the response. Values are obtained for the coefficients of the characteristic equation of any order making L a minimum. An approximate method is given for correcting these coefficients to enable the response to be improved to give equal damping in the least damped modes of oscillation. For the zero-velocity-error and zero-acceleration -error systems the method in extended to allow for the requirement of a zero-displacement-error in the final a toady state. The method is extended to linear systems with any member of degrees of freedom.The response of a linear first order system with two degrees of freedom is considered in detail, two overall response functions R and R1 being defined in a similar manner to L a and L1. It is shown that, in the optimum system, there is no coupling; the damping in each mode is the same. A first order system with integral control is also considered; in this case the binomial response is the optimum.

Item Type: Thesis (PhD) Doctoral Mathematics 1955 Enlighten Team glathesis:1955-79159 Copyright of this thesis is held by the author. 05 Mar 2020 11:36 05 Mar 2020 11:36 http://theses.gla.ac.uk/id/eprint/79159 View Item