Calculation of the Integrated Squared Error for Systems with Time Lag

Abstract

Calculation of the integrated squared error (ISE) for linear systems with time lag is studied by using the Lyapunov functional method of Repin, and an error in his method is corrected. The method described here makes it possible to obtain the exact values of the ISE through the solutions of certain linear differential equations and algebraic equations.
The analytical forms of the ISE for the first-order system K_cKe^-Ls/1+sT and the second-order system K_c(1+1/T_I^s)Ke^-Ls/1+sT are obtained. In numerical examples, the relationship between the ISE and the system parameters is given, which is, for reference, compared with the parameter setting proposed by Ziegler and Nichols.
It is also shown that the instability condition for the first-order system, which can be obtained from this Lyapunov functional, is identical with the necessary and sufficient condition for the stability of the system.