Abstract
This paper presents a method of reduction for linear structured uncertain system using the integral squared error criterion. The four fixed Kharitonov's polynomials associated with the numerators n
(s), n
(s) and denominators d
(s), d
(s) of the original uncertain system and uncertain reduced model are obtained. By taking all combinations of the n
(s), n
(s) and d
(s),d
(s) for (k,h = 1,2,3,4), respectively, we obtain sixteen fixed Kharitonov's systems and sixteen fixed Kharitonov's reduced models. Stability equation method and integral squared error criterion are used together to find the uncertain reduced order model. The stability equation method is used to preserve the stability of the sixteen fixed Kharitonov's systems and original uncertain system by first determining the denominator coefficients of the sixteen fixed Kharitonov's reduced models and uncertain reduced model respectively. The numerators of the sixteen fixed Kharitonov's reduced models are determined so that the integral squared error between the unit step responses of the sixteen fixed Kharitonov's reduced models and the corresponding sixteen fixed Kharitonov's systems are minimum. The sixteen fixed Kharitonov's reduced models tend to approximate the transient portions of the corresponding sixteen fixed Kharitonov's systems in the sense of minimum squared error, while the steady portions of the sixteen fixed Kharitonov's reduced models are matched exactly with that of the corresponding sixteen fixed Kharitonov's systems. Instead of actually evaluating time responses of the sixteen fixed Kharitonov's systems and reduced models, a matrix formulas are used for calculating the integral squared error from the coefficients of the error transfer functions. Finally the lower and upper bounds c
,c
for (i=0,1,...r,-1) and d
,d
for (j=1,2...,r) of the uncertain reduced model are found from the coefficients of the sixteen fixed Kharitonov's reduced models. An illustrative example is included in order to demonstrate the main points.