مرکز منطقه ای اطلاع رساني علوم و فناوري - Further study of the linear regulator with disturbances--The case of vector disturbances satisfying a linear differential equation

Abstract :

In a previous paper [1], the conventional optimal linear regulator theory was extended to accommodate the case of external input disturbances $\\omega (t)$ which are not directly measurable but which can be assumed to satisfy $d^{m+1}\\omega (t)/dt^{m+1} = 0$ , i.e., represented as $m$ th-degree polynomials in time $t$ with unknown coefficients. In this way, the optimal controller $u^{0}(t)$ was obtained as the sum of: 1) a linear combination of the state variables $x_{i}, i = 1,2,...,n$ , plus 2) a linear combination of the first $(m + 1)$ time integrals of certain other linear combinations of the state variables. In the present paper, the results obtained in [1] are generalized to accommodate the case of unmeasurable disturbances $\\omega (t)$ which are known only to satisfy a given $\\rho$ th-degree linear differential equation $D: d^{\\rho}\\omega (t)/dt^{\\rho} + \\beta _{\\rho}d^{\\rho-1}\\omega (t)/dt^{\\rho-1}+...+\\beta _{2}d\\omega /dt + \\beta _{1}\\omega =0$ where the coefficients $\\beta _{i}, i = 1,...,\\rho$ , are known. By this means, a dynamical feedback controller is derived which will consistently maintain state regulation $x(t) \\approx 0$ in the face of any and every external disturbance function $\\omega (t)$ which satisfies the given differential equation $D$ -even steady-state periodic or unstable functions $\\omega (t)$ . An essentially different method of deriving this result, based on stabilization theory, is also described, In each cases the results are extended to the case of vector control and vector disturbance.