Tính bền vững của các hệ điều khiển nhiều đầu vào nhiều đầu ra

Abstract

The time domain description of a multi-input multi-output  plant to be controlled has the form

Ex(t) = Fx(t) + Gu(t), x|_t=0 =x₀ (1)

y(t) = Hx(t),                                               (2)

where x Î Rⁿ is state vector, u Î R¹ is the vector, y Î R^m is the output vector, and E Î R ^{n x n}, FÎR ^nxn, GÎR^nxl, HÎR^mxm denote the system matrices. A control law

u(t) = K₁ y(t) + K₂ y’(t) + r(t)                                 (3)

will be chosen to stabilize the plant (1) and (2), where K_i, i=1,2 are constant matrices and r(t) is a external input (reference input). If E=I, there is no great difference between the single-input single-output system and the multi-input multi-output one in the state space description. That will be the reason why we restrict our attention to the system model of the form (1) and (2). Furthermore, we will study the cases where the plant models contain some uncertainties. An interesting approach dealing with modeluncertainties is the robust control concept. It is well known that there are several algorithms os robust control to be found at international publications. The main difference of them is how to characterize the uncertainties being taken into consideration. In this paper it may be seen that the uncertainties only appear in the matrix E. If E is nonsingular, the uncertainties in E imply the uncertainties in F and G. The singularity of E carries the uncertainties of E to the transfer function matrix of the plant, Since the continuous variation of E cause a continuous variation of closed-loop system parameters, we can utilize all the well known mathematical results on the topology for  studying the robustness of control systems.