State feedback design considering overexcitation

Abstract

The state equation describing the relationship between the input signal u(t), the state variable x(t) and the output signal y(t) of a linear, time invariant n^th order SISO process is:\textit dx/dt=Ax+Bu, y=Cx+Du. The transfer function between the output signal and the input signal of the process is: \textity(s)/u(s)=W_p(s) and the time constants characterizing the delays of signals due to energy storage elements result from the eigenvalues of the state matrix A. In the classical feedback control system, the controller computes the control signal according to the expression u(s)= W_c(s)[u_a(s)-y(s)]. The reduction of signal delay in the process is implemented by the PID algorithm described by the transfer function W_c(s) that accelerates the feedback system by \textitoverexciting the control signal to a specified extent. The reduction of signal delay in the process can also be implemented by negative feedback of the state variables x. If the process is state controllable and the control signal is computed according to the algorithm u=k_cu_a-Fx, the time constants of the feedback system can be freely specified by appropriate selection of F and k_c. The design of the feedback gain F can be performed using the \textitAckermann formula; the system is accelerated by means of \textitoverexcitation of the control signal to an appropriate extent even in this case. The paper presents the fact that the gain can be chosen according to k_c=[C(A-BF^-1B]^-1CA^-1B, and the overexcitation ratio of the control signal can be calculated using the relationship u(0)/u(∞ )=[1+F(A-BF)^-1B]^-1. This overexcitation ratio is in connection with the rate of pole transfers that can be expressed analytically. It occurs frequently that the state variables x of the process cannot play any part in the computation of the control signal since the state variables cannot be measured. In such cases, the state feedback can be implemented from the state variables x*(t) of a state observer according to the expression u=k_cu_aFx*. The paper presents the fact that the state feedback implemented based on the state observer - as opposed to the common concept - can also be interpreted as a state feedback of the process model, with the task of computing the control signa l that fulfils the requirements of acceleration. This signal is applied at the input of both the process model and the real process.