1 research outputs found
State feedback design considering overexcitation
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
nth 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)=Wp(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)= Wc(s)[ua(s)-y(s)]. The reduction of signal delay in the process is
implemented by the PID algorithm described by the transfer function
Wc(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=kcua-Fx, the time constants of the feedback system can be
freely specified by appropriate selection of F and kc. 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 kc=[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=kcuaFx*. 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