Optimal compensation of linear time-invariant systems

Optimal control theory can be applied to linear, time-invariant control systems to determine the compensating feedback gains necessary to: 1. Cause zero system state error from a specified steady state value in response to a step input, 2. Minimize the transient deviation of the system state from the desired steady state value. The state equation of the system to be compensated is: ẋ = Ax + Bu The final state value, x(∞) is expressed as linear in r, the magnitude of the step input: z = x (∞) = Cr where C is an nxl matrix of constants. The analytical measure of performance chosen to minimize system response is: J (u) = ½ ʃ ∞o((x-z)T Q(x-z) + R(u- d)2)dt where d is the steady state value of u required to force x to z, and Q and R are weighting matrices. An nth order, linear, time-invariant system described by: ẋ = Ax + Bu is forced to the desired steady state value response to a step input while minimizing the measure of performance if it is compensated to achieve: ẋ = (A - BR-l BT K)x + (BR-l BT K - A)Cr where K satisfies the matrix equation: KBR-1 BT K – KA – ATK – Q = 0