I have a question about LQR. I apply optimal controller by solving Ricatti equation based on normal plant. Suppose that I have one or few parameter variations in the plant that changes some values in matrix $A$, and let me name it $A_{\text{new}}$, then the cost will be different from normal value, could either increase or decrease the value of cost function $$ J=trace(x_0'\, P\, x_0) $$ where $P$ is solved from new Ricatti equation $$ (A_{\text{new}}-BK)'P+P(A_{\text{new}}-BK)+Q+K'RK $$
Now my question is, if now I face the new plant $A_{\text{new}}$ and I still keep the same optimal controller solved from normal plant, I want to know what level of variations, or what value of matrix $A_{\text{new}}$ will make the new cost function achieve some certain value I specified $J_0$.
Assume that I don't need to worry about stability, then $P$ will be symmetric. But the problem is, with given initial condition $x_0$, there are still infinite solutions about $P$ from equation $trace(x_0'Px_0)=J_0$. I know there are various numerical approach to solve this Lyapunov equation, but none of them specifies the target value of $P$.
To make it easier, you can assume only one or two elements in matrix $A$ change from normal.
Any help will be appreciated!
Clark
