LMIs in Control/SolutionInTermsofRiccatiInequalities

Let us now come back to the H∞-problem by output feedback control. This amounts to finding the matrices $$(A_K, B_K, C_K, D_K)$$ such that the conditions K stabilizes P and achieves $$ \bar S(P, K)_\infty < 1$$. are satisfied. We proceed as in the state-feedback problem. We use the Bounded Real Lemma to rewrite the H∞ norm bound into the solvability of a Riccati inequality, and we try to eliminate the controller parameters to arrive at verifiable conditions.

The System
Matrices to controller that solve the the H∞-problem by output feedback control

$$(A_K, B_K, C_K, D_K)$$

The Data
Following matrices are needed as Inputs:.


 * $$ (A_K, B_K, C_K, D_K)$$.
 * $$ (A, B_1, B_2, C_1,C_2)$$.

The Optimization Problem
In control systems theory, many analysis and design problems are closely related to Riccati algebraic equations or inequalities. Find X and Y

The LMI:
There exist $$A_K, B_K, C_K, D_K$$ that solve the output-feedback H∞ problem (K that stabilizes P and achieves $$ \bar S(P, K)_\infty < 1$$) if and only if there exist X and Y that satisfy the two ARIs

$$A^TX+XA+XB_1B_1^TX+C_1^T-C_2^TC_2<0$$

$$AY+YA^T+YC_1^TC_1Y+B_1B_1^T-B_2B_2^T<0$$

and the coupling condition

\begin{align} {\displaystyle {\begin{aligned}{\begin{bmatrix}X&I\\\ I&Y\end{bmatrix}}<0\end{aligned}}}\\ \end{align}$$

Conclusion:
Let U and V be square and non-singular matrices with $$UV^T=I-XY$$, and set $$L=-X^{-1}C^T_2$$ ,$$F:=-B^T_2 Y^{-1}$$. Then $$A_K, B_K, C_K$$ as defined in

$$A_k=-U^{-1}[A^T+X(A+LC_2+B_2F)Y+X(B_1+LD_{21})B_1^T+C_1^T(C_1+D_{12}F)Y]V^{-T}$$

$$B_K=U^{-1}XL$$

$$C_K=FYV^{-T}$$

lead to a controller K that stabilizes P and achieves $$ \bar S(P, K)_\infty < 1$$