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 $\overline{S}(P,K{)}_{\mathrm{\infty }}<1$. 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 $\overline{S}(P,K{)}_{\mathrm{\infty }}<1$) if and only if there exist X and Y that satisfy the two ARIs

$A^{T}X+XA+X{B}_1{B}_1^{T}X+C_1^T-C_2^T{C}_2<0$

$AY+Y{A}^T+Y{C}_1^T{C}_{1}Y+B_1{B}_1^T-B_2{B}_2^T<0$

and the coupling condition

$\begin{array}{c}\begin{array}{c}\left[\begin{array}{cc}X& I\\ I& Y\end{array}\right]<0\end{array}\\ \end{array}$

Conclusion:

Let U and V be square and non-singular matrices with $U{V}^T=I-XY$, and set $L=-X^{-1}{C}_2^T$ ,$F:=-B_2^T{Y}^{-1}$. Then $A_K,B_K,C_K$ as defined in

$A_k=-U^{-1}[A^T+X(A+L{C}_2+B_{2}F)Y+X(B_1+L{D}_{21})B_1^T+C_1^T(C_1+D_{12}F)Y]V^{-T}$

$B_K=U^{-1}XL$

$C_K=FY{V}^{-T}$

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

External links

• http://control.asu.edu/MAE598_frame.htm LMI Methods in Optimal and Robust Control- A course on LMIs in Control by Matthew Peet.
• https://https://arxiv.org/abs/1903.08599/ LMI Properties and Applications in Systems, Stability, and Control Theory] - A List of LMIs by Ryan Caverly and James Forbes.
• https://web.stanford.edu/~boyd/lmibook/ LMIs in Systems and Control Theory- A downloadable book on LMIs by Stephen Boyd.