Structured adaptive control for solving LMIs

Numerical problems such as finding eigenvalues, singular value decomposition, linear programming, are traditionally solved with algorithms that can be interpreted as discrete-time processes. One can also find in the literature continuous-time methods for these same problems where solutions are the equilibrium points to which converge stable differential equations. The paper exposes one such continuous-time method for solving linear matrix inequalities. The proposed differential equations are those of an adaptive control feedback loop on an LTI system. The adaptive law is passivity-based with additional structural constraints of two types. The first constraint imposes the gain to be block-diagonal at all times. It can be interpreted as a decentralized control structure. The second constraint is only required asymptotically. It for example reads as requiring the feedback gain to be symmetric when time goes to infinity. Point-wise global stability is proved with quadratic Lyapunov functions. Results are illustrated on LMIs related to an H∞ norm computation problem. Solutions to the LMIs are obtained by simulations in Simulink.