Quality Diversity Optimization Method for Bilinear Matrix Inequality Problems in Control System Design

In this paper, a quality diversity optimization method (QDOM) based on an adaptive bound-searching algorithm and diversity-selecting immune algorithm is proposed for solving bilinear matrix inequality (BMI) problems in control system design. By using the proposed adaptive bound-searching algorithm, appropriate bound values can be obtained for the entries of controller gain matrices or the eigenvalues of closed-loop systems represented by a state space model. Given the bound values, the proposed diversity-selecting immune algorithm can produce the best-so-far controller gain for a given BMI problem. To find the global optimum efficiently and avoid being trapped in a local optimum, the concept of quality diversity is employed in the proposed method. The proposed method was validated through solving some spectral abscissa,
$H_{2}$
, and
$H_{\infty} $
optimization problems. The simulation results show that the proposed QDOM achieved better or similar performance in many benchmark problems as compared with existing BMI solution methods.