Memetic Covariance Matrix Adaptation Evolution Strategy for Bilinear Matrix Inequality Problems in Control System Design

TL;DR

This work tackles the nonconvex BMI-constrained control design problem, aiming to minimize the $H_ fty$ norm of the closed-loop transfer while satisfying bilinear matrix inequalities. It introduces a memetic CMA-ES that couples global search with a deterministic $(1+1)$-CMA-ES local refinement and a penalty mechanism to bound controller gains, enabling effective exploration without predefined bounds. Empirical results on 47 BMI benchmarks from COMPleib show superior performance in $H_ fty$ optimization and spectral abscissa reduction, with the memetic approach achieving 85.11% success in $H_ fty$ problems and 73.33% in spectral abscissa tasks, outperforming established solvers. The method demonstrates strong robustness and practical relevance for control problems requiring reliable nonconvex optimization, albeit with higher offline cost that does not affect real-time control deployment. Potential extensions include experimental validation and handling of sensor degradation, model mismatch, and actuation delays to further validate applicability in safety-critical applications.

Abstract

Bilinear Matrix Inequalities (BMIs) are fundamental to control system design but are notoriously difficult to solve due to their nonconvexity. This study addresses BMI-based control optimization problems by adapting and integrating advanced evolutionary strategies. Specifically, a memetic Covariance Matrix Adaptation Evolution Strategy (memetic CMA-ES) is proposed, which incorporates a local refinement phase via a (1+1)-CMA-ES within the global search process. While these algorithmic components are established in evolutionary computing, their tailored integration and specific tuning for control design tasks represent a novel application in this context. Experimental evaluations on $H_{\infty}$ controller synthesis and spectral abscissa optimization demonstrate that the proposed method achieves superior performance compared to existing BMI solvers in terms of both solution quality and robustness. This work bridges the gap between evolutionary computation and control theory, providing a practical and effective approach to tackling challenging BMI-constrained problems.

Figures (5)

• Figure 1: Illustration of the feasible region and objective value distribution for the BMI benchmark AC4, where $k_1$ and $k_2$ denote the two entries of the controller gain. The red region indicates the set of points for which the objective value in (\ref{['BMI problems2_ori']}) is less than 1, while the overall feasible region comprises both the red and gray areas.
• Figure 2: Visualization of population dynamics in CMA-ES and memetic CMA-ES. The black crosses are the mean in the algorithms, the black circles are the one standard deviation of the covariance matrix, the green points are the offsprings in current generation, the blue points are the better solution than original offsprings (green points) from memetic CMA-ES, the red cross-star is the global optimal point, and the contour value is from the right side color bar. There is a number besides the blue points. The smaller the number, the greater the objective value. The blue line is the baseline of mean, and the required mean for the next generation will be on the blue line, influenced by weighted contributions. (a) The standard CMA-ES generates the children with the mean and covariance matrix. (b) The next generation of the mean and covariance matrix in standard CMA-ES. (c) The memetic CMA-ES generates the children with the mean and covariance matrix. (d) The next generation of mean and covariance matrix in memetic CMA-ES. This shows how the refinement of (1+1)-CMA-ES allows the memetic CMA-ES to lead the global search in a more appropriate direction.
• Figure 3: Boxplots of $H_{\infty}$ performance results of HIFOO and memetic CMA-ES on the AC9 model over 32 independent runs (each with 850 iterations). memetic CMA-ES achieves a lower median with smaller variance.
• Figure 4: Boxplots of computation time for HIFOO and memetic CMA-ES on the AC9 model over 32 independent runs. HIFOO achieves a lower median execution time with smaller variance, while memetic CMA-ES shows longer runtime and greater variability. It is worth noting that this computational cost is incurred entirely during the offline controller design phase and does not affect the real-time performance of the closed-loop system. Once a feasible controller gain is obtained, it can be directly implemented without further computation, making the offline cost a minor concern in practical applications.
• Figure 5: Closed-loop step responses of the AC9 system to an exogenous step disturbance, showing that memetic CMA-ES achieves superior disturbance rejection and faster convergence compared to HIFOO.