linearizebmi & solvebmi

Overbounding Approximation Method

Many practical and important controller design problems, such as multi-objective controller design, structured controller design, and lower order controller design, are bilinear matrix inequality (BMI) problems and cannot be reduced to LMI problems. Accordingly, methods to solve BMI problems are still important, even though the BMI problems are known to be NP-hard. The overbounding approximation method is a one of the numerical methods to obtain approximate solutions to BMI problems. This method successively approximates BMI constraints to sufficient LMI constraints and solve the approximated LMI problem iteratively. This approach has following features.

This approach only requires a general SDP solver.
This approach can be applied to any types of BMI problems.
The performance index to be optimized is monotonically decreasing during the successive design procedure.

References

Noboru Sebe,
Sequential Convex Overbounding Approximation Method for Bilinear Matrix Inequality Problems,
9th IFAC Symposium on Robust Control Design (ROCOND'18), Florianopolis, Brazil, September 3-5, pp.175-182, 2018.
(IFAC-PapersOnLine, 51-25, pp.102-109, 2018)
doi:10.1016/j.ifacol.2018.11.089

Noboru Sebe,
A New Dilated LMI Characterization and Iterative Control System Synthesis,
11th IFAC/IFORS/IMACS/IFIP Symposium on Large Scale Systems: Theory and Applications (LSS 2007), Gdańsk, July 23-25, 6 pages, 2007.
(IFAC Proceedings Volumes, 40-9, pp.250-255, 2007)
doi:10.3182/20070723-3-PL-2917.00040

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