LMIs in Control/pages/MatrixNormMinimization

LMI for Matrix Norm Minimization This problem is a slight generalization of the eigenvalue minimization problem for a matrix. Calculating norm of a matrix is necessary in designing an or an optimal controller for linear time-invariant systems. In those cases, we need to compute the norm of the matrix of the closed-loop system. Moreover, we desire to design the controller so as to minimize the closed-loop matrix norm. The System Assume that we have a matrix function of variables : where are symmetric matrices. The Data The symmetric matrices () are given. The Optimization Problem The optimization problem is to find the variables in order to minimize the following cost function: where is the cost function and indicates the norm of the matrix function . According to Lemma 1.1 in LMI in Control Systems Analysis, Design and Applications (page 10), the following statements are equivalent: The LMI: LMI for matrix norm minimization This optimization problem can be converted to an LMI problem. The mathematical description of the LMI formulation can be written as follows: Conclusion: As a result, the variables after solving this LMI problem and we obtain that is the norm of matrix function . Implementation A link to Matlab codes for this problem in the Github repository: https://github.com/asalimil/LMI-for-Matrix-Norm-Minimization Related LMIs LMI for Matrix Norm Minimization LMI for Generalized Eigenvalue Problem LMI for Maximum Singular Value of a Complex Matrix LMI for Matrix Positivity External Links A list of references documenting and validating the LMI. - LMI in Control Systems Analysis, Design and Applications Return to Main Page LMIs in Control/Tools