Sparse feedback stabilization in high-order linear systems

We consider the column sparsity of the feedback stabilization gain matrix in high-order linear systems. By means of a special matrix norm and the state transition matrix quadratic cost function (SQF) of the systems, the sparse feedback stabilization controller design problem is formulated as a regularized SQF optimization problem. We further derive the proximal mapping of the special matrix norm, and then based on the gradient descent of the SQF part of the objective function, the proximal gradient method is introduced to develop an algorithm for solving the non-smooth optimization problem. Numerical examples are given to illustrate the effectiveness of the proposed method.