On the Complexity of SOS Programming and Applications in Control Systems

The minimization of a linear cost function subject to the condition that some matrix polynomials depending linearly on the decision variables are sums of squares of matrix polynomials (SOS) is known as SOS programming. This paper proposes an analysis of the complexity of SOS programming, in particular of the number of linear matrix inequality (LMI) scalar variables required for establishing whether a matrix polynomial is SOS. This number is analyzed for real and complex matrix polynomials, in the general case and in the case of some exact reductions achievable for some classes of matrix polynomials. An analytical formula is proposed in each case in order to provide this number as a function of the number of variables, degree and size of the matrix polynomials. Some tables reporting this number are also provided as reference for the reader. Two applications in control systems are presented in order to show the usefulness of the proposed results.