LMI conditions for the existence of polynomially parameter-dependent Lyapunov functions assuring robust stability

The robust stability of uncertain systems in polytopic domains is investigated by means of homogeneous polynomially parameter-dependent Lyapunov (HPPDL) functions which are quadratic with respect to the state variables. A systematic procedure to construct linear matrix inequality (LMI) conditions whose solutions assure the existence of HPPDL functions of increasing degree is given. For each degree, a sequence of relaxations based on real algebraic methods provides sufficient LMI conditions of increasing precision for the existence of an HPPDL function which tend asymptotically to the necessity. As a result, families of LMI conditions parametrized on the degree of the HPPDL functions and on the relaxation level provide efficient numerical tests of different complexities to assess the robust stability of both continuous and discrete-time uncertain systems.