On the robust stability of 2D mixed continuous-discrete-time systems with uncertainty

This paper addresses the problem of establishing robust exponential stability of 2D mixed continuous-discrete-time systems affected by uncertainty. Specifically, it is supposed that the matrices of the system are polynomial functions of an uncertain vector constrained over a semialgebraic set. First, it is shown that robust exponential stability is equivalent to the existence of a complex Lyapunov functions depending polynomially on the uncertain vector and an additional parameter of degree not greater than a known quantity. Second, a condition for establishing robust exponential stability is proposed via convex optimization by exploiting sums-of-squares (SOS) matrix polynomials. This condition is sufficient for any chosen degree of the complex Lyapunov function candidate, and is also necessary for degrees sufficiently large.