Preservation of Common Quadratic Lyapunov Functions and Pade Approximations

It is well known that the bilinear transform, or first order diagonal Pade approximation to the matrix exponential, preserves quadratic Lyapunov functions between continuous-time and corresponding discrete-time linear time invariant (LTI) systems, regardless of the sampling time. It is also well known that this mapping preserves common quadratic Lyapunov functions between continuous-time and discrete-time switched systems. In this note we show that while diagonal Pade approximations do not in general preserve other types of Lyapunov functions (or even stability), it is true that diagonal Pade approximations of the matrix exponential, of any order and sampling time, preserve quadratic stability. A consequence of this result is that the quadratic stability of switched systems is robust with respect to certain discretization methods.