On Padé Approximations and the Preservation of Quadratic Stability for Switched Linear Systems

It is well known that the bilinear transform, or first order diagonal Padé 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. The analagous result also holds for switched linear systems. In this note we show that, for any sampling time, diagonal Padé approximations of any order preserve quadratic Lyapunov functions for LTI systems and switched linear systems. Also, for a certain type of switching system where a quadratic Lyapunov function does not necessarily exist, yet the system is stable, we show that certain even ordered diagonal Padé transformations also preserve stability regardless of the sampling time. Other Padé approximations also preserve stability provided the sampling time is less than a readily computable upper bound.