A limiting property of the matrix exponential with application to multi-loop control

A limiting property of the matrix exponential is proven: For a real square matrix, where the log norm of the upper-left n by n block approaches negative infinity in a limiting process, the matrix exponential goes to zero in the first n rows and n columns. This property is useful for simplification of dynamic systems that exhibit modes with sufficiently different time scales; for example, in multi-loop control systems with fast inner and slow outer feedback loops. For this case, we derive a time scale separation algorithm for a linear continuous-time model under the assumption of high-gain inner loop feedback, which yields a simplified discrete-time model at the slow time scale. The proposed technique is applied to the design of a two-loop control system for stabilizing an inverted pendulum. Experimental results are provided.