Parametrization of All Stabilizing LTI Controllers for Multirate SISO Systems

Abstract

In this work we provide a solution for the problem of parametrization of all stabilizing LTI controllers for multirate SISO systems. This problem has been previously addressed with the use of the lifting technique which can be utilized to convert a linear multirate system into a single-rate higher-order system. We propose a new procedure to solve the problem without lifting the multirate system. The new approach reduces the computational effort and restrict the solution to the set of all stabilizing LTI controllers. The non-commutative property of the down-sampling operation is the main obstacle to solve the problem. To overcome this difficulty we transform the problem by rewriting the controller transfer function using the modified Z-transform. This facilitates the separation of plant and controller dynamics in the closed-loop characteristic polynomial. Tools from the modified Z-transform method and control synthesis using the Diophantine equation are utilized to solve the problem for multirate systems. Moreover, by using the Diophantine equation the result is extended to the set of all stabilizing controllers for which the multirate system exhibits a dead-beat response.