On the Use of Lypunov Equations in Continuization and Discretization of Linear Systems

In this paper Lyapunov equations are used to convert a discrete time system (DTS) into a continuous time system (CTS). It is shown that with the appropriate choice of the input matrix B and output matrix C, in balanced coordinates, and knowledge of the Hankel singular values of the DTS, one can solve the two CT Lyapunov equations for the CTS´s dynamic matrix A. Different choices of B and C result in different CT models that correspond to different discretization techniques. For example, if B and C were chosen as in Euler´s method, the resulting CT would be equivalent to the inverse of Euler´s method, while the choice of B and C as in the bilinear transform method would result in the CT model normally obtained using the direct bilinear transform method. The proposed technique preserves stability and both the Hankel and the H-infinity norms of the system. Unlike the bilinear transform method, the continuized system is strictly proper and thus guarantees a zero initial error in the system response. Moreover, the steady state error can be set to zero without requiring a direct feedthrough term as in the bilinear transform method.