System transformation of unstable systems induced by a shift-invariant subspace

Given an inner function, the orthogonal complement of the corresponding shift invariant subspace induces a system transformation for linear time-invariant systems, which is a generalization of the lifting technique for the sample-data control and Hambo-transform in the sense the inner function is arbitrary. This paper extends the transformation for systems with unstable eigenvalues, and derives a unified formula for transformation operators for both stable and antistable systems. A potential application is in the area of closed-loop system identification, where an unstable system is identified under the stabilizing feedback connection. The application to closed-loop system identification will be presented elsewhere.