Disturbance rejection and robustness for LTV Systems

In this paper, we consider the optimal disturbance rejection problem for (possibly infinite dimensional) linear time-varying (LTV) systems using a framework based on operator algebras of classes of bounded linear operators. In particular, after reducing the problem to a shortest distance minimization in a space of bounded linear operators, duality theory is applied to show existence of optimal solutions, which satisfy a "time-varying" allpass or flatness condition. With the use of the theory of M-ideals of operators, it is shown that the computation of time-varying (TV) controllers reduces to a search over compact TV Youla parameters. This involves the norm of a TV compact Hankel operator and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is also derived. The results are generalized to the mixed sensitivity problem for TV systems as well, where it is shown that the optimum is equal to the operator induced of a TV mixed Hankel-Toeplitz operator generalizing analogous results known to hold in the LTI case. Finally, a numerical algorithm to compute optimal TV controllers is proposed