Optimal 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. 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. Under mild assupmtions the optimal TV controller is shown to be essentially unique. With the use of Mideals 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 defined on the space of causal trace-class 2 operators and its maximal vectors. Moreover, an operator identity to compute the optimal TV Youla parameter is provided. These 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. The final outcome of the approach developed here is that it leads to two tractable finite dimensional convex optimizations producing estimates to the optimum within desired tolerances, and a method to compute optimal time-varying controllers. ∗S.M. Djouadi is with the Electrical & Computer Engineering Department, University of Tennessee, Knoxville, TN 37996-2100. [email protected] 1 Definitions and Notation • B(E, F ) denotes the space of bounded linear operators from a Banach space E to a Banach space F , endowed with the operator norm ‖A‖ := sup x∈E, ‖x‖≤1 ‖Ax‖, A ∈ B(E, F ) • ` denotes the usual Hilbert space of square summable sequences with the standard norm ‖x‖2 := ∞ ∑ j=0 |xj|, x := ( x0, x1, x2, · · · )∈ ` • Pk the usual truncation operator for some integer k, which sets all outputs after time k to zero. • An operator A ∈ B(E, F ) is said to be causal if it satisfies the operator equation: PkAPk = PkA, ∀k positive integers and stricly causal if it satisfies Pk+1APk = Pk+1A, , ∀k positive integers (1) The subscript “c” denotes the restriction of a subspace of operators to its intersection with causal (see [30, 10] for the definition) operators. “⊗” denotes for the tensor product. “” stands for the adjoint of an operator or the dual space of a Banach space depending on the context [6, 9].