Optimal robustness in the gap metric

Normalized doubly coprime factorizations for infinite-dimensional linear systems

“…In the proof of Theorem 4.5 we pointed out that the observability Lyapunov equation for the right factor system is precisely (12). That L 1 = P (I +P Q) −1 is a solution to the controllability Lyapunov equation can be verified algebraically as in [6,Theorem 9.4.10].…”

“…Theorem 4.5 Suppose that the state linear system Σ(A, B, C, D) is output stabilizable, and let Q opt denote the smallest bounded nonnegative solution to the control Riccati equation (12). Then the optimal right factor system…”

“…Lemma 4.9 Suppose that the state linear system Σ(A, B, C, D) is input and output stabilizable, P opt is the smallest bounded nonnegative solution of the filter Riccati equation (14) and Q is an arbitrary bounded nonnegative solution of the control Riccati equation (12). Then L 1 := P opt (I+QP opt ) −1 = L B , the controllability Gramian of right factor system (13).…”

“…Another important result in finite-dimensional systems theory is that of robustly stabilizing controllers with respect to factor perturbations. This relies on the existence of normalized left-or right-coprime factorizations (see Glover and McFarlane [13] and Georgiou and Smith [12]). Meyer and Franklin [22] obtained formulas for normalized coprime factorizations for any stabilizable and detectable realization Σ(A, B, C, D) of a rational transfer function.…”

A pointwise criterion for controller robustness

“…Of course, this definition depends on the measure that is chosen for the perturbations. Several distance notions for linear time-invariant systems have been proposed, of which the so-called gap metric [9,38) has gained much popularity because it is relatively easy to compute (13) and lends itself well to optimization (14,15]. However, the gap between two systems is a single number, whereas the uncertainty of a model is often seen as a frequencydependent quantity.…”