Newton's approach to gain-controlled robust pole placement

Robust repeated pole placement

“…Since the spectral condition number κ 2 (X) is nondifferentiable, it is not amenable to optimisation via gradient search methods. The Frobenius condition number κ f ro (X) = X f ro X −1 f ro is differentiable, and since κ 2 (X) ≤ κ f ro (X), many authors, including [4], [11], [15], [18], have used this as their robustness measure. Note it is possible to reduce the Frobenius condition number of a matrix X by suitably scaling the lengths of its column vectors, yet when X is a matrix of eigenvectors, such scaling does not improve the eigenvalue conditioning.…”

Performance survey of robust pole placement methods

“…Subsequently two of the present authors revisited the heuristic methods of [3] and offered a range of improvements [4]. Byers and Nash [5], Tam and Lam [6] and Varga [7] took κ fro (X) as their robustness measure and cast the REPP problem as an unconstrained nonlinear optimization problem, to be solved by gradient iterative search methods. Recent contributors in this area include Li et al [8], who introduced a method for minimizing the 'departure from normality' robustness measure, and Ait Rami et al [9], who introduced a global constrained nonlinear optimal problem for the minimisation of the Frobenius condition number and showed that the solution could be approximated by a global optimization problem under LMI constraints, for which the authors gave an LMI-based algorithm.…”

Feedback norm minimisation with regional pole placement

“…For multi-input systems such an F is non-unique (Wonham, 1974) and several algorithms (e.g. see Chu, 2001;Kautsky, Nichols, & Dooren, 1985;Mehrmann & Xu, 1997;Miminis & Paige, 1982;Tam & Lam, 1997;Varga, 2000aVarga, , 2000b, and references therein) optimise various aspects of the F matrix and associated quantities based on the degrees of freedom available in F. It was shown in Mehrmann and Xu (1997) that relevant quantities for such optimisation formulations are the state feedback matrix norm (Keel, Fleming, & Bhattacharyya, 1985;Kouvaritakis & Cameron, 1980;Varga, 2000aVarga, , 2000bWang & Chow, 2000), the condition number of the associated eigenvector matrix (Chu, 2001;Kautsky et al, 1985;Mehrmann & Xu, 1997;Miminis & Paige, 1982;Rami, Faiz, Benzaouia, & Tadeo, 2009;Tam & Lam, 1997;Tits & Yang, 1996;Varga, 2000aVarga, , 2000b and the distance to uncontrollability (Mehrmann & Xu, 1997). These optimisation problems are rarely convex and have varying numerical properties (Rami et al, 2009).…”