Positive polynomials and robust stabilization with fixed-order controllers

Abstract

Recent results on positive polynomials are used to obtain a convex inner approximation of the stability domain in the space of coefficients of a polynomial. An application to the design of fixed-order controllers robustly stabilizing a linear system subject to polytopic uncertainty is then proposed, based on LMI optimization. The key ingredient in the design procedure resides in the choice of a central polynomial, or desired nominal closed-loop characteristic polynomial. Several numerical examples illustrate the relevance of the approach.

Polynomial LPV synthesis applied to turbofan engines

“…We have described an extension of the fixedorder controller design procedure of (Henrion et al, 2003b) to SISO gain-scheduling design with guaranteed H ∞ performance over the whole parameter range. We used the idea of central polynomial developed in (Henrion et al, 2003b) in order to derive an LMI formulation with the following characteristics:…”

“…Roughly speaking, as explained in (Henrion et al, 2003b), the central polynomial enforces the pole dynamics desired in closed-loop. For a given choice of a central polynomial, if the design LMI are infeasible, then it may mean that the desired closed-loop dynamics are not achievable.…”

“…Sufficient conditions for stability of this transfer function are now given, based on Lyapunov's stability theory (de Oliveira and Skelton, 2001). An alternative proof based on polynomials positivity can be found in (Henrion et al, 2003a;Henrion et al, 2003b).…”

“…As mentioned in (Henrion et al, 2003b), it is hard to give a general formula for the choice of the central polynomial. Somehow, it can be compared to the choice of the weighting filters of the mixed sensitivity problem, see (Skogestad and Postlethwaite, 1996) for example.…”

Sensitivity shaping with degree constraint by nonlinear least-squares optimization

“…Much effort has been made for such development, e.g., classical control methodologies such as PID-based control and lead-lag compensations (Horowitz, 1992), both open-loop (McFarlane and Glover, 1992) and closed-loop shaping techniques in H ∞ control (e.g., (Doyle et al, 1992)), an approach based on positive polynomials (Henrion et al, 2003), to name a few. However, these tools heavily require designers' engineering knowledge and intuition in manual selection of design parameters such as weighting functions.…”