Convex Optimization Approaches to Robust ℒ1 Fixed-Order Filtering for Polytopic Systems with Multiple Delays

Abstract

This paper investigates the robust L 1 fixed-order filtering problem for continuous polytopic systems with multiple state delays. Attention is focused on the design of robust full-order and reduced-order filters that guarantee the filtering error system to be asymptotically stable and satisfy the worst case peak-to-peak gain of the filtering error system for all admissible uncertainties and time delays. In particular, we concentrate on the delay-dependent case, and the peak-to-peak performance criterion is first established for polytopic systems with multiple state delays. Two different convex optimization approaches are proposed to solve this problem. One is the parameter-dependent Lyapunov approach in which the filter is not only dependent on the parameters (residing in a polytope), but also the Lyapunov matrices are different over the entire polytope domain. The other is the quadratic stability approach which obtains an admissible filter in the quadratic framework. Computational algorithms in terms of linear matrix inequalities (LMIs) are provided. It is shown that the parameter-dependent Lyapunov approach turns out to be less conservative than the quadratic stability approach, but the quadratic stability approach is computationally less demanding. Two numerical examples are presented to illustrate the proposed theory.

Gain-scheduled L-one control for linear parameter-varying systems with parameter-dependent delays

“…Thus, inequalities (7), (8) and (16) can guarantee that the closedloop system is asymptotically stable and has a guaranteed L 1 noise attenuation level γ. Remark 4 Compared with Theorem 1, Theorem 2 is more conductive to performing the numerical calculation and is easier to apply to control system analysis and synthesis.…”

“…In this paper, we use the L 1 norm as the optimization objective in the gain-scheduled controller syn-618 Y. LI et al / J Control Theory Appl 2011 9 (4) 617-623 thesis that reflects the worst case peak value of outputs due to persistent input signals (bounded in magnitude), where the signal size (norm) is taken to be the signal's peak value. Very recently, based on the theory of positively invariant sets and the linear matrix inequality technique, we have resolved the L 1 optimal problem for linear systems [16,17], and this technique has also been extended to multiobjective situations [18].…”

L1 impedance control for bilateral teleoperation containing model uncertainty

“…To exemplify, Zhu and Chen developed small-gain theorem for time-delayed systems based upon L 1 theory (Zhu and Chen, 2015). L 1 filtering methods for systems with time delays have also been investigated by Li et al (2008, 2011). Designing controller for delayed systems is proposed by Li and Liang (2011), Maiti et al (2021), and Yungang et al (2000) by means of optimal L 1 theory, and by Nguyen and Dankowicz (2017) by making use of L 1 adaptive control.…”

Model Order Reduction for Neutral Systems by Moment Matching

“…In many physical, industrial and circuit systems, time delays occur due to the finite capability of information processing, data transmission among various parts of the systems and some essential simplification of the corresponding process models [9,27,29,37,39,49,[55][56][57]. The delaying effect is often detrimental to the performance, and even renders instability.…”