[Paper Review] On Robust Stability and Performance with a Fixed-Order Controller Design for Uncertain Systems

Jun Ma, Haiyue Zhu+2 more|arXiv (Cornell University)|Dec 8, 2019

Stability and Control of Uncertain Systems39 references4 citations

TL;DR

This paper proposes a fixed-order controller design method for uncertain systems that ensures robust stability and performance within a restricted frequency range, avoiding the computational burden of checking all extreme models. By leveraging positive realness for stability and bounded realness for performance, the approach formulates conditions as linear matrix inequalities (LMIs), enabling efficient, single-solution computation that outperforms full-domain and nominal-only methods in both efficiency and robustness.

ABSTRACT

Typically, it is desirable to design a control system that is not only robustly stable in the presence of parametric uncertainties but also guarantees an adequate level of system performance. However, most of the existing methods need to take all extreme models over an uncertain domain into consideration, which then results in costly computation. Also, since these approaches attempt (rather unrealistically) to guarantee the system performance over a full frequency range, a conservative design is always admitted. Here, taking a specific viewpoint of robust stability and performance under a stated restricted frequency range (which is applicable in rather many real-world situations), this paper provides an essential basis for the design of a fixed-order controller for a system with bounded parametric uncertainties. A Hurwitz polynomial is used in the design and the robust stability is characterized by the notion of positive realness, such that the required robust stability condition is then suitably successfully constructed. Also, the robust performance criteria in terms of sensitivity shaping under different frequency ranges are constructed based on an approach of bounded realness analysis. Necessary and sufficient conditions are provided for both the robust stability and robust performance criteria. Furthermore, these conditions are expressed in the framework of linear matrix inequality (LMI) constraints, and thus can be efficiently solved. Comparative simulations are provided to illustrate the effectiveness and efficiency of the proposed approach.

Motivation & Objective

• To address the high computational cost of traditional robust control methods that require evaluating all extreme models in uncertain systems.
• To reduce conservatism in performance guarantees by focusing on a restricted frequency range relevant to real-world applications.
• To develop a fixed-order controller design that maintains robustness without exhaustive verification of all uncertain system variations.
• To formulate robust stability and performance conditions using positive realness and bounded realness within a linear matrix inequality (LMI) framework.

Proposed method

• Uses a Hurwitz polynomial as the basis for controller structure to ensure inherent stability properties.
• Applies the concept of positive realness to characterize robust stability under parametric uncertainties.
• Employs bounded realness analysis to define robust performance criteria within a specified frequency range.
• Translates stability and performance conditions into linear matrix inequality (LMI) constraints for efficient numerical solution.
• Solves the LMI problem once for the entire uncertain domain, avoiding the need to check all 2^n extreme systems.
• Validates the method through comparative simulations using a 4-parameter uncertain plant, contrasting with full-domain and nominal-only approaches.

Experimental results

Research questions

• RQ1 Can robust stability and performance be guaranteed for a fixed-order controller without evaluating all extreme models in the uncertain domain?
• RQ2 How can the conservatism of full-frequency robust control be reduced in practical applications where performance is only required over a limited frequency band?
• RQ3 Can a unified LMI-based framework be developed to simultaneously ensure robust stability and performance under parametric uncertainties?
• RQ4 What is the computational advantage of the proposed method compared to traditional approaches that require checking all vertex systems?
• RQ5 How does the performance of the proposed controller compare to nominal and full-domain designs under parametric perturbations?

Key findings

• The proposed method achieves robust stability and performance with only 5 LMIs, significantly reducing computational cost compared to checking all 16 extreme systems (Case II).
• Computational time for the proposed method (Case I) is 1.0883 seconds, less than half of the 2.6000 seconds required by the full-domain method (Case II).
• The proposed method yields the best tracking performance with an RMSE of 0.0325 and MaxAE of 0.0488, outperforming both the full-domain (RMSE: 0.0794, MaxAE: 0.1205) and nominal-only (RMSE: 0.1652, MaxAE: 0.2498) approaches.
• The nominal-only design (Case III) is fastest but performs worst under perturbations, confirming the necessity of considering uncertainties in controller design.
• The method maintains high robustness despite parametric uncertainties, as demonstrated by superior performance in time-domain tracking simulations.
• The LMI-based formulation enables a single solution for the entire uncertain domain, eliminating the exponential growth in computation with increasing uncertain parameters.