On Robust Stability and Performance with a Fixed-Order Controller Design for Uncertain Systems

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 fixedorder controller for a system with bounded parametric uncertainties. A Schur 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. A numerical example is provided to illustrate the effectiveness of the proposed approach.

Convex Parameterization and Optimization for Robust Tracking of a Magnetically Levitated Planar Positioning System

“…By defining the set C Ri , all the extreme systems are suitably taken into consideration. Additionally, it is notable that the robust stability for the entire uncertain domain is checked at only the vertices of the convex polyhedron as defined in (31). On the other hand, for the robust performance, γ * is the upper bound to H ∞ -norm by taking all the extreme models into consideration.…”

“…A disturbance compensation scheme is proposed in [28], which alters the reference profile and compensates for the disturbance using the DOB, such that improved tracking performance is attained in the tray indexing application. Also, considerable attention is commonly drawn towards the use of parameterization techniques, for example, the Youla-Kucera parameterization [29], and alternative parameterization techniques based on the positive real lemma and the bounded real lemma [30], [31]. However, these methods still require various further developments and modifications before they can be accepted as effective solutions in the presence of parametric uncertainties and structural constraints, and especially also as solutions with attendant manageable numerical computation.…”

Robust Fixed-Order Controller Design for Uncertain Systems with Generalized Common Lyapunov Strictly Positive Realness Characterization

“…It is thus worthwhile to highlight that while the KYP lemma accounts for the infinite frequency range, the generalized counterpart is capable of limiting the frequency range under consideration to be finite. However, it needs to be noted that the use of the GKYP lemma for sensitivity shaping in a specific finite frequency range only admits a sufficient but not necessary condition for the controller design [22], due to the use of the convex separation lemma; and it also introduces hyper-parameters that are required be manually defined before the optimization.…”