Nonlinear parameter‐varying state‐feedback design for a gyroscope using virtual control contraction metrics

Abstract

In this article, we present a virtual control contraction metric (VCCM) based nonlinear parameter-varying approach to design a state-feedback controller for a control moment gyroscope (CMG) to track a user-defined trajectory set. This VCCM based nonlinear (NL) stabilization and performance synthesis approach, which is similar to linear parameter-varying (LPV) control approaches, allows to achieve exact guarantees of exponential stability and  2 -gain performance on NL systems with respect to all trajectories from the predetermined set, which is not the case with the conventional LPV methods. Simulation and experimental studies conducted in both fully-and under-actuated operating modes of the CMG show effectiveness of this approach compared with standard LPV control methods.

Virtual control contraction metrics: Convex nonlinear feedback design via behavioral embedding

“…For control realization, we introduce VRGs to relax the strong requirement on virtual feedforward control design. [32][33][34] Recently, we applied the proposed method to various stabilization tasks of a control moment gyroscope. 34 Compared to Reference 34, besides introducing VRGs and integrating it into the VCCM method, we provide a comprehensive description of the proposed VCCM approach, comparison with the CCM based approach, and an extensive discussion on the connection of the VCCM approach to local and global LPV methods.…”

“…For virtual feedforward design, we choose y * = x * ∶= [𝜃 * , 0] ⊤ and v * = 𝜅 ff (x, x * , u * ) ∶= u * where u * is the nominal input torque satisfying Ru * = G(𝜃 * ). Finally, by applying the embedding rule y(t) = x(t) and u(t) = v(t) for all t, we obtain the overall controller for the (34) as…”

“…In this article, a parametric‐LMI based computational method is proposed to remove the structural assumption. For control realization, we introduce VRGs to relax the strong requirement on virtual feedforward control design 32–34 . Recently, we applied the proposed method to various stabilization tasks of a control moment gyroscope 34 .…”

Corrigendum

“…In the paper under consideration, 1 the formulas below the Equation (2) were published as: …”