Global Smooth Time-Varying Dynamic Gain Scheduled Control of Nonholonomic Systems With Application to Spacecraft Systems

Abstract

We propose a globally smooth time-varying dynamic gain scheduled control scheme for chained nonholonomic systems with linear and bilinear chained subsystems. Key properties of parametric Lyapunov equations are exploited to design an exponentially convergent controller for the linear subsystem. Subsequently, a linear time-varying state transformation converts the bilinear subsystem into a time-varying system, based on which smooth dynamic gain scheduled controllers are constructed. In contrast to traditional static gain-scheduled methods for nonholonomic systems, which only consider kinematic nonholonomic systems and require solving complex nonlinear equations online at each time step, the proposed approach considers nonholonomic systems with both kinematics and dynamics, and generates gains through scalar differential equations. Meanwhile, the output-feedback case is also considered. This leads to a smooth and computationally efficient solution, significantly simplifying practical implementation. Additionally, we introduce dynamic gain scheduled strategies to adaptively adjust design parameters in real time, thereby enhancing the convergence rates of the closed-loop systems. As a practical application, we design a smooth time-varying dynamic gain scheduled control for an underactuated axisymmetric spacecraft system. The effectiveness of the proposed methods is demonstrated through comprehensive simulation results. Note to Practitioners—Gain scheduled methods play a crucial role in aerospace applications. However, the emergence of novel aerospace vehicles with diverse structural designs often leads to nonholonomic characteristics, posing significant challenges to traditional gain scheduled control algorithms. This paper aims to develop a gain scheduled method tailored for a class of nonholonomic systems to enhance control performance. Existing gain scheduled approaches for such systems are not only limited in scope—often applicable only to nonholonomic systems with a scalar first subsystem (i.e., considering only kinematics)—but are also typically complex in form, hindering their practical use in engineering. In contrast, the method proposed in this paper is applicable to systems considering both kinematics and dynamics, is linear in structure, and is relatively concise in design. In addition, the method proposed in this paper can also be extended to the observer case, which is more suitable for practical scenarios where not all states can be precisely obtained. To facilitate practical implementation by engineers, the proposed approach is illustrated through the example of an underactuated nonholonomic attitude system of a spacecraft.