Adaptive Dynamic Inversion Control with Actuator Saturation Constraints Applied to Tracking Spacecraft Maneuvers

This paper presents an adaptive control methodology for nonlinear plants that prevents the adaptive parameters from drifting due to trajectory errors arising from control saturation. The reference trajectory is modified upon saturation, so that the modified trajectory approximates the original reference closely and can be tracked within saturation limits. The adaptive parameters are updated by the error between the plant trajectory and this modified reference. Asymptotic tracking of the modified reference is guaranteed with the assumption that the modified reference remains bounded and asymptotic convergence of the modified reference to the original reference is guaranteed whenever the control becomes unsaturated. Also, bounded learning of the adaptive parameters is guaranteed. A numerical example of attitude tracking for a rigid spacecraft is presented which shows that, in the presence of persistent excitation, the adaptive parameters converge to the true parameters, even when the actuator saturates. Introduction Structured Adaptive Model Inversion (SAMI) [1] is based on the concepts of Feedback Linearization [2], Dynamic Inversion, and Structured Model Reference Adaptive Control (SMRAC) [3, 4]. In SAMI, dynamic inversion is used to linearize the nonlinear dynamics and solve for the control. The dynamic inversion is approximate, as the system parameters are not assumed to be modeled accurately. An adaptive control structure is then wrapped around the dynamic inverter to account The Journal of the Astronautical Sciences, Vol. 52, No. 4, October–December 2004, pp. 000–000 1 Presented at the 6th International Conference on Dynamics and Control of Systems and Structures in Space, Riomaggiore, Italy. 18–22 July, 2004. Graduate Research Assistant, Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141. Email: [email protected]. Associate Professor and Director, Flight Simulation Laboratory, Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141. Email: [email protected], Web Page: http://jungfrau.tamu.edu/valasek/. for the uncertainties in the system parameters [5, 6, 7]. This controller is designed to drive the error between the output of the actual plant and that of the desired reference trajectories to zero, with prescribed error dynamics. Most dynamic systems can be represented as two sets of differential equations: an exactly known kinematic level part, and a momentum level part with uncertain system parameters. The adaptation included in the SAMI framework can be limited to only the uncertain momentum level equations. This restricts the adaptation only to a subset of the statespace, making it efficient. SAMI has been shown to be effective for tracking spacecraft [8] and aggressive aircraft maneuvers [9]. The SAMI approach has also been extended to handle actuator failures [10, 11]. Control saturation in linear as well as nonlinear control systems is an important area of research and has attracted a lot of attention in recent years [12, 13, 14, 15]. Adaptive control usually assumes full authority control, and generally lacks an adequate theoretical treatment for control in the presence of actuator saturation limits. Saturation becomes more critical for adaptive systems than non-adaptive systems, since adaptation is based on the tracking error. Assuming that the dynamics are modeled perfectly and only parametric uncertainties exist in the system, the tracking error has contributions due to the initial error conditions, parametric uncertainties, and saturation. The adaptation scheme adapts only the uncertain parameters, so the error driving the adaptation scheme should not include the error due to saturation. Including the error component due to saturation can cause adaptation problems if the control system attempts to correct for this error by adapting the learning parameters. In earlier adaptive control formulations the adaptation rate was reduced or adaptation was completely stopped on saturation [16]. Correct adaptation in the presence of saturation is ensured by using the method of pseudo control hedging. The pseudo control hedging methodology has been successfully demonstrated by Johnson, E. N., Calise, A. J. et al. in a neural network based direct adaptive control law [17, 18, 19, 20]. The difference between the calculated and the applied control effort due to saturation results in a lack of acceleration produced in the plant as compared to the demanded reference acceleration. This is called the hedging signal, and if the hedge is removed from the reference, the resulting modified reference can be tracked within saturation limits. The tracking error seen will only be due to the initial error and the parametric uncertainty, and not due to saturation. Hence the controller will adapt correctly. This paper presents an application of Structured Adaptive Model inversion with Pseudo Control Hedging to the tracking of an attitude trajectory for a rigid spacecraft. Mathematical Formulation Formulation of the Mathematical Model with the Minimal Parameterization of the Inertia Matrix The equations of motion of a rigid spacecraft can be cast into a structured form, as an exactly known kinematic differential equation and a momentum level equation with uncertain parameters. The kinematic differential equation is (1) (2) T 1 4 1 T I3 3 2̃ 2 T ̇ T 2 Tandale and Valasek