Solutions for Handling Control Magnitude Bounds in Adaptive Dynamic Inversion Controlled Satellites

Traditional adaptive control assumes full control authority and lacks an adequate theoretical treatment for control in the presence of actuator saturation limits. The adaptive dynamic inversion control methodology, which uses dynamic inversion to calculate the control and adaptation to compensate for errors in the inversion due to model uncertainties, also lacks an adequate theoretical treatment for saturation. This paper investigates the problems introduced in adaptive dynamic inversion control schemes due to bounds on the control, and develops a three component control scheme to overcome them. The main contribution of the paper is determination of the maximum possible domain of attraction with respect to the control position limit, and development of a control switching strategy to contain the plant within the maximum possible domain of attraction. This strategy ensures boundedness of the state by restricting it within the Domain of Control Authority. A direction consistent control constraint mechanism was also developed, to maintain the resultant direction of the rate of change of state to be the same as that of the desired, even in the presence of control saturation. Finally, a modified adaptation mechanism was implemented to prevent incorrect adaptation arising from trajectory errors due to control saturation. Mathematical development of the control laws and the adaptation mechanisms is presented, along with proofs for convergence of the tracking error and stability of the overall control scheme. To demonstrate the control scheme, two different numerical simulations for rigid spacecraft attitude tracking with uncertain inertias and saturated controls are presented. Results show that the control scheme successfully handles adaptive dynamic inversion control of systems with dynamics that are nonlinear in terms of the state, with uncertain parameters that appear linearly, in the presence of initial condition errors and control position bounds, and nonlinear saturation constraints on the components of the control. Introduction Actuator saturation is a major consideration for all practical control systems, and the literature shows that much effort has been expended on strategies to overcome The Journal of the Astronautical Sciences, Vol. 55, No. 2, April–June 2007, pp. 171–194 171 Research Scientist, Optimal Synthesis Inc., Palo Alto, CA 94303. E-mail: [email protected]. Associate Professor and Director, Flight Simulation Laboratory, Aerospace Engineering Department, Texas A&M University, College Station, Texas 77843-3141. E-mail: [email protected], Web page: http://jungfrau. tamu.edu/valasek/. its effects. Hu and Lin have done seminal work in analyzing the controllability and stabilization of unstable, linear time invariant systems with input saturation [1–3]. They have explicitly identified the null controllable region of the state-space for linear systems with saturated linear feedback. However, their work does not address nonlinear systems, or adaptative systems. Traditionally, adaptive control assumes full control authority and thus lacks a complete theoretical basis for control in the presence of actuator saturation limits. Saturation is a problem for adaptive systems, since continued adaptation in the presence of saturation may lead to instability. In recent years, much effort has been expended for adaptive control design in the presence of input saturation constraints [4]. A modification to the standard adaptive control structure to address the adverse effects of control saturation was suggested by Monopoli [5], but no formal proof of stability was provided. Karason and Annaswamy presented the concept of modifying the error, proportional to the control deficiency [6]. They laid out a rigorous mathematical proof of asymptotic stability for a model reference framework, and identified the largest set of initial conditions of the plant and the controller for which a stable controller could be realized. Akella, Junkins and Robinett devised a methodology to impose actuator saturation constraints on the adaptive control law analogous to Pontryagin’s principle for optimal control, to make the error energy rate as negative as possible with admissible controls [7]. They identify a boundary layer term which is the difference between the calculated and the applied control, and impose conditions on the adaptive update laws to bound the boundary layer thickness. Johnson and Calise applied the concept of “pseudo-control hedging” to adaptive control, which is a fixed gain adjustment to the reference model that is proportional to the control deficiency [8]. More recently, Lavretsky and Hovakimyan have proposed a new design approach called “positive -modification” that guarantees the control never incurs saturation [9]. Dynamic Inversion is an approach which has been widely used in recent years for the control of nonlinear systems, especially in the field of aerospace engineering [10–16]. A fundamental assumption in this approach is that the inherent plant dynamics are modeled accurately, and therefore can be canceled exactly by the feedback functions. In practice this assumption is not realistic: the dynamic inversion controller requires some level of robustness to suppress undesired behavior due to plant uncertainties. To overcome this robustness problem, an adaptive model of the plant dynamics is sometimes used to facilitate the inversion, which is then updated in real-time based on the response of the system. This gives rise to an entire class of controllers which may be referred to as Adaptive Dynamic Inversion Controllers [17–20]. In control system design, it is good practice to incorporate all known information about the kinematic and dynamic characteristics of the system in the mathematical formulation of the controller. The dynamics of most mechanical systems can usually be cast into a “structured form” with an exactly known kinematic part, and a dynamic part with uncertain plant parameters. Therefore, it makes sense to restrict the adaptation to only the unknown part, the dynamics. The resulting control systems are simpler (lower order) and easier to implement because all of the learning and adaptation due to the uncertainty can be restricted to a subspace of the entire state-space. Structured Model Reference Adaptive Control (SMRAC) [21] and Structured Adaptive Model Inversion (SAMI) [22] are model following and dynamic inversion control architectures respectively, which follow this structured 172 Tandale and Valasek