Structured Adaptive Model Inversion Control to Simultaneously Handle Actuator Failure and Actuator Saturation

Traditional adaptive control lacks rigorous theoretical treatment for control in the presence of actuator saturation. Generally, adaptation is stopped as soon as the control saturates to avoid incorrect adaptation. Adaptation in the presence of saturation may be critical, especially when the controller is recovering from a failure. This paper presents an Adaptive Control methodology that facilitates correct adaptation in the presence of actuation saturation limits. The central idea is to modify the reference trajectory on saturation, in such a way that the modified trajectory approximates the original reference as close as possible, and can be tracked within saturation limits. Nonlinear six degree of freedom simulations of an F-16 type aircraft are shown to demonstrate this control scheme. INTRODUCTION In actual practice, dynamic systems that are being controlled may be poorly modelled or the parameters of the system may be varying with respect to the operating environment. To compensate for poor modelling or parameter variations, an adaptive controller is used whose parameters are updated online, based on the signals of the system. Structured Adaptive Model Inversion (SAMI) is based on the concepts of Feedback Linearization, ∗Graduate Research Assistant, Aerospace Engineering Department. Student Member AIAA. [email protected] †Associate Professor and Director, Flight Simulation Laboratory, Aerospace Engineering Department, Associate Fellow AIAA. [email protected]. Copyright c ©2003 by M. Tandale, and J. Valasek. Published by the American Institute of Aeronautics and Astronautics, Inc. with permission. Dynamic Inversion, and Structured Model Reference Adaptive Control (SMRAC). In SAMI, dynamic inversion is used to solve for the control. The dynamic inversion is approximate, as the system parameters are not modelled accurately. An adaptive control structure is wrapped around the dynamic inverter to account for the uncertainties in the system parameters. This controller is designed to drive the error between the output of the actual plant and that of a model reference to zero, as in model reference adaptive control. Most dynamic systems can be broken into an exactly known kinematic level part, and a momentum level part with uncertain system parameters. The adaptation included in this framework can be limited to only the uncertain momentum level equations, to simplify the adaptation. The closed-loop system is shown to be globally stable for trajectories without singularities. However, the adaptively estimated parameters do not converge to the actual parameters of the system. SAMI has been shown to be effective for tracking spacecraft and aggressive aircraft maneuvers. The SAMI approach has been extended to handle actuator failures. REDUNDANT CONTROL AND ACTUATION FAILURE PROBLEM In high performance dynamic systems, the total number of actuators used may be greater than the number of states to be closely controlled or tracked. This control redundancy generally exists to achieve optimality with respect to control effort. Some modern high performance aircraft have thrust vectoring in addition to the aerodynamic control surfaces. Thrust vectoring is mainly used for vertical take off and landing, but these controls can be used to augment the aerodynamic controls in flight. In this case of redundant actuation it is still possible to closely track the desired states even if some of the actuators fail, as long as the number 1 American Institute of Aeronautics and Astronautics of active actuators is greater than or equal to the number of states to be tracked. Thus, if it is possible to reconfigure the control after failure, the stability and performance of the system can in theory be maintained. Strategies to Handle Actuation Failure Most actuator failure schemes employ some form of failure detection algorithm to detect the failure. A new control effectiveness matrix is estimated and the controller is redesigned by recalculating the control gains. This approach depends strongly on the efficacy of the failure detection algorithm. The algorithm may fail to detect a failure, or may give false warning when there is no actuation failure. In contrast, the SAMI controller is constantly updating its parameters so it does not need to specifically detect the failure. The failure is implicitly identified as a change in the parameters of the control effectiveness matrix, and the adaptation mechanism adapts to this change. Therefore, SAMI is a good candidate to address the problem of actuator failure. Mathematical Modelling of Actuator Freezes The actuator failures commonly encountered in aircraft and re-entry vehicles such as the X-38 are called control freezes, in which the control surface freezes or remains fixed at a position that may or may not be zero. In such a case, the remaining active actuators must not only compensate for the lack of the desired control effort of the failed actuator, but also cancel the undesired control effect produced if the actuator freezes at any position other than zero. Control freezes can be modelled by the following mathematical model uapplied = Ducalculated + E (1) where D ∈ Rm×m is a constant matrix and E ∈ R is a constant vector for a particular control configuration, but they change and settle to other constant values if a control freeze failure occurs. Here m is the number of controls. If one wants to model only control freezes then D should be strictly diagonal as the cross coupling between the calculated value of one control and the applied value of the other control is zero. Since uapplied = ucalculated in the absence of failure, the D is initialized an identity matrix, and the E is a null vector.   ua1 ua2 ua3   =   D11 0 0 0 D22 0 0 0 D33     uc1 uc2 uc3   +   E1 E2 E3  (2) If an actuator freezes, the corresponding diagonal term in the D matrix should go to zero, and the corresponding element in the vector E should go to the constant value at which the control surface has frozen. This mathematical model can also model damaged control surfaces. Consider a case in which a control surface such as an elevator is damaged by gunfire or some other cause, such that only half of the surface is left. Since the control effectiveness is reduced by half, the corresponding diagonal term in the D matrix should go to 0.5 and the corresponding element in the vector E should remain at zero. By adding this model to the SAMI formulation, a framework can be created to accommodate actuator failures and damage as changes in the parameters of the system. CONTROL SATURATION LIMITS Adaptive control usually assumes full authority control, and 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 the adaptation is based on the tracking error. Assuming that the dynamics are modelled 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 will cause incorrect adaptation. To get around this problem two approaches are usually followed: • Reduce the adaptation rate in the presence of saturation, or completely stop adaptation when the control is saturated. This stops the incorrect adaptation, but adaptation may be critical when the input is saturated. Consider a case where the parametric uncertainty is high and the control saturates because of these uncertainties. The system may diverge unless the uncertainties are corrected and may not recover from saturation at all. 2 American Institute of Aeronautics and Astronautics • If the input is saturated due to an aggressive reference trajectory, the reference command is adjusted so that the input does not saturate. This paper presents an adaptive control methodology which follows the second school of thought. After separating the system into structured kinematic and dynamic parts, it is noticed that the control directly affects the acceleration. So, 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 hedge signal. If the hedge signal is removed from the reference, the resulting modified reference can be tracked within saturation limits. The tracking error seen will be only due to the initial error and the parametric uncertainty, hence the controller will adapt correctly. The SAMI formulation will always remain unsaturated, but the focus now shifts to the stability of the reference model as the reference model now gets dynamically coupled with the plant and the adaptive law. The hedge signal now acts as a disturbance input to the reference model. But, the reference model is a hypothetical mathematical model selected by the designer and is free from the input rate and position saturation constraints. So ensuring stability in the presence of bounded disturbances is simplified. MATHEMATICAL FORMULATION This section mathematically formulates a fault tolerant SAMI controller that can handle actuator saturation. Definition of the Plant and the Reference Model Consider the mathematical model of the system as follows M(q, q̇)q̈ = G(q, q̇) + Cuapp (3) q̈ = M−1G + M−1Cuapp (4)