An Output Feedback Based Adaptive Robust Fault Tolerant Control Scheme for a Class of Nonlinear Systems

In this paper we present an output feedback based Adaptive Robust Fault Tolerant Control (ARFTC) strategy to solve the problem of output tracking in presence of actuator failures, disturbances and modeling uncertainties for a class of nonlinear systems. The class of faults addressed here include stuck actuators, actuator loss of efficiency or a combination of the two. We assume no a priori information regarding the instant of failure, failure pattern or fault size. The ARFTC combines the robustness of sliding mode controllers with the online learning capabilities of adaptive control to accommodate sudden changes in system parameters due to actuator faults. Comparative simulation studies are carried out on a nonlinear hypersonic aircraft model, which shows the effectiveness of the proposed scheme over backstepping based robust adaptive fault-tolerant control. INTRODUCTION Reliability and performance are twin objectives of many complex systems like chemical plants, nuclear plants, flight control system etc., and one cannot be sacrificed for the other. For such systems, it is desirable to have a certain degree of fault tolerance with respect to various faults. In this work, we focus on the problem of fault accommodation for unknown actuator failures for a class of nonlinear systems with unknown parameters and uncertain nonlinearities. We address two types of fault scenarios: actuator loss in efficiency and stuck actuators. We do not assume the knowledge of failed actuators or failure pattern in the present work. Fortunately, adaptive schemes, by virtue of its on-line learning capability can bypass this problem. Consequently, many adaptive schemes have been developed to solve this problem. Tao et al. proposed a model reference adaptive control (MRAC) based direct scheme to solve the problem of actuator fault-accommodation for linear system in [1]. Such approaches are inherently limited as they rely on conventionalMRAC, which suffers from poor transients during the learning phase and offers difficulty in checking stability and robustness bounds in presence of exogenous disturbances. They also addressed various classes of nonlinear systems using backstepping based adaptive control in [2] and [3]. Robust Adaptive backstepping based fault compensation scheme for a class of nonlinear systems was proposed in [4]. Note that even when robustness modifications are made to backstepping based adaptive control, there is still no transparent way to attenuate the effect of disturbances and modeling uncertainties on the transient response and steady-state tracking error. Robust control based schemes, on the other hand, can handle such disturbances and unstructured uncertainties with guaranteed transient performance and attenuate their effect on the steadystate error. LMI based fault-tolerant control was proposed in [5] and sliding mode control based approaches were used in [6]. But, in presence of large parametric uncertainties, the robust control based direct fault-accommodation schemes can result in input chattering or large steady-state errors when smoothing techniques are used. Thus, both adaptive and robust control based schemes can solve one part of the problem, but cannot address all the issues associated with actuator faults, viz., desired transient response and small steady-state tracking error in presence of parametric and non-parametric uncertainties. In spite of the inherent limitations of adaptive control based techniques, it has been realized that adaptation is of key importance in dealing with large parametric uncertainties introduced due to actuator faults in safety-critical missions like flight control systems. Consequently, the idea of safe adaptive control is coming to forefront, which ensures certain stability properties even without adaptation [7, 8]. In this respect, we would like 1 Copyright © 2009 by ASME DSCC2009-2740 Proceedings of the ASME 2009 Dynamic Systems and Control Conference DSCC2009 October 12-14, 2009, Hollywood, California, USA to point out that ARC based schemes have already resolved this issue [9, 10] and may be classified as the so-called safe adaptive control. Switching the adaptation off at any instant converts the adaptive robust controller into a deterministic robust controller with guaranteed transient performance. Moreover, the design procedure allows us to calculate explicit upper bound for tracking errors over the entire time history in terms of certain controller parameters and achieve prespecified final tracking accuracy. Thus, ARC based schemes are natural choices for safety sensitive systems over conventional adaptive and robust schemes. In the present work, we develop an output feedback ARC based scheme for accommodation of unknown actuator faults. The technique used here is a combination of adaptive backstepping [11] and discontinuous projection based ARC proposed in [10] and differs significantly from the techniques presented in [4] and [2] which relies on backstepping based direct adaptive control. The fundamental difference between the two schemes is due to the fact that ARC uses robust filter structures as the baseline controller, and adaptation is used only as a means to reduce the extent of parametric uncertainties. This is the reason that switching the adaptation off at any instant converts the ARC controller to a deterministic robust controller with guaranteed performance. On the other hand, in direct adaptive designs, adaptation is the underlying mechanism which makes the controller work. Furthermore, in backstepping based adaptive designs, tuning functions are used to compensate the for parameter-estimation error dynamics. But, as discontinuous projection is used in our approach, tuning functions cannot be used. In order to compensate for the effects of parameter-estimation error dynamics, the robust component of the control law is strengthened in ARC. In order to show the superior performance of the proposed scheme, comparative studies are performed using a hypersonic aircraft model. PROBLEM STATEMENT We will consider systems in the following form ẋ1 = x2 + φ0,1(y)+ p ∑ j=1 a jφ1, j + ∆1(x,t) .. ẋρ−1 = xρ + φ0,ρ−1(y)+ p ∑ j=1 a jφρ−1, j + ∆ρ−1(x,t) ẋρ = xρ+1 + φ0,ρ(y)+ p ∑ j=1 a jφρ, j + q ∑ j=1 bm, jβ j(y)u j(t)+ ∆ρ(x,t) .. ẋn = φ0,n(y)+ p ∑ j=1 a jφn, j + q ∑ j=1 b0, jβ j(y)u j(t)+ ∆n(x,t) (1) where ρ = n−m is the relative degree, u j is the control input, y = x1 is the measured output, φ0,i(y) and β j(y) are known smooth functions of y and β j(y) 6= 0 for any y. ∆i(x,t) represents uncertain nonlinearities and ai, bi, j are unknown constants such that sign of the high frequency gain (sgn(bm, j)) is known. We will make the following realistic assumptions regarding the uncertainties present in the system A1 The extent of parametric uncertainties and uncertain nonlinearities satisfy ai ∈ Ωa , {ai : (ai)min < ai < (ai)max} bi, j ∈ Ωb , {bi, j : (bi, j)min < bi, j < (bi, j)max} ∆i ∈ Ω∆ , {∆i : |∆i(x,t)| ≤ δi(t)} (2) where (ai)min, (ai)max, (bi, j)min, (bi, j)max are known and δi(t) is a bounded but unknown function. In this work, we will consider faults which can be modeled as