An Adaptive Robust Scheme for Multiple Actuator Fault Accommodation

In this paper, we solve the problem of output tracking for linear systems in presence of unknown actuator failures using discontinuous projection based output feedback adaptive robust control (ARC) scheme. The faulty actuators are characterized as unknown inputs stuck within certain bounds at unknown instants of time. This problem is of prime importance for safety critical missions like flight control system. Many existing techniques to solve this problem use model reference adaptive control (MRAC), which is not well suited for handling various disturbances and modeling errors inherent to any realistic system model. In comparison, the backstepping based output feedback ARC approach used here can effectively deal with such uncertainties. Simulation studies are carried out on a linearized Boeing 747 model, which shows the effectiveness of the proposed scheme. Furthermore, we compare our simulation results with that of MRAC in presence of disturbances, which clearly illustrates the superior performance of the proposed ARC based actuator fault compensation scheme. INTRODUCTION For many safety critical missions like flight control 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 in a linear system with unknown parameters and subjected to bounded disturbances. The faults are modeled as actuators getting stuck at unknown instants of time, within certain bounds and are allowed to vary within the bound. Furthermore, we do not assume the knowledge of failed actuators, as fault isolation for systems with redundant actuators is a fairly difficult problem, if not impossible. 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. A novel approach for solving the problem of unknown actuator failure compensation was posed and solved in [1] for linear systems. They further extended their technique to nonlinear systems in [2]. These approaches are inherently limited as they rely on conventional MRAC, which suffers from poor transients during the learning phase and offers difficulty in checking stability and robustness bounds in presence of exogenous disturbances. Robust schemes for actuator fault accommodation, which can handle such disturbances and unstructured uncertainties with guaranteed transient performance, include LMI based techniques [3] and sliding mode control based approaches [4]. But, in presence of large parametric uncertainties, the robust control laws can result in very high gain controllers and may not guarantee good final tracking accuracy. One approach which potentially alleviates these problems is multiple model adaptive control (MMAC), switching and tuning [5, 6]. It is worth noting that in multiple model based approaches, the problem of covering the state-space with a finite set of nominal models is not a trivial one, especially in presence of unstructured uncertainties. Furthermore, as pointed out in [7], MMAC based techniques are not intrinsically stable and a safe switching rule needs to be designed. Given the need for stability in safety critical missions, the large parametric uncertainties introduced due to unknown actuator failures and the inherent limitations of conventional adaptive control, 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 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 conProceedings of DSCC2008 2008 ASME Dynamic Systems and Control Conference October 20-22, 2008, Ann Arbor, Michigan, USA 1 Copyright © 2008 by ASME DSCC2008-2237 troller 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 [11] is a combination of adaptive backstepping [12] and discontinuous projection based ARC proposed in [10]. The paper is organized as follows. In the next section, we describe the problemwe are trying to solve and certain assumptions that are needed to solve the problem. In the third section, we describe the output feedback based ARC approach to unknown actuator fault accommodation. In the fourth section, we present comparative simulation results to demonstrate the superior performance achievable using the proposed scheme and finally, we conclude the paper by summarizing the main contributions. PROBLEM STATEMENT In the present work, we consider systems which can be represented in the input-output form as follows, y(t) = k ∑ j=1 B j(s) A(s) u j(t)+ D(s) A(s) Δ(y,t)+dy(t) (1) where, A(s) = sn+ an−1s+ . . .+ a1s+ a0, Bj(s) = b jms+ . . .+b j1s+b j0 and D(s) = dls + . . .+d1s+d0, and m≤ l ≤ n. The plant parameters ai and bi are unknown constants. The coefficients di corresponding to the disturbance distribution are assumed to be known but, the results can be readily extended to the case where they are unknown constants. dy(t) represents the output disturbance, and Δ(y,t) represents any disturbance coming from the intermediate channels of the plant. An implicit assumption in the system representation (1) is, A1: The relative degree ρ = n−m is known and same for any input u j. In this work, we will consider actuator failures which can be modeled as [1], u j(t) = ū j(t), t ≥ t j, j ∈ {1,2, . . . ,m} and ū j,min ≤ ū j(t)≤ ū j,max (2) where, ū j,min and ū j,max are known bounds and t j is the unknown instant of failure for each j. We will also describe in a remark how to deal with other fault scenarios where this bound in unknown. Without actuator redundancy, actuator faults cannot be accommodated and this is formally stated in the following assumption, A2: System (1) can fulfill the desired control objective with up to m− 1 failed actuators, when implemented with unknown parameters. Thus, in presence of actuator failures, the input vector can be represented as, u(t) = u(t)+σ(ū(t)−u(t)) (3) where u∗(t) is the control input to be designed and, ū = [ū1, ū2, . . . , ūk] T , σ= diag{σ1,σ2, . . . ,σk} (4) σi = { 1 if the ith actuator fails 0 otherwise (5) Now, the problem we attempt to solve in this work can be stated precisely as follows. For the system described by (1), subjected to unknown actuator failures (3-5) and bounded disturbances, the goal is to design an output feedback control law such that the output tracking error converges exponentially to a prespecified bound and has a guaranteed transient performance. In addition to actuator fault compensation, it is also desirable that the closed-loop system posses good disturbance rejection properties. In the present approach, such properties are achieved by explicitly taking into account Δ(y,t): we use prior information about the nature of disturbance to construct a nominal disturbance model Δn(y, t) = q(y, t)T c, where q(y,t) = [qp(y, t), . . . ,q1(y, t)]T ∈ Rp represents the vector of known basis shape functions and c= [cp, . . . ,c1] represents the vector of unknown magnitudes. Thus, the disturbance can be represented as, Δ = Δn+ Δ̃, where Δ̃ is the modeling error. Adaptation will be used to compensate for the effect of Δn on the output tracking performance and Δ̃ will be dealt with via certain robust feedback for robust performance. OUTPUT FEEDBACK BASED ARC Observer Canonical Form In the present work, we will assume that control signals to all the actuators are same [1], i.e., u1 = . . . = u ∗ k = u ∗. With this choice of control input, the system with p∈ {1, . . . ,m−1} failed actuators can be represented as, y(t) = ∑ j =Jp B j(s) A(s) uj (t)+ ∑ j∈Jp B j(s) A(s) ū j(t) + D(s) A(s) Δ(y,t)+dy(t) (6) = b ms+ . . .+b p 1 s+b p 0 s+an−1s+ . . .+a1s+a0 u(t) + ∑ j∈Jp b jms+ . . .+b j1s+b j0 s+an−1s+ . . .+a1s+a0 ū j(t) + dls + . . .+d1s+d0 s+an−1s+ . . .+a1s+a0 Δ(y, t)+dy(t) (7) where, Jp = { j1, . . . , jp} is a set of subscripts such that [ū j1 , . . . , ū jp ] represents the set of unknown failed actuators. 2 Copyright © 2008 by ASME Also, b i = ∑ j =Jp b ji and the superscript p represents that this value corresponds to the Jp failure pattern. The following assumption will be made which is standard in adaptive control, A3: The polynomial ∑ j =Jp B j(s) is Hurwitz and the sign of the high frequency gain (sign(b m)) is known, irrespective of the failure patter Jp. Now we present an observer canonical realization of the above input-output model which is more suitable for the controller design technique presented here, ẋ1 = x2−an−1x1 .. ẋn−l−1 = xn−l −al+1x1 ẋn−l = xn−l+1−alx1+dlq T (y,t)c+dl Δ̃ .. ẋρ−1 = xρ−am+1x1+dm+1q T (y,t)c+dm+1Δ̃ ẋρ = xρ+1−amx1+b p mu (t)+ ∑ j∈Jp b jmū j(t) +dmq (y,t)c+dmΔ̃ .. ẋn = −a0x1+b p 0u (t)+ ∑ j∈Jp b j0ū j(t)+d0q T (y,t)c+d0Δ̃