Adaptive Fault-Tolerant Control for Flight Systems with Input Saturation and Model Mismatch

A novel scheme for fault-tolerant control is proposed in this paper, in which model reference adaptive control method is incorporated with control allocation to cope with simultaneous actuator failures, input saturation, and model mismatch in the flight system. In order to reduce performance degradation caused by actuator failures, the proposed scheme redistributes the control signal to healthy actuators and updates the weighting matrix based on actuator effectiveness. Because of saturation errors resulting from actuator constraints and model mismatch caused by abnormal changes in the system, the original reference model may not be appropriate. Under this circumstance, an adaptive reference model which can also provide satisfactory performance is designed. Simulations of a flight control example are given to illustrate the effectiveness of the proposed scheme.


Introduction
Actuator failures may adversely affect the stability and performance of flight systems.Fault-tolerant control (FTC) has been studied as a method for accommodating failures.In the past decades, a number of approaches have been proposed for FTC, including sliding mode control-based designs [1,2], variable structure control-based designs [3], learning-based approaches [4], and adaptive control-based designs [5].
The availability of actuator redundancy is an important element of FTC.Control allocation (CA) is an approach that can effectively manage redundancy by distributing virtual control law requirements to redundant actuators in the most efficient manner [6].There is an extensive literature on CA approaches and applications.Enns presented the linear and quadratic programming approaches in detail in [7].Härkegård and Glad compared control allocation with optimal control for solving actuator redundancy in [8].Several CA approaches taking into account actuator limits have been discussed [9][10][11].Kishore et al. gave an algorithm that updated the weighting matrix to deal with actuator limits in [9].However, in the event of failure, only depending on updating the weighting matrix sometimes may be invalid.
In addition, flight systems may experience other disruptive factors such as model mismatch.Model mismatch can be caused by icing or damage to the aircraft.Joshi et al. first considered this problem in [12] and further developed his work in [13].Although the previous work has achieved satisfactory tracking performance, none of them considered actuator limits.
In this paper, we develop a novel scheme combining CA with model reference adaptive control (MRAC) for flight systems with simultaneous actuator failures, model mismatch, and input saturation.On the basis of the existing literature, this paper proposes an improved control allocation (ICA) method that updates the weighting matrix based on actuator effectiveness.In addition to updating the weighting matrix, this paper employs the positive -modification method [14] to impose constraints on actuators.MRAC can guarantee the closed-loop stability of the system and reduce the error caused by actuator failures.Because of input saturation and model mismatch, the original reference model may not be appropriate.We have designed an adaptive reference model that can provide satisfactory performance under these circumstances.Compared with the traditional CA methods, simulation results show that the new scheme provides better tracking performance.The control structure is shown in Figure 1.This paper focuses on the tracking control on the single system.In the recent study, the coordination control of multiagent systems has attracted considerable attention.Several approaches and applications have been developed in [15][16][17].We will explore the FTC of multiple flight systems in our future study.
This paper is organized as follows.Section 2 presents the problem statement.Section 3 gives a detailed description of the novel adaptive fault-tolerant control scheme.Simulation results of a flight control example are given in Section 4. Section 5 is concluding remarks.

Problem Statement
Consider a linear time-invariant system in the presence of model mismatch and actuator failures described by where () ∈ R  is the state vector, () ∈ R  is the actual control input, () ∈ R  is the output, and

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Assumption 2. There is sufficient actuator redundancy in the system so that the input matrix   can be factorized into a full column rank matrix  V ∈ R × and a full row rank matrix  ∈ R × ; that is, Assumption 3. (,   Δ) is controllable.
In addition, the saturation error can be defined as which may lead to performance degradation or even system instability.
Based on Assumption 4, we can define the conventional reference model as where   () ∈ R  is the reference model state and () ∈ R  is a bounded reference input.Optimal and robust control theory can be employed to design the gain matrices  1 ,  2 .However, due to model mismatch   , ( +   +  V   1 ) will not have the same structure as   for any  1 .Therefore, stability and asymptotic tracking performance can no longer be guaranteed.
In response, this paper designs an adaptive reference model while taking saturation error ũ and model mismatch   into consideration.The control objective is to develop a novel adaptive fault-tolerant control scheme, for the faulty system given by ( 1) with simultaneous actuator failures, model mismatch, and input saturation, to achieve the desired tracking performance.

Adaptive Fault-Tolerant Control
In order to achieve the desired control objective, we propose an adaptive fault-tolerant control scheme in this section.The scheme mainly incorporates an improved control allocation method and an adaptive virtual controller design.
3.1.Improved Control Allocation Method.Substituting ( 6) and ( 7) into (1) yields the following closed-loop system dynamics: where Based on Assumption 2, we can get where V() ∈ R  is a virtual control input that represents the total control effort produced by actuators.
As distinct from conventional CA problem, the matrix   incorporates fault information.Generally, CA can be formulated as an optimization problem as follows [1]: where  = diag( 1 ,  2 , . . .,   ) is a positive definite weighting matrix and the scalar   represents the weight of the th actuator.
In contrast to the preceding CA results, the weighting matrix  can be updated online based on actuator effectiveness in this paper.Suppose that the th actuator has partially lost effectiveness; that is, Δ = diag(1, . . ., 1,   , 1, . . ., 1), 0 <   < 1.Let the th diagonal element of  be perturbed from   to  2    , where the scalar   can be obtained by   = 1/  .Then the new weighting matrix becomes Ŵ = , where  = diag(1, . . ., 1,   , 1, . . ., 1).Consequently, the solution to (11) can be formulated as Usually the initial value of the weighting matrix  is selected based on the priority of actuators.Once actuator failures occur,  will be updated.As   → 0,   → ∞, the associated component   in (12) is therefore weighted heavily since   becomes large.If the th actuator has damaged completely, the value of   will be infinity.Equation ( 13) is able to distribute the virtual control signal to the remaining physical actuators based on actuator effectiveness.However, due to the failure, only depending on the adaption of  to avoid saturation sometimes may be invalid.Therefore, this paper employs a positive -modification method to ensure that all control inputs lie within the bounds.
The following lemma gives the algorithm of the positive -modification method.
Lemma 5 (see [14]).If  1 , . . .,   ≥ 0, then the solution to (6) is given by a convex combination of   () and   max sat(  ()/  max ) for all  > 0: which can also be represented component-wise in the following form: ( Remark 6. Detailed discussion of the above lemma can be found in [14].

Adaptive Virtual Controller
Design.The adaptive controller is designed as where K1 and K2 are the estimates of  1 and  2 , respectively.Substituting ( 11) and ( 16) into ( 9) yields the closed-loop system where In order to compensate for the saturation error ũ , we design an adaptive compensation controller K3 ∈ R × .The initial value of K3 is defined as  3 and satisfies Using ( 4) and ( 18), the closed-loop system in (17) can be modified as The modified closed-loop system dynamics lead to consideration of the following adaptive reference model system: where   is a time-varying matrix incorporating the model mismatch estimate δ , denoted as Following [18], the system matrix   can be described as an affine function of a parameter vector  ∈    that lies in a convex polytope S having vertices   ,  = 1, 2, . . . V ; that is, where    are constant matrices and   () ∈ [  min ,   max ].Suppose that   (  ) denotes the value of the reference model system matrix at vertex   and there exist positive definite matrices  =   ,  =   ∈ R × satisfying Based on (23), we can get the following Lyapunov inequality for all () ∈ S: Thus the autonomous part of the adaptive reference model in (20) is exponentially stable for all () ∈ S [18].Note that this paper only considers the case that the true value of model mismatch lies within the stable region Figure 6.The model mismatch   is assumed to satisfy Subtracting ( 20) from ( 19) yields the following error dynamics: where  =  −   and δ = δ −   denote the tracking error and model mismatch estimate error, respectively.
The following theorem gives the adaption and parameter estimation laws which ensure that () → 0 and all signals remain bounded.We will first assume that the initial conditions satisfy Theorem 4.1 in [14].
Theorem 7.For the system given by (1), the adaptive controller (16) with gain adaption laws and parameter estimation laws where Γ 1 , Γ 2 , Γ 3 ∈ R × and Γ  ∈ R × are positive definite matrices and  is determined by (24), guarantees that all the closed-loop signals remain bounded and that the tracking error () converges to 0 as  → ∞.
Proof.In order to analyze the closed-loop stability and tracking performance, choose a Lyapunov function as where tr(⋅) denotes the trace of a matrix and δ denotes the th column of δ .Differentiating (29), we can get As K1 = K1 , K2 = K2 , K3 = K3 , and δ  = δ  , substituting ( 27) and ( 28) into (30) yields Since () is a positive definite function and V() ≤ 0, the signals (), K1 (), K2 (), K3 (), and δ () are all bounded.However, due to the modification of the reference model, the stability of the system cannot be directly proven.It is necessary to prove that one of the signals () or   () is bounded.
Then, we define the following candidate Lyapunov function: Substituting ( 4) into (1) yields Based on (25),   +   is assumed to satisfy the following Lyapunov inequality: where   =    ,   =    ∈ R × are both positive definite matrices.
Differentiating (32), we can get Based on the previous results in [14], we know that there always exists a nonempty annulus region such that Ẇ() < 0 holds for all  from that region.The boundedness of  can be guaranteed as long as the initial conditions satisfy Theorem 4.1 in [14].Since () is bounded and   () = () − (), it is easy to conclude that   () is bounded.
Remark 8. Since the reference model is time varying, it is necessary to ensure that the parameter estimate δ remains within the stable region.If δ exceeds the stable region, a parameter projection method [18] will be utilized to ensure that the adaptive reference model has satisfactory stability and performance.

Application Example
To investigate the effectiveness of the proposed adaptive faulttolerant control scheme, we utilize a flight control example based on the ADMIRE model [19].A low-speed flight case is considered, where the control surface efficiency is poor.

System Description.
The state space of the model is given by ( 1), where the nominal values of the system matrices are in the appendix and System states , , , , and  represent the angle of attack, the sideslip angle, the roll rate, the pitch rate, and the yaw rate, respectively.Control inputs   ,   ,   , and   represent the positions of the canard wings, the right elevons, the left elevons, and the rudder, respectively.The virtual control input V contains the angular accelerations in roll, pitch, and yaw produced by the control surface.
The actuator position constraints are given by It is assumed that the actuator   loses 99% of effectiveness at 1.01 s while the remaining actuators are functional throughout and the aircraft experiences damage at 0 s; therefore, a model mismatch exists in the system matrix from the outset.The model mismatch   in the system is The reference inputs are given in Figure 2. ] .
The initial value of the weighting matrix Ŵ is chosen as diag{0.01, 1, 1, 1}.Once actuator failures occur, the weighting matrix will be updated.

Simulation Results
Case 1.In the first case, all adaption is turned off; that is, the weighting matrix, the gains, and the parameter estimates are not updated.The plant states are shown in Figure 3. Obviously, the stability of the system cannot be guaranteed due to the presence of actuator failures, model mismatch, and input saturation.
Case 2. We utilize the adaptive fault-tolerant scheme in this case.The controller gain and parameter estimates are generated by the adaptive laws (27)-(28).At  = 1.01 s, the first actuator fails while the remaining actuators continue to be functional, adaptively compensating for the failed first actuator.Unfortunately, actuators  2 and  4 have exceeded the input constraints as shown in Figure 4. Using the antisaturation algorithm, all of the actuators can be rigorously kept within the bounds, as shown in Figure 5.During the simulation, the parameter estimates δ 14 and δ 25 lie in the quadratic stable region throughout and approach the true value of model mismatch adaptively as shown in Figure 7.The plant with the adaptive reference model states and the tracking errors are illustrated in Figures 7 and 8, respectively.Simulation results show that the tracking errors remain small and converge to zero asymptotically.

Conclusion
This paper proposed a novel fault-tolerant control scheme for flight systems in the presence of simultaneous actuator failures, model mismatch, and input saturation.The scheme combined CA and MRAC to guarantee the system's stability and tracking performance.Practical conditions such as model mismatch and input saturation were taken into consideration in this paper.Due to model mismatch and input saturation, the original reference model might no longer be appropriate; therefore, we designed an adaptive reference model that also provides satisfactory performance.Simulation results indicated that mismatch estimation, input compensation, and satisfactory tracking performance could be obtained in the presence of failures, model mismatch, and input saturation.Future investigation will be necessary to achieve adaptive fault estimation and test more complex and realistic models.The FTC of multiple flight systems will be explored in our future study.

Figure 1 :
Figure 1: The overall control structure.

Figure 4 :
Figure 4: The adaptive control input   of Case 2.

Figure 8 :
Figure 8: The tracking errors of Case 2.
Design.The matrices  1 ,  2 are designed using the linear quadratic regulator (LQR) method.The reference model is designed based on (4).The quadratic stable region is obtained as   14 and   25 ∈ [−1.1564, 1.1564].The security threshold is arbitrarily set at 80% of the maximum stable region.The corresponding Lyapunov matrix  in (25) is obtained as