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Journal of Control Science and Engineering
Volume 2012 (2012), Article ID 502149, 16 pages
Research Article

Adaptive Control Allocation in the Presence of Actuator Failures

National Institute of Aerospace, Hampton, VA 23666, USA

Received 3 November 2011; Revised 17 February 2012; Accepted 2 March 2012

Academic Editor: Yunjun Xu

Copyright © 2012 Yu Liu and Luis G. Crespo. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


This paper proposes a control allocation framework where a feedback adaptive signal is designed for a group of redundant actuators and then it is adaptively allocated among all group members. In the adaptive control allocation structure, cooperative actuators are grouped and treated as an equivalent control effector. A state feedback adaptive control signal is designed for the equivalent effector and adaptively allocated to the member actuators. Two adaptive control allocation algorithms, guaranteeing closed-loop stability and asymptotic state tracking when partial and total loss of control effectiveness occur, are developed. Proper grouping of the actuators reduces the controller complexity without reducing their efficacy. The implementation and effectiveness of the strategies proposed is demonstrated in detail using several examples.

1. Introduction

Actuator redundancy is highly desirable for fault-tolerant control. This redundancy yields multiple ways to implement the forces and moments prescribed by the controller. However, this freedom creates the need for properly allocating the control inputs among all individual actuators. While multiple actuator configurations do generate the desired forces and moments, some of them may yield unintended outcomes, for example, the effect of some control surface deflections counteracts the effect of other ones. Redundancy management is the need for properly allocating the control inputs among functionally redundant actuators when some of them may not be fully functional.

The purpose of control allocation is to distribute the control signals to the available actuators such that the desired moments and forces are efficiently generated. Traditional control allocation methods include explicit ganging [1], daisy chaining [2], pseudinverse [3, 4], and error and control minimization [510]. Explicit ganging performs control allocation by finding a fixed relation between the desired control moments and forces and the designed control signals. Multiple actuators (e.g., two aileron surfaces) can be combined to generate the desired effects. Daisy chaining allocates inputs in a prioritized fashion. It utilizes the actuators in sequence to generate certain effect. If an actuator is unable to generate such an effect, say due to actuator saturation, the next actuator in the sequence will be used. The pseudoinverse approach, which accounts for input saturation and failure, performs control allocation by solving a linear optimization problem in real time. Error and control minimization is another common control allocation approach. This approach minimizes the error between the desired and generated control moments subject to control constraints. Several approaches can accommodate for actuator failure and saturation compensation, provided that the actuator failure or saturation has been identified. Thus one obvious drawback of these approaches is that they require a fault detection system. These systems, which are usually complex, require the characterization of several failure modes a priori. Furthermore, they may require solving optimization problems in real time; thus they can substantially increase the software and hardware demands of the flight control system.

Adaptive control, on the other hand, does not require knowing which controllers have failed nor the class or severity of the failure. This is due to its ability to change control parameters according to the existing flight condition. Due to parameter adaptation, it is also able to accommodate for parametric uncertainties in the vehicle dynamics. Substantial developments in adaptive control for actuator failures have been made in the last decade [11, 12]. Adaptive control’s ability to seamlessly compensate for actuator failures requires for the system’s built-in actuation redundancy to be sufficient. This is usually described as a rank condition on the input matrix 𝐵 [11, 13]. To take advantage of all redundancy available, one common approach in multivariable adaptive control is to generate a control signal for each control surface. Such an approach endows the controller the maximum degree of freedom to compensate for failure. When certain control surfaces are stuck or have reduced control effectiveness, the remaining control surfaces will adaptively cooperate until a new combination of inputs for the remaining control surfaces is found. This will occur automatically without knowing which surfaces have failed, or when such failures occur. Although adaptive control ensures closed-loop stability and tracking performance, it does not differentiate between control generation and control allocation, and the resulting actuation scheme may be unacceptable. For instance, separately designed control signals for multiple control surface segments may cancel each other’s effects, for example, it has been observed that the steady-state deflection of both rudder segments for a direct adaptive control had opposite angles, resulting in a wings leveled flight with increased drag.

The lack of control allocation in the current direct adaptive control framework motivates this research effort. In this study, we aim at separating control generation from control allocation. In the adaptive control allocation framework, a key step is to combine redundant control surfaces similar to explicit ganging. For each group of combined actuators, we design an adaptive control signal, which is then allocated among group members by an adaptive gain. If no failure occurs, a nonadaptive control allocation scheme set in advance is enforced. In the presence of uncertainty of actuator failure, the adaptive flight controller modifies the allocation of input accordingly.

The structure of the adaptive control allocation framework is illustrated in Figure 1. This is a simple aircraft control example with the elevator controlling the pitch motion. The aircraft longitudinal state, denoted as 𝑥(𝑡), should track the state of a reference system for a given reference input 𝑟(𝑡). The elevator consists of four segments, namely, left outboard, left inboard, right outboard, and right inboard segments. For pitch control, they can be grouped together and considered as an equivalent elevator by adding the four columns of the input matrix. A “virtual’’ elevator signal 𝑣0 is generated by the adaptive controller for the desired pitching maneuver. This elevator signal is then allocated by the adaptive allocation gains 𝛼𝑖(𝑡),𝑖=1,,4. The resulting elevator signals 𝑣0𝑖(𝑡)=𝛼𝑖(𝑡)𝑣0(𝑡),𝑖=1,,4 will be fed to the four elevator segments. The allocation gains can be updated on-line based on the knowledge of the nominal plant and 𝑣0 to mitigate the uncertainties of actuator failures. Conversely, in a fixed allocation scheme 𝛼𝑖 are constant, for example, 𝛼𝑖=1/4, 𝑖=1,,4.

Figure 1: Aircraft pitch control with adaptive control allocation.

One advantage of this adaptive control allocation structure is the ease in solving problems such as the counteracting actuation. To remedy this problem, the signs of the adaptive allocation gains 𝛼𝑖 can be made the same, say, via the projection algorithm [14] so that the allocated control signals cannot go in opposite directions. Another advantage of this structure is the reduction of the controller complexity. For instance, let us consider a state feedback adaptive control design for an 𝑛-state system with 𝑚 controls. The number of controller parameters to be updated is 𝑚×𝑛. If the problem is solved by having all actuators in one group, there are 𝑚+𝑛 adaptive parameters. For example, if we try to stabilize a 5-state system with 3 controls, the total adaptive parameters using a direct model reference adaptive controller would be 15. With a grouping of control inputs for which the system remains controllable (i.e., the system is controllable for the control input matrix associated with the grouping), the total number of adaptive parameters required is 8 (1×5 gain vector and 3𝛼’s). The larger the family of redundant actuators, the bigger the benefit. Another advantage of this approach is that the virtual control signal can be designed using a nonadaptive (fixed) state feedback gain, and failure compensation is achieved by only adapting the control allocation gains 𝛼’s. This implies that the proposed structure could be added to a conventionally designed state feedback controllers without having to use fault detection and isolation.

In this paper, we develop the mathematical foundation of this adaptive control allocation structure for a single group of actuators. Two adaptive control allocation algorithms are presented for both loss of effectiveness and constant-magnitude actuator failures. Technical issues such as design conditions, adaptive law designs, and stability analysis are addressed. The proposed schemes are shown to guarantee stability and asymptotic state tracking in the presence of unknown failures. Simulation-based examples are used to demonstrate the strategy proposed.

The paper is organized as follows. Section 2 presents the adaptive control allocation algorithm for loss of control effectiveness. This is followed by Section 3, where a scheme for constant-magnitude actuator failure compensation is developed in Section 3. Finally, a few concluding remarks close the paper.

2. Adaptive Control Allocation Design for Loss of Effectiveness Failures

2.1. Problem Formulation
2.1.1. Plant

Consider the linear time-invariant (LTI) systeṁ𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝑢(𝑡),(1) where 𝑥𝑅𝑛 and 𝑢𝑅𝑚 are the system state and the control input. The matrices 𝐴𝑅𝑛×𝑛 and 𝐵𝑅𝑛×𝑚 are constant and known. The matrix 𝐵 is the control gain matrix for the group of actuators.

The control signal 𝑢(𝑡) can be expressed as𝑢(𝑡)=Λ𝑣(𝑡),(2) where 𝑣(𝑡) is the allocated control signal (i.e., input to the actuator) and Λ is a piecewise constant uncertain diagonal control effectiveness matrix with Λ=diag{𝜆1,𝜆2,,𝜆𝑚}, and 0<𝜆𝑖1, 𝑖=1,,𝑚. The 𝑖th actuator is fully functional when 𝜆𝑖=1 and has a loss of effectiveness failure when 0<𝜆𝑖<1. For the design in this section, we assume that 𝜆𝑖0, that is, no actuator outage occurs. For the reminder of this section we assume that Λ is always positive definite.

The control input 𝑣(𝑡)=[𝑣1(𝑡),𝑣2(𝑡),,𝑣𝑚(𝑡)] is given by𝑣𝑗(𝑡)=𝛼𝑗(𝑡)𝑣0(𝑡),𝑗=1,2,,𝑚,(3) where 𝑣0(𝑡) is a control signal designed for the group, and 𝛼𝑗(𝑡) is the adaptive allocation gain for 𝑗th actuator. We assume that a nominal allocation scheme has been prescribed. This scheme will be enforced under nominal operating conditions. The nominal allocation gain vector is denoted as 𝛼𝑅𝑚. We also define the equivalent control gain vector as 𝑏0𝐵𝛼. The vector 𝑏0 can be seen as the equivalent control gain matrix for the equivalent control effector representing the actuator group. We further assume that the pair (𝐴,𝑏0) is stabilizable.

2.1.2. Reference Model

For the adaptive control, the desired closed-loop dynamics is given bẏ𝑥𝑚(𝑡)=𝐴𝑚𝑥𝑚(𝑡)+𝐵𝑚𝑟(𝑡),(4) where 𝐴𝑚𝑅𝑛×𝑛 is a Hurwitz matrix, and 𝐵𝑚𝑅𝑛. The signal 𝑟(𝑡)𝑅 is a bounded piecewise continuous reference input, and 𝑥𝑚(𝑡) is the desired state. For a given symmetric positive definite matrix 𝑄𝑅𝑛×𝑛, there exists a unique 𝑃𝑅𝑛×𝑛 that satisfies𝑃𝐴𝑚+𝐴𝑚𝑃=𝑄,𝑃=𝑃>0.(5) For the adaptive control design, we need the following standard plant model matching condition.

Assumption 1. There exist constant 𝐾1𝑅𝑛, 𝐾2𝑅 such that 𝐴+𝑏0𝐾1=𝐴𝑚,𝑏0𝐾2=𝐵𝑚.(6)

2.1.3. Control Objective

The control objective is to design the virtual control signal 𝑣0(𝑡) and adaptive allocation gains 𝛼𝑗,𝑗=1,,𝑚, such that all the closed-loop signals are bounded and the system state 𝑥(𝑡) tracks the desired state 𝑥𝑚(𝑡) asymptotically in the presence of uncertain control effectiveness Λ.

2.2. Adaptive Control Allocation Design
2.2.1. Nominal Controller

The plant model matching condition in Assumption 1 indicates the existence of a nominal controller 𝑣(𝑡) for the system without failures and a nominal constant allocation gain vector 𝛼 such that the closed-loop response is identical to that of the reference model when the responses of any unmatched initial conditions vanish exponentially. This signal takes the form𝑣0(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑟(𝑡)𝜃𝑣𝜔(𝑡),(𝑡)=𝛼𝑣0(𝑡),(7) where 𝜃[𝐾1,𝐾2]𝑅𝑛+1, and 𝜔(𝑡)=[𝑥(𝑡),𝑟(𝑡)]. The above state feedback control 𝑣0(𝑡) together with a prespecified distribution 𝛼 ensures the state tracking error 𝑒(𝑡)=𝑥(𝑡)𝑥𝑚(𝑡) approaches zero exponentially.

2.2.2. Adaptive Controller

When a failure occurs, the allocation gain will be adaptively adjusted to accommodate for failure, but 𝛼 may no longer be effective. For this, we use the adaptive versions of control signal and allocation gain𝑣0(𝑡)=𝐾1(𝑡)𝑥(𝑡)+𝐾2(𝑡)𝑟(𝑡)=𝜃(𝑡)𝜔(𝑡),𝑣(𝑡)=𝛼(𝑡)𝑣0(𝑡),(8) where 𝐾1(𝑡) and 𝐾2(𝑡) are estimates of 𝐾1 and 𝐾2, and 𝜃(𝑡)=[𝐾1(𝑡),𝐾2(𝑡)]𝑅𝑛+1. The updated 𝛼(𝑡)=[𝛼1(𝑡),,𝛼𝑚(𝑡)] is an estimate of 𝛼.

2.2.3. Error Dynamics

With the plant and control in (1) and (2), nominal controller in (7), and adaptive controller in (8), the closed-loop dynamics can be expressed as ̇𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵Λ𝛼(𝑡)𝑣0(𝑡)=𝐴𝑥(𝑡)+𝐵𝛼𝑣0(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡)=𝐴𝑥(𝑡)+𝑏0𝑣0(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡)=𝐴𝑥(𝑡)+𝑏0𝑣0(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡)+𝐵𝛼(𝑡)𝑣0=(𝑡)𝐴+𝑏0𝐾1𝑥(𝑡)+𝑏0𝐾2𝑟(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡)=𝐴𝑚𝑥(𝑡)+𝐵𝑚𝑟(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡),(9) where 𝛼(𝑡)Λ𝛼(𝑡)𝛼(𝑡) and ̃𝜃(𝑡)=𝜃(𝑡)𝜃.

From the closed-loop dynamics in (9) and reference model in (4), the error dynamics can be obtained aṡ𝑒(𝑡)=𝐴𝑚𝑒(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡).(10) It can be seen that the error dynamics in (10) is suitable for adaptive law design in that the latter half of its right-hand side is linear in ̃𝜃(𝑡) and 𝛼(𝑡).

2.2.4. Adaptive Laws

Based on the error dynamics in (10), we can design the adaptive laws for the control parameter 𝜃(𝑡) and allocation gain 𝛼(𝑡) aṡ𝜃(𝑡)=Γ𝜃𝜔(𝑡)𝑒(𝑡)𝑃𝑏0,(11)̇𝛼(𝑡)=Γ𝛼𝐵𝑃𝑒(𝑡)𝑣0(𝑡),(12) where Γ𝜃 and Γ𝛼 are symmetric positive definite matrices and 𝑃 is determined by (5). From the adaptive laws, we can see that the controller parameters of the virtual control signal for the actuator group are updated using the information of the equivalent control gain vector 𝑏0. The allocation gains are updated with the 𝐵 matrix since successful allocation of the virtual control signal to each actuator requires the knowledge of each column of 𝐵.

The properties of the adaptive control allocation scheme can be summarized in the following theorem whose proof is presented in the appendix.

Theorem 1. For the system in (1), the adaptive controller and allocation scheme in (8), and the adaptive laws in (11) and (12) guarantee that the all the closed-loop signals are bounded and lim𝑡[𝑥(𝑡)𝑥𝑚(𝑡)]=0 in the presence of uncertain loss of effectiveness actuator failures in (2).

Remark 2. From the definition of 𝛼(𝑡) in (9), we can see that the adaptation of 𝛼(𝑡) is essential for compensating the uncertainties in Λ, while 𝐾1(𝑡) and 𝐾2(𝑡) can be fixed to their nominal values 𝐾1 and 𝐾2. In this case, with ̃𝜃(𝑡)=0 in (10) and (A.1), the closed-loop stability and asymptotic state tracking results still hold.

2.3. Examples
2.3.1. Linear Plants

Two case studies based on linearized aircraft models are presented next.

Consider the linearized lateral dynamic model of a large transport aircraft flying in a steady wings-level cruise condition with 𝑢=778ft/s [15]. The aircraft model is 𝑣̇𝑥=𝐴𝑥+𝐵𝑢,(13)𝑥=𝑏,𝑝𝑏,𝑟𝑏,𝜙,𝜓𝛿,𝑢=𝑎,𝛿𝑟.(14) The state includes the lateral velocity 𝑣𝑏 (ft/s), roll rate 𝑝𝑏 (rad/s), yaw rate 𝑟𝑏 (rad/s) (all in body-axis frame), roll angle 𝜙 (rad), and yaw angle 𝜓 (rad). The control inputs are aileron 𝛿𝑎 and rudder 𝛿𝑟 deflections (deg). The system matrices 𝐴 and 𝐵 are ,.𝐴=0.12928.328774.9232.14500.0121.44190.9409000.0040.04090.17570.00010010.037200001.000700𝐵=0.05420.46690.04430.02000.00250.03820000(15) Hereafter we will group the aileron and rudder into a single group. Note that these surfaces have related but different functionalities. If adaptive control signals are designed separately for the two control inputs with the state feedback structure, the total updated parameters would be (5+1)×2=12. They include 5 state feedback parameters and 1 feed-forward parameter (for tracking the reference input) for each control surface. The adaptive control allocation scheme, on the other hand, requires 5+1+2=8 parameters since only one adaptive control signal is designed and allocated with two updated allocation gains.

Nominal Parameters
The nominal allocation gain vector is selected as 𝛼=[8.6103,1], thus the equivalent control gain vector is 𝑏0=𝐵𝛼=[0,0.3614,0.0594,0,0]. The nominal allocation gain is chosen so that the first element in 𝑏0, that is, the control gain for the lateral acceleration is zero. The purpose of this choice is to attain a coordinated turn.
The nominal controller is designed based on the LQR approach for (𝐴,𝑏0). The resulting gains are 𝐾1=[]0.8963,22.1655,73.6645,28.8488,4.0825,𝐾2=1.(16)

Reference Model
For this example, the reference model is chosen as the closed-loop dynamics with the above LQR controller, that is, 𝐴𝑚=𝐴+𝑏0𝐾1,𝐵𝑚=𝑏0𝐾2.(17) The reference input to the reference model is chosen as 𝑟(𝑡)=2.1376 for 𝑡0, which leads to a desired state trajectory with steady-state values 𝑥𝑚[]()=0,0,0,0,0.5236.(18) This reference command yields a right turn with a change in the yaw angle of 30 degrees.

Actuator Failure
We will consider the 80% loss in control effectiveness: 𝑢1(𝑡)=0.2𝑣1(𝑡),for𝑡10seconds,(19) where 𝑢1(𝑡) is the output of the aileron actuator and 𝑣1(𝑡) is the designed control input to aileron.

Simulation Results
The following cases will be studied.(i)Case 1: adaptive allocation of the adaptive control designed in Section 2.2 ((11) and (12)).(ii)Case 2: adaptive allocation (as in (12)) of a nonadaptive control signal with fixed gains (as in (16)).

Case 1. The time history of the system state under failure and the desired state is shown in Figure 2. It can be seen that the asymptotic state tracking is achieved after failure.

Figure 2: Time history of plant state (solid) and reference model state (dashed) (Case 1).

Figure 3 shows the designed control signal 𝑣0(𝑡), allocated control signals 𝑣(𝑡) and actuator outputs 𝑢(𝑡). The designed control signal is allocated by the adaptive allocation gains. The actuator outputs are different from the allocated control signals due to the failure.

Figure 3: Time history of control signals and actuator outputs (Case 1).

The adaptive parameters are shown in Figure 4. We can see that 𝐾2(𝑡) also plays an important role in the compensation of actuator failures. The adaptive allocation gains are shown in Figure 5. Both allocations gains adapt immediately after the failure occurs and settle to a new combination of steady-state values that, together with 𝐾2(𝑡) and 𝐾1(𝑡), guarantee state tracking after failure.

Figure 4: Time history of controller parameters (Case 1).
Figure 5: Time history of allocation gains (Case 1).

Case 2. The system state and desired state are shown in Figure 6. Despite the uncertain actuator failure, asymptotic state tracking is achieved. The designed control signal, allocated control signals and the actuator outputs are shown in Figure 7. Similar to the previous case, the allocated control signals and the actuator outputs are not identical due to the actuator failure. The allocation gains are shown in Figure 8. The allocation gains are updated autonomously for failure compensation.
In this simulation case, we have achieved similar results to Case 1 with a slightly degraded transient response (see Figures 2 and 6). A possible explanation for it is that the controller parameters are fixed in this case and cannot contribute to the failure compensation and trajectory tracking as they do in Case 1. However, closed-loop stability and asymptotic state tracking are achieved even though only the allocation gains are adapted. The successful demonstration of failure compensation in Case 2 implies that the proposed adaptive allocation unit can be added to a control loop having any state feedback controller. The added adaptive allocation to the nonadaptive controller ensures closed-loop stability and asymptotic state tracking despite uncertain actuator failures.

Figure 6: Time history of plant state (solid) and reference model state (dashed) (Case 2).
Figure 7: Time history of control signals and actuator outputs (Case 2).
Figure 8: Time history of allocation gains (Case 2).
2.3.2. NASA Generic Transport Model

In this section, we apply the control allocation strategy to the NASA Generic Transport Model (GTM). The NASA GTM is a high-fidelity model of the NASA AirSTAR UAV testbed [16, 17]. The purpose of this example is to show that this adaptive scheme, with the grouping of actuators having different physical functions, can be applied to the nonlinear plant to achieve closed-loop stability and asymptotic tracking.

LTI Model
For this simulation study, we trim and linearize the GTM at a wings-level flight for an aerodynamic speed of 92.09 knots. The same states and controls of (14) are used.

Flight Conditions
As in the previous example, the aircraft is commanded to turn right from the initial wings-level horizontal flight. The turn starts at 10 seconds and at steady state the heading angle will increase 60 degrees.

Actuator Failure
We let the left aileron lose 90% of its effectiveness at 12 seconds. The failure magnitude and its onset time instant are unknown to the controller. The control objective is for the aircraft to achieve an accurate turn in the presence of the failure.

Simulation Results
The simulation results are shown in Figures 914. Figure 9 shows the relevant states of the reference model and of the plant. The states track the reference trajectories asymptotically in spite of the disturbance caused by the failure. The yaw angle is shown to reach 60 degrees accurately. This accurate turning is also shown in Figure 11 by the ground track of the aircraft.
The longitudinal states are shown in Figure 10. There are fluctuations in the longitudinal states since they are not controlled by aileron and rudder.
Figures 12 and 13 show the time history of the virtual control signal, allocated control signals, and actual aileron and rudder deflections. The discontinuity in Figure 13 at the top is a consequence of failure.
Figure 14 shows the control allocation gains and controller parameters. The parameters adjust autonomously after the failure occurs. Note that the allocation gains start to update at the beginning of the turn at 𝑡=10 seconds, before the failure occurs. The reason for this phenomenon is that the allocation gains are improving the tracking performance beyond of what the adaptive gains can do alone.

Figure 9: Time history of lateral states (solid) and reference model states (dashed).
Figure 10: Time history of longitudinal states.
Figure 11: Ground track of the aircraft.
Figure 12: Time history of virtual control signal and actuator inputs.
Figure 13: Time history of actuator deflections.
Figure 14: Time history of allocation gains and adaptive parameters.

3. Adaptive Control Allocation Design for Constant Failures

3.1. Problem Formulation

When constant failures occur, the control signal can be rewritten as [11]𝑢(𝑡)=𝑣(𝑡)+𝜎𝑓,𝑢𝑣(𝑡)(20) where 𝑢=[𝑢1,,𝑢𝑚] is the failure vector whose elements are unknown constants, and 𝜎𝑓 represents the failure pattern, which is defined as𝜎𝑓𝜎=diag1,𝜎2,,𝜎𝑚,(21) with 𝜎𝑖=1 if the 𝑖th actuator has failed, that is, 𝑢𝑖=𝑢𝑖, and 𝜎𝑖=0 otherwise. The failures are assumed to occur instantaneously, that is, 𝜎𝑖 are piecewise constant function of time. An example of such actuator failures is when a control surface (such as the rudder or an aileron) is stuck at some unknown fixed angle at an unknown time instant. This type of failures could be caused by failed hydraulic systems or mechanical linkages.

The plant dynamics can then be rewritten aṡ𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝐼𝜎𝑓𝑣(𝑡)+𝐵𝜎𝑓𝑢.(22) The constant failure 𝑢 introduces an uncertain disturbance that needs to be accommodated for. The control objective for this adaptive control allocation scheme is to design 𝑣(𝑡) to guarantee closed-loop stability and asymptotic state tracking when uncertain constant failures occur.

For adaptive control of constant actuator failures, sufficient built-in actuation redundancy is required. The redundancy condition is described in the following assumption.

Assumption 2. The rank of 𝐵 matrix satisfies that rank[𝐵]=1, and there is at least one operable actuator in the system.

The rank condition characterizes the redundancy of actuation which is necessary for a successful constant failure compensation. This rank condition suggests that the system remains controllable after a failure, and the effect of the constant failure can be properly matched and canceled by the allocated control signals through other columns of 𝐵. Based on this condition, there can be up to 𝑚1 constant actuator failures at any given time.

3.1.1. Nominal Controller

Consider the nominal controller structure𝑣0(𝑡)=𝐾1𝑥(𝑡)+𝐾2𝑟(𝑡)+𝐾3𝜃𝑣𝜔(𝑡),(𝑡)=𝛼𝑣0(𝑡),(23) where 𝜃=[𝐾1,𝐾2,𝐾3] and 𝜔=[𝑥(𝑡),𝑟(𝑡),1]. When there is no failure in the system, 𝐾1, 𝐾2, and 𝛼 are chosen as in (6), and 𝐾3 is set to be zero. In this way, the controller ensures the match between the reference model and the nominal plant without failures. Similar to the previous section, we also define 𝐵𝛼=𝑏0 which is known in advance for controller design.

Next we will show that the controller in (23) attains model matching when failures occur. When there are failures in the system, 𝐾3 cannot generally be zero, and a new set of allocation gains, denoted as 𝛼, may be needed. From (22) and (23) we obtaiṅ𝑥=𝐴𝑥+𝐵𝐼𝜎𝑓𝛼𝐾1𝑥+𝐾2𝑟+𝐾3+𝐵𝜎𝑓𝑢.(24) We assume that at most 𝑝 actuators can fail with 𝑝𝑚1 so that there is at least one operable actuator left. Define an index set for failed actuators as ={𝑖1,,𝑖𝑝} such that 𝜎𝑘=1 for any 𝑘. Then 𝐵(𝐼𝜎𝑓)𝛼 can be expressed as𝐵𝐼𝜎𝑓𝛼=𝑗𝑏𝑗𝛼𝑗,(25) where 𝑏𝑗 is the 𝑗th column of 𝐵 and 𝛼𝑗 is the 𝑗th element of 𝛼.

Based on the rank condition in Assumption 2, we know that the columns of 𝐵 are linearly dependent, that is, for any two columns of 𝐵, 𝑏𝑖 and 𝑏𝑗, 𝑏𝑖=𝑐𝑖𝑗𝑏𝑗 where 𝑐𝑖𝑗 is a constant scalar. Thus we know that the vector 𝑏0=𝐵𝛼, which is the linear combination of all columns of 𝐵, is also parallel to any column in 𝐵. So for each column 𝑏𝑘 in 𝐵, 𝑘=1,2,,𝑚, we can find a scalar 𝑐𝑘 satisfying 𝑏𝑘=𝑐𝑘𝑏0. Therefore (25) can be expressed as𝐵𝐼𝜎𝑓𝛼=𝑗𝑏𝑗𝛼𝑗=𝑗𝑐𝑗𝛼𝑗𝑏0.(26) If   𝛼𝑗, 𝑗 are chosen such that𝑗𝑐𝑗𝛼𝑗=1,(27) then we may obtain𝐵𝐼𝜎𝑓𝛼=𝑏0.(28) One possible choice for 𝛼𝑗, 𝑗 is𝛼𝑗=1𝑐𝑗.(𝑚𝑝)(29) Equation (28) indicates that, under constant failures, the plant model condition in (6) can still be satisfied. Note that for the system without failures𝐵𝛼=𝑚𝑗=1𝛼𝑗𝑏𝑗=𝑚𝑗=1𝑐𝑗𝛼𝑗𝑏0=𝑏0(30) which implies that𝑚𝑗=1c𝑗𝛼𝑗=1.(31) Comparing (27) and (31), we can see that 𝛼 is generally different from 𝛼.

For 𝐵𝜎𝑓𝑢, we can also get𝐵𝜎𝑓𝑢=𝑘𝑏𝑘𝑢𝑘=𝑘𝑐𝑘𝑢𝑘𝑏0𝑑𝑏0.(32)

With (28) and (32), (24) can be expressed aṡ𝑥(𝑡)=𝐴𝑥(𝑡)+𝑏0𝐾1𝑥(𝑡)+𝑏0𝐾2𝑟(𝑡)+𝑏0𝐾3+𝑑𝑏0.(33) By choosing 𝐾3=𝑑, we have 𝑏0𝐾3=𝐵𝜎𝑓𝑢, and (33) can be reduced tȯ𝑥(𝑡)=𝐴𝑥(𝑡)+𝑏0𝐾1𝑥(𝑡)+𝑏0𝐾2𝑟=(𝑡)𝐴+𝑏0𝐾1𝑥(𝑡)+𝑏0𝐾2𝑟(𝑡).(34) Therefore when failures are present, a nominal controller can always be found, with 𝐾1 and 𝐾2 specified in (6), 𝛼 characterized in (27), and 𝐾3=𝑑 ensures that the closed-loop system is stable under constant failures, and the state converges to the desired state 𝑥𝑚 in (4) exponentially.

3.2. Adaptive Control Allocation Design
3.2.1. Adaptive Controller

Due to the uncertain nature of the failures, the controller parameters must be adjusted. Consider the adaptive control allocation scheme𝑣0(𝑡)=𝐾1(𝑡)𝑥(𝑡)+𝐾2(𝑡)𝑟(𝑡)+𝐾3(𝑡)𝜃(𝑡)𝜔(𝑡),𝑣(𝑡)=𝛼(𝑡)𝑣0(𝑡),(35) where 𝐾1(𝑡), 𝐾2(𝑡), and 𝐾3(𝑡) are the estimates of 𝐾1, 𝐾2, and 𝐾3 in (23). The signal 𝜔(𝑡)=[𝑥(𝑡),𝑟(𝑡),1]. With the adaptive controller in (35) and the plant dynamics in (22), the closed-loop system iṡ𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝐼𝜎𝑓𝛼×𝐾(𝑡)1(𝑡)𝑥(𝑡)+𝐾2(𝑡)𝑟(𝑡)+𝐾3(𝑡)+𝐵𝜎𝑓𝑢.(36)

3.2.2. Error Dynamics

For this adaptive control scheme design, we define 𝛼(𝑡)𝛼(𝑡)𝛼, and (36) becomeṡ𝑥(𝑡)=𝐴𝑥(𝑡)+𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝐵𝜎𝑓𝑢=𝐴𝑥(𝑡)+𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝐵𝐼𝜎𝑓𝛼𝑣0(𝑡)+𝐵𝜎𝑓𝑢=𝐴𝑥(𝑡)+𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝐵𝐼𝜎𝑓𝛼𝑣0(𝑡)+𝐵𝐼𝜎𝑓𝛼̃𝑣0(𝑡)+𝐵𝜎𝑓𝑢,(37) where ̃𝑣0(𝑡)=(𝜃(𝑡)𝜃)̃𝜃𝜔(𝑡)(𝑡)𝜔(𝑡).

With (28), 𝐵(𝐼𝜎𝑓)𝛼𝑣0(𝑡) in (37) becomes𝐵𝐼𝜎𝑓𝛼𝑣0(𝑡)=𝑏0𝑣0(𝑡)=𝑏0𝐾1𝑥(𝑡)+𝑏0𝐾2𝑟(𝑡)+𝑏0𝐾3.(38) With (38), (37), and the plant model matching condition in (6), we havė𝑥(𝑡)=𝐴𝑚𝑥(𝑡)+𝐵𝑚𝑟(𝑡)+𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡).(39) From the closed-loop dynamics in (39) and the reference model in (4), we obtain the error dynamicṡ𝑒(𝑡)=𝐴𝑚𝑒(𝑡)+𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡).(40)

3.2.3. Adaptive Laws

From the error dynamics in (40), the following adaptive laws are derived:̇𝜃(𝑡)=Γ𝜃𝜔(𝑡)𝑒(𝑡)𝑃𝑏0,(41)̇𝛼𝑗(𝑡)=𝛾𝑗𝑒(𝑡)𝑃𝑏𝑗𝑣0(𝑡),𝑗=1,2,,𝑚,(42) where 𝑏𝑗 is the 𝑗th column of 𝐵, 𝛾𝑗>0 and Γ𝜃=Γ𝜃>0 are adaptive gains.

The following theorem summarizes the properties of the adaptive control allocation scheme and the proof is provided in the Appendix:

Theorem 3. The adaptive control allocation scheme in (35) with the adaptive laws in (41) and (42) applied to the plant in (22) in the presence of constant failures guarantees that all closed-loop signals are bounded and lim𝑡(𝑥(𝑡)𝑥𝑚(𝑡))=0.

3.3. Example
3.3.1. Linear Plant

Consider the longitudinal LTI model of the NASA GTM given by𝑉̇𝑥=𝐴𝑥+𝐵𝑣,𝑥=𝑇,𝛼𝑎,𝑞,𝜃,𝛿𝑣=elob,𝛿elib,𝛿erob,𝛿erib,(43) where the state includes the true airspeed 𝑉𝑇 (ft/s), angle of attack 𝛼𝑎 (rad), pitch rate 𝑞 (rad/s), and pitch angle 𝜃 (rad). The control inputs are the deflections of the four elevator segments: left outboard elevator 𝛿elob, left inboard elevator 𝛿elib, right outboard elevator 𝛿erob, and right inboard elevator 𝛿erib (deg). The system matrices 𝐴 and 𝐵 are,.𝐴=0.04508.96320.034932.17400.00352.74290.951400.005642.62333.561600010𝐵=0.01100.01100.01100.01100.00120.00120.00120.00120.19620.19620.19620.19620000(44) Obviously, rank[𝐵]=1. For the simulation study, we will include the four elevator surfaces into one group, for which an elevator control signal will be designed and allocated.

Nominal Parameters
The nominal allocation gain 𝛼 is chosen as 𝛼=[0.25,0.25,0.25,0.25] and 𝑏0=𝐵𝛼=[0.011,0.0012,0.1962,0] so the deflection of the four segments will be the same if no failure occurs. The nominal controller is designed using the LQR approach for (𝐴,𝑏0). The resulting gains are 𝐾1=[]0.1494,25.0761,3.2841,27.1293,𝐾2=1.(45)

Reference Model
Similar to the simulation study in Section 2, the reference model is chosen as the closed-loop dynamics of the LQR controller, that is, 𝐴𝑚=𝐴+𝑏0𝐾1,𝐵𝑚=𝑏0𝐾2.(46) The reference input to the reference model is chosen as 𝑟(𝑡)=4.1841 for 𝑡0, which leads to a reference trajectory with steady-state values 𝑥𝑚[]()=10,0.0139,0,0.0110,(47) whose physical meaning is that the aircraft speed is increased by 10ft/s; its angle of attack is reduced by 0.0139 rad; and its pitch angle is reduced by 0.0110 rad.

Actuator Failure
The left outboard elevator is stuck at −5 degrees after 1 second, that is, 𝑢1(𝑡)=5deg,for𝑡𝑡𝑓second,(48) where 𝑡𝑓=1 second. The signal 𝑢1(𝑡) is the output of the left outboard elevator. For 𝑡>𝑡𝑓 the elevator does not respond to the elevator input 𝑣1(𝑡).

Simulation Results
The results are shown in Figures 15 and 16. From Figure 15, we can see that the plant state tracks the reference state after failure. Figure 16 shows the designed elevator signal, elevator inputs, and elevator outputs. The failure occurs at 𝑡𝑓=1 second and the other elevators can be seen to accommodate the failure with the adaptation.

Figure 15: Time history of plant state (solid) and reference model state (dashed).
Figure 16: Time history of control signals and actuator outputs.
3.3.2. NASA Generic Transport Model

Here we apply the adaptive control scheme to the nonlinear NASA GTM.

LTI Model for Controller Design
For adaptive control design, we consider a LTI model for a wings levelled flight having a 3 deg angle of attack at trim. Similar to the linear simulation study, we study the compensation for constant elevator failures and will consider multiple elevator failures in which both the lateral and longitudinal are active. The state and control vectors are given by [],𝛿𝑥=𝑢,𝑞,𝜃,𝑝,𝑟,𝜙,𝜓𝑢=elob,𝛿elib,𝛿erob,𝛿erib,𝛿tl,𝛿tr,𝛿al,𝛿ar,𝛿ru,𝛿rl,(49) where 𝛿elob, 𝛿elib, 𝛿erob, and 𝛿erib are the four elevator segments, 𝛿tl and 𝛿tr are the two engine throttles, 𝛿al and 𝛿ar are left and right aileron segments, and 𝛿ru and 𝛿rl are upper and lower rudder segments. The actuators will be grouped into elevator, engine, aileron, and rudder groups. A virtual control signal is designed for each group and allocated to its members adaptively. The design is based on that in Section 3.2 and is similar to that in the linear simulation study. We include lateral states and lateral actuators to regulate the disturbance of the elevator failures propagated to the lateral dynamics.

Flight Conditions
A set of commands that aim to make the aircraft climb at 4-degree pitch angle is applied at 5 seconds. The nominal parameters and reference inputs are obtained as in the linear simulation example.

Actuator Failures
The right outboard elevator is locked at 10 degrees at 10 seconds and the right inboard elevator at 5 degrees at 20 seconds. The control objective is to maintain the climbing flight in the presence of these failures.
The simulation results are shown in Figures 1723. Figure 17 shows the time history of the longitudinal states and reference model states. It can be seen that the tracking of the desired longitudinal attitude is accurately achieved.
The lateral states are shown in Figure 18. Some disturbances can be observed after the occurrence of asymmetric elevator failures. These disturbances are regulated by the ailerons and asymptotic tracking of the lateral states can be also achieved.
Figure 19 shows the flight trajectory, ground track, and altitude of the aircraft. From the ground track plot, we can see the effect of asymmetric failure on the lateral dynamics. The deviation is corrected by the lateral actuators. Figure 20 shows the lumped elevator signal, allocated elevator signals, and the actual elevator deflections. The elevator failures appear as constant (step signals) on the lower plot, and the designed and allocated elevator signals are shown to accommodate for the failures after their occurrence. The input signals and outputs of other actuators are shown in Figure 21. Note the actuation of the aileron and rudder required to compensate for the asymmetric failure. The time history of the control allocation gains and some selected adaptive parameters are shown in Figures 22 and 23. It can be seen that the allocation gains and controller parameters are updated autonomously to ensure the desired flight.

Figure 17: Time history of longitudinal states (solid) and reference model states (dashed).
Figure 18: Time history of lateral states (solid) and reference model states (dashed).
Figure 19: Flight trajectory, ground track, and altitude of the aircraft.
Figure 20: Time history of lumped and allocated elevators and elevator deflections.
Figure 21: Time history of the inputs and outputs of ailerons, rudders, and engines.
Figure 22: Time history of control allocation gains.
Figure 23: Time history of selected adaptive controller parameters.

4. Conclusions

A novel adaptive control allocation framework is proposed herein. The adaptive allocation scheme includes an adaptive control signal and a control allocation unit with adaptively updated allocation gains. Two adaptive control allocation algorithms have been proposed for the compensation of uncertain failures. The proposed algorithms have been shown to guarantee closed-loop stability and asymptotic state tracking. It has also been shown that the proposed adaptive control allocation framework reduces the controller complexity with proper grouping of the actuators. In this framework, the control signal to be adaptively allocated can be actuated by nonadaptive controllers. The simulation results demonstrate the performance of the proposed algorithms and their applicability to aircraft flight control. Some future research topics in this direction include the extension of the adaptive control allocation framework to systems with multiple groups of actuators and the strict enforcement of a pure allocation, that is, exactly enforcing the designed control signal with 𝛼𝑖=1.


Proof of Theorem 1. When a failure occurs, the control effectiveness matrix Λ has a discontinuity. It would lead to a finite jump in 𝛼(𝑡) as well, which could result in a a finite jump in the error dynamics. Let us assume that there are 𝑙 failures occurring in the system, and the actuator failures occur at time instants 𝑡𝑘, with 𝑡𝑘<𝑡𝑘+1, 𝑘=1,2,,𝑁. For the closed-loop stability and state tracking analysis, we choose the following Lyapunov-like function: 1𝑉(𝑡)=2𝑒1(𝑡)𝑃𝑒(𝑡)+2𝛼(𝑡)Γ𝛼1Λ11𝛼(𝑡)+2̃𝜃(𝑡)Γ𝜃1̃𝜃(𝑡),(A.1) for each time interval (𝑡𝑘,𝑡𝑘+1), 𝑘=0,1,,𝑁 with 𝑡0=0 and 𝑡𝑁+1=. 𝑉(𝑡) is thus discontinuous with finite jumps at 𝑡𝑘, 𝑘=1,,𝑁. Taking the time derivative of 𝑉(𝑡) and substituting the adaptive laws in (11) and (12) into the result for each (𝑡𝑘,𝑡𝑘+1), we obtain  ̇𝑉(𝑡)=𝑒(𝑡)𝑃̇𝑒(𝑡)+𝛼(𝑡)Γ𝛼1̃𝜃̇𝛼(𝑡)+(𝑡)Γ𝜃1̇𝜃(𝑡)=𝑒𝐴(𝑡)𝑃𝑚𝑒(𝑡)+𝑏0̃𝜃(𝑡)𝜔(𝑡)+𝐵𝛼(𝑡)𝑣0(𝑡)𝛼(𝑡)𝐵𝑃𝑒(𝑡)𝑣0̃𝜃(𝑡)(𝑡)𝜔(𝑡)𝑒(𝑡)𝑃𝑏01=2𝑒𝑄𝑒0.(A.2) Thus we can conclude that for each (𝑡𝑘,𝑡𝑘+1), 𝑘=0,1,,𝑁, 𝑉(𝑡) is bounded. Since 𝑉(𝑡) only has finite jumps at 𝑡𝑘, 𝑘=1,,𝑁, we can conclude that 𝑉(𝑡) is bounded for 𝑡[0,), and 𝑒(𝑡)𝐿, 𝛼(𝑡)𝐿, ̃𝜃(𝑡)𝐿, 𝛼(t)𝐿, 𝜃(𝑡)𝐿, 𝑥(𝑡)𝐿, and 𝜔(𝑡)𝐿. Integrating both sides of (A.2), we can obtain 𝑉𝑡+𝑘𝑡𝑉𝑘+1=12𝑡𝑘+1𝑡𝑘𝑒(𝜏)𝑄(𝜏)𝑒(𝜏)𝑑𝜏.(A.3) For 𝑁+1 intervals: [0,𝑡1), (𝑡1,𝑡2), , (𝑡𝑁1,𝑡𝑁), and (𝑡𝑁,), (A.3) holds. Summing both sides of (A.3) for 𝑘=0,1,,𝑁, we obtain 120𝑒𝑡(𝜏)𝑄(𝜏)𝑒(𝜏)𝑑𝜏=𝑉(0)𝑉1𝑡+𝑉+1𝑡𝑉2𝑡+𝑉+2𝑡𝑉𝑘𝑡+𝑉+𝑘𝑡𝑉𝑁𝑡+𝑉+𝑁𝑉()=𝑉(0)+𝑁𝑖=1𝑉𝑡+𝑖𝑡𝑉𝑖𝑉()<,(A.4) because the jumps 𝑉(𝑡+𝑖)𝑉(𝑡𝑖) are finite and the number 𝑁 of jumps is also finite. Thus we have 𝑒(𝑡)𝐿2. We can also conclude from (10) that ̇𝑒(𝑡)𝐿. So from 𝑒(𝑡)𝐿2𝐿, and ̇𝑒(𝑡)𝐿, we have lim0𝑒(𝑡)=0.

Proof of Theorem 3. In Section 3.1, we have assumed that there are at most 𝑝𝑚1 constant actuator failures and defined an index set for failed actuators as ={𝑖1,,𝑖𝑝} such that 𝜎𝑘=1 for all 𝑘. Here we further assume that the failures occur at instants 𝑡𝑘, with 𝑡𝑘<𝑡𝑘+1, 𝑘=1,2,,𝑁 with 1𝑁𝑝. The number of failure instants may be smaller than the total number of failures since multiple failures may happen at the same time. For the stability proof, we choose the following Lyapunov-like function 1𝑉(𝑡)=2𝑒1(𝑡)𝑃𝑒(𝑡)+2𝑖𝛾𝑖1𝛼2𝑖1(𝑡)+2̃𝜃(𝑡)Γ𝜃1̃𝜃(𝑡),(A.5) for each time interval (𝑡𝑘,𝑡𝑘+1), 𝑘=0,1,,𝑁, with 𝑡0=0 and 𝑡𝑁+1=. The time derivative in each time interval (𝑡𝑘,𝑡𝑘+1) is ̇𝑉(𝑡)=𝑒(𝑡)𝑃̇𝑒(𝑡)+𝑖𝛾𝑖1𝛼𝑖(𝑡)̇𝛼𝑖̃𝜃(𝑡)+(𝑡)Γ𝜃1̇𝜃(𝑡)=𝑒(𝑡)𝑃𝐴𝑚𝑒(𝑡)+𝑒(𝑡)𝑃𝐵𝐼𝜎𝑓𝛼(𝑡)𝑣0(𝑡)+𝑒(𝑡)𝑃𝑏0̃𝜃(𝑡)𝜔(𝑡)𝑒(𝑡)𝑃𝑣0(𝑡)𝑖𝛼𝑖(𝑡)𝑏𝑖̃𝜃(𝑡)𝜔(𝑡)𝑒(𝑡)𝑃𝑏0=𝑒(𝑡)𝑃𝐴𝑚1𝑒(𝑡)=2𝑒(𝑡)𝑄𝑒(𝑡)0(A.6) with the fact that 𝐵(𝐼𝜎𝑓)𝛼(𝑡)=𝑖𝛼𝑖(𝑡)𝑏𝑖. Following a similar approach to the stability analysis in Section 2, we can conclude that for any 𝑡[0,), 𝑉(𝑡)𝐿, 𝑒(𝑡)𝐿, 𝑥(𝑡)𝐿, 𝜔(𝑡)𝐿, 𝜃(𝑡)𝐿, 𝑣0(𝑡)𝐿, 𝑒(𝑡)𝐿2. Since 𝑉(𝑡) only includes 𝛼𝑖(𝑡) with 𝑖, we can only conclude the boundedness of 𝛼𝑖(𝑡) and 𝛼𝑖(𝑡) for 𝑖. To show the boundedness of 𝛼𝑗(𝑡), 𝑗, we note that for any 𝑗𝛼𝑗(𝑡)=𝛼𝑗(0)𝑡0𝛾𝑗𝑒(𝜏)𝑃𝑣0(𝜏)𝑏𝑗𝑑𝜏,(A.7) based on the adaptive law in (42). Note that all the columns of 𝐵 are linearly dependent based on the rank condition in Assumption 2. So we can have 𝑏𝑗=𝑐𝑗𝑏𝑘,𝑗,(A.8) where 𝑏𝑘 is the column of 𝐵 that corresponds to an arbitrary healthy actuator, that is, 𝑘, and 𝑐𝑗 is a nonzero constant. Equation (A.7) can thus be expressed as 𝛼𝑗(𝑡)=𝛼𝑗(0)𝑡0𝛾𝑗𝑒(𝑡)𝑃𝑣0(𝑡)𝑐𝑗𝑏𝑘𝑑𝑡=𝛼𝑗𝛾(0)𝑗𝛾𝑘𝑐𝑗𝑡0𝛾𝑘𝑒(𝑡)𝑃𝑣0(𝑡)𝑏𝑘𝑑𝑡=𝛼𝑗𝛾(0)+𝑗𝛾𝑘𝑐𝑗𝑡0𝛾𝑘𝑒(𝑡)𝑃𝑣0(𝑡)𝑏𝑘𝑑𝑡=𝛼𝑗𝛾(0)+𝑗𝛾𝑘𝑐𝑗𝑡0̇𝛼𝑘(𝑡)𝑑𝑡=𝛼𝑗𝛾(0)+𝑗𝛾𝑘𝑐𝑗𝛼𝑘(𝑡)𝛼𝑘(0),𝑗.(A.9) Since 𝛼𝑘(𝑡), 𝑘 has been proved to be bounded, we have 𝛼𝑗(𝑡)𝐿, for 𝑗.
We can further obtain that ̇𝑒(𝑡)𝐿 from (40). With 𝑒(𝑡)𝐿𝐿2 and ̇𝑒(𝑡)𝐿, we can have lim𝑡𝑒(𝑡)=0.


This work was supported by the NRA NNX08AC62A of the IRAC project of NASA. The authors would like to thank Drs. Suresh M. Joshi and Sean P. Kenny at the NASA Langley Research Center for their valuable comments.


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