Adaptive Output Tracking Control for Nonlinear Systems with Failed Actuators and Aircraft Flight System Applications

An adaptive failure compensation scheme using output feedback is proposed for a class of nonlinear systems with nonlinearities depending on the unmeasured states of systems. Adaptive high-gain K-filters are presented to suppress the nonlinearities while the proposed backstepping adaptive high-gain controller guarantees the stability of the closed-loop system and small tracking errors. Simulation results verify that the adaptive failure compensation scheme is effective.


Introduction
Actuator failure compensation has significant impact on control systems, such as aircraft flight systems and nuclear power systems.Due to the unknown failure patterns, times, and values, it has been an important and challenging research problem.Remarkable progress has been made in the area, such as adaptive control [1][2][3][4][5][6][7][8][9][10], sliding-mode based designs [11][12][13][14], switching based designs [15][16][17], fault-detection diagnosis based designs, fuzzy systems based designs, and neuralnetwork based designs.Since adaptive designs use an adaptive controller to accommodate the uncertainties, it has been extensively employed.An adaptive state feedback controller is proposed to guarantee the performance requirements in presence of actuator failures [1][2][3][4].Adaptive actuator failure compensation using output feedback is studied for a class of nonlinear systems [5-9, 14, 16, 18-21].Unfortunately, if the systems have nonlinearities that cannot be bounded by any function of output, the existing methods fail to compensate the control systems.Therefore, new techniques need to be developed.
In this paper, an adaptive output feedback controller for a class of nonlinear systems with actuator failures is discussed.Motivated by the fact that the nonlinearities satisfying the growth condition [15,[22][23][24][25] can be suppressed by a dynamic high-gain output feedback controller, a modified compensation scheme is proposed where adaptive high-gain K-filters and an adaptive high-gain controller are introduced.The main contribution of our paper is as follows.
(1) We relax the condition imposed by Tang et al. in [5][6][7].(2) An adaptive output feedback controller is developed with switching laws [15,16].(3) The parameters of the Kfilters [26] and the controller are adaptively tuned online depending on system nonlinearities and actuator failures.By applying the backstepping technique, the robust controller is recursively constructed step by step.Parameter update laws are addressed to ensure closed-loop signal boundedness and small tracking errors.The simulation results are presented to demonstrate the effectiveness of the scheme proposed in this paper.
The rest of the paper is outlined as follows.In Section 2, the control problem is formulated.In Section 3, a robust adaptive compensation scheme with switching laws via the backstepping design is proposed.In Section 4, the stability analysis is presented.Two detailed simulation examples show the proposed scheme is effective in Section 5. Finally, this paper is concluded by Section 6.
Remark 2. The values of   can change only from   = 1 to some values with 0 ≤   < 1.This means that possible changes from normal to any one of the failure cases are unidirectional.The uniqueness of   indicates that a failure occurs only once on the th actuator.Our objective is to design an output feedback controller for the nonlinear systems (1) with  unknown actuator failures when  changes at time instants   ,  = 1, 2, . . ., , such that the plant output  () tracks a given reference signal   () with up to th-order derivatives bounded as close as possible and that all closed-loop signals are bounded despite the presence of unknown actuator failures and unknown plant parameters.

Adaptive Compensation Control Scheme
The zero dynamics of system (1) with actuator failures are only dependent on the failure pattern [1].For a fixed failure pattern, there is a resulting pattern of zero dynamics.Since the failure pattern  =  1 , . . .,   , 0 ≤  ≤ −1 is unknown, a desirable adaptive design is expected to achieve the control objective for any possible failure pattern.For the closedloop stability, all zero dynamics corresponding to the possible failure patterns need to be stable.To derive a suitable adaptive control scheme, the following assumptions are made.Assumption 3. When TOLE type of actuator failures is up to  − 1, the remaining actuators can still achieve a desired control objective. where ( = 1, 2, . . ., ) . ( Assumption 5.The sign of   * , is known for  = 1, 2, . . ., .
For adaptive actuator failure compensation, the proportional actuation scheme is where V 0 is a control signal generated by a backstepping design procedure to be given in Section 3.3.

Parameterized Model with Actuator Failures.
To obtain a compact form of system (1) with actuator failures, we define Let where  = 0, 1, . . .,  * ,  = 1, 2, . . ., ,  = 1, 2, . . ., .Substituting (9) into (8), we have where We rewrite the system (10) as where Remark 6.It is significant to point that the parametric model ( 12) is very similar to (8.3) and (8.7) in [26] and (38) in [9].However, the system has nonlinearity terms   () which are functions of state vector.In other words, the design procedures in [9,26] are not applicable for the system.In this paper, we will develop adaptive high-gain K-filters to deal with this problem.

State Observation and Switching
Laws.Since the states of system (1) are not available, an observer is needed to provide auxiliary signals for handling the unmeasured states in control design.The conventional K-filters cannot deal with the nonlinearities depending on the system state.Motivated by [15,16,25], we develop adaptive high-gain K-filters which can be expressed as where , and  ≥ 1.As a result, x =  + Ω   is an observer for  with the estimation error which satisfies Remark 7. Since  ≥ 1 and it varies with the system nonlinearities and actuator failures, we call it adaptive highgain K-filters.But  is so complicated that we cannot solve an analytic value.Moreover, when actuator failures occur, the uncertainties will be also brought into the system.In the following, an adaptive controller is proposed which can tune the high-gain parameters online according to system nonlinearities and actuator failures.
Tuning mechanism and switching signal (): motivated by [15,16], we consider the integral of ‖()‖ as where The tuning steps are shown as follows.
Remark 8. We briefly explain the idea of the tuning mechanism with the switching law.First, () is a monotonically nondecreasing function on the interval [ 0 ,  0 + ] by taking integral value of ‖()‖.Obviously,  stops increasing only when ‖()‖ = 0.Under this modified rule, once the system is stabilized, any integral of errors smaller than the prespecified value will not cause further switching.Also, from the switching logic, it is obvious that () is a piecewise constant function of time and increases stepwise with some time interval.

Adaptive Output Feedback Controller.
To prepare for the backstepping procedure, we consider the equation for the output  =  1 : where the regressor  and truncated regressor  are defined as Define a change of coordinates where ρ0 is an estimate of  0 = 1/ 1, * .
Step 1.Consider where  1 is the first stabilizing function and  1 is the first tuning function,  * , Δ 0 are design parameters, and () ≥ 1 is an adaptive tuning function.
Finally, the actual control signal and parameter adaptive laws are, respectively, designed as where  =   > 0 is the adaptive gain.
Theorem 9.The adaptive output feedback control scheme consisting of the controller (25) and the filters (14) along with the parameter update laws (26)

applied to the system (1) based on Assumptions 1-5 ensures global boundedness of all closedloop signals and desired output tracking performance:
for any  2 >  1 ≥ 0, where  > 0,  0 > 0,  0 > 0 are design parameters.The tracking error can be made as small as desired by choosing sufficiently large  0 and .
Note that Substituting (34) and (36) into (33), we have It should be noted that  can be made as small as desired by choosing sufficiently larger .Recalling (39), we can obtain where Starting from the first interval, we conclude that () ∈  ∞ ∀ ∈ [ 0 ,  1 ), so that , θ, and  are all bounded ∀ ∈ [ 0 ,  1 ).From ( 14) and ( 25), it follows that ‖ x‖ is bounded.At time  =  1 , there occur  1 actuator failures, which results in the abrupt change of the parameters.Since the change of values is finite, we have where  1 is bounded.Therefore, it can be concluded from ( 41) that () ∈  ∞ ∀ ∈ [ 1 ,  2 ).By repeating the argument above, the boundedness of all the signals is proved for the time interval ( 1 ,  2 ).Continuing in the same manner, finally we have that ∀ ∈ (  , ∞), () ∈  ∞ and so are the closed-loop signals.
To prove tracking performance, consider the last time interval (  , ∞).From (41), we see that . Since () ∈  ∞ , there exists  0 > 0 such that the desired output tracking given by ( 32) is achieved.This completes the proof.

Examples and Simulations
In this section, we present two examples to illustrate the applications of Theorem 9.
Example 1.The nonlinear longitudinal dynamics of the twin otter aircraft [1,7,27] is used for our actuator failure compensation study.Choosing the velocity, angle of attack, pitch angle, and pitch rate as the states  1 ,  2 ,  3 , and  4 and the elevator angles of an augmented two-piece elevator as the inputs  1 ,  2 , the system can be modeled as where For this application,  3 is considered as the output; that is,  =  3 ,  1 , and the velocity is measured while  2 ,  4 are unmeasured.Apparently, [ 3 ,  4 ]  subsystem can be described as with some unknown parameters  ∈  2 and  = [ 1 ,  2 ]  .
The [ 1 ,  2 ]  subsystem constructs the zero dynamics, which is input-to-state stability [1,7].The unmeasured state  4 exhibits the appearance of a linear form (), which satisfies Assumption 1. Taking advantage of this property, we design adaptive high-gain K-filters to estimate  4 .The control objective is to design an adaptive scheme to control the elevator angles such that the pitch angle  =  3 tracks a reference signal   as close as desired even if one piece of the elevator is stuck at an unknown angle at an unknown time instant.For simulation, we consider two actuator failure cases.
By Theorem 9, we can obtain the actual control law, the high-gain K-filters, and the update laws.The initial conditions are set as The switching law parameters are chosen as The simulation results including output (), reference output   (), and tracking error () are shown in Figure 1, the control inputs  1 ,  2 are shown in Figure 2, the adaptive parameter  and unmeasured state  4 are shown in Figure 3, and the zero dynamics  1 and  2 are shown in Figure 4.The system responses are as expected.When one of the actuators fails, there is a transient response in tracking errors.But as time goes on, the tracking errors become smaller and ultimately vanish.The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters also change their value if necessary.Furthermore, it is easily seen that the unmeasured state  4 and the zero dynamics  1 ,  2 are stable even though there is an actuator failing during operation.
Case 2. We consider the case where  2 (TLOE) fails at the 100th second and  1 (PLOE) loses 50% of its effectiveness from  = 160 s.That is, where The other parameters are the same as those in Case 1.
The simulation results including output (), reference output   (), and tracking error () are shown in Figure 5, the control inputs  1 ,  2 are shown in Figure 6, the adaptive parameter  and unmeasured state  4 are shown in Figure 7, and the zero dynamics  1 and  2 are shown in Figure 8.The system responses are as expected.When one of the actuators fails, there is a transient response in tracking errors.But as time goes on, the tracking errors become smaller and ultimately vanish.The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters also change their value if necessary.Furthermore, it is easily seen that the unmeasured state  4 and the zero dynamics  1 ,  2 are stable even though there is an actuator failing during operation.
Example 2. A rigid-body longitudinal model of a hypersonic aircraft cruising at a velocity of 15 Mach and at an altitude of 110,000 ft [1,11] is considered: where We rewrite the plant (51) as where  1 is the angle of attack,  2 is the pitch rate and unmeasured,  3 is the flight-path angle and unmeasured, and  The control objective for actuator failure compensation of aircraft model is the angle of attack  1 that tracks a reference signal   with a sufficiently small error, while the closed-loop system is stabilized in the presence of an actuator failure in either one of the elevator segments.Since  =  1 is the output, equation ẋ 3 =  10 cos( 3 ) −  1 () constructs the zero dynamics.For simulation, we also consider two actuator failure cases.
Case 1.We consider the case where  2 (TLOE) fails at the 50th second.That is, where  2 = 1,   2 = 0, and  ∈ [0, 50);  2 = 0,   2 = 0.1, and  ∈ [50, ∞).From Theorem 9, we can obtain the actual control law, the high-gain K-filters, and the update laws.The simulation parameters are as follows: (56) The simulation results including output (), reference output   (), and tracking error () are shown in Figure 9, the control inputs  1 ,  2 are shown in Figure 10, the adaptive parameter  is shown in Figure 11, and the adaptive parameter  is shown in Figure 12.The system responses are as expected.When one of the actuators fails, there is a transient response in tracking errors.But as time goes on, the tracking errors become smaller and ultimately vanish.The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters do not change their value.Furthermore, it is easily seen that the unmeasured state  2 is stable even though there is an actuator failing during operation.
Case 2. We consider the case where  1 (TLOE) fails at the 100th second and  2 (PLOE) loses 40% of its effectiveness from  = 150 s: where (58) The other parameters are the same as those in Case 1.
The simulation results including output (), reference output   (), and tracking error () are shown in Figure 13, the control inputs  1 ,  2 are shown in Figure 14, the adaptive parameter  is shown in Figure 15, and the adaptive parameter  is shown in Figure 16.The system responses are as expected.When one of the actuators fails, there is a transient  response in tracking errors.But as time goes on, the tracking errors become smaller and ultimately vanish.The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters do not change their value.Furthermore, it is easily seen that the unmeasured state  2 is stable even though there is an actuator failing during operation.

Conclusions
In this paper, adaptive high-gain K-filters and adaptive highgain controller are applied to handle a class of nonlinear systems uncertainties in the presence of uncertain actuator failures.The adaptive high-gain K-filters can suppress the nonlinearities while the adaptive controller guarantees the closed-loop signals stability and small tracking errors.Simulation results verify the effectiveness of the adaptive actuator failure compensation for desired performance.