Abstract
A comprehensive adaptive compensation control strategy based on feedback linearization design is proposed for multivariable nonlinear systems with uncertain actuator fault and unknown mismatched disturbances. Firstly, the linear dynamic system is obtained through nonlinear feedback linearization, and the dynamic model of the mismatched disturbances as well as its relevance to the nonlinear system is given. The effect of disturbances on the system output is suppressed with the basic controller of the linearized system. Then, a direct adaptive controller is developed for the multiple uncertain actuator faults. Finally, an integrated algorithm based on adaptive weighted fusion could provide an effective compensation for the effect of multiple uncertain faults and mismatched disturbances. Thus, the stability and asymptotic tracking performance of the closed-loop system are ensured. The feasibility and performance of the proposed control strategy are validated by the numerical simulation results.
1. Introduction
Actuator faults are common in performance-critical systems. The occurrence of faults will cause severe deterioration in performance or even catastrophic problems of system instability. Actuator faults are featured with multiple essential uncertainties, including the fault mode, time, value, and type. Therefore, it is necessary to develop the effective fault-tolerant control technology to address the problem associated with the multiple uncertainties of actuator faults, so as to sustain reliability and safety of the closed-loop system.
In recent years, the problem of actuator faults compensation control has attracted more and more attention. A variety of control methods are tested with several profound achievements. Many effective fault-tolerant control methods were reviewed in literatures [1–5]. Multimodel adaptive control methods were employed as a fault compensation in literatures [6–8]. Literatures [9–11] applied neural network to the design of reconfigurable aircraft control in the case of sensors or actuator faults. The fault recognition and fault-tolerant control strategies of the near space vehicle are designed base on the adaptive sliding mode control method in reference [12]. For the spacecraft attitude control system with external disturbances, two kinds of effective fault-tolerant control method were proposed in literature [13]. To enhance the overall performance of the multisensor measurement system and reduce the influence of faults of each sensor on the system, a new multisensor information fusion design framework was proposed in reference [14]. Fault detection and diagnosis methods are also widely used to for the problems of component faults in the control system [15]. In literatures [16, 17], the adaptive observer design was used to reconstruct actuator faults and a fault-tolerant controller was designed based on estimated information for fault. Besides, adaptive control is also an effective tool with widespread application in fault-tolerant control for both linear and nonlinear systems [18–21]. Although great practical progress has been made in actuator fault compensation for the nonlinear system, there are still many unresolved problems for control system with uncertain dynamics and actuator faults. For example, the problems of multiple-actuator fault compensation control in the general nonlinear system can be further investigated to improve closed-loop system stability and asymptotic tracking control.
The so-called feedback linearized system refers to a kind of nonlinear system linearized by appropriate nonlinear feedback control [22]. Based on the feedback linearization, the control objectives such as models match, pole assignment, and tracking can be further realized. References [23, 24] combined feedback linearization theory with adaptive control and effectively solved the parameter uncertainty and fault-tolerant control problems of nonlinear systems. In addition, the performance of the controlled system suffers from quite different influences due to the various disturbances during the actual operation of the nonlinear system. Therefore, the disturbance suppression problems should be given adequate attention. In literatures [25–27], disturbance decoupling for the measurable disturbances in linear systems provides a potential approach for disturbance suppression problems. However, this method is not suitable for nonmeasurable disturbances. The robust control method is proposed for the nonmeasurable disturbances in literatures [28, 29], without implementation for control objective of asymptotic tracking. The disturbances suppression method based on adaptive control design can effectively estimate the unknown system parameters and disturbance parameters. In literature [30], the adaptive internal model control method was applied in the spacecraft system to realize the attitude tracking with external disturbances. For general hypersonic vehicles with uncertain system parameters and external disturbances, a new sliding mode control method was proposed in literature [31]. The problem of asymptotic tracking of nonlinear systems under sinusoidal disturbances was investigated in literature [32]. And a disturbance suppression algorithm was proposed for single-input single-output nonlinear systems, but the algorithm is inappropriate for multi-input multioutput nonlinear systems with mismatched disturbances. In addition, the suppression of mismatched disturbances in multi-input and multioutput nonlinear systems were studied in literatures [33–35].
Unknown disturbances and uncertain actuator faults may occur simultaneously in the actual operation, which increases the difficulties in asymptotic tracking control for multi-input and multioutput nonlinear systems. Although some theoretical achievements have been made in disturbance suppression and actuator fault compensation for multi-input and multioutput nonlinear systems, some critical problems are left open. The problem of unmatched disturbance suppression in nonlinear systems with uncertain multivariable is solved in literature [35]. On this basis, the problem of multiple uncertain actuator fault compensation and mismatched input disturbance suppression is further studied in this paper for the case of a feedback linearized multivariable nonlinear systems. Compared with some available fault-tolerant control methods, the currently proposed control method presents the following improvement: (1) a new adaptive actuator failure compensation and disturbance rejection scheme with relaxed design conditions is designed for general multivariable nonlinear systems; (2) a new composite fault-tolerant control approach is developed to handle a set of uncertain actuator failures, by using a complete parametrization for estimation of both the failure pattern parameters and the failure value parameters; (3) an adaptive disturbance rejection scheme is developed in details, including error equations, adaptive laws, and stability analysis, for multivariable nonlinear systems with uncertainties from both the actuator failure and unmatched disturbances, such that desired closed-loop performances are ensured including signals boundedness and asymptotic output tracking; and (4) an important aircraft flight control application is conducted.
2. Problem Description and Knowledge Preparation
This chapter first describes the problem of actuator fault compensation and disturbance suppression of the systems with redundant actuators and then introduces some basic concepts involved in this paper.
2.1. Control Problem Statement
Consider the nonlinear system as below where is state vector, is system output, is system input, and is the uncertain external disturbance. , , and are known.
2.1.1. Actuator Fault Model
The classical model of the actuator fault can be represented as [19] where , , ¯ and represent the parameters of the uncertain fault. are known. The fault model (3) is written in the following parameterized form where , When the uncertain actuator fault occurs in the system, the actual input acting on the system can be expressed as where is the control input signal to be designed. . is the corresponding actuator fault mode matrix. If the actuator fails, then ; otherwise, . Considering actuator fault (5), the system model can be expressed as
2.1.2. External Disturbance Model
The disturbance term in this paper has the following characteristics: (1) indicates that the disturbance signal is incompatible with the control signal ; (2) the component of the disturbance vector can be expressed as [36]: and it also can be rewritten in the parameterized form as where , , , and are unknown while are known. By selecting appropriate and basic function , the disturbance model (7) can offer an approximate description for many practical disturbance signals, such as constant value, sinusoidal signal, and nonsinusoidal time-varying disturbance.
Remark 1. When the disturbance is consistent with the control input, i.e., and , the control signal can be derived as , where is the basic control variable that can stabilize the nonlinear multivariable system, and is the disturbance suppression component. Without such match, i.e., and , the above control method cannot eliminate the influence of disturbances. Therefore, a new control input needs to be designed to suppress disturbance.
2.1.3. Control Objective
For system (1) with uncertain actuator faults (3) up to and unmatched external disturbance , the number of the faults depends on the actual application. In this paper, . That is, the total actuator faults are no more than , but it is impossible to identify in advance the exact amount of faults. The actuator fault compensation method designed in this case can be applied to the problem of simultaneous or alternating faults of multiple actuators. The mathematical expressions of the corresponding fault modes are
In this paper, a fault compensation control algorithm is developed based on the following assumptions to achieve the above control objectives.
Assumption 2. When at most one actuator of system (1) fails and the fault information is available, it is still possible to design effective control methods to adjust the residual actuators adaptively so that the system still fulfills the desired control objective.
The goal of this paper is to design an adaptive controller to solve the issue due to multiple uncertainties of faults and disturbances, especially the uncertain fault mode, in order to guarantee the stability of the closed-loop system and asymptotical tracking performance of system output.
2.2. Feedback Linearization
For a multi-input and multioutput nonlinear system where , .
Assumption 3. Supposing the correlation vector as in a neighborhood at , if for , , , , , and for .
Similarly, for a nonlinear system with input disturbances where , , and it has a correlation set , . The disturbance suppression design of the multivariable nonlinear system in this paper involves the following assumption:
Assumption 4. If , then ; If , then .
2.2.1. Strict Feedback Linearization
If , the system (1) can be transformed into a strict feedback subsystem through strict feedback linearization and differential homeomorphic mapping , where
Then the dynamic equation becomes where , , , and are the th row of , , and , respectively, , , with
The system output is expressed as
2.2.2. Partial Feedback Linearization
If , only partial feedback linearization can be carried out in system (1) by means of coordinate transformation within the neighborhood of . Supposing is a smooth function with the following form
Literature [24] indicates that there is always smooth mapping To form the differential homeomorphism , , , system (1) is converted into where , , , have the same definition with the equation (13). , is the unmatched disturbance. , , and . The system (19) is termed as the zero dynamic form of the multivariable nonlinear system (1).
2.2.3. Nonlinear Feedback Control Law
Based on Assumptions 3 and 4, if the system parameters and fault parameters of nonlinear system (1) are accessible, feedback linearization design can be used to design an ideal controller. By taking derivatives of in system (1), we can obtain the following equation: where . We can further obtain When and assuming is nonsingular in , the control input signal could be rearranged as
The linearized system can be obtained where is the linear feedback control law to be designed.
2.2.4. Linear Feedback Control
The control law from the linearized system provides the possibility to guarantee the output tracking performance of the system. The control law is where . With substitution of and equation (24) into equation (23), the dynamic equation of the tracking error is obtained as
By selecting appropriate value for , , becomes the Hurwitz polynomial. The output error and its higher derivative asymptotically approach to zero as . If is bounded, then boundedness could be expected for .
Remark 5. If for all , , then , in equation (26), i.e., , . Thus, equation (21) can be simplified as
Linearized system becomes disturbance-free, and disturbance suppression is unnecessary in the basic feedback control system. In addition, in combination with equation (24), equation (21) can be further expressed as , where can assume the following simplification
Remark 6. If , , then in equation (20) is related to the disturbance term and its differential term , and could be expressed as a function of the differential term . In this case, in order to achieve disturbance suppression and asymptotic tracking control, it is necessary to acquire the differential information of the disturbance in advance. However, the derivation process is rather complicated. Therefore, such situation is not considered in this design.
3. Actuator Fault Compensation and Disturbance Suppression Design
If the relevance of the system satisfies , the system with uncertain actuator fault can be linearized by strict feedback and converted into
Based on , in equation (24), the control signal of the system (27) could be determined through nonlinear feedback, if
The control signal can guarantee asymptotic output tracking, i.e., . With occurrence of uncertain actuator fault, the control input signal could be calculated according to equation (28).
3.1. Adaptive Disturbance Suppression Design
The control signal and the feedback linearization are determined in this chapter. The detailed derivation includes adaptive controller, error equation, parameter adaptive updating law, and stability analysis.
3.1.1. Adaptive Feedback Linearization Design
In the disturbance model , are known functions while are unknown parameters. The unknown parameters could be estimated with in the disturbance suppression design, where is the estimate of the disturbance parameter , . Based on the estimation, the adaptive linear control law is obtained as where is the estimated value of , and its estimated component is
3.1.2. Error Model
Let , , , , , , , . Combining the system output in equation (15): , , , and , one can obtain , , , , , , , . And the state error equation of the multi-input multioutput system is calculated by where , , and ,
3.1.3. Adaptive Laws
Based on error system (31), an adaptive law is incorporated to update unknown disturbance parameters . Lyapunov function is designed in following form where adaptive gain matrix , is positive definite symmetric matrix and satisfies the following equation where . Taking the derivative with respect to gives where , , are the components of , . Design control equation is given by and the adaptive law of the parameter is
With a substitution into equation (33), the following could be obtained
So it is ensured that , . The stability of the closed-loop system can be determined from the negative definition of and , . It indicates that a desired performance is achieved with the control system.
3.2. Adaptive Fault Compensation Control Design
Supposing the fault information (fault mode, fault value, and fault time) is known. Two ideal controllers and are designed for the two cases (without fault and actuator fault). Through weighted fusion design, an integrated controller is obtained, which can deal with the simultaneous coexistence of two fault modes mentioned above.
3.2.1. Fault Free Condition
In this case, , the control equation (28) is . By selecting an appropriate , the following equation could be satisfied
By solving the equation , we could obtain the following equation where .
3.2.2. Fault Condition
In case of fault condition, , , , , , where , , where , by selecting the appropriate matrix equation , one could have . And the equation could be solved
The ideal controller under this condition is
3.2.3. Integrated Control Law
The fault index function is defined as
With a weighted fusion of controller and , an ideal integrated controller structure is achieved. where
3.2.4. Adaptive Controller Structure
From equation (44), the structure of adaptive controller can be deduced as where
and are the estimated value of and , , , , , .
Remark 7. As the number of increase, the parameters of the actuator failure (including the parameters of failure indicator function and , failure model also increases. In our proposed actuator failure compensation design, all the unknown parameters will be estimated multiple ( or ) times based on , , , , . With the development of science and technology, the computers have become more advanced, the computation complexity can be solved effectively.
3.2.5. Error Equations
Equation (21) could be rewritten as
To obtain the output error and the parameter estimation error , , the dynamic formula between and is reformulated as
If where , , .
If , equation (49) can be expressed as
The state error equation can be obtained from equations ((50), (51)). where , is the th component of , , .
3.2.6. Adaptive Laws
Based on state error equation (52), the adaptive laws could be derived with projection algorithm, parameters , , and , , are estimated as where , and are the adaptive gains, is the projection algorithm. Consequently, according to adaptive laws , we can derive that and
3.2.7. Performance Analysis
(I) For time period , , . The Lyapunov function is defined as
Combining equations ((37), (50), (51), (52), (53), (54), (55)) one would have the derivative of
Thus, it can be proved that the designed adaptive controller and its parameter adaptive laws could ensure the desired system performance under the free fault condition, i.e., , , , , and are all bounded, and the output error asymptotes to zero as time going on.
(II) If actuator has faults in time period (), i.e., , the Lyapunov function is defined as
Combining equations ((37), (50), (51), (52), (53), (54), (55)) gives the derivative of ,
The above equation indicates that , , , , , and , are bounded when actuator is failed. In addition, the adaptive projection algorithm can ensure . Thus, it can be verified that with increase in , the closed-loop system is stable and the output asymptotically approaches zero: . In conclusion, the following theorem can be obtained.
Theorem 8. For the multivariable nonlinear system (1) with potential uncertain actuator fault (3) and mismatched disturbance , controller (45) and its parameter adaptive laws can ensure the closed-loop system stability and asymptotic tracking output: , ifand the equivalent control matrix in uncertain fault condition has full rank in the domain (definition is ).
3.3. Fault Compensation Design of Zero Dynamic System
If , there is differential homeomorphism
The nonlinear system with uncertain actuator fault , is converted into and zero dynamic subsystem where , definitely exists and is nonunique. , is related to the fault mode .
3.3.1. Stable Zero Dynamic Assumption
To ensure the stability of the closed-loop system and output asymptotic tracking reference signal , the differentials of , of are bounded and piecewise continuous. In this paper, the controller is developed based on the following assumption:
Assumption 9. The nonlinear system (1) still belongs to the minimum phase system under condition of centralized arbitrary fault, which is considered as the fault mode of this paper. That is, with input of , and , the zero dynamic subsystem given by could guarantee input state stability.
Remark 10. Based on Assumption 9, if in any fault case, the state , fault signal , and the designed feedback control signal are all bounded while is bounded disturbance. According to the input state stability condition of the zero dynamic system, is bounded. Combined with the performance analysis results in Section 3.2, it can be inferred that the nonlinear feedback control signal designed in this paper is bounded.
Combined with Assumption 3, the signal of adaptive fault compensation designed for the partial feedback linearization system (18) is similar to that for full feedback linearization system in Section 3.2. The detailed derivation is not rendered. The closed-loop system has the following desired control performance.
Theorem 11. Based on the input state stability condition of zero dynamic (Assumption 9) and the equivalent control matrix in uncertain fault with row full ranks in domain of U, the adaptive controller (45) and its parameter adaptive law can achieve desired stability for closed-loop system (3) and asymptotic tracking output: in the case of multiple uncertain actuator faults (3) and unknown disturbances.
Proof. Assuming one of the actuators failed at time , and the system has no fault during time period , it can be derived according to the performance analysis in Section 3.2 that the estimated parameters , , , and are bounded and the state error asymptotically approaches to zero as tends towards infinity. Boundedness of could be further confirmed from equations ((45), (46), (47)). Input state stability and row full rank constitute an estimation criterion for performance of a closed-loop system in terms of stability and asymptotical tracking capability.
4. Applications in Aircraft Control System
In this section, the proposed control method is applied to the aircraft control system, so that the developed control algorithm could be comprehensively validated. The numerical simulation results show that this method can offer an effective compensation for uncertain actuator fault in the case of gust disturbance.
4.1. Aircraft Dynamics in Turbulent Flow
The research of aircraft dynamic model under turbulence conditions in reference [37] shows that the longitudinal nonlinear dynamic model of the aircraft can be expressed as [38, 39] where is the aircraft speed, is the attack angle, is the angle of pitch, is the pitch rate, is the mass, is the rotational inertia, is the pitch moment, and , , and are turbulence disturbance signals is the dynamic pressure, is the air density, is the density of the wing, is the average chord, and and are thrusters. , , and are given by where and are the two actuators that require fault compensation.
4.1.1. State Space Representation
The state variables , , , and are represented by , , , and , respectively. The input variables , , , and are represented by , , , and . Nonlinear system (1) can be expressed as where , , , , , , , , , and . , , , , , , , , , , , and are known constants.
4.1.2. Control Objectives
For the aircraft control system (68) with uncertain turbulent disturbance and actuator faults, an adaptive fault compensation controller is designed to ensure that the stability of the closed-loop system is satisfied and that the system output could track the desired control instruction . According to Theorem 11, , , , and . The system satisfies Assumption 3 without zero dynamic subsystem after feedback linearization. The following fault modes corresponding to the requirements of fault compensation can be obtained:
4.1.3. Numerical Simulation Conditions
The aircraft parameters in reference [36] are as follows: kg, , , , , , , , , , , , , , , , , , , , , and . The disturbances are given by , , and .
During the simulation verification, the following fault conditions are incorporated: (i) When , the system is in the absence of faults: , ; (ii) When , actuator is stuck: deg, = , ; (iii) When , actuator returns to normal: , ; (iv) When , actuator is stuck: N, , .
4.2. Simulation Results
In the simulation, the parameters of the adaptive controller are , and the other design parameters are as follows: (1)Initial state: (2)The base function in disturbance model (8), initial disturbance parameter, and the adaptive gain are , , and , respectively(3)The base function in actuator failure model (3), initial failure parameter, and the adaptive gain are , , , , , , , , and
Simulation results are shown in Figures 1–3, including a comparison between the actual output of the system and the corresponding reference signal, the tracking error of the system, and the control input signals of four actuators acting on the system in the aircraft.
It can be seen from Figures 1 and 2 that during the actual operation, the designed control algorithm can always fulfill the control objective of system stability and asymptotic tracking, irrespective of normal operation or uncertainties in time, value, or fault model. The results in Figure 3 show that the system has external disturbance and no actuator fault during the period . In the process of the asymptotic tracking of a given instruction, a transient response appears and decreases with time. The robustness of the proposed control method is verified through the results. When the actuator fails at and actuator fails at (shown in Figure 3), the simulation results demonstrate the effectiveness of the proposed adaptive compensation algorithm for both actuator fault and the disturbance. Moreover, the estimates of the adaptive controller parameters , , , , and of , , , and for are shown in Figures 4–8, which indicate that all signals in the adaptive control system are bounded, and the desired performance is met.
5. Conclusions
For multivariable nonlinear systems with multiple uncertain actuator faults and mismatched input disturbances, a control method of adaptive fault and disturbance compensation is proposed in this paper, with the following main conclusions. (1) An adaptive algorithm is adopted to establish the relation, and a set of adaptive fault compensation controllers is constructed based on parameter estimation. Then, a weighted algorithm is used to fuse multiple controllers into a comprehensive controller, so as to solve multiple uncertain actuator faults. (2) Under the condition of uncertain fault, a new parametric design method is adopted to obtain the parameter adaptive law of the fault compensation controller, so that the desired performance of the closed-loop system can be guaranteed. (3) The effectiveness of the proposed theoretical method is verified by the simulation results of aircraft control under fault and disturbance conditions. The problem of fault compensation control for multivariable nonlinear system with known parameters is studied in this paper. (4) The proposed method can be further extended to solve the problem of fault compensation of the system with unknown parameters.
Data Availability
The data (System parameters and Simulation parameters) used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by Changzhou Science and Technology Support Programme (Social Development) of China under grant CE20195027, Jiangsu University Natural Science Foundation of China under grant 18KJB580006, Jiangsu Natural Science Foundation of China under grant BK20170318, and National Natural Science Foundation of China under grant 61903165.