Robust Finite-Time Tracking for Uncertain Linear Systems with Actuator Faults

Fang, Xinpeng; Fan, Huijin; Liu, Lei

doi:https://doi.org/10.1155/2020/2463790

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Abstract Introduction Preliminaries Conclusions Appendix Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Theoretical and Applied Contributions to Robust Stability Analysis of Complex Systems

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Research Article | Open Access

Volume 2020 | Article ID 2463790 | https://doi.org/10.1155/2020/2463790

Robust Finite-Time Tracking for Uncertain Linear Systems with Actuator Faults

Xinpeng Fang,¹Huijin Fan,¹and Lei Liu¹

Guest Editor: Baltazar Aguirre-Hernandez

Received22 Jul 2019

Revised25 Sept 2019

Accepted25 Oct 2019

Published11 Jan 2020

Abstract

In this paper, a robust finite-time fault-tolerant control (FTC) scheme is developed for uncertain linear systems in the presence of actuator faults. Since the system uncertainties and actuator faults are unknown, the controller parameters are updated online by the adaptive laws without the need for fault detection and isolation. It is proved that the proposed state-feedback model reference adaptive finite-time FTC scheme can guarantee that the tracking error converges to a small neighborhood of the origin in finite time. An application example for an aircraft lateral-directional dynamic system is presented to show the effectiveness of the proposed control scheme.

1. Introduction

In a practical system, e.g., networked control system and flight control system, the actuator component usually suffers from a partial loss of effectiveness (LOE) or even a total loss of control (LOC) due to the increasing complexity of system itself and operating environment [1–5]. See for example, the rudder of an aircraft may experience a loss of gain in the control channel or be stuck in a fixed position. Such actuator faults may degrade the system performance, cause system instability, and even lead to catastrophic accidents. In order to improve system reliability and security, it is significantly important to consider the fault-tolerant control (FTC) problem. Numerous FTC strategies have been proposed which can be classified into two main types: (1) passive method and (2) active one. Passive FTC is actually a robust scheme by designing a fixed controller which ensures the closed-loop system be insensitive to specific preconsidered faults, see for example [6–9]. Unlike the passive control method, the active FTC method can guarantee the stability of the entire closed-loop system by adjusting the parameters or structure of the controller corresponding to the occurred fault. Therefore, compared with the passive control method, the active method is more flexible and practical. Fault detection and diagnosis (FDD) can provide fault information, for example, a sliding mode observer was developed in [10] for detecting and reconstructing actuator and sensor faults. Several active FTC methods based on FDD have been proposed in literatures, such as switching-based design [11], sliding mode control-based design [12, 13], pseudo-inverse approach [14], and model predictive control-based design [15]. It is noted that the performance of these active methods depends heavily on the accuracy of FDD.

On the contrary, adaptive FTC which is a kind of robust technique and a main class of active FTC has been widely used due to its flexibility and diversity in design and its ability to handle unknown actuator faults without the need for FDD modules [16–28]. By considering that the stuck fault is bounded, a robust adaptive FTC scheme was proposed for uncertain linear systems in [16]. In [17], a direct adaptive control scheme was designed to compensate for spacecraft systems with multiple actuator faults and inertia matrix uncertainties. A robust adaptive control strategy based on the generalized restricted potential function was proposed for linear systems in [18] to achieve the desired tracking error norm bound. By employing the adaptive backstepping technique, an adaptive control strategy was presented to deal with stochastic LOE faults for half-car active suspension systems in [19]. In [20, 21], two adaptive fault-tolerant controllers were developed with LOC faults only. By designing the auxiliary systems, the fault-tolerant constrained controllers were designed to achieve FTC in the presence of input amplitude saturation in [22, 23]. Moreover, as a promising adaptive control method, model reference adaptive control (MRAC) has been widely applied to guarantee the desired tracking performance and handle unknown actuator faults and system uncertainties. In [29, 30], state-feedback FTC schemes were presented to ensure the plant state asymptotically tracks the reference model state, and in [31, 32], output tracking property was guaranteed by designing state-feedback MRAC schemes.

It is noted that the aforementioned control laws only guarantee the asymptotic stability of the system, that is, they only promise the stability of infinite time. However, in practice, we expect the desired system performance to be achieved in a finite time. For example, the actual attitude of the aircraft must recover the desired attitude within a finite time once fault occurring. Since finite-time control has a faster convergence rate, better robustness against uncertainties and disturbances, it has drawn a lot of attention. Therefore, designing a finite-time FTC scheme is of great significance. In [33–38], the issues of finite-time stability were investigated for fault-free cases. Two finite-time command filtered backstepping-based controllers were proposed in [33, 34], which guarantee the finite-time tracking property. In order to solve the problem of finite-time FTC, a number of control schemes have been developed in [39–41] and applied to spacecraft attitude control. In [39], a finite-time attitude controller was designed based on sliding mode control technology. A finite-time terminal sliding mode controller was proposed in [40] by using the reconstruction information of the adaptive terminal sliding mode observer. It is worth mentioning that the designed controllers in [36, 39, 40] contain the sign functions, which may cause the undesired controller chattering problem. Therefore, it is more valuable to design the chattering-free finite-time FTC scheme. In addition, due to the inevitable existence of uncertainties, it is necessary and a key challenge to develop control strategies for uncertain systems. In [42], a robust adaptive hierarchical insensitive tracking controller was designed for linear systems with uncertainties in the system and input matrices.

Motivated by the above discussion, this paper will study the finite-time tracking control of uncertain linear systems with unknown actuator faults. The main contributions are given as follows:(1)A robust model reference adaptive fault-tolerant tracking control scheme with finite-time convergence property for linear systems is developed to compensate for time-varying system matrix uncertainty and unknown actuator faults without requiring fault detection and isolation(2)The unknown parameters caused by system uncertainty and actuator faults can be estimated by the designed tracking error-driven adaptive laws, which promises the adaptivity of the proposed controller(3)The proposed control law is chattering-free, which is more practical for engineering applications

The rest of this paper is organized as follows. In Section 2, the problem to be addressed is formulated and some preliminaries are introduced. A robust adaptive finite-time FTC scheme is developed in Section 3. In Section 4, an application to an aircraft lateral-directional dynamic system is presented to illustrate the effectiveness of the proposed scheme. Finally, the conclusion is given in Section 5.

Notations. represents the n-dimensional Euclidean space, while refers to the set of all real matrices. denotes that x is an n-dimensional vector, and means that x is an matrix. denotes the Euclidean norm of a vector or matrix. The superscript stands for matrix transposition. represents a block-diagonal matrix. and denote the minimum and maximum eigenvalues of a matrix, respectively. stands for the m-dimensional identity matrix.

2. Preliminaries and Problem Statement

2.1. Preliminaries

Definition 1 (see [43]). Consider an autonomous dynamic system:where is continuous on an open neighborhood of the origin. The equilibrium of the system is uniformly finite-time stable if it is uniformly Lyapunov stable and for any initial condition where , if there is a settling time function , such that every solution of system (1) satisfies for with the properties: and , .
If , then the origin is a uniformly globally finite-time stable equilibrium.

Lemma 1 (see [44]). For , , and , the following relation holds:

Lemma 2 (see [45]). For a positive definite matrix and any , the following inequality holds:

Lemma 3 (see [36]). For any scalar , matrices K, , , and , the inequality in (4) holds:

Lemma 4 (see [46]). For some constants , , and , the following relation holds:where y and z are real variables.

Lemma 5 (see [36]). Consider the system . If there exists a continuous positive definite function , real numbers , , and , such thatthen the system is practical finite-time stable. The trajectories of the system can reach the set in a finite time , which is defined aswhere and is the initial value of .

2.2. System Model

Consider the linear systems with time-varying system matrix uncertainty described bywhere is the state vector which can be available at every instant and is the control input vector, which means that there are m actuators in the system. and represent the nominal system and input matrices, respectively, where B is full-row rank. represents the time-varying system matrix uncertainty which satisfies the following condition:where is an unknown continuous matrix function with and is an unknown positive constant.

Remark 1. For a practical system, e.g., aircraft system [47] and industrial system [48], it can always be linearized and decoupled into the form of system (8). The full-row rank condition of B is a general requirement for the system, see for example [18, 49], which means that there must be enough control channels in the system, especially for actuator redundancy purpose. Then, the establishment of condition (9) is obvious, since any n-dimensional square matrix can be expressed in the form of (9) by a full-row rank matrix B.

2.3. Actuator Fault Model

The actuator fault model considered in this article is described aswhere denotes the designed control input for the actuator, represents the actuator’s uncontrollable time-varying fault, and are unknown fault indicators for the actuator, and and are piecewise constant functions of time. More specifically(1)The actuator is fault-free when and (2)The actuator is partial LOE when and (3)The actuator corresponds to outage when and (4)The actuator is stuck at when and

For convenience, the actual input vector of system (8) can be formulated aswhere

Remark 2. The compact fault model (11) includes normal (case 1), partial LOE faults (case 2) and total LOC faults (cases 3 and 4), which can represent most of the possible occurrences of an actuator in a practical system.

2.4. Control Objective

The reference model is given aswhere is the reference state vector which can be available at every instant, is the bounded reference input, and is a Hurwitz matrix, that is, there exist positive definite matrices such that

Some assumptions are to be introduced.

Assumption 1. .

Assumption 2. There exist and such that

Remark 3. As discussed in [21], Assumption 1 is an actuator redundancy condition which ensures that the system remains capable to be stabilized even with actuator faults. In other words, the actuator faults satisfying Assumption 1 can be completely compensated by the proposed algorithm. Assumption 2 is a standard state-feedback state tracking matching condition, i.e., [29, 50]. In our scheme, Assumption 2 ensures that system (8) can match reference model (13) through state-feedback, so as to ensure the performance of state tracking. More specifically, according to Assumptions 1 and 2, it can be concluded that there exist and such that

Lemma 6 (see [51]). The rank relation in Assumption 1 is a necessary and sufficient condition for the existence of a matrix such thatOur control objective is to design a robust adaptive finite-time FTC scheme for system (8) in the presence of unknown actuator faults and system matrix uncertainty, such that the finite-time tracking is ensured for the given reference model (13).

3. Robust Adaptive Finite-Time Fault-Tolerant Control Scheme

In this section, a robust adaptive finite-time FTC scheme is developed to achieve the desired state tracking property in the presence of unknown actuator faults and system matrix uncertainty.

3.1. Controller Structure

Denote the tracking error as

Let , , and be the estimates of , , and , respectively. Then, the finite-time FTC scheme in this article is constructed aswhereand will be designed later.

By considering actuator faults (11), substituting (11) and (19) into (8), the closed-loop system can be formulated as

Together with (13), (16), (18), and (21) and using Lemma 6, the tracking error system can be derived as

Lemma 7 (see [16]). For the diagonal matrix σ in (11), there exists a constant , such thatNow, introducing two unknown positive scalars and aswhere is a chosen constant and is the upper bound of .
Then, term in (19) can be designed aswhere is the estimate of unknown constant and and are chosen constants.
Term in (19) is thus designed aswhere is the estimate of and is a chosen positive constant.

Remark 4. The designed controller (19) consists of six items: , , , , , and . is inherited from the typical state-feedback model reference controller structure. is the estimate of given in (17), which is to eliminate the effects of the stuck fault . is designed especially for stability analysis purpose. is actually an exponential term which is related to a finite-time control. is designed to eliminate the effects of uncertainty.

3.2. Adaptive Laws

To construct , for , the adaptive laws to update , , , , and are chosen aswhere represents the ith column of B, , and are chosen constant diagonal positive definite matrices, and and are chosen constants. The initial values of and are positive, that is, and .

So far we have obtained the state-feedback model reference adaptive finite-time FTC scheme. For clarity, it is summarized in Appendix A.

3.3. Stability Analysis

Theorem 1. Consider the linear systems described by (8) with unknown actuator faults (11) and system matrix uncertainty, if Assumptions 1 and 2 are satisfied, the control signal in (19) updated by the adaptive laws (27)–(31) can guarantee that the tracking errors converge to a small neighborhood of the origin in finite time.

Proof. See Appendix B.

4. Application: Aircraft Attitude Control

In this section, simulation studies on a lateral-directional dynamic model of the F-18 high-angle-of-attack research vehicle (HARV) system [47] are given to illustrate the effectiveness of the proposed robust adaptive finite-time FTC scheme. To present the superiority of the proposed control scheme, the developed controller (19) is compared with the one given in [28].

4.1. F-18 HARV Model

The state variables for the linear lateral-directional dynamic model of the F-18 HARV system are side-slip angle β (), roll rate p (), and yaw rate r (), that is, . The control input vector , where , , , , and represent the differential tail deflection (), aileron deflection (), rudder deflection (), roll thrust vector deflection (), and yaw thrust vector deflection (), respectively.

As in [47], the linear plant is described by (8) with

The time-varying matrix is given as

4.2. Simulation Scenarios

The reference model is chosen as (13) with

To demonstrate the effectiveness of the proposed adaptive finite-time FTC scheme, the following actuator faults are simulated:

In (35), denotes the differential tail deflection loses its effectiveness; denotes that the second actuator corresponds to outage, that is, the aileron deflection stuck at the fixed value ; and denotes the roll thrust vector deflection cannot be influenced by control action and stuck at the fixed value . Table 1 shows the fault indication matrices σ and ρ in different time intervals.

The initial conditions of the proposed controller (19) are given as follows:

The reference input signal . In order to obtain a good FTC effect of the designed controller (19), after repeatedly debugging, the controller parameters are chosen as

Although the parameters are chosen by trial and error, there are still some rules to be referred to. The larger the adaptive gains and in (27) are, the more drastic the change of is, which leads to stronger control effect. Similar phenomenon lies in (28)–(31). However, too-large adaptive gains will affect the smoothness of the controller. and have direct impacts on , which means that they will affect the strength of the controller directly. is a constant relating to the smoothness of the controller, while a large will lead to a large and thus reduce the finite-time convergence accuracy. If α chosen by the designer is too small, then will exhibit a fast rate of change when the tracking error turns small, which is not expected in a practical system. In sum, to obtain a more practical FTC performance, the designers should choose the parameters with a compromise among the convergence rate, convergence accuracy, and the amplitude and smoothness of the control signals.

Through trial and error, the initial conditions and parameters of the controller given in [28] are chosen as follows to obtain a good FTC effect:

4.3. Simulation Results

Figure 1 shows the plant state and reference model state by using the proposed controller (19) and the one given in [28]; Figure 2 shows the state tracking error by using controller (19) and the one given in [28]; Figure 3 shows the actual control signal and designed control signal of controller (19); and Figure 4 shows the actual control signal and designed control signal of the controller given in [28].

By using the proposed controller (19), it can be seen that (1) during , there is only system matrix uncertainty and no fault, and the tracking performance can be guaranteed in finite time; (2) whenever a new fault occurs at 10, 30, and 70 s and the fault recovers at 50 and 90 s, after the transient response, the tracking performance recovers. Similar conclusions can be obtained when using the controller given in [28], while it only guarantees the asymptotic tracking property, that is, it only promises the stability of infinite time. It is obvious that the proposed robust finite-time FTC scheme possesses a better transient response and faster convergence rate.

Summarizing all the above results, it is clear that the proposed robust adaptive finite-time FTC scheme can effectively ensure the state practical tracking property in finite time in the presence of unknown actuator faults and system matrix uncertainty.

5. Conclusions

In this article, a state-feedback model reference adaptive finite-time FTC scheme has been developed for linear systems in the presence of unknown actuator faults and system matrix uncertainty. The designed robust controller updated by the adaptive laws ensures that the tracking errors converge to a small neighborhood of the origin in finite time. Finally, the performance of the proposed control scheme is illustrated by simulation results for a lateral-directional dynamic model of the F-18 HARV system.

Our future work will focus on the control problem for the system under practical constrains, such as actuator amplitude saturation, rate limitation, and input matrix uncertainty.

Appendix

A. The Control Algorithm

The proposed control algorithm is summarized as follows: Step 1. Give the initial state , , the reference model input , and initial values of parameter estimates , , , , and . Step 2. For , choose the following adaptive gains:and constants Step 3. Estimate the unknown parameters by adaptive laws: Step 4. Obtain the controller:where

B. Proof of Theorem 1

Proof. Construct the Lyapunov function aswhere , , , , and .
The time derivative of V can be derived as follows:Now, discuss the term in two cases.

Case 1. If , by using Lemma 7, it has

Case 2. If , due to the fact of B being full-row rank, for positive definite matrix P, it follows that is full-row rank, that is, in this case , hence,Combining the above discussions, we getSubstituting (20), (26), and (B.5) into (B.2), we obtainAccording to the definition of and in (24), it hasThen, using Lemma 7 and substituting (B.7) and (B.8) into (B.6) yields thatSubstituting (14) and adaptive laws in (27)–(31) into (B.9) yieldsThen, according to the inequalityit hasThen, substituting (B.12) and (B.13) into (B.10), we obtainFrom Lemma 3, for any and , one hasNow, substituting (B.15)–(B.19) into (B.14), we haveBy applying Lemma 4, letit haswhere .
Similarly, we can conclude thatSubstituting (B.22)–(B.26) into (B.20) and applying Lemma 2, we havewhereFurthermore, according to Lemma 1, one hasTherefore, according to Lemma 5, the decrease of V can drive the trajectories of the closed-loop system into in a finite time withwhere .
This completes the proof.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This work was supported by the National Natural Science Foundation of China under grant nos. 61573161 and 61873319.

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Copyright © 2020 Xinpeng Fang et al. 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.

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