#### Abstract

In this study, an adaptive output feedback fault tolerant control (FTC) scheme is proposed for a class of multi-input and multioutput (MIMO) nonlinear systems with multiple constraints. The neural network (NN) is adopted to handle the unknown nonlinearity by means of its superior approximation capability. Based on it, the state observer is designed to estimate the unmeasured states, and the nonlinear disturbance observer is constructed to tackle the external disturbances. In addition, the Nussbaum function is utilized to cope with the actuator faults, which are coupled with the unknown control directions. Combining with the Lyapunov theory, a NN-based output feedback FTC law is developed for the MIMO nonlinear systems, and the boundedness of all closed-loop system error signals is proved. Simulation results on the unmanned helicopter are performed to demonstrate the effectiveness of the proposed controller.

#### 1. Introduction

During these years, the control issue of multi-input and multioutput (MIMO) nonlinear systems has been drawing great attention due to their generalized and natural description for many practical applications, such as ocean vessel system, mobile robot system, and unmanned aerial vehicle system [1–4]. Since existing linear control methods do not match with the nonlinear systems, many effective nonlinear control approaches have been proposed and considerable results have been received. In [5–9] and the references therein, the adaptive backstepping control and sliding mode control schemes were, respectively, developed for a class of nonlinear systems to achieve acceptable control performance. Nevertheless, as the system structure becomes more and more complex, the unknown nonlinearity caused by nonlinear elements always appears and is worthy of further study.

Nowadays, in order to handle the unknown nonlinearity in the nonlinear systems, numerous control strategies have been presented, such as the neural network (NN) [10–13], extended state observer [14, 15], and fuzzy logic system [16–18]. In particular, the NNs are often combined with other control techniques to deal with the specific nonlinear systems due to their unique properties. In [10, 11], the adaptive backstepping-based neural control schemes were severally proposed to handle the unknown nonlinear functions for the helicopter and near space vehicle (NSV) systems. In [12], the NNs were combined with the model predictive control to overcome the so-called quadratic programming problem for a class of nonholonomic chained systems. In [13], a robust NN sliding mode control law was developed for a two-axis motion control system with unknown nonlinear terms and external disturbances. However, plenty of the above-mentioned literature are based on the available states, which are strict for some practical control systems. Actually, when the sensors malfunction, the signal usually cannot be accurately measured, and the state feedback approach is unreliable. Therefore, the high-quality controller needs to be further designed when the MIMO systems suffer from unmeasurable states.

The state observer is a common method to estimate the unmeasurable states in the academic community [19]. Meanwhile, fruitful and significant results can be found in the recent literature [20–26]. In [20], an adaptive NN output feedback optimized controller was designed for the nonlinear systems with unknown dead zones. In [21, 22], the output feedback control problem for uncertain stochastic nonlinear systems was investigated by employing the state observer approach. In [23], an observer-based finite time controller was developed for a class of second-order nonlinear homogenous systems. Considering the objective existence of performance constraint, the output feedback control schemes were separately presented for the single-input and single-output and MIMO nonlinear systems in [24, 25]. In [26], a stable fuzzy output feedback controller was constructed for a class of nonlinear systems by means of the small-AAIN approach. Nevertheless, it is worth noting that the control directions are usually unknown, and actuator faults occur frequently in many practical application [27, 28]. Therefore, for the purpose of maintaining admissible system performance, more attentions and deeper considerations should be paid to the issues of unknown control directions and actuator faults.

At present, Nussbaum gain theory, which was first proposed by Nussbaum in 1983 [29], has become one of the most valid methods to cope with the unknown control directions. Meanwhile, existing achievements show that the combination of Nussbaum gain control technology with nonlinear control method attains favorable control performance in practice [30–33]. In [30], a distributed consensus controller for uncertain nonlinear systems with unknown control directions was designed to complete control task. In [31], the adaptive protocol was presented for nonlinear systems when both control directions and parameters were all unknown. In [32], a predefined performance-based adaptive controller was proposed for MIMO nonlinear systems in presence of unknown control directions and unknown hysteresis nonlinearities. In [33], an adaptive fuzzy tracking controller was designed to assure the stability of the stochastic nonlinear systems. However, the actuator faults are neglected in the above literature, whose occurrence will worsen the system performance and even lead to instability. Over the years, a quantity of attentions have been focused on fault tolerant control (FTC), and some effective control strategies have been implemented to faulty system. In [34], a robust adaptive sliding mode FTC scheme was proposed for coaxial helicopter to deal with actuator faults. In [35], a fuzzy adaptive nonlinear FTC strategy was presented for hypersonic vehicles with actuator stuck and loss of effectiveness faults. In [36], an adaptive decentralized FTC algorithm was developed for NSV attitude dynamics with actuator faults and control surface damage. Nonetheless, besides the negative factors mentioned above, the time-varying disturbances derived from the outside world also should be further considered.

The problem of robust control has a long history, and a number of disturbance rejection methods have been developed in past decades [15, 19, 37–40]. Especially, the nonlinear disturbance observer (NDO) has been extensively employed in practice since it does not depend on complete information of the disturbance model. In [15], the NDO-based composite fuzzy control issue was investigated when the uncertain nonlinear systems suffered from unknown dead zone. In [19], a NDO-based output feedback control scheme was developed for uncertain nonlinear systems with unknown hysteresis and external disturbance. By combining the NDO and asymptotic tracking control techniques, a composed control approach was proposed for the spacecraft formation flying system under nonzero disturbances in [38]. However, when the unknown nonlinearity, unknown control directions, unmeasurable states, actuator faults, and external disturbances appear simultaneously, the control performance of the MIMO nonlinear system faces severe challenges, and it is of great significance to design high-quality control algorithms.

Motivated by above analysis, the radial basis function neural network (RBFNN), state observer, Nussbaum function, and NDO are combined with the backstepping technique to achieve satisfactory tracking control property. The main contributions can be summarized as follows:(1)Different from the traditional state feedback control [7], an output feedback controller is designed to tackle the unmeasured states and unknown nonlinearity by means of the RBFNN and state observer;(2)Compared with some direct adaptive fault estimation method using projection function [11], the presented Nussbaum-based FTC approach can overcome the singularity problem in a simpler way and reduce the complexity of controller design;(3)The developed output feedback FTC scheme can guarantee satisfactory tracking performance for the MIMO nonlinear systems under multiple negative effects.

The rest of this paper is organized as follows. Problem formulation and preparation knowledge are presented in Section 2. Section 3 derives the main results. Simulation studies on unmanned helicopter are carried out in Section 4. Section 5 draws the conclusion.

#### 2. Problem Formulation and Preparation

Consider a class of MIMO nonlinear systems with actuator faults and external disturbances as follows:where and with being the system state vectors. and denote the input vector and output vector, respectively. defines an unknown smooth nonlinear function. is the unknown nonzero constant control gain matrix. , refers to the actuator fault factor, which describes the unknown remaining control efficiency of actuator. represents external time-varying disturbance. In this paper, it is assumed that only the system output is available for measurement.

*Remark 1. *The considered MIMO nonlinear systems can be employed to describe many practical plants, such as ocean vessel system [2], unmanned helicopter system [4]__,__ and NSV system [11]. Moreover, it is worth mentioning that the control directions are unknown in many application requirements and usually coupled with actuator faults. Hence, for reflecting the system dynamics more practically, the unknown control directions and actuator faults are considered simultaneously in this work.

The control objective of this study is to design a robust adaptive NN output feedback FTC scheme, such that all error signals of the closed-loop system are convergent, and the output can follow the desired trajectory . To this end, the following definition, lemmas, and assumptions are introduced.

*Definition 1. *(see [33]). If a continuous function fulfills the performances as follows:Then is called a Nussbaum function. At present, the functions , , , and have been proven to be the Nussbaum-type functions in [41, 42]. In this paper, is used.

Lemma 1. *(see [33]). Let and be smooth functions defined on , and let be smooth Nussbaum-type function. If the following inequality holds:where , and are suitable positive constants, then, , and must be bounded on .*

Lemma 2. *(see [10]). Owing to the powerful nonlinear approximation capability, RBFNN is frequently employed to approximate any unknown smooth nonlinear function , which can be expressed aswhere is the input vector, is the weight matrix, is the approximation error, and is the basis function vector with beingwhere and denote the center and width of the basis function, respectively.*

Then, the RBFNN (4) can approach any unknown smooth function in the form ofwhere is the optimal weight matrix, is the approximation error satisfying , and is a positive constant.

*Assumption 1. *(see [7]). For the bounded desired trajectory and its derivatives and , there exists an unknown positive constant making holds.

*Assumption 2. *(see [11]). The unknown external disturbance is supposed to satisfy and , where and are unknown positive constants.

*Assumption 3. *(see [11, 33]). The control gain is assumed to have the unknown but same sign with each other, and it satisfies , where is the known constant. Moreover, the fault factor is assumed to be unknown constant and satisfies , where is the known lower bound.

*Remark 2. *For a practical system, the tracking mission should be realizable, and there should exist a feasible controller to achieve it, which means that assumption 1 is reasonable. In addition, assumption 2 is provided to illustrate that if the external disturbance is unbounded, it may result in that the system cannot provide enough energy to accomplish the specified object. Similarly, if the actuator loses too much effectiveness, the whole system may lose the FTC capacity. Therefore, it is reasonable for assumptions 1–3__,__ and they have been extensively used in existing literature such as [4, 5, 7, 16–18, 40–42].

#### 3. Control Design and Stability Analysis

In this section, a backstepping-based robust adaptive output feedback FTC control scheme will be developed for the MIMO nonlinear systems to deal with the unknown nonlinearity, unavailable states, unknown control directions, actuator faults, and external disturbances. The block diagram of the design thread is given in Figure 1.

##### 3.1. Model Transformation and State Observer Design

Since the control gain and fault factor of the actuator are all unknown and coupled with the corresponding control input, it is difficult to design the controller directly. According to the properties of and , it can be obtained that . By defining , it is known that is an invertible matrix. In order to promote the control design, we introduce the new state variable , the new smooth function , and the new disturbance . Then, the MIMO nonlinear system (1) can be transformed into the following system:where and are the new system state vector and output vector.

*Remark 3. *Considering assumption 2 and assumption 3, we obtain that the new disturbance and its first derivative are also bounded. That is, there exist unknown positive constants and such that and . Moreover, we should note that the state variable and the output vector of the new MIMO nonlinear system (7) are all unavailable.

Here, the following RBFNN is adopted to approximate the unknown nonlinear term :where is the designed constant matrix, is the basis function, and is the weight matrix.

Due to the unavailability of the system states , the above function approximations (8) are invalid. Hence, we use the following approximation to design the state observer:where is the estimation of , and is the estimation of .

It is obvious that the approximation depends on the available and . In order to cope with the unknown system states in the MIMO nonlinear systems (7), the following state observer is designed [19]:where with , with , is the estimation of . The corresponding updating laws with respect to and will be given gradually along with the controller design process in the next subsection.

Define , , and . Considering (7–10) and differentiating yieldwhere is the designed diagonal matrix, and is the estimation error of the auxiliary variable , which will be given in the following.

Then, the observation error dynamics (11) and (12) can be rewritten aswhere , , , , , , , , , , is the unit matrix, , .

Here, proper parameters should be selected to ensure that is Hurwitz. In other words, for a given matrix , there exists positive definite matrix such thatDue to the unknown disturbance , the following auxiliary variable is introduced:In light of (15), the NDOs are designed asDefine and . Then, we haveOn the one hand, by invoking (15) and (16), we obtainOn the other hand, by invoking (15) and (17), we obtainConsider the following Lyapunov function candidate:Invoking (13), (14), (19) and (20), one obtainsBased on Young’s inequality, the following inequalities can be obtained:where and are the designed positive constants, , , , , , is the tracking error, which will be given subsequently.

Substituting the above inequalities into (22) produceswhere , .

##### 3.2. Robust Adaptive Neural Output Feedback Control Design

*Step 1. *Define the following tracking errors:where is the designed virtual control law.

Considering (7) and taking the time derivative of yieldBecause of the unknown smooth function , the following RBFNN is used to approximate it:where is the designed matrix, is Gaussian function, and is the weight matrix.

Substituting (27) into (26) followsConsider the Lyapunov function candidate aswhere and are the designed matrixes, .

Invoking (24) and (28), we obtainwhere , .

Design the virtual control function and parameter adaptive laws aswhere , , is the element of , , is Nussbaum-type function, which is chosen as , , , and are designed positive constants.

Considering (31)–(35), the following facts can be obtained:where with being the diagonal element.

Substituting (36)–(38) into (30) giveswhere , , , .

*Step 2. *Considering (10) and differentiating giveIn particular, the dynamic surface control technique is employed to avoid the repeated computation of and achieve its available derivative. Let pass the first-order filter [7].where is the time constant matrix of the filter.

By defining , we havewhere is smooth function vector in regard to . Since the set is compact, the smooth function has a maximum on set for the given initial conditions.

Considering (41), the virtual control law is designed aswhere , is the designed positive constant.

The parameter adaptive law is proposed aswhere is the designed matrix, and is the designed constant.

Invoking (43), we obtainChoose the following Lyapunov function candidate asThe time derivative of isConsidering (44), we obtainSubstituting (39), (45) and (48) into (47), we havewhere , .

*Step 3. *: Considering (10) and differentiating giveSimilar to Step 2, let pass the following first-order filter [7]:where is the time constant of the filter.

Define . Differentiating yieldswhere is a smooth function vector in regard to . Since the set is compact, the smooth function has a maximum on set for the given initial conditions.

Considering (51), the virtual control law is developed aswhere , is the designed positive constant.

The parameter adaptive law is designed aswhere is the designed matrix, and is the designed constant.

Invoking (53), we obtainSelect the Lyapunov function candidate asBy invoking (54), it outputsDifferentiating (56), we obtainwhere , .

*Step 4. *Taking the derivative of yieldsSimilarly, let pass the following filter [7]:where is the time constant of the filter.

Define . Taking the derivative of yieldswhere is smooth function vector in regard to . Since the set is compact, the smooth function has a maximum on set for the given initial conditions.

Design the actual control input and parameter adaptive law aswhere with being positive constant, is the designed matrix, and is the designed constant.

Define the Lyapunov function candidate asDifferentiating outputsConsidering (60)–(63), we obtainwhere is the designed constant.

Substituting (66) and (67) into (65), we have