Robust Fault Tolerant Control for a Class of Time-Delay Systems with Multiple Disturbances

Abstract

A robust fault tolerant control (FTC) approach is addressed for a class of nonlinear systems with time delay, actuator faults, and multiple disturbances. The first part of the multiple disturbances is supposed to be an uncertain modeled disturbance and the second one represents a norm-bounded variable. First, a composite observer is designed to estimate the uncertain modeled disturbance and actuator fault simultaneously. Then, an FTC strategy consisting of disturbance observer based control (DOBC), fault accommodation, and a mixed 𝐻₂/𝐻_∞ controller is constructed to reconfigure the considered systems with disturbance rejection and attenuation performance. Finally, simulations for a flight control system are given to show the efficiency of the proposed approach.

1. Introduction

To reduce the influence of model uncertainties and system disturbances, there are several control approaches focusing on nonlinear systems with unknown disturbances (see the survey paper [1] and references therein). The methodologies can be mainly classified into disturbance attenuation methods (such as 𝐻_∞ and 𝐻₂ control) and disturbance rejection approaches (such as output regulation theory, disturbance observer based control). The disturbance attenuation approaches have conservativeness for bounded stochastic disturbance. The disturbance rejection methods are established based on model-matching conditions. It has been shown that multiple disturbances exist in most practical systems. The idea of disturbance observer based control (DOBC) is to construct an observer to estimate and compensate some external disturbances [2]. A composite control scheme combining DOBC and PD (proportional derivative) control for flexible spacecraft attitude control was proposed in the presence of model uncertainty, elastic vibration, and external disturbances [3]. For nonlinear systems with multiple disturbances, it has been seen that the 𝐻_∞ and variable structure control have been integrated with DOBC in [4, 5]. In [6], a composite DOBC and adaptive control approach were proposed for a class of nonlinear systems with multiple disturbances. By constructing a disturbance compensation gain vector in the composite control law, a nonlinear robust DOBC was proposed to attenuate the mismatched disturbances and the influence of parameter variations from output channels [7].

In order to increase the reliability and safety of practical engineering, the issues of fault diagnosis and fault tolerant control (FTC) have become an attractive topic and have been paid much attention in recent years (see [8–13] and references therein). It is difficult to accommodate faults if the disturbances and faults exist simultaneously in the controlled systems. In [14], an optimal fault tolerant control approach was proposed for the nonlinear systems, where generalized 𝐻_∞ optimization was applied to estimate the fault and attenuate the disturbances. In [15], a robust observer was proposed to simultaneously estimate system states, faults, and their finite time derivatives and attenuate disturbances; then an FTC approach was designed based on their estimations. For systems with modeled disturbance, a fault diagnosis approach based on disturbance observer was firstly proposed in [16] with disturbance rejection performance. For the nonlinear system with multiple disturbances, [17] addressed a fault tolerant control approach with disturbance rejection and attenuation performances. It is well known that time delay frequently occurs in many practical systems, such as manufacturing systems, telecommunication, and economic systems. Therefore, the problem of fault accommodation for time-delay systems has been a hot topic in the control field. Many important results have been reported in the literature (see [18–21] and references therein). In [18], an adaptive fuzzy fault accommodation control approach was proposed for nonlinear time-delay system. In [19], a fault accommodation was addressed for time-varying delay system using adaptive fault diagnosis observer. [20] dealt with fault tolerant guaranteed cost controller design problem for linear time-delay system against actuator faults.

In this paper, FTC problem is discussed for a class of time-delay systems with actuator fault and multiple disturbances. The first part of multiple disturbances is modeled disturbance formulated by an exogenous system and the second one is norm bounded uncertain variable. A composite observer is designed to estimate the modeled disturbance and time-varying fault. Then, an FTC scheme is addressed with disturbance rejection and attenuation performance by combining fault accommodation and DOBC with a robust 𝐻₂/𝐻_∞ controller.

2. Model Description

In this paper, we consider the following nonlinear system with time-varying faults, time-delay, and multiple disturbances simultaneously: 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥()(𝑥(𝑡[𝑢(𝑡)(𝑡𝑡−𝜏+𝐺𝑔))+𝐽+𝐹)]+𝐽₁𝑑₁(𝑡)+𝐽₂𝑑₂(𝑡)𝑦(𝑡)=𝐶𝑥(𝑡), (1) where 𝑥(𝑡)∈𝑅^𝑛 is system state, 𝑢(𝑡) is control input, and 𝑦(𝑡)∈𝑅^𝑚 represents output variable. 𝐹(𝑡) is time-varying actuator fault to be diagnosed. 𝐴,𝐴_𝑑,𝐶,𝐺,𝐽,𝐽₁, and 𝐽₂ represent coefficient matrices of the system with suitable dimensions. 𝜏 is a known constant delay. The modeled external disturbance 𝑑₁(𝑡) is supposed to be generated by a linear exogenous system described by 𝜔̇(𝑡)=𝑊𝜔(𝑡)+𝐽₃𝛿(𝑡),𝑑₁(𝑡)=𝑉𝜔(𝑡), (2) where 𝜔(𝑡) is state variable, 𝑊∈𝑅^𝑝×𝑝, and 𝑉 and 𝐽₃ are known parameter matrices of the exogenous system. 𝛿(𝑡) is additional disturbance which results from perturbations and uncertainties in the exogenous system. The disturbances 𝑑₂(𝑡) and 𝛿(𝑡) are supposed to have the bounded 𝐻₂ norm.

For a known matrix 𝑈₁,𝑔(𝑥(𝑡)) is a known nonlinear vector function that is supposed to satisfy 𝑔(0)=0 and the following norm condition: 𝑔𝑥₁(𝑡)−𝑔𝑥₂(𝑡)≤𝑈₁𝑥₁(𝑡)−𝑥₂(𝑡) (3) for any 𝑥₁(𝑡) and 𝑥₂(𝑡).

The following assumptions are required so that the considered problem can be well-posed in this paper.

Assumption 1. (𝐴,𝐽) is controllable; (𝑊,𝐽₁𝑉) is observable.

Assumption 2. rank(𝐽,𝐽₁)=rank(𝐽).

Remark 1. In practical engineering, the exogenous model (2) can represent many kinds of disturbances including harmonic disturbance signal caused by vibration, unknown constant load in the motor, inertial sensor drift represented by first-order Gaussian Markov process, and so on. Compared with the previous works [17, 19, 20], both the time-delay and multiple disturbances are simultaneously considered in this paper. Furthermore, the modeled disturbance 𝑑₁(𝑡) and control input are assumed in different channels, while in [17] are in the same channel.

3. Robust Fault Tolerant Controller Design

3.1. Disturbance Observer

In order to reject the modeled external disturbance, disturbance observer should be designed in this subsection. In this paper, we only consider the case of available states. The disturbance observer is formulated as 𝜔(𝑡)(𝑡)(𝑡),𝑑=𝜉−𝐿𝑥₁(𝑡)=𝑉𝜔(𝑡), (4) where 𝜉(𝑡) is auxiliary variable generated by 𝜉̇(𝑡)=𝑊+𝐿𝐽₁𝑉𝜉(𝑡)(𝑡×(𝑡)−𝐿𝑥)+𝐿𝐴𝑥+𝐴_𝑑𝑥()(𝑥(𝑡(𝑡)𝑡−𝜏+𝐺𝑔))+𝐽𝑢+𝐽𝑢_𝑓𝑐(𝑡) (5)𝜔(𝑡) is estimation of 𝜔(𝑡), 𝑑₁(𝑡) is estimation of modeled disturbance 𝑑₁(𝑡), matrix 𝐿 is the disturbance observer gain to be determined later, and 𝑢_𝑓𝑐(𝑡)=𝐹(𝑡) is compensation term to be designed in fault diagnosis observer, where 𝐹(𝑡) is denoted as an estimation of fault 𝐹(𝑡).

By defining 𝑒_𝜔(𝑡)=𝜔(𝑡)−𝜔(𝑡) and 𝑒_𝐹(𝑡)=𝐹(𝑡)−𝐹(𝑡), estimation error system can be obtained from (1), (2), (4), and (5) to show the following: 𝑒_𝜔̇(𝑡)=𝑊+𝐿𝐽₁𝑉𝑒_𝜔(𝑡)+𝐿𝐽𝑒_𝐹(𝑡)+𝐿𝐽₂𝑑₂(𝑡)+𝐽₃𝛿(𝑡). (6)

In the following subsection, we will construct a fault diagnosis observer with disturbance estimation so that the modeled disturbance can be rejected and fault can be diagnosed.

3.2. Fault Diagnosis Observer

The following fault diagnosis observer is constructed to diagnose the time-varying actuator fault: 𝐹(𝑡)(𝑡)(𝑡),𝜂̇(𝑡)𝜂(𝑡)(𝑡(𝑡)=𝜂−𝐾𝑥=𝐾𝐽−𝐾𝑥)+𝐾𝐴𝑥+𝐴_𝑑𝑥()(𝑥(𝑡(𝑡)𝑡−𝜏+𝐺𝑔))+𝐽𝑢+𝐽₁𝑢_𝑑𝑐(𝑡), (7) where 𝐹(𝑡) is estimation of 𝐹(𝑡). The disturbance observer based control term 𝑢_𝑑𝑐(𝑡)=𝑑₁(𝑡) is applied to reject modeled disturbance 𝑑₁(𝑡) by its estimation from disturbance observer. 𝐾 is the fault diagnosis observer gain to be determined later.

The fault estimation error system yields 𝑒_𝐹̇(𝑡)=𝐹̇(𝑡)−𝐹̇(𝑡)=𝐹̇(𝑡)+𝐾𝐽𝑒_𝐹(𝑡)+𝐾𝐽₁𝑉𝑒_𝜔(𝑡)+𝐾𝐽₂𝑑₂(𝑡). (8)

In the next subsection, a composite fault tolerant controller should be determined for reconfiguring the systems with disturbance rejection and attenuation performance.

3.3. Composite Fault Tolerant Controller

In this section, the object is to construct a control approach to guarantee that the system (1) is stable in the presence (or absence) of faults and multiple disturbances simultaneously for the considered time-delay system. The structure of composite fault tolerant controller is formulated as 𝑢(𝑡)=𝑢_𝑠𝑐(𝑡)−𝑢_𝑓𝑐(𝑡)−𝐽^∗𝐽₁𝑢_𝑑𝑐(𝑡), (9) where 𝑢_𝑓𝑐(𝑡)=𝐹(𝑡),𝑢_𝑑𝑐(𝑡)=𝑑₁(𝑡), and 𝑢_𝑠𝑐(𝑡)=𝑆𝑥(𝑡), 𝑆 is the state feedback controller gain to be determined later. Substituting (9) into (1), it can be seen that 𝑥̇(𝑡)=(𝐴+𝐽𝑆)𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐺𝑔(𝑥(𝑡))+𝐽𝑒_𝐹(𝑡)+𝐽₁𝑑₁(𝑡)−𝐽𝐽^∗𝐽₁𝑑₁(𝑡)+𝐽₂𝑑₂(𝑡). (10)

From Assumption 2, it can be seen that the vector space spanned by the columns of 𝐽₁ is a subset of the space spanned by the column vectors of 𝐽 [19]; that is, span(𝐽₁)⊂span(𝐽), which is equivalent to the existence of 𝐽, such that 𝐽₁−𝐽𝐽^∗𝐽₁=0. (11)

Then, it can be concluded that 𝑥̇(𝑡)=(𝐴+𝐽𝑆)𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐺𝑔(𝑥(𝑡))+𝐽𝑒_𝐹(𝑡)+𝐽₁𝑉𝑒_𝜔(𝑡)+𝐽₂𝑑₂(𝑡). (12)

Combing estimation error equations (6) and (8) with (12) yields𝑥̇‌(𝑡)=𝐴𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐺𝑔(𝑥(𝑡))+𝐽𝑑(𝑡),𝑧₂(𝑡)=𝐶₂𝑥(𝑡)+𝐶_𝑑2𝑥(),𝑧𝑡−𝜏_∞(𝑡)=𝐶_∞𝑥(𝑡)+𝐶_𝑑∞𝑥(𝑡−𝜏)+𝐷𝑑(𝑡), (13) where 𝑥(𝑡)=𝑥(𝑡)𝑒_𝜔(𝑡)𝑒_𝐹(𝑡),𝐴=𝐴+𝐽𝑆𝐽₁𝑉𝐽0𝑊+𝐿𝐽₁𝑉𝐿𝐽0𝐾𝐽₁,𝑉𝐾𝐽𝐺00𝐺=,𝐴_𝑑=𝐴_𝑑,00000000𝐽𝐽=²00𝐿𝐽₂𝐽₃0𝐾𝐽₂,0𝐼𝑑(𝑡)=𝑑₂(𝑡)𝛿(𝑡)𝐹̇(𝑡),𝑔(𝑥(𝑡))=𝑔(𝑥(𝑡)) (14)𝑧₂(𝑡) is 𝐻₂ reference output, 𝑧₂(𝑡)=𝐶₂₁𝑥(𝑡)+𝐶₂₂𝑒^𝑇_𝜔(𝑡)𝑒^𝑇_𝐹(𝑡)^𝑇+𝐶_𝑑2𝑥(𝑡−𝜏) (15)𝑧_∞(𝑡) is 𝐻_∞ reference output, 𝑧_∞(𝑡)=𝐶_∞1𝑥(𝑡)+𝐶_∞2𝑒^𝑇_𝜔(𝑡)𝑒^𝑇_𝐹(𝑡)^𝑇+𝐶_𝑑∞𝑥(𝑡−𝜏)+𝐷𝑑(𝑡), (16) where 𝐶₂₁,𝐶₂₂,𝐶_𝑑2,𝐶_∞1,𝐶_∞2,𝐶_𝑑∞, and 𝐷 are the selected weighting matrices.

Definition 2. For constants 𝛾₁>0,𝛾₂>0,and𝛾₃>0, the 𝐻_∞ performance is denoted as follows: 𝐽_∞=𝑧_∞(𝑡)²−𝛾²₁𝑑₂(𝑡)²−𝛾²₂‖𝛿(𝑡)‖²−𝛾²₃𝐹̇(𝑡)²−𝛿𝑃₁,𝑄, (17) where 𝛿𝑃₁,𝑄=𝜙^𝑇(0)𝑃₁𝜙(0)+⁰_−𝑑𝜙^𝑇(𝜏)𝑄𝜙(𝜏)𝑑𝜏. (18)

Definition 3. The 𝐻₂ performance measure for (13) is defined as 𝐽₂=‖𝑧₂(𝑡)‖².

Remark 4. Compared with [17], 𝐻₂/𝐻_∞ mixed multiobjective optimization technique is used for the composite system (13). In the proposed approach, the modeled disturbance and fault are rejected by their estimations, while 𝐻_∞ performance is adopted to attenuate norm bounded uncertain disturbances and 𝐻₂ performance index is applied to optimize estimation error.
At this stage, the objective is to find 𝐾,𝐿, and 𝑆 such that system (13) is stable. The following result provides a design method based on convex optimization technology [22].

Theorem 5. If for the parameter 𝜆>0,𝑟_𝑖(𝑖=1,…,4), matrices 𝐶₂₁,𝐶₂₂,𝐶_𝑑2,𝐶_∞1,𝐶_∞2,𝐶_𝑑∞, and 𝐷, there exist matrices 𝑃₀>0,𝑃₂>0,𝑄₀>0,𝑅₀, and 𝑅₂ and constants 𝛾₁>0,𝛾₂>0, and 𝛾₃>0, if the following LMI-based optimization problem holds: min𝑟₁𝛾₁+𝑟₂𝛾₂+𝑟₃𝛾₃+𝑟₄𝜙^𝑇(0)𝑃₀⁻¹𝜙(0) (19) subject to Ξ₁𝐴_𝑑𝑄₀𝐺𝑉Ξ₁₅𝑃₀𝑈^𝑇₁𝑃₀𝐶^𝑇_∞1𝑃₀𝐶^𝑇₂₁𝑃₀∗−𝑄₀0000𝐶^𝑇_𝑑∞𝐶^𝑇_𝑑201∗∗−𝜆𝐼000000∗∗∗Ξ₄Ξ₄₅0𝐶^𝑇_∞2𝐶^𝑇₂₂0∗∗∗∗Ξ₅0𝐷^𝑇00∗∗∗∗∗−𝜆𝐼000∗∗∗∗∗∗−𝐼00∗∗∗∗∗∗∗−𝐼0∗∗∗∗∗∗∗∗−𝑄₀<0, (20) where Ξ₁=sym𝐴𝑃₀+𝐽𝑅₀,Ξ₁₅=𝐽₂,Ξ00₄=sym𝑃₂𝑊+𝑅₂𝑉,Ξ₄₅=𝑅₂𝐽₂𝑃₂𝐽₃𝑃₂𝐽₁,Ξ₅=−𝛾²₁𝐼000−𝛾²₂𝐼000−𝛾²₃𝐼,𝑊=,𝑊000𝑉=𝐽₁𝑉𝐽,𝐽₁=0𝐼,𝐽₃=𝐽₃0,𝐶=𝐶₁𝐶₂, (21) then with gains 𝑆=𝑅₀𝑃₀⁻¹ and 𝐿=^𝐿_𝐾=𝑃₂⁻¹𝑅₂, error system (13) is stable and satisfies 𝐽_∞<0 and 𝐽₂≤𝛿(𝑃₁,𝑄). The symmetric terms in a symmetric matrix are denoted by ∗. The symbol sym() represents sym(Θ)∶=Θ+Θ^𝑇.

Proof. Consider the following Lyapunov function: Π(𝑡)=𝑥^𝑇(𝑡)𝑃₁𝑥(𝑡)+^𝑡_𝑡−𝜏𝑥^𝑇(𝜏)𝑄𝑥(𝜏)𝑑𝜏+𝑒^𝑇(𝑡)𝑃₂𝑒(𝑡)+𝜆^𝑡₀‖𝑈𝑥(𝜏)‖²−𝑔(𝜏)²𝑑𝜏. (22)
It is verified that Π(𝑡)≥0 holds for all arguments. Along with the trajectories of (13), it can be shown thatΠ̇(𝑡)=2𝑥^𝑇(𝑡)𝑃₁𝑥̇(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥(𝑡−𝜏)^𝑇𝑄𝑥(𝑡−𝜏)+2𝑒^𝑇(𝑡)𝑃₂𝑒̇‌(𝑡)+𝜆‖𝑈𝑥(𝑡)‖²−𝑔(𝑥(𝑡))²=2𝑥^𝑇(𝑡)𝑃₁(𝐴+𝐽𝑆)𝑥(𝑡)+𝐴_𝑑𝑥(𝑡−𝜏)+𝐺𝑔(𝑥(𝑡))+𝐽₁𝑉𝑒_𝜔(𝑡)+𝐽𝑒_𝐹(𝑡)+𝐽₂𝑑₂(𝑡)+𝑥^𝑇(𝑡)𝑄𝑥(𝑡)−𝑥(𝑡−𝜏)^𝑇𝑄𝑥(𝑡−𝜏)+𝜆𝑥^𝑇(𝑡)𝑈^𝑇𝑈𝑥(𝑡)−𝜆𝑔(𝑥(𝑡))²+2𝑒^𝑇(𝑡)×𝑃₂𝑊+𝐿𝑉𝑒(𝑡)+𝐿𝐽₂𝑑₂(𝑡)+𝐽₃𝛿(𝑡)+𝐽₁𝐹̇(𝑡). (23)
In the absence of 𝑑(𝑡) (i.e., 𝑑(𝑡)=0), it can be seen that 𝑉̇(𝑡)=𝑠^𝑇(𝑡)Φ−𝜁𝜁^𝑇𝑠(𝑡), (24) where 𝑠^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏)𝑔^𝑇(𝑥(𝑡))𝑒^𝑇(𝑡),𝜁^𝑇=𝐶₂₁𝐶_𝑑20𝐶₂₂,ΦΦ=¹¹𝑃₁𝐴_𝑑+𝐶^𝑇₂₁𝐶_𝑑2𝑃₁𝐺𝑃₁𝑉+𝐶^𝑇₂₁𝐶₂₂∗−𝑄+𝐶^𝑇_𝑑2𝐶_𝑑200∗∗−𝜆𝐼0∗∗∗Φ₄₄,Φ₁₁=sym𝑃₁(𝐴+𝐽𝑆)+𝜆𝑈^𝑇𝑈+𝑄+𝐶^𝑇₂₁𝐶₂₁,Φ₄₄=sym𝑃₂𝑊+𝐿𝑉+𝐶^𝑇₂₂𝐶₂₂. (25)
From (20), it can be seen thatΞ₁+𝑃₀𝑄₀⁻¹𝑃₀+1𝜆𝑃₀𝑈^𝑇₀𝑈₀𝑃₀𝑄₀𝐴_𝑑𝐺𝑉Ξ₁₅𝑃₀𝐶^𝑇_∞1𝑃₀𝐶^𝑇₂₁∗−𝑄₀000𝐶^𝑇_𝑑∞𝐶^𝑇_𝑑21∗∗−𝜆𝐼0000∗∗∗Ξ₄Ξ₄₅𝐶^𝑇_∞2𝐶^𝑇₂₂∗∗∗∗Ξ₅00∗∗∗∗∗−𝐼0∗∗∗∗∗∗−𝐼<0. (26)
From the first, second, third, fourth, and seventh columns and rows of the left matrix in inequality (26) it can be verified that Φ₀=Ξ₁+𝑃₀𝑄₀⁻¹𝑃₀+1𝜆𝑃₀𝑈^𝑇₀𝑈₀𝑃₀𝑄₀𝐴_𝑑𝐺𝑉𝑃₀𝐶^𝑇₂₁∗−𝑄₀00𝐶^𝑇_𝑑21∗∗−𝜆𝐼00∗∗∗Ξ₄𝐶^𝑇₂₂∗∗∗∗−𝐼<0. (27)
Defining 𝑃₀=𝑃₁⁻¹,𝑄⁻¹=𝑄₀, premultiplied and postmultiplied simultaneously by diag{𝑃₀⁻¹,𝑄₀⁻¹,𝐼,𝐼,𝐼}, it can be seen by using Schur complement formula that Φ₀<0 leads to Φ<0. When Φ<0 holds, we have 𝑉̇(𝑡)<−𝑠^𝑇(𝑡)𝜁𝜁^𝑇𝑠(𝑡)=−𝑧^𝑇₂𝑧₂≤0. (28)
It follows that (13) is asymptotically stable in the absence of the exogenous input 𝑑(𝑡). Next, we consider the performances to be optimized for (13). Consider two auxiliary functions as follows: 𝐽₀=𝑧^𝑇₂(𝑡)𝑧₂(𝑡)+𝑉̇(𝑡),𝐽₁=𝑧^𝑇_∞(𝑡)𝑧_∞(𝑡)−𝛾²₁𝑑₂(𝑡)^𝑇𝑑₂(𝑡)−𝛾²₂𝛿(𝑡)^𝑇𝛿(𝑡)−𝛾²₃𝐹̇(𝑡)^𝑇𝐹̇(𝑡)+𝑉̇(𝑡). (29)
Following the definition of the 𝐻₂ performance, we only consider the case in the absence of 𝑑(𝑡). It can be verified that 𝐽₀≤𝑠^𝑇(𝑡)Φ𝑠(𝑡).
In the presence of 𝑑(𝑡), it can be seen that 𝐽₁=𝑞^𝑇(𝑡)Ψ𝑞(𝑡), where 𝑞^𝑇(𝑡)=𝑥^𝑇(𝑡)𝑥^𝑇(𝑡−𝜏)𝑔^𝑇(𝑥(𝑡))𝑒^𝑇(𝑡)𝑑^𝑇(𝑡),ΨΨ=¹¹𝑃₁𝐴_𝑑𝑃₁𝐺𝑃₁𝑉Ψ₁₅1∗−𝑄000∗∗−𝜆𝐼00∗∗∗Ψ₄₄Ψ₄₅∗∗∗∗Ξ₅+𝐶^𝑇_∞1𝐶^𝑇_𝑑∞0𝐶^𝑇_∞20𝐶^𝑇_∞1𝐶^𝑇_𝑑∞0𝐶^𝑇_∞20^𝑇, (30) where Ψ₁₁=sym𝑃₁𝐴+𝑃₁+1𝐽𝑆𝜆𝑈^𝑇Ψ𝑈+𝑄,₁₅=𝑃₁𝐽₂,Ψ00₄₄=sym𝑃₂𝑊+𝑃₂𝐿𝑉,Ψ₄₅=𝑃₂𝐿𝐽₂𝑃₂𝐽₃𝑃₂𝐽₁. (31)
Denote Ψ₀=Ψ+𝐶₂₁𝐶_𝑑20𝐶₂₂0^𝑇𝐶₂₁𝐶_𝑑20𝐶₂₂0. (32)
It can be seen by using Schur complement formula that (20) leads to Ψ₀<0, and then Ψ<0 holds. It can be verified that both 𝐽₀<0 and 𝐽₁<0 hold. This completes the proof.

4. Simulation Examples

In this section, we consider the longitudinal dynamics of A4D aircraft at a flight condition of 15000 ft altitude and 0.9 Mach given in [2]. The longitudinal dynamics can be denoted as 𝑥̇(𝑡)=𝐴𝑥(𝑡)+𝐺𝑔(𝑥(𝑡))+𝐽(𝑢(𝑡)+𝐹(𝑡))+𝐽₁𝑑₁(𝑡)+𝐽₂𝑑₂(𝑡), (33) where 𝑥(𝑡) are measurable by using sensors technique. The coefficient matrices of aircraft model are given by ,00𝐴=−0.060532.37032.2−0.00014−1.47510−0.0111−34.72−2.79300010𝐽=−0.1064−33.8,𝐽₁=00−0.0532−16.9000,𝐽₂=0.0.1−30.1 (34)

It is supposed that 𝐺=00500^𝑇,𝑔(𝑥(𝑡))=sin(2𝜋5𝑡)𝑥₂(𝑡). (35) then, the matrix 𝑈 can be selected as 𝑈=diag{0,1,0,0} and the norm condition (3) can be satisfied. 𝐴_𝑑=0.5×𝐴,𝜏=2. Periodic disturbance 𝑑₁(𝑡) caused by rotating aerial propeller is assumed to be an unknown harmonic disturbance described by (2) with 𝑊=05−50,𝐸=250,𝐻₃=0.10.1, (36)𝛿(𝑡) is the additional disturbance signal resulting from the perturbations and uncertainties in the exogenous system (2) and satisfies 2-norm boundedness. In simulation, we select 𝛿(𝑡) as the random signal with upper 2-norm bound 1. Wind gust and system noises 𝑑₂(𝑡) can also be considered as the random signal with upper 2-norm bounded.

The initial values of the states are supposed to be 𝑥^𝑇(0)=[2−232]. For the reference output, it is denoted that 𝐶_∞1=1111,𝐶_∞2=,𝐶111_𝑑∞=,𝐶0.10.10.10.1₂₁=0.10.10.10.1,𝐶₂₂=,𝐶0.10.10.1_𝑑2=0.010.010.010.01. (37)

For 𝜆=1,𝛾₁=1,𝛾₂=1, and 𝛾₃=1, it can be solved via LMI related to (20) that the gain of fault diagnosis observer (7) is 𝐾=20.929910.02311.355320.9682, (38) the gain of disturbance observer (4) is 𝐿=−0.2584−0.5402−0.0142−0.26102.32460.75230.15262.3269, (39) and the gain of state feedback controller is 𝑆=89.6396471.067074.2419465.7941. (40)

When the disturbance observer is constructed based on (4) and (5), the estimation error of exogenous disturbances is shown in Figure 1. The actuator bias fault is supposed to occur at 15th second as 𝐹=4. The estimation and its error of fault with system disturbances are demonstrated in Figure 2, where the solid line represents the real fault signal, the dash-dotted line is its estimation, and the dash line denotes its estimation error. In Figure 3, the state response signals of the control system are illustrated. It can be seen that the proposed fault tolerant controller has a good control ability for configuring fault and rejecting and attenuating disturbances simultaneously. In Figure 4, the ramp fault is assumed to occur at 15th second with slope 0.1. From Figure 4, it is shown that the time-varying fault can also be well estimated.

Figure 1 

Disturbances estimation error in disturbance observer.

Figure 2 

Bias fault, its estimation, and error.

Figure 3 

The responses of state variable.

Figure 4 

Ramp fault, its estimation, and error.

5. Conclusion

In this paper, a robust antidisturbance fault tolerant control problem is investigated for nonlinear time delay systems with faults and multiple disturbances. There are the following features of the proposed algorithm compared with the previous results. First, the multiple disturbances and state time-delay are considered simultaneously in this paper. Second, an FTC scheme is addressed with disturbance rejection and attenuation performance by combining fault accommodation and DOBC with a mixed 𝐻₂/𝐻_∞ controller, with which the fault can be accommodated and the disturbances can be rejected and attenuated simultaneously. Finally, simulation for a flight control system is given to show the efficiency of the proposed approach.

Acknowledgment

This work is partially supported by the National Natural Science Foundation of China (Grant no. 61203195), Universities Natural Science Foundation of Jiangsu Province (Grant no. 12KJB510033), and China Postdoctoral Science Foundation (Grant no. 2013M530508).