Adaptive PID Fault-Tolerant Tracking Controller for Takagi- Sugeno Fuzzy Systems with Actuator Faults: Application to Single-Link Flexible Joint Robot

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http://pubs2.ascee.org/index.php/ijrcs

https://doi.org/10.31763/ijrcs.v2i3.762 ijrcs@ascee.org

Adaptive PID Fault-Tolerant Tracking Controller for Takagi- Sugeno Fuzzy Systems with Actuator Faults: Application to Single-Link Flexible Joint Robot

Mohamed Elouni â,1,*, Habib Hamdi â,2, Bouali Rabaoui â,3, Naceur BenHadj Braiek â,4

a Advanced Systems Laboratory, Tunisian Polytechnic School, Tunis, Tunisia

1 mohamed.elouni@isetb.rnu.tn; ² habibhemdi@gmail.com; ³ rabaaouibouali@gmail.com;⁴ naceur.benhadj@ept.rnu.tn

* Corresponding Author

1. Introduction

In recent decades, with the challenge of increasing demand for reliability and safety of various industrial control processes, Fault Diagnosis (FD) and Fault Tolerant Control (FTC) has attracted more and more attention. More precisely, the FTC system is a control system that can accommodate system component faults and is able to maintain system stability and performance despite the presence of faults [1]–[4]. Indeed, real-world physical systems are generally nonlinear in nature.

Moreover, it is well known that it is difficult to directly extend the existing FTC methods of linear systems to nonlinear ones.

Furthermore, a large class of nonlinear systems can be described by using suitable Takagi- Sugeno (T-S) fuzzy models [5]. The main idea of the T-S fuzzy modeling method is to decompose the model of a nonlinear system into several locally linearized models involving nonlinear weighting functions.

Accordingly, several researchers have been interested in developing new FTC approaches based on T-S fuzzy representation. For example, in [6], a fuzzy-reduced-order robust state and fault estimation observer has been proposed to estimate the system states, sensor, and actuator faults. In addition, an observer-based fault-tolerant controller has been designed to guarantee the robustly asymptotically stability of the closed-loop system. Also, authors in [7] developed a method for

ARTICLE INFO ABSTRACT

Article history Received June 26, 2022 Revised August 08, 2022 Accepted August 18, 2022

This paper considers the problem of Fault Tolerant Tracking Control (FTTC) strategy design for nonlinear systems using Takagi-Sugeno (T-S) fuzzy models with measurable premise variables affected by actuator faults subject to unknown bounded disturbances (UBD). Firstly, the Adaptive Fuzzy Observer (AFO) is proposed to estimate the faults. Based on the information provided by this observer, an active fault tolerant tracking controller described by an adaptive Proportional-Integral-Derivative (PID) structure has been developed to compensate for the actuator fault effects and to guarantee the trajectory tracking of desired outputs to the reference model despite the presence of actuator faults. The stability and the trajectory tracking performances of the proposed approach are analyzed based on the Lyapunov theory. Sufficient conditions can be obtained and solved for the design of the controller, and the observer gains using Linear Matrix Inequalities (LMIs). Finally, the effectiveness of the proposed technique is illustrated by using a single-link flexible joint robot.

Keywords T-S fuzzy systems;

PID Fault-Tolerant Tracking Control;

Adaptive Fuzzy Observer;

Single-link flexible joint robot;

LMIs Constraints

This is an open-access article under the CC–BY-SA license.

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Mohamed Elouni (Adaptive PID Fault-Tolerant Tracking Controller for Takagi-Sugeno Fuzzy Systems with Actuator Faults: Application to Single-Link Flexible Joint Robot)

designing a robust fault-tolerant tracking control technique for nonlinear uncertain systems approximated by Takagi–Sugeno fuzzy models with unmeasurable premise variables subject to sensor faults. The authors in [8] have proposed a fault-tolerant tracking control for uncertain T-S fuzzy systems affected simultaneously by sensor faults, actuator faults, and unknown bounded disturbance. A robust descriptor adaptive observer-based FTTC has been proposed to estimate unmeasured system states and both sensor and actuator fault vectors simultaneously, which have then been used in order to compensate for the effects of the faults and to track the reference model states despite the presence of external disturbances and uncertainties. A robust adaptive observer- based fault tolerant control is presented in [9] for nonlinear systems described by a T-S fuzzy model affected by both sensor and actuator faults in the presence of external disturbances. So, a robust adaptive observer has been developed to simultaneously estimate the state, sensor, and actuator faults vectors. Based on this observer, the fuzzy control law can preserve the stability and the performance of the closed-loop system and compensate for fault effects. The authors [10] have presented a combined state, faults estimation, and fault tolerant control method based on Proportional Multi-Integral Observer for fuzzy descriptor systems affected by actuator faults.

Today, the standard Proportional-Integral-Derivative (PID) control schemes continue to provide the simplest yet most effective solutions to most control engineering applications. The popularity of the PID controllers is mainly due to their simplicity and easy adaptation to control a large number of systems. Recently, the PID controller has been used for FTC in order to compensate for the fault effects and stabilize the closed-loop system. Indeed, authors in [11] have designed an adaptive PID actuator fault tolerant control based on the sliding mode approach for a single-link flexible manipulator. In [12], an adaptive PID-FTTC for DC series Motor represented by T-S model subject to actuator faults has been proposed using the adaptive fuzzy observer. Moreover, in [13], the authors consider the problem of model reference tracking based on the design of an Active Fault Tolerant Control for polytopic Linear Parameter Varying (LPV) systems affected by actuator faults and unknown inputs. The authors in [14] have designed a descriptor observer-based sensor and actuator fault-tolerant tracking control for LPV systems. The descriptor observer is able de estimates sensor and actuator faults affecting systems simultaneously, and a proposed PID-Active FTTC law which allows for tolerance of actuator faults and ensures the stability and trajectory tracking performances, especially in terms of accuracy and rapidity.

Until now, there are very few works dealing with the tracking FTC design with PID controllers for T-S fuzzy models in the realm of control, and it is more difficult and more general than the stabilizing FTC. This field remains very important, which motivates our study in this work.

Therefore, the main contribution of this paper consists in designing a PID-FTTC closed loop system scheme based on the synthesis of an Adaptive Fuzzy Observer (AFO), which estimates states and actuator faults. The principle of the proposed method is to use a T-S fuzzy model to represent the nonlinear dynamics of the system. Based on the estimated actuator faults, the designed controller is automatically reconfigured in order to compensate for the fault effects, keep the system stable, and to ensure tracking trajectories with good accuracy performances. However, in this work, we compute both controller and observer gains by solving a set of LMIs in an integrated and in one step. Moreover, a single-link flexible joint robot system with disturbance and actuator faults is used to validate the effectiveness of the developed PID-FTTC scheme.

The remaining of this paper is structured in the following order. Section 2 provides the system description and the problem statement. Section 3 is dedicated to the design of an observer-based PID-FTTC to stabilize the closed loop system in the presence of actuator faults. In section 4, simulation results are given, using the single-link flexible joint robot example in order to illustrate the importance and the effectiveness of the proposed PID-FTTC scheme with actuator faults. To close the paper, some conclusions are given in section 5.

Notations: In this paper, we note 𝐼 as an identity matrix with an appropriate dimension. In a matrix, * denotes the transposed element in the symmetric position.

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2. Preliminaries and Problem Formulation 2.1. T-S System Modeling

Generally, most physical systems are inherently nonlinear. To overcome these non-linearities, they can be described by fuzzy models based on the T-S approach. The T-S fuzzy model is a set of linear time-invariant systems connecting via membership functions. Actually, three techniques have been used to build a T-S fuzzy model from existing nonlinear models. An interesting method is well-known in the nonlinear sector transformation [15]. These transformations allow obtaining an exact T-S representation of a nonlinear system model without information loss on a compact set of the state space.

Consider the following nonlinear system:

{𝑥̇(𝑡) = 𝑓(𝑥(𝑡)) + 𝑔(𝑥(𝑡))𝑢(𝑡)

𝑦(𝑡) = 𝐶𝑥(𝑡) (1)

where 𝑥(𝑡) ∈ℝ^𝑛 is the state, 𝑢(𝑡) ∈ℝ^𝑝 is the input vector, 𝑦(𝑡) ∈ℝ^𝑚 represents the output measurement vectors and 𝐶 ∈ℝ^𝑚×𝑛 is the output matrix. In addition, 𝑓(. )and 𝑔(. ) represent the nonlinear functions.

The T-S fuzzy model uses a set of fuzzy if-then rules, which represent local linear input-output relations of a nonlinear system. The i^th rule of the T-S model is given as follows:

Rule i:

If 𝜉₁(𝑡) is 𝐹₁^𝑖(𝜉₁(𝑡)) and 𝜉₂(𝑡) is 𝐹₂^𝑖(𝜉₂(𝑡))….and 𝜉_𝑝(𝑡) is 𝐹_𝑝^𝑖(𝜉_𝑝(𝑡)) THEN {𝑥̇(𝑡) = 𝐴_𝑖𝑥(𝑡) + 𝐵_𝑖𝑢(𝑡)

𝑦(𝑡) = 𝐶_𝑖𝑥(𝑡) (2)

where 𝐹_𝑝^𝑖 are the membership functions of fuzzy sets, 𝑖 ∈ {1,2, . . . ,ℎ}, 𝑟 is the number of rules, 𝐴_𝑖 ∈ℝ^𝑛×𝑛, 𝐵_𝑖 ∈ℝ^𝑛×𝑚, 𝐶_𝑖 ∈ℝ^𝑝×𝑛 and 𝜉₁(𝑡), 𝜉₂(𝑡), . . . , 𝜉_𝑝(𝑡) are the premise variables which can be dependent on the inputs, the outputs, or the states. The global T-S structure is given by:

{𝑥̇(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝐴_𝑖𝑥(𝑡) + 𝐵_𝑖𝑢(𝑡)) 𝑦(𝑡) = 𝐶𝑥(𝑡)

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where 𝜇_𝑖(𝜉(𝑡)) are the weighting functions indicate the activation degree of the i^th associated local model. These nonlinear functions verify all time the convex sum propriety:

{∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

= 1 0 ≤ 𝜇_𝑖(𝜉(𝑡)) ≤ 1

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2.2. Problem Formulation

Our goal in this work is to synthesize an FTC law based on an adaptive PID controller. The controller design ensures automatic reconfiguration to the nominal control input after the apparition of actuator faults to guarantee the asymptotic stability in a closed loop and to minimize the tracking trajectory error between the reference model outputs and faulty system outputs. The bloc scheme of the proposed PID-FTTC closed loop system is given in Fig. 1. The elements of the FTTC loop include the Adaptive Fuzzy Observer, the adaptive PID controller, and the model reference. All

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these elements are described using the T-S fuzzy technique. Firstly, the AFO is developed in order to estimate states and actuator faults for T-S fuzzy systems with external disturbances. Then, a PID- FTTC is designed to compensate for the actuator faults effects, stabilize the closed-loop system, and ensure very good tracking performances.

Fig. 1. PID-FTTC closed loop system scheme

The model reference is considered given by the following T-S fuzzy healthy model defined as:

{𝑥̇𝑟(𝑡) = ∑ 𝜇𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝐴_𝑖𝑥_𝑟(𝑡) + 𝐵_𝑖𝑢(𝑡)) 𝑦𝑟(𝑡) = 𝐶𝑥𝑟(𝑡)

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Where 𝑥_𝑟(𝑡) ∈ℝ^𝑛, 𝑢(𝑡) ∈ℝ^𝑝 and 𝑦_𝑟(𝑡) ∈ℝ^𝑚 represent respectively state vector, the bounded control input vector, and the measured output vector of the reference model.

In the case of actuator faults and unknown bounded disturbances, the faulty nonlinear system described by the T-S fuzzy model can be expressed as follows:

{

𝑥̇_𝑓(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))(𝐴_𝑖𝑥_𝑓(𝑡) + 𝐵_𝑖(𝑢_𝑓(𝑡) + 𝑓(𝑡)) + 𝑅_𝑖𝑑(𝑡))

ℎ

𝑖=1

𝑦_𝑓(𝑡) = 𝐶𝑥𝑓(𝑡)

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where 𝑥𝑓(𝑡) ∈ℝ^𝑛 represents the faulty state vector, 𝑢𝑓(𝑡) ∈ℝ^𝑝 is the FTC law vector to be designed, 𝑓(𝑡) ∈ℝ^𝑝 is the actuator fault vector, 𝑑(𝑡) ∈ℝ^𝑞 represents the unknown bounded disturbance vector and 𝑦_𝑓(𝑡) ∈ℝ^𝑚 stands the measured output vector.

Firstly, an Adaptive Fuzzy Observer is developed in order to estimate faulty states and actuator faults for T-S fuzzy systems with external disturbances. The fault estimation 𝑓̂(𝑡) provided by an AFO that is given as follows:

𝑓(𝑡) 𝑑(𝑡)

𝑓̂(𝑡)

𝑢(𝑡) 𝑢_𝑓(𝑡) 𝑦_𝑓(𝑡)

𝑦_𝑟(𝑡) Adaptive PID

Controller Nonlinear

System

Adaptive T-S Fuzzy Observer + +

T-S Reference

Model

- +

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{

𝑧̇(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝑁_𝑖𝑧(𝑡) + 𝐺_𝑖𝑢_𝑓(𝑡) + 𝐿_𝑖𝑦_𝑓(𝑡) + 𝐵_𝑖𝑓̂(𝑡)) 𝑥̂_𝑓(𝑡) = 𝑧(𝑡) + 𝑇₂𝑦_𝑓(𝑡)

𝑦̂_𝑓(𝑡) = 𝐶𝑥̂_𝑓(𝑡) 𝑓̂̇(𝑡) = 𝛤 ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝛷_𝑖(𝑒̇_𝑦(𝑡) + 𝜎𝑒_𝑦(𝑡))) 𝑒_𝑦(𝑡) = 𝑦_𝑓(𝑡) − 𝑦̂_𝑓(𝑡)

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where 𝑧(𝑡) ∈ℝ^𝑛 is the observer state vector, 𝑥̂(𝑡) ∈ℝ^𝑛 the estimated state vector, 𝑦̂(𝑡) ∈ℝ^𝑚is the estimated output vector, 𝑓̂(𝑡) ∈ℝ^𝑞is the estimated actuator faults and 𝑁_𝑖∈ℝ^𝑛×𝑛, 𝐺_𝑖 ∈ℝ^𝑛×𝑝, 𝐿_𝑖 ∈ ℝ^𝑛×𝑚, 𝛷_𝑖 ∈ℝ^𝑞×𝑚 and 𝑇₂ ∈ℝ^𝑛×𝑚 are the observer gain matrices to be determined. The matrix 𝛤 ∈ ℝ^𝑝×𝑝 is a symmetric positive definite learning rate matrix and 𝜎 is a positive scalar.

Now, based on the presented strategy in [9], the designed PID-FTTC control law has a structure given by:

𝑢_𝑓(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝐾_𝑃𝑖𝑒_𝑦𝑟(𝑡) + 𝐾_𝐼𝑖∫ 𝑒_𝑦𝑟(𝑡)

𝑡 0

𝑑𝑡 + 𝐾_𝐷𝑖𝑑𝑒_𝑦𝑟(𝑡)

𝑑𝑡 − 𝐾_𝑓𝑖𝑓̂(𝑡)) + 𝑢(𝑡) (8) where 𝐾_𝑃𝑖 ∈ℝ^𝑝×𝑚, 𝐾_𝐼𝑖∈ℝ^𝑝×𝑚, 𝐾_𝐷𝑖 ∈ℝ^𝑝×𝑚 and 𝐾_𝑓𝑖 ∈ℝ^𝑝×𝑞 are the gain matrices to be determined.

Adaptive Fuzzy Observer (7) based fault tolerant tracking control (8) has been designed for nonlinear systems represented by T-S fuzzy models affected by actuator faults and external disturbances. First of all, the AFO can estimate system states and actuator faults accurately. After that, the fuzzy controller can compensate for fault effects, guarantee the stabilization of the closed- loop fuzzy system and ensure the tracking of the desired reference trajectories with good accuracy and rapid responses.

Furthermore, to develop the PID-FTTC closed loop system, the following assumptions are necessary:

Assumption 1 [16]. We assume that the actuator fault distribution matrices 𝐵_𝑖 are of full column rank ∀𝑖 = 1, . . . ,ℎ:

𝑟𝑎𝑛𝑘(𝐶𝐵_𝑖) = 𝑟𝑎𝑛𝑘(𝐵_𝑖) = 𝑝 (9)

Assumption 2 [16]. The pair (𝐴_𝑖, 𝐶) is observable ∀𝑖 = 1, . . . ,ℎ.

𝑟𝑎𝑛𝑘 [ 𝐶 𝐶𝐴_𝑖

⋮ 𝐶𝐴_𝑖^𝑛−1

] = 𝑛 (10)

Assumption 3 [16]. The input vector 𝑢(𝑡) is bounded as ‖𝑢(𝑡)‖ ≤ 𝛼₁. The fault 𝑓(𝑡) satisfies

‖𝑓(𝑡)‖ ≤ 𝛼₂, its derivative is bounded such that ‖𝑓̇(𝑡)‖ ≤ 𝛼₃and the unknown input vector 𝑑(𝑡) verifies ‖𝑑(𝑡)‖ ≤ 𝛼₄, where 𝛼₁, 𝛼₂, 𝛼₃, 𝛼₄≥ 0.

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3. Design and Analysis of PID-FTTC for T-S Fuzzy System 3.1. System Reconfiguration

To analyze the reconfigured system, let's consider the following augmented model 𝑥̃(𝑡) which includes the faulty state vector 𝑥_𝑓(𝑡), the state of the tracking error 𝑒_𝑡(𝑡), the state of the estimation error 𝑒_𝑠(𝑡) and the fault estimation error 𝑒_𝑓(𝑡) as follows:

𝑥̃(𝑡) = (

𝑥_𝑓(𝑡) 𝑒_𝑡(𝑡) 𝑥_𝑡(𝑡) 𝑒_𝑠(𝑡) 𝑒_𝑓(𝑡))

=

(

𝑥_𝑓(𝑡) 𝑥_𝑟(𝑡) − 𝑥_𝑓(𝑡)

∫ 𝑒_𝑡(𝑡)

𝑡 0

𝑥_𝑓(𝑡) − 𝑥̂_𝑓(𝑡) 𝑓(𝑡) − 𝑓̂(𝑡) )

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Now, a new state vector 𝑥_𝑡(𝑡) is used such that its dynamic is given by [17]

𝑥̇_𝑡(𝑡) = 𝑒_𝑡(𝑡) (12)

By using the state space representations (5) and (6) and the expression of 𝑒_𝑡(𝑡) in (11), the error 𝑒_𝑦𝑟(𝑡) can be written as:

𝑒_𝑦𝑟(𝑡) = 𝑦_𝑟(𝑡) − 𝑦_𝑓(𝑡) = 𝐶𝑒_𝑡(𝑡) (13) Knowing the expressions of 𝑒_𝑡(𝑡), 𝑥̇_𝑡(𝑡) and using (11), the FTC law (8) can be written as:

𝑢_𝑓(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

(𝐾_𝑃𝑖𝐶𝑒_𝑡(𝑡) + 𝐾_𝐼𝑖𝐶𝑥_𝑡(𝑡) + 𝐾_𝐷𝑖𝐶𝑒̇_𝑡(𝑡) + 𝐾_𝑓𝑖𝑒_𝑓(𝑡) − 𝐾_𝑓𝑖𝑓(𝑡)) + 𝑢(𝑡) (14) The dynamic of the tracking error is:

𝑒̇_𝑡(𝑡) = 𝑥̇_𝑟(𝑡) − 𝑥̇_𝑓(𝑡) (15) Now, by substituting (5) and (6) on the one hand and (14) and (10) on the other hand into (15), we can obtain:

𝑒̇_𝑡(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡))[𝐴_𝑖𝑗𝑒_𝑡(𝑡) − 𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) − 𝐵_𝑖𝐾_𝐷𝑗𝐶𝑒̇_𝑡(𝑡) − 𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 𝐻_𝑖𝑗𝑓(𝑡) − 𝑅_𝑖𝑑(𝑡)]

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with 𝐴_𝑖𝑗= 𝐴_𝑖− 𝐵_𝑖𝐾_𝑃𝑗𝐶 and 𝐻_𝑖𝑗 = 𝐵_𝑖𝐾_𝑓𝑗− 𝐵_𝑖.

Using the dynamic equation (16), generating 𝑒_𝑡(𝑡), the following equality is obtained ∀𝑖, 𝑗 = 1, . . . ,ℎ:

∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡))𝛥_𝑖𝑗𝑒̇_𝑡(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡)) [𝐴_𝑖𝑗𝑒_𝑡(𝑡) − 𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) − 𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 𝐻_𝑖𝑗𝑓(𝑡) − 𝑅_𝑖𝑑(𝑡)]

(17)

with 𝛥_𝑖𝑗 = 𝐼 + 𝐵_𝑖𝐾_𝐷𝑗𝐶.

Under the following Assumption

Assumption 4.∀𝑖, 𝑗 = 1, . . . ,ℎ, the matrices 𝛥_𝑖𝑗 = 𝐼 + 𝐵_𝑖𝐾_𝐷𝑗𝐶 ∈ℝ^𝑛×𝑛 are invertible; i.e.:

𝑑𝑒𝑡( 𝛥_𝑖𝑗) ≠ 0.

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Equation (17) can be expressed as follows:

𝑒̇_𝑡(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡))[𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗𝑒_𝑡(𝑡) − 𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) − 𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 𝛥_𝑖𝑗⁻¹𝐻_𝑖𝑗𝑓(𝑡) − 𝛥_𝑖𝑗⁻¹𝑅_𝑖𝑑(𝑡)]

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Substituting (8) into 𝑥̇_𝑓(𝑡) defined in the state space (6), it may be expressed as follows:

𝑥̇_𝑓(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡)) [𝐴_𝑖𝑥_𝑓(𝑡) + 𝐵_𝑖𝐾_𝑃𝑗𝐶𝑒_𝑡(𝑡) + 𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) + 𝐵_𝑖𝐾_𝐷𝑗𝐶𝑒̇_𝑡(𝑡)

+𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) − 𝐻_𝑖𝑗𝑓(𝑡) + 𝐵_𝑖𝑢(𝑡) + 𝑅_𝑖𝑑(𝑡) ] (19) After that, by using (16) and by taking into account the following relationship:

𝐵_𝑖𝐾_𝐷𝑗𝐶𝛥_𝑖𝑗⁻¹= 𝐼 − 𝛥_𝑖𝑗⁻¹ (20) Expression (19) becomes:

𝑥̇_𝑓(𝑡)

= ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗=1

𝜇_𝑗(𝜉(𝑡)) [𝐴_𝑖𝑥_𝑓(𝑡) + (𝐴_𝑖− 𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)𝑒_𝑡(𝑡) + 𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡)

+𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) − 𝛥_𝑖𝑗⁻¹𝐻_𝑖𝑗𝑓(𝑡) + 𝐵_𝑖𝑢(𝑡) + 𝛥_𝑖𝑗⁻¹𝑅_𝑖𝑑(𝑡)] (21) Knowing that 𝑒_𝑠(𝑡) = 𝑥_𝑓(𝑡) − 𝑥̂_𝑓(𝑡) and using the expression of 𝑥̂_𝑓(𝑡) in (7), we can write

𝑒_𝑠(𝑡) = 𝑇₁𝑥_𝑓(𝑡) − 𝑧(𝑡) (22) Since that, for 𝑟𝑎𝑛𝑘[𝐼𝑛 𝐶]^𝑇 = 𝑛, there exists nonsingular matrices 𝑇₁∈ ℝ^𝑛×𝑛 and 𝑇₂∈ ℝ^𝑛×𝑚 such that

[𝑇1 𝑇2] [𝐼_𝑛

𝐶] = 𝐼𝑛 (23)

Then, a particular solution of (23) using the generalized inverse matrix is given by [𝑇₁ 𝑇₂] = [𝐼_𝑛

𝐶]

+ (24)

From (22), the dynamic of the state estimation error obeys the differential equation

𝑒̇_𝑠(𝑡) = 𝑇₁𝑥̇_𝑓(𝑡) − 𝑧̇(𝑡) (25) Substituting the equations of 𝑥̇_𝑓(𝑡) and 𝑧̇(𝑡) respectively from (6) and (7), the equation (25) becomes after some calculations

𝑒̇_𝑠(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

[(𝑇₁𝐴_𝑖+ 𝐸_𝑖𝐶 − 𝑁_𝑖)𝑥_𝑓(𝑡) + (𝑇₁𝐵_𝑖− 𝐺_𝑖)𝑢(𝑡) + 𝑁_𝑖𝑒_𝑠(𝑡) + 𝐵_𝑖𝑒_𝑓(𝑡) + 𝑀_𝑖𝑓(𝑡) + 𝑅̄_𝑖𝑑(𝑡)]

(26)

where

𝑀_𝑖 = 𝑇₁𝐵_𝑖− 𝐵_𝑖 (27)

𝐸_𝑖 = 𝑁_𝑖𝑇₂− 𝐿_𝑖 (28)

(8)

𝑅̄_𝑖 = 𝑇₁𝑅_𝑖 (29)

If the following conditions hold ∀𝑖 = 1, . . . ,ℎ :

𝑁_𝑖 = 𝑇₁𝐴_𝑖+ 𝐸_𝑖𝐶 (30)

𝐺_𝑖 = 𝑇₁𝐵_𝑖 (31)

The dynamic of the state estimation error reduces to:

𝑒̇_𝑠(𝑡) = ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖=1

[𝑁_𝑖𝑒_𝑠(𝑡) + 𝐵_𝑖𝑒_𝑓(𝑡) + 𝑀_𝑖𝑓(𝑡) + 𝑅̄_𝑖𝑑(𝑡)] (32) Finally, the fault estimation error dynamic is defined as follows:

𝑒̇_𝑓(𝑡) = 𝑓̇(𝑡) − 𝑓̂̇(𝑡) (33)

3.2. Stability Analysis

To obtain our result, we need the following lemmas:

Lemma1 [18]. For a positive definite matrixQ; that is: 𝑄 = 𝑄^𝑇 > 0 and a positive scalar 𝜇; the following inequality is true:

2𝑥^𝑇𝑦 ≤1

𝜇𝑥^𝑇𝑄𝑥 + 𝜇𝑦^𝑇𝑄⁻¹𝑦; 𝑥, 𝑦 ∈ℝ^𝑛 (34) Lemma2 [19]. Given real matrices 𝛬, 𝑌 and 𝑍 of appropriate dimensions and a matrix 𝐽 satisfying 𝐽^𝑇𝐽 ≤ 𝐼, then:

𝛬 + 𝑌𝐽𝑍 + 𝑍^𝑇𝐽^𝑇𝑌^𝑇 < 0 (35) If and only if there exists a scalar 𝜂 > 0 verifying the inequality:

𝛬 + 𝜂𝑍^𝑇𝑍 +1

𝜂𝑌𝑌^𝑇 < 0 (36)

which is equivalent to:

(

𝛬 𝑌 𝜂𝑍^𝑇 𝑌^𝑇 −𝜂𝐼 0

𝜂𝑍 0 −𝜂𝐼

) < 0 (37)

Now, the goal is to calculate the observer and the controller gains conjointly in one step to ensure the asymptotic convergence of the faulty T-S fuzzy system outputs to the reference model ones. A basic result is summarized in the following theorem.

Theorem 1. Given positive scalars 𝜎, 𝜇, 𝛽, and 𝜓 and a positive definite matrix 𝛤, the PID-FTTC closed loop system ensures the tracking trajectories of the faulty system (6) to the T-S reference model (5) if there exist symmetric and positive definite matrices 𝑋₁= 𝑃₁⁻¹, 𝑋₂= 𝑃₂⁻¹, 𝑋₃ = 𝑃₃⁻¹, 𝑄₁, 𝑄₂, 𝑄₃ and 𝑄₄ and matrices 𝑊_𝑃𝑗 = 𝐾_𝑃𝑗𝐶𝑋₂, 𝑊_𝐼𝑗 = 𝐾_𝐼𝑗𝐶𝑋₂ , 𝑊_𝐷𝑖𝑗 = (𝐼 + 𝐵_𝑖𝐾_𝐷𝑗𝐶)⁻¹, 𝑆_𝑖= 𝐸_𝑖𝐶𝑋₃, 𝐾_𝑓𝑖 and 𝛷_𝑖 such that the following LMIs are satisfied for all 𝑖, 𝑗, 𝑘 = 1, . . . ,ℎ:

(

𝑋̄_𝑖𝑘 𝑌_𝑖𝑗 𝛹𝑍_𝑖𝑗𝑘^𝑇 𝑃

∗ −𝛹𝐼 0 0

∗ ∗ −𝛹𝐼 0

∗ ∗ ∗ −𝑄)

< 0 (38)

(9)

where

𝑋̄_𝑖𝑘 = (

𝛱_𝑖 𝐴_𝑖𝑋₂ 0 0 0

∗ −𝛽𝐼 𝑋₂ 0 0

∗ ∗ −𝛽𝐼 0 0

∗ ∗ ∗ 𝛩_𝑖 𝐵_𝑖

∗ ∗ ∗ ∗ 𝛴_𝑖𝑘)

(39)

𝑌_𝑖𝑗 = (

𝑊_𝐷𝑖𝑗 0 0 0 0

−𝑊_𝐷𝑖𝑗 0 0 0 0

0 0 0 0 0

0 0 −𝛷_𝑖𝐶 0 0)

(40)

𝑍_𝑖𝑗𝑘^𝑇 = (

0 0 0 0 0

−𝑋₂𝐴_𝑖^𝑇+ 𝑊_𝑃𝑗^𝑇𝐵_𝑖^𝑇 0 0 0 0 𝑊_𝐼𝑗^𝑇𝐵_𝑖^𝑇 0 0 0 0

0 0 𝛯_𝑘^𝑇 0 0

𝐾_𝑓𝑗^𝑇𝐵_𝑖^𝑇 0 0 0 0)

(41)

𝑃 = (

𝑋₁^𝑇 0 0 0 0 0 𝑋₂^𝑇 0 𝐼 0

0 0 0 0 𝐼

0 0 𝑋₃^𝑇 0 0

0 0 0 0 0)

(42)

and

𝑄 =

( 3

𝜇𝑄₁ 0 0 0 0

0 2

𝜇𝑄₂ 0 0 0

0 0 2

𝜇𝑄₃ 0 0

0 0 0 𝛽𝐼 0

0 0 0 0 𝛽𝐼)

(43)

with

𝛱_𝑖= 𝐴_𝑖𝑋₁+ 𝑋₁𝐴_𝑖^𝑇 (44)

𝛩_𝑖 = 𝑇₁𝐴_𝑖𝑋₃+ 𝑋₃𝐴_𝑖^𝑇𝑇₁^𝑇+ 𝑆_𝑖+ 𝑆_𝑖^𝑇 (45) 𝛴_𝑖𝑘 = −1

𝜎(𝛷_𝑖𝐶𝐵_𝑘+ 𝐵_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇) + 3

𝜇𝜎𝑄₄ (46)

𝛯_𝑘 = 𝑋₃+1

𝜎(𝑇₁𝐴_𝑘𝑋₃+ 𝑆_𝑘) (47)

The controller gain matrices are defined by:

𝐾_𝑃𝑗 = 𝑊_𝑃𝑗𝑃₂𝐶⁻¹ (48)

(10)

𝐾_𝐼𝑗 = 𝑊_𝐼𝑗𝑃₂𝐶⁻¹ (49)

𝐾_𝐷𝑗 = 𝐵_𝑖⁻¹(𝑊_𝐷𝑖𝑗⁻¹− 𝐼)𝐶⁻¹ (50) The observer gain matrices can be obtained by:

𝑁_𝑖 = 𝑇₁𝐴_𝑖+ 𝐸_𝑖𝐶 (51)

𝐿_𝑖 = 𝑁_𝑖𝑇₂− 𝐸_𝑖 (52)

𝐺_𝑖 = 𝑇₁𝐵_𝑖 (53)

Proof of Theorem 1. In this proof, we use the theory of Lyapunov to demonstrate the stability of the closed-loop system illustrated in Fig. 1. Then, the problem is turned into an optimization problem through the use of LMI so as to determine the unknown matrices of both the proposed FTC controller and the observer.

Thus, let us consider the following quadratic Lyapunov function:

𝑉(𝑥̃(𝑡)) = 𝑥̃^𝑇(𝑡)𝑃𝑥̃(𝑡) (54)

where 𝑥̃(𝑡) is the augmented state vector defined by (11) and 𝑃 = 𝑑𝑖𝑎𝑔 (𝑃₁, 𝑃₂, 𝑃₂, 𝑃₃,¹

𝜎𝛤⁻¹) > 0 is a positive definite matrix with appropriate dimensions.

The time derivative of the quadratic Lyapunov function (54) leads to

𝑉̇(𝑥̃(𝑡)) = 𝑥̇_𝑓^𝑇(𝑡)𝑃₁𝑥_𝑓(𝑡) + 𝑥_𝑓^𝑇(𝑡)𝑃₁𝑥̇_𝑓(𝑡) + 𝑒̇_𝑡^𝑇(𝑡)𝑃₂𝑒_𝑡(𝑡) + 𝑒_𝑡^𝑇(𝑡)𝑃₂𝑒̇_𝑡(𝑡) + 𝑥̇_𝑡^𝑇(𝑡)𝑃₂𝑥_𝑡(𝑡) + 𝑥_𝑡^𝑇(𝑡)𝑃₂𝑥̇_𝑡(𝑡) + 𝑒̇_𝑠^𝑇(𝑡)𝑃₃𝑒_𝑠(𝑡) + 𝑒_𝑠^𝑇(𝑡)𝑃₃𝑒̇_𝑠(𝑡) +1

𝜎𝑒̇_𝑓^𝑇(𝑡)𝛤⁻¹𝑒_𝑓(𝑡) +1

𝜎𝑒_𝑓^𝑇(𝑡)𝛤⁻¹𝑒̇_𝑓(𝑡) (55)

By considering the expressions (21) of 𝑥̇_𝑓(𝑡), (18) of 𝑒̇_𝑡(𝑡), (12) of 𝑥̇_𝑡(𝑡), (32) of 𝑒̇_𝑠(𝑡) and (33) of 𝑒̇𝑓(𝑡) and by taking into account the fault estimation of 𝑓̂̇(𝑡) in (7), 𝑉̇(𝑥̃(𝑡)) becomes

𝑉̇(𝑥̃(𝑡)) = ∑ 𝜇_𝑖𝑗𝑘(𝜉(𝑡))

ℎ

𝑖,𝑗,𝑘=1

[𝑥_𝑓^𝑇(𝑡)(𝑃₁𝐴_𝑖+ 𝐴_𝑖^𝑇𝑃₁)𝑥_𝑓(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁(𝐴_𝑖

− 𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)𝑒_𝑡(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁𝐵_𝑖𝑢(𝑡)

− 2𝑥𝑓𝑇(𝑡)𝑃1𝛥𝑖𝑗−1𝐻𝑖𝑗𝑓(𝑡) + 2𝑥𝑓𝑇(𝑡)𝑃1𝛥𝑖𝑗−1𝑅𝑖𝑑(𝑡) + 𝑒𝑡𝑇(𝑡)(𝑃2𝛥𝑖𝑗−1𝐴𝑖𝑗

+ (𝛥𝑖𝑗−1𝐴𝑖𝑗)^𝑇𝑃2)𝑒𝑡(𝑡) + 2𝑒𝑡𝑇(𝑡)(𝑃2− 𝑃2𝛥𝑖𝑗−1𝐵𝑖𝐾𝐼𝑗𝐶)𝑥𝑡(𝑡)

− 2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝐻_𝑖𝑗𝑓(𝑡)

− 2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝑅_𝑖𝑑(𝑡) + 𝑒_𝑠^𝑇(𝑡)(𝑃₃𝑁_𝑖+ 𝑁_𝑖^𝑇𝑃₃)𝑒_𝑠(𝑡) + 2𝑒𝑠𝑇(𝑡)𝑃3𝑀𝑖𝑓(𝑡) + 2𝑒𝑠𝑇(𝑡)(𝑃3𝐵𝑖− 𝐶^𝑇𝛷𝑖𝑇 −1

𝜎𝑁𝑘𝑇𝐶^𝑇𝛷𝑖𝑇)𝑒𝑓(𝑡)

−2

𝜎𝑒_𝑓^𝑇(𝑡)𝛷_𝑖𝐶𝑀_𝑘𝑓(𝑡) + 2𝑒_𝑠^𝑇(𝑡)𝑃₃𝑅̄_𝑖𝑑(𝑡) −1

𝜎𝑒_𝑓^𝑇(𝑡)(𝛷_𝑖𝐶𝐵_𝑘 + 𝐵_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇)𝑒_𝑓(𝑡) +2

𝜎𝑒_𝑓^𝑇(𝑡)𝛷_𝑖𝐶𝑅̄_𝑘𝑑(𝑡) +2

𝜎𝑒_𝑓^𝑇(𝑡)𝛤⁻¹𝑓̇(𝑡)]

(56)

(11)

Under Assumption 3, we apply Lemma 1 so as to get the following term inequalities from (56):

2𝑥𝑓𝑇(𝑡)𝑃1𝐵𝑖𝑢(𝑡) ≤ 1

𝜇𝑥𝑓𝑇(𝑡)𝑄1𝑥𝑓(𝑡) + 𝜂1𝑖𝑗 (57)

−2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝐻_𝑖𝑗𝑓(𝑡) ≤1

𝜇𝑥_𝑓^𝑇(𝑡)𝑄₁𝑥_𝑓(𝑡) + 𝜂_2𝑖𝑗 (58) 2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝑅_𝑖𝑑(𝑡) ≤1

𝜇𝑥_𝑓^𝑇(𝑡)𝑄₁𝑥_𝑓(𝑡) + 𝜂_3𝑖𝑗 (59) 2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝐻_𝑖𝑗𝑓(𝑡) ≤1

𝜇𝑒_𝑡^𝑇(𝑡)𝑄₂𝑒_𝑡(𝑡) + 𝜂_4𝑖𝑗 (60)

−2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝑅_𝑖𝑑(𝑡) ≤1

𝜇𝑒_𝑡^𝑇(𝑡)𝑄₂𝑒_𝑡(𝑡) + 𝜂_5𝑖𝑗 (61) 2𝑒_𝑠^𝑇(𝑡)𝑃₃𝑀_𝑖𝑓(𝑡) ≤1

𝜇𝑒_𝑠^𝑇(𝑡)𝑄₃𝑒_𝑠(𝑡) + 𝜂_6𝑖 (62) 2𝑒_𝑠^𝑇(𝑡)𝑃₃𝑅̄_𝑖𝑑(𝑡) ≤1

𝜇𝑒_𝑠^𝑇(𝑡)𝑄₃𝑒_𝑠(𝑡) + 𝜂_7𝑖 (63)

−2

𝜎𝑒_𝑓^𝑇(𝑡)𝛷_𝑖𝐶𝑀_𝑘𝑓(𝑡) ≤1

𝜇𝑒_𝑓^𝑇(𝑡)𝑄₄𝑒_𝑓(𝑡) + 𝜂_8𝑖𝑘 (64) 2

𝜎𝑒_𝑓^𝑇(𝑡)𝛷_𝑖𝐶𝑅̄_𝑘𝑑(𝑡) ≤1

𝜇𝑒_𝑓^𝑇(𝑡)𝑄₄𝑒_𝑓(𝑡) + 𝜂_9𝑖𝑘 (65) 2

𝜎𝑒_𝑓^𝑇(𝑡)𝛤⁻¹𝑓̇(𝑡) ≤1

𝜇𝑒_𝑓^𝑇(𝑡)𝑄₄𝑒_𝑓(𝑡) + 𝜂₁₀ (66) where the scalars 𝜂_1𝑖𝑗, 𝜂_2𝑖𝑗, 𝜂_3𝑖𝑗, 𝜂_4𝑖𝑗, 𝜂_5𝑖𝑗, 𝜂_6𝑖, 𝜂_7𝑖, 𝜂_8𝑖𝑘, 𝜂_9𝑖𝑘 and 𝜂₁₀ are expressed as follows:

𝜂_1𝑖𝑗 = 𝜇𝛼₁²𝜆_𝑚𝑎𝑥(𝐵_𝑖^𝑇𝑃₁𝑄₁⁻¹𝑃₁𝐵_𝑖) (67) 𝜂_2𝑖𝑗= 𝜇𝛼₂²𝜆_𝑚𝑎𝑥(𝐻_𝑖𝑗^𝑇(∆_𝑖𝑗⁻¹)^𝑇𝑃₁𝑄₁⁻¹𝑃₁∆_𝑖𝑗⁻¹𝐻_𝑖𝑗) (68) 𝜂_3𝑖𝑗 = 𝜇𝛼₄²𝜆_𝑚𝑎𝑥(𝑅_𝑖^𝑇(∆_𝑖𝑗⁻¹)^𝑇𝑃₁𝑄₁⁻¹𝑃₁∆_𝑖𝑗⁻¹𝑅_𝑖) (69) 𝜂_4𝑖𝑗 = 𝜇𝛼₂²𝜆_𝑚𝑎𝑥(𝐻_𝑖𝑗^𝑇(∆_𝑖𝑗⁻¹)^𝑇𝑃₂𝑄₂⁻¹𝑃₂∆_𝑖𝑗⁻¹𝐻_𝑖𝑗) (70) 𝜂_5𝑖𝑗 = 𝜇𝛼₄²𝜆_𝑚𝑎𝑥(𝑅_𝑖^𝑇(∆_𝑖𝑗⁻¹)^𝑇𝑃₂𝑄₂⁻¹𝑃₂∆_𝑖𝑗⁻¹𝑅_𝑖) (71) 𝜂_6𝑖 = 𝜇𝛼₂²𝜆_𝑚𝑎𝑥(𝑀_𝑖^𝑇𝑃₃𝑄₃⁻¹𝑃₃𝑀₁) (72) 𝜂_7𝑖 = 𝜇𝛼₄²𝜆_𝑚𝑎𝑥(𝑅̅_𝑖^𝑇𝑃₃𝑄₃⁻¹𝑃₃𝑅̅₁) (73) 𝜂_8𝑖𝑘 = 𝜇𝛼₂²𝜆_𝑚𝑎𝑥(𝑀_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇𝑄₄⁻¹𝛷_𝑖𝑀_𝑘) ⁽⁷⁴⁾

(12)

𝜂_9𝑖𝑘 = 𝜇𝛼₄²𝜆_𝑚𝑎𝑥(𝑅̅_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇𝑄₄⁻¹𝛷_𝑖𝐶𝑅̅_𝑘) ⁽⁷⁵⁾

𝜂₁₀ = 𝜇𝛼₂²𝜆_𝑚𝑎𝑥(Γ^−1𝑇𝑄₄⁻¹Γ⁻¹) (76)

By taking into consideration the inequalities (57)-(66) and the equation (56), 𝑉̇(𝑥̃(𝑡)) can be bounded as follows:

𝑉̇(𝑥̃(𝑡)) ≤ ∑ 𝜇_𝑖𝑗𝑘(𝜉(𝑡))

ℎ

𝑖,𝑗,𝑘=1

[𝑥_𝑓^𝑇(𝑡)(𝑃₁𝐴_𝑖+ 𝐴_𝑖^𝑇𝑃₁+3

𝜇𝑄₁)𝑥_𝑓(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁(𝐴_𝑖

−𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)𝑒_𝑡(𝑡) + 2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶𝑥_𝑡(𝑡) +2𝑥_𝑓^𝑇(𝑡)𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) + 𝑒_𝑡^𝑇(𝑡)(𝑃₂𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗+ (𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)^𝑇𝑃₂+2

𝜇𝑄₂) 𝑒_𝑡(𝑡) +2𝑒_𝑡^𝑇(𝑡)(𝑃₂− 𝑃₂𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶)𝑥_𝑡(𝑡) − 2𝑒_𝑡^𝑇(𝑡)𝑃₂𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗𝑒_𝑓(𝑡) +𝑒_𝑠^𝑇(𝑡) (𝑃₃𝑁_𝑖+ 𝑁_𝑖^𝑇𝑃₃+2

𝜇𝑄₃) 𝑒_𝑠(𝑡) + 2𝑒_𝑠^𝑇(𝑡) (𝑃₃𝐵_𝑖− 𝐶^𝑇𝛷_𝑖^𝑇−1

𝜎𝑁_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇) 𝑒_𝑓(𝑡)

−1

𝜎𝑒_𝑓^𝑇(𝑡) (𝛷_𝑖𝐶𝐵_𝑘+ 𝐵_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇+ 3

𝜇𝜎𝑄₄) 𝑒_𝑓(𝑡)] + 𝛿

(77)

where the scalar 𝛿 is the maximum value over 𝑖, 𝑗, and 𝑘 such that:

𝛿 = 𝑚𝑎𝑥

𝑖,𝑗,𝑘 (𝜂_1𝑖𝑗+ 𝜂_2𝑖𝑗+ 𝜂_3𝑖𝑗+ 𝜂_4𝑖𝑗+ 𝜂_5𝑖𝑗+ 𝜂_6𝑖+ 𝜂_7𝑖+ 𝜂_8𝑖𝑘+ 𝜂_9𝑖𝑘+ 𝜂₁₀) ₍₇₈₎ At this stage, we can rewrite the inequality (77) under the following reduced form:

𝑉̇(𝑥̃(𝑡)) ≤ 𝑥̃^𝑇(𝑡) ( ∑ 𝜇_𝑖𝑗𝑘(𝜉(𝑡))𝛬_𝑖𝑗𝑘

ℎ

𝑖,𝑗,𝑘=1

) 𝑥̃(𝑡) + 𝛿 (79)

where 𝛬_𝑖𝑗𝑘 is a matrix defined as follows

𝛬_𝑖𝑗𝑘 = (

𝛱̃_𝑖 𝑃₁(𝐴_𝑖− 𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗) 𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶 0 𝑃₁𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗

∗ 𝛺̃_𝑖𝑗 𝑃₂− 𝑃₂𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶 0 −𝑃₂𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗

∗ ∗ 0 0 0

∗ ∗ ∗ 𝛩̃_𝑖 𝛯̃_𝑖𝑘

∗ ∗ ∗ ∗ 𝛴_𝑖𝑘 )

(80)

with

𝛱̃_𝑖 = 𝑃₁𝐴_𝑖+ 𝐴_𝑖^𝑇𝑃₁+3

𝜇𝑄₁ (81)

𝛺̃_𝑖𝑗= 𝑃₂𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗+ (𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)^𝑇𝑃₂+2

𝜇𝑄₂ (82)

𝛩̃_𝑖 = 𝑃₃𝑁_𝑖+ 𝑁_𝑖^𝑇𝑃₃+2

𝜇𝑄₃ (83)

𝛯̃_𝑖𝑘 = 𝑃₃𝐵_𝑖− 𝐶^𝑇𝛷_𝑖^𝑇−1

𝜎𝑁_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇 (84)

(13)

𝛴_𝑖𝑘 = −1

𝜎(𝛷_𝑖𝐶𝐵_𝑘+ 𝐵_𝑘^𝑇𝐶^𝑇𝛷_𝑖^𝑇) + 3

𝜇𝜎𝑄₄ (85)

If the following constraint is satisfied:

∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗,𝑘=1

𝜇_𝑗(𝜉(𝑡))𝜇_𝑘(𝜉(𝑡))𝛬_𝑖𝑗𝑘 < 0 (86) and if there exists a positive scalar 𝜀 defined by:

𝜀 = 𝑚𝑖𝑛 𝜆 (− ∑ 𝜇_𝑖(𝜉(𝑡))

ℎ

𝑖,𝑗,𝑘=1

𝜇_𝑗(𝜉(𝑡))𝜇_𝑘(𝜉(𝑡))𝛬_𝑖𝑗𝑘) < 0 (87) and satisfying the constraint: 𝜀‖𝑥̃(𝑡)‖²> 𝛿;∀𝑡 ≥ 0

we deduce that the time derivative of the Lyapunov function 𝑉̇(𝑥̃(𝑡)) will be bounded as follows:

𝑉̇(𝑥̃(𝑡)) ≤ −𝜀‖𝑥̃(𝑡)‖²+ 𝛿 ⁽⁸⁸⁾

Consequently, we can say that 𝑉̇(𝑥̃(𝑡)) is negative.

Based on the Lyapunov stability theory, this proves that the augmented system 𝑥̃(𝑡) is stable, meaning that the faulty state vector 𝑥_𝑓(𝑡), the state vector 𝑥_𝑡(𝑡), the state of the tracking error 𝑒_𝑡(𝑡), the state of the estimation error 𝑒_𝑠(𝑡), and the fault estimation error 𝑒_𝑓(𝑡) are bounded.

By considering the constraint (), we define a matrix:

𝑋 = (

𝑃₁⁻¹ 0 0 0 0

0 𝑃₂⁻¹ 0 0 0 0 0 𝑃₃⁻¹ 0 0

0 0 0 𝐼 0

0 0 0 0 𝐼)

> 0 (89)

such that 𝛹_𝑖𝑗𝑘= 𝑋𝛬_𝑖𝑗𝑘𝑋 < 0.

Then, consider the following change of variables

𝑋₁= 𝑃₁⁻¹ (90)

𝑋2= 𝑃2−1 (91)

𝑋₃= 𝑃₃⁻¹ (92)

and computing 𝛹_𝑖𝑗𝑘 < 0 that holds

𝛹_𝑖𝑗𝑘 = (

𝑋₁𝛱̃_𝑖𝑋₁ (𝐴_𝑖− 𝛥_𝑖𝑗⁻¹𝐴_𝑖𝑗)𝑋₂ 𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝐼𝑗𝐶𝑋₂ 0 𝛥_𝑖𝑗⁻¹𝐵_𝑖𝐾_𝑓𝑗

∗ 𝑋2𝛺̃𝑖𝑗𝑋2 𝑋2− 𝛥𝑖𝑗−1

𝐵𝑖𝐾𝐼𝑗𝐶𝑋2 0 𝛥𝑖𝑗−1

𝐵𝑖𝐾𝑓𝑗

∗ ∗ 0 0 0

∗ ∗ ∗ 𝑋₃𝛩̃_𝑖𝑋₃ 𝑋₃𝛯̃_𝑖𝑘

∗ ∗ ∗ ∗ 𝛴_𝑖𝑘 )

(93)

where

𝑋₁𝛱̃_𝑖𝑋₁= 𝛱_𝑖+3

𝜇𝑋₁𝑄₁𝑋₁ (94)