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Abstract

Over the past few decades, there have been increasing research activities in fault diagnosis (FD) and fault-tolerant control (FTC) for switched hybrid systems. This paper addresses the problem of active-fault tolerant control (AFTC) for switched hybrid systems subject to actuator faults to enhance system security and keep system stability. The proposed FTC is designed by adding the state feedback control with integral action to an additive control law which requires accurate fault estimation to compensate for the fault effect. Thus, a data-based projection method (DPM) is extended (EDPM) based on inputs and outputs measures to estimate the fault without using mathematical models. The synthesis of the state feedback control with integral action is proposed for recovering the desired performances. It integrates a set of controllers corresponding to a set of partial models to design a set of switching control laws. Indeed, new linear matrix inequalities (LMIs) using Lyapunov stability analysis are proposed to find the optimal values of the control gains matrices and keeping system stability. A comparative study of the proposed FTC with existing work is given to show the effectiveness of the proposed technique.

Keywords

AFTC switched hybrid system actuator fault EDPM LMIs

Introduction

Switched systems belong to hybrid systems, which are characterized by many subsystems (modes) and a switching rule specifying the switching between them.¹ Over the past few decades, the synthesis and the study of the control of switched systems received much attention because of their applications in several practical engineering systems, namely, power electronics, chemical processes, network systems, DC/DC converters, and so on.² Many achievements have been proposed, readers can refer to Sun and Ge,³ Ocampo-Martinez and Puig,⁴ and the references therein. However, real industrial processes are unavoidably affected by unexpected faults in actuators or sensors. When a fault appears, the characteristics of the system causes changes, and enduring the instability.⁵ In this context, several techniques and strategies have been proposed to deal with fault diagnosis (FD) and fault-tolerant control (FTC) theories.^6,7

Most current studies works have targeted the improvement of FTC for continuous-time systems. In Ladel et al.,⁸ a sturdy fault tolerant control (RFTC) design is proposed for continuous-time switched systems. The goal is to synthesize the couple (controller, observer) to stabilize the switched systems issue to actuator faults. A novel LMIs is proposed to decide the controller and observer gains in a single step. The instance given suggests that the acquired consequences are powerful withinside the presence of risky modes and simultaneous faults. Due to the speedy advances in computers, numerous works have focused on the improvement of the discrete-time model of the switched hybrid device. However, the discretization of the continuous switched system can harm the invariance properties. On the only hand, to make certain fault compensation for the discreet switched system, the observer approach has been utilized in current years. In Zhu et al.,⁹ an AFTC for discrete-time systems withinside the finite-frequency area is proposed. The observers are designed via way of means of thinking about the descriptor device approach to examine the stability in dynamic error systems primarily based totally on GKYP lemma. In Liao et al.,¹⁰ the problem of an observer primarily based totally on FTC for discrete-time switched systems issue to a combined switching scheme is addressed. A combined switching regulation is proposed to keep away from the chattering phenomenon of augmented error switched affine system, and a Lyapunov function is used to make certain system stability. Furthermore, those methods can be increase system complexity, in particular while fault and state estimators are concurrently used. On the other hand, the additive FTC is an effective technique to deal with fault compensation. The significance of the usage of the additive FTC arises via their inherent potential to react efficiently at some stage in a brief duration among the fault incidence and the overall performance restoration.¹¹

Thus, the main goal of this work is to design an additive FTC for a discreet-time switched system (Table 1). The principal benefits are as follows: miss assumption for the information of the parameter values to estimate the fault and no previous information approximately the dynamical evolution of the actuator fault. This makes the proposed estimate technique intrinsically strong to parameters values and easy to apply. Besides, a brand new LMIs is proposed the use of the Lyapunov characteristic to compute the control gains which lets in system stabilization and faults compensation. In Yang et al.,¹² a hybrid data-driven FD strategy is proposed based on multi-sequence residual analysis and OC-SVM, which is applied to navigation sensors. In Sun et al.,¹³ a fault observer-based hierarchical integrated control algorithm is presented to process the actuator faults and coupling properties of vehicle chassis model. In survey papers, two kinds of FTC are mentioned^14,15: passive approaches (PFTC) and active approaches (AFTC). PFTC uses the same controller for normal and faulty cases while the AFTC adjusts the controller’s structure to reconfigure the controller. AFTC requires fault identification as well as an updated control mechanism^16,17. Generally speaking, fault identification are classified into two kinds: data-driven method and model-based method.^18,19

Table 1.

Table of acronyms.

$abbreviations$	$explanation$
FTC	fault-tolerant control
AFTC	active fault-tolerant control
PFTC	passive fault-tolerant control
FD	fault diagnosis
DPM	data-based projection method
EDPM	extended data-based projection method
LMIs	linear matrix inequalities

The former uses the available input-output data to have some behaviors properties such as neural networks and classification techniques and so on. However, the latter uses the model of the studied system to find the fault indicator, such as the observer based approach, parity space approach, parameter estimation techniques, and so on.^20,21.

To the best of our knowledge, the overall performance reached via way of means of a PFTC can in no way be most reliable as compared with an AFTC for all design scenarios.²² If we strive to synthesize a PFTC to deal with an immoderate variety of faults, the general conservatism will increase.²³ Indeed, the AFTC is greater efficient in treating distinctive kinds of faults.²⁴ Motivated through this situation, the novelties of this paintings is to design an AFTC of a discreet-time switched system that offers an amazing compromise among simplicity of the control system design and the closed-loop favored performances. In Noura et al.,¹¹ an additive FTC is evolved for a linear system. Indeed, its design is simple and appropriate for the purpose of our paintings as compared with others current approaches. The idea is to mix an additive control regulation to the nominal control. Thus, the first part of our proposed FTC regulation is designed to assure the stability by an most efficient computation of the control profits the usage of a set of novel LMIs generated through Lyapunov function. Then, the goal of the second one element is to compensate for the fault impact the usage of the additive control primarily based totally at the proposed fault estimation method and including it to the nominal one. In different words, the important thing function of this paintings may be recapitulated as follows: first, a brand new LMIs is evolved to compute the manage profits for flawlessly tracking overall performance. Switched system introduces partial models tailored to every modes “operating zone”. Thanks to the functions of switched hybrid systems, a bank of controllers related to every mode are designed to generate a set of control laws to account for the fault. Secondly, the estimation module includes residual “fault indicator” technology the usage of most effective the inputs-outputs data is advanced. The computation of the additive control begins involved primarily based totally on fault estimation module that’s capable of compensate for the fault impact at the system as soon as the fault is located. The proposed fault estimation approach is reached as quickly as feasible to keep away from large losses in system performances. Then, we show the effectiveness of the proposed FTC approach via a numerical instance and we evaluate its overall performance with present works (additive state feedback control) proposed in Yahia et al.^25,26 Many papers dealt with the design of the additive FTC. In the literature, the control gains are obtained using different strategies such as, the pole placement, LQ optimization^11,27, Lambert W method,²⁸ Lyapunov redesign principle²⁹ or assumed that the gain of the integral term is equal to one.²⁶ To the best of our knowledge, to tolerate for the fault effect these techniques require a significant response time and an important deviation of the control signal when of fault occurrence is notified. Motivated by this situation, this papers’ the main challenge consist in the design of an AFTC to compensate for the fault effect with a minimum reconfiguration time and ensure perfectly tracking performances. Besides, most of these works address the attitude tracking control issue and few results treated the stability of switched hybrid in the presence of faults. So, an important additional difficulties and challenge is the study of the stability of switched system subject to faults. An effective solution is to develop a novel LMIs using the Lyapunov function to elaborate the necessary conditions to improve the closed-loop performances in terms of trajectory tracking and fault compensation. The splendor of the proposed FTC structure arises via its inherent capacity to react correctly at some point of a brief duration among the fault incidence and the overall performance recovery. Then, computation of an additive control, based on the estimation fault to ensure fault compensation goals. Indeed, the fast fault estimation procedure is proposed using a data-based projection method presented in previous work by the authors for sensor fault estimation.³⁰ So, another challenge is to extend the procedure to deal with actuator fault. The proposed approach (EDPM) for actuator fault estimation has an additional constraint compared to the sensor fault estimation approach which is the input inversion analysis. The concept of an input inverse with delay $v$ is defined, and necessary conditions are derived for the existence of such an inverse. Construction technique to input inverse with delay is formulated in recursive form by analogy with known result in Masseyy and Sain.³¹ This work is organized as follows. In section 2, the problem statement is presented. In section 3, we present the additive state feedback control proposed in our previous work. In section 4, the proposed FTC via LMIs technique is developed. In section 5, the extended data-based projection method is developed. To prove the effectiveness of the proposed method, a comparative study is given in section 6. Section 7 concludes the paper.

Problem statement

Consider the dynamic switched system with linear discrete-time modes described by:

{\begin{matrix} x (k + 1) = A_{j} x (k) + B_{j} (u (k) + f_{a} (k)) \\ y (k) = C_{j} x (k) \end{matrix}

(1)

where, $x (k) \in ℜ^{n}$ , $y (k) \in ℜ^{p}$ and $u (k) \in ℜ^{m}$ are respectively the state, the output and the control input. $f_{a} (k)$ represents the actuator fault. $j$ is the mode index and $M$ is the number of modes ( $j \in μ = {1, 2, . . . M}$ ). $A_{j} \in ℜ^{n \times n}, B_{j} \in ℜ^{n \times m}, C_{j} \in ℜ^{p \times n}$ are the parameter matrices associated with the mode indexed by $j$ .

The following assumptions are performed:

A.1: The switching signal is known.

A.3: Matrices $A_{j}$ are stables.

Consider the switched system described by (1), this paper proposes a new AFTC based on the design of nominal control law, fault estimation, and modification of the control law to allow actuator fault compensation. Fault estimation approach is proposed using a data-based approach. A bank of controllers, corresponding to a set of modes to generate a set of switching control is built. Using the Lyapunov function, a novel adequate criterion is developed by means of linear matrix inequalities (LMIs). The obtained LMIs are then solved for obtaining the controller gain matrices to compensate for the fault effect.

Additive state feedback control

Given the system described by (1), in our previous works^25,26 an additive state feedback control is proposed to treat the problem of fault-tolerant control for switched systems.

u (k) = K_{j} x (k) + G_{j} Z (k) - B_{j}^{+} B_{j} {\hat{f}}_{a} (k)

(2)

where, $Z (k)$ is un integrator, ${\hat{f}}_{a}$ is the estimated fault, $B_{j}$ is the parameter matrices associated with the mode indexed by $j$ and $B_{j}^{+}$ is the pseudo inverse of the matrix $B_{j}$ . In fact, the gain matrices $G_{j}$ of the integral term $Z (k)$ can be obtained by the pole placement,¹¹ Lyapunov redesign principle,²⁹ or assumed that they are equal to one.^27,28 In a previous works,^25,26 we assumed that $G_{j}$ are equal to one and the gain matrices $K_{j}$ of each mode are computed by the following LMIs developed:

Theorem 1: If there is definite symmetric positive matrix $P$ , and $H_{j}$ with appropriate dimension solution of following LMIs:

(\begin{matrix} - P & A_{j} P + B_{j} H_{j} \\ {(A_{j} P + B_{j} H_{j})}^{T} & - P \end{matrix}) < 0

(3)

Then, $K_{j}$ are given by:

K_{j} = H_{j} P^{- 1}

In fact, the proposed additive state feedback control don’t always shows a satisfactory results for closed-loop performance in terms of trajectory tracking and fault tolerance. Therefore, it is necessary to modify the algorithm that takes into account the gain matrices $G_{j}$ of the integral term $Z (k)$ .

Proposed fault tolerant control

The proposed FTC law is composed of two parts: state feedback integral control and additive control laws performed using the fault estimation approach called the “data-based projection method”. First, we focused on the design of a nominal tracking controller. Thus, a state feedback integral control is proposed to allow trajectory tracking.

State feedback control with integral action

A state feedback control with integral action is developed to ensure the trajectory tracking. Consider the following state feedback control:

u (k) = K_{j} x (k)

(4)

where, $K_{j}$ are the gain matrices of each mode which should be determined to stabilize the system. The goal is to design a partial controller to make the system output follows the reference input as close as possible. Therefore, a comparator integrator $Z (k)$ is added to (4) according to the following relation:

Z (k + 1) = Z (k) + T_{e} (y_{r} (k) - y (k))

(5)

where, $y_{r} (k)$ and $T_{e}$ are the reference input and the sampling interval respectively.

Hence, the feedback control law, which guarantees both stability and dynamic behavior of the closed-loop system, is modified as following:

u (k) = {\tilde{K}}_{j} \tilde{x} (k)

(6)

where, ${\tilde{K}}_{j}$ and $\tilde{x} (k)$ are the new feedback gain and the augmented state respectively.

{\tilde{K}}_{j} = [\begin{matrix} K_{j} & G_{j} \end{matrix}]

\tilde{x} (k) = [\begin{matrix} x (k) \\ Z (k) \end{matrix}]

$G_{j}$ denotes the gain matrices of $Z (k)$ of each mode. The stability of the mode indexed by $j$ is studied with the Lyapunov approach in terms of linear matrix inequalities. $K_{i}$ and $G_{i}$ are computed if a symmetric definite positive matrix $P = P^{T} > 0$ , matrices $F_{j}$ and $H_{j}$ exist and satisfying the following theorem.

Theorem 2: If there is symmetric definite positive matrix $P$ , matrices $F_{j}$ and $H_{j}$ solution of following LMIs:

{\begin{matrix} P > 0 \\ [\begin{matrix} Q & 0 & {QA}_{j}^{T} + F_{j} B_{j}^{T} & 0 \\ 0 & Q & H_{j} B_{j}^{T} & Q \\ A_{j} Q^{T} + B_{j} F_{j}^{T} & B_{j} H_{j}^{T} & Q & 0 \\ 0 & Q^{T} & 0 & Q \end{matrix}] \end{matrix} > 0

(7)

where, $P^{- 1} = [\begin{matrix} Q & 0 \\ 0 & Q \end{matrix}]$ then, the switched system (1) is asymptotically stable and the gain matrices $K_{j}$ and $G_{j}$ of the state feedback control with integral action of each mode (6) are given by:

\begin{matrix} K_{j} = (Q_{j}^{- 1} F_{j})^{T} \\ G_{j} = (Q_{j}^{- 1} H_{j})^{T} \end{matrix}

Proof: Using (1) and (6) the augmented state model is as follows:

\tilde{x} (k + 1) = [\begin{matrix} A_{j} + B_{j} K_{j} & B_{j} G_{j} \\ - T_{e} C_{j} & I \end{matrix}] \tilde{x} (k) + [\begin{matrix} 0 \\ T_{e} y_{r} (k) \end{matrix}]

(8)

The equation (8) is equivalent to the following equation:

\tilde{x} (k + 1) = [\begin{matrix} A_{j} + B_{j} K_{j} & B_{j} G_{j} \\ 0 & I \end{matrix}] \tilde{x} (k) + [\begin{matrix} 0 \\ T_{e} (y_{r} (k) - C_{j} x (k)) \end{matrix}]

(9)

Or, when $k \to + \infty$ the outputs of the system converge to the reference inputs $(y_{r} (k) - C_{j} x (k) \to 0)$ . Then, the augmented state model is given by:

\tilde{x} (k + 1) = {\tilde{A}}_{j} \tilde{x} (k)

(10)

where, ${\tilde{A}}_{j} = [\begin{matrix} A_{j} + B_{j} K_{j} & B_{j} G_{j} \\ 0 & I \end{matrix}]$ .

In order to find the optimal values of $K_{j}$ and $G_{j}$ , the following candidate Lyapunov function is used:

V (\tilde{x} (k), k) = {\tilde{x}}^{T} (k) P \tilde{x} (k)

(11)

where, $Δ V (\tilde{x})$ is given by:

Δ V (\tilde{x}) = V (\tilde{x} (k + 1)) - V (\tilde{x} (k))

(12)

So,

Δ V (\tilde{x}) = {\tilde{x}}^{T} (k + 1) P \tilde{x} (k + 1) - {\tilde{x}}^{T} (k) P \tilde{x} (k)

(13)

Replacing (10) in (13):

Δ V (\tilde{x}) = ({\tilde{A}}_{j} \tilde{x} (k))^{T} P ({\tilde{A}}_{j} \tilde{x} (k)) - {\tilde{x}}^{T} (k) P \tilde{x} (k)

(14)

Then, the following equation is obtained:

Δ V (\tilde{x}) = {\tilde{x}}^{T} (k) [{\tilde{A}}_{j}^{T} P {\tilde{A}}_{j} - P] \tilde{x} (k)

(15)

Or, the switched system is asymptotically stable, if $\forall \tilde{x} (k) \neq 0$ , there are symmetric definite positive matrix $P = P^{T} > 0$ , matrices $K_{j}$ and $G_{j}$ which verify:

Δ V (\tilde{x}) < 0

(16)

The inequality (16) is equivalent to:

{\tilde{A}}_{j}^{T} P {\tilde{A}}_{j} - P < 0

(17)

Using the Schur Complement, the following LMIs is obtained to find $K_{j}$ and $G_{j}$ :

{\begin{matrix} P > 0 \\ [\begin{matrix} P & {\tilde{A}}_{j}^{T} \\ {\tilde{A}}_{j} & P^{- 1} \end{matrix}] \end{matrix} > 0

(18)

It can be seen from the inequality (18), there exist a nonlinear term which can not be solved using the “MATLAB” toolbox. In order to avoid this problem, we supposed to do the post multiplying and pre multiplying of (18) with the diagonal ${P^{- 1}, I}$ . $I$ denotes the identity matrix.

Performing the post multiplying with the diagonal ${P^{- 1}, I}$ :

[\begin{matrix} P^{- 1} & 0 \\ 0 & I \end{matrix}] \times [\begin{matrix} P & {\tilde{A}}_{j}^{T} \\ {\tilde{A}}_{j} & P^{- 1} \end{matrix}] = [\begin{matrix} I & P^{- 1} {\tilde{A}}_{j}^{T} \\ {\tilde{A}}_{j} & P^{- 1} \end{matrix}]

(19)

Performing the pre multiplying of (19) with the diagonal ${P^{- 1}, I}$ :

[\begin{matrix} I & P^{- 1} {\tilde{A}}_{j}^{T} \\ {\tilde{A}}_{j} & P^{- 1} \end{matrix}] \times [\begin{matrix} P^{- 1} & 0 \\ 0 & I \end{matrix}] = [\begin{matrix} P^{- 1} & P^{- 1} {\tilde{A}}_{j}^{T} \\ {\tilde{A}}_{j} P^{- 1} & P^{- 1} \end{matrix}]

(20)

Using $P = P^{T} > 0$ and $P^{- 1} = [\begin{matrix} Q & 0 \\ 0 & Q \end{matrix}]$ , the following LMIs is obtained:

[\begin{matrix} Q & 0 & {QA}_{j}^{T} + {QK}_{j}^{T} B_{j}^{T} & 0 \\ 0 & Q & {QG}_{j}^{T} B_{j}^{T} & Q \\ A_{j} Q^{T} + B_{j} K_{j} Q^{T} & B_{j} G_{j} Q^{T} & Q & 0 \\ 0 & Q^{T} & 0 & Q \end{matrix}] > 0

(21)

Supposing that there exist matrices $F_{j} = {QK}_{j}^{T}$ and $H_{j} = {QG}_{j}^{T}$ then, the following LMIs is obtained:

{\begin{matrix} P > 0 \\ [\begin{matrix} Q & 0 & {QA}_{j}^{T} + F_{j} B_{j}^{T} & 0 \\ 0 & Q & H_{j} B_{j}^{T} & Q \\ A_{j} Q^{T} + B_{j} F_{j}^{T} & B_{j} H_{j}^{T} & Q & 0 \\ 0 & Q^{T} & 0 & Q \end{matrix}] \end{matrix} > 0

(22)

The obtained LMIs (22) allows to optimize simultaneous the values of the gain matrices $K_{j}$ and $G_{j}$ which insure the stability of the system and the fault compensation correctly.

Additive control of actuator fault

In most industrial systems, controllers are synthesized, neglecting that faults can occur. The nominal control is up to date in accordance to the incidence of the fault. For this reason, it is considerable to take into account the reality that direct actuator fault lodging have to be considered to make sure the system stability. Therefore, whilst an actuator fault happens an additive control is introduced to the nominal one. Then, the novel partial control law is computed as following.¹¹

u (k) = K_{j} x (k) + G_{j} Z (k) + u_{ada} (k)

(23)

So, the closed loop state equation will be as follows:

x (k + 1) = (A_{j} + B_{j} K_{j}) x (k) + B_{j} G_{j} Z (k) + B_{j} f_{a} (k) + B_{j} u_{ada} (k)

(24)

In fact, in order to compensate the effect of the fault, the system should be as close as soon as possible to the (6). In other words, the additive control should verify the following equation:

B_{j} f_{a} (k) + B_{j} u_{ada} (k) = 0

(25)

Using the estimated fault (48) “theorem 5”, the additive control is then generated based on the EDPM:

u_{ada} (k) = - B_{j}^{+} B_{j} {\hat{f}}_{a} (k)

(26)

where, the matrix $B_{j}^{+}$ is the pseudo inverse of the matrix $B_{j}$ .

Extended data based projection method

In a previous work, the data-based projection method is used for sensor fault estimation.³⁰ This section extends this technique for actuator fault estimation. In fact, it has an additional constraint compared to the sensor fault estimation approach that is the inversion analysis. It uses only the inputs and outputs measured data. The main idea is to project an input matrix (generated using input data collected in a time interval T) on the orthogonal of an output matrix (collected using output measurements synthesized in the same time interval T). To detect and estimate the actuator fault, the following steps should be performed:

The state model (1) is converted to the switched model (27) firstly to consider the actuator fault as sensor fault in the system and then, to use the DPM approach developed for sensor fault estimation in Yahia et al.³⁰

The obtained model (27) is used to generate the databases from the inputs-outputs measured data.

The residual is computed using the databases already constructed to find the estimated fault.

Switched model of the input left inverse

In this part, we are interested in the switched model of the input left inverse.³² So, the definition of input inversion is first given.

Definition: The system (1) has an input inverse with delay $v$ if the input $u (k)$ can be uniquely generated from $y (k)_{v}$ , where $y (k)_{v} = [y^{T} (k) | y^{T} (k + 1) | . . . | y^{T} (k + v)]$ . In the following, a essential and enough situation is given for the existence of an input left inverse with delay $v$ in phrases of the structural matrices that represent the system (1).

Theorem 3: The system (1) has an input inverse with delay $v$ if there is matrices $K_{j}$ such that $K_{j} M_{j, v} = [I_{m} | 0]$ and the switched model of the input left inverse is given by:

{\begin{matrix} x (k + 1) = A_{j}^{*} x (k) + B_{j}^{*} y (k)_{v} \\ u (k) = C_{j}^{*} x (k) + D_{j}^{*} y (k)_{v} + f_{a} (k) \end{matrix}

(27)

where, $A_{j}^{*} = A_{j} - B_{j} K_{j} O_{j, v} \in ℜ^{n \times n}$ and $A_{j}^{*}$ is supposed stable.

\begin{matrix} B_{j}^{*} = B_{j} K_{j} \in ℜ^{n \times p (v + 1)}, C_{j}^{*} = - K_{j} O_{j, v} \in ℜ^{m \times n}, \\ D_{j}^{*} = K_{j} \in ℜ^{m \times p (v + 1)}, O_{j, v} = {[C_{j}^{T} | (C_{j} A_{j})^{T} | . . . | (C_{j} A_{j}^{v})^{T}]}^{T}, \\ M_{j, v} = [\begin{matrix} 0_{(p \times m)} & 0_{(p \times m)} & \dots & 0_{(p \times m)} \\ C_{j} B_{j} & 0_{(p \times m)} & \dots & 0_{(p \times m)} \\ C_{j} A_{j} B_{j} & C_{j} B_{j} & \dots & 0_{(p \times m)} \\ ⋮ & ⋮ & \dots & ⋮ \\ C_{j} A_{j}^{v - 1} B_{j} & C_{j} A_{j}^{v - 2} B_{j} & \dots & 0_{(p \times m)} \end{matrix}] . \end{matrix}

Proof: To present the inversion conditions, the Massey Sain algorithm is used.³¹

The equation of the outputs in term of the control inputs are given by substituting the state into the system outputs (1):

\begin{matrix} y (k + 1) = C_{j} x (k + 1) \\ y (k + 1) = C_{j} A_{j} x (k) + C_{j} B_{j} u (k) \end{matrix}

(28)

Then, by repeating $v$ times the previous procedure, the following expression is obtained:

y (k)_{v} = O_{j, v} x (k) + M_{j, v} U (k)_{v}

(29)

where,

\begin{matrix} - y (k)_{v} = [y^{T} (k) | y^{T} (k + 1) | . . . | y^{T} (k + v)], \\ - O_{j, v} = {[C_{j}^{T} | (C_{j} A_{j})^{T} | . . . | (C_{j} A_{j}^{v})^{T}]}^{T}, \end{matrix}

- U (k)_{v} = [u^{T} (k) | u^{T} (k + 1) | . . . | u^{T} (k + v)],

- M_{j, v} = [\begin{matrix} 0_{(p \times m)} & 0_{(p \times m)} & \dots & 0_{(p \times m)} \\ C_{j} B_{j} & 0_{(p \times m)} & \dots & 0_{(p \times m)} \\ C_{j} A_{j} B_{j} & C_{j} B_{j} & \dots & 0_{(p \times m)} \\ ⋮ & ⋮ & \dots & ⋮ \\ C_{j} A_{j}^{v - 1} B_{j} & C_{j} A_{j}^{v - 2} B_{j} & \dots & 0_{(p \times m)} \end{matrix}] .

Supposed that $rank (M_{j, v}) - rank (M_{j, v - 1}) = m$ and there is matrix $K_{j}$ such that $K_{j} M_{j, v} = [I_{m} | 0]$ , where $m$ is the number of inputs for the switched system.

Then, using (29), the following expression is given:

K_{j} y (k)_{v} = K_{j} O_{j, v} x (k) + u (k)

(30)

The inverse can be constructed as follows:

u (k) = - K_{j} O_{j, v} x (k) + K_{j} y (k)_{v}

(31)

By substituting (31) into the state expression of (1), we get:

\begin{matrix} x (k + 1) = A_{j} x (k) + B_{j} u (k) \\ x (k + 1) = A_{j} x (k) - B_{j} K_{j} O_{j, v} x (k) + B_{j} K_{j} y (k)_{v} \end{matrix}

(32)

Finally, the expressions (31) and (32) together form the state-space model of the input left inverse of system (1).

{\begin{matrix} x (k + 1) = (A_{j} - B_{j} K_{j} O_{j, v}) x (k) + B_{j} K_{j} y (k)_{v} \\ u (k) = - K_{j} O_{j, v} x (k) + K_{j} y (k)_{v} + f_{a} (k) \end{matrix}

(33)

Inputs and outputs databases

The acquired state model of the input left inverse (33) is taken into consideration to synthesis the residual for actuator fault estimation. This allows us to address actuator fault estimation as sensor fault estimation. In fact, the technique is similar. However, the expression of the acquired residual is different. The expression of the control input database is given by:

\bar{U} (k) = {[u (k - L + 1) . . . u (k)]}^{T} \in ℜ^{m \times L}

(34)

The expression of the output database is given by:

\bar{Y} (k) = [\bar{y} {(k - L + 1)}_{i} \bar{y} {(k - L + 2)}_{i} . . . \bar{y} {(k)}_{i}] \in ℜ^{p (v + 1) (i + 1) \times L}

(35)

where,

\bar{y} (k)_{i} = [y (k - i)_{v}^{T} y (k - i + 1)_{v}^{T} . . . y (k)_{v}^{T}]

(36)

where, $L$ and $i$ are two integers which verify:

L > p (v + 1) (i + 1)

Estimated actuator fault

The residual is the fault indicator used to find the estimated fault. It is obtained from the input-output databases ( $\bar{U} (k), \bar{Y} (k)$ ) already constructed and the projection matrix of $\bar{Y} (k)$ ( $Π (k)$ ).

$Π (k)$ is defined by:

Π (k) = I_{L} - \bar{Y} (k)^{T} {[\bar{Y} (k) \bar{Y} {(k)}^{T}]}^{- 1} \bar{Y} (k)

(37)

\bar{Y} (k) Π (k) = 0

(38)

Theorem 4: Consider the switched version of the input left inverse given with the aid of using (33) and the use of the input and the output databases, the residual $r (k)$ is given through matrix projection:

r (k) = \bar{U} (k) Π (k)

(39)

Proof: We will give a theoretical analysis on how ${\hat{f}}_{a} (k)$ is close to the target signal $f_{a} (k)$ . So, the model parameters is used to show how (48) can be obtained and why the residual $r (k)$ could be used as residual for actuator fault estimation.

Consider the obtained switched model of the input left inverse:

{\begin{matrix} x (k + 1) = A_{j}^{*} x (k) + B_{j}^{*} y (k)_{v} \\ u (k) = C_{j}^{*} x (k) + D_{j}^{*} y (k)_{v} + f_{a} (k) \end{matrix}

(40)

The system parameters are defined as follows: $A_{j}^{*} = A_{j} - B_{j} K_{j} O_{j, v}$ , $B_{j}^{*} = B_{j} K_{j}$ , $C_{j}^{*} = - K_{j} O_{j, v}$ , $D_{j}^{*} = K_{j}$ .

By substituting the state equation (40), the following expression can be obtained:

u (k) = C_{j}^{*} A_{j}^{*^{i}} x (k - i) + H_{j, i} \bar{y} (k)_{i} + f_{a} (k)

(41)

where,

H_{j, i} = [\begin{matrix} C_{j}^{*} A_{j}^{*^{i - 1}} B_{j}^{*} & \dots & C_{j}^{*} B_{j}^{*} & D_{j}^{*} \end{matrix}]

(42)

is the Markov parameters matrix of order $i$ .

\bar{y} (k)_{i} = [\begin{matrix} y {(k - i)}_{v} \\ ⋮ \\ [y {(k - 1)}_{v}] \\ y {(k)}_{v} \end{matrix}]

Under stability hypothesis of matrix $A_{j}^{*}$ , $‖ A_{j}^{*^{i}} ‖ = 0$ , when $i$ tends to infinity:

\lim_{i \to + \infty} ‖ A_{j}^{*^{i}} ‖ = 0

(43)

Consequently, $C_{j}^{*} A_{j}^{*^{i}} x (k - i)$ in (41) becomes negligible for $i > \tilde{i}$ . Therefore equation (41) becomes:

u (k) ≃ H_{j, i} \bar{y} (k)_{i} + f_{a} (k)

(44)

Concatenating equation (44) on a interval of size $L$ leads to:

\bar{U} (k) ≃ H_{j, i} \bar{Y} (k) + F_{a} (k)

(45)

where, $F_{a} (k)$ is constructed similarly as $\bar{U} (k)$ . So, $F_{a} (k) = [\begin{matrix} f_{a} (k - L + 1) & \dots & f_{a} (k) \end{matrix}]$ .

Let us define $Π (k)$ the orthogonal of the extended Hankel matrix $\bar{Y} (k)$ . The proposed residual $r (k)$ in discrete time switched system is obtained by right multiplying equation (45) by $Π (k)$ in order to eliminate the effect of inputs and the matrix $H_{j, i}$ of system parameters.

r (k) = \bar{U} (k) Π (k) ≃ \underset{= 0}{\underset{︸}{H_{j, i} \bar{Y} (k) Π (k)}} + F_{a} (k) Π (k)

(46)

Then, the computational form of $r (k)$ vector is given by the following expression:

r (k) = \bar{U} (k) Π (k)

(47)

Theorem 5: Using the residual (47), the actuator fault estimated is given by:

{\hat{f}}_{a} (k) = r (k) W

(48)

where, $W = {(0 . . . 01)}^{T}$ is a selective vector. Any column of $W$ can be chosen as residual.

Proof: The main idea to get the estimated actuator fault is to impose a special structure of $Π (k)$ .The matrix $Π (k)$ may be chosen as follows:

Π (k) = [\begin{matrix} Π_{a} (k) \\ I \end{matrix}]

(49)

where $I$ is the identity matrix and $Π_{a} (k)$ is a matrix to be found such that:

\bar{Y} (k) Π (k) = 0

(50)

Once the matrix $Π (k)$ is calculated, the proposed residual can be obtained base on a computation form (47). In order to prove that the residual $r (k)$ is able to estimate the actuator fault, the evaluation form of the residual is given. Two cases are considered:

*No fault occurs:

r (k) = \bar{U} (k) Π (k) = H_{j, i} \bar{Y} (k) Π (k)

(51)

Using condition (50), the residual evaluation form is zero $(r (k) = 0)$ .

*Actuator fault occurs:

\begin{matrix} r (k) = \bar{U} (k) Π (k) \\ r (k) = H_{j, i} \bar{Y} (k) Π (k) + [0 | F_{a} (k)] Π (k) \end{matrix}

(52)

Using condition (50), the proposed residual becomes:

r (k) = [0 | F_{a} (k)] Π (k)

(53)

Replacing the expression of (49) in (53), then the residual is given by:

\begin{matrix} r (k) = [0 | F_{a} (k)] [\begin{matrix} Π_{a} (k) \\ I \end{matrix}] \\ r (k) = F_{a} (k) \end{matrix}

(54)

From equation (54) and using a selective vector $W = {(0 . . . 01)}^{T}$ , one can deduce that the actuator fault $f_{a} (k) = F_{a} (k) W$ is estimated by:

{\hat{f}}_{a} (k) = r (k) W

(55)

Remark: From equation (48), it is simple to observation that no model parameter is wanted to compute the expected fault and simplest the input and output information are used. In the following, the acquired predicted fault is used so one can synthesis the additive control.

Comparative study

In this part, we illustrate the effectiveness of the proposed FTC with an example and we compare its performance with the additive state feedback control proposed^25,26,30 (see section 3). It will be shown that, the proposed FTC control performance is better than the additive state feedback control. Let us consider a switched system including two partial models:

Partial model 1:

A_{1} = [\begin{matrix} 0.6 & 0.45 \\ - 0.2 & 0.3 \end{matrix}] B_{1} = [\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix}], C_{1} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .

Partial model 2:

A_{2} = [\begin{matrix} 0.4 & - 0.3 \\ 0.11 & 0 \end{matrix}] B_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] C_{2} = [\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix}] .

Then, the system is defined by two inputs $(u_{1} (k), u_{2} (k))$ and two outputs $(y_{1} (k), y_{2} (k))$ . It is composed of two modes. The Table 2 summarized the interval of activation of modes.

Table 2.

Simulated switching sequence.

Mode number	1	2
$k \in$	[0,800]	[800,1500]

Actuators faults may be evoked with the aid of using numerous extraordinary styles of problems, are associated with incorrect readings because of a failure of their additives which reasons the lack of effectiveness. In fact, they’re taken into consideration additive indicators at the measurements.

Scenario 1: The additive fault is considered as follows:

\begin{matrix} f_{a} (k) = \\ {\begin{matrix} [\begin{matrix} 0 & 0 \end{matrix}]^{T}; k \in [0, 100] \cup [250, 450] \cup {[600, 1200]}^{T} \\ {[\begin{matrix} 0.2 & 0.2 \end{matrix}]}^{T}; k \in {[100, 250]}^{T} \\ {[\begin{matrix} - 0.25 & - 0.25 \end{matrix}]}^{T}; k \in [450, 600] \\ [\begin{matrix} 0.25 & 0.25 \end{matrix}]; k \in [1200, 1500] \end{matrix} \end{matrix}

Using theorem 2 (novel LMIs developed), $G_{j}$ and $K_{j}$ _j each mode are computed (see Table 2)

The evolution of the outputs using the additive state feedback control and the FTC proposed in this paper are compared in Figures 1 and 2. The additive actuator fault appears at $k$ = 100, $k$ = 450, and $k$ = 1200. Compared with the additive state feedback control, the proposed FTC shows its capacity to improve the closed-loop performances in terms of actuator fault compensation. It is easy to remark that, using the additive state feedback control (green plot), the fault is not compensated correctly. Indeed, this compensation needs a significant reconfiguration time. Based on the simulations for comparisons (see Tables 3 and 4), the proposed FTC is claimed to have faster reaction. Indeed, the control performance relies on the actuator fault estimation quality, that should be rapid and accurate. In the present paper, the proposed data based projection approach is efficient in estimating actuator fault accurately. Then, the estimation gives an efficient control law which allows, reference fault compensation. Fault estimation is achieved by the proposed EDPM as illustrated in Figure 3. Figures 4 and 5 describe the evolution of controls using the proposed FTC. It is easy to notice that the proposed FTC reacts quickly to cancel for the effect of the fault.

Scenario 2: The additive fault is considered as follows:

fa (k) = {\begin{matrix} {[\begin{matrix} 0 & 0 \end{matrix}]}^{T}; k \in [0, 1200] \\ {[\begin{matrix} 0.25 & 0 \end{matrix}]}^{T}; k \in [1200, 1500] \end{matrix}

Figure 1.

Evolution of the first output.

Figure 2.

Evolution of the second output.

Table 3.

Optimized values of gains.

modes	1	2
$K_{j}$	$[\begin{matrix} 0.2 & - 0.3 \\ - 0.6 & - 0.45 \end{matrix}]$	$[\begin{matrix} - 0.4 & 0.3 \\ - 0.11 & 0 \end{matrix}]$
$G_{j}$	$[\begin{matrix} - 0.883 & 0 \\ 0 & - 0.883 \end{matrix}]$	$[\begin{matrix} - 0.883 & 0 \\ 0 & - 0.883 \end{matrix}]$

Table 4.

Response time.

Approach	Proposed FTC	Additive state feedback control
Response time (second)	0.0310	0.1421

Figure 3.

Actuator fault estimation.

Figure 4.

Evolution of the first control using the proposed FTC.

Figure 5.

Evolution of the second control using the proposed FTC.

This means that the switched system is working in nominal non faulty conditions until $k = 1200$ . At $k = 1200$ , there appears a bias fault. The simulations results are given in Figures 6 to 10. In Figure 6, it can easily be seen that the estimated fault (red plot) converges with a good accuracy to the real one (blue plot). In Figures 7 and 8, it is easy to remark that using the proposed FTC, the outputs reach quickly the reference input and the performances are guaranteed. In fact, the proposed FTC has faster reaction and the time response of fault compensation (red plot) is smaller compared to the use of additive state feedback control (green plot). Figures 9 and 10 describe the evolution of the controls using the proposed FTC. It is clear that the proposed FTC react quickly to compensate for the effect of the fault.

Figure 6.

Actuator fault estimation.

Figure 7.

Evolution of the first output.

Figure 8.

Evolution of the second output.

Figure 9.

Evolution of the first control using the proposed FTC.

Figure 10.

Evolution of the second control using the proposed FTC.

To evaluate the accuracy of the proposed FTC, the following different performances criteria are used. The first performance index is the mean square error (MSE) and it has the following expression:

MSE = \frac{1}{N_{e}} \sum_{k = 1}^{N_{e}} (y (k) - y_{r} (k))^{2}

(56)

The second is the variance accounted for (VAF) criterion and it has the following expression:

V A F = \max {1 - \frac{var (y (k) - y_{r})}{var (y (k))}, 0} 100 %

(57)

Where $y (k)$ is the system outputs, $y_{r} (k)$ is the reference signal and $N_{e}$ is the number of measures. Using the proposed FTC, a good adequacy between the system outputs and the reference signal illustrated by the two following indices: $MSE = 0.0043$ , $VAF = 99.05$ $percent$ .

Conclusion

This paper investigates the problem of actuator fault estimation and compensation of switched hybrid system. A data-projection method is extended (EDPM) for actuator fault estimation. The fault estimation size and time profile are close to the real one. Fault compensation is performed based on the FTC. The proposed FTC is designed by adding of the state feedback control with integral action to an additive control that is evoked using the EDPM. In fact, a novel LMIs is developed to optimize the values of the gain matrices of the state feedback control with integral action. This allows to keep the stability and compensate for the fault effect with a minimum reconfiguration time. Besides, industrial systems are generally complex and issue now no longer simplest to faults however additionally to external disturbances (ramp, harmonic, white/gaussian noise signals) that may degrade the system performances. However, with inside the presence of disturbances, the proposed FTC can not make certain disturbance attenuation which can result in the instability. Therefore, it’s far essential to adjust the proposed FTC algorithm, considering the presence of disturbance to enhance the closed-loop overall performance in phrases of trajectory tracking and disturbance attenuating.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the ministry of Higher Education and Scientific Research in Tunisia.

ORCID iD

Yahia Salwa

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Diagnosis and fault tolerant control against actuator fault for a class of hybrid dynamic systems

Abstract

Keywords

Introduction

Problem statement

Additive state feedback control

Proposed fault tolerant control

State feedback control with integral action

Additive control of actuator fault

Extended data based projection method

Switched model of the input left inverse

Inputs and outputs databases

Estimated actuator fault

Comparative study

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References