International Journal of Automation and Computing  2018, Vol. 15 Issue (6): 736-746   PDF    
Adaptive Fault Tolerant Control of Multi-time-scale Singularly Perturbed Systems
Adel Tellili1,3, Nouceyba Abdelkrim1,2, Amina Challouf1,2, Mohamed Naceur Abdelkrim1,4     
1 Research Unit MACS, University of Gabes, Gabes, Tunisia;
2 Higher Institute of Industrial Systems at Gabes, University of Gabes, Gabes, Tunisia;
3 Higher Institute of Technological Studies at Djerba, Djerba, Tunisia;
4 National School of Engineers of Gabes, University of Gabes, Gabes, Tunisia
Abstract: This paper studies the fault tolerant control, adaptive approach, for linear time-invariant two-time-scale and three-time-scale singularly perturbed systems in presence of actuator faults and external disturbances.First, the full order system will be controlled using ε-dependent control law.The corresponding Lyapunov equation is ill-conditioned due to the presence of slow and fast phenomena.Secondly, a time-scale decomposition of the Lyapunov equation is carried out using singular perturbation method to avoid the numerical stiffness.A composite control law based on local controllers of the slow and fast subsystems is also used to make the control law ε-independent.The designed fault tolerant control guarantees the robust stability of the global closed-loop singularly perturbed system despite loss of effectiveness of actuators.The stability is proved based on the Lyapunov stability theory in the case where the singular perturbation parameter is sufficiently small.A numerical example is provided to illustrate the proposed method.
Key words: Singularly perturbed systems     time scale decomposition     adaptive fault tolerant control     actuator fault     Lyapunov equations    
1 Introduction

Systems containing dynamics which operate on two or more time-scales are called singularly perturbed systems. Such systems are constituted of multiple subsystems and distinguished by the presence of slow and fast dynamics and weak and strong interconnections between state variables. A system represented in explicit singularly perturbed form is characterized by the presence of a small parameter $\varepsilon$ appearing as a multiplier to the derivative of the "fast" variables. Many research results in singularly perturbed systems exhibit that the two time-scale analysis concept is a useful method for large-scale systems. This approach is both achievable and easy to apply to decompose a system into two smaller order subsystems, reducing the complexity of the filtering, stability analysis or control problem. The singularly perturbed systems are widely used in control engineering due to their ability to model multi-time scale systems like power system dynamics, chemical reactors and similar large scale processes. They are characterized by the presence of small positive parameter called singular perturbation parameter. If the parameters are of the same order of magnitude, the system is considered as a two time scales singularly perturbed system. If they have different orders of magnitude, the system gets three-time or multi-time scale structure. The multi-modeling singularly perturbed systems are a specific structure of multiple time scales and parameters singularly perturbed systems. They describe dynamics of several real physical systems like power systems and appear in some dynamic models of physical as well as biochemical systems. It generally results from the presence of small parameters in various physical systems like machine reactance in power systems, wire inductance and capacitance in electronic control systems. The multi-modeling problems appear in large-scale dynamic systems. The presence of high dimensionality and ill-conditioning in such systems, gives rise to difficulties in their fault diagnosis and control. These problems are considerably simplified if a decomposition of fast and slow dynamics is carried out. The time scales separation is realized through the singular perturbation method by setting the singular perturbation parameter to zero, or using a variable transformation, known as Chang$'$s transformation, yielding a bloc diagonal form of the full order system[1-4]. The complexity of the singularly perturbed systems makes them susceptible to faults being able to affect actuators, sensors, the process itself, or the controller. Such faults cause undesired or intolerable behavior of the process. The increasing requirement to ensure higher system performance, product quality and process safety needs the fault detection, diagnosis and management. The fault detection is the step to observe that a fault arises, early fault detection allows precious warning on emerging problems in order to avoid serious process damages. Fault diagnosis is aimed to determine the cause and the location of the fault. This step is necessary to the reduction or elimination of the fault effect. The main part of the model based fault diagnosis scheme is a process model running parallel to the real process. The comparison between outputs of the real process and the corresponding mathematical model makes it possible, using a signal called residual, to detect the presence of defects. Most systems operate in a feedback form, where defects may be worsened by the closed-loop control systems and can develop into malfunction of the loop. The closed-loop control action may hide a fault from being observed, which will easily cause production to stop or system malfunction at a plant level. Therefore, the fault management is an important field. It serves to avoid system failures and process breakdown by early detection of fault occurrence. If process dysfunction cannot be avoided, hardware redundancy is used to treat failures mostly in critical cases. This approach consists to provide multiple components that accomplish the same function. When any process constituent fails, the damaged part will be replaced by the redundant healthy components. Nevertheless, this solution is expensive and requires to much space in the process. It can not be applied to each element of the system. In such situations, it is necessary to design an adequate control scheme, called fault-tolerant control, to guarantee system stability even in faulty case. Fault tolerant control (FTC) contributes to increase human and system safety and efficiency, it is a way for increasing the reliability of a controlled dynamical system that is subject to faults which can affect the system, the actuators, the sensors or the controllers. Reconfigurable control modifies the control law after the faults occurrence in real time and without human interaction so that the reconfigured closed-loop system continues to achieve its mission and to avoid process shut-down. The aim of fault-tolerant control is to ensure the specified performance of a system in faulty case and to overcome the limitation of conventional feedback control[5-7]. Most of the fault-tolerant design approaches can be classified into two main classes, namely passive approaches[8-10] and active approaches[11, 12]. In active fault-tolerant control methods, the principle is to use a fault diagnosis block in the control system to detect the failure occurrence and to make available the information about system faults in order to exploit such information in the controller conception. Active fault tolerant control scheme is achievable in two steps. The first one is the fault detection and isolation procedure which goal is to detect the presence of fault and to recognize the faulty part in the process, whereas the second step is to compensate the fault effect through the fault accommodation module using the information provided by the fault detection and isolation part. In the faulty case, a reconfiguration system modifies the structure and/or the parameters of the feedback control system. Consequently, a new set of control parameters is determined such that the faulty system reaches the nominal system performance. In contrast, in the passive approach, a fixed parameter controller is designed to maintain (at least) stability if a fault occurs in the system, the controller structure remains unchanged for both normal and faulty cases. This approach, also called reliable control, uses usually robust control techniques to ensure that a closed-loop system remains insensitive to a priori known faults without the need for system reconfiguration and fault detection, the treated faults are considered as specific class of disturbance or uncertainties[6, 13]. Several approaches have been developed to design fault tolerant controller in order to guarantee closed-loop stability not only when all control components are operational, but also under various component faults. Veillette[8] used a Riccati equation based method to elaborate a state-feedback controller, which could tolerate the outage within a selected subset of actuators while preserving the stability and the known quadratic performance bound. Yang et al.[9, 14] and Veillette et al.[15] proposed a controller that could ensure locally asymptotic stability and $H_{\infty}$ performance even in presence of faults. Tellili et al.[16] proposed a reliable $H_{\infty}$ controller for linear time-invariant multi-parameter singularly perturbed system against sensor failures. The global controller is then simplified to three reduced reliable $H_{\infty}$ sub-controllers based on the slow and fast problems through the manipulation of the algebraic Riccati equations. However, the passive methods have a limited fault tolerant capability. On the contrary, an active fault-tolerant control system can compensate for defects either by selecting a precomputed control law or by generating a new control strategy online. Another typical technique for fault compensation is based on adaptive method (see [17-20]). Several authors were interested in this subject. Ye and Yang[11] established an adaptive reconfigurable control law by actuator defects with application to flight control. Chandler et al.[21] and Smith et al.[22] have developed system identification schemes appropriate to adaptive and reconfigurable control. The results are applied in flight control systems. Tao et al.[18, 23] have focused on adaptive fault tolerant control laws based on model reference tracking. Jin and Yang[24] develop a direct adaptive state feedback control schemes on the basis of Lyapunov stability theory. The resulting closed-loop system is then asymptotically stable in the presence of actuator faults and external disturbances. The result is extended and improved in [25]. Wang et al.[26] used an adaptive output feedback control to compensate actuator fault including outage, loss of effectiveness and stuck. For the singularly perturbed systems, Ioannou and Kokotovic[27] developed reduced order adaptive controller for interconnected two-time scale singularly perturbed systems to ensure stability in the presence of disturbances and unmodeled interconnections, whereas, Shao and Sawan[4] studied the robust stabilization problem of a linear time-invariant singularly perturbed system with nonlinear uncertainties. In both cases, faulty actuators or sensors are not considered. The main goal of this paper is to design an adaptive fault tolerant control law for singularly perturbed systems in presence of external disturbances and actuator faults characterized by a loss of effectiveness. The concerned systems are two-time-scale and three-time-scale, also called multi-modeling structure, singularly perturbed systems. This work is an extension of Jin$'$s results presented in [24] to the fault tolerant control of singularly perturbed systems with actuator faults. The outline of this paper is as follows. Section 2 describes the problem formulation, the control objectives and the plant model. In Section 3, the adaptive laws are developed to update the controller parameters in presence of actuator faults and external disturbances. Section 4 treats the stability analysis after controller simplification. The results developed for two-time-scale perturbed systems are extended, in Section 5, to multi-modeling structure. A numerical example and simulation results are given in Section 6. The conclusion is then presented in the last section.

2 System description and problem formulation

The considered system is a two-time scale singularly perturbed system under external disturbances:

$ \begin{eqnarray} {\left[\begin{array}{c} {\dot{x}(t)} \\ {\varepsilon \, \dot{z}(t)} \end{array}\right]}=\left[\begin{array}{cc} {A_{11} }&{A_{12} } \\ {A_{21} }&{A_{22} } \end{array}\right] \left[\begin{array}{c} {x(t)} \\ {z(t)} \end{array}\right]+ \nonumber \\[5mm] ~~~~~~~~~~~~\left[\begin{array}{c} {B_{1x} } \\ {B_{1z} } \end{array}\right]\, w(t) { +\left[\begin{array}{c} {B_{2x} } \\ {B_{2z} } \end{array}\right]\, u(t)} \end{eqnarray} $ (1)

where $x\in {\bf R}^{n1} $ and $z\in {\bf R}^{n2} $ are state vectors, $u\in {\bf R}^{m} \, $is the control vector, $w\in {\bf R}^{q} $ models piecewise continuous bounded external disturbances acting on the system and verifying $\left\|w\right\|\le \overline{w}$ with $\overline{w}$ is unknown positive constant. $A_{11} \in {\bf R}^{n1{\rm x}n1} $, $A_{12} \in {\bf R}^{n1{\rm x}n2} $, $A_{21} \in {\bf R}^{n2{\rm x}n1} $, $A_{22} \in {\bf R} ^{n2{\rm x}n2} $, $B_{1x} \in {\bf R}^{n1{\rm x}q} $, $B_{2x} \in {\bf R}^{n1{\rm x}m} $, $B_{1z} \in {\bf R}^{n2{\rm x}q} $, and $B_{2z} \in {\bf R}^{n2{\rm x}m} $ are constant matrices. $A_{22}$ is assumed to be nonsingular (standard singularly perturbed system). The parameter $\varepsilon $, called singular perturbation parameter, is a positive scalar taking values between 0 and 1. Denote throughout the paper: $B_{i} =\left[\begin{array}{cc} {B_{ix}^{\rm T} }&{B_{iz}^{\rm T} } \end{array}\right]^{\rm T} $ for $i=1, \ 2$ and $\left\|(\cdot)\right\|$ is the Euclidian norm of ($\cdot$).

According to the time-scale property of the singularly perturbed system, the slow and the fast subsystems related to the full-order system (1) can be derived by introducing a change of coordinates in which the system appears in block-triangular forms or by using the singular perturbation method which formally sets the singular perturbation parameter $\varepsilon $ to zero (see [1, 28]). The second approach will be used in this paper. To define the slow subsystem, the fast dynamics will be neglected, which is equivalent to suppose that $\varepsilon = 0 $. The resulting equation for the fast variables is

$ \begin{equation} \label{GrindEQ__1a_} 0=A_{21} x_{s}+ A_{22} z_{s}+B_{1x} w_{s} +B_{2x} u_{s}. \end{equation} $ (2)

In the case of standard singularly perturbed systems, $A_{22}$ is supposed to be nonsingular and the slow part of the second state can be expressed as

$ \begin{equation} \label{GrindEQ__1b_} z_{s}=-A^{-1}_{22} \, A_{21} x_{s}-A^{-1}_{22}\, B_{1x} w_{s} -A^{-1}_{22}\, B_{2x} u_{s}. \end{equation} $ (3)

So, the slow subsystem is obtained as

$ \begin{equation} \label{GrindEQ__2_} \dot{x}_{s} =A_{s} x_{s} +B_{1s} w_{s} +B_{2s} u_{s} \end{equation} $ (4)

where $A_{s} =A_{11} -A_{12} A_{22}^{-1} A_{21} $, $B_{is} =B_{ix} -A_{12} A_{22}^{-1} B_{iz} $ for $i=1, \ 2$. ${x}_{s}$, ${u}_{s}$ and ${w}_{s}$ are respectively, the slow parts of the states, the control input $u$, and the disturbance input $w$. If $(A_s, B_{2s})$ is stabilizable, then there exists a symmetric and positive definite matrix $P_s$ satisfying the following slow Lyapunov equation

$ \begin{equation} \label{GrindEQ__3_} (A_{s} +B_{2s} K_{s} )^{\rm T} \, P_{s} +P_{s} \, (A_{s} +B_{2s} K_{s} )=-Q_{s} \end{equation} $ (5)

with $Q_s$ as any given positive definite symmetric matrix and $K_s$ as the slow controller gain stabilizing the pair $(A_s, B_{2s})$ such that $u_s=K_sx_s$. The closed-loop slow subsystem is then defined by

$ \begin{equation} \label{GrindEQ__4_} \dot{x}_{s} =(A_{s} +B_{2s} K_{s} )\, x_{s} +B_{1s} w_{s}. \end{equation} $ (6)

Using the following approximations $A_s=A_{11}$, $B_{1s}=B_{1x}$ and $B_{2s}=B_{2x}$ (see [29, 30]), the slow subsystem (4), the slow Lyapunov equation (5) and the closed-loop slow subsystem (6) can be approached respectively, by (7)-(9):

$ \begin{equation} \label{GrindEQ__5_} \dot{x}_{s} =A_{11} x_{s} +B_{1x} w_{s} +B_{2x} u_{s} \end{equation} $ (7)
$ \begin{equation} \label{GrindEQ__6_} (A_{11} +B_{2x} K_{s} )^{\rm T} \, P_{s} +P_{s} \, (A_{11} +B_{2x} K_{s} )=-Q_{s} \end{equation} $ (8)
$ \begin{equation} \label{GrindEQ__7_} \dot{x}_{s} =(A_{11} +B_{2x} K_{s} )\, x_{s} +B_{1x} w_{s}. \end{equation} $ (9)

The fast subsystem is described by

$ \begin{equation} \label{GrindEQ__8_} \varepsilon \, \dot{z}_{f} =A_{22} z_{f} +B_{1z} w_{f} +B_{2z} u_{f} \end{equation} $ (10)

where ${z}_{f}$, ${u}_{f}$ and ${w}_{f}$ are respectively, the fast parts of the states, the control input $u$ and the disturbance input $w$. The fast subsystem can also be expressed in the stretching (fast) time scale $\displaystyle\frac {t}{\varepsilon}$ by

$ \begin{equation} \label{GrindEQ__9_} \dot{z}_{f} =A_{22} z_{f} +B_{1z} w_{f} +B_{2z} u_{f}. \end{equation} $ (11)

Assuming that $(A_{22}, B_{2z})$ is stabilizable, there exists a symmetric and positive definite matrix $P_f$ satisfying the following fast Lyapunov equation,

$ \begin{equation} \label{GrindEQ__10_} (A_{22} +B_{2z} K_{f} )^{\rm T} \, P_{f} +P_{f} \, (A_{22} +B_{2z} K_{f} )=-Q_{f} \end{equation} $ (12)

with $Q_f$ as any given positive definite symmetric matrix and $K_f$ as the fast controller gain stabilizing the pair $(A_{22}, B_{2z})$ such that $u_f=K_fz_f$. The closed-loop fast subsystem is then defined by

$ \begin{equation} \label{GrindEQ__11_} \dot{z}_{f} =(A_{22} +B_{2z} K_{f} )\, z_{f} +B_{1z} w_{f}. \end{equation} $ (13)

The considered actuator faults in this paper include loss of actuator efficiency. For the control $u_i$, $i=1, \cdots, m$, let $u^f_i$ be the signal generated by the $i$-th actuator that has failed. Consequently, the actuator failure model is as

$ \begin{equation} \label{GrindEQ__12_} u^f_i\left(t\right)={\rho }_i{\ u}_i(t) \end{equation} $ (14)

where ${\rho }_i$ represents the actuator efficiency factor and verifies $0\leq \underline{\rho}_{i}\leq \rho_{i}\leq \overline{\rho}_{i}\leq 1$. $\underline{\rho}_{i}$ and $\overline{\rho}_{i}$ denote the known lower and upper bounds of ${\rho }_i$, respectively. The case ${\underline{\rho }}_i=\ {\overline{\rho }}_i=0$ indicates the outage of the actuator $i$. ${\underline{\rho }}_i>0$ corresponds to the case of partial failure of $u_i$. If $\ {\underline{\rho }}_i={\overline{\rho }}_i=1$, then we have $u^f_i\left(t\right)={\ u}_i(t)$, which corresponds to the case of no failure. Denoting $\rho ={\rm diag}({\rho }_i)$, $i=1, \cdots, m$, the uniform actuator fault model becomes

$ \begin{equation} \label{GrindEQ__13_} u^{f}(t)=\rho \, u(t). \end{equation} $ (15)

The assumption that the control signals and disturbances use identical channels, which is satisfied for many systems to solve robust control problems (see [24, 26]), leads to suppose that $B_{1x}=\ B_{2x}\ F$ and $B_{1z}=\ B_{2z}\ F$, where $F$ is a matrix with appropriate dimension.

3 Adaptive fault tolerant controller design

In this section, adaptive laws will be developed to update the controller parameters in presence of actuator faults and external disturbances in order to guarantee the asymptotic stability of the closed-loop system via state feedback.

3.1 The full-order problem

Introduce the following notations

$ X(t)=\left[\begin{array}{c} x(t) \\ z(t) \\\end{array} \right], A(\varepsilon)=\left[\begin{array}{cc} {A_{11} }&{A_{12} } \\ \varepsilon^{-1}\, A_{21}&\varepsilon^{-1}\, A_{22} \end{array}\right]\\[3mm] ~~~~B_{i} (\varepsilon )=\left[\begin{array}{c} {B_{ix} } \varepsilon^{-1}\, B_{iz} \end{array}\right], \; {\rm{for}} \; i=1, \ 2. $

System (1) can be rewritten as

$ \begin{equation} \label{GrindEQ__14_} \dot{X}(t)=A(\varepsilon )\, X(t)+B_{1} (\varepsilon )\, w(t)+B_{2} (\varepsilon )\, u(t). \end{equation} $ (16)

Considering the actuator fault model (15) and the assumption held for the disturbances, system (16) becomes

$ \begin{equation} \label{GrindEQ__15_} \dot{X}(t)=A(\varepsilon )\, X(t)+B_{2} (\varepsilon )\, F\, w(t)+B_{2} (\varepsilon )\, \rho \, u(t). \end{equation} $ (17)

The proposed controller model to stabilize the system (16) is given by

$ \begin{equation} \label{GrindEQ__16_} u(t)=K_1(t)\ X(t)+K_2(t) \end{equation} $ (18)

where $K_{1}$ is chosen such that $(A(\varepsilon )+B_{2} (\varepsilon )\, K_{1} )$ is Hurwitz and $K_{2}(t)$ is designed using the following update law

$ \begin{equation} \label{GrindEQ__17_} K_{2} (t)=-\frac{\beta \, B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X}{\alpha \left\| B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X\right\| } \, \, (\hat{k}_{3} (t)+\left\| K_{1} X\right\| ) \end{equation} $ (19)

where $\alpha $ and $\beta $ are suitable positive constants satisfying (see [24-26])

$ \begin{equation} \label{GrindEQ__18_} \alpha \left\| B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X\right\| ^{2} \le \beta \left\| B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X\sqrt{\rho } \right\| ^{2}. \end{equation} $ (20)

And ${\hat{k}}_3$ is updated using the following adaptive law

$ \begin{equation} \label{GrindEQ__19_} \frac{{\rm d}\hat{k}_{3} (t)}{{\rm d}t} =\varepsilon \, \gamma \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \end{equation} $ (21)

where $\gamma $ is an appropriate positive constant and $\varepsilon $ is the singular perturbation parameter. Let ${\tilde{k}}_3(t)={\hat{k}}_3(t)-k_3$, then the update law (21) can be expressed as $\dot{\tilde{k}}_{3} (t)=\varepsilon \, \gamma \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| $.

The constant $k_{3}$ is involved in the inequality (27) which will be used later to bound the disturbance.

To solve the fault tolerant control problem (17), Theorem 1 is proposed,

Theorem 1. Consider the system described by (16). Suppose that

1) There exists a singular perturbation parameter ${\varepsilon }^*>0$ such that $(A(\varepsilon), \, B_2(\varepsilon))$ is stabilizable for all $\varepsilon \in \, [0, \, {\varepsilon }^*]$.

2) For any given positive definite symmetric matrix $Q$, there exists a symmetric and positive definite matrix $P(\varepsilon )$ satisfying the following Lyapunov equation

$ \begin{eqnarray} {(A(\varepsilon )+B_{2} (\varepsilon )\, K_{1} )^{\rm T} \, P(\varepsilon )}+\quad\quad\quad \nonumber \\ {P(\varepsilon )\, (A(\varepsilon )+B_{2} (\varepsilon )\, \, K_{1} )=-Q}. \end{eqnarray} $ (22)

3) $K_{1}$ is chosen such that $(A(\varepsilon )+B_{2} (\varepsilon )\, K_{1} )$ is Hurwitz.

There exists ${\varepsilon }^*>0$ such that for every ${\varepsilon \in ]0, \varepsilon }^*]$, the controller described by (18), with the adaptive laws (19) and (21) and the controller gain $K_{1}$, stabilize asymptotically the system (16) subject to actuator fault (15).

Proof. In the light of equations (17) and (18), the closed-loop fault tolerant control system becomes

$ \begin{align} \label{GrindEQ__21_} %\begin{eqnarray} &{\dot{X}(t)=A(\varepsilon )X(t)+B_{2} (\varepsilon )\, F\, w(t)}+\nonumber \\ &\quad\quad\quad {B_{2} (\varepsilon )\, \rho \, K_{1} \, X(t)+\, B_{2} (\varepsilon )\, \rho \, K_{2} (t)} %\end{eqnarray} \end{align} $ (23)

which can be expressed as

$ \begin{align} \label{GrindEQ 22} %\begin{eqnarray} & \dot{X}(t)=(A(\varepsilon)+B_{2}(\varepsilon )\, K_{1} )\, X(t) +B_{2} (\varepsilon )\, F\, w(t)+ \nonumber \\ &\quad\quad\quad B_{2} (\varepsilon )\, (\rho -I)\, K_{1} \, X(t)+\, B_{2} (\varepsilon )\, \rho \, K_{2}(t). %\end{eqnarray} \end{align} $ (24)

For the above mentioned closed-loop system, a Lyapunov function candidate is defined as

$ \begin{equation} \label{GrindEQ__23_} V\left(\varepsilon \right)=X^{\rm T} P(\varepsilon) X+{\varepsilon }^{-1}\ {\gamma }^{-1}{\ \tilde{k}}^2_3\ >0. \end{equation} $ (25)

Then, according to (24) and the above mentioned assumptions (about the disturbances and the Lyapunov function), the time derivative of $V(\varepsilon)$ for $t>0$ is

$ \begin{align} \label{GrindEQ_24} %\begin{split} &\dot{V}(\varepsilon)=X^{\rm T} [(A(\varepsilon )+B_{2}(\varepsilon )\, K_{1})^{\rm T} \, P(\varepsilon ) + \nonumber\\ &\quad\quad\quad P(\varepsilon )\, (A(\varepsilon )+B_{2}(\varepsilon )\, K_{1} )]\, X+2\, \varepsilon ^{-1} \gamma ^{-1} \tilde{k}_{3} \, \dot{\tilde{k}}_{3}+\nonumber \\ &\quad\quad\quad 2\, X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \rho \, K_{2} +2\, X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, F\, w+\nonumber \\ &\quad\quad\quad 2X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, (\rho -I)K_{1} X. %\end{split} \end{align} $ (26)

Since $\overline{w}$ is unknown bounded constant, there exists a constant $k_3$ such that (see [24])

$ \begin{align} \label{GrindEQ__25_} %\begin{array}{l} &{\left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \left\| F\right\| \, \bar{w}\le }\nonumber \\& \quad{\left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \, k_{3}}. %\end{array} \end{align} $ (27)

It can be shown that (see [25])

$ \begin{align} \label{GrindEQ__26_} &{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, (\rho -I)K_{1} X\le } \nonumber\\ &\quad{\left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, }. \end{align} $ (28)

Using the update law (19) and the expressions (20) and (22), (24) can be rewritten as

$ \begin{align} \label{GrindEQ__27_} &{\dot{V}(\varepsilon )\le -2\, X^{\rm T} Q\, X+2\, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \, k_{3}}+\nonumber \\& \quad\qquad\;{2\, \left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, }-\nonumber \\& \quad\qquad\;{\, 2\, \, \beta \dfrac{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \rho \, B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X}{\alpha \left\| B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X\right\| } \, \, (\hat{k}_{3} (t)}+\nonumber \\& \quad\qquad\;{\left\| K_{1} X\right\| )+2\, \varepsilon ^{-1} \, \gamma ^{-1} \tilde{k}_{3} \, \dot{\tilde{k}}_{3}}. \end{align} $ (29)

Therefore, from (20) it holds that

$ \begin{align} \label{GrindEQ__28_} &{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \rho \, B_{2}^{\rm T} (\varepsilon )P(\varepsilon )X\ge } \nonumber\\ &\quad\quad\displaystyle{\frac{\alpha }{\beta } \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| ^{2}}. \end{align} $ (30)

This leads to

$ \begin{align} \label{GrindEQ__29_} &{\dot{V}(\varepsilon )\le -2\, X^{\rm T} Q\, X+2\, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \, k_{3} }+\nonumber \\& \quad\quad{2\, \left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, }-\nonumber \\& \quad\quad{2\, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \right\| \, (\hat{k}_{3} (t)+\left\| K_{1} X\right\|)}+\nonumber \\& \quad\quad{2\, \varepsilon ^{-1} \, \gamma ^{-1} \tilde{k}_{3} \, \dot{\tilde{k}}_{3}}. \end{align} $ (31)

Using the adaptation law (21), it is easy to see that $\dot{V}(\varepsilon)\le0$, which guarantees the asymptotic stability of the faulty closed-loop system for sufficiently small singular perturbation parameter $\varepsilon $.

3.2 Controller simplification

It is well known that if both the slow subsystem and the fast subsystem are stable, then there exists ${\varepsilon }^*>0$ such that for every ${\varepsilon \in \ [0, \, \varepsilon }^*]$, the full-order singularly perturbed system (1) is stable[1, 4]. Thus, to stabilize the full-order singularly perturbed system (1), it is adequate to stabilize the reduced subsystems independently and then to design a composite control law using the local feedback gains. That means the feedback controller gain $K_1$ can be designed using the slow and fast subsystems controllers $K_s$ and $K_f$. Let $K_1$ be of the form $K_1=[K_{11} \, \, \, K_{12}]$, it is shown in [4] that the composite controller element $K_{11}$ is an ${O}(\varepsilon)$-approximation of the slow controller gain $K_s$ stabilizing the slow subsystem (4) and $K_{12}$ is an ${O}(\varepsilon )$-approximation of the fast controller gain $K_{f}$ stabilizing the fast subsystem (11). The control law (19) is simplified by multiplying $B_2(\varepsilon )$ in the numerator and denominator by $\varepsilon $ and letting $\varepsilon \to 0$, the following form is then obtained

$ \begin{equation} \label{GrindEQ__30_} K_{2} (t)=-\frac{\beta \, B_{2*}^{\rm T} P(\varepsilon )X}{\alpha \left\| B_{2*}^{\rm T} P(\varepsilon )X\right\| } \, \, (\hat{k}_{3} (t)+\left\| K_{1} X\right\| ) \end{equation} $ (32)

where $B^{\rm T}_{2*}=\left[\begin{array}{cc} 0&B^{\rm T}_{2z} \end{array} \right]$. Multiplying $B_2(\varepsilon )$ by $\varepsilon $ in the adaptive law (21) and letting $\varepsilon \to 0$ yields the following $\varepsilon $-independent form

$ \begin{equation} \label{GrindEQ__31_} \frac{{\rm d}\hat{k}_{3} (t)}{{\rm d}t} =\, \gamma \left\| X^{\rm T} P(\varepsilon )B_{2*} \right\| \, \end{equation} $ (33)

where $B_{2z}$ is the dominant part in $B_2\left(\varepsilon \right)$ when $\varepsilon \to 0$.

As mentioned in [25], to avoid discontinuity which can be caused by the term $\left\| X^{\rm T} P(\varepsilon )B_{2*} \right\| \, $in the case where $X=0$, it is sufficient to add a small constant in the denominator of the control law (32).

3.3 Solving Lyapunov equation using reduced order models

The Lyapunov equation given by expression (22) is numerically ill-conditioned due to the small positive singular perturbation parameter $\varepsilon $ which affects the coefficient matrices. This numerical stiffness, attributed to the simultaneous occurrence of "slow" and "fast" phenomena (see [31, 32]) is removed using the decomposition of the full-order Lyapunov equation (22) into slow and fast subsystem components. The structure of $P(\varepsilon)$ is assumed to be of the form

$ \begin{equation} \label{GrindEQ__32_} P(\varepsilon )=\left[\begin{array}{cc} {P_{1} }&{P_{2} } \\ {P_{2}^{\rm T} }&{P_{3} } \end{array}\right]. \end{equation} $ (34)

The solution $P(\varepsilon)$ is $\varepsilon$-dependent, because (22) contains ${\varepsilon }^{-1}$-order matrices. Let $Q(\varepsilon)$ be of the form

$ \begin{equation} \label{GrindEQ__32a_} Q=\left[\begin{array}{cc} {I_{n1} }&{0} \\ {0}&{\varepsilon ^{-1} I_{n2} } \end{array}\right] \end{equation} $ (35)

where $I_{n1}$ and $I_{n2}$ are identity matrices of dimension $n_{1}$ and $n_{2}$ respectively, and $K_1=[K_{11} \, \, \, K_{12}]$. Substituting $P(\varepsilon )$ into Lyapunov equation (22) leads to the following partitioned three equations

$ \begin{equation} \label{GrindEQ__33_} a_{11}^{\rm T} P_{1} +\varepsilon ^{-1} a_{21}^{\rm T} P_{2}^{\rm T} +P_{1} \, a_{11} +\varepsilon ^{-1} P_{2} \, a_{21} =-I_{n1} \end{equation} $ (36)
$ \begin{equation} \label{GrindEQ__34_} \varepsilon ^{-1} a_{21}^{\rm T} P_{3} +a_{11}^{\rm T} P_{2} +P_{1} \, a_{12} +\varepsilon ^{-1} P_{2} \, a_{22} =0 \end{equation} $ (37)
$ \begin{equation} \label{GrindEQ__35_} a_{12}^{\rm T} P_{2} +\varepsilon ^{-1} a_{22}^{\rm T} P_{3} +P_{2}^{\rm T} a_{12} +\varepsilon ^{-1} P_{3} \, a_{22} =-\varepsilon ^{-1} I_{n2} \end{equation} $ (38)

where $a_{11} =A_{11} +B_{2x} \, K_{11} $, $a_{12} =A_{12} +B_{2x} \, K_{12} $, $a_{21} =A_{21} +B_{2z} \, K_{11} $ and $a_{22} =A_{22} +B_{2z} \, K_{12} $.

By limiting solutions of equations (33-35) as $\varepsilon \to 0$, the following zero-order equations are obtained

$ \begin{equation} \label{GrindEQ__36_} a_{11}^{\rm T} P_{1} +P_{1} \, a_{11} =-I_{n1} \end{equation} $ (39)
$ \begin{equation} \label{GrindEQ__37_} a_{21}^{\rm T} P_{3} +P_{2} \, a_{22} =0 \end{equation} $ (40)
$ \begin{equation} \label{GrindEQ__38_} a_{22}^{\rm T} P_{3} +P_{3} \, a_{22} =-I_{n2}. \end{equation} $ (41)

Since $K_{11}$ and $K_{12}$ are approached respectively, by $K_{s}$ and $K_{f}$ (see [4]), it is clear that equations (39) and (41) correspond respectively, to the Lyapunov equations of the slow subsystem approximation described by (7) and the fast subsystem described by (11) of the singularly perturbed system (1). Hence, $P_1$ and $P_3$ correspond respectively, to $P_s$ which is solution of slow Lyapunov equation (5) (approached by (8)) and $P_f$ which is solution of fast Lyapunov equation (12). Since $a_{22}$ is Hurwitz (see (11)), equation (41) can be solved for $P_3$. Consequently, $P_2$ can be obtained by solving (40): $P_{2} \, =-a_{21}^{\rm T} P_{3} \, a_{22}^{-1} $. Finally, $P\left(\varepsilon \right)$ can be approached with $P(\varepsilon )\approx \left[\begin{array}{cc} {P_{s} }&{P_{2} } \\ {P_{2}^{T} }&{P_{f} } \end{array}\right]$ where $P_{2} \, \approx -a_{21}^{\rm T} P_{f} \, a_{22}^{-1} $.

4 Stability analysis after controller simplification

The technique used to simplify the controller parameters $K_{1}$ and $K_{2}$ is the singular perturbation method where the principle is to set the singular perturbation parameter to zero and to consider the dominant part. It is well known that this technique preserves stability (see [1, 4, 29, 30]).

Additionally, the theoretical proof of the stability with the simplified controller will be presented below.

4.1 Composite state-feedback control

To stabilize the full-order singularly perturbed system (1) with a state-feedback control, it is sufficient to stabilize the reduced subsystems independently and then to design a composite control law using the local feedback gains $K_{s}$ and $K_{f}$ provided that $(A_s, B_s)$ and $(A_f, B_f)$ are controllable and $A^{-1}_{22}$ exists[1, 4]. The composite control is of the following form:

$ \begin{equation} \label{GrindEQ__38a_} u(t)=K_1(t)\, X(t)=\left[\begin{array}{cc} K_{11}&K_{12} \end{array} \right]\left[\begin{array}{c} x \\ z \end{array} \right] \end{equation} $ (42)

where $K_{12}=K_f$ and $ K_{11}=( I_{f}+K_{f}\, A_{22}^{ -1}\, B_{ 2})\, K_{s}+K_{f}\, A_{22}^{ -1}\, A_{21}$.

Thus, the simplified feedback controller conserves the stability of the full-order singularly perturbed system.

4.2 Stability analysis with the simplified adaptive laws

The system equation is expressed as in (24) and the Lyapunov function candidate is defined as

$ \begin{equation} \label{GrindEQ__39_} V_{*}(t)=X^{\rm T} P(\varepsilon) X+ {\gamma }^{-1}{\ \tilde{k}}^2_3 >0. \end{equation} $ (43)

Compared with the form in (25), the singular perturbation parameter is omitted from the second term of the expression. The assumptions (27) and (28) can be transformed as follows.

There exists a singular perturbation parameter $\varepsilon $ such that

$ \begin{align} \label{GrindEQ__40_} &{\left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \left\| F\right\| \, \bar{w}\le }\nonumber \\&\quad\quad {\varepsilon \, \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \, k_{3} } \end{align} $ (44)

and

$ \begin{align} \label{GrindEQ__41_} &{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, (\rho -I)K_{1} X\le }\nonumber \\&\quad\quad {\varepsilon \, \, \left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, .} \end{align} $ (45)

The simplification of the expressions (44) and (45) using the same method employed in (32) and (33), yields the following expressions

$ \begin{align} \label{GrindEQ__42_} &{\left\| X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\right\| \, \left\| F\right\| \, \bar{w}\le }\nonumber \\& \quad\quad{\left\| X^{\rm T} P(\varepsilon )B_{2*} (\varepsilon )\right\| \, \, k_{3} } \end{align} $ (46)

and

$ \begin{align} \label{GrindEQ__43_} &{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, (\rho -I)K_{1} X\le }\nonumber \\& \quad\quad{\left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2*} (\varepsilon )\right\| \, .} \end{align} $ (47)

Using the simplified adaptive law (32) and the same assumptions (46) and (47), the time derivative of $V_{*}(t)$ for $t>0$ becomes

$ \begin{align} \label{GrindEQ__44_} &{\dot{V}_{*} (t)\le -2\, X^{\rm T} Q\, X+2\, \left\| X^{{\rm T}} P(\varepsilon )B_{2*} (\varepsilon )\right\| \, \, k_{3} \, }+\nonumber \\& \quad\quad\quad{2\, \left\| K_{1} X\right\| \, \, \left\| X^{\rm T} P(\varepsilon )B_{2*} (\varepsilon )\right\| \, }-\nonumber \\& \quad\quad\quad{2\, \, \beta \dfrac{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \rho \, B_{2*}^{\rm T} (\varepsilon )P(\varepsilon )X}{\alpha \left\| B_{2*}^{\rm T} (\varepsilon )P(\varepsilon )X\right\| } \, (\hat{k}_{3} (t)}+\nonumber \\& \quad\quad\quad{\left\| K_{1} X\right\| )+2\, \, \gamma ^{-1} \tilde{k}_{3} \, \dot{\tilde{k}}_{3}}. \end{align} $ (48)

From (20) and (30), it holds that

$ \begin{align} \label{GrindEQ__45_} &{X^{\rm T} P(\varepsilon )B_{2} (\varepsilon )\, \rho \, B_{2*}^{\rm T} (\varepsilon )P(\varepsilon )X\ge }\nonumber \\& \quad\quad{\dfrac{\alpha }{\beta } \, \left\| X^{\rm T} P(\varepsilon ) B_{2} (\varepsilon )\right\| \, \left\| X^{\rm T} P(\varepsilon )B_{2*} (\varepsilon )\right\|}. \end{align} $ (49)

This leads to

$ \begin{align} \label{GrindEQ__46_} &{\dot{V}_{*} (t)\le -2\, X^{\rm T} Q\, X+2\, \left\| X^{\rm T} P(\varepsilon )B_{2*} (\varepsilon )\right\| \, \, (k_{3}}+\nonumber \\& \quad\quad\quad{\, \left\| K_{1} X\right\| \, -\, \hat{k}_{3} (t)-\left\| K_{1} X\right\|)}+\nonumber \\& \quad\quad\quad{2\, \, \gamma ^{-1} \tilde{k}_{3} \, \dot{\tilde{k}}_{3}}. \end{align} $ (50)

Knowing that ${\tilde{k}}_3\left(t\right)={\hat{k}}_3\left(t\right)-k_3$ and considering the simplified form (33), it is easy to see that $\dot{V}\left(\varepsilon \right)\le 0$, which guarantees the asymptotic stability of the faulty closed-loop system using the simplified gain $K_{{1}}$ and the adaptive laws (32) and (33).

5 Extension to multi-modeling structures

The multi-modeling structure (three time scales singularly perturbed systems) was introduced to the control community by Khalil and Kokotovic[33]. This concept is defined by a linear dynamic system that has one slow and two fast subsystems. The fast subsystems are strongly connected to the slow subsystem and weakly connected (or not connected) among themselves[33-35].

5.1 Modeling and reduction of the full order system

The considered multi-modeling structure with disturbance input is defined by

$ \label{GrindEQ__47_} \begin{eqnarray}{\left[\begin{array}{c} {\dot{x}_{1} } \\ {\begin{array}{l} {\varepsilon _{1} \, \dot{x}_{2} } \\ {\varepsilon _{2} \, \dot{x}_{3} } \end{array}} \end{array}\right]} =\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \nonumber\\ \left[\begin{array}{ccc} {A_{1} }&{A_{2} }&{A_{3} } \\ {A_{4} }&{A_{5} }&{\varepsilon _{3} \, A_{6} } \\ {A_{7} }&{\varepsilon _{3} \, A_{8} }&{A_{9} } \end{array}\right]\, \left[\begin{array}{c} {x_{1} } \\ {\begin{array}{l} {x_{2} } \\ {x_{3} } \end{array}} \end{array}\right]+B_{1} \, w + B_{2} \, u \end{eqnarray} $ (51)

where for ${i\; =\; 1, \; 2}$, $B_{i} =\left[\begin{array}{c} {B_{i1} } \\ {\begin{array}{l} {B_{i2} } \\ {B_{i3} } \end{array}} \end{array}\right]$ and $C_{i} =\left[\begin{array}{ccc} {C_{i1} }&{C_{i2} }&{C_{i3} } \end{array}\right]$ are matrices with appropriate dimensions; $x_{1} \in {\bf R}^{n1} $, $x_{2} \in {\bf R}^{n2} $, $x_{3} \in {\bf R}^{n3} $, $u\in {\bf R}^{m} $, $w\in {\bf R}^{p} $ and $Y\in {\bf R}^{q} $ are respectively, the slow state variables, the fast state variables, the control input, the disturbance input and the measured output; $\varepsilon $${}_{1}$ and $\varepsilon $${}_{2}$ are small positive singular perturbation parameters of the same order of magnitude such that: ${0\; < \; \; k}_{{\rm 1}} \le \frac{\varepsilon _{{\rm 2}} }{\varepsilon _{{\rm 1}} } \le {\; k}_{{\rm 2}} {\rm \; < \; }\infty $. $\varepsilon $${}_{3}$ is a small weak coupling parameter. In order to simplify the considered model, without loss of generality, we assume that the fast state variables are not connected among themselves, i.e., we set the weak coupling parameter $\varepsilon $${}_{3}$ to zero. The multi-model singularly perturbed system is then transformed as follows:

$ \label{GrindEQ__48_} \begin{eqnarray}{\left[\begin{array}{c} {\dot{x}_{1} } \\ {\begin{array}{l} {\varepsilon _{1} \, \dot{x}_{2} } \\ {\varepsilon _{2} \, \dot{x}_{3} } \end{array}} \end{array}\right]}=\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\nonumber\\ \left[\begin{array}{ccc} {A_{1} }&{A_{2} }&{A_{3} } \\ {A_{4} }&{A_{5} }&{0} \\ {A_{7} }&{0}&{A_{9} } \end{array}\right]\, \left[\begin{array}{c} {x_{1} } \\ {\begin{array}{l} {x_{2} } \\ {x_{3} } \end{array}} \end{array}\right]+B_{1} \, w + B_{2} \, u . \end{eqnarray} $ (52)

The multi-model singularly perturbed system (52) can be iteratively decomposed into slow and fast subsystems by considering $\varepsilon_{1}=0$ and $\varepsilon_{2}=0$ respectively to give rise to an overall reduced system, $\varepsilon_{1}$-reduced system and $\varepsilon_{2}$-reduced systems in respective "stretched" time scales. Substituting $\varepsilon_{1}=0$, we obtain the reduced equation

$ \begin{equation} \label{GrindEQ__48a_} 0=A_{7}\, \bar{x}_{1}+A_{9}\, \bar{x}_{3}+B_{13}\, \bar{w}+B_{23}\, \bar{u} \end{equation} $ (53)

where $\bar{x}_{1}$, $\bar{x}_{3}$, $\bar{w}$ and $\bar{u}$ denote the values of the respective variables for the approximation $\varepsilon_{2}=0$. The assumption that $A_{9}$ is non-singular, leads to the following expression

$ \begin{equation} \label{GrindEQ__48b_} \bar{x}_{3}=-A^{-1}_{9}\, A_{7}\, \bar{x}_{1}-A^{-1}_{9}\, B_{13}\, \bar{w}-A^{-1}_{9}\, B_{23}\, \bar{u}. \end{equation} $ (54)

Thus, we get the $\varepsilon_{2}$-reduced subsystem by substituting the expression of $\bar{x}_{3}$ in the main system

$ \begin{equation} \label{GrindEQ__49_} \left\{\begin{array}{l} {\dot{x}_{1} =(A_{1} -A_{3} \, A_{9}^{-1} \, A_{7} )\, \, x_{1} +A_{2} \, x_{2} }+ \\ \quad\quad~{(B_{11} -A_{3} \, A_{9}^{-1} \, B_{13} )\, w}+ \\ \quad\quad~{(B_{21} -A_{3} \, A_{9}^{-1} \, B_{23} )\, u} \\ {\varepsilon _{1} \, \dot{x}_{2} =A_{4} \, x_{1} +A_{5} \, x_{2} +B_{12} \, w+B_{22} \, u}. \end{array}\right. \end{equation} $ (55)

The second stage of reduction is carried out by assuming that $A_{5}$ is non-singular. Then, the $\varepsilon $${}_{1}$-reduced subsystem is obtained by letting $\varepsilon_{1} \rightarrow 0$, the resulting subsystem is known as the overall reduced subsystem or the slowest subsystem

$ \begin{equation} \label{GrindEQ__50_} \dot{x}_{s} =A_{s} \, x_{s} +B_{1s} \, w_{s} +B_{2s} \, u_{s} \end{equation} $ (56)

where $x_{s}$, $u_{s}$ and $w_{s}$ are respectively, the slow parts of the states, the control input $u$ and the disturbance input $w$ and

$A_{s} =A_{1} -A_{3} \, A_{9}^{-1} \, A_{7} -A_{2} \, A_{5}^{-1} \, A_{4} \\ B_{1s} =B_{11} -A_{3} \, A_{9}^{-1} \, B_{13} -A_{2} \, A_{5}^{-1} \, B_{12} \\ B_{2s} =B_{21} -A_{3} \, A_{9}^{-1} \, B_{23} -A_{2} \, A_{5}^{-1} \, B_{22}.$

The slowest subsystem provides an approximation of the singularly perturbed system behavior.

With the following approximations used for $H_{\infty}$ control of two time scales singularly perturbed system in [30, 36]: $A_{s} \approx A_{1} $, $B_{1s} \approx B_{11} $, and $B_{2s} \approx B_{21} $, the slowest subsystem (56) can be approached by the following subsystem:

$ \begin{equation} \label{GrindEQ__51_} \dot{x}_{s} =A_{1} \, x_{s} +B_{11} \, w_{s} +B_{21} \, u_{s}. \end{equation} $ (57)

The fast subsystems (FSS1 and FSS2) are defined in the fast time scales by

$ \begin{equation} \label{GrindEQ__52_} \dot{x}_{f1} =A_{5} \, x_{f1} +B_{12} \, w_{f1} +B_{22} \, u_{f1} \end{equation} $ (58)
$ \begin{equation} \label{GrindEQ__53_} \dot{x}_{f2} =A_{9} \, x_{f2} +B_{13} \, w_{f2} +B_{23} \, u_{f2} \end{equation} $ (59)

where for $i=1, \, 2$, $x_{fi}$, $u_{fi}$ and $w_{fi}$ are respectively, the fast part of the states, the control input $u$ and the disturbance input $w.$

Assuming that $(A_s, B_{2s})$, $(A_5, B_{22})$ and $(A_9, B_{23})$ are stabilizable, and taking into account the approximations of the slow system matrices, there exist symmetric and positive definite matrices $P_s$, $P_{f1}$ and $P_{f2}$ satisfying respectively, the following one slow and two fast Lyapunov equations:

$ \begin{equation} \label{GrindEQ__54_} (A_{1} +B_{21}\, K_{s} )^{\rm T} \, P_{s} +P_{s} \, (A_{1} +B_{21}\, K_{s} )=-Q_{s} \end{equation} $ (60)
$ \begin{equation} \label{GrindEQ__55_} (A_{5} +B_{22}\, K_{f1} )^{\rm T} \, P_{f1} +P_{f1} \, (A_{5} +B_{22}\, K_{f1} )=-Q_{f1} \end{equation} $ (61)
$ \begin{equation} \label{GrindEQ__56_} (A_{9} +B_{23}\, K_{f2} )^{\rm T} \, P_{f2} +P_{f2} \, (A_{9} +B_{23}\, K_{f2} )=-Q_{f2} \end{equation} $ (62)

where $Q_s$, $Q_{f1}$ and $Q_{f2}$ are any given positive definite symmetric matrices. $K_s$, $K_{f1}$ and $K_{f2}$ are controller gains stabilizing respectively, the pairs $(A_s, B_{2s})$, $(A_5, B_{22})$ and $(A_9, B_{23})$. The controller gains $K_s$, $K_{f1}$ and $K_{f2}$ satisfy respectively, the local control laws $u_{s}=K_{s}\, x_{s}$, $u_{f1}=K_{f1}\, x_{f1}$ and $u_{f2}=K_{f2}\, x_{f2}$.

5.2 Controller design

The full order problem in the multi-time-scale case is similar to the two-time-scale case. It is sufficient to put the (52) in the form (16). For that purpose, we consider the following notations:

$ \begin{align} A(\varepsilon )=\left[\begin{array}{cc} {A_{1} }&{A_{2} \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, A_{3} } \\ {\begin{array}{l} \varepsilon^{-1}_{1} \, A_{4} \\ \varepsilon^{-1}_{2} \, A_{7} \end{array}}&{\begin{array}{l} {\, \, \varepsilon^{-1}_{1}\, A_{5} \, \, \, \, \, \, \, \, \, \, ~~ 0\, } \\ {\, \, \, \, \, \, \, 0\, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \varepsilon^{-1}_{2}\, A_{9} \, \, } \end{array}} \end{array}\right] \end{align} $
$B_{i} (\varepsilon )=\left[\begin{array}{c} {B_{i1} } \\ {\begin{array}{l} \varepsilon^{-1}_{1} \, B_{i2} \\ \varepsilon^{-1}_{2} \, B_{i3} \end{array}} \end{array}\right],\;{\rm{for}}\; {i\; =\; 1, \; 2} \; {\rm{and}}\; X=\left[\begin{array}{c} {x_{1} } \\ {\begin{array}{l} {x_{2} } \\ {x_{3} } \end{array}} \end{array}\right].$

Consequently, the system (52) can be rewritten as

$ \begin{equation} \label{GrindEQ__57_} \dot{X}=A(\varepsilon )\, \, X+B_{1} (\varepsilon )\, w+B_{2} (\varepsilon )\, u. \end{equation} $ (63)

This form is similar to (16). For the same fault model (15) and the control law (18), Theorem 1 is then valid in this case. In the proof, the parameter $\varepsilon $ can take the value $\varepsilon $${}_{1}$ or $\varepsilon $${}_{2}$ where ${\varepsilon }_1\in \ ]0, {\varepsilon }^*_1]$ and ${\varepsilon }_2\in ]0, {\varepsilon }^*_2]$.

5.3 Controller simplification

The same control law (18) is considered. The controller gain $K_{1}$ will be designed using composite control which takes the following form in the case of vanishing disturbance[37]:

$ \begin{equation} \label{GrindEQ__58_} \begin{array}{l} {u(t)_{composite} =u_{s} (t)+\, u_{f1} (t)+u_{f2} (t)\, }=\quad\quad\quad \\ \quad\quad{\, K_{s} \, x_{s} (t)+K_{f1} \, x_{f1} (t)+K_{f2} \, x_{f2} (t)\, }= \\ \quad\quad{K_{1} \, X(t)\, =\, \left[\begin{array}{ccc} {g_{1} }&{g_{2} }&{g_{3} } \end{array}\right]\left[\begin{array}{c} {x_{1} (t)} \\ {x_{2} (t)} \\ {x_{3} (t)} \end{array}\right]}. \end{array} \end{equation} $ (64)

The local control laws $u_s(t)$, $u_{f1}(t)$ and $u_{f2}(t)$ are separately designed for respectively, the slow, the first fast and the second fast subsystems, see Section 5.1. Nevertheless, a realizable composite control must be expressed in terms of the full order system states $x_1$, $x_2$ and $x_3$. Thereafter, $g_1$, $g_2$ and $g_3$ should be expressed as a function of $K_s$, $K_{f1\ }$ and $K_{f2}$. This can be obtained by substituting $x_s$ by $x_1$, $x_{f1}$ by $x_2-\ {\tilde{x}}_2$ and $x_{f2}$ by $x_3-\ {\hat{x}}_3$, where ${\tilde{x}}_2$ and ${\hat{x}}_3$ denote the variables corresponding respectively to the approximations ${\varepsilon }_1=0$ and ${\varepsilon }_2=0$. This leads to the following composite control law:

$ \begin{align} \label{GrindEQ__59_} &{u(t)_{composite} =K_{s} \, x_{1} (t)}+\nonumber \\& \quad\quad{K_{f1} \, [x_{2} (t)+A_{5}^{-1} \, (A_{4} \, +B_{22} \, K_{s} \, )\, x_{1} (t)]}+\nonumber \\&\quad\quad {K_{f2} \, [x_{3} (t)+A_{9}^{-1} \, (A_{7} \, +B_{23} \, K_{s} \, )\, x_{1} (t)]}. \end{align} $ (65)

Equation (65) can be rearranged as follows:

$ \begin{align} \label{GrindEQ__60_} &{u(t)_{composite} =[K_{s} \, +K_{f1} \, A_{5}^{-1} \, (A_{4} \, +B_{22} \, K_{s} \, )\, }+\nonumber \\& \quad\quad{K_{f2} \, A_{9}^{-1} \, (A_{7} \, +B_{23} \, K_{s} \, )]\, \, x_{1} (t)\, }+\nonumber \\& \quad\quad{\, K_{f1} \, x_{2} (t)\, +K_{f2} \, x_{3} (t)} \end{align} $ (66)

where $g_{2} = K_{f1} $, $g_{3} = K_{f2} $ and $g_{1} =K_{s} +K_{f1} \, A_{5}^{-1} \, (A_{4} +B_{22} \, K_{s} \, ) +K_{f2} \, A_{9}^{-1} \, (A_{7} +B_{23} \, K_{s} \, )$.

Thus, the controller gain $K_{1 }$ of the control law (18) can be designed using the local gains $K_s$, $K_{f1\ }$ and $K_{f2}$ which are derived based on slow and fast subsystem performance specifications. The second term of the control law (18) depends on the matrix $P(\varepsilon)$. However, the presence of time scales in the dynamics of the singularly perturbed system (63) causes ill-conditioning. Subsequently, it is difficult to determine $P(\varepsilon)$ directly from (22) in the multi-modeling case. In order to obviate this problem, the singular perturbation approach will be applied on the full-order Lyapunov equation (22). Thus, the slow and fast subsystem components will be used. Consider $P(\varepsilon)$ and $Q(\varepsilon)$ of the form:

$ \begin{equation} \label{GrindEQ__60a_} P(\varepsilon )=\left[\begin{array}{ccc} {P_{1} }&{P_{2} } & {P_{3} } \\ {P_{2}^{\rm T} }&{P_{4} }&{P_{5} } \\ {P_{3}^{\rm T} }&{P_{5}^{\rm T} }&{P_{6} } \end{array}\right] \end{equation} $ (67)

and

$ \begin{equation} \label{GrindEQ__60b_} Q(\varepsilon )=\left[\begin{array}{ccc} {I_{n1} }&{0}&{0} \\ {0}&{\varepsilon _{1}^{-1} I_{n2} }&{0} \\ {0}&{0}&{\varepsilon _{2}^{-1} I_{n3} } \end{array}\right] \end{equation} $ (68)

where $I_{ni}$, $i = 1, 2, 3$ is identity matrix. Expanding the full-order Lyapunov equation (22) in the multi-modeling case, we obtain the following main equations (diagonal terms of the expanded form):

$ \begin{align} \label{GrindEQ__61_} &{a_{1}^{\rm T} P_{1} +\varepsilon _{1}^{-1} \, a_{4}^{\rm T} P_{2}^{T} +\varepsilon _{2}^{-1} \, a_{7}^{\rm T} P_{3}^{\rm T}}+\nonumber \\& \quad\quad{P_{1} \, a_{1} +\varepsilon _{1}^{-1} P_{2} \, a_{4} +\varepsilon _{2}^{-1} P_{3} \, a_{7} =-I_{n1} } \end{align} $ (69)
$ \begin{align} \label{GrindEQ__62_} &{a_{2}^{\rm T} P_{2} +\varepsilon _{1}^{-1} a_{5}^{\rm T} P_{4} +\varepsilon _{2}^{-1} a_{8}^{\rm T} P_{5}^{\rm T}}+\nonumber \\& \quad\quad{P_{2}^{\rm T} a_{2} +\varepsilon _{1}^{-1} P_{4} \, a_{5} +\varepsilon _{2}^{-1} P_{5} \, a_{8} =-\varepsilon _{1}^{-1} I_{n2} } \end{align} $ (70)
$ \begin{align} \label{GrindEQ__63_} &{a_{3}^{\rm T} P_{3} +\varepsilon _{1}^{-1} a_{6}^{\rm T} P_{5} +\varepsilon _{2}^{-1} a_{9}^{\rm T} P_{6}}+\nonumber \\& \quad\quad{P_{3}^{\rm T} a_{3} +\varepsilon _{1}^{-1} P_{5}^{\rm T} \, a_{6} +\varepsilon _{2}^{-1} P_{6} \, a_{9} =-\varepsilon _{2}^{-1} I_{n3} } \end{align} $ (71)

where $a_{1} =A_{1} +B_{21} \, g_{1} $, $a_{2} =A_{2} +B_{21} \, g_{2} $, $a_{3} =A_{3} +B_{21} \, g_{3} $, $a_{4} =A_{4} +B_{22} \, g_{1} $, $a_{5} =A_{5} +B_{22} \, g_{2} $, $a_{6} =B_{22} \, g_{3} $, $a_{7} =A_{7} +B_{23} \, g_{1} $, $a_{8} =B_{23} \, g_{2} $ and $a_{9} =A_{9} +B_{23} \, g_{3} $.

Multiplying equations (69) by the singular perturbation parameter $\varepsilon $${}_{1}$ and letting $\varepsilon_{1} \to 0$, then multiplying the same equations by $\varepsilon $${}_{2}$ and letting $\varepsilon_{2} \to 0$ leads respectively to $P_{2} \, =0$ and $P_{3} \, =0$, which reduces equation (69) to

$ \begin{equation} \label{GrindEQ__64_} a_{1}^{\rm T} P_{1} +P_{1} \, a_{1} =-I_{n1}. \end{equation} $ (72)

Multiplying equations (70) by $\varepsilon $${}_{1}$ and (71) by $\varepsilon $${}_{2 }$ and ${}_{ }$ letting respectively, $\varepsilon_{1} \to 0$ and $\varepsilon_{2} \to 0$ generate the following zero-order equations:

$ \begin{equation} \label{GrindEQ__65_} a_{5}^{\rm T} P_{4} +P_{4} \, a_{5} =-I_{n2} \end{equation} $ (73)
$ \begin{equation} \label{GrindEQ__66_} a_{9}^{\rm T} P_{6} +P_{6} \, a_{9} =-I_{n3}. \end{equation} $ (74)

Considering the approximations $g_{1} =K_{s} $, $g_{2} =\, K_{f1} $ and $g_{3} =\, K_{f2} $ (used for two-time-scale singularly perturbed systems in [4]), and setting $Q_{s} =I_{n1} $, $\, Q_{f1} =I_{n2} $ and $\, Q_{f2} =I_{n3} $ leads respectively, to the Lyapunov equations of the slowest subsystem approximation described by (60), the fast subsystem (FSS1) described by (61) and the fast subsystem (FSS2) described by (62) of the singularly perturbed system (52). Hence, $P_{1}$, $P_{4}$ and $P_{6}$ correspond respectively to ${P}_{s}$, ${P}_{f1}$ and ${P}_{f2}$. Finally, $P\left(\varepsilon \right)$ can be approached with $P(\varepsilon )\approx \left[\begin{array}{ccc} {P_{s} }&{0}&{0} \\ {0}&{P_{f1} }&{0} \\ {0}&{0}&{P_{f2} } \end{array}\right]$. Thereby, the slow and fast subsystems of the full order singularly perturbed system will be used, for small singular perturbation parameters $\varepsilon $${}_{i}$, to approximate the Lyapunov matrix $P(\varepsilon) $, by solving three smaller order problems. This approach avoids the ill-conditioning due to the presence of slow and fast phenomena.

6 Example of application

To illustrate the effectiveness of the proposed method, a numerical example is given below. The state space representation described by (1) is considered with the following system matrices

$ \begin{align} A_{11} =\left[\begin{array}{cc} {-5}&{0.2} \\ {-0.5}&{6} \end{array}\right], \, A_{12} =\left[\begin{array}{cc} {0}&{0.1} \\ {-1}&{1} \end{array}\right] \end{align} $
$ \begin{align} ~~~~A_{21} =\left[\begin{array}{cc} {-9}&{-8} \\ {0.3}&{0.1} \end{array}\right], \, A_{22} =\left[\begin{array}{cc} {-7}&{1} \\ {-0.5}&{-6} \end{array}\right] \end{align} $
$ \begin{align} ~~~~{\it B}_{2} ={\it B}_{1} =\, \left[\begin{array}{cc} {1}&{1.55} \\ {1}&{-0.5} \\ {0.9}&{0.8} \\ {0.2}&{0.11} \end{array}\right], \, {\it w=}\, \left[\begin{array}{c} {0.01\, \, \sin (5\, t)} \\ {0.5} \end{array}\right]. \end{align} $

The considered actuator fault is a 10% loss of effectiveness in the first actuator and 50% in the second actuator, i.e., $\rho={\rm diag}(0.9, 0.5)$. The system operates normally until the time instant $t$ = 100 s. From that moment, the actuators are attacked by the above mentioned faults. After checking that the pairs $(A_s, B_s)$ and $(A_f, B_f)$ are each controllable and $A^{-1}_{22}$ exists, the slow gain $K_s=\left[\begin{array}{cc} {\rm 2.24}&{\rm 13.5} \\ {\rm 1.78}&{\rm-}{\rm 8.58} \end{array} \right]$ and the fast gain $K_{f}=\left[\begin{array}{cc} {\rm-}{\rm 48.03}&{\rm 312.9} \\ {\rm 82.78}&{\rm-}{\rm 350.8} \end{array} \right]$ are designed to assign the eigenvalue pairs (-10, -9) and (-30, -31) to the closed loop slow and fast subsystems respectively. This leads to the first composite control gain $K_{1}=\left[\begin{array}{cc} {\rm-}{\rm 109.05}&{\rm-}{\rm 80.07} \\ {\rm 1}{\rm 60.5}&{\rm 124.06} \end{array} \right]$. The control design parameters are chosen as $\gamma ={\rm diag}(10, 10, 10, 10)$ and $\frac{\beta }{\alpha }=5$. By setting the initial values $X_0={[0.2, \ 0.03, \ 0.02, \ 0.01]}^{\rm T}$ and $\varepsilon =0.05$, the simulation results in presence of actuator fault and external disturbances are given in Figs. 1-3. The first one represents the states responses and depicts that the closed-loop fault tolerant control system becomes asymptotically stable in the presence of actuators faults and external disturbances. It is clear in Fig. 2 that the gain ${\hat{k}}_3(t)$ reacts instantly further to fault occurrence. Fig. 3 shows the trajectory of the controller $K(t)$.

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Fig. 1. States trajectories in the faulty case

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Fig. 2. Controller gain ${\hat{k}}_3(t)$ in the faulty case

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Fig. 3. Controller $K$ in the faulty case

7 Conclusions

This paper presented an adaptive reconfigurable control for linear time-invariant two-time-scale and three-time-scale singularly perturbed system subject to actuator faults and external disturbances. The adopted actuator fault model covers the cases of normal operation and loss of effectiveness. The controller designed for the full-order model is $\varepsilon $-dependent. The first part is simplified using a composite controller based on local controllers stabilizing slow and fast subsystems. The second part is designed through adaptive laws, where the corresponding Lyapunov equation is ill-conditioned caused by the presence of different dynamics. In the two-time-scale singularly perturbed system case and the multi-modeling case, the numerical stiffness of the Lyapunov equation is avoided using singular perturbation method, where the Lyapunov matrix $P(\varepsilon) $ is expressed utilizing the solutions of slow and fast Lyapunov equations.

The adaptive fault tolerant controller yields the required stability properties for the global singularly perturbed system not only in the fault-free case, but also in the presence of actuator faults and disturbances. The assessment of the upper bound of singular perturbation parameter $\varepsilon $ remains an interesting theme of research perspective.

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