^{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
Systems containing dynamics which operate on two or more timescales 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
The considered system is a twotime 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
According to the timescale property of the singularly perturbed system, the slow and the fast subsystems related to the fullorder system (1) can be derived by introducing a change of coordinates in which the system appears in blocktriangular forms or by using the singular perturbation method which formally sets the singular perturbation parameter
$ \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,
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \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
$ \begin{equation} \label{GrindEQ__12_} u^f_i\left(t\right)={\rho }_i{\ u}_i(t) \end{equation} $  (14) 
where
$ \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
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 closedloop system via state feedback.
3.1 The fullorder problemIntroduce 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
$ \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
$ \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
$ \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
The constant
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
2) For any given positive definite symmetric matrix
$ \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)
There exists
Proof. In the light of equations (17) and (18), the closedloop 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 closedloop 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
$ \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
$ \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
It is well known that if both the slow subsystem and the fast subsystem are stable, then there exists
$ \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
$ \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
As mentioned in [25], to avoid discontinuity which can be caused by the term
The Lyapunov equation given by expression (22) is numerically illconditioned due to the small positive singular perturbation parameter
$ \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
$ \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
$ \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
By limiting solutions of equations (3335) as
$ \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
The technique used to simplify the controller parameters
Additionally, the theoretical proof of the stability with the simplified controller will be presented below.
4.1 Composite statefeedback controlTo stabilize the fullorder singularly perturbed system (1) with a statefeedback control, it is sufficient to stabilize the reduced subsystems independently and then to design a composite control law using the local feedback gains
$ \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
Thus, the simplified feedback controller conserves the stability of the fullorder singularly perturbed system.
4.2 Stability analysis with the simplified adaptive lawsThe 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
$ \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
$ \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
The multimodeling 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^{[3335]}.
5.1 Modeling and reduction of the full order systemThe considered multimodeling 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
$ \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 multimodel singularly perturbed system (52) can be iteratively decomposed into slow and fast subsystems by considering
$ \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
$ \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
$ \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
$ \begin{equation} \label{GrindEQ__50_} \dot{x}_{s} =A_{s} \, x_{s} +B_{1s} \, w_{s} +B_{2s} \, u_{s} \end{equation} $  (56) 
where
$A_{s} =A_{1} A_{3} \, A_{9}^{1} \, A_{7} A_{2} \, A_{5}^{1} \, A_{4} \\ B_{1s} =B_{11} A_{3} \, A_{9}^{1} \, B_{13} A_{2} \, A_{5}^{1} \, B_{12} \\ B_{2s} =B_{21} A_{3} \, A_{9}^{1} \, B_{23} A_{2} \, A_{5}^{1} \, B_{22}.$ 
The slowest subsystem provides an approximation of the singularly perturbed system behavior.
With the following approximations used for
$ \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
Assuming that
$ \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
The full order problem in the multitimescale case is similar to the twotimescale 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
The same control law (18) is considered. The controller gain
$ \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
$ \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
Thus, the controller gain
$ \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
$ \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
Multiplying equations (69) by the singular perturbation parameter
$ \begin{equation} \label{GrindEQ__64_} a_{1}^{\rm T} P_{1} +P_{1} \, a_{1} =I_{n1}. \end{equation} $  (72) 
Multiplying equations (70) by
$ \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
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.,
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Fig. 1. States trajectories in the faulty case 
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Fig. 2. Controller gain 
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Fig. 3. Controller 
7 Conclusions
This paper presented an adaptive reconfigurable control for linear timeinvariant twotimescale and threetimescale 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 fullorder model is
The adaptive fault tolerant controller yields the required stability properties for the global singularly perturbed system not only in the faultfree case, but also in the presence of actuator faults and disturbances. The assessment of the upper bound of singular perturbation parameter
[1] 
P. Y. Kokotovic, H. K. Khalil, J. O'Reilly. Singular Perturbation Methods in Controlanalysis and Design, USA: Academic Press, 1999.

[2] 
H. K. Khalil. Output feedback control of linear twotimescale systems. IEEE Transactions on Automatic Control, vol.32, no.9, pp.784792, 1987. DOI:10.1109/TAC.1987.1104720 
[3] 
Z. Gajic, M. Lim. Optimal Control of Singularly Perturbed Linear Systems and Applications, New York, USA: Marcel Dekker, 2001.

[4] 
Z. H. Shao, M. E. Sawan. Stabilization of uncertain singularly perturbed systems. IEE Proceedings:Control Theory and Applications, vol.153, no.1, pp.99103, 2006. DOI:10.1049/ipcta:20045155 
[5] 
R. J. Patton. Faulttolerant control systems: The 1997 situation. In Proceedings of the 3rd IFAC Symposium on Fault Detection Supervision and Safety for Technical Processes, Kingston Upon Hull, UK, vol.2, pp.10331055, 1997.

[6] 
H. Noura, D. Sauter, F. Hamelin, D. Theilliot. Fault tolerant control in dynamic systems:Application to a winding machine. IEEE Control Systems Magazine, vol.20, no.1, pp.3349, 2000. DOI:10.1109/37.823226 
[7] 
M. Y. Zhao, H. P. Liu, Z. J. Li, D. H. Sun. Fault tolerant control for networked control systems with packet loss and time delay. International Journal of Automation and Computing, vol.8, no.2, pp.244253, 2011. DOI:10.1007/s116330110579z 
[8] 
R. J. Veillette. Reliable linearquadratic statefeedback control. Automatica, vol.31, no.1, pp.137143, 1995. DOI:10.1016/00051098(94)E0045J 
[9] 
G. H. Yang, S. Y. Zhang, J. Lam, J. L. Wang. Reliable control using redundant controllers. IEEE Transactions on Automatic Control, vol.43, no.11, pp.15881593, 1998. DOI:10.1109/9.728875 
[10] 
R. Wang, G. Jin, J. Zhao. Robust faulttolerant control for a class of switched nonlinear systems in lower triangular form. Asian Journal of Control, vol.9, no.1, pp.6872, 2007. 
[11] 
D. Ye, G. H. Yang. Adaptive faulttolerant tracking control against actuator faults with application to flight control. IEEE Transactions on Control Systems Technology, vol.14, no.6, pp.10881096, 2006. DOI:10.1109/TCST.2006.883191 
[12] 
M. L. Corradini, G. Orlando. Actuator failure identification and compensation through sliding modes. IEEE Transactions on Control Systems Technology, vol.15, no.1, pp.184190, 2007. DOI:10.1109/TCST.2006.883211 
[13] 
J. Stoustrup, D. Blondel. Fault tolerant control:A simultaneous stabilization result. IEEE Transactions on Automatic Control, vol.49, no.2, pp.305310, 2004. DOI:10.1109/TAC.2003.822999 
[14] 
G. H. Yang, J. L. Wang, Y. C. Soh. Reliable H_{∞} controller design for linear systems. Automatica, vol.37, no.5, pp.717725, 2001. DOI:10.1016/S00051098(01)000073 
[15] 
R. J. Veillette, J. V. Medanic, W. R. Perkins. Design of reliable control systems. IEEE Transactions on Automatic Control, vol.37, no.3, pp.290304, 1992. DOI:10.1109/9.119629 
[16] 
A. Tellili, M. N. Abdelkrim, M. Benrejeb. Reliable H_{∞} control of multiple time scales singularly perturbed systems with sensor failure. International Journal of Control, vol.80, no.5, pp.659665, 2007. DOI:10.1080/00207170601009634 
[17] 
M. Bodson, J. E. Groszkiewicz. Multivariable adaptive algorithms for reconfigurable flight control. IEEE Transactions on Control Systems Technology, vol.5, no.2, pp.217229, 1997. DOI:10.1109/87.556026 
[18] 
G. Tao, S. M. Joshi, X. L. Ma. Adaptive state feedback and tracking control of systems with actuator failures. IEEE Transactions on Automatic Control, vol.46, no.1, pp.7895, 2001. DOI:10.1109/9.898697 
[19] 
X. N. Zhang, J. L. Wang, G. H. Yang. Adaptive faulttolerant tracking controller design against actuator stuck faults. In Proceedings of the 24th Chinese Control and Decision Conference, IEEE, Taiyuan, China, pp.40984103, 2012.

[20] 
B. K. Sahu, B. Subudhi. Adaptive tracking control of an autonomous underwater vehicle. International Journal of Automation and Computing, vol.11, no.3, pp.299307, 2014. DOI:10.1007/s1163301407927 
[21] 
P. Chandler, M. Pachter, M. Mears. System identification for adaptive and reconfigurable control. Journal of Guidance, Control, and Dynamics, vol.18, no.3, pp.516524, 1995. DOI:10.2514/3.21417 
[22] 
L. Smith, P. Chandler, M. Pachter. Regularization techniques for realtime identification of aircraft parameters. In Proceedings of the AIAA Guidance, Navigation, and Control Conference, AIAA, New Orleans, USA, pp.14661480, 1997.

[23] 
G. Tao, X. D. Tang, S. H. Chen, J. T. Fei, S. M. Joshi. Adaptive failure compensation of twostate aircraft morphing actuators. IEEE Transactions on Control Systems Technology, vol.14, no.1, pp.157164, 2006. DOI:10.1109/TCST.2005.859639 
[24] 
X. Z. Jin, G. H. Yang. Robust adaptive faulttolerant compensation control with actuator failures and bounded disturbances. Acta Automatica Sinica, vol.35, no.3, pp.305309, 2009. 
[25] 
L. L. Fan, Y. D. Song. On faulttolerant control of dynamic systems with actuator failures and external disturbances. Acta Automatica Sinica, vol.36, no.11, pp.16201625, 2010. 
[26] 
J. Wang, H. L. Pei, N. Z. Wang. Adaptive output feedback control using fault compensation and fault estimation for linear system with actuator failure. International Journal of Automation and Computing, vol.10, no.5, pp.463471, 2013. DOI:10.1007/s1163301307438 
[27] 
P. Ioannou, P. Kokotovic. Decentralized adaptive control of interconnected systems with reducedorder models. Automatica, vol.21, no.4, pp.401412, 1985. DOI:10.1016/00051098(85)900767 
[28] 
K. B. Datta, A. RaiChaudhuri. H_{2}/H_{∞} control of singularly perturbed systems:The state feedback case. European Journal of Control, vol.16, no.1, pp.5469, 2010. 
[29] 
H. K. Khalil, F. C. Chen. H_{∞} control of twotimescale systems. Systems Control Letter, vol.19, no.1, pp.3542, 1992. DOI:10.1016/01676911(92)90037S 
[30] 
W. Tan, T. Leungs, Q. Tu. H_{∞} control for singularly perturbed systems. Automatica, vol.34, no.2, pp.255260, 1998. DOI:10.1016/S00051098(97)001830 
[31] 
S. D.Naidu. Singular Perturbation Methodology in Control Systems, England: Bath Press, 1988.

[32] 
S. Koskie, C. Coumarbatch, Z. Gajic. Exact slowfast decomposition of the singularly perturbed matrix differential Riccati equation. Applied Mathematics and Computation, vol.216, no.5, pp.14011411, 2010. DOI:10.1016/j.amc.2010.02.040 
[33] 
H. Khalil, P. V. Kokotovic. Control strategies for decision makers using different models of the same system. IEEE Transactions on Automatic Control, vol.23, pp.289298, 1978. DOI:10.1109/TAC.1978.1101712 
[34] 
G. S. Ladde, S. G. Rajalakshmi. Diagonalization and stability of multitimescale singularly perturbed linear systems. Applied Mathematics and Computation, vol.16, pp.115140, 1985. DOI:10.1016/00963003(85)900037 
[35] 
Z. Gajic, M. Lim. Optimal control of singularly perturbed linear systems and applications, New York, USA: Marcel Dekker, 2001.

[36] 
S. M. Shahruz. H_{∞} optimal compensators for singularly perturbed systems. In Procedings of IEEE Conference on decision and control, IEEE, Brighton, England, pp. 23972398, 1989.

[37] 
S. R. Shimjith, P. A.Tiwari, B. Bandyopadhyay. Modeling and Control of a Large Nuclear ReactorA ThreeTimeScale Approach, Berlin, Germany: Springer, 2013.
