**1. Introduction**

In recent decades, industrial processes are becoming more and more complex; thus, ensuring the operational reliability of these processes is an important task. Among them, fault detection and isolation (FDI) methods play a pivotal role in making the process reliable. The sensor and actuator faults are known as the most frequent faults that occur in many control systems such as satellite/aircraft [1,2], wind turbines [3,4], vehicles suspension system [5,6], offshore platforms [7], motor drives [8], power systems, and renewable energies [9,10]. In the event of a fault occurrence, the reliability and efficiency of the system are severely affected, and thus, the fault reconstruction is an important issue in the context of FDI approaches, and various types of research have been done in this field. However, when the system is subject to the uncertainty and disturbance, at the same time, identifying and reconstructing simultaneous sensor and actuator faults are still challenging issues that need to be addressed carefully.

In [11], a PI observer is proposed for fault estimation purposes based on convex structures and by employing nonquadratic Lyapunov functions. As a result, less conservative conditions in the form of LMIs are obtained. In [12], the sensor and actuator faults reconstruction problem is addressed by only considering the uncertainty in the model.

**Citation:** Taherkhani, A.; Bayat, F.; Hooshmandi, K.; Bartoszewicz, A. Generalized Sliding Mode Observers for Simultaneous Fault Reconstruction in the Presence of Uncertainty and Disturbance. *Energies* **2022**, *15*, 1411. https:// doi.org/10.3390/en15041411

Academic Editor: Sergio Nesmachnow

Received: 8 January 2022 Accepted: 4 February 2022 Published: 15 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

In [13], the actuator fault reconstruction (AFR) problem is investigated by introducing two observers: one to estimate unknown inputs and the other to facilitate fault reconstruction. A particular kind of actuator faults in manipulator systems, i.e., joint luck failure, is considered in [14], and two kinds of reconfiguration schemes are proposed to cope with this issue. In [15], a fault-tolerant control technique is studied for electro-hydraulic actuators. In this reference, an unknown input observer is used to reconstruct sensor faults in the presence of disturbances. In [16], a fault-tolerant sliding mode controller was designed for a class of fuzzy T-S systems subject to actuator saturation, external disturbances, and time-varying delay. Sliding mode control is a variable structure control method that is well known in nonlinear system control. In [17], integral sliding mode control is proposed to a new five-dimensional four-wing hyper chaotic system with hidden attractor. An adaptive finite-time sliding mode control is proposed in [18] to construct a family of nine new chameleon chaotic systems subjected to uncertainties and disturbances. In [19], a new synchronous quasi-sliding mode control (QSMC) is studied for Rikitake chaotic system. A selection on switching surface and the existing of QSMC is also considered in this reference. A composite sliding mode observer is proposed in [20] to study multi-sensor fault diagnosis and active fault-tolerant control in a PMSM drive system. For the FDI of a class of uncertain Lipschitz nonlinear systems, an adaptive robust sliding mode observer (SMO) is proposed in [21], where both external disturbance and faults are considered. A second-order sliding mode observer is considered in [22] to reconstruct sensor faults in an air-path system of a heavy-duty diesel engine in the presence of disturbance. In [23], an adaptive estimation approach is proposed to recover the bias fault of sensors in a class of nonlinear systems subject to unstructured uncertainty. In [9], the fault detection and fault-tolerant control problem for multi-area power systems with sensor failures were considered using a descriptor form SMO. In [24], an adaptive SMO and a descriptive form observer are combined to reconstruct the sensor and actuator faults where the stability analysis was performed by the Lyapunov method. For a linear system with disturbance and time-varying delay, an adaptive estimation approach is presented in [25] for AFR. The problem of fault-tolerant controller design for a synchronization problem of complex dynamical networks subject to actuator faults and saturation was investigated in [26,27]. In [28], a time shift approach for AFR with a time-delay of output is introduced by using an SMO. For wind turbine faults detecting, a new technique is proposed in [29] as a signal reconstruction modeling technique. In the mentioned paper, to detect faults at an early stage, multiple indicators are also calculated. A new data-driven sensor FDI technique is presented in [30] using interval-valued data and an enhanced reconstruction approach to develop fault isolation. Various methods for a simultaneous actuator and sensor faults reconstruction have been proposed in the literature. Inspired by a singular system theory, a descriptor observer design is presented in [31] to reconstruct the actuator fault based on the transformed coordinate system. In [32], both faults are simultaneously reconstructed in a special class of nonlinear system described by the Takagi–Sugeno model. In [33], a new robust and simultaneous actuator and sensor faults estimation is proposed for a class of LPV systems described with polytopic representation where the parameters evolve in the hypercube domain.

Discrepancies between the actual process and its model such as model uncertainties and disturbances cause misleading of fault detection and reconstruction. The problem of simultaneous fault detection and reconstruction of sensors and actuators in the presence of both uncertainties and unknown disturbances has been addressed in this paper. A noticeable feature of the proposed approach is that the inherent differences between the effect of uncertainty and disturbance on the system have been considered in the design of sliding mode observers in a generalized state-space form when faults occur at both sensors and actuators coincidentally. This problem is efficiently addressed in this paper, where two different distribution matrices are incorporated to represent perturbations and uncertainties in the system. Then, LMI and the equivalent output error injection (EOEI) methods have been used to design a robust SMO. Since the state trajectories of SMO do

not leave the sliding manifold in the presence of the uncertainties and disturbances, then the sensor and actuator faults are reconstructed based upon information retrieved from the equivalent output error injection signal. In order to verify the robust performances of the proposed approach, we applied it to a 5 MW wind turbine system. The wind energy conversion system (WECS) is a typical large and complex nonlinear system with random and intermittent wind force. In the electrical power system, the safety of electrical equipment is the basis for ensuring the stability and reliability. Fault reconstruction's aim is to guarantee the security of electrical power system operation and industry production. For this reason, we proposed fault reconstruction to ensure the safe and efficient operation of wind turbines. The wind turbine systems actuators and sensors have the highest probability of failure, which has the greatest impact on the WECS safe and efficient operation. A robust fault-tolerant control for a Takagi–Sugeno fuzzy model is studied for the wind energy conversion system in [34].

The rest of this paper is organized as follows. Description of the system in the presence of an actuator and sensor fault, disturbances and uncertainties, and design of the proposed SMO are presented in Section 2. A robust AFR employing the EOEI approach is presented in Section 3. Sensor fault reconstruction is studied similar to the actuator fault method by introducing a new state in Section 4. Simulation results and concluding remarks are provided in Section 5.

### **2. Description of the Problem**

We consider a class of uncertain systems in the presence of fault and disturbance given as:

$$\begin{aligned} \dot{z}(t) &= Az(t) + Bu(t) + M\partial(t, y, u) + Dd(t) + Ff\_d(t) \\ y(t) &= \mathbb{C}z(t) + F\_sf\_s(t) \end{aligned} \tag{1}$$

where *<sup>B</sup>* <sup>∈</sup> *<sup>R</sup>n*×*m*, *<sup>A</sup>* <sup>∈</sup> *<sup>R</sup>n*×*n*, *<sup>C</sup>* <sup>∈</sup> *<sup>R</sup>p*×*n*, *<sup>M</sup>* <sup>∈</sup> *<sup>R</sup>n*×*k*, *<sup>D</sup>* <sup>∈</sup> *<sup>R</sup>n*×*q*, *<sup>F</sup>* <sup>∈</sup> *<sup>R</sup>n*×*<sup>r</sup>* , and *Fs* ∈ *Rp*×*<sup>l</sup>* denote the matrices of inputs, states, outputs, unknown bounded uncertainties, disturbances, actuator faults, and sensor faults, respectively. We assume *p* ≥ *q*, *p* ≥ *l*, *n* > *p* ≥ *r*, and *F* and *C* are full column rank matrices. We also assume that *fa*(*t*) is a bounded unknown function indicating the fault of actuators, where *fa*(*t*) ≤ *α*(*t*), and *α* is a known function. Furthermore, the unknown bounded function *∂*(*t*, *y*, *u*) denotes the system's uncertainty and *∂*(*t*, *y*, *u*) ≤ *β*, where *β* > 0 is a known parameter. Moreover, *d*(*t*) denotes the disturbance signal, which is bounded *d*(*t*) ≤ *d*0, where *d*<sup>0</sup> is a positive constant.

**Assumption 1.** *It is assumed that rank*(*CF*) = *rank*(*F*) = *r and the system with* (*A*, *F*, *C*) *matrices has all its invariant zeros in the LHP.*

It is important to note that *p* < *n* implies that some states may not be observable. To cope with this issue, the following theorem is utilized to extract the observable and unobservable parts of the system in (1) with *fs*(*t*) = 0 where the matrix *F* only appears in the observable subsystem.

**Theorem 1.** *Assuming the conditions of Assumption 1 are satisfied and fs*(*t*) = 0*, then, there exist linear nonsingular transformations z*˜ = T*bz and* ¯ *z*¯ = T*cz such that:*

$$
\vec{A} = \begin{bmatrix} A\_1 & A\_2 \\ A\_3 & A\_4 \end{bmatrix}, \quad \vec{\mathcal{C}} = [\mathbf{0}\_{p \times (n-p)}, \mathcal{T}]\_\prime \quad \mathcal{F} = \begin{bmatrix} \mathbf{0}\_{r \times r} \\ \mathbf{F\_2} \end{bmatrix} \tag{2}
$$

*where F*<sup>2</sup> = <sup>0</sup>(*p*−*r*)×*<sup>r</sup> <sup>F</sup>*<sup>22</sup> <sup>∈</sup> *<sup>R</sup>r*×*<sup>r</sup> and* T ∈ *<sup>R</sup>p*×*<sup>p</sup> is orthogonal and nonsingular, <sup>A</sup>*<sup>1</sup> <sup>∈</sup> *R*(*n*−*p*)×(*n*−*p*)*, B*˜*<sup>T</sup>* = [*B<sup>T</sup>* <sup>1</sup> , *<sup>B</sup><sup>T</sup>* <sup>2</sup> ], *<sup>D</sup>*˜ *<sup>T</sup>* = [*D<sup>T</sup>* <sup>1</sup>(*n*−*p*)×*<sup>q</sup>* , *D<sup>T</sup>* 2*p*×*<sup>q</sup>* ], *M*˜ *<sup>T</sup>* = [*M<sup>T</sup>* <sup>1</sup>(*n*−*p*)×*<sup>k</sup>* , *M<sup>T</sup>* 2*p*×*<sup>k</sup>* ]*.*

**Proof.** First, consider <sup>T</sup>*<sup>c</sup>* = [*Nc*, *<sup>C</sup>T*] *<sup>T</sup>*, where columns of *Nc* span the null space of *C* and are orthonormal. Then, one obtains:

$$\begin{aligned} \dot{z}(t) &= \underbrace{\mathcal{T}\_{\varepsilon} A \mathcal{T}\_{\varepsilon}^{-1}}\_{A\_{\varepsilon}} z(t) + \underbrace{\mathcal{T}\_{\varepsilon} B}\_{B\_{\varepsilon}} u(t) + \underbrace{\mathcal{T}\_{\varepsilon} M}\_{B\_{\varepsilon}} \delta(t, y, u) \\ &+ \underbrace{\mathcal{T}\_{\varepsilon} D}\_{D\_{\varepsilon}} d(t) + \underbrace{\mathcal{T}\_{\varepsilon} F}\_{F\_{\varepsilon}} f\_{a}(t) \\ \bar{y} &= \underbrace{\mathcal{T}\_{\varepsilon}^{-1} C}\_{C\_{\varepsilon}} \bar{z} = \begin{bmatrix} \ 0\_{(n-p)} & I\_{p} \end{bmatrix} \bar{z}. \end{aligned} \tag{3}$$

It can be seen that only the last *p* states are present at the output. Now, considering *Fc* = " *<sup>f</sup>*1(*r*×*r*) *<sup>f</sup>*2(*p*×*r*) # , we define a nonsingular linear transformation matrix T*<sup>b</sup>* as:

$$\mathcal{T}\_b = \begin{bmatrix} I\_{(n-p)} & -f\_1 \left( f\_2^T f\_2 \right)^{-1} f\_2^T \\ \mathbf{0}\_{p \times (n-p)} & \mathcal{T}^T \end{bmatrix} \tag{4}$$

where the QR decomposition of *f*<sup>2</sup> is used to obtain T . Then, by using *z*˜ = T*bz*¯, one obtains:

$$\begin{cases}
\dot{\tilde{z}}(t) = \tilde{A}\tilde{z}(t) + \mathcal{B}u(t) + \mathcal{M}\partial(t, y, u) + \mathcal{D}d(t) + \mathcal{F}f\_a(t) \\
\tilde{y}(t) = \tilde{\mathbb{C}}\tilde{z}(t)
\end{cases} \tag{5}$$

where

$$\begin{aligned} \vec{A} &= \begin{bmatrix} A\_1 & A\_2 \\ A\_3 & A\_4 \end{bmatrix}, \vec{F} = \begin{bmatrix} 0\_{r \times r} \\ F\_2 \end{bmatrix}, \vec{D} = \begin{bmatrix} D\_1 \\ D\_2 \end{bmatrix} \\ \mathcal{K} &= \begin{bmatrix} 0\_{p \times (n-p)} \mathcal{T} \end{bmatrix}, \mathcal{B} = \begin{bmatrix} B\_1 \\ B\_2 \end{bmatrix}, \mathcal{M} = \begin{bmatrix} M\_1 \\ M\_2 \end{bmatrix}. \end{aligned} \tag{6}$$

The following SMO is considered:

$$\begin{aligned} \dot{\tilde{z}}(t) &= \tilde{A}\dot{\tilde{z}}(t) + \mathcal{B}u(t) - \mathcal{G}\_l v\_{\mathcal{G}}(t) + \mathcal{G}\_n v \\ \dot{\tilde{y}}(t) &= \tilde{\mathcal{C}}\dot{\tilde{z}}(t) . \end{aligned} \tag{7}$$

where ˆ *y*˜(*t*) and ˆ *z*˜(*t*) denote the estimation of outputs and states, respectively. The output estimation error is represented by *ey*˜(*t*) = ˆ *y*˜(*t*) − *y*˜(*t*). Furthermore, the observer gains *G*˜*n*, *G*˜ *<sup>l</sup>* <sup>∈</sup> *<sup>R</sup>n*×*<sup>p</sup>* will be defined in the following.

The sliding variable *v* has a nonlinear discontinuous term to maintain the sliding motion, which is given as:

$$w = \begin{cases} 0, & \forall eg = 0 \\ -\rho(t\_\prime y\_\prime u) \left||eg\right||^{-1} eg\_\prime \,\forall eg \neq 0 \end{cases} \tag{8}$$

where the upper bound for the fault plus uncertainty and disturbance is represented by the gain factor *ρ*(*t*, *y*, *u*) ∈ *R*.

The gain *G*˜*<sup>n</sup>* is chosen as:

$$
\tilde{\mathbf{G}}\_{\rm II} = \begin{bmatrix} -L\boldsymbol{\mathcal{T}}^T \\ \boldsymbol{\mathcal{T}}^T \end{bmatrix} \boldsymbol{P}\_0^{-1} \tag{9}
$$

where *P*<sup>0</sup> = *P<sup>T</sup>* <sup>0</sup> <sup>∈</sup> *<sup>R</sup>p*×*<sup>p</sup>* is a PDF design matrix that will be calculated in the following and *L* is defined as:

$$L = \begin{bmatrix} \ L\_0 & 0 \ \end{bmatrix} \in \mathcal{R}^{(n-p)\times p} \tag{10}$$

where *<sup>L</sup>*<sup>0</sup> <sup>∈</sup> *<sup>R</sup>*(*n*−*p*)×(*p*−*r*) is adjusted such that (*L*0*A*<sup>31</sup> <sup>+</sup> *<sup>A</sup>*1) is Hurwitz, where *<sup>A</sup>*<sup>31</sup> represents the first *p* − *q* rows of *A*3.

Now, the following theorem is recalled from [35].

**Theorem 2.** *Assume an observer dynamic as given in (7), a Lyapunov matrix P*˜*, and a matrix G*˜ *<sup>l</sup> satisfying:*

$$\begin{array}{l} \mathcal{P} = \begin{bmatrix} P\_1 & P\_1 L \\ L^T P\_1 & \mathcal{T}^T P\_0 \mathcal{T} + L^T P\_1 L \end{bmatrix} \\ (\tilde{A} - \tilde{G}\_I \tilde{\mathbf{C}})^T \tilde{P} + \tilde{P} (\tilde{A} - \tilde{G}\_I \tilde{\mathbf{C}}) < 0 \end{array} \tag{11}$$

*where <sup>L</sup> defined in (10) and <sup>P</sup>*<sup>1</sup> <sup>∈</sup> *<sup>R</sup>*(*n*−*p*)×(*n*−*p*)*. Then, the observation error <sup>e</sup>*(*t*) <sup>Δ</sup> = ˆ *z*˜(*t*) − *z*˜(*t*) *is asymptotically stable.*

Considering Assumption 1, it can be shown that there exists a stable sliding motion on the sliding surface given as [36]:

$$S = \left\{ \boldsymbol{e}(t) | \bar{\mathbb{C}} \boldsymbol{e}(t) = 0 \right\}. \tag{12}$$

Then, one obtains:

$$
\begin{aligned}
\dot{e}(t) &= (\bar{A} - \bar{G}\_l \bar{\mathbb{C}})e(t) - \tilde{M}\partial(t, y, u) \\ &- \mathcal{D}d(t) - \mathbb{F}f\_a(t) + \bar{\mathbb{G}}\_n v
\end{aligned}
\tag{13}
$$

**Lemma 1.** *The error dynamics in* (13) *is bounded in the region* Ω *defined as:*

$$
\Omega = \left\{ \varepsilon || \varepsilon \| \, \le \left. 2(\mu\_2 \beta + \mu\_1 d\_0) \right\} / \mu\_0 \right\} \tag{14}
$$

*where <sup>μ</sup>*<sup>0</sup> <sup>=</sup> <sup>−</sup>*λ*max(*A*˜ *<sup>c</sup>*)*, <sup>μ</sup>*<sup>1</sup> <sup>=</sup> \$ \$*P*˜*D*˜ \$ \$, *<sup>μ</sup>*<sup>2</sup> <sup>=</sup> \$ \$*P*˜*M*˜ \$ \$*, <sup>A</sup>*˜ *<sup>c</sup>* <sup>=</sup> <sup>−</sup>(*G*˜ *lC*˜ <sup>−</sup> *<sup>A</sup>*˜)*TP*˜ <sup>−</sup> *<sup>P</sup>*˜(*G*˜ *lC*˜ <sup>−</sup> *<sup>A</sup>*˜)*.*

**Proof.** Define *V* = *eTPe*˜ . Then

$$\begin{aligned} \dot{V} &= \varepsilon^T \bar{A}\_c \varepsilon - 2\varepsilon^T \bar{P} \tilde{M} \partial(t, y, u) - 2\varepsilon^T \bar{P} \tilde{D} d(t) \\ &- 2\varepsilon^T \bar{P} \mathcal{F} f\_a(t) + 2\varepsilon^T \mathcal{P} \mathcal{G}\_n v \end{aligned} \tag{15}$$

From (16) and considering *∂*(*t*, *y*, *u*) ≤ *β* and *d*(*t*) ≤ *d*<sup>0</sup> and utilizing the Cauchy– Schwartz inequality, yield:

$$\dot{V} \le -\mu\_0 \|\boldsymbol{e}\|\boldsymbol{e}\|\boldsymbol{\varepsilon}^2 + 2\|\boldsymbol{e}\|\mu\_1 \boldsymbol{d}\_0 + 2\|\boldsymbol{e}\|\mu\_2 \boldsymbol{\beta} - 2\boldsymbol{e}^T \boldsymbol{P} \mathbf{F} f\_a(\boldsymbol{t}) + 2\boldsymbol{e}^T \boldsymbol{P} \mathbf{G}\_n \boldsymbol{v}.\tag{16}$$

Using (6), (9), and (11), it is simply verified that *P*˜*F*˜ = *C*˜*TP*0*C*˜*F*˜ and *P*˜*G*˜*<sup>n</sup>* = *C*˜*T*. Then, considering *ey*˜(*t*) = *Ce*˜ (*t*) = *<sup>C</sup>*˜(*z*˜(*t*) <sup>−</sup> *<sup>z</sup>*(*t*)), *fa*(*t*) <sup>≤</sup> *<sup>α</sup>* and (8), one obtains:

$$\begin{array}{l} \dot{V} \le 2||\boldsymbol{\varepsilon}||\mu\_{1}d\boldsymbol{o} - \mu\_{0}||\boldsymbol{\varepsilon}||^{2} + 2||\boldsymbol{\varepsilon}||\mu\_{2}\boldsymbol{\beta} \\ \quad - 2\left(\boldsymbol{\rho}(\boldsymbol{t},\boldsymbol{y},\boldsymbol{u}) - \boldsymbol{\alpha}(\boldsymbol{t},\boldsymbol{u})||\boldsymbol{P}\_{0}\boldsymbol{\zeta}\boldsymbol{F}||\right)||\boldsymbol{\varepsilon}\_{\mathcal{G}}|| \\ \leq -||\boldsymbol{\varepsilon}||\left(\mu\_{0}||\boldsymbol{\varepsilon}|| - 2\mu\_{1}d\boldsymbol{o} - 2\mu\_{2}\boldsymbol{\beta}\right). \end{array} \tag{17}$$

Therefore, if *e* > 2(*μ*2*β* + *μ*1*d*0) 0 *μ*0, then *V*˙ < 0, and this implies that *e*(*t*) will converge to the following bounded region:

$$
\Omega = \left\{ \varepsilon || \varepsilon || \varepsilon \right\| \, \left\{ \, \left( 2\mu\_1 d\rho + 2\mu\_2 \beta \right) / \mu\_0 \right\}. \tag{18}
$$

Now, we show that with the proper selection of *ρ*(*y*, *u*, *t*), the sliding surface in (13) is reached in finite time. Define:

$$\mathcal{T}\_{\rm L} = \begin{bmatrix} I\_{n-p} & L \\ 0 & \mathcal{T}\_{\rm L} \end{bmatrix}. \tag{19}$$

Using this transformation, the matrices in (6), (9), and (11) are converted to the following form:

$$\begin{aligned} \mathcal{A} &= \mathcal{T}\_{L} \mathcal{A} \mathcal{T}\_{L}^{-1} = \begin{bmatrix} \mathcal{A}\_{1} & \mathcal{A}\_{2} \\ \mathcal{A}\_{3} & \mathcal{A}\_{4} \end{bmatrix}, \quad \mathcal{M} = \mathcal{T}\_{L} \tilde{\mathcal{M}} = \begin{bmatrix} \mathcal{M}\_{1} \\ \mathcal{M}\_{2} \end{bmatrix} \\ \mathcal{G}\_{\mathcal{U}} &= \mathcal{T}\_{L} \tilde{\mathcal{G}}\_{\mathcal{U}} = \begin{bmatrix} 0 \\ P\_{0}^{-1} \end{bmatrix}, \quad \mathcal{F} = \mathcal{T}\_{L} \tilde{\mathcal{F}} = \begin{bmatrix} 0\_{(n-p)\times r} \\ \mathcal{F}\_{2} \end{bmatrix} \\ \mathcal{D} &= \mathcal{T}\_{L} \mathcal{D} = \begin{bmatrix} \mathcal{D}\_{1} \\ \mathcal{D}\_{2} \end{bmatrix}, \quad \mathcal{C} = \mathcal{C} \mathcal{T}\_{L}^{-1} = \begin{bmatrix} 0\_{p\times(n-p)} & I\_{p} \end{bmatrix} \\ \mathcal{P} &= (\mathcal{T}\_{L}^{-1})^{T} P \mathcal{T}\_{L}^{-1} = \begin{bmatrix} P\_{1} & 0 \\ 0 & P\_{0} \end{bmatrix} \end{aligned} \tag{20}$$

where:

$$\begin{aligned} \mathcal{A}\_1 &= A\_1 + L A\_3, & \mathcal{M}\_1 &= M\_1 + L M\_2, & \mathcal{D}\_1 &= D\_1 + L D\_2 \\ \mathcal{A}\_3 &= \mathcal{T} A\_3, & \mathcal{M}\_2 &= \mathcal{T} M\_2, & \mathcal{D}\_2 &= \mathcal{T} D\_2, & \mathcal{F}\_2 &= \mathcal{T} \mathcal{F}\_2. \end{aligned} \tag{21}$$

Therefore, the error in (14) becomes:

$$\begin{aligned} \dot{e}\_l(t) &= (\mathcal{A} - \mathcal{G}\_L \mathcal{C})e\_l(t) - \mathcal{M}\partial(t, y, u) - \mathcal{D}d(t) \\ &- \mathcal{F}f\_d(t) + \mathcal{G}\_ll v \end{aligned} \tag{22}$$

where

$$
\varepsilon\_{l} = \mathcal{T}\_{L}\varepsilon = \begin{bmatrix} \varepsilon\_{1} \\ \varepsilon\_{\mathcal{G}} \end{bmatrix}, \quad \mathcal{G}\_{L} = \mathcal{T}\_{L}\mathcal{G}\_{l} = \begin{bmatrix} \mathcal{G}\_{L1} \\ \mathcal{G}\_{L2} \end{bmatrix}. \tag{23}
$$

Using this, (23) can be decomposed as:

$$\begin{array}{l} \dot{e}\_{1}(t) = \mathcal{A}\_{1}e\_{1}(t) + (\mathcal{A}\_{2} - \mathcal{G}\_{L1})e\_{\mathcal{Y}}(t) \\ \quad - \mathcal{M}\_{1}\ddot{\mathcal{O}}(t, \mathcal{y}, u) - \mathcal{D}\_{1}d(t)s \\ \dot{e}\_{\mathcal{Y}}(t) = \mathcal{A}\_{3}e\_{1}(t) + (\mathcal{A}\_{4} - \mathcal{G}\_{L2})e\_{\mathcal{Y}}(t) + P\_{0}^{-1}v \\ \quad - \mathcal{M}\_{2}\ddot{\mathcal{O}}(t, \mathcal{y}, u) - \mathcal{D}\_{2}d(t) - \mathcal{F}\_{2}f\_{a}(t). \end{array} \tag{24}$$

The following theorem proposes a proper choice of *ρ* to guarantee finite time convergence to the sliding surface *S*.

**Theorem 3.** *The error dynamic (23) reaches the sliding surface S in finite-time Ts* ≤ √*V*(0) *η*0 *λ*min(*P*−<sup>1</sup> <sup>0</sup> ) *and stays there forever, if:*

$$\begin{array}{lcl}\rho(t,y,u) \ge \|\|P\_0\mathcal{D}\_2\|\|d\_0 + \|\|P\_0\mathcal{M}\_2\|\|\beta + \|\|P\_0\mathcal{F}\_2\|\|u+\tag{25} \\ 2\|\|P\_0\mathcal{A}\_3\|\| (\mu\_1 d\_0 + \mu\_2 \beta) / \mu\_0 + \eta\_0. \end{array} \tag{25}$$

**Proof.** Define the candidate Lyapunov function *V* = *e<sup>T</sup> <sup>y</sup>*˜ *P*0*ey*˜. Then:

$$\begin{aligned} \dot{V} &= \varepsilon\_{\mathcal{g}}^{T} \Big( P\_{0} (\mathcal{A}\_{4} - \mathcal{G}\_{L2}) + (\mathcal{A}\_{4} - \mathcal{G}\_{L2})^{T} P\_{0} \Big) \varepsilon\_{\mathcal{g}} + \\ &2 \varepsilon\_{\mathcal{g}}^{T} P\_{0} \mathcal{A}\_{3} \varepsilon\_{1} - 2 \varepsilon\_{\mathcal{g}}^{T} P\_{0} \mathcal{F}\_{2} f - 2 \varepsilon\_{\mathcal{g}}^{T} P\_{0} \mathcal{D}\_{2} d - \\ &2 \varepsilon\_{\mathcal{g}}^{T} P\_{0} \mathcal{M}\_{2} \partial + 2 \varepsilon\_{\mathcal{g}}^{T} v \end{aligned} \tag{26}$$

where 2*e<sup>T</sup> <sup>y</sup>*˜ *v* = −2*ρ* \$ \$*ey*˜ \$ \$. Then, by using the Cauchy–Schwartz inequality, one gets:

$$\dot{\mathcal{V}} \le -2||\boldsymbol{\varepsilon\_{\mathcal{G}}}|| \left( \frac{\rho - ||P\_{\rm O} \mathcal{A}\_{\rm 3}|| ||\boldsymbol{\varepsilon\_{1}}|| - ||P\_{\rm O} \mathcal{F}\_{\rm 2}|| \kappa - }{||P\_{\rm O} \mathcal{D}\_{2}|| d\_{0} - ||P\_{\rm O} \mathcal{M}\_{2}|| \beta} \right). \tag{27}$$

From (15), (24), and (25) we conclude that *ρ* − *P*0A3*e*1 − *P*0M2*β* − *P*0D2*d*<sup>0</sup> − *P*0F2*α* = *η*<sup>0</sup> > 0 and *e*1 < 2(*μ*2*β* + *μ*1*d*0) 0 *μ*0. This results:

$$\dot{V} \le -2\eta\_0 \|e\_{\mathcal{G}}\| \le -2\eta\_0 \sqrt{\lambda\_{\text{min}} \left(P\_0^{-1}\right)} \sqrt{V}.\tag{28}$$

Therefore, using (29) and the finite-time stability theorem (see Theorem 4.2 of [37]), we conclude that the estimation error converges to zero, and the sliding motion reaches *S* in finite-time *Ts* <sup>≤</sup> <sup>1</sup> *η*0 ! *<sup>V</sup>*(0) *λ*min(*P*−<sup>1</sup> <sup>0</sup> ) .

Now, an LMI-based approach is proposed to obtain an appropriate gain matrix *G*˜ *<sup>l</sup>*. In this regard, Theorem 2.2 requires finding a matrix *P*˜ that satisfies:

$$(\boldsymbol{A} - \boldsymbol{G}\_l \mathbf{C})^T \boldsymbol{P} + \mathcal{P}(\boldsymbol{A} - \mathbf{G}\_l \mathbf{C}) < 0. \tag{29}$$

As discussed in [36], an inequality of the form (30) can be alternatively solved by the following set of inequalities:

$$\mathcal{P} > 0, \quad \mathcal{A}^T \mathcal{P} + \mathcal{P}\mathcal{A} - \mathcal{C}^T \mathcal{U}^{-1} \mathcal{C} + \mathcal{P}Q\mathcal{P} < 0 \tag{30}$$

where *<sup>U</sup>* <sup>∈</sup> *<sup>R</sup>p*×*<sup>p</sup>* and *<sup>Q</sup>* <sup>∈</sup> *<sup>R</sup>n*×*<sup>n</sup>* are PSD matrices. Applying the Schur lemma, (31) is converted to the following LMI:

$$
\begin{bmatrix}
\tilde{P}\tilde{A} + \tilde{A}^T \tilde{P} - \tilde{C}^T \mathcal{U}^{-1} \tilde{\mathcal{C}} & \tilde{P} \\
\mathcal{P} & -\mathcal{Q}^{-1}
\end{bmatrix} < 0. \tag{31}
$$

The matrix *P*˜ is obtained by solving the LMI (32), and then:

$$
\vec{G}\_l = \mathcal{P}^{-1} \vec{\mathcal{C}}^T \mathcal{U}^{-1}.\tag{32}
$$
