A Fuzzy Design for a Sliding Mode Observer-Based Control Scheme of Takagi-Sugeno Markov Jump Systems under Imperfect Premise Matching with Bio-Economic and Industrial Applications

Abstract

Fuzzy theory is widely studied and applied. This article introduces an adaptive control scheme for a class of non-linear systems with Markov jump switching. The introduced scheme supposes that the system is submitted to external disturbances under imperfect premise matching. By using discrete-time Takagi–Sugeno fuzzy models, a sliding mode observer-based control scheme is utilized to estimate unmeasured states of the system. We build two fuzzy switching manifolds for the disturbance and sliding mode observer systems. Then, a linear matrix inequality-based criterion is developed using slack matrices. This criterion proves that the sliding mode dynamics are robustly admissible under an H-infinity performance often used in control theory. Hence, new adaptive sliding mode controllers are synthesized for the disturbance and sliding mode observer systems. This allows the reachability of pre-designed sliding surfaces to be guaranteed. Finally, experimental numerical illustrations on a bio-economic system and a tunnel diode circuit are presented to show potential applications, as well as validating the effectiveness of the scheme proposed in the present investigation.

1. Introduction, Notations, and Outline

In this section, we provide the literature review, abbreviations, notations, and objectives of the present article.

1.1. Bibliographical Review

Fuzzy models have been largely analyzed derived and applied [1]. The Takagi–Sugeno (TS) fuzzy models [2] are used for approximating non-linear systems by means of local linear structures. The TS models are connected by fuzzy membership functions. Results for this class of systems based on the parallel distribution compensation (PDC) method are reported in [3,4,5,6,7]. This method requires the same grades of membership between the fuzzy controller and TS fuzzy system.

Instead of utilizing the PDC method, an imperfect premise matching was employed to enhance the robustness of fuzzy control. In [8], this matching was used for event-triggered control of a suspension system. In [9], a non-PDC control algorithm was proposed for networked TS fuzzy systems with limited data transmission and imperfect premise matching. In [10], an imperfect control design for TS fuzzy systems was described.

Descriptor models are used for complex systems by means of differential equations [11]. Thus, when analyzing such systems, regularity and causality, for discrete and continuous systems, respectively, need to be guaranteed [12,13,14,15,16]. TS fuzzy approaches are applied to non-linear descriptor systems and control problems. In [17], state-feedback and output feedback controls were analyzed with TS fuzzy models. Synthesis and dissipativity for TS fuzzy descriptor schemes were treated with slack matrices in [18]. Actuator saturation was investigated in [19,20] for TS fuzzy systems. In [21,22], a state control design was solved for non-linear descriptor systems with TS fuzzy models. Due to various environmental factors, such as random faults and abrupt mutations of subsystem interconnections, the controlled scheme and system parameters are heavily affected. Markov jump systems (MJS) are often adopted to model such systems. MJS find applications in control engineering, such as fault tolerant, target tracking, manufacture, networked control, and multiagent systems.

Studies on control/filtering designs and stability analysis on MJS were published in [23,24]. Networked Markov switching systems using quantized non-stationary filtering were analyzed in [25]. A sliding mode control (SMC) scheme is examined in [26] for switching systems under semi-Markov processes using adaptive event triggering. In [27], control of Markov switching systems with finite-time output feedback was considered assuming quantization effects from the actuator to the controller.

The SMC scheme is used as a robust controller of complex systems. SMC utilizes a discontinuous controller to conduct the system trajectories onto a sliding surface and force them to move along the surface. The main advantages of the SMC approach are: (i) fast response; (ii) good transient performance; and (iii) invariance against matched uncertainties and external disturbances. Much work has been devoted to SMC approaches for different systems [28,29,30,31,32,33]. In addition, because of the finite sampling rate, an SMC law designed for continuous-time systems could be inappropriate for discrete-time systems. Thus, the analysis of discrete-time sliding mode control (DSMC) has inspired a new line of research and some works on D$H_{\infty }$SMC were published in [34,35] and references therein.

In the mentioned works, DSMC control laws are synthesized for discrete-time systems with external disturbances and time-varying delays. In [36], the resilient SMC problem with H-infinity performance (often denoted as $H_{\infty }$ and utilized in control theory) was investigated for discrete-time descriptor fuzzy systems with multiple delays. The $H_{\infty }$ techniques are used in control theory to reach stabilization for a guaranteed performance.

Note that the results associated with the SMC approaches cited in the aforementioned works are obtained upon the premise that all the state variables are available. However, in practice, the full system states are generally unavailable for measurement and the observers have been introduced to determine the state variables for controller design. Moreover, benefiting from the advantages of the observer and SMC approaches, the observer-based SMC and sliding mode observer (SMO) techniques have been studied to cope with the state estimation problems for different classes of systems. Consequently, the SMO technique has attracted considerable attention with some remarkable results [37,38]. To name a few, SMO was employed to design a controller for singular semi-Markov jumping systems with delay [39]. The adaptive SMO-based control for non-linear MJS was studied in [40]. The control problem using SMO for Itô stochastic frameworks was analyzed in [41].

Therefore, on the basis of the above detailed bibliographical review, we detect little progress has been made on the state estimation problem with DSMC for non-linear MJS. Specifically, no SMO-based control schemes for TS Markov jump descriptor systems under imperfect premise matching have been proposed until now. Consequently, the contribution we present in this investigation and its objectives are established in Section 1.3 after describing in the following subsection the abbreviations and notations employed here.

1.2. Abbreviations and Notations

Table 1. Abbreviations, acronyms, and notations used in the present document.

Symbol	Abbreviation/Acronym/Notation
$\mathbb{N}$	set of the positive integer numbers
$\mathbb{R}$	set of the real numbers
n	dimension of the Euclidean space
$\mathit{X}\in {\mathbb{R}}^{n\times m}$	$n\times m$ real matrix
$\mathit{X}>0$	real symmetric positive definite matrix $\mathit{X}$
$\parallel \mathit{X}\parallel$	norm of the matrix $\mathit{X}$
${\mathit{X}}^{\top }$	transpose of the matrix $\mathit{X}$
$\mathrm{sym}\left(\mathit{X}\right)$	$\mathit{X}+{\mathit{X}}^{\top }$
∗	term that is induced by symmetry of a matrix
⊗	Kronecker product
${\lambda }_{\max }\left(\right)$	the maximal eigenvalue of a matrix
$\mathbb{E}$	mathematical expectation
sat	saturation function
${\pi }_{pq}=P({\overline{r}}_{k+1}=q|{\overline{r}}_k=p)$	transition probability from states p to q
r	number of if-then rules
${\overline{r}}_k$	discrete-time Markov process ${}^{•}$
${\theta }_j$	premise variable j
$\mathbf{\theta }$	${[{\theta }_1,\dots ,{\theta }_s]}^{\top }$
s	number of premise variables
${\mathcal{M}}_j^i$	fuzzy set i of ${\theta }_j$
${\mathcal{M}}_j^i\left({\theta }_j\right)$	grade of membership of ${\theta }_j$ to ${\mathcal{M}}_j^i$
$h_i\left(\mathbf{\theta }\right)$	normalized membership
$h_i\left(\mathbf{\theta }\right)$	${\prod }_{j=1}^s{\mathcal{M}}_j^i\left({\theta }_j\right)/{\sum }_{i=1}^r{\prod }_{j=1}^s{\mathcal{M}}_j^i\left({\theta }_j\right)$
$f_i$	bounded non-linear function i
$\mathit{e}\left(k\right)=\mathit{x}\left(k\right)-\widehat{\mathit{x}}\left(k\right)$	output error
${\mathit{X}}_h$	convex combination h
${\sum }_{i=1}^r{h}_i{X}_i$	convex combination ${\sum }_{i=1}^r{\sum }_{j=1}^r{h}_i{h}_j{X}_{ij}$
DSMC	discrete-time sliding mode control
LMI	linear matrix inequalities
MJS	Markov jump systems
N/A	not applicable or not available
SMC	sliding mode control
SMO	sliding mode observer
TS	Takagi–Sugeno

${}^{•}$ For the notational sake, in each ${\overline{r}}_k=p\in \mathbb{N}$, the matrices or vectors related to ${\overline{r}}_k$ are denoted using the index p.

presents the abbreviations, acronyms, and notations to be used in this article.

1.3. Objective and Outline

The contribution of this article is in introducing an adaptive control scheme for a class of non-linear systems with Markov jump switching. This scheme is based on an SMO technique, assumes that the system is exposed to disturbances under imperfect premise matching, and is employed to estimate unmeasured states of the system considering discrete-time TS fuzzy models. Specifically, we first construct two fuzzy switchings for the disturbance and sliding mode observer systems. Then, we derive a linear matrix inequality-based criterion using slack matrices to prove that the sliding mode dynamics are robust under $H_{\infty }$ performance. Hence, new adaptive sliding mode controllers are synthesized for the disturbance and sliding mode observer systems. This allows the reachability of pre-designed sliding surfaces to be guaranteed. Therefore, the main objectives of this investigation are the following:

(i)

To propose a general model of fuzzy descriptor systems with Markov jump switching, uncertainties, external disturbance, and unknown premise variables.

(ii)

To design a novel SMO technique for reconstructing the unmeasured state variables using the imperfect premise matching approach combined with the non-PDC control scheme, and to then design two sliding surfaces that drive with the estimated and error trajectories for guaranteeing the stability of the closed-loop system.

(iii)

To show that the system is admissible with an $H_{\infty }$ performance level by developing a linear matrix inequality (LMI) criterion, guaranteeing robustness of the control scheme against uncertainties, and designing controller and observer gains in one step.

(iv)

To synthesize new adaptive sliding mode control laws for the error and SMO system ensuring the reachability of the pre-designed sliding surfaces.

We illustrate our scheme with experimental numerical results on a bio-economic system and a tunnel diode circuit to show potential applications and validate the effectiveness of the proposed scheme.

After stating the introduction, abbreviations, notations, and objectives of our investigation, the article is organized as follows. Section 2 provides essential preliminary results presenting the model and assumptions, as well as establishing the problem under study. In Section 3, we propose and derive the mains results of the article. In Section 4, the computational framework of this work, an algorithm that summarizes the proposed methodology, and experimental numerical illustrations on a single-species bio-economic system and a tunnel diode circuit are introduced to show potential applications, as well as validating the effectiveness of the proposed scheme. We use the Matlab programming language to conduct this numerical application. Finally, we make some conclusions on the obtained results and provide some limitations of our proposal, as well as ideas for possible future work in Section 5.

2. Preliminaries and Problem Statement

In this section, we introduce some preliminaries, which facilitate the comprehension of our proposal, and state the problem under study.

2.1. The Model

Consider a class of non-linear discrete-time descriptor MJS, which can be described by the following TS fuzzy model defined on a probability space:

$$\begin{array}{ccc}\hfill & \mathrm{Rule}i:\mathtt{if}{\theta }_1\left(\mathit{x}\left(k\right)\right)\mathrm{is}{\mathcal{M}}_1^i,{\theta }_2\left(\mathit{x}\left(k\right)\right)\mathrm{is}{\mathcal{M}}_2^i,\dots ,\mathrm{and}{\theta }_s\left(\mathit{x}\left(k\right)\right)\mathrm{is}{\mathcal{M}}_s^i,\mathtt{then}\hfill & \hfill \\ & \left\{\begin{array}{cc}\hfill \mathit{E}\mathit{x}(k+1)& ={\mathit{A}}_i(k,{\overline{r}}_k)\mathit{x}\left(k\right)+{\mathit{B}}_{1i}\left({\overline{r}}_k\right)\mathit{w}\left(k\right)+{\mathit{B}}_2\left({\overline{r}}_k\right)\left(\mathit{u}\left(k\right)+\mathit{f}(k,{\overline{r}}_k,\mathit{x}\left(k\right))\right),\hfill \\ \hfill \mathit{y}\left(k\right)& ={\mathit{C}}_2\left({\overline{r}}_k\right)\mathit{x}\left(k\right),\hfill \\ \hfill \mathit{z}\left(k\right)& ={\mathit{C}}_{1i}\left({\overline{r}}_k\right)\mathit{x}\left(k\right)+{\mathit{D}}_i\left({\overline{r}}_k\right)\mathit{w}\left(k\right),\hfill \end{array}\right.\hfill \end{array}$$

(1)

where ${\mathcal{M}}_d^i$ are the fuzzy sets, for $i\in \{1,\dots ,r\}$ and $d\in \{1,\dots ,s\}$, with d and r being the numbers of premise variables and if-then rules, respectively. In addition, we have that $\mathbf{\theta }\left(\mathit{x}\right(k\left)\right)={[{\theta }_1\left(\mathit{x}\left(k\right)\right),\dots ,{\theta }_s\left(\mathit{x}\left(k\right)\right)]}^{\top }$ is the premise variable vector, $\mathit{x}\left(k\right)\in {\mathbb{R}}^n$ is the state vector, $\mathit{u}\left(k\right)\in {\mathbb{R}}^m$ is the control input, $\mathit{y}\left(k\right)\in {\mathbb{R}}^{ny}$ is the measured output, $\mathit{z}\left(k\right)\in {\mathbb{R}}^v$ is the controlled output, and $\mathit{w}\left(k\right)\in {\mathbb{R}}^w$ is the disturbance input vector that belongs to $L_2[0,\infty )$, that is, the Euclidean norm. Furthermore, observe that $\mathit{f}(k,{\overline{r}}_k,\mathit{x}\left(k\right))$ is an unknown and bounded non-linear real-valued vector function, which establishes the model uncertainties in the system considering external disturbances. Note that the matrix $\mathit{E}\in {\mathbb{R}}^{n\times n}$ is singular, with $\mathrm{rank}\left(\mathit{E}\right)=q_0<n$. In addition, ${\mathit{A}}_i(k,{\overline{r}}_k)={\mathit{A}}_i\left({\overline{r}}_k\right)+\Delta {\mathit{A}}_i(k,{\overline{r}}_k)$ is a time-varying matrix, where $\Delta {\mathit{A}}_i(k,{\overline{r}}_k)={\mathit{M}}_i\left({\overline{r}}_k\right)\Delta \left(k\right){\mathit{N}}_i\left({\overline{r}}_k\right)$, with ${\Delta }^{\top }\left(k\right)\Delta \left(k\right)\le \mathit{I}$, for $i\in \{1,\dots ,r\}$. Matrices ${\mathit{A}}_i\left({\overline{r}}_k\right)$, ${\mathit{B}}_{1i}\left({\overline{r}}_k\right)$, ${\mathit{B}}_2\left({\overline{r}}_k\right)$, ${\mathit{C}}_2\left({\overline{r}}_k\right)$, ${\mathit{M}}_i\left({\overline{r}}_k\right)$, ${\mathit{N}}_i\left({\overline{r}}_k\right)$, ${\mathit{C}}_{1i}\left({\overline{r}}_k\right)$, and ${\mathit{D}}_i\left({\overline{r}}_k\right)$ are constant with appropriate dimensions. Here, ${\overline{r}}_k$ is a discrete-time Markov process taking values in $\{1,\dots ,N\}$ with a probability matrix $\overline{\mathbf{\Pi }}={\left[{\pi }_{pq}\right]}_{N\times N}$, for $p,q\in \mathbb{N}$. The transition probabilities are ${\pi }_{pq}=P({\overline{r}}_{k+1}=q|{\overline{r}}_k=p)$ and satisfy ${\pi }_{pq}\ge 0$ and ${\sum }_{q=1}^N{\pi }_{pq}=1$ for each p.

2.2. Assumptions and Resulting Model

The following assumptions are necessary for future discussion in the present article:

(A1)

The external disturbance $\mathit{w}\left(k\right)$ is bounded and satisfies $\parallel \mathit{w}\left(k\right)\parallel \le \overline{w}$, where $\overline{w}$ is an unknown positive real constant.

(A2)

Matched non-linearity functions $\mathit{f}(k,{\overline{r}}_k,\mathit{x}\left(k\right))$ are bounded and satisfy $\parallel \mathit{f}(k,{\overline{r}}_k,\mathit{x}\left(k\right))\parallel$$\le \overline{\eta }\parallel \mathit{y}\parallel$, where $\overline{\eta }>0$ is an unknown positive real constant.

Note that the assumptions (A1) and (A2) are reasonable, since, in practice, the disturbances are generally unknown but bounded. The adaptive approach is used here to estimate these unknown disturbances.

Remark 1.

It is recognized that the transition probabilities play a crucial role in Markov processes. In this study, the transition probabilities are supposed to be known. However, in many practical situations, these probabilities are unknown. Thus, the determination of time-varying and partly transition probabilities in our framework are still open problems which can be considered in future works [26].

For simplicity, denote ${\mathbf{\theta }}_s\left(\mathit{x}\left(k\right)\right)={\mathbf{\theta }}_s\left(\mathit{x}\right)$. Based on the expressions defined in (1) and the above aspects, for $\mathbf{\theta }\left(\mathit{x}\right)={[{\theta }_1\left(\mathit{x}\right),\dots ,{\theta }_s\left(\mathit{x}\right)]}^{\top }$, the resulting Markov jump TS fuzzy model is stated as:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill \mathit{E}\mathit{x}(k+1)& =\sum _{i=1}^r{h}_i\left(\mathbf{\theta }\left(\mathit{x}\right)\right)\left({\mathit{A}}_i^p\left(k\right)\mathit{x}\left(k\right)+{\mathit{B}}_{1i}^p\mathit{w}\left(k\right)+{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)+{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))\right)\right),\hfill \\ \hfill \mathit{y}\left(k\right)& ={\mathit{C}}_2^p\mathit{x}\left(k\right),\hfill \\ \hfill \mathit{z}\left(k\right)& =\sum _{i=1}^r{h}_i\left(\mathbf{\theta }\left(\mathit{x}\right)\right)\left({\mathit{C}}_{1i}^p\mathit{x}\left(k\right)+{\mathit{D}}_i^p\mathit{w}\left(k\right)\right),\hfill \end{array}\right.\hfill \end{array}$$

(2)

with $h_i\left(\mathbf{\theta }\left(\mathit{x}\right)\right)={\prod }_{d=1}^s{\mathcal{M}}_d^i\left({\theta }_d\left(\mathit{x}\right)\right)/{\sum }_{i=1}^r{\prod }_{d=1}^s{\mathcal{M}}_d^i\left({\theta }_d\left(\mathit{x}\right)\right)$, in which ${\mathcal{M}}_d^i\left({\theta }_d\left(\mathit{x}\right)\right)$ is the grade of membership of ${\theta }_d\left(\mathit{x}\right)$ to ${\mathcal{M}}_d^i$. The normalized membership function $h_i\left(\theta \left(\mathit{x}\right)\right)$ satisfies the following property:

$$\forall i\in \{1,\dots ,r\},h_i\left(\mathbf{\theta }\left(\mathit{x}\right)\right)\ge 0,\sum _{i=1}^r{h}_i\left(\theta \left(\mathit{x}\right)\right)=1.$$

Then, the system formulated in (2) can be written as:

$$\begin{array}{cc}\hfill & \left\{\begin{array}{cc}\hfill \mathit{E}\mathit{x}(k+1)& ={\mathit{A}}_h^p\left(k\right)\mathit{x}\left(k\right)+{\mathit{B}}_{1h}^p\mathit{w}\left(k\right)+{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)+{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))\right),\hfill \\ \hfill \mathit{y}\left(k\right)& ={\mathit{C}}_2^p\mathit{x}\left(k\right),\hfill \\ \hfill \mathit{z}\left(k\right)& ={\mathit{C}}_{1h}^p\mathit{x}\left(k\right)+{\mathit{D}}_h^p\mathit{w}\left(k\right).\hfill \end{array}\right.\hfill \end{array}$$

(3)

For the subsequent development, we introduce a nominal descriptor MJS formulated as:

$$\begin{array}{cc}\hfill \mathit{E}\mathit{x}(k+1)& ={\mathit{A}}^p\mathit{x}\left(k\right),\hfill \end{array}$$

(4)

and the following definition is recalled.

Definition 1.

Based on [12], note that: (i) the pair $(\mathit{E},{\mathit{A}}^p)$ is regular if $det(\mathit{z}\mathit{E}-{\mathit{A}}^p)$ is different from zero for each $p\in \mathbb{N}$; (ii) the pair $(\mathit{E},{\mathit{A}}^p)$ is causal if $\mathrm{deg}det(\mathit{z}\mathit{E}-{\mathit{A}}^p)=\mathrm{rank}\left(\mathit{E}\right)$ for each $p\in \mathbb{N}$; (iii) the system formulated in (4) is stochastically stable if for any initial state $(r_0,x_0)$, the condition $\mathbb{E}({\sum }_{k=0}^{\infty }\parallel \mathit{x}\left(k\right){\parallel }^2|r_0,x_0)<\infty$ is satisfied; and (iv) the system stated in (4) is stochastically admissible if it is regular, causal, and stochastically stable.

2.3. Problem Statement

For a given non-linear descriptor MJS expressed by a TS fuzzy model, such as stated in (3), the main problem addressed in this article is to design an SMO and adaptive sliding mode controllers for the error and SMO systems, which must hold the following requirements simultaneously:

(i)

The reachability of the sliding surfaces is ensured.

(ii)

The sliding motions are stochastically admissible with $H_{\infty }$ performance, that is, under a zero initial condition, $\mathbb{E}\left({\sum }_{k=0}^{\infty }{\mathit{z}}^{\top }\left(k\right)\mathit{z}\left(k\right)\right)<{\gamma }^2{\sum }_{k=0}^{\infty }{\mathit{w}}^{\top }\left(k\right)\mathit{w}\left(k\right)$, for all $\mathit{w}\left(k\right)\ne$ 0 $\in L_2[0,\infty )$.

We end this section by recalling two lemmas proposed in [42,43], respectively.

Lemma 1.

Let $\mathit{M},\mathit{N}$ be constant matrices, and $\Delta \left(k\right)$ be a matrix function satisfying ${\Delta }^{\top }\left(k\right)\Delta \left(k\right)\le \mathit{I}$. For a positive scalar $\epsilon >0$, the inequality $\mathit{M}\Delta \left(k\right)\mathit{N}+{\mathit{N}}^{\top }{\Delta }^{\top }\left(k\right){\mathit{M}}^{\top }\le \epsilon \mathit{M}{\mathit{M}}^{\top }+{\epsilon }^{-1}{\mathit{N}}^{\top }\mathit{N}$ holds.

Lemma 2.

For any matrices $\mathit{Q}$, $\mathit{X}$, $\mathit{Y}$, $\mathit{A}$ with appropriate dimensions, and a positive scalar α, the inequality $\mathit{Q}+sym\left(\mathit{X}\mathit{A}\right)<\mathbf{0}$ is fulfilled if the following condition holds:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\mathit{Q}& \mathit{X}+\alpha {\mathit{A}}^{\top }{\mathit{Y}}^{\top }\\ {(\mathit{X}+\alpha {\mathit{A}}^{\top }{\mathit{Y}}^{\top })}^{\top }& -\alpha sym\left(\mathit{Y}\right)\end{array}\right]<\mathbf{0}.\end{array}$$

3. Main Results

In this section, the different analytical steps are presented for the development of the main results for the SMO enhanced adaptive control.

3.1. Sliding Mode Observer Design under Imperfect Premise Matching

The design of an observer can be challenging when state variables cannot be completely acquired in engineering systems. To reconstruct the state variables, we consider the following SMO of the system under consideration for which the premise variables are unmatched.

$$\begin{array}{ccc}& \mathrm{Rule}j:\mathtt{if}{\mu }_1\left(\widehat{\mathit{x}}\left(k\right)\right)\mathrm{is}{\mathcal{W}}_1^j,{\mu }_2\left(\widehat{\mathit{x}}\left(k\right)\right)\mathrm{is}{\mathcal{W}}_2^j,\dots ,\mathrm{and}{\mu }_{s_{\mathrm{o}}}\left(\widehat{\mathit{x}}\left(k\right)\right)\mathrm{is}{\mathcal{W}}_{s_{\mathrm{o}}}^j,\mathtt{then}\hfill & \hfill \\ & \left\{\begin{array}{cc}\hfill \mathit{E}\widehat{\mathit{x}}(k+1)& ={\mathit{A}}_j^p\widehat{\mathit{x}}\left(k\right)+{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)-\mathbf{\nu }\left(k\right))\right)+{\mathit{L}}_j^p(\mathit{y}\left(k\right)-\widehat{\mathit{y}}\left(k\right)),\hfill \\ \hfill \widehat{\mathit{y}}\left(k\right)& ={\mathit{C}}_2^p\widehat{\mathit{x}}\left(k\right),\hfill \end{array}\right.\hfill \end{array}$$

where $\mathbf{\mu }\left(\widehat{\mathit{x}}\left(k\right)\right)={[{\mu }_1\left(\widehat{\mathit{x}}\left(k\right)\right),\dots ,{\mu }_{s_o}\left(\widehat{\mathit{x}}\left(k\right)\right)]}^{\top }$ is the premise variable vector depending on the estimated states $\widehat{\mathit{x}}\left(k\right)$; $\widehat{\mathit{y}}\left(k\right)$ is the measured output; and $\mathbf{\nu }\left(k\right)$ is the observer input vector to attenuate the effect of the matched non-linearity. Note that ${\mathit{L}}_j$ is the observer gain to be determined. The global dynamics of the observer can be inferred as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathit{E}\widehat{\mathit{x}}(k+1)\hfill & {\displaystyle =\sum _{j=1}^r{\varpi }_j\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)\left({\mathit{A}}_j^p\widehat{\mathit{x}}\left(k\right)+{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)-\mathbf{\nu }\left(k\right))\right)+{\mathit{L}}_j^p(\mathit{y}\left(k\right)-\widehat{\mathit{y}}\left(k\right))\right),}\hfill \\ \widehat{\mathit{y}}\left(k\right)\hfill & ={\mathit{C}}_2^p\widehat{\mathit{x}}\left(k\right),\hfill \end{array}\right.\end{array}$$

(5)

whereas the normalized fuzzy membership function for the observer is given by:

$${\varpi }_j\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)=\prod _{d_{\mathrm{o}}=1}^{s_{\mathrm{o}}}{\mathcal{W}}_{d_{\mathrm{o}}}^j\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)/\sum _{j=1}^r\prod _{d_{\mathrm{o}}=1}^{s_{\mathrm{o}}}{\mathcal{W}}_{d_{\mathrm{o}}}^j\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)\ge 0,\sum _{j=1}^r{\varpi }_j\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)=1,$$

with the compact form of (5) being expressed as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathit{E}\widehat{\mathit{x}}(k+1)\hfill & ={\mathit{A}}_{\varpi }^p\widehat{\mathit{x}}\left(k\right)+{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)-\mathbf{\nu }\left(k\right)\right)+{\mathit{L}}_{\varpi }^p{\mathit{e}}_{\mathit{y}}\left(k\right),\hfill \\ \widehat{\mathit{y}}\left(k\right)\hfill & ={\mathit{C}}_2^p\widehat{\mathit{x}}\left(k\right).\hfill \end{array}\right.\end{array}$$

(6)

Define the estimation error as $\mathit{e}\left(k\right)=\mathit{x}\left(k\right)-\widehat{\mathit{x}}\left(k\right)$, and ${\mathit{e}}_{\mathit{y}}\left(k\right)=\mathit{y}\left(k\right)-\widehat{\mathit{y}}\left(k\right)$ as the output estimation error. By subtracting (6) from (2), the error dynamic system is defined as:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathit{E}\mathit{e}(k+1)=& ({\mathit{A}}_{\varpi }^p-{\mathit{L}}_{\varpi }^p{\mathit{C}}_{2\varpi }^p)\mathit{e}\left(k\right)+{\mathit{B}}_2^p\left(\mathbf{\nu }\left(k\right)+{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))\right)+{\mathit{B}}_{1h}^p\mathit{w}\left(k\right)\hfill \\ & +\Delta {\mathit{A}}_h^p\mathit{x}\left(k\right)+({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\mathit{x}\left(k\right).\hfill \end{array}\end{array}$$

(7)

Remark 2.

From (7), note that the proposed SMO is essential to eliminate the effect of the matched non-linearity. Fast convergence and easy implementation are two important features that drive us to use such an observer.

3.2. Switching Manifolds Design

Next, we focus on the switching manifolds design as the first step of SMC. For the error system established in (7), the switching manifold is suggested by means of

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathit{s}}_{\mathit{e}}\left(k\right)\hfill & ={\mathit{G}}_{\mathit{e}}^p\mathit{E}\mathit{e}\left(k\right)+{\mathit{b}}_{\mathit{e}}\left(k\right),\hfill \\ {\mathit{b}}_{\mathit{e}}(k+1)\hfill & ={\mathit{s}}_{\mathit{e}}\left(k\right)+{\mathit{G}}_{\mathit{e}}^p{\mathit{L}}_{\varpi }^p{\mathit{e}}_{\mathit{y}}\left(k\right),\hfill \\ {\mathit{b}}_{\mathit{e}}\left(0\right)\hfill & =-{\mathit{G}}_{\mathit{e}}^p\mathit{E}\mathit{e}\left(0\right),\hfill \end{array}\right.\end{array}$$

(8)

where ${\mathit{G}}_{\mathit{e}}^p\in {\mathbb{R}}^{m\times n}$ is a matrix, such that ${\mathit{G}}_e^p{\mathit{B}}_2^p$ is invertible and satisfies a constraint stated as:

$$\begin{array}{c}\hfill {\mathit{G}}_{\mathit{e}}^p\mathit{E}={\mathit{H}}^p{\mathit{C}}_2^p,\end{array}$$

(9)

with ${\mathit{H}}^p\in {\mathbb{R}}^{m\times ny}$ being the matrix to be determined.

Remark 3.

(i) The constraint presented in (9) is introduced because ${\mathit{s}}_{\mathit{e}}\left(k\right)$ is undefined, since $\mathit{e}\left(k\right)$ is unknown. However, ${\mathit{e}}_{\mathit{y}}\left(k\right)=\mathit{y}\left(k\right)-\widehat{\mathit{y}}\left(k\right)={\mathit{C}}_2^p\mathit{e}\left(k\right)$ is available and so the sliding surface defined as ${\mathit{s}}_{\mathit{e}}\left(k\right)={\mathit{H}}^p{\mathit{e}}_{\mathit{y}}\left(k\right)+{\mathit{b}}_{\mathit{e}}\left(k\right)$ is well stated for the observer design; and (ii) the existence of the constraint formulated in (9) is guaranteed by the assumption that:

$$\begin{array}{c}\hfill \mathrm{rank}\left[\begin{array}{c}{\mathit{G}}_{\mathit{e}}^p\mathit{E}\\ {\mathit{C}}_2^p\end{array}\right]=\mathrm{rank}\left({\mathit{C}}_2^p\right).\end{array}$$

(10)

Note that the following switching manifold can be constructed for the SMO system given in (6):

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathit{s}\left(k\right)\hfill & ={\mathit{G}}^p\mathit{E}\widehat{\mathit{x}}\left(k\right)-\mathit{b}\left(k\right),\hfill \\ \mathit{b}(k+1)\hfill & ={\mathit{G}}^p\left({\mathit{A}}_{\varpi }^p+{\mathit{B}}_2^p{\mathit{K}}_{\varpi }^p\right)\widehat{\mathit{x}}\left(k\right)+{\mathit{G}}^p{\mathit{L}}_{\varpi }^p{\mathit{e}}_y\left(k\right),\hfill \\ \mathit{b}\left(0\right)\hfill & ={\mathit{G}}^p\mathit{E}\widehat{\mathit{x}}\left(0\right),\hfill \end{array}\right.\end{array}$$

(11)

where ${\mathit{G}}^p\in {\mathbb{R}}^{m\times n}$ is a matrix, such that ${\mathit{G}}^p{\mathit{B}}_2^p$ is invertible, and ${\mathit{K}}_{\varpi }^p={\sum }_{l=1}^r{\varpi }_l\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right){\mathit{K}}_l^p$ is the controller gain to be designed.

3.3. $H_{\infty }$ and Admissibility Analysis

Here, we analyze the dynamics of the error and observer systems based on the sliding surfaces designed in the previous section.

The SMC theory states that the system trajectories reach the sliding surface if the ideal condition ${\mathit{s}}_{\mathit{e}}(k+1)={\mathit{s}}_{\mathit{e}}\left(k\right)=\mathbf{0}$ is verified. Thus, we obtain from (7) and (8) that:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \Delta {\mathit{s}}_{\mathit{e}}\left(k\right)=& {\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\left(\mathbf{\nu }\left(k\right)+{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))\right)\hfill \\ \hfill & +{\mathit{G}}_{\mathit{e}}^p\left({\mathit{A}}_{\varpi }^p\mathit{e}\left(k\right)+\Delta {\mathit{A}}_h^p\mathit{x}\left(k\right)+({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\mathit{x}\left(k\right)+{\mathit{B}}_{1h}^p\mathit{w}\left(k\right)\right)=\mathbf{0}.\hfill \end{array}\end{array}$$

(12)

Therefore, the corresponding control input is calculated as:

$$\begin{array}{cc}\hfill \mathbf{\nu }\left(k\right)=& -{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))-{\left({\mathit{G}}_e^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{G}}_{\mathit{e}}^p\left({\mathit{A}}_{\varpi }^p\mathit{e}\left(k\right)+\Delta {\mathit{A}}_h^p\mathit{x}\left(k\right)+({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\mathit{x}\left(k\right)+{\mathit{B}}_{1h}^p\mathit{w}\left(k\right)\right).\hfill \end{array}$$

(13)

By substituting (13) into (7), the dynamics of the sliding mode is expressed by:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill \mathit{E}\mathit{e}(k+1)& =({\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{A}}_{\varpi }^p-{\mathit{L}}_{\varpi }^p{\mathit{C}}_{2\varpi }^p)\mathit{e}\left(k\right)+{\overline{\mathit{G}}}_{\mathit{e}}^p\Delta {\mathit{A}}_h^p\mathit{x}\left(k\right)+{\overline{\mathit{G}}}_{\mathit{e}}^p({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\mathit{x}\left(k\right)+{\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{B}}_{1h}^p\mathit{w}\left(k\right),\hfill \end{array}\hfill \end{array}$$

(14)

where ${\overline{\mathit{G}}}_{\mathit{e}}^p=\mathit{I}-{\widehat{\mathit{G}}}_{\mathit{e}}^p$ and ${\widehat{\mathit{G}}}_{\mathit{e}}^p={\mathit{B}}_2^p{\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{G}}_{\mathit{e}}^p$. In addition, when the system formulated in (6) is driven to the sliding surface, then $\mathit{s}(k+1)=\mathit{s}\left(k\right)=\mathbf{0}$. Thus, from (6) and (11), we obtain that:

$$\begin{array}{cc}\hfill \mathit{s}(k+1)& ={\mathit{G}}^p{\mathit{B}}_2^p\left(\mathit{u}\left(k\right)-\mathbf{\nu }\left(k\right)-{\mathit{K}}_{\varpi }^p\widehat{\mathit{x}}\left(k\right)\right)=\mathbf{0}.\hfill \end{array}$$

(15)

Then, the resultant control input is given by:

$$\mathit{u}\left(k\right)={\mathit{K}}_{\varpi }^p\widehat{\mathit{x}}\left(k\right)+\mathbf{\nu }\left(k\right).$$

(16)

Substituting (16) into (6), the sliding mode dynamics can be written as:

$$\mathit{E}\widehat{\mathit{x}}(k+1)=({\mathit{A}}_{\varpi }^p+{\mathit{B}}_2^p{\mathit{K}}_{\varpi }^p)\widehat{\mathit{x}}\left(k\right)+{\mathit{L}}_{\varpi }^p{\mathit{C}}_2^p\mathit{e}\left(k\right).$$

(17)

From (14) and (17), the augmented closed-loop system is built by a form established as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\overline{\mathit{E}}\overline{\mathit{x}}(k+1)\hfill & =({\overline{\mathit{A}}}_{h\varpi \varpi }^p+\Delta {\overline{\mathit{A}}}_h^p)\overline{\mathit{x}}\left(k\right)+{\overline{\mathit{B}}}_{1h}^p\mathit{w}\left(k\right),\hfill \\ \mathit{z}\left(k\right)\hfill & ={\overline{\mathit{C}}}_{1h}^p\overline{\mathit{x}}\left(k\right)+{\mathit{D}}_h^p\mathit{w}\left(k\right),\hfill \end{array}\right.\end{array}$$

(18)

where

$$\overline{\mathit{x}}\left(k\right)={[{\widehat{\mathit{x}}}^{\top }\left(k\right),{\mathit{e}}^{\top }\left(k\right)]}^{\top },\Delta {\overline{\mathit{A}}}_h^p={\overline{\mathit{M}}}_h^p\Delta \left(k\right){\overline{\mathit{N}}}_h^p,\overline{\mathit{E}}=\left[\begin{array}{cc}\mathit{E}& \mathbf{0}\\ \mathbf{0}& \mathit{E}\end{array}\right],$$

$${\overline{\mathit{B}}}_{1h}^p=\left[\begin{array}{c}\mathbf{0}\\ {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{B}}_{1h}^p\end{array}\right],{{\overline{\mathit{C}}}_{1h}}^p=\left[\begin{array}{cc}{\mathit{C}}_{1h}^p& {\mathit{C}}_{1h}^p\end{array}\right],{\overline{\mathit{M}}}_h^p=\left[\begin{array}{c}\mathbf{0}\\ {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{M}}_h^p\end{array}\right],{\overline{\mathit{N}}}_h^p=\left[\begin{array}{cc}{\mathit{N}}_h^p& {\mathit{N}}_h^p\end{array}\right],$$

$$\begin{array}{cc}\hfill {\overline{\mathit{A}}}_{h\varpi \varpi }^p& =\left[\begin{array}{cc}{\mathit{A}}_{\varpi }^p+{\mathit{B}}_2^P{\mathit{K}}_{\varpi }^p& {\mathit{L}}_{\varpi }^p{\mathit{C}}_2^p\\ {\overline{\mathit{G}}}_{\mathit{e}}^p({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)& {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{A}}_{\varpi }^p-{\mathit{L}}_{\varpi }^p{\mathit{C}}_2^p\end{array}\right]=\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right){\varpi }_l\left(\widehat{\mathit{x}}\right){\overline{\mathit{A}}}_{ijl}^p,\hfill \\ \hfill {\overline{\mathit{A}}}_{ijl}^p& =\left[\begin{array}{cc}{\mathit{A}}_j^p+{\mathit{B}}_2^P{\mathit{K}}_l^p& {\mathit{L}}_j^p{\mathit{C}}_2^p\\ {\overline{\mathit{G}}}_{\mathit{e}}^p({\mathit{A}}_i^p-{\mathit{A}}_j^p)& {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{A}}_j^p-{\mathit{L}}_j^p{\mathit{C}}_2^p\end{array}\right].\hfill \end{array}$$

Theorem 1.

Given a scalar $\gamma >0$, the closed-loop system stated in (18) is stochastically admissible with $H_{\infty }$ disturbance attenuation level γ, if there exist matrices ${\mathit{P}}^p>0,{\mathit{S}}^p$, and a scalar ${\epsilon }_p>0$ such that, for $i,j,l\in \{1,\dots ,r\}$, and $p\in \{1,\dots ,N\}$, we have that

$$\begin{array}{c}\hfill {\overline{\mathbf{\Phi }}}_{ijl}^p=\left[\begin{array}{cccccc}sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{ijl}^p\right)-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}& {\mathit{S}}^{p^{\top }}{\overline{\mathit{R}}}_{}{\overline{\mathit{B}}}_{1i}^p& {\overline{\mathit{C}}}_{1i}^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\overline{\mathit{A}}}_{ijl}^{p^{\top }}& {\epsilon }_p{\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{M}}}_i^p& {\overline{\mathit{N}}}_i^p\\ \ast & -{\gamma }^2\mathit{I}& {\mathit{D}}_i^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\overline{\mathit{B}}}_{1i}^{p^{\top }}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & -\mathit{I}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & \ast & -\mathbf{\psi }& {\epsilon }_p{\mathit{I}}_{\pi i}^{{}^{\top }}\otimes {\overline{\mathit{M}}}_i^p& \mathbf{0}\\ \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}& \mathbf{0}\\ \ast & \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}\end{array}\right]<0,\end{array}$$

(19)

where the symbol “∗” is the same value as the opposite corner in the symmetric matrix,${\mathit{I}}_{\pi }^p=[\sqrt{{\pi }_{p1}}\mathit{I},\dots ,\sqrt{{\pi }_{pN}}\mathit{I}]$, $\mathbf{\psi }=diag({{\mathit{P}}^1}^{-1},\dots ,{{\mathit{P}}^N}^{-1})$,$\overline{\mathit{R}}$is any matrix satisfying$\overline{\mathit{R}}\overline{\mathit{E}}=\mathbf{0}$, and$\mathrm{rank}\left(\overline{\mathit{R}}\right)=2n-2q_0$, recalling that$q_0=\mathrm{rank}\left(\mathit{E}\right)<n$.

Proof. 

First, we prove the regularity and causality characterization of (18) with $\Delta {\overline{\mathit{A}}}_h^p=\mathbf{0}$. Assume that there exist two non-singular matrices $\widehat{\mathit{M}}$ and $\widehat{\mathit{N}}$, such that:

$$\widehat{\mathit{E}}=\widehat{\mathit{M}}\overline{\mathit{E}}\widehat{\mathit{N}}=\left[\begin{array}{cc}{\mathit{I}}_{2q_0}& \mathbf{0}\\ \mathbf{0}& \mathbf{0}\end{array}\right].$$

From (19), it can be verified that:

$$\begin{array}{c}\hfill -{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}+sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{ijl}^p\right)<0.\end{array}$$

(20)

Then, the following inequality holds:

$$-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}+sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i{\varpi }_j{\varpi }_l{\overline{\mathit{A}}}_{ijl}^p\right)=-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}+sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)<0.$$

Now, define that:

$$\begin{array}{cc}\hfill & \begin{array}{cccc}\hfill {\widehat{\mathit{A}}}_{h\varpi \varpi }^p& =\widehat{\mathit{M}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p\widehat{\mathit{N}}=\left[\begin{array}{cc}{\widehat{\mathit{A}}}_{11h\varpi \varpi }^p& {\widehat{\mathit{A}}}_{12h\varpi \varpi }^p\\ {\widehat{\mathit{A}}}_{21h\varpi \varpi }^p& {\widehat{\mathit{A}}}_{22h\varpi \varpi }^p\end{array}\right],\hfill & \hfill \widehat{\mathit{R}}& =\widehat{\mathit{M}}\overline{\mathit{R}}{\widehat{\mathit{M}}}^{-1}\left[\begin{array}{cc}{\widehat{\mathit{R}}}_{11}& {\widehat{\mathit{R}}}_{12}\\ {\widehat{\mathit{R}}}_{21}& {\widehat{\mathit{R}}}_{22}\end{array}\right],\hfill \\ \hfill {\widehat{\mathit{S}}}^p& ={\widehat{\mathit{M}}}^{-1\top }{\mathit{S}}^p\widehat{\mathit{N}}=\left[\begin{array}{cc}{\widehat{\mathit{S}}}_{11}^p& {\widehat{\mathit{S}}}_{12}^p\\ {\widehat{\mathit{S}}}_{21}^p& {\widehat{\mathit{S}}}_{22}^p\end{array}\right],\hfill & \hfill {\widehat{\mathit{P}}}^p& ={\widehat{\mathit{M}}}^{-1\top }{\mathit{P}}^p{\widehat{\mathit{M}}}^{-1}=\left[\begin{array}{cc}{\widehat{\mathit{P}}}_{11}^p& {\widehat{\mathit{P}}}_{12}^p\\ \ast & {\widehat{\mathit{P}}}_{22}^p\end{array}\right].\hfill \end{array}\hfill \end{array}$$

(21)

From $\overline{\mathit{R}}\overline{\mathit{E}}=\mathbf{0}$, we have ${\widehat{\mathit{R}}}_{}\widehat{\mathit{E}}=\mathbf{0}$ and so ${\widehat{\mathit{R}}}_{11}={\widehat{\mathit{R}}}_{21}=\mathbf{0}$.

By performing the congruence transformation to the expression stated in (20) by $\widehat{\mathit{N}}$, it follows according to (21) that: $\mathrm{sym}({\widehat{\mathit{S}}}_{12}^{p^{\top }}{\widehat{\mathit{R}}}_{12}{\widehat{\mathit{A}}}_{22h\varpi \varpi }^p+{\widehat{\mathit{S}}}_{22}^{p^{\top }}{\widehat{\mathit{R}}}_{22}{\widehat{\mathit{A}}}_{22h\varpi \varpi }^p)<0.$

It is readily concluded that ${\widehat{\mathit{A}}}_{22h\varpi \varpi }^p$ is non-singular and, according to Definition 1, $(\overline{\mathit{E}},{\overline{\mathit{A}}}_{h\varpi \varpi }^p)$ is regular and causal.

Now, select $V(\overline{\mathit{x}}\left(k\right),{\overline{r}}_k)={\overline{\mathit{x}}}^{\top }\left(k\right){\overline{\mathit{E}}}^{\top }\mathit{P}\left({\overline{r}}_k\right)\overline{\mathit{E}}\overline{\mathit{x}}\left(k\right)$ as a stochastic Lyapunov function to prove the stochastic stability of the closed-loop system stated in (18), with $\mathit{w}\left(k\right)=\mathbf{0}$. To simplify the notation, we use $V(\overline{\mathit{x}}\left(k\right),{\overline{r}}_k)=V\left(k\right)$.

Let $\Delta V\left(k\right)$ be the forward difference of $V\left(k\right)$. Then, along the trajectories of the system given in (18), we have that: $\mathbb{E}(\Delta V\left(k\right))=\mathbb{E}\left(V(k+1)-V\left(k\right)|\overline{\mathit{x}}\left(k\right),{\overline{r}}_k=p\right)$, where

$$\mathbb{E}(\Delta V\left(k\right))=\mathbb{E}\left({\overline{\mathit{x}}}^{\top }\left(k\right)\left({\overline{\mathit{A}}}_{h\varpi \varpi }^{p^{\top }}{\mathit{X}}^p{\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)\overline{\mathit{x}}\left(k\right)\right)-{\overline{\mathit{x}}}^{\top }\left(k\right){\overline{\mathit{E}}}^{\top }{\mathit{P}}_{}^p\overline{\mathit{E}}\overline{\mathit{x}}\left(k\right),$$

(22)

with ${\mathit{X}}^p={\sum }_{q=1}^N{\pi }_{pq}{\mathit{P}}^q$. Furthermore, in view of the constraint $\overline{\mathit{R}}\overline{\mathit{E}}=\mathbf{0}$, the following equation is easily obtained:

$$\begin{array}{c}\hfill 2{\overline{\mathit{x}}}^T\left(k\right){\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p\overline{\mathit{x}}\left(k\right)=\mathbf{0}.\end{array}$$

(23)

Substituting (23) into (22), results:

$$\begin{array}{c}\hfill \mathbb{E}(\Delta V\left(k\right))=\mathbb{E}\left({\overline{\mathit{x}}}^{\top }\left(k\right)\left({\overline{\mathit{A}}}_{h\varpi \varpi }^{p^{\top }}{\mathit{X}}^p{\overline{\mathit{A}}}_{h\varpi \varpi }^p-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}+sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)\right)\overline{\mathit{x}}\left(k\right)\right).\end{array}$$

(24)

By the Schur complement, it can be verified from (19) that: ${\Psi }_{h\varpi \varpi }^p={\overline{\mathit{A}}}_{h\varpi \varpi }^{p^{\top }}{\mathit{X}}^p{\overline{\mathit{A}}}_{h\varpi \varpi }^p-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}+sym({\mathit{S}}^{p\top }\overline{\mathit{R}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p)<0.$ Thus, we deduce that: $\mathbb{E}(\Delta V\left(k\right))\le \phi \mathbb{E}\left({\overline{\mathit{x}}}^{\top }\left(k\right)\overline{\mathit{x}}\left(k\right)\right)$, where $\phi <0$ denotes ${max}_p\left({\lambda }_{\max }\left({\Psi }_{h\varpi \varpi }^p\right)\right)$, for all $p\in \mathbb{N}$. Then, from (24), we obtain that $\mathbb{E}\left({\sum }_0^{\infty }{\overline{\mathit{x}}}^{\top }\left(k\right)\overline{\mathit{x}}\left(k\right)\right)\le (1/\phi )\mathbb{E}\{{\sum }_0^{\infty }\Delta V\left(k\right)\}\le -(1/\phi )V\left(0\right)<\infty$. Hence, in accordance with Definition 1, the system stated in (18) is stochastically admissible. To investigate the $H_{\infty }$ performance for the system defined in (18), the following index is introduced: $J=\mathbb{E}\left({\sum }_{k=0}^{\infty }({\mathit{z}}^{\top }\left(k\right)\mathit{z}\left(k\right)-{\gamma }^2{\mathit{w}}^{\top }\left(k\right)\mathit{w}\left(k\right))\right)$.

Define ${\psi }_0\left(k\right)=col(\overline{\mathit{x}}\left(k\right),\mathit{w}\left(k\right))$, ${\mathcal{A}}_{h\varpi \varpi }^p=[{\overline{\mathit{A}}}_{h\varpi \varpi }^p,{\overline{\mathit{B}}}_{1h}^p]$, ${\mathcal{C}}_h=[{\overline{\mathit{C}}}_{1h}^p,{\mathit{D}}_h^p]$ and $J_{zw}\left(k\right)={\mathit{z}}^{\top }\left(k\right)\mathit{z}\left(k\right)-{\gamma }^2{\mathit{w}}^{\top }\left(k\right)\mathit{w}\left(k\right)$. Based on the similar procedure as above and considering the equation $2{\overline{\mathit{x}}}_k^{\top }{\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\mathit{A}}_{h\varpi \varpi }^p{\mathbf{\psi }}_0\left(k\right)=\mathbf{0}$, it follows that

$$\begin{array}{c}\hfill \mathbb{E}(\Delta V\left(k\right))+J_{zw}\left(k\right)={\mathbf{\psi }}_0^{\top }\left(k\right)({\overline{\mathbf{\Phi }}}_{1h\varpi \varpi }^p\left({\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)+{\mathcal{C}}_h^{\top }{\mathcal{C}}_h+{\mathcal{A}}_{h\varpi \varpi }^{p^{\top }}{\mathit{X}}^p{\mathcal{A}}_{h\varpi \varpi }^p){\mathbf{\psi }}_{\mathbf{0}}\left(k\right),\end{array}$$

(25)

where

$${\overline{\mathbf{\Phi }}}_{1h\varpi \varpi }^p\left({\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)=\left[\begin{array}{cc}sym\left({\mathit{S}}^{p^{\top }}\overline{\mathit{R}}{\overline{\mathit{A}}}_{h\varpi \varpi }^p\right)-{\overline{\mathit{E}}}^{\top }{\mathit{P}}^p\overline{\mathit{E}}& {\mathit{S}}^{p^{\top }}{\overline{\mathit{R}}}_{}{\overline{\mathit{B}}}_{1h}^p\\ \ast & -{\gamma }^2\mathit{I}\end{array}\right].$$

Summing (25) with respect to k from 0 to ∞, under a zero initial condition, it yields the expression given by $J\le {\sum }_{k=0}^{\infty }(\mathbb{E}(\Delta V\left(k\right))+J_{zw}\left(k\right))<0$. Hence, the system formulated in (18) is stochastically admissible with $H_{\infty }$ performance. Suppose that $\Delta {\overline{\mathit{A}}}_h^p\ne 0$. By defining ${\overline{\mathit{A}}}_{h\varpi \varpi }^p\left(k\right)={\overline{\mathit{A}}}_{h\varpi \varpi }^p+\Delta {\overline{\mathit{A}}}_{h\varpi }^p$, ${\mathcal{A}}_{h\varpi \varpi }^p\left(k\right)=[{\overline{\mathit{A}}}_{h\varpi \varpi }^p\left(k\right),{\overline{\mathit{B}}}_{1h}^p]$, and applying the Schur complement accordingly with Lemma 1 to (19), we obtain: ${\overline{\mathbf{\Phi }}}_{1h\varpi \varpi }^p\left({\overline{\mathit{A}}}_{h\varpi \varpi }^p\left(k\right)\right)+{\mathcal{C}}_h^{\top }{\mathcal{C}}_h+{\mathcal{A}}_{h\varpi \varpi }^{p^{\top }}\left(k\right){\mathit{X}}^p{\mathcal{A}}_{h\varpi \varpi }^p\left(k\right)<0.$ By the same reasoning as above, we can conclude that the closed-loop system given in (18) is robustly admissible with $H_{\infty }$ performance.    □

3.4. Sliding Mode Dynamics Synthesis

In the following theorem, we provide a method to determine the gains ${\mathit{K}}_l^p$ and ${\mathit{L}}_j^p$.

Theorem 2.

The closed-loop system stated in (18) is stochastically admissible, with attenuation level γ of $H_{\infty }$ performance if, for a set of scalars α, β, ${\epsilon }_p>0$, and matrices ${\mathit{P}}^p>0$, ${\mathbb{S}}_{11}^p\in {\mathbb{R}}^{n\times n}$, ${\mathbb{S}}_{22}^p\in {\mathbb{R}}^{n\times n}$, ${\mathit{W}}^p\in {\mathbb{R}}^{n_y\times n_y}$, ${\mathit{Y}}_l^p\in {\mathbb{R}}^{m\times n}$, ${\mathit{F}}_j^p\in {\mathbb{R}}^{n\times n_y}$, and $J^p>0$, the following conditions can be satisfied for $i,j,l\in \{1,\dots ,r\}$ under the constraint ${\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right)\ge 0$, where ${\tau }_l$ is a positive scalar:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\widehat{\mathbf{\Phi }}}_{ijl}^p-{\mathbf{\Lambda }}_i<0,\hfill \\ \hfill & {\tau }_i{\widehat{\mathbf{\Phi }}}_{iji}^p-{\tau }_i{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i<0,\hfill \\ \hfill & {\tau }_l{\widehat{\mathbf{\Phi }}}_{ijl}^p+{\tau }_i{\widehat{\mathbf{\Phi }}}_{lji}^p-{\tau }_l{\mathbf{\Lambda }}_i-{\tau }_i{\mathbf{\Lambda }}_l+{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_l<0,l>i,\hfill \end{array}\right.\end{array}$$

(26)

where

$${\widehat{\mathbf{\Phi }}}_{ijl}^p=\left[\begin{array}{ccc}{\widehat{\Psi }}_{ijlt}^p& \beta {\mathbf{{\rm Y}}}_{1i}^p& {\mathbf{{\rm Y}}}_2^{p\top }\\ \ast & -\beta sym\left({\mathit{W}}^p\right)+{\mathit{J}}^p& \mathbf{0}\\ \ast & \ast & -{\mathit{J}}^p\end{array}\right],$$

$${\widehat{\mathbf{\Psi }}}_{ijl}^p=\left[\begin{array}{cccccc}{\widehat{\mathbf{\Phi }}}_{11}& {\overline{\mathit{R}}}_{}{\overline{\mathit{B}}}_{1i}^p& {\mathbb{S}}^{p^{\top }}{\overline{\mathit{C}}}_{1i}^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\mathit{A}}_{ijl}^{{\mathit{p}}^{\top }}& {\epsilon }_p\overline{\mathit{R}}{\overline{\mathit{M}}}_i^p& {{\mathbb{S}}_{}}^{p^{\top }}{\overline{\mathit{N}}}_i^p\\ \ast & -{\gamma }^2\mathit{I}& {\mathit{D}}_i^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\overline{\mathit{B}}}_{1i}^{{\mathit{p}}^{\top }}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & -\mathit{I}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & \ast & -{\widehat{\mathbf{\Phi }}}_{44}& {\epsilon }_p{\mathit{I}}_{\pi }^{p\top }\otimes {\overline{\mathit{M}}}_i^p& \mathbf{0}\\ \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}& \mathbf{0}\\ \ast & \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}\end{array}\right],$$

with${\widehat{\mathbf{\Phi }}}_{11}=sym(\overline{\mathit{R}}{\mathit{A}}_{ijl}^p+\alpha \overline{\mathit{E}}{{\mathbb{S}}_{}}^p)+{\alpha }^2{\mathit{P}}^p$,${\mathbf{{\rm Y}}}_{1i}^p={[{\left(\overline{\mathit{R}}{\mathbf{\Gamma }}_{1i}^p\right)}^{\top },\mathbf{0},\mathbf{0},{({\mathit{I}}_{\pi }^p\otimes {\mathbf{\Gamma }}_{1i}^p)}^{\top },\mathbf{0},\mathbf{0}]}^{\top }$,${\mathbf{{\rm Y}}}_2^p=[{\mathbf{\Gamma }}_2^p,\mathbf{0},\mathbf{0},\mathbf{0},\mathbf{0},\mathbf{0}]$,

$$\begin{array}{cc}\hfill {\mathit{A}}_{ijl}^p& =\left[\begin{array}{cc}{\mathit{A}}_j^p{\mathbb{S}}_{11}^p+{\mathit{B}}_2^p{\mathit{Y}}_l^p& {\mathit{F}}_j^p{\mathit{C}}_2^p\\ {\overline{\mathit{G}}}_{\mathit{e}}^p({\mathit{A}}_i^p-{\mathit{A}}_j^p){\mathbb{S}}_{11}^p& {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{A}}_j^p{\mathbb{S}}_{22}^p-{\mathit{F}}_j^p{\mathit{C}}_2^p\end{array}\right],{\mathbb{S}}_{}^p=\left[\begin{array}{cc}{\mathbb{S}}_{11}^p& \mathbf{0}\\ \mathbf{0}& {\mathbb{S}}_{22}^p\end{array}\right]{\mathbf{\Gamma }}_{1j}^p=\left[\begin{array}{c}{\mathit{F}}_j^p\\ -{\mathit{F}}_j^p\end{array}\right],\hfill \end{array}$$

${\mathbf{\Gamma }}_2^p=[\mathbf{0},{\mathit{C}}_2^p{\mathbb{S}}_{22}^p-{\mathit{W}}^p{\mathit{C}}_2^p]$, and${\widehat{\mathbf{\Phi }}}_{44}=diag({\mathit{P}}^1,\dots ,{\mathit{P}}^N)$. Moreover, the parameters${\mathit{K}}_l^p$and${\mathit{L}}_j^p$are given by${\mathit{K}}_l^p={\mathit{Y}}_l^p{\left({\mathbb{S}}_{11}^p\right)}^{-1}$and${\mathit{L}}_j^p={\mathit{F}}_j^p{\left({\mathit{W}}^p\right)}^{-1}$, respectively.

Proof. 

For a feasible solution of Theorem 2, it can be verified that: $sym(\overline{\mathit{R}}{\mathit{A}}_{ijl}^p+\alpha \overline{\mathit{E}}{\mathbb{S}}^p)+{\alpha }^2{\mathit{P}}^p<0$ and so ${\mathbb{S}}^p$ is non-singular. Moreover, using ${(\overline{\mathit{E}}{\mathbb{S}}^p+\alpha {\mathit{P}}_{}^p)}^{\top }{\left({\mathit{P}}_{}^p\right)}^{-1}(\overline{\mathit{E}}{\mathbb{S}}^p+\alpha {\mathit{P}}_{}^p)\ge 0$, it can be verified that:

$$\begin{array}{c}\hfill -{\mathbb{S}}^{p\top }{\overline{\mathit{E}}}^{\top }{\left({\mathit{P}}_{}^p\right)}^{-1}\overline{\mathit{E}}{\mathbb{S}}^p\le sym\left(\alpha \overline{\mathit{E}}{\mathbb{S}}^p\right)+{\alpha }^2{\mathit{P}}_{}^p.\end{array}$$

(27)

To obtain less conservative results, a slack matrix, defined as:

$$\sum _{i=1}^r\sum _{j=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(\sum _{l=1}^r{h}_l\left(\widehat{\mathit{x}}\right)-\sum _{l=1}^r{\varpi }_l\left(\widehat{\mathit{x}}\right)\right){\mathbf{\Lambda }}_i=\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(h_l\left(\widehat{\mathit{x}}\right)-{\varpi }_l\left(\widehat{\mathit{x}}\right)\right){\mathbf{\Lambda }}_i=\mathbf{0},$$

is used, where ${\mathbf{\Lambda }}_i$ is an arbitrary matrix with appropriate dimensions. Then, we have that:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill & \sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right){\varpi }_l\left(\widehat{\mathit{x}}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p\hfill \\ \hfill =& \sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left({\varpi }_l\left(\widehat{\mathit{x}}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p+\left(h_l\left(\widehat{\mathit{x}}\right)-{\varpi }_l\left(\widehat{\mathit{x}}\right)\right){\mathbf{\Lambda }}_i\right)\hfill \\ \hfill =& \sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(h_l\left(\widehat{\mathit{x}}\right)({\tau }_l{\widehat{\mathbf{\Phi }}}_{ijl}^p-{\tau }_l{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i)+({\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right))({\widehat{\mathbf{\Phi }}}_{ijl}^p-{\mathbf{\Lambda }}_i)\right)\hfill \\ \hfill \le & \sum _{i=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right)h_i\left(\widehat{\mathit{x}}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(({\tau }_i{\widehat{\mathbf{\Phi }}}_{iji}^p-{\tau }_i{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i)\right)\hfill \\ \hfill & +\left(\sum _{i=1}^r\sum _{j=1}^r\sum _{l>i}^{}{h}_i\left(\mathit{x}\right)h_l\left(\widehat{\mathit{x}}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(({\tau }_l{\widehat{\mathbf{\Phi }}}_{ijl}^p-{\tau }_l{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i)+({\tau }_i{\widehat{\mathbf{\Phi }}}_{lji}^p-{\tau }_i{\mathbf{\Lambda }}_l+{\mathbf{\Lambda }}_l)\right)\right)\hfill \\ \hfill & +\left(\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)({\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right))({\widehat{\mathbf{\Phi }}}_{ijl}^p-{\mathbf{\Lambda }}_i)\right).\hfill \end{array}\end{array}$$

(28)

From Theorem 2 and (28), we reach:

$$\begin{array}{c}\hfill \sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right){\varpi }_l\left(\widehat{\mathit{x}}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p<0.\end{array}$$

(29)

Now, define that:

$$\mathbb{Z}={\left[\begin{array}{ccc}\mathit{I}& \mathbf{0}& \mathbf{0}\\ \mathbf{0}& \mathit{I}& \mathit{I}\end{array}\right]}^{\top }.$$

Performing a congruence transformation to (29) by $\mathbb{Z}$, we obtain:

$$\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathbf{x}\right){\varpi }_j\left(\widehat{\mathbf{x}}\right){\varpi }_l\left(\widehat{\mathbf{x}}\right)\left[\begin{array}{cc}{\widehat{\mathbf{\Psi }}}_{ijl}^p& \beta {\mathbf{{\rm Y}}}_{1i}^p+{\mathbf{{\rm Y}}}_2^{p\top }\\ \ast & -\beta sym\left({\mathbf{W}}^p\right)\end{array}\right]<0.$$

Note that:

$$\begin{array}{c}\hfill {\overline{\mathit{A}}}_{ijl}^p{{\mathbb{S}}_{}}^p=\left[\begin{array}{cc}{\mathit{A}}_j^p{\mathbb{S}}_{11}^p+{\mathit{B}}_2^p{\mathit{Y}}_l^p& {\mathit{L}}_j^p{\mathit{C}}_2^p{\mathbb{S}}_{22}^p\\ {\overline{\mathit{G}}}_{\mathit{e}}^p({\mathit{A}}_i^p-{\mathit{A}}_j^p){\mathbb{S}}_{11}^p& {\overline{\mathit{G}}}_{\mathit{e}}^p{\mathit{A}}_j^p{\mathbb{S}}_{22}^p-{\mathit{L}}_j^p{\mathit{C}}_2^p{\mathbb{S}}_{22}^p\end{array}\right]={\mathit{A}}_{ijl}^p+\left[\begin{array}{c}{\mathit{L}}_j^p\\ -{\mathit{L}}_j^p\end{array}\right]\left[\begin{array}{cc}0& {\mathit{C}}_2^p{\mathbb{S}}_{22}^p-{\mathit{W}}^p{\mathit{C}}_2^p\end{array}\right].\end{array}$$

(30)

Thus, considering (27) and (30), the following condition holds according to Lemma 2:

$$\begin{array}{c}\hfill \sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i{\varpi }_j{\varpi }_l\left[\begin{array}{cccccc}(1,1)& {\overline{\mathit{R}}}_{}{\overline{\mathit{B}}}_{1i}^p& {\mathbb{S}}^{p^{\top }}{\overline{\mathit{C}}}_{1i}^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\left({\overline{\mathit{A}}}_{ijl}^p{{\mathbb{S}}_{}}^p\right)}^{\top }& {\epsilon }_p\overline{\mathit{R}}{\overline{\mathit{M}}}_i^p& {\mathbb{S}}^{p^{\top }}{\overline{\mathit{N}}}_i^p\\ \ast & -{\gamma }^2\mathit{I}& {\mathit{D}}_i^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\overline{\mathit{B}}}_{1i}^{p^{\top }}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & -\mathit{I}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & \ast & -{\mathbf{\psi }}_t& {\epsilon }_p{\mathit{I}}_{\pi }^{p\top }\otimes {\overline{\mathit{M}}}_i^p& \mathbf{0}\\ \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}& \mathbf{0}\\ \ast & \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}\end{array}\right]<0,\end{array}$$

(31)

with $(1,1)=sym\left({\overline{\mathit{A}}}_{ijl}^p{{\mathbb{S}}_{}}^p\right)-{\mathbb{S}}^{p\top }{\overline{\mathit{E}}}^{\top }{\left({\mathit{P}}_{}^p\right)}^{-1}\overline{\mathit{E}}{\mathbb{S}}^p$.

Let ${\mathit{S}}^p={\left({{\mathbb{S}}_{}}^p\right)}^{-1}$ and ${\mathbf{\Pi }}^p=diag({\mathbb{S}}_{}^p,\mathit{I},\mathit{I},\dots ,\mathit{I})$. By performing a congruence transformation to (31) by ${\mathbf{\Pi }}^p$, condition (19) holds by the Schur complement lemma. Then, in view of Theorem 1, the closed-loop system stated in (18) is stochastically admissible and the $H_{\infty }$ performance is fulfilled for all $\mathit{w}\left(k\right)\ne \mathbf{0}\in L_2[0,\infty )$.    □

Remark 4.

To address the equality condition, we establish the LMI condition stated as:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}-\eta \mathit{I}& {({\mathit{G}}_{\mathit{e}}^p\mathit{E}-{\mathit{H}}^p{\mathit{C}}_2^p)}^{\top }\\ \ast & -\mathit{I}\end{array}\right]<0,\end{array}$$

(32)

where η is a small positive coefficient.

Remark 5.

Note that the conditions of Theorem 2 are not strict LMIs. However, this can be transformed into an mathematical optimization problem for fixed parameters α and β formulated as:

$$\begin{array}{cc}\hfill & min\{\gamma +\eta \}\hfill \\ \hfill \mathrm{subject}\mathrm{to}& \left(\mathrm{i}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p-{\mathbf{\Lambda }}_i<0,\hfill \\ \hfill & \left(\mathrm{ii}\right){\tau }_i{\widehat{\mathbf{\Phi }}}_{iji}^p-{\tau }_i{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i<0,\hfill \\ \hfill & \left(\mathrm{iii}\right){\tau }_l{\widehat{\mathbf{\Phi }}}_{ijl}^p+{\tau }_i{\widehat{\mathbf{\Phi }}}_{lji}^p-{\tau }_l{\mathbf{\Lambda }}_i-{\tau }_i{\mathbf{\Lambda }}_l+{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_l>0,l>i,\hfill \\ \hfill & \left(\mathrm{iv}\right)-{\mathbb{S}}^{p\top }{\overline{\mathit{E}}}^{\top }{\left({\mathit{P}}_{}^p\right)}^{-1}\overline{\mathit{E}}{\mathbb{S}}^p\le sym\left(\alpha \overline{\mathit{E}}{\mathbb{S}}^p\right)+{\alpha }^2{\mathit{P}}_{}^p,\hfill \\ \hfill & \left(\mathrm{v}\right)\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right){\varpi }_l\left(\widehat{\mathit{x}}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p\hfill \\ \hfill & \le \sum _{i=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right)h_i\left(\widehat{\mathit{x}}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(({\tau }_i{\widehat{\mathbf{\Phi }}}_{iji}^p-{\tau }_i{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i)\right)\hfill \\ \hfill & +\left(\sum _{i=1}^r\sum _{j=1}^r\sum _{l>i}^{}{h}_i\left(\mathit{x}\right)h_l\left(\widehat{\mathit{x}}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)\left(({\tau }_l{\widehat{\mathbf{\Phi }}}_{ijl}^p-{\tau }_l{\mathbf{\Lambda }}_i+{\mathbf{\Lambda }}_i)+({\tau }_i{\widehat{\mathbf{\Phi }}}_{lji}^p-{\tau }_i{\mathbf{\Lambda }}_l+{\mathbf{\Lambda }}_l)\right)\right)\hfill \\ \hfill & +\left(\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right)({\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right))({\widehat{\mathbf{\Phi }}}_{ijl}^p-{\mathbf{\Lambda }}_i)\right),\hfill \\ \hfill & \left(\mathrm{vi}\right)\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i\left(\mathit{x}\right){\varpi }_j\left(\widehat{\mathit{x}}\right){\varpi }_l\left(\widehat{\mathit{x}}\right){\widehat{\mathbf{\Phi }}}_{ijl}^p<0,\hfill \\ \hfill \left(\mathrm{vii}\right)\sum _{i=1}^r\sum _{j=1}^r\sum _{l=1}^r{h}_i{\varpi }_j{\varpi }_l& \left[\begin{array}{cccccc}(1,1)& {\overline{\mathit{R}}}_{}{\overline{\mathit{B}}}_{1i}^p& {\mathbb{S}}^{p^{\top }}{\overline{\mathit{C}}}_{1i}^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\left({\overline{\mathit{A}}}_{ijl}^p{{\mathbb{S}}_{}}^p\right)}^{\top }& {\epsilon }_p\overline{\mathit{R}}{\overline{\mathit{M}}}_i^p& {\mathbb{S}}^{p^{\top }}{\overline{\mathit{N}}}_i^p\\ \ast & -{\gamma }^2\mathit{I}& {\mathit{D}}_i^{p^{\top }}& {\mathit{I}}_{\pi }^p\otimes {\overline{\mathit{B}}}_{1i}^{p^{\top }}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & -\mathit{I}& \mathbf{0}& \mathbf{0}& \mathbf{0}\\ \ast & \ast & \ast & -{\mathbf{\psi }}_t& {\epsilon }_p{\mathit{I}}_{\pi }^{p\top }\otimes {\overline{\mathit{M}}}_i^p& \mathbf{0}\\ \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}& \mathbf{0}\\ \ast & \ast & \ast & \ast & \ast & -{\epsilon }_p\mathit{I}\end{array}\right]<0,\hfill \\ \hfill & \left(\mathrm{viii}\right)\left[\begin{array}{cc}-\eta \mathit{I}& {({\mathit{G}}_{\mathit{e}}^p\mathit{E}-{\mathit{H}}^p{\mathit{C}}_2^p)}^{\top }\\ \ast & -\mathit{I}\end{array}\right]<0.\hfill \end{array}$$

(33)

Note that above constraints stated in (i)-(viii) are related to Theorem 2 and expressions (26)–(32).

Furthermore, by performing thefminsearchfunction of the optimization toolbox ofMatlab, the parameters α and β can be determined with a locally convergent solution.

3.5. Adaptive SMC Law Design

Now, a method is presented to synthesize an adaptive SMC law that may ensure the reachability of the designed sliding surface. As the system states $\mathit{x}\left(k\right)$ are not completely available, it follows that their error term $\mathit{e}\left(k\right)$ cannot be precisely estimated. Based on the relationships among the system states $\mathit{x}\left(k\right)$, the error $\mathit{e}\left(k\right)$, the outputs $\mathit{y}\left(k\right)$, and $\widehat{\mathit{y}}\left(k\right)$, we can consider that there exist unknown scalars holding $\parallel \mathit{e}\left(k\right)\parallel \le {\iota }_1\left(k\right)\parallel \mathit{y}\left(k\right)\parallel +{\iota }_2\left(k\right)\parallel \widehat{\mathit{y}}\left(k\right)\parallel$. Then, according to the assumptions stated in (A1) and (A2), the estimation may be produced where the unknown scalars ${\rho }_l>0$, for $l\in \{1,2,3\}$, can be determined to confirm that the expression, given by:

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill \mathsf{\Lambda }=& \parallel {\mathit{X}}^p\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)\parallel (\parallel {\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))\parallel +\parallel {\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{G}}_{\mathit{e}}^p\parallel (\parallel {\mathit{A}}_{\varpi }^p\parallel \parallel \mathit{e}\left(k\right)\parallel \hfill \\ \hfill & +\parallel \Delta {\mathit{A}}_h^p\parallel \parallel \mathit{x}\left(k\right)\parallel +\parallel ({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\parallel \parallel \mathit{x}\left(k\right)\parallel \parallel {\mathit{B}}_{1h}^p\parallel \parallel \mathit{w}\left(k\right)\parallel \left)\right),\hfill \end{array}\end{array}$$

must be less than or equal to ${\rho }_1\parallel \mathit{y}\left(k\right)\parallel +{\rho }_2\parallel \widehat{\mathit{y}}\left(k\right)\parallel +{\rho }_3$, with ${\mathbf{\Pi }}^p={\sum }_{q=1}^N{\pi }_{pq}{\left({\mathit{G}}_{\mathit{e}}^q{\mathit{B}}_2^q\right)}^{-1}$.

The following theorem establishes the state trajectories of the controlled system.

Theorem 3.

Consider the fuzzy descriptor MJS defined in (2). Suppose that the sliding functions stated in (8)–(11) are well designed. Then, the state trajectories of the controlled system is driven onto the sliding surface $\mathit{s}\left(k\right)=\mathbf{0}$ almost surely, if we apply the adaptive SMC law given by:

$$\begin{array}{cc}\hfill \mathit{u}\left(k\right)& =\sum _{l=1}^r{\varpi }_l\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)\left({\mathit{K}}_l^p\widehat{\mathit{x}}\left(k\right)+\mathbf{\nu }\left(k\right)+{\displaystyle \frac{1}{2}}{\left({\mathit{G}}^p{\mathit{B}}_2^p\right)}^{-1}\mathit{s}\left(k\right)\right),\hfill \end{array}$$

(34)

where the discontinuous switched term$\mathbf{\nu }\left(k\right)$is stated as:$\mathbf{\nu }\left(k\right)=-{\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\left({\mathbf{\Pi }}^p\right)}^{-1}({\mathbf{\kappa }}^p{\mathit{s}}_{\mathit{e}}\left(k\right)+\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)+\widehat{\alpha }\left(k\right)sat\left({\mathit{s}}_{\mathit{e}}\left(k\right)\right))$, with

$$sat\left({\mathit{s}}_{\mathit{e}}\left(k\right)\right)=\left\{\begin{array}{c}{\displaystyle \frac{{\mathit{s}}_{\mathit{e}}\left(k\right)}{\sigma }},\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel \le \sigma ,\hfill \\ sign\left({\mathit{s}}_{\mathit{e}}\left(k\right)\right),\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel >\sigma ,\hfill \end{array}\right.$$

$\widehat{\alpha }\left(k\right)={\widehat{\rho }}_1\left(k\right)\parallel \mathit{y}\left(k\right)\parallel +{\widehat{\rho }}_2\left(k\right)\parallel \widehat{\mathit{y}}\left(k\right)\parallel +{\widehat{\rho }}_3\left(k\right)$,Σbeing a positive-definite matrix, and${\kappa }^p$holding

$$\begin{array}{c}\hfill {\kappa }^p-{\mathsf{\Lambda }}_1^p\ge 0,{\mathsf{\Lambda }}_1^p={\displaystyle \frac{1}{2}}\parallel {\Pi }^p-{\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}\parallel ,\end{array}$$

(35)

which guarantees that state trajectories of the error system established in (7) are attracted to a region around the sliding surface${\mathit{s}}_{\mathit{e}}\left(k\right)$. The updating laws are designed as:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill \Delta {\widehat{\rho }}_1\left(k\right)& ={\widehat{\rho }}_1(k+1)-{\widehat{\rho }}_1\left(k\right)=q_1(-{\epsilon }_1{\widehat{\rho }}_1\left(k\right)+\parallel \mathit{y}\left(k\right)\parallel \parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel ),\hfill \\ \hfill \Delta {\widehat{\rho }}_2\left(k\right)& ={\widehat{\rho }}_2(k+1)-{\widehat{\rho }}_2\left(k\right)=q_2(-{\epsilon }_2{\widehat{\rho }}_2\left(k\right)+\parallel \widehat{\mathit{y}}\left(k\right)\parallel \parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel ),\hfill \\ \hfill \Delta {\widehat{\rho }}_3\left(k\right)& ={\widehat{\rho }}_2(k+1)-{\widehat{\rho }}_3\left(k\right)=q_3(-{\epsilon }_3{\widehat{\rho }}_3\left(k\right)+\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel ),\hfill \end{array}\hfill \end{array}$$

(36)

in which$q_1$,$q_2$,$q_3$,${\epsilon }_1$,${\epsilon }_2$, and${\epsilon }_3$are positive scalars chosen by the designer, and${\widehat{\rho }}_1\left(k\right)$,${\widehat{\rho }}_2\left(k\right)$, and${\widehat{\rho }}_3\left(k\right)$are the estimates of${\rho }_1$,${\rho }_2$, and${\rho }_3$, respectively.

Proof. 

Let us choose the Lyapunov function $V_s({\mathit{s}}_{\mathit{e}}\left(k\right),{\overline{r}}_k)=(1/2){\mathit{s}}_{\mathit{e}}^{\top }\left(k\right){\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{s}}_{\mathit{e}}\left(k\right)+(1/2q_1){\tilde{\rho }}_1^2+(1/2q_2){\tilde{\rho }}_2^2+(1/2q_3){\tilde{\rho }}_3^2$ as a candidate for the system formulated in (14), where ${\tilde{\rho }}_1={\rho }_1-{\widehat{\rho }}_1\left(k\right)$, ${\tilde{\rho }}_2={\rho }_2-{\widehat{\rho }}_2\left(k\right)$, and ${\tilde{\rho }}_3={\rho }_3-{\widehat{\rho }}_3\left(k\right)$. Then, we have that:

$$\begin{array}{cc}\hfill \Delta V_s({\mathit{s}}_{\mathit{e}}\left(k\right),p)=& \mathbb{E}\left(V_s({\mathit{s}}_{\mathit{e}}(k+1),{\overline{r}}_{k+1})-V_s({\mathit{s}}_{\mathit{e}}\left(k\right),{\overline{r}}_k)\right)\hfill \\ \hfill =& {\mathit{s}}_{\mathit{e}}^{\top }\left(k\right){\mathbf{\Pi }}^p\Delta {\mathit{s}}_{\mathit{e}}\left(k\right)+{\displaystyle \frac{1}{2}}{\mathit{s}}_{\mathit{e}}^{\top }\left(k\right)\sum _{q=1}^N{\pi }_{pq}{\left({\mathit{G}}_{\mathit{e}}^q{\mathit{B}}_2^q\right)}^{-1}{\mathit{s}}_{\mathit{e}}\left(k\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{2}}{\mathit{s}}_{\mathit{e}}^{\top }\left(k\right){\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{s}}_{\mathit{e}}\left(k\right)-{\displaystyle \frac{1}{q_1}}{\tilde{\rho }}_1\Delta {\widehat{\rho }}_1\left(k\right)-{\displaystyle \frac{1}{q_2}}{\tilde{\rho }}_2\Delta {\widehat{\rho }}_2\left(k\right)-{\displaystyle \frac{1}{q_3}}{\tilde{\rho }}_3\Delta {\widehat{\rho }}_3\left(k\right)+{\mathsf{\Psi }}_0\hfill \\ \hfill =& {\mathit{s}}_{\mathit{e}}^{\top }\left(k\right){\Pi }^p\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)(\nu \left(k\right)+{\mathit{f}}_{}^p(k,\mathit{x}\left(k\right))+{\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{G}}_{\mathit{e}}^p({\mathit{A}}_{\varpi }^p\mathit{e}\left(k\right)\hfill \\ \hfill & +\Delta {\mathit{A}}_h^p\mathit{x}\left(k\right)+({\mathit{A}}_h^p-{\mathit{A}}_{\varpi }^p)\mathit{x}\left(k\right)+{\mathit{B}}_{1h}^p\mathit{w}\left(k\right)\left)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{2}}{\mathit{s}}_{\mathit{e}}^{\top }\left(k\right)\sum _{q=1}^N{\pi }_{pq}{\left({\mathit{G}}_{\mathit{e}}^q{\mathit{B}}_2^q\right)}^{-1}{\mathit{s}}_{\mathit{e}}\left(k\right)-{\displaystyle \frac{1}{2}}{\mathit{s}}_{\mathit{e}}^{\top }\left(k\right){\left({\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p\right)}^{-1}{\mathit{s}}_{\mathit{e}}\left(k\right)\hfill \\ \hfill & -{\displaystyle \frac{1}{q_1}}{\tilde{\rho }}_1\Delta {\widehat{\rho }}_1\left(k\right)-{\displaystyle \frac{1}{q_2}}{\tilde{\rho }}_2\Delta {\widehat{\rho }}_2\left(k\right)-{\displaystyle \frac{1}{q_3}}{\tilde{\rho }}_3\Delta {\widehat{\rho }}_3\left(k\right)+{\mathsf{\Psi }}_0\hfill \\ \hfill \le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)-\left({\kappa }^p-{\mathsf{\Lambda }}_1^p\right)\parallel {\mathit{s}}_{\mathit{e}}{\left(k\right)\parallel }^2+\mathsf{\Lambda }\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel \hfill \\ \hfill & -{\tilde{\rho }}_1\left(-{\epsilon }_1{\widehat{\rho }}_1\left(k\right)+\parallel \mathit{y}\left(k\right)\parallel \parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel \right)-{\tilde{\rho }}_2\left(-{\epsilon }_2{\widehat{\rho }}_2\left(k\right)+\parallel \widehat{\mathit{y}}\left(k\right)\parallel \parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel \right)\hfill \\ \hfill & -{\tilde{\rho }}_3\left(-{\epsilon }_3{\widehat{\rho }}_3\left(k\right)+\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel \right)-\widehat{\alpha }\left(k\right){{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)sat\left({\mathit{s}}_{\mathit{e}}\left(k\right)\right)+{\mathsf{\Psi }}_0\hfill \\ \hfill \le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)+{\epsilon }_1{\tilde{\rho }}_1{\widehat{\rho }}_1\left(k\right)+{\epsilon }_2{\tilde{\rho }}_2{\widehat{\rho }}_2\left(k\right)+{\epsilon }_3{\tilde{\rho }}_3{\widehat{\rho }}_3\left(k\right)\hfill \\ \hfill & +\widehat{\alpha }\left(k\right)\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel -\widehat{\alpha }\left(k\right){{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)sat\left({\mathit{s}}_{\mathit{e}}\left(k\right)\right)+{\mathsf{\Psi }}_0.\hfill \end{array}$$

If $\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel >\sigma$, we get that:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill \mathbb{E}(\Delta V_{\mathit{s}}\left(k\right))\le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)+{\epsilon }_1{\tilde{\rho }}_1{\widehat{\rho }}_1\left(k\right)+{\epsilon }_2{\tilde{\rho }}_2{\widehat{\rho }}_2\left(k\right)+{\epsilon }_3{\tilde{\rho }}_3{\widehat{\rho }}_3\left(k\right)+{\mathsf{\Psi }}_0\hfill \\ \hfill \le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)-{\epsilon }_1{({\widehat{\rho }}_1\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_1)}^2-{\epsilon }_2{({\widehat{\rho }}_2\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_2)}^2-{\epsilon }_3{({\widehat{\rho }}_3\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_3)}^2+{\mathsf{\Psi }}_1.\hfill \end{array}\hfill \end{array}$$

If $\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel <\sigma$, we reach at:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill \mathbb{E}(\Delta V_{\mathit{s}}\left(k\right))\le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)+{\epsilon }_1{\tilde{\rho }}_1{\widehat{\rho }}_1\left(k\right)+{\epsilon }_2{\tilde{\rho }}_2{\widehat{\rho }}_2\left(k\right)+{\epsilon }_3{\tilde{\rho }}_3{\widehat{\rho }}_3\left(k\right)\hfill \\ \hfill & +{\displaystyle \frac{\widehat{\alpha }\left(k\right)}{\sigma }}\left(\sigma \parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel -\parallel {\mathit{s}}_{\mathit{e}}{\left(k\right)\parallel }^2\right)+{\mathsf{\Psi }}_0\hfill \\ \hfill \le & -{{\mathit{s}}_{\mathit{e}}}^{\top }\left(k\right)\mathsf{\Sigma }{\mathit{s}}_{\mathit{e}}\left(k\right)-{\epsilon }_1{\left({\widehat{\rho }}_1\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_1\right)}^2-{\epsilon }_2{\left({\widehat{\rho }}_2\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_2\right)}^2-{\epsilon }_3{\left({\widehat{\rho }}_3\left(k\right)-{\displaystyle \frac{1}{2}}{\rho }_3\right)}^2\hfill \\ \hfill & -{\displaystyle \frac{\widehat{\alpha }\left(k\right)}{\sigma }}{\left(\parallel {\mathit{s}}_{\mathit{e}}\left(k\right)\parallel -\frac{1}{2}\sigma \right)}^2+{\mathsf{\Psi }}_2.\hfill \end{array}\hfill \end{array}$$

If ${\mathit{s}}_{\mathit{e}}\left(k\right)$ exceeds a certain bounded region that contains the equilibrium point, $\mathsf{\Sigma }$ can be selected appropriately, such that $\Delta V_s<0$, where ${\mathsf{\Psi }}_0=(1/2)\Delta {\mathit{s}}^{\top }\left(k\right){\mathit{X}}^p\Delta \mathit{s}\left(k\right)$, ${\mathsf{\Psi }}_1=(1/4){\epsilon }_1{\rho }_1^2+(1/4){\epsilon }_2{\rho }_2^2+(1/4){\epsilon }_3{\rho }_2^3+{\mathsf{\Psi }}_0$, and ${\mathsf{\Psi }}_2={\mathsf{\Psi }}_1+(1/4){\sigma }^2$.

Selecting the Lyapunov function $V_c(\mathit{s}\left(k\right),{\overline{r}}_k)={\mathit{s}}^{\top }\left(k\right)\mathit{s}\left(k\right)$ as a candidate for the system defined in (15), we attain at:

$$\Delta V_c(\mathit{s}\left(k\right),i)=\mathbb{E}(V_c(\mathit{s}(k+1),{\overline{r}}_{k+1})-V_c(\mathit{s}\left(k\right),{\overline{r}}_k))={\mathit{s}}^{\top }(k+1)\mathit{s}(k+1)-{\mathit{s}}^{\top }\left(k\right)\mathit{s}\left(k\right).$$

Therefore, using (15) and (34), we obtain:

$$\begin{array}{cc}\hfill & \begin{array}{cc}\hfill \Delta V_c(\mathit{s}\left(k\right),i)& ={\displaystyle \frac{1}{4}}{\mathit{s}}^{\top }\left(k\right)\mathit{s}\left(k\right)-{\mathit{s}}^{\top }\left(k\right)\mathit{s}\left(k\right)=-{\displaystyle \frac{3}{4}}{\parallel \mathit{s}\left(k\right)\parallel }^2<0,\forall \parallel \mathit{s}\left(k\right)\parallel \ne 0.\hfill \end{array}\hfill \end{array}$$

   □

4. Numerical Applications

In this section, we detail the computational framework used, an algorithm and its corresponding flowchart, and demonstrate the effectiveness and advantages of the proposed method by mean of a descriptor single-species bio-economic system, and a tunnel diode circuit.

4.1. Computational Framework and Algorithm

The computational experiments were carried out in the Matlab programming language using a computer with the following characteristics: (i) [OS] Windows 10 Enterprise for 64 bits; (ii) [RAM] 8 Gigabytes; and (iii) [Processor] Intel(R) Core(TM) i7-4790T CPU @ 2.70 GigaHertz.

After the rigorous development for designing the adaptive SMC law, the detailed procedure is summarized in Algorithm 1, whereas Figure 1 displays a flowchart that gives a clear description of the proposed control scheme. Algorithm 1 was executed by using the Yalmip software with the optimization toolbox mosek  [44].

Algorithm 1 Procedure design

1: Choose the membership functions of SMO such that ${\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right)\ge 0$ is verified. 2: Select ${\mathit{G}}_{\mathit{e}}^p$, ${\mathit{G}}_{}^p$ such that ${\mathit{G}}_{\mathit{e}}^p{\mathit{B}}_2^p$ is non-singular and the rank of the constraint stated in (9) is verified. 3: Determine the gains ${\mathit{K}}_l^p$, ${\mathit{L}}_j^p$ of the LMI established in Theorem 2. 4: Design the observer indicated in (6) and the sliding functions given in (8)–(11). 5: Select suitable parameters ${\kappa }^p$ such that the condition formulated in (35) is verified. 6: Obtain the controller variables $\mathbf{\Sigma }$, ${\epsilon }_1$, $q_1$, ${\epsilon }_2$, $q_2$, ${\epsilon }_3$, and $q_3$ presented in (34)–(36). 7: Apply the designed control law expressed in (34)–(36) to the model.

4.2. Bio-Economic System

Consider the bio-economic system described in [45] and stated as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{z}}_1\left(t\right)\hfill & =-0.5z_1\left(t\right)+0.15z_2\left(t\right)-0.01z_1^2\left(t\right)-E\left(t\right)z_1\left(t\right)+u_1\left(t\right),\hfill \\ {\dot{z}}_2\left(t\right)\hfill & =0.5z_1\left(t\right)-0.1z_2\left(t\right),\hfill \\ 0\hfill & =E\left(t\right)(d_{\overline{r}\left(t\right)}{z}_1\left(t\right)-50)+u_2\left(t\right),\hfill \end{array}\right.\end{array}$$

(37)

with $z_1\left(t\right)$ and $z_2\left(t\right)$ being the population density of immature and mature species at time t. In addition, $E\left(t\right)$ corresponds to the harvest effort on the immature population, $u_1\left(t\right)$ and $u_2\left(t\right)$ state, respectively, the capture of an immature population and the government regulation, by means of a tax or subsidy, of a biological resource. Furthermore, $d_{r\left(t\right)}$ represents a coefficient governed by a Markov process related to $r\left(t\right)$ at the time t, for the price per the individual population. Based on the model established in (37), by translating the positive equilibriums to zero [46], we obtain the model stated as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\dot{x}}_1\left(t\right)\hfill & =a_{\overline{r}\left(t\right)}{x}_1\left(t\right)+0.15x_2\left(t\right)+b_{\overline{r}\left(t\right)}{x}_3\left(t\right)-0.01x_1^2\left(t\right)-x_1\left(t\right)x_3\left(t\right)+u_1\left(t\right),\hfill \\ {\dot{x}}_2\left(t\right)\hfill & =0.5x_1\left(t\right)-0.1x_2\left(t\right),\hfill \\ 0\hfill & =c_{\overline{r}\left(t\right)}{x}_1\left(t\right)+d_{\overline{r}\left(t\right)}{x}_1\left(t\right)x_3\left(t\right)+u_2\left(t\right).\hfill \end{array}\right.\end{array}$$

(38)

By employing the Euler discretization method, the discrete-time mathematical model of the bio-economic system formulated in (38) is obtained as:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x_1(k+1)=& (1+a_{\overline{r}\left(k\right)}{T}_s)x_1\left(k\right)+0.15T_s{x}_2\left(k\right)+b_{\overline{r}\left(k\right)}{T}_s{x}_3\left(k\right)\hfill \\ & -0.01T_s{x}_1^2\left(k\right)-T_s{x}_1\left(k\right)x_3\left(k\right)+T_s{u}_1\left(k\right),\hfill \\ \hfill x_2(k+1)=& 0.5T_s{x}_1\left(k\right)(1-0.1T_s)x_2\left(k\right),\hfill \\ \hfill 0=& (c_{\overline{r}\left(k\right)}+0.01\sin \left(k\right))x_1\left(k\right)+d_{\overline{r}\left(k\right)}{x}_1\left(k\right)x_3\left(k\right)+u_2\left(k\right),\hfill \end{array}\right.\end{array}$$

(39)

where $T_s=0.1s$, $\overline{r}\left(k\right)\in \{1,2\}$, with $a_1=-1.25$, $a_2=-1.67$, $b_1=-50$, $b_2=-41.67$, $c_1=-0.75$, $c_2=-0.8$, $d_1=1$, and $d_2=1.2$. The Markov transition matrix is given by:

$$\overline{\mathbf{\Pi }}=\left[\begin{array}{cc}0.8& 0.2\\ 0.35& 0.65\end{array}\right].$$

Assume that $|x_1\left(k\right)|\le 10$. Using the sector non-linearity technique, the non-linear system stated in (39) may be described by a Markov jump TS fuzzy descriptor model, where the related matrices are expressed by:

$$\begin{array}{c}\hfill {\mathit{A}}_i^p=\left[\begin{array}{ccc}1+(a_p+0.1)T_s& 0.15T_s& {\mathit{A}}_{13i}^p\\ 0.5T_s& 1-0.1T_s& 0\\ c_p& 0& {\mathit{A}}_{33i}^p\end{array}\right],{\mathit{B}}_2^p=\left[\begin{array}{cc}{T}_s& 0\\ 0& 0\\ 0& 1\end{array}\right],{\mathit{B}}_{1i}^p=\left[\begin{array}{c}{T}_s\\ T_s\\ T_s\end{array}\right],{\mathit{C}}_2^p=\left[\begin{array}{ccc}1& 0& -1\\ 1& 1& 0\\ 2& 1& 1\end{array}\right],\end{array}$$

${\mathit{A}}_{131}^p=(b_p+10)T_s$, ${\mathit{A}}_{132}^p=(b_p-10)T_s$, ${\mathit{A}}_{331}^p=-10d_p$, ${\mathit{A}}_{332}^p=10d_p$, and $\mathit{E}=diag(1,1,0)$. The remaining parameters are selected as:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left[\begin{array}{cc}{\mathit{M}}_1^p& {\mathit{M}}_2^p\\ \\ {\mathit{N}}_1^p& {\mathit{N}}_2^p\\ \\ {\mathit{C}}_{11}^p& {\mathit{C}}_{12}^p\\ \\ {\mathit{D}}_1^p& {\mathit{D}}_2^p\end{array}\right]=\left[\begin{array}{cc}{\left[\begin{array}{ccc}0.1,& 0,& 0\end{array}\right]}^{\top }& {\left[\begin{array}{ccc}0.1,& 0,& 0\end{array}\right]}^{\top }\\ \\ \left[\begin{array}{ccc}0,& 0,& 0.1\end{array}\right]& \left[\begin{array}{ccc}0,& 0,& 0.12\end{array}\right]\\ \\ \left[\begin{array}{ccc}0.15,& 0.1,& 0.02\end{array}\right]& \left[\begin{array}{ccc}0.2,& 0.2,& -0.2\end{array}\right]\\ \\ 0.01& -0.02\end{array}\right].\hfill \end{array}\end{array}$$

The membership functions of the fuzzy subsystems are $h_1\left(x_1\left(k\right)\right)=(1-0.1x_1\left(k\right))/2$ and $h_2\left(x_1\left(k\right)\right)=(1+0.1x_1\left(k\right))/2$. The membership functions are chosen as: ${\varpi }_1\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)={\mathrm{e}}^{-x_1^2/5}$ and ${\varpi }_2\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)=1-{\mathrm{e}}^{-x_1^2/5}$. For ${\tau }_1={\tau }_2=0.9$, ${\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right)\ge 0$ is verified. Let

$${\mathit{G}}_{\mathit{e}}^p=\left[\begin{array}{ccc}5& -5& 5\\ -6& 18& -6\end{array}\right],{\mathit{R}}_0=diag(0,0,1).$$

Then, by solving the problem formulated in (33), a feasible solution can be obtained with minimum values of $\alpha =-0.51039$, $\beta =0.10231$, and the design parameters given by:

$${\mathit{K}}_1^1=\left[\begin{array}{ccc}25.659& 24.774& 85.668\\ -722.04& -723.54& -741.05\end{array}\right],$$

$${\mathit{K}}_2^1=\left[\begin{array}{ccc}27.179& 26.294& 87.171\\ -755.34& -756.92& -775.67\end{array}\right],$$

$${\mathit{K}}_1^2=\left[\begin{array}{ccc}35.494& 34.172& 86.864\\ -758.3& -759.11& -777.53\end{array}\right],$$

$${\mathit{K}}_2^2=\left[\begin{array}{ccc}37.046& 35.717& 88.396\\ -795.79& -796.81& -816.49\end{array}\right],$$

$${\mathit{L}}_1^1=\left[\begin{array}{ccc}2.3369& 0.90341& -1.0727\\ -2.466& -1.0971& 1.2566\\ -9.2263& -10.825& 7.8098\end{array}\right],$$

$${\mathit{L}}_2^1=\left[\begin{array}{ccc}2.4901& 0.94202& -1.2913\\ -2.394& -0.8324& 1.1523\\ -10.553& -11.718& 9.0278\end{array}\right],$$

$${\mathit{L}}_1^2=\left[\begin{array}{ccc}2.1987& 0.7992& -0.90291\\ -2.241& -0.85986& 0.99334\\ -8.3025& -10.25& 6.8911\end{array}\right],$$

$${\mathit{L}}_2^2=\left[\begin{array}{ccc}2.3267& 0.79629& -1.1006\\ -2.2353& -0.67524& 0.96386\\ -9.7363& -11.196& 8.2478\end{array}\right].$$

Set $\sigma =0.1$, $\mathsf{\Sigma }=0.1$, ${\epsilon }_1=0.1$, ${\epsilon }_2=0.15$, ${\epsilon }_3=0.25$, and $q_1=q_2=q_3=0.00001$. Then, the sliding surface functions and SMC laws are designed according to (8), (11), and (34)–(36). The simulation is conducted with the exogenous disturbance $\mathit{w}\left(k\right)={\mathrm{e}}^{-1.5k}\sin \left(2k\right)$ and the non-linear function defined as ${\mathit{f}}_i^p(k,\mathit{x}\left(k\right))=[{\delta }^{p}sin\left(x_2\left(k\right)\right)$, ${\delta }^p{\mathrm{e}}^{-k}{)cos(\parallel \mathit{x}\left(k\right)\parallel \left)\right]}^{\top }$, for $i\in \{1,2\}$ and $p\in \{1,2\}$. Assume ${\delta }^1=0.1$ and ${\delta }^2=0.2$. The results of the numerical simulation under 50 independent trials are provided in Figure 2, from where we represent the system dynamics and its related observer state variables. The dynamics of the sliding surface functions and the control input signal defined as in (34)–(36) are provided in Figure 3. This study assumes that the initial conditions are such that $\mathit{x}\left(0\right)={[0.9,-0.4,0]}^{\top }$ and the mode jump time sequence is represented in Figure 4a. However, the adaptive variable is shown in Figure 4b. We observe that the system dynamics are stabilized despite external uncertainties and disturbances. Moreover, these results confirm the relevance of the proposed robust SM control scheme for the considered class of systems where the state variables are inaccessible.

4.3. A Tunnel Diode Circuit

A tunnel diode circuit with external disturbance, as shown in Figure 5a, is adopted in this example with parameters: $C=20$ mF (capacitor in milli Farad), $L=1000$ mH (inductor in milli Henry), and $R=25\mathsf{\Omega }$ (resistor in ohm).

The tunnel diode is characterized by a model expressed as:

$$i_D\left(t\right)=0.002v_D\left(t\right)+{\alpha }_0{v}_D^3\left(t\right),$$

where ${\alpha }_0$ is the characteristic parameter.

As in [47], by setting $x_1\left(k\right)=v_C\left(k\right)$, $x_2\left(k\right)=i_L\left(k\right)$, and $x_3\left(k\right)=-50{\alpha }_0{T}_s(x_1^3\left(k\right)+w\left(k\right))$ as the state variables, the discrete-time mathematical model of the circuit is given by:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill x_1(k+1)& =(1-0.1T_s)x_1\left(k\right)+50T_s{x}_2\left(k\right)+x_3\left(k\right),\hfill \\ \hfill x_2(k+1)& =-T_s{x}_1\left(k\right)+(1-10T_s)x_2\left(k\right)+T_{s}u\left(k\right)+0.1w\left(k\right),\hfill \\ \hfill 0& =50T_s{\alpha }_0{x}_1^3\left(k\right)+x_3\left(k\right)+50T_s\alpha w\left(k\right)),\hfill \end{array}\right.\end{array}$$

(40)

where $T_s$ is the sampling time and $w\left(k\right)$ denotes the disturbance input, with the parameter $\alpha$ varying according to

Table 2. Variation of $\alpha$.

Mode p	${\mathbf{\alpha }}_0$
1	0.01
3	0.05

.

Accordingly, the nominal transition probability matrix relating the operation modes is defined as:

$$\overline{\mathbf{\Pi }}=\left[\begin{array}{cc}0.625& 0.375\\ 0.315& 0.685\end{array}\right].$$

Suppose that $|x_1\left(k\right)|\le 3$. Using the sector non-linearity technique, the non-linear circuit model defined in (40) can be transformed into an equivalent Markov jump TS fuzzy descriptor model with two rules ($r\in \{1,2\}$) for each one of three modes ($p\in \{1,2,3\}$). The membership functions are depicted in Figure 5b, and with $T_s=0.01$, the data of the model are given by:

$${\mathit{A}}_i^p=\left[\begin{array}{ccc}(1-0.1T_s)& 50T_s& 1\\ -T_s& (1-10T_s)& 0\\ a_i& 0& 1\end{array}\right],$$

$${\mathit{B}}_2^p=\left[\begin{array}{c}0\\ T_s\\ 0\end{array}\right],$$

$${\mathit{B}}_{1i}^p=\left[\begin{array}{c}0\\ 0.1\\ 50T_s{\alpha }_0\end{array}\right],$$

$\mathit{E}=diag(1,1,0)$, $a_1=0$, and $a_2=-450T_s{\alpha }_0$.

The membership functions are $h_1\left(x_1\left(k\right)\right)=\left(3-|x_1\left(k\right)|\right)/3$ and $h_2\left(x_1\left(k\right)\right)=\left(|x_1\left(k\right)|\right)/3$, whereas the other parameters are set as:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left[\begin{array}{cc}{\mathit{C}}_2^1& {\mathit{C}}_2^p\\ \\ {\mathit{M}}_1^p& {\mathit{M}}_2^p\\ \\ {\mathit{N}}_1^p& {\mathit{N}}_2^p\\ \\ {\mathit{C}}_{11}^p& {\mathit{C}}_{12}^p\\ \\ {\mathit{D}}_1^p& {\mathit{D}}_2^p\end{array}\right]=\left[\begin{array}{cc}\left[\begin{array}{ccc}1& 1& -1\\ 0& 1& 0\end{array}\right]& \left[\begin{array}{ccc}0& 1& -1\\ -1& 1& 0\end{array}\right]\\ \\ {\left[\begin{array}{ccc}0,& 0.5,& 0\end{array}\right]}^{\top }& {\left[\begin{array}{ccc}0,& 0.5,& 0\end{array}\right]}^{\top }\\ \\ \left[\begin{array}{ccc}0,& 0.5,& 0\end{array}\right]& \left[\begin{array}{ccc}0,& -0.6,& 0\end{array}\right]\\ \\ \left[\begin{array}{ccc}0.1,& 0.15,& 0.1\end{array}\right]& \left[\begin{array}{ccc}0.1,& 0.15,& 0.1\end{array}\right]\\ \\ 0.2& 0.2\end{array}\right].\hfill \end{array}\end{array}$$

The membership functions for the SMO are stated as: ${\varpi }_1\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)={\mathrm{e}}^{-x_1^2/5}$ and ${\varpi }_2\left(\mathbf{\mu }\left(\widehat{\mathit{x}}\right)\right)=1-{\mathrm{e}}^{-x_1^2/5}$. Note that, for ${\tau }_1={\tau }_2=0.6$, ${\varpi }_l\left(\widehat{\mathit{x}}\right)-{\tau }_l{h}_l\left(\widehat{\mathit{x}}\right)\ge 0$ is verified. Let

$${\mathbb{G}}_e^1=\left[\begin{array}{ccc}0,& {\displaystyle \frac{1}{T_s}},& 0\end{array}\right],{\mathbb{G}}_e^2=\left[\begin{array}{ccc}-{\displaystyle \frac{1}{T_s}},& {\displaystyle \frac{1}{T_s}},& 0\end{array}\right],$$

and $R_0=diag(0,0,10)$. By solving the problem stated in (33), a feasible solution can be obtained for $\alpha =-1$, $\beta =0.12$ and the design parameters defined as:

$$\begin{array}{cc}& \begin{array}{cccc}\hfill {\mathit{K}}_1^1& =\left[\begin{array}{ccc}73.892& -129.53& -74.22\end{array}\right],\hfill & \hfill {\mathit{K}}_2^1& =\left[\begin{array}{ccc}-71.951& -128.28& -75.066\end{array}\right],\hfill \\ \hfill {\mathit{K}}_1^2& =\left[\begin{array}{ccc}-82.05& -131.38& -60.526\end{array}\right],\hfill & \hfill {\mathit{K}}_2^2& =\left[\begin{array}{c}-68.338-126.9-61.994\end{array}\right],\hfill \\ \hfill {\mathit{L}}_1^1& =\left[\begin{array}{cc}-0.22298& -0.21216\\ 0.13906& 0.28188\\ -0.38721& -0.2313\end{array}\right],\hfill & \hfill {\mathit{L}}_2^1& =\left[\begin{array}{cc}-0.23296& -0.24552\\ 0.14305& 0.30993\\ -0.37305& -0.22916\end{array}\right],\hfill \\ \hfill {\mathit{L}}_1^2& =\left[\begin{array}{cc}-0.32382& -0.085824\\ 0.20298& 0.11664\\ -0.50337& 0.19951\end{array}\right],\hfill & \hfill {\mathit{L}}_2^2& =\left[\begin{array}{cc}-0.24674& -0.10949\\ 0.16607& 0.10029\\ -0.41125& 0.17809\end{array}\right].\hfill \end{array}\hfill \end{array}$$

In addition, the associated minimal $H_{\infty }$ performance index is computed as $\gamma =0.20087$.

Set ${\mathit{H}}^p=\left[\begin{array}{cc}0& 1/T_s\end{array}\right]$, and ${\mathit{G}}^p=\left[\begin{array}{ccc}0& 1/T_s& 0\end{array}\right]$. Then, the adaptive SMC law can be established referring to (34)–(36) with the adjustable parameters selected as $\mathsf{\Sigma }=0.15$, $\sigma =0.01$, ${\epsilon }_1=0,15$, ${\epsilon }_2=0.25{\epsilon }_3=0.25$, $q_1=0.00001$, $q_2=0.00001$, and $q_3=0.0001$. The exogenous disturbance and non-linear functions are, respectively, set as $w\left(k\right)=\mathrm{rand}\left(1\right)sin\left(k\right)/(k^2+5)$ and $f^p(k,x\left(k\right))=0.2sin(x_1\left(k\right)-2x_3\left(k\right))$, for $p\in \{1,2\}$.

Now, by using the control input signal stated in (34)–(36), the simulation results are shown in Figure 6 and Figure 7 for a possible switching signal $r\left(k\right)$ displayed in Figure 8a, and the initial condition $x\left(0\right)={[-2,2,1]}^{\top }.$ Among them, Figure 6a–c show the response times of the system and observer states. The evolution of the control signal, and sliding surface function are plotted in Figure 7a–c. The response time of the adaptive value is provided by Figure 8b (for the first 50 steps) in a trial. From Figure 6, Figure 7 and Figure 8, which are obtained based on 50 random independent trials, note that the circuit state variables converge to the steady state in 0.3 s, despite of the hard non-linear behavior that characterizes the model dynamics. Moreover, under the existence of external disturbances, and model parameter uncertainties, the asymptotic stability of the system is ensured. Note that an adequate selection of the adjustable parameters $\mathbf{\Sigma }$, ${\epsilon }_1$, $q_1$, ${\epsilon }_2$, $q_2$, ${\epsilon }_3$, and $q_3$ as described in (34)–(36) makes the system stable and attenuates the chattering phenomenon affecting the input signal. This selection can be achieved through an intensive numerical analysis. Observe that this delicate step can be defined in terms of an optimization problem, where a meta-heuristic technique can be implemented to compute the above mentioned parameters accurately.

5. Discussion, Conclusions, Limitations, and Future Work

This section gathers a comparative discussion, conclusions, potential future perspectives, and limitations of the developed control strategy.

5.1. Comparative Discussion

There are numerous studies on control of descriptor systems in the literature so that several methods can be seen as comparable to the approach designed in the present article. However, our approach differs in different ways as follows:

• In [48], a method was presented to design an observer-based controller for fuzzy descriptors with partially measured states. However, in our approach, the problem is tackled in a more general way, since it allows us to design a sliding mode observer when the system is subject to constraints related to Markov jump switching and matched uncertainties. Indeed, we believe that the described approach in [48] does not achieve a satisfactory performance under the previous mentioned constraints.
• As discussed in [49], a non-PDC approach cannot be applied to discrete-time fuzzy descriptor systems with model specifications related to unmeasurable premise variables and imperfect membership functions. Nevertheless, in our work, these specifications are considered in the approach designed in the present study. Here, we believe that a numerical comparative study will not provide additional conclusion to the superiority of our scheme regarding the controller defined in [49].
• In [50,51], it was solved the problem of observer-based and reduced-order robust sliding mode control schemes in the continuous time domain. Nonetheless, this controller cannot be implemented in discrete-time fuzzy descriptor systems with uncertainties, whose discrete time domain was considered in our proposal. Both time domains are different, so a comparative study with our method is not viable.

5.2. Concluding Remarks

In the present study, a novel approach to stabilize the output of a descriptor system was developed. This approach is based on a sliding mode observer-based control for non-linear descriptor systems described by Markov jumping TS fuzzy models. The hypothesis of existence of uncertainties, perturbations, and unknown premise variables, on the dynamics of the studied system was successfully addressed. Additionally, a non-PDC scheme was used to handle the imperfectly matched premise between the SMO and TS fuzzy model. Considering the SMO gain matrix, two fuzzy switching manifolds were developed. Then, a relaxed MLI-based criterion was addressed to show that the sliding mode dynamics are robustly admissible under an $H_{\infty }$ performance. Our study allowed the synthesis of new adaptive sliding mode control laws for the error and SMO systems ensuring the reachability conditions. The proposed control scheme was validated by numerical simulations considering a bio-economic system and a tunnel diode circuit.

5.3. Limitations

Despite the power and performance of the synthesized technique, this imposes that the conservative assumption stated in (10) must be satisfied imperatively. Moreover, identifying the Markov chain transition probability is a delicate task in practice. Equally important, the synchronization between the modes of the controller, observer and model is a task that may be difficult to achieve in practice.

5.4. Future Work

Further research that may be focused from the present study is related to sliding mode control triggered by output feedback events for more complex systems. The class of semi-Markov non-linear jump systems with random time-varying delays, sensor/actuator saturation, and/or failure should be further investigated. As mentioned, it is known that the transition probabilities play an essential role in the analysis and modeling of Markov processes. In the present study, the transition probabilities were assumed to be known. However, in practical situations, such probabilities are unknown. Thus, the determination of partial and time-varying transition probabilities in our framework are an open problem to be addressed in the future.

As mentioned, a key point in controller scheme design is selecting the control parameters and so efficiently computing them in an accurate way. An adequate selection of the adjustable parameters makes the system stable and attenuates the chattering phenomenon affecting the input signal. This selection can be achieved through an intensive numerical analysis. Observe that this delicate step can be defined in terms of an optimization problem, where a meta-heuristic technique may be implemented to compute the above mentioned parameters accurately. Additionally, meta-heuristic learning algorithms [52,53] can be considered to improve the performance of our proposal. These algorithms are widely utilized in the digital era to solve several real-world problems, including neuroadaptive control of saturated non-linear systems with disturbance compensation [54], and neuroadaptive learning algorithm for constrained non-linear systems with disturbance rejection [55]. Implementing a meta-heuristic optimization technique is a promising perspective to this work. We hope to report findings on these issues in future works.