Observer-Based Robust H_∞ Control for Stochastic Markov Jump Delay Systems Through Dual Adaptive Sliding Mode Approach

Abstract

This study presents an approach to enhancing the robustness of $H_{\infty }$ control in It$\widehat{o}$-type stochastic Markov jump systems, addressing uncertainties in parameters, time-varying delays, and nonlinear perturbations. In this study, the nonlinearities are not exactly known, so an adaptive control strategy is employed. Firstly, an adaptive state observer of full dimension is constructed along with the derivation of error dynamics. Subsequently, different from traditional methods, two linear sliding surfaces are designed, respectively, for the state observer system and error dynamics, resulting in two sliding mode dynamics. Secondly, by employing the linear matrix inequality (LMI) method, sufficient conditions are established to ensure mean-square exponential stability of the closed-loop systems, including observer sliding mode dynamics and error sliding mode dynamics, along with an $H_{\infty }$ attenuation performance index $\gamma$. Thirdly, adaptive sliding mode controllers are proposed, ensuring the finite-time arrival and maintenance of the established sliding surfaces. Finally, the efficacy of the derived outcomes is illustrated utilizing the Tunnel Diode circuit model as a demonstrative case study. In this example, the system’s state responses, sliding surface functions, control input, and estimated parameters are simulated under different operating modes and external disturbances. The results demonstrate that the proposed adaptive sliding mode control strategy ensures faster and better convergence compared to error dynamics without control.

1. Introduction

Recently, Markov jump systems (MJSs) [1] have garnered considerable interest due to their wide real-world applications; for instance, RLC circuit [2], virus mutation treatment [3], wheeled mobile manipulators [4], and networked control systems [5]. MJSs serve as a versatile framework for modeling various physical systems that undergo sudden structural changes, such as those caused by unexpected environmental disturbances or shifts in the operational parameters of nonlinear systems. There has been considerable research conducted within the realm of MJSs, encompassing areas like robust $H_{\infty }$ filtering [6], security control [7], passivity analysis [8], state synchronization control [9], resilient sampled-data control [10], and stability analysis [11], among other topics. However, despite their widespread application in engineering, MJSs have certain limitations. In previous research, it is commonly assumed that the duration of stay within each subsystem follows an exponential distribution; that is, the sojourn time for MJSs relies on the constant transition rates (TRs). Recently, semi-Markov jump systems [12] offer a more general framework since the TRs are dependent on the sojourn time, rather than remaining constant. However, it remains challenging to address the unresolved issues in MJSs, and the reported results for MJSs are still representative and beneficial for the study of S-MJSs.

Sliding mode control (SMC) [13] is a prominent and widely recognized field within the domain of dynamic systems research. It has found widespread application in various complex systems, including robots [14], underwater propellers [15], and power systems [16]. A key benefit of SMC is its capacity to adeptly counteract unpredictable fluctuations and extrinsic interferences by incorporating discontinuous terms within the variable structure controllers. This feature has attracted significant attention and has been extensively explored, as shown in [17] and its references. Moreover, the application of SMC to MJSs or S-MJSs has also garnered considerable interest due to its favorable properties. Some notable contributions to the field include the theoretical approach to stochastic admissibility analysis for singular MJSs shown in [18]. The study of exponential stability analysis for S-MJSs by employing a sliding mode observer framework was discussed in [19]. Particularly noteworthy is the work presented in [19], where phase-type S-MJSs were formulated in a manner similar to MJSs, thus enabling a parallel investigation based on the same principles applied to MJSs. However, most existing literature on SMC design focuses on a single sliding surface, and the convergence of the error dynamics state is often negatively affected. Therefore, to ensure a rapid transient response of the system state, it is beneficial to control the error dynamics directly. This motivation has led us to investigate the design of dual SMC, both for the state observer and the error dynamics.

The robust $H_{\infty }$ SMC method has been widely studied due to its effectiveness in dealing with complex systems with nonlinearity, and is frequently applied to deal with practical engineering problems. Therefore, the combination of the robust $H_{\infty }$ SMC method with other techniques has emerged as a topic of considerable interest within the community in recent years. For example, in the context of discrete-time stochastic MJSs prone to packet dropouts, the authors have explored the robust $H_{\infty }$ SMC problem by designing a state observer in [20]. In [21], the authors have tackled the robust $H_{\infty }$ stability challenge for stochastic T-S fuzzy singular MJSs by devising an integral-type sliding surface function. It is well established that unknown perturbations or environmental noise can destabilize a system to a certain extent, with even minor vibrations potentially causing significant fluctuations. To date, for stochastic systems subject to perturbations, a number of excellent research methodologies have been proposed and applied across various fields; see, for instance, refs. [22,23,24] and references therein. In [25], the authors explored the issue of mean-square asymptotic stability for MJSs, addressing a less conservative approach with partially known transition probabilities by designing event-triggered SMC schemes. Concurrently, within the framework of the sliding mode observer, the author considered the controller design problem for MJSs in [26]. Recently, significant attention has been given to the development of suitable generalized functions for sliding mode surface, leading to the proposal of several outstanding methodologies. To the best of the authors’ knowledge, the challenge of designing adaptive controllers for $H_{\infty }$ sliding mode observers in the context of nonlinear stochastic MJSs remains largely unexplored, especially when it comes to the simultaneous design of dual sliding mode surface functions.

Motivated by the above discussion, for the It$\widehat{o}$-type stochastic MJSs involving parameter uncertainties, time-varying delays and nonlinear perturbations, we aim to design a novel adaptive SMC strategy to address the problem of $H_{\infty }$ robust control, employing a dual sliding mode surface approach. The primary contributions of this paper are listed as follows: (1) The problem of observer-based $H_{\infty }$ adaptive control for a class of It$\widehat{\mathrm{o}}$-type stochastic MJSs is explored along with system parameter uncertainties, nonlinearities and unknown state components; (2) Different from the existing results, a novel dual SMC scheme with two sliding surfaces is employed by the aid of a full-order state observer, which ensures better system performance; (3) Adaptive compensators are incorporated into the SMC law to remove the constrain of unknown variables, ensuring the finite-time reachability of state components onto the sliding surface and remaining sliding motion; (4) Sufficient conditions are developed for the closed-loop system, comprising both sliding mode dynamics and estimation error dynamics, to guarantee mean square exponential stability with an $H_{\infty }$ performance level $\gamma$. The organization of the remaining paper is as follows: Section 2 provides the preliminaries; Section 3 presents the main results, including the design of the state observer, dual sliding surfaces, $H_{\infty }$ performance analysis, and the design of the dual adaptive sliding mode controllers. A practical example is demonstrated in Section 4, and Section 5 concludes the paper.

Notation 1.

In the following, the symbol $\mathcal{E}$ represents the mathematical expectation; The asterisk ∗ signifies a symmetric element within a symmetric matrix; The heaviside matrix notation $\mathit{sym}\{\mathit{X}\}$ is used to express $\mathit{X}+{\mathit{X}}^T$; ${\lambda }_{max}\{\mathit{A}\}$ means the maximum eigenvalue of matrix $\mathit{A}$.

2. Problem Statement and Preliminaries

In this paper, $\{r_t,t\ge 0\}$ is employed to represent a continuous-time Markov process characterized by right-continuous paths, where $r_t$ takes discrete values within a set $\mathcal{S}=\{1,2,\cdots ,s\}$, accompanied by the transition probabilities as

$$\mathbf{Pr}\{r_{t+h}=j|r_t=i\}=\left\{\begin{array}{ccc}{\pi }_{ij}h+o(h),\hfill & if\hfill & i\ne j,\hfill \\ 1+{\pi }_{ii}h+o(h),\hfill & if\hfill & i=j,\hfill \end{array}\right.$$

(1)

where $h>0$ and the condition $\underset{h\to 0}{lim}o(h)/h=0$ holds, with ${\pi }_{ij}>0$ for $(i\ne j)$ indicating the TR from state i at time t to state j at the subsequent time increment $t+h$, and for each $i\in \mathcal{S}$, it follows ${\pi }_{ii}=-{\displaystyle \sum _{j\ne i}}{\pi }_{ij}<0$.

Now, let us examine the following stochastic MJSs:

$$\left\{\begin{array}{c}\mathrm{d}\mathit{x}(t)=[\mathit{A}(t,r_t)\mathit{x}(t)+{\mathit{A}}_{\tau }(t,r_t)\mathit{x}(t-\tau (t))+\mathit{B}(\mathit{\varphi }(\mathit{u}(t,r_t))+\mathit{f}(\mathit{x}(t),r_t))\hfill \\ +\mathit{D}(r_t)\mathit{v}(t)]\mathrm{d}t+\mathit{E}(r_t)\mathit{x}(t)\mathrm{d}\omega (t),\hfill \\ \mathit{y}(t)=\mathit{C}(r_t)\mathit{x}(t),\hfill \\ \mathit{x}(s)=\mathit{\psi }(s),s\in [-{\tau }_2,0],\hfill \end{array}\right.$$

(2)

where $\mathit{x}(t)\in {\mathbb{R}}^n$ denotes the system state vector, $\mathit{y}(t)\in {\mathbb{R}}^q$ signifies the measured output, $\mathit{\varphi }(\mathit{u}(t,r_t))\in {\mathbb{R}}^m$ embodies the system’s control input which is nonlinear, and $\mathit{v}(t)$, an element of ${\mathcal{L}}_2[0,+\infty ]$, stands for the exogenous disturbance. The matrices $\mathit{A}(t,r_t)=\mathit{A}(r_t)+\Delta \mathit{A}(r_t)$, ${\mathit{A}}_{\tau }(t,r_t)={\mathit{A}}_{\tau }(r_t)+\Delta {\mathit{A}}_{\tau }(r_t)$, $\mathit{C}(r_t)$, $\mathit{D}(r_t)$, $\mathit{E}(r_t)$ are all function matrices, and it is postulated that the matrix $\mathit{B}$ possesses full column rank. In addition, $\Delta \mathit{A}(r_t)$ and $\Delta {\mathit{A}}_{\tau }(r_t)$ are the system model uncertainties satisfying

$$[\Delta \mathit{A}(r_t)\Delta {\mathit{A}}_{\tau }(r_t)]=\mathit{M}(r_t)\mathit{F}(t)\left[\mathit{N}(r_t){\mathit{N}}_{\tau }(r_t)\right],$$

(3)

where $\mathit{M}(r_t)$, $\mathit{N}(r_t)$ and ${\mathit{N}}_{\tau }(r_t)$ are defined as known matrices, while $\mathit{F}(t$) is an unknown time-varying matrix that fulfills the condition ${\mathit{F}}^{\mathrm{T}}(t)\mathit{F}(t)\le \mathit{I}$. $\tau (t)$ represents the system time-varying delay that satisfies

$$0\le {\tau }_1\le \tau (t)\le {\tau }_2<+\infty ,\dot{\tau }(t)\le \mu <1,$$

(4)

where ${\tau }_1$, ${\tau }_2$, and $\mu$ are identified as constants that are already established. The term $\omega (t)$ represents a Brownian motion that is one-dimensional, which is characterized by the properties that its differential expectation $\mathcal{E}\{\mathrm{d}\omega (t)\}$ equals zero, and the expectation of its squared differential $\mathcal{E}\{\mathrm{d}{\omega }^2(t)\}$ is equivalent to $\mathrm{d}t$. The nonlinear function $\mathit{f}(\mathit{x}(t),r_t)$ satisfies

$$\parallel \mathit{f}(\mathit{x}(t),r_t)\parallel \le \alpha +\beta \parallel \mathit{y}(t)\parallel ,$$

(5)

with $\alpha$ and $\beta$ characterized as real-valued constants that remain unspecified.

Moving forward, for each potential mode $r_t=i$, the matrix $\mathit{A}(r_t)$ is referred to as ${\mathit{A}}_i$, the uncertainty $\Delta \mathit{A}(r_t)$ is characterized as $\Delta {\mathit{A}}_i$, and so on for the corresponding matrices.

Assumption 1.

Given an unknown constant $\delta >0$ that meets ${\mathbf{x}}^{\mathrm{T}}(t){\mathit{E}}^{\mathrm{T}}\mathit{E}\mathbf{x}(t)\le \delta \parallel \mathbf{y}(t)\parallel$, where δ is to be estimated by $\widehat{\delta }(t)$ with the estimation error defined by $\tilde{\delta }(t)=\widehat{\delta }(t)-\delta$.

Definition 1

([27]). The system (2), when not subjected to external forces with $\mathbf{v}(t)$ = 0, is deemed to be mean-square exponentially (MSE) stable for any prescribed initial condition $\mathit{\psi }(t)$ and initial operational mode $r_0$, provided that certain constants $a>0$ and $b>0$ can be identified to fulfill the subsequent criteria

$$\mathcal{E}\left\{\parallel \mathbf{x}(t,\mathit{\psi }(s),r_0){\parallel }^2\right\}<a{e}^{-bt}\underset{-{\tau }_2\le \theta \le 0}{sup}{\parallel \mathit{\varphi }(\theta )\parallel }^2.$$

(6)

Lemma 1

([28]). Given the matrices $\mathit{D}$ and $\mathit{H}$, with $\mathit{Y}$ being symmetric and $\mathit{F}$ fulfilling the condition ${\mathit{F}}^{\mathrm{T}}\mathit{F}⩽\mathit{I}$, the subsequent analysis can be conducted.

$$\mathit{Y}+\mathit{D}\mathit{F}\mathit{H}+{\mathit{H}}^{\mathrm{T}}{\mathit{F}}^{\mathrm{T}}{\mathit{D}}^{\mathrm{T}}<0,$$

provided that a positive scalar $\epsilon >0$ can be identified, the subsequent condition holds true.

$$\mathit{Y}+\epsilon \mathit{D}{\mathit{D}}^{\mathrm{T}}+{\epsilon }^{-1}\mathit{H}{\mathit{H}}^{\mathrm{T}}<0.$$

3. Main Results

In this part, particular attention will be paid to devising the state observer and analyzing the MSE stability for a closed-loop system with a given disturbance suppression level $\gamma$. Subsequently, the design of adaptive sliding mode controllers is discussed, ensuring the sliding motion.

3.1. Observer Design

Considering the system (2), a corresponding state observer is formulated as follows:

$$\left\{\begin{array}{c}\mathrm{d}\widehat{\mathit{x}}(t)=[{\mathit{A}}_i\widehat{\mathit{x}}(t)+{\mathit{A}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+\mathit{B}(\mathit{\varphi }({\mathit{u}}_{1i}(t))+{\mathit{v}}_s(t))\hfill \\ +{\mathit{L}}_i(\mathit{y}(t)-{\mathit{C}}_i\widehat{\mathit{x}}(t))]\mathrm{d}t,\hfill \\ \widehat{\mathit{y}}(t)={\mathit{C}}_i\widehat{\mathit{x}}(t),\hfill \\ \widehat{\mathit{x}}(s)=\mathit{\phi }(s),s\in [-{\tau }_2,0].\hfill \end{array}\right.$$

(7)

where $\widehat{\mathit{x}}(t)\in {\mathbb{R}}^n$ represents the estimated state of $\mathit{x}(t)$, $\widehat{\mathit{y}}(t)\in {\mathbb{R}}^q$ corresponds to the actual measured output, $\mathit{\varphi }({\mathit{u}}_{1i}(t))$ denotes the input function, ${\mathit{L}}_i$ is recognized as the gain matrix associated with the observer. ${\mathit{v}}_s(t)$ is an external feedback compensating control that can be used to counteract the negative effects of the unknown nonlinear variable ${\mathit{f}}_i(\mathit{x}(t))$.

According to systems (2) and (7), denote the state error vector as $\mathit{e}(t)=\mathit{x}(t)-\widehat{\mathit{x}}(t)$. Then, the dynamics of the estimation error can be derived as follows:

$$\left\{\begin{array}{c}\mathrm{d}\mathit{e}(t)=[({\mathit{A}}_i+\Delta {\mathit{A}}_i-{\mathit{L}}_i{\mathit{C}}_i)\mathit{e}(t)+({\mathit{A}}_{\tau i}+\Delta {\mathit{A}}_{\tau i})\mathit{e}(t-\tau (t))\hfill \\ +\Delta {\mathit{A}}_i\widehat{\mathit{x}}(t)+\Delta {\mathit{A}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+\mathit{B}\mathit{\varphi }({\mathit{u}}_{2i}(t))-\mathit{B}({\mathit{v}}_s(t)\hfill \\ -{\mathit{f}}_i(\mathit{x}(t))+{\mathit{D}}_i\mathit{v}(t)]\mathrm{d}t+{\mathit{E}}_i(\widehat{\mathit{x}}(t)+\mathit{e}(t))\mathrm{d}\omega (t),\hfill \\ {\mathit{y}}_e(t)={\mathit{C}}_i\mathit{e}(t),\hfill \\ \mathit{e}(s)=\mathit{\psi }(s)-\mathit{\phi }(s),s\in [-{\tau }_2,0],\hfill \end{array}\right.$$

(8)

where $\mathit{\varphi }({\mathit{u}}_{2i}(t))=\mathit{\varphi }({\mathit{u}}_i(t))-\mathit{\varphi }({\mathit{u}}_{1i}(t))$.

3.2. Sliding Surface Design

Firstly, the following sliding surface function is proposed for the observer system (7):

$$\mathit{s}(t)=\mathit{G}\widehat{\mathit{x}}(t)-\mathit{G}{\int }_0^t({\mathit{A}}_i+\mathit{B}{\mathit{K}}_i)\widehat{\mathit{x}}(s)\mathrm{d}s,$$

(9)

wherein the matrix $\mathit{G}\in {\mathbb{R}}^{m\times n}$ is selected to ensure that $\mathit{G}\mathit{B}$ is nonsingular, and the gain matrix ${\mathit{K}}_i\in {\mathbb{R}}^{m\times n}$ is to be specified subsequently.

It follows from (9) that

$$\dot{\mathit{s}}(t)=\mathit{G}\dot{\widehat{\mathit{x}}}(t)-\mathit{G}({\mathit{A}}_i+\mathit{B}{\mathit{K}}_i)\widehat{\mathit{x}}(t).$$

(10)

In accordance with the SMC theory [13], it has $\dot{\mathit{s}}(t)$ = 0 on the sliding surface. Then, combining system (7) with (10) derives the following equivalent control input:

$${\mathit{\varphi }}_{\mathrm{eq}}({\mathit{u}}_{1i}(t))=-{(\mathit{G}\mathit{B})}^{-1}\mathit{G}[{\mathit{A}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+{\mathit{L}}_i{\mathit{C}}_i\mathit{e}(t)]+{\mathit{K}}_i\widehat{\mathit{x}}(t)-{\mathit{v}}_s(t).$$

(11)

Substituting equivalent control (11) into system (7), one has the following sliding mode dynamics (SMD) for the observer (7):

$$\mathrm{d}\widehat{\mathit{x}}(t)=[({\mathit{A}}_i+\mathit{B}{\mathit{K}}_i)\widehat{\mathit{x}}(t)+{\tilde{\mathit{A}}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+{\tilde{\mathit{L}}}_i{\mathit{C}}_i\mathit{e}(t)]\mathrm{d}t.$$

(12)

where ${\tilde{\mathit{A}}}_{\tau i}={\mathit{B}}_I{\mathit{A}}_{\tau i}$, ${\tilde{\mathit{L}}}_i={\mathit{B}}_I{\mathit{L}}_i$, ${\mathit{B}}_I=\mathit{I}-\mathit{B}{(\mathit{G}\mathit{B})}^{-1}\mathit{G}$.

Remark 1.

Generally speaking, as mentioned in [29], although $\mathit{G}({\mathit{A}}_i+\mathit{B}{\mathit{K}}_i)$ is a switching item for every mode i, the integral $\mathit{G}{\int }_0^t({\mathit{A}}_i+\mathit{B}{\mathit{K}}_i)\widehat{\mathit{x}}(s)\mathrm{d}s$ is continuous at the jumping point. Therefore, it knows that $\mathit{s}(t)$ is still continuous when the switching occurs.

Secondly, for the state error dynamics (8), let us construct the following sliding surface function:

$${\mathit{s}}_e(t)=\mathit{X}\mathit{C}\mathit{e}(t),$$

(13)

where $\mathit{C}$ is the weighted sum of the output matrices, which is defined as $\mathit{C}\triangleq {\displaystyle \sum _{i=1}^s}{\alpha }_i{\mathit{C}}_i$, where $\mathit{X}$ is chosen such that $\mathit{X}\mathit{C}\mathit{B}$ is nonsingular.

Following error dynamics (8) and sliding surface funciton (13), it has

$$\mathrm{d}{\mathit{s}}_e(t)=\mathit{X}\mathit{C}\mathrm{d}\mathit{e}(t)=\mathcal{L}{\mathit{s}}_e(t)\mathrm{d}t+\mathit{X}\mathit{C}{\mathit{E}}_i[\widehat{\mathit{x}}(t)+\mathit{e}(t)]\mathrm{d}\omega (t),$$

(14)

where $\mathcal{L}{\mathit{s}}_e(t)=\mathit{X}\mathit{C}[({\mathit{A}}_i+\Delta {\mathit{A}}_i-{\mathit{L}}_i{\mathit{C}}_i)\mathit{e}(t)+({\mathit{A}}_{\tau i}+\Delta {\mathit{A}}_{\tau i})\mathit{e}(t-\tau (t))+\Delta {\mathit{A}}_i\widehat{\mathit{x}}(t)+\Delta {\mathit{A}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+\mathit{B}\mathit{\varphi }({\mathit{u}}_{2i}(t))-\mathit{B}({\mathit{v}}_s(t)-{\mathit{f}}_i(\mathit{x}(t)))+{\mathit{D}}_i\mathit{v}(t)]$.

Similarly, by the condition $\mathcal{E}\{\mathrm{d}{\mathit{s}}_e(t)/\mathrm{d}t\}=0$ according to SMC theory [13], one has the following equivalent control variable:

$$\begin{array}{cc}\hfill & {\mathit{\varphi }}_{\mathrm{eq}}({\mathit{u}}_{2i}(t))={\mathit{v}}_s(t)-{\mathit{f}}_i(\mathit{x}(t))-{(\mathit{X}\mathit{C}\mathit{B})}^{-1}\mathit{X}\mathit{C}[({\mathit{A}}_i+\Delta {\mathit{A}}_i-{\mathit{L}}_i{\mathit{C}}_i)\mathit{e}(t)\hfill \\ \hfill & +({\mathit{A}}_{\tau i}+\Delta {\mathit{A}}_{\tau i})\mathit{e}(t-\tau (t))+\Delta {\mathit{A}}_i\widehat{\mathit{x}}(t)\hfill \\ \hfill & +\Delta {\mathit{A}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t))+{\mathit{D}}_i\mathit{v}(t)]\hfill \end{array}$$

(15)

Substituting equivalent control (15) into error dynamics (8), the error SMD is derived as follows:

$$\mathrm{d}\mathit{e}(t)=[{\mathit{g}}_i(t)+{\overline{\mathit{D}}}_i\mathit{v}(t)]\mathrm{d}t+{\mathit{h}}_i(t)\mathrm{d}\omega (t),$$

(16)

where ${\mathit{g}}_i(t)=({\overline{\mathit{A}}}_i+\Delta {\overline{\mathit{A}}}_i-{\overline{\mathit{L}}}_i{\mathit{C}}_i)\mathit{e}(t)+({\overline{\mathit{A}}}_{\tau i}+\Delta {\overline{\mathit{A}}}_{\tau i})\mathit{e}(t-\tau (t))+\Delta {\overline{\mathit{A}}}_i\widehat{\mathit{x}}(t)+\Delta {\overline{\mathit{A}}}_{\tau i}\widehat{\mathit{x}}(t-\tau (t)),$ ${\mathit{h}}_i(t)={\mathit{E}}_i(\widehat{\mathit{x}}(t)+\mathit{e}(t)),$ with ${\overline{\mathit{A}}}_i={\mathit{C}}_I{\mathit{A}}_i$, $\Delta {\overline{\mathit{A}}}_i={\mathit{C}}_I\Delta {\mathit{A}}_i$, ${\overline{\mathit{A}}}_{\tau i}={\mathit{C}}_I{\mathit{A}}_{\tau i}$, $\Delta {\overline{\mathit{A}}}_{\tau i}={\mathit{C}}_I\Delta {\mathit{A}}_{\tau i}$, ${\overline{\mathit{D}}}_i={\mathit{C}}_I{\mathit{D}}_i$, ${\overline{\mathit{L}}}_i={\mathit{C}}_I{\mathit{L}}_i$, ${\mathit{C}}_I=\mathit{I}-\mathit{B}{(\mathit{X}\mathit{C}\mathit{B})}^{-1}\mathit{X}\mathit{C}$.

Thus far, the objectives of this paper can be summarized as follows:

(1)

The closed-loop system, comprising SMDs (12) and (16), is shown to be robustly MSE stable under the condition $\mathit{v}(t)$ = 0, encompassing all inaccuracies that are deemed allowable.

(2)

The subsequent $H_{\infty }$ performance criterion with index $\gamma$ is maintained for every $\mathit{v}(t)$ that belongs to the ${\mathcal{L}}_2[0,+\infty ]$ space, that is

$$\mathcal{E}\left\{{\int }_0^{+\infty }{\mathit{y}}_e^{\mathrm{T}}(s){\mathit{y}}_e(s)\mathrm{d}s\right\}\le {\gamma }^2{\int }_0^{+\infty }{\mathit{v}}^{\mathrm{T}}(s)\mathit{v}(s)\mathrm{d}s.$$

(17)

Now, this section presents an important lemma before proceeding to the main results.

Lemma 2

([30]). Considering the SMD (16) and given that $\mathbf{v}(t)$ = 0, it is established that there are specific symmetric matrices ${\mathbf{R}}_1$ > 0, ${\mathbf{R}}_2$ > 0, as well as matrices ${\tilde{\mathbf{U}}}_{ji}(j=1,2,3)$, which ensure the fulfillment of the ensuing inequality across all operational modes $i\in \mathcal{N}$.

$$\begin{array}{cc}\hfill & -{\int }_{t-{\tau }_2}^t{\mathbf{g}}^{\mathrm{T}}(s,r_s){\mathbf{R}}_1\mathbf{g}(s,r_s)\mathrm{d}s\hfill \\ \hfill & \le 2{\eta }^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{1i}^{\mathrm{T}}(\mathbf{e}(t)-\mathbf{e}(t-{\tau }_1))+2{\eta }^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{2i}^{\mathrm{T}}(\mathbf{e}(t-{\tau }_1)-\mathbf{e}(t-\tau (t)))\hfill \\ \hfill & +2{\eta }^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{3i}^{\mathrm{T}}(\mathbf{e}(t-\tau (t))-\mathbf{e}(t-{\tau }_2))+{\tau }_1{\mathit{\eta }}^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{1i}^{\mathrm{T}}{\mathbf{R}}_1^{-1}{\tilde{\mathbf{U}}}_{1i}\mathit{\eta }(t)\hfill \\ \hfill & +(\tau (t)-{\tau }_1){\mathit{\eta }}^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{2i}^{\mathrm{T}}{\mathbf{R}}_1^{-1}{\tilde{\mathbf{U}}}_{2i}\eta (t)+({\tau }_2-\tau (t)){\mathit{\eta }}^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{3i}^{\mathrm{T}}{\mathbf{R}}_1^{-1}{\tilde{\mathbf{U}}}_{3i}\mathit{\eta }(t)\hfill \\ \hfill & +\sum _{j=1}^3{\mathit{\eta }}^{\mathrm{T}}(t){\tilde{\mathbf{U}}}_{ji}^{\mathrm{T}}{\mathbf{R}}_2^{-1}{\tilde{\mathbf{U}}}_{ji}\mathit{\eta }(t)+\sum _{i=1}^3{\tilde{h}}_i^{\mathrm{T}}(t){\mathbf{R}}_2{\tilde{h}}_i(t),\hfill \end{array}$$

(18)

where $\mathit{\eta }(t)={[{\mathbf{x}}^{\mathrm{T}}(t),{\mathbf{x}}^{\mathrm{T}}(t-\tau (t)),{\mathbf{e}}^{\mathrm{T}}(t),{\mathbf{e}}^{\mathrm{T}}(t-{\tau }_1),{\mathbf{e}}^{\mathrm{T}}(t-\tau (t)),{\mathbf{e}}^{\mathrm{T}}(t-{\tau }_2)]}^{\mathrm{T}}$, with

$$\begin{array}{c}{\tilde{h}}_1(t)={\int }_{t-{\tau }_1}^t\mathbf{h}(s,r_s)\mathrm{d}\omega (s),\hfill \\ {\tilde{h}}_2(t)={\int }_{t-\tau (t)}^{t-{\tau }_1}\mathbf{h}(s,r_s)\mathrm{d}\omega (s),\hfill \\ {\tilde{h}}_3(t)={\int }_{t-{\tau }_2}^{t-\tau (t)}\mathbf{h}(s,r_s)\mathrm{d}\omega (s),\hfill \\ {\tilde{\mathbf{U}}}_{1i}=[{\mathbf{U}}_{11i}, {\mathbf{U}}_{12i},{\mathbf{U}}_{13i},{\mathbf{U}}_{14i},{\mathbf{U}}_{15i},{\mathbf{U}}_{16i}],\hfill \\ {\tilde{\mathbf{U}}}_{2i}=[{\mathbf{U}}_{21i},{\mathbf{U}}_{22i},{\mathbf{U}}_{23i},{\mathbf{U}}_{24i},{\mathbf{U}}_{25i},{\mathbf{U}}_{26i}],\hfill \\ {\tilde{\mathbf{U}}}_{3i}=[{\mathbf{U}}_{31i},{\mathbf{U}}_{32i},{\mathbf{U}}_{33i},{\mathbf{U}}_{34i},{\mathbf{U}}_{35i},{\mathbf{U}}_{36i}].\hfill \end{array}$$

Remark 2.

Comparing the sliding surface functions (9) and (13) reveals that the sliding surface designed here differs from the linear ones presented in [17,31]. If the state trajectories can converge to the sliding surfaces introduced in [17,31], this defines the behavior of the dynamic systems. Otherwise, it is worth noting that this strategy not only enhances the system’s robustness, but also improves its dynamic response through the design of dual sliding surfaces.

3.3. $H_{\infty }$ Performance Analysis

Based on the above preliminary, the following theorem is proposed:

Theorem 1.

The SMDs (12) and (16) are MSE stable with a disturbance attenuation level γ, provided that matrices ${\mathbf{P}}_i$ > 0, ${\mathbf{Q}}_j$ > 0, $(j=1,\dots ,4)$, ${\mathbf{R}}_1$ > 0, ${\mathbf{R}}_2$ > 0, and ${\mathbf{U}}_{jki},(j=1,2,3;k=1,2,\dots ,6)$, ${\mathit{X}}_i$ and ${\mathit{Y}}_i$ along with scalars ${\epsilon }_i>0$, exist to meet the ensuing LMIs for every $i\in \mathcal{S}$,

$$\left[\begin{array}{ccc}{\left[{\Gamma }^i\right]}_{11\times 11}& {\mathcal{U}}_i& {\mathcal{U}}_i\\ \ast & -{\mathcal{R}}_1& \mathsf{0}\\ \ast & \ast & -{\mathcal{R}}_2\end{array}\right]<\mathsf{0},$$

(19)

where

${\mathcal{U}}_i=\left[\begin{array}{ccc}{\tilde{\mathbf{U}}}_{1i}^{\mathrm{T}}& {\tilde{\mathbf{U}}}_{2i}^{\mathrm{T}}& {\tilde{\mathbf{U}}}_{3i}^{\mathrm{T}}\\ \mathsf{0}& \mathsf{0}& \mathsf{0}\end{array}\right], {\mathcal{R}}_1=\mathrm{diag}\left\{{\displaystyle \frac{1}{{\tau }_1}}{\mathbf{R}}_1,{\displaystyle \frac{1}{\overline{\tau }}}{\mathbf{R}}_1,{\displaystyle \frac{1}{\overline{\tau }}}{\mathbf{R}}_1\right\},$

${\mathcal{R}}_2=\mathrm{diag}\left\{{\mathbf{R}}_2,{\mathbf{R}}_2,{\mathbf{R}}_2\right\},$

$\begin{array}{cc}\hfill {\Gamma }_{11i}& =\mathrm{sym}\left\{{\mathbf{P}}_i{\mathbf{A}}_i+{\mathbf{X}}_i\right\}+{\mathbf{Q}}_1+\sum _{j=1}^s{\pi }_{ij}{\mathbf{P}}_j+{\mathbf{E}}_i^{\mathrm{T}}({\mathbf{P}}_i+{\tau }_2{\mathbf{R}}_2){\mathbf{E}}_i+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_i,\hfill \end{array}$

${\tilde{\Gamma }}_{11i}={\Gamma }_{11i}+{\mathbf{C}}_i^{\mathrm{T}}{\mathbf{C}}_i, {\Gamma }_{12i}={\mathbf{P}}_i{\tilde{\mathbf{A}}}_{\tau i}+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_{\tau i},$

${\Gamma }_{13i}={\mathbf{E}}_i^{\mathrm{T}}({\mathit{P}}_i+{\tau }_2{\mathbf{R}}_2){\mathbf{E}}_i+{\mathbf{U}}_{11i}^{\mathrm{T}}+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_i, {\Gamma }_{14i}=-{\mathbf{U}}_{11i}^{\mathrm{T}}+{\mathbf{U}}_{21i}^{\mathrm{T}},$

${\Gamma }_{15i}=-{\mathbf{U}}_{21i}^{\mathrm{T}}+{\mathbf{U}}_{31i}^{\mathrm{T}}+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_{\tau i}, {\Gamma }_{16i}=-{\mathbf{S}}_{31i}^{\mathrm{T}}, {\Gamma }_{1,10i}={\mathbf{P}}_i{\mathbf{B}}_I{\mathbf{C}}_I^{†},$

${\Gamma }_{22i}=-(1-\mu ){\mathbf{Q}}_1+{\epsilon }_i{\mathbf{N}}_{\tau i}^{\mathrm{T}}{\mathbf{N}}_{\tau i}, {\Gamma }_{23i}={\mathbf{U}}_{12i}^{\mathrm{T}}+{\epsilon }_i{\mathbf{N}}_{\tau i}^{\mathrm{T}}{\mathbf{N}}_i,$

${\Gamma }_{24i}=-{\mathbf{U}}_{12i}^{\mathrm{T}}+{\mathbf{U}}_{22i}^{\mathrm{T}}, {\Gamma }_{25i}=-{\mathbf{U}}_{22i}^{\mathrm{T}}+{\mathbf{U}}_{32i}^{\mathrm{T}}+{\epsilon }_i{\mathbf{N}}_{\tau i}^{\mathrm{T}}{\mathbf{N}}_{\tau i}, {\Gamma }_{26i}=-{\mathbf{U}}_{32i}^{\mathrm{T}},$

$\begin{array}{cc}\hfill {\Gamma }_{33i}& =\mathrm{sym}\left\{{\mathbf{P}}_i{\overline{\mathbf{A}}}_i-{\mathbf{Y}}_i{\mathbf{C}}_i+{\mathbf{U}}_{13i}\right\}+\sum _{k=2}^4{\mathbf{Q}}_k+\sum _{j=1}^s{\pi }_{ij}{\mathbf{P}}_j+{\mathbf{E}}_i^{\mathrm{T}}({\mathbf{P}}_i+{\tau }_2{\mathbf{R}}_2){\mathbf{E}}_i+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_i,\hfill \end{array}$

${\Gamma }_{34i}=-{\mathbf{U}}_{13i}^{\mathrm{T}}+{\mathbf{U}}_{23i}^{\mathrm{T}}+{\mathbf{U}}_{14i},$

${\Gamma }_{35i}=-{\mathbf{U}}_{23i}^{\mathrm{T}}+{\mathbf{U}}_{33i}^{\mathrm{T}}+{\mathbf{P}}_i{\overline{\mathbf{A}}}_{\tau i}+{\mathbf{U}}_{15i}+{\epsilon }_i{\mathbf{N}}_i^{\mathrm{T}}{\mathbf{N}}_{\tau i},$

${\Gamma }_{36i}=-{\mathbf{U}}_{33i}^{\mathrm{T}}+{\mathbf{U}}_{16i}, {\Gamma }_{37i}={\mathbf{P}}_i{\overline{\mathbf{D}}}_i, {\Gamma }_{38i}={\overline{\mathbf{A}}}_i^{\mathrm{T}}{\mathbf{P}}_i-{\mathbf{C}}_i^{\mathrm{T}}{\mathit{Y}}_i^{\mathrm{T}},$

${\Gamma }_{39i}={\mathbf{P}}_i{\overline{\mathbf{M}}}_i, {\Gamma }_{311i}={\mathbf{C}}_i^{\mathrm{T}}{\mathbf{Y}}_i^{\mathrm{T}},$

${\Gamma }_{44i}=-{\mathbf{Q}}_2+\mathrm{sym}\left\{-{\mathbf{U}}_{14i}^{\mathrm{T}}+{\mathbf{U}}_{24i}^{\mathrm{T}}\right\},$

${\Gamma }_{45i}=-{\mathbf{U}}_{24i}^{\mathrm{T}}+{\mathbf{U}}_{34i}^{\mathrm{T}}-{\mathbf{U}}_{15i}+{\mathbf{U}}_{25i},$

${\Gamma }_{46i}=-{\mathbf{U}}_{34i}^{\mathrm{T}}-{\mathbf{U}}_{16i}+{\mathbf{U}}_{26i},$

${\Gamma }_{55i}=-(1-\mu ){\mathbf{Q}}_3+\mathrm{sym}\left\{-{\mathbf{U}}_{25i}^{\mathrm{T}}+{\mathbf{U}}_{35i}^{\mathrm{T}}\right\}+{\epsilon }_i{\mathbf{N}}_{\tau i}^{\mathrm{T}}{\mathbf{N}}_{\tau i},$

${\Gamma }_{56i}=-{\mathbf{U}}_{35i}^{\mathrm{T}}-{\mathbf{U}}_{26i}+{\mathbf{U}}_{36i},$

${\Gamma }_{66i}=-{\mathbf{Q}}_4-\mathrm{sym}\left\{{\mathbf{U}}_{36i}^{\mathrm{T}}\right\},$

${\Gamma }_{77i}=-{\gamma }^2\mathbf{I}, {\Gamma }_{88i}=-2{\mathbf{P}}_i+\tau {\mathbf{R}}_1, {\Gamma }_{89i}={\mathbf{P}}_i{\overline{\mathbf{M}}}_i,$

${\Gamma }_{99i}=-{\epsilon }_i\mathbf{I}, {\Gamma }_{10,10i}={\Gamma }_{11,11,i}=-{\mathbf{P}}_i.$

Furthermore, the gain matrix ${\mathbf{K}}_i$ is defined by ${\mathbf{K}}_i={\mathbf{B}}^{†}{\mathbf{P}}_i^{-1}{\mathbf{X}}_i$, while the observer gain matrix ${\mathbf{L}}_i$ is derived as ${\mathbf{L}}_i={\mathbf{C}}_I^{†}{\mathbf{P}}_i^{-1}{\mathbf{Y}}_i$, utilizing the Moore–Penrose pseudoinverse denoted by †.

Proof. 

Please refer to the proof in Appendix A, starting from page 14. □

Remark 3.

By employing the LMI method, sufficient conditions have been established to ensure ETM stability for the SMDs (12) and (16). The LMI approach is highly valued for its efficiency and maturity in solving convex optimization problems, offering a powerful mathematical tool for controller design. Furthermore, the applicability of the LMI method also indicates that the proposed control strategy possesses good generality and adaptability, making it suitable for a wide range of different systems. Additionally, it is worth mentioning that, when solving inequality (19) in the LMI toolbox, it is recommended to use the solver [tmin,xfeas] = feasp(lmisys).

3.4. SMC Law Synthesise

In this section, the finite-time reachabililty of sliding surfaces $\mathit{s}(t)=\mathsf{0}$ and ${\mathit{s}}_e(t)=\mathsf{0}$ is guaranteed by employing two adaptive laws. On one hand, in order to obtain the state observer, ${\mathit{v}}_s(t)$ in (7) must be designed simultaneously. However, the nonlinear function ${\mathit{f}}_i(\mathit{x}(t))$ is unknown, and the bounds $\alpha$ and $\beta$ are also unavailable. In this discourse, the variables $\alpha$ and $\beta$ are, respectively, estimated as $\widehat{\alpha }(t)$ and $\widehat{\beta }(t)$, with the relevant estimation errors delineated by $\tilde{\alpha }(t)=\widehat{\alpha }(t)-\alpha$ and $\tilde{\beta }(t)=\widehat{\beta }(t)-\beta$. Then, one can design the compensation control law ${\mathit{v}}_s(t)$ as follows:

$${\mathit{v}}_s(t)=\left(\widehat{\alpha }(t)+\widehat{\beta }(t)\parallel \mathit{y}(t)\parallel +ϵ\right)\mathrm{sgn}({\mathit{s}}_e(t)),$$

(20)

with adaptive law

$$\dot{\widehat{\alpha }}(t)=c_1\parallel {\mathit{s}}_e(t)\parallel ,\dot{\widehat{\beta }}(t)=c_2\parallel {\mathit{s}}_e(t)\parallel \parallel \mathit{y}(t)\parallel ,$$

(21)

where $ϵ>0$ is a small scalar, $c_1>0$ and $c_2>0$ are both given scalars.

On the other hand, in an effort to guarantee the finite-time attainability of sliding surfaces ${\mathit{s}}_e(t)=\mathsf{0}$ by controller $\mathit{\varphi }({\mathit{u}}_{2i}(t))$, all the state components need to be known. However, the fact is that $\mathit{e}(t)$ is also unavailable; while considering the relationship of $\mathit{e}(t)$, $\mathit{x}(t)$, $\mathit{y}(t)$ and $\widehat{\mathit{y}}(t)$, it is reasonable to assume that there exist two scalars such that $\parallel \mathit{e}(t)\parallel \le l_1\parallel \mathit{y}(t)\parallel +l_2\parallel \widehat{\mathit{y}}(t)\parallel$, where $l_i$ $(i=1,2)$ are unknown scalars. So the following assumption is made.

Assumption 2.

Scalar values $l_i$, although currently unspecified, can be determined to satisfy the following estimation:

$$\begin{array}{cc}\hfill \mathcal{Z}(t)& {= \parallel (\mathbf{X}\mathbf{C}\mathbf{B})}^{-1}\mathbf{X}\mathbf{C}({\mathbf{A}}_i+\Delta {\mathbf{A}}_i)-{\mathbf{L}}_i{\mathbf{C}}_i\parallel \parallel \mathbf{e}(t)\parallel \hfill \\ \hfill & {+\parallel (\mathbf{X}\mathbf{C}\mathbf{B})}^{-1}\mathbf{X}\mathbf{C}({\mathbf{A}}_{\tau i}+\Delta {\mathbf{A}}_{\tau i})\parallel \parallel \mathbf{e}(t-\tau (t))\parallel \hfill \\ \hfill & \le l_1\parallel \mathbf{y}(t)\parallel +l_2\parallel \widehat{\mathbf{y}}(t)\parallel .\hfill \end{array}$$

(22)

Also, noticing that the bounds $l_1$ and $l_2$ are unknown. In this section, $l_1$ and $l_2$ are, respectively, estimated by ${\widehat{l}}_1(t)$ and ${\widehat{l}}_2(t)$, while the relevant estimation errors are defined by ${\tilde{l}}_1(t)={\widehat{l}}_1(t)-l_1$ and ${\tilde{l}}_2(t)={\widehat{l}}_2(t)-l_2$. The adaptive laws are designed as follows:

$${\dot{\widehat{l}}}_1(t)=d_1\parallel \mathbf{y}(t)\parallel \parallel {\mathbf{s}}_e(t)\parallel ,{\dot{\widehat{l}}}_2(t)=d_2\parallel \widehat{\mathbf{y}}(t)\parallel \parallel {\mathbf{s}}_e(t)\parallel ,$$

(23)

with $d_i>0$ $(i=1,2)$ chosen by the designer.

Theorem 2.

Assuming the existence of sliding surface functions as delineated in Equations (9) and (13), and given that the gain matrix ${\mathbf{K}}_i$ fulfills the conditions of Theorem 1, the attainment of the desired sliding surfaces $\mathbf{s}(t)=\mathsf{0}$ and ${\mathbf{s}}_e(t)=\mathsf{0}$ within a finite time frame is ascertained through the formulation of adaptive SMC laws designed below:

$$\begin{array}{cc}\hfill & \mathit{\varphi }({\mathbf{u}}_{1i}(t))={\mathbf{K}}_i\widehat{\mathbf{x}}(t)-{(\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{A}}_{\tau i}\widehat{\mathbf{x}}(t-\tau (t))\hfill \\ \hfill & -\left(\widehat{\alpha }(t)+\widehat{\beta }(t)\parallel \mathbf{y}(t)\parallel +ϵ+{\rho }_i(t)\right)\mathrm{sgn}(\mathbf{s}(t)),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill \mathit{\varphi }({\mathbf{u}}_{2i}(t))& {=-[\parallel (\mathbf{X}\mathbf{C}\mathbf{B})}^{-1}\mathbf{X}\mathbf{C}{\mathbf{M}}_i\parallel (\parallel {\mathbf{N}}_i\widehat{\mathbf{x}}(t)\parallel +\parallel {\mathbf{N}}_i\widehat{\mathbf{x}}(t-\tau (t))\parallel )\hfill \\ \hfill & +{\widehat{l}}_1(t)\parallel \mathbf{y}(t)\parallel +{\widehat{l}}_2(t)\parallel \widehat{\mathbf{y}}{(t)\parallel +\parallel (\mathbf{X}\mathbf{C}\mathbf{B})}^{-1}\mathbf{X}\mathbf{C}{\mathbf{D}}_i\parallel \parallel \mathbf{v}(t)\parallel \hfill \\ \hfill & +\sigma +{\displaystyle \frac{\overline{\lambda }\widehat{\delta }(t){\parallel \mathbf{y}(t)\parallel }^2}{2\parallel {\mathbf{s}}_e(t)\parallel }}]\mathrm{sgn}({\mathbf{s}}_e(t)),\hfill \end{array}$$

(25)

in which the gains are designed in (21) and (23), and

${\rho }_i(t)={max}_{i\in \mathcal{S}}{\{\parallel (\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{L}}_i{\mathbf{y}(t)\parallel +\parallel (\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{L}}_i{\mathbf{C}}_i\widehat{\mathbf{x}}(t)\parallel \}$,

$\overline{\lambda }={\lambda }_{max}\{{\mathbf{C}}^{\mathrm{T}}{\mathbf{X}}^{\mathrm{T}}{(\mathbf{X}\mathbf{C}\mathbf{B})}^{-1}\mathbf{X}\mathbf{C}\}$, $\dot{\widehat{\delta }}=c_3\overline{\lambda }{\parallel \mathbf{y}(t)\parallel }^2$, $c_3>0$,

$\begin{array}{cc}\hfill \sigma & =\underset{i\in \mathcal{S}}{max}{\{\parallel (\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{L}}_i{\mathbf{y}(t)\parallel +\parallel (\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{L}}_i{\mathbf{C}}_i\widehat{\mathbf{x}}{(t)\parallel -\parallel (\mathbf{G}\mathbf{B})}^{-1}\mathbf{G}{\mathbf{L}}_i{\mathbf{C}}_i\mathbf{e}(t)\parallel \}.\hfill \end{array}$

Proof. 

Please refer to the proof in Appendix B, starting from page 17. □

Remark 4.

The proposed dual SMC approach offers several advantages. Firstly, it enhances robustness against variations in system parameters and external disturbances; secondly, it provides a faster dynamic response due to its dual control strategy, which is particularly beneficial in scenarios that require rapid tracking or regulation; thirdly, it improves system stability through more refined control mechanisms, especially in complex dynamic systems. Additionally, in practical engineering applications, this means the system can more quickly return to its desired operating state when faced with uncertainties and external disturbances.

The overall implementation of the control strategy in the MATLAB R2018b environment is given as follows:

Step 1

Giving all the system parameters $A_i$, $A_{\tau i}$, B, $C_i$, $D_i$, $E_i$, $M_i$, $N_i$, and a prescribed $H_{\infty }$ performance index $\gamma$;

Step 2

Using the MATLAB LMI Toolbox to solve the LMIs in Theorem 1 based on the parameters given in step 1 so as to obtain gain matrices $K_i$ and $L_i$;

Step 3

Parameter initialization before simulation, including step size h, $\varphi (s)$, $\phi (s)$, $ϵ$, $c_1$, $c_2$, $l_1$ and $l_2$, etc.

Step 4

Employing the Euler’s method to develop MATLAB code,

for j = 1:N

$x(j+1)=f_1(x(j),\varphi (u(j)),v(j),\omega (j),\tau (j),h);$

$\widehat{x}(j+1)=f_2(\widehat{x}(j),\varphi (u_1(j)),v_s(j),y(j),\tau (j),h);$

$s(j+1)=f_3(\widehat{x}(j+1),\widehat{x}(j),h);$

$s_e(j+1)=f_4(y(j+1),\widehat{y}(j+1));$

$\widehat{\alpha }(j+1)=f_5(s_e(j),h);$

⋮

$v_s(j+1)=f_6(s_e(j+1),y(j+1),\widehat{\alpha }(j+1),\widehat{\beta }(j+1));$

$\varphi (u_1(j+1))=f_7(\widehat{x}(j+1),y(j+1),s(j+1),\widehat{\alpha }(j+1),\widehat{\beta }(j+1),\tau (j+1));$

$\varphi (u_2(j+1))=f_8(\widehat{x}(j+1),y(j+1),\widehat{y}(j+1),s_e(j+1),\cdots ,\widehat{\delta }(j+1),\tau (j+1));$

end;

in which $f_k(·)$ denotes the mapping after discretization.

4. Numerical Example

Taking the modeling of a Tunnel Diode circuit (Described in Figure 1) into consideration, assume that the resistor $R_{r_t}$ switching behavior can be consistent with the Markov process $r_t$. Then, the following system dynamics can be derived according to the Kirchoff current law:

$$\left\{\begin{array}{c}\dot{V_C}(t)={\displaystyle \frac{1}{C}}{i}_L(t)+{\displaystyle \frac{1}{C}}{i}_{TD}(t)\hfill \\ \dot{i_L}(t)=-{\displaystyle \frac{1}{L}}{V}_C(t)-{\displaystyle \frac{1}{L}}{R}_{r_t}{i}_L(t)+{\displaystyle \frac{1}{L}}u(t),\hfill \end{array}\right.$$

(26)

where $i_{TD}(t)=-0.2V_{TD}(t)-0.01V_{TD}^3(t)$.

It is well known that, in practical matters, the circuits usually encounter time lag and packet loss; hence, the controlled plant can also be represented by (1), incorporating the process of nonlinearization as detailed in reference [7]. Then, let $x_1(t)=V_C(t)$, $x_2(t)=i_L(t)$, and choose $R_1=1\Omega$, $R_2=2\Omega$, $R_3=3\Omega$, $L=0.5\mathrm{H}$ and $C=0.2\mathrm{F}$ in Figure 1. Consider the TD circuit model with the TR matrix

$$\left[\begin{array}{ccc}-0.6& 0.4& 0.2\\ 0.5& -0.6& 0.1\\ 0.2& 0.5& -0.7\end{array}\right],$$

among the following modes:

$${\mathit{A}}_1=\left[\begin{array}{cc}0& 5\\ -2& -2\end{array}\right],{\mathit{A}}_{\tau 1}=\left[\begin{array}{cc}-4.2877& -4.0701\\ -3.1802& -0.3651\end{array}\right],{\mathit{D}}_1=\left[\begin{array}{c}1\\ 0\end{array}\right],$$

$${\mathit{E}}_1=\left[\begin{array}{cc}0.2& 0.5\\ 0& 0.1\end{array}\right],{\mathit{M}}_1=\left[\begin{array}{cc}-0.1& 0\\ 0.1& 0\end{array}\right],{\mathit{N}}_1=\left[\begin{array}{cc}0& 0.1\\ 0.1& 0\end{array}\right],$$

$${\mathit{N}}_{\tau 1}=\left[\begin{array}{cc}0& 0.1\\ 0.1& 0\end{array}\right],\mathit{B}=\left[\begin{array}{c}0\\ 2\end{array}\right],{\mathit{C}}_1=\left[\begin{array}{cc}0& 1\end{array}\right],$$

$${\mathit{A}}_2=\left[\begin{array}{cc}0& 5\\ -2& -4\end{array}\right],{\mathit{A}}_{\tau 2}=\left[\begin{array}{cc}-4.9067& 1.4274\\ 4.1503& -4.9858\end{array}\right],{\mathit{D}}_2=\left[\begin{array}{c}0.5\\ 0\end{array}\right],$$

$${\mathit{E}}_2=\left[\begin{array}{cc}0& 0.1\\ -0.1& 0.2\end{array}\right],{\mathit{M}}_2=\left[\begin{array}{cc}-0.1& 0\\ 0& 0.1\end{array}\right],{\mathit{N}}_2=\left[\begin{array}{cc}-0.1& 0\\ 0.2& 0\end{array}\right],$$

$${\mathit{N}}_{\tau 2}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.2\end{array}\right],\mathit{B}=\left[\begin{array}{c}0.8\\ 1.3\end{array}\right],{\mathit{C}}_2=\left[\begin{array}{cc}0& 1\end{array}\right],$$

$${\mathit{A}}_3=\left[\begin{array}{cc}0& 5\\ -2& -6\end{array}\right],{\mathit{A}}_{\tau 3}=\left[\begin{array}{cc}-4.6961& -0.4503\\ -2.9153& -3.7273\end{array}\right],{\mathit{D}}_3=\left[\begin{array}{c}-0.5\\ 0\end{array}\right],$$

$${\mathit{E}}_3=\left[\begin{array}{cc}0& 0.1\\ 0& -0.1\end{array}\right],{\mathit{M}}_3=\left[\begin{array}{cc}-0.2& 0\\ 0& 0\end{array}\right],{\mathit{N}}_3=\left[\begin{array}{cc}0& 0.1\\ 0.2& 0\end{array}\right],$$

$${\mathit{N}}_{\tau 3}=\left[\begin{array}{cc}0& -0.2\\ 0.1& 0.1\end{array}\right],\mathit{B}=\left[\begin{array}{c}0.8\\ 1.3\end{array}\right],{\mathit{C}}_3=\left[\begin{array}{cc}0& 1\end{array}\right].$$

Letting $\tau (t)=0.25sin(2t)+0.35$, then ${\tau }_1=0.1$, ${\tau }_2=0.6$, $\mu =0.5$. The unknown nonlinear functions are given by $f_1(\mathit{x}(t))=sin(x_1(t))$, $f_2(\mathit{x}(t))=cos(x_2(t))$, and $f_3(\mathit{x}(t))=cos(x_1(t))sin(x_2(t))$. The exogenous disturbance is chosen as $v(t)=-0.2x_1(t)-0.01x_1^3(t)+e^{-t}$. According to Theorem 1, let $\gamma =1.2$, by solving the LMIs (19) with MALAB LMI Toolbox, one can acquire the following solutions:

$${\mathit{P}}_1=\left[\begin{array}{cc}0.8814& -0.0823\\ -0.0823& 49.0636\end{array}\right],{\mathit{P}}_2=\left[\begin{array}{cc}0.9290& -0.3728\\ -0.3728& 45.2001\end{array}\right],$$

$${\mathit{P}}_3=\left[\begin{array}{cc}1.1298& -0.3646\\ -0.3646& 45.8597\end{array}\right],{\mathit{Q}}_1=\left[\begin{array}{cc}36.3844& -0.0128\\ -0.0128& 36.0497\end{array}\right],$$

$${\mathit{Q}}_2=\left[\begin{array}{cc}0.3146& -0.1224\\ -0.1224& 3.2604\end{array}\right],{\mathit{Q}}_3=\left[\begin{array}{cc}0.2070& -0.0446\\ -0.0446& 2.4994\end{array}\right],$$

$${\mathit{Q}}_4=\left[\begin{array}{cc}0.2292& -0.0815\\ -0.0815& 3.9047\end{array}\right],{\mathit{R}}_1=\left[\begin{array}{cc}2.6326& 0.5600\\ 0.5600& 6.7218\end{array}\right],$$

$${\mathit{R}}_2=\left[\begin{array}{cc}30.1774& 1.5620\\ 1.5620& 26.8296\end{array}\right],{\mathit{X}}_1=\left[\begin{array}{cc}-32.2613& 42.7861\\ 42.8172& 59.6739\end{array}\right],$$

$${\mathit{X}}_2=\left[\begin{array}{cc}-33.3296& 45.2317\\ 45.2602& 136.9182\end{array}\right],{\mathit{X}}_3=\left[\begin{array}{cc}-30.7448& 42.1553\\ 42.1826& 282.6200\end{array}\right],$$

$${\mathit{Y}}_1=\left[\begin{array}{c}3.7081\\ 28.6319\end{array}\right],{\mathit{Y}}_2=\left[\begin{array}{c}3.2501\\ 26.9858\end{array}\right],{\mathit{Y}}_3=\left[\begin{array}{c}3.5559\\ 23.9109\end{array}\right],$$

$${ϵ}_1=15.5177,{ϵ}_2=8.0218,{ϵ}_3=16.8514.$$

Based on the above feasible solutions, one can easily obtain that the gain matrix ${\mathit{K}}_i$ and observer gain matrix ${\mathit{L}}_i$ are

$${\mathit{K}}_1=\left[\begin{array}{cc}0.4057& 0.6489\end{array}\right], {\mathit{K}}_2=\left[\begin{array}{cc}0.3539& 1.7210\end{array}\right], {\mathit{K}}_3=\left[\begin{array}{cc}0.3526& 3.2380\end{array}\right],$$

$${\mathit{L}}_1=\left[\begin{array}{c}4.2620\\ 0\end{array}\right], {\mathit{L}}_2=\left[\begin{array}{c}3.7504\\ 0\end{array}\right], {\mathit{L}}_3=\left[\begin{array}{c}3.3242\\ 0\end{array}\right].$$

Choose the initial conditions $\mathit{x}(t)={[0.3-0.2]}^{\mathbf{T}}$, $\widehat{\mathit{x}}(t)={[-0.50.5]}^{\mathbf{T}}$ where $t\in [-{\tau }_2,0]$, and $\widehat{\alpha }(0)=0.1$, $\widehat{\beta }(0)=0.05$, $\widehat{\delta }(0)=0.1$, ${\widehat{l}}_1(0)=0.2$, ${\widehat{l}}_2(0)=0.15$. Meanwhile, choose scalars $c_1=0.05$, $c_2=0.08$, $c_3=0.1$, $d_1=0.02$, $d_2=0.05$, $ϵ=0.01$. Then, the simulation outcomes are depicted in Figure 2, Figure 3 and Figure 4. Figure 2 plots the state responses of $\mathit{x}(t)$ and $\widehat{\mathit{x}}(t)$ in the closed-loop system with the adaptive controllers (24) and (25) under the same individual jump mode $r_t$, respectively. The curves of $s(t)$ and $s_e(t)$ are offered in Figure 3. Meanwhile, the adaptive sliding mode controller $\mathit{\varphi }({\mathit{u}}_i(t))$ and its estimated parameters for $\alpha (t)$, $\beta (t)$, $\delta (t)$, $l_1(t)$, and $l_2(t)$ are exhibited in Figure 4. To demonstrate the advantages of the dual SMC approach, simulation results are derived for error dynamics (8) in a scenario without control. These results are compared with the error SMD (16) obtained by SMC. As seen from Figure 5, the state properties of the error SMD (16) are superior to those of error dynamics (8) in both transient and steady-state behavior. This indicates that the dual SMC approach ensures faster and better convergence. Furthermore, a comparative study was conducted with traditional state feedback control without switching signals. The simulation results, as depicted in Figure 6, demonstrate the superior system performance achieved by the SMC method. Above all, it is known that the proposed adaptive SMC strategy is effective and feasible.

Remark 5.

In this example, the Tunnel Diode circuit is considered to demonstrate the effectiveness of the proposed control strategy. By simulating different operating modes and external disturbances, we have validated the performance of the adaptive sliding mode controller in practical applications. This example not only proves the feasibility of the theoretical method but also provides insights in the future application of such control strategies to more complex real-world systems.

Remark 6.

Although dual SMC offers several advantages, this paper is not without potential limitations. For example, the control strategies depend on specific assumptions; thus, easing these assumptions could enhance the approach’s applicability to a wider array of systems. In the controller design process, some tuning parameters are assigned based on prior experience rather than being dynamically optimized, which suggests potential for further enhancement. Combining the approach with other control theories, such as fuzzy logic or neural networks, could lead to more robust and intelligent control systems. Moreover, this paper primarily relies on simulation results, and incorporating experimental results from real-world systems would be more helpful.

5. Conclusions

This paper has delved into the development and analysis of a robust $H_{\infty }$ control strategy for Itô-type stochastic MJSs with parameter uncertainties, time-varying delays, and nonlinear perturbations. The proposed approach employs a dual adaptive SMC strategy, designing to enhance the system’s robustness and stability. The key findings are as follows: (1) This paper has introduced a novel dual SMC scheme that incorporates two sliding surfaces, one for the state observer system and another for the error dynamics. This approach differs from traditional single surface methods and has been shown to improve system performance by directly controlling the error dynamics; (2) By utilizing the LMI method, sufficient conditions for the MSE stability of the closed-loop systems have been established. (3) This paper proposes adaptive sliding mode controllers that ensure finite-time arrival and maintenance on the established sliding surfaces, which has been proven to be robust against parameter variations, time-varying delays, and external disturbances; (4) Through a Tunnel Diode circuit model, the effectiveness of the proposed control strategy has been demonstrated, and the simulation results show that the dual SMC approach ensures faster and better convergence compared to error dynamics without control. However, there are areas for future research. These include extending the approach to more complex nonlinear systems, improving computational efficiency, and validating the approach through experimental studies.