Observer-Based Local Stabilization of State-Delayed Quasi-One-Sided Lipschitz Systems with Actuator Saturation

Abstract

This paper addresses the problem of asymptotic stabilization for a class of systems composed of linear and nonlinear parts, both of which are affected by a common state delay that increases the complexity of the dynamics. Within this class of systems, the nonlinear component depends on unmeasurable states and satisfies a quasi-one-sided Lipschitz (QL) condition, which allows for tractable analysis. Moreover, the control input is subject to saturation, further complicating the stabilization task. The proposed remedy involves three key components: an observer to estimate the unmeasurable states, a Lyapunov–Krasovskii (LK) functional to handle the delay, and a dead-zone model to represent the saturation nonlinearity. This combined approach allows for the derivation of sufficient conditions that ensure the local asymptotic stabilization of an augmented system comprising the state and the estimation error. Furthermore, the domain of attraction is estimated. The obtained conditions are not LMIs. This arises from a shared matrix variable that is required to simultaneously verify the weak QL Lipschitz condition and appear within the LK functional, creating a nonlinear coupling. In the existing literature, this matrix is typically fixed and not treated as a decision variable to simplify the problem. In contrast, this work proposes a novel approach by employing an appropriate decoupling technique, which allows this matrix to remain a decision variable and provides greater flexibility in the design. To validate the proposed design, we provide a numerical simulation.

1. Introduction

The combined presence of time delay and actuator saturation (AS) in dynamical systems can have a profoundly deleterious impact on closed-loop stability. Specifically, the time delay generates infinitely many roots, which makes exact stability analysis computationally intractable [1]. This challenge has driven the creation of efficient time-domain methods. Notably, methods utilizing the Lyapunov–Krasovskii (LK) framework have become a powerful tool for analyzing stability and designing controllers for these systems. On the other hand, AS, resulting from the inherent limitation on control input, can further compromise system stability and performance [2]. To address the intrinsic nonlinearity of AS, two principal analytical approaches have been developed. The first one, known as the polytopic approach, models the saturated control input by representing it as a convex combination of $2^j$ linear feedback laws, where j is the number of actuators [2]. This formulation leads to a set of $2^j$ LMIs. The second approach, called the dead-zone nonlinearity approach, requires only j LMIs, in contrast to the $2^j$ LMIs of the polytopic approach. Furthermore, a comparison in [3] indicates that this method achieves a region of stability that is comparable to or even larger than that of the polytopic approach.

Given the critical importance of addressing both AS and time delay, considerable research has focused on various classes of dynamical systems subject to these constraints. For instance, based on an improved LK functional, the authors of [4] proposed enhanced delay-dependent stabilization criteria for linear systems. The study employed both polytopic and dead-zone approaches to model AS. In [5], regional stabilization of discrete-time linear systems was investigated through the polytopic approach. The authors of [6] focused on the stabilization problem of neutral systems through a descriptor formulation together with a dead-zone approach to handle the nonlinearity. The study in [7] explored the stability of Takagi–Sugeno fuzzy descriptor systems by constructing an augmented LK functional and adopting a delay-partitioning method. In [8], the authors proposed two controllers for polynomial fuzzy models, where a polytopic representation was employed to address AS. In [9], a static anti-windup compensator was proposed for a class of nonlinear systems, where the nonlinear term $f(t,\vartheta )$ satisfies $\parallel f(t,\vartheta )-f(t,\hat{\vartheta })\parallel \le \parallel \varphi (\vartheta -\hat{\vartheta })\parallel$, with $\varphi$ being a known constant matrix. An event-triggered control approach for linear singular systems subject to time-varying delay was established by Jafari and Binazadeh [10]. However, despite these advances, only a limited amount of work has focused on Lipschitz nonlinear systems affected by saturated inputs and time delay.

The Lipschitz nonlinearity, depending on unmeasurable states, is a key property in observer design, as it provides a strict bound on the nonlinear growth and thus enables rigorous guarantees on the convergence of the estimation error for unmeasurable states. In contrast, approximating such nonlinearities with a Takagi–Sugeno fuzzy model often introduces unmeasurable premise variables, complicating the establishment of formal stability conditions for the observer. Given this advantage, observer design for Lipschitz systems has been addressed using a full-order observer [11,12] and a reduced-order observer for the delay-free case in [13] and the time-delay case in [14]. While observer designs for standard Lipschitz (SL) systems are often limited to small Lipschitz constants, the one-sided Lipschitz (OL) condition, initially presented by Hu in [15], offers a less conservative alternative. Hence, several observer strategies have been developed for nonlinear plants satisfying the OL condition. For instance, the observer-based control (OC) problem for OL systems under parametric uncertainties was studied in [16] using the quadratically inner-bounded conditions. Accounting for input saturation, an OC framework for OL systems was developed in [17]. The work in [18] eliminated the need for the quadratic inner-boundedness condition, developing a less conservative OC scheme for generalized OL systems with delayed outputs.

Building on the concepts of SL and OL systems, a further generalization, the quasi-one-sided Lipschitz (QL) condition, was proposed in [19]. The main advantage of this function is that it provides more detailed information on the system dynamics, which systematically leads to less conservative results. As a result, over the past few years, much attention has been devoted to OC for QL systems. For example, a separation principle was derived in [20] for the delay-free case and in [21] for the time-delay case. Using a reduced-order observer, the work in [22] addressed the time-delay case. In [20,21,22], the OC design was carried out in two steps. First, LMIs were used to estimate the system states. Then, using this estimated information, the controller was designed. We note that the combined effects of time delay and AS in QL systems have only been addressed in [23]. In that study, the authors derived sufficient conditions using an LK approach and employing the dead-zone representation of the saturation function. The primary limitation of that work is that the domain of attraction (DOA) is not determined, although this aspect is crucial under AS. Furthermore, the adopted decoupling technique simplifies the problem, reducing the matrices $(R_1,U_1,V_1)$ that define the weak QL condition to a set of tuning parameters. Motivated by these observations, this study aims to further investigate the problem of OC for QL systems with time delay and AS. The key contributions of this work are summarized as follows:

• This is the first study to provide an estimate of the DOA for the OC problem associated with this class of systems. Furthermore, we employ an optimization procedure to maximize the estimated DOA.
• The design is achieved in only one step, which reduces implementation complexity.
• Using a suitable decoupling approach, the matrices $(R_1,U_1,V_1)$ are treated as decision variables, where $U_1$ and $V_1$ vary linearly with $R_1$. This constitutes a significant advantage, since in [23], these matrices are constrained to act only as tuning variables.

This paper is structured as follows. Section 2 presents the necessary preliminaries, including the notations, problem description, and key mathematical tools. Section 3 details the main results. By leveraging the LK methodology, we derive sufficient conditions that achieve the local asymptotic stability of the system and quantify the DOA. This bilinearity stems from the dual role of the matrix $R_1$, which is employed in both the construction of the LK functional and the requisite weak QL condition. Finally, an appropriate decoupling technique is employed to transform the BMI into a set of Linear Matrix Inequalities (LMIs). Section 4 presents a numerical example to validate the proposed observer-based controller (OCtrl). Finally, Section 5 concludes this work and suggests avenues for future research.

2. Preliminaries

2.1. Notations

The following sets and notations are introduced for $(\mathsf{ı},\mathit{j},\ell )\in {\mathbb{N}}_{>0}^3$, an $\mathsf{ı}\times \mathsf{ı}$ matrix $\mathrm{\Theta }\in {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$, ı-dimensional vectors $(\theta ,\zeta )\in {\mathbb{R}}^{\mathsf{ı}}\times {\mathbb{R}}^{\mathsf{ı}}$, and $\hslash \in {\mathbb{R}}_{>0}$:

• ${\mathbb{I}}_{\mathsf{ı}}:=\{1,\dots ,\mathsf{ı}\}$; ⊕ indicates the symmetric elements in the matrix expression; $\langle \theta ,\zeta \rangle ={\theta }^T\zeta$; $\parallel \theta \parallel =\sqrt{{\theta }^T\theta }$; ${\left\{\mathrm{\Theta }\right\}}_{★}=\mathrm{\Theta }+{\mathrm{\Theta }}^T$; and $I_{\mathsf{ı}}$ (resp. $0_{\mathsf{ı}\times j}$) denote the $\mathsf{ı}\times \mathsf{ı}$ identity matrix (resp. the $\mathsf{ı}\times j$ zero matrix).
• $\mathrm{\Theta }[\ell ]$ (resp. $\theta [\ell ]$) denotes the ℓ-th row of $\mathrm{\Theta }$ (resp. the ℓ-th component of $\theta$); $\mathrm{\Theta }\in {\mathbb{R}}_{\mathcal{S}}^{\mathsf{ı}\times \mathsf{ı}}$ (resp. $\mathrm{\Theta }\in {\mathbb{R}}_{\mathcal{D}}^{\mathsf{ı}\times \mathsf{ı}}$) indicates that $\mathrm{\Theta }$ is symmetric (resp. diagonal).
• For $\mathrm{\Theta }\in {\mathbb{R}}_{\mathcal{S}}^{\mathsf{ı}\times \mathsf{ı}}$, $\mathrm{\Theta }⪰0$ (resp. $\prec 0$) signifies that $\mathrm{\Theta }$ is positive semi-definite (resp. negative definite); $\overline{\lambda }\left(\mathrm{\Theta }\right)$ (resp. $\underline{\lambda }\left(\mathrm{\Theta }\right)$) is the largest (resp. the smallest) eigenvalue of $\mathrm{\Theta }$.
• $\theta >0$ (resp. ≥, <, ≤) means that $\theta [\ell ]>0$ (resp. ≥, <, ≤), $\forall \ell \in {\mathbb{I}}_{\mathsf{ı}}$;$\theta >\zeta$ (resp. ≥, <, ≤) means that $\theta [\ell ]>\zeta [\ell ]$ (resp. ≥, <, ≤), $\forall \ell \in {\mathbb{I}}_{\mathsf{ı}}$.
• $\mathcal{C}\left(\left[\begin{array}{cc}-\hslash & 0\end{array}\right],{\mathbb{R}}^{\mathsf{ı}}\right)$ stands for the space of continuous functions from $\left[\begin{array}{cc}-\hslash & 0\end{array}\right]$ to ${\mathbb{R}}^{\mathsf{ı}}$.
• For $\psi \left(t\right)\in \mathcal{C}\left(\left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right],{\mathbb{R}}^{\mathsf{ı}}\right)$, ${\parallel \psi \parallel }_s=ma{x}_{t\in \left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right]}\parallel \psi \left(t\right)\parallel$.

2.2. Problem Description

Consider the nonlinear time-delay system described in state-space form as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\vartheta }\left(t\right)=A\vartheta \left(t\right)+D\vartheta (t-\hslash \left(t\right))+f(\vartheta \left(t\right),\vartheta (t-\hslash \left(t\right)))+B\sigma (\upsilon \left(t\right),\overline{\upsilon }),\hfill \\ m\left(t\right)=M\vartheta \left(t\right),\hfill \\ \vartheta \left(t\right)=\phi \left(t\right),t\in [-\overline{\hslash },0],\hfill \end{array}\right\}.\end{array}$$

(1)

where $\vartheta \left(t\right)\in {\mathbb{R}}^{\mathsf{ı}}$, $\upsilon \left(t\right)\in {\mathbb{R}}^j$, $m\left(t\right)\in {\mathbb{R}}^{\ell }$ are, in order, the system states, control input, and measured output, respectively; $(A\in {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}},D\in {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}},B\in {\mathbb{R}}^{\mathsf{ı}\times j},M\in {\mathbb{R}}^{\ell \times \mathsf{ı}})$ are the system matrices defining the state-space model; and $f(\vartheta \left(t\right),\vartheta (t-\hslash \left(t\right)))\in {\mathbb{R}}^{\mathsf{ı}}$ represents the nonlinear part of the system dynamics and satisfies:

$$\begin{array}{c}\hfill f(0,0)=0,\end{array}$$

(2)

$\hslash \left(t\right)$ is a time-dependent delay satisfying

$$\begin{array}{c}\hfill 0\le \hslash \left(t\right)\le \overline{\hslash },\dot{\hslash }\left(t\right)\le \check{\hslash }<1,\end{array}$$

(3)

where $\overline{\hslash }\in {\mathbb{R}}_{>0}$ and $\check{\hslash }\in {\mathbb{R}}_{\ge 0}$, and $\phi \left(t\right)\in \mathcal{C}\left(\left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right],{\mathbb{R}}^{\mathsf{ı}}\right)$ is the initial condition. $\sigma :{\mathbb{R}}^j\to {\mathbb{R}}^j$ is the AS, which is described as follows:

$$\begin{array}{ccc}\hfill \sigma (\upsilon \left(t\right),\overline{\upsilon })\left[l\right]& =& \left\{\begin{array}{c}\overline{\upsilon }\left[l\right],\mathrm{if}\upsilon \left(t\right)\left[l\right]>\overline{\upsilon }\left[l\right],\hfill \\ \upsilon \left(t\right)\left[l\right],\mathrm{if}-\overline{\upsilon }\left[l\right]\le \upsilon \left(t\right)\left[l\right]\le \overline{\upsilon }\left[l\right],\hfill \\ -\overline{\upsilon }\left[l\right],\mathrm{if}\upsilon \left(t\right)\left[l\right]<-\overline{\upsilon }\left[l\right],\hfill \end{array}\right.\};\forall l\in {\mathbb{I}}_j,\hfill \end{array}$$

(4)

where the saturation limits are assumed to be symmetric, with $\overline{\upsilon }\left[l\right]$ representing both the positive and negative bounds of the l-th actuator.

Remark 1. 

In (3), $\overline{\hslash }$ is the maximum allowable time delay, and $\check{\hslash }$ is an upper bound on the rate of change of the delay.

Definition 1 

([21]). System (1) is referred to as a QL system if $\exists \{R,U,V\}\subset {\mathbb{R}}_{\mathcal{S}}^{\mathsf{ı}\times \mathsf{ı}}$, where the following conditions are fulfilled $\forall \{x_{\kappa },{\hat{x}}_{\kappa }\}\subset {\mathbb{R}}^{\mathsf{ı}}$ for $\kappa \in {\mathbb{I}}_2$:

$$\begin{array}{c}\mathrm{R}\succ 0,\mathrm{V}⪰0,\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill & \langle \mathrm{R}(\mathrm{f}(x_1,x_2)-\mathrm{f}({\hat{x}}_1,{\hat{x}}_2)),x_1-{\hat{x}}_1\rangle \le {\left[\begin{array}{c}{x}_1-{\hat{x}}_1\\ x_2-{\hat{x}}_2\end{array}\right]}^T\left[\begin{array}{cc}\mathrm{U}& 0\\ 0& \mathrm{V}\end{array}\right]\left[\begin{array}{c}{x}_1-{\hat{x}}_1\\ x_2-{\hat{x}}_2\end{array}\right].\hfill \end{array}$$

(6)

Given that $f(0,0)=0$, (6) reduces to the weak QL condition:

$$\begin{array}{ccc}\hfill & & \langle \mathrm{R}\mathrm{f}(x_1,x_2),x_1\rangle \le {\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right]}^T\left[\begin{array}{cc}\mathrm{U}& 0\\ 0& \mathrm{V}\end{array}\right]\left[\begin{array}{c}{x}_1\\ x_2\end{array}\right].\hfill \end{array}$$

(7)

We consider the following OCtrl:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\hat{\vartheta }}\left(t\right)=A\hat{\vartheta }\left(t\right)+D\hat{\vartheta }(t-\hslash \left(t\right))+f(\hat{\vartheta }\left(t\right),\hat{\vartheta }(t-\hslash \left(t\right)))+B\sigma (\upsilon \left(t\right),\overline{\upsilon })+\mathcal{X}(m\left(t\right)-\hat{m}\left(t\right))\hfill \\ +\mathcal{Y}(m(t-\hslash \left(t\right))-\hat{m}(t-\hslash \left(t\right))),\hfill \\ \hat{m}\left(t\right)=M\hat{\vartheta }\left(t\right),\hfill \\ \upsilon \left(t\right)=\mathcal{Z}\hat{\vartheta }\left(t\right),\hfill \\ \hat{\vartheta }\left(t\right)=\psi \left(t\right),t\in [-\overline{\hslash },0],\hfill \end{array}\right\}.\end{array}$$

(8)

where variables denoted by a hat, such as $\hat{\vartheta }\left(t\right)$, indicate the estimates of their associated variables, and ($\mathcal{X}\in {\mathbb{R}}^{\mathsf{ı}\times \ell },\mathcal{Y}\in {\mathbb{R}}^{\mathsf{ı}\times \ell },\mathcal{Z}\in {\mathbb{R}}^{j\times \mathsf{ı}}$) represent the OCtrl gains.

In the sequel, each vector depends only on t, such that $\vartheta \left(t\right)$ is denoted by $\vartheta$, and its delayed term is denoted by $\vartheta (t-\hslash (t\left)\right)$ by $\mathit{\vartheta }$.

Application of the OCtrl (8) to system (1) yields

$$\begin{array}{c}\hfill \left\{\begin{array}{c}\dot{\mu }=\mathcal{A}\mu +\mathcal{D}\mathit{\mu }+\mathcal{F}\left(\tilde{\vartheta }\right)+\mathcal{B}\beta (\upsilon ,\overline{\upsilon }),\hfill \\ \mu =\mathrm{\Psi },t\in [-\hslash (t),0],\hfill \end{array}\right\}.\end{array}$$

(9)

where

$$\begin{array}{ccc}\hfill \mu & =& \left[\begin{array}{c}\vartheta \\ \vartheta -\hat{\vartheta }\end{array}\right],\mathcal{A}=\left[\begin{array}{cc}A+B\mathcal{Z}& -B\mathcal{Z}\\ 0& A-\mathcal{X}M\end{array}\right],\mathcal{D}=\left[\begin{array}{cc}D& 0\\ 0& D-\mathcal{Y}M\end{array}\right],\hfill \\ \hfill \mathcal{F}\left(\tilde{\vartheta }\right)& =& \left[\begin{array}{c}f(\vartheta ,\mathit{\vartheta })\\ f(\vartheta ,\mathit{\vartheta })-f(\hat{\vartheta },\hat{\mathit{\vartheta }})\end{array}\right],\mathcal{B}=\left[\begin{array}{c}-B\\ 0\end{array}\right],\beta (\upsilon ,\overline{\upsilon })=\upsilon -\sigma (\upsilon ,\overline{\upsilon }),\mathrm{\Psi }=\left[\begin{array}{c}\phi \\ \phi -\psi \end{array}\right],\hfill \end{array}$$

in which $\tilde{\vartheta }=\left[\begin{array}{cccc}\vartheta & \mathit{\vartheta }& \hat{\vartheta }& \hat{\mathit{\vartheta }}\end{array}\right]$.

Under AS, the OCtrl gains are designed to achieve the local asymptotic stability of system (9) within a state-space region. The next subsection provides the necessary mathematical tools for defining regional stability and designing the OCtrl.

2.3. Essential Definitions and Lemmas

Definition 2 

([4]). The set $\mathcal{S}$ is considered invariant if every trajectory starting within it remains contained within $\mathcal{S}$ for all time.

Definition 3 

([4]). For given $R\in {\mathbb{R}}^{2\mathsf{ı}\times 2\mathsf{ı}}$, a convex, compact ellipsoid $\wp (R,1)\subset \mathcal{S}$ is represented by

$$\wp (R,1)=\{\nu \in {\mathbb{R}}^{2\mathsf{ı}};{\nu }^{T}R\nu \le 1\},$$

(10)

where $R\succ 0$.

Definition 4 

([4]). Let $\mu (t,\mathrm{\Psi })$ represent the state trajectory of system (9) corresponding to $\mathrm{\Psi }\in \mathcal{C}\left(\left[\begin{array}{cc}-\hslash & 0\end{array}\right],{\mathbb{R}}^{2\mathsf{ı}}\right)$. The DOA is given by

$$\aleph =\{\mathrm{\Psi }\in \mathcal{C}\left(\left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right],{\mathbb{R}}^{2\mathsf{ı}}\right)|\underset{t\to \infty }{lim}\mu (t,\mathrm{\Psi })=0\}.$$

(11)

Determining the exact DOA is generally intractable [24]. An approximation of this domain, ${\nabla }_{\alpha }\subset \aleph$, can be expressed as

$${\nabla }_{\alpha }=\{\mathrm{\Psi }\in \mathcal{C}\left(\left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right],{\mathbb{R}}^{2\mathsf{ı}}\right){|\parallel \mathrm{\Psi }\parallel }_s\le \alpha \},$$

(12)

where $\alpha \in {\mathbb{R}}_{>0}$.

Lemma 1 

([25]). Let

$$\begin{array}{c}\hfill \mathrm{\Xi }=\{\xi \in {\mathbb{R}}^j\mid -\overline{\upsilon }\le \upsilon -\xi \le \overline{\upsilon }\}.\end{array}$$

(13)

Then, $\forall \xi \in \mathrm{\Xi }$, we have

$$\begin{array}{c}\hfill {\beta }^T(\upsilon ,\overline{\upsilon })\mathrm{\Lambda }\left(\xi -\beta (\upsilon ,\overline{\upsilon })\right)\ge 0,\forall \mathrm{\Lambda }\in {\mathbb{R}}_{\mathcal{D}}^{j\times j},\end{array}$$

(14)

where $\mathrm{\Lambda }\succ 0$.

Lemma 2 

([26]). Let $\mathcal{G}$, $\mathcal{H}$, $\mathcal{K}$, and $\mathcal{J}$ be dimensionally compatible matrices, and let $\rho \in \mathbb{R}$.

If

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\mathcal{G}& \rho \mathcal{H}+{\mathcal{K}}^T{\mathcal{J}}^T\\ \oplus & -\rho \left({\left\{\mathcal{J}\right\}}_{★}\right)\end{array}\right]\prec 0,\end{array}$$

(15)

then

$$\begin{array}{c}\hfill \mathcal{G}+{\left\{\mathcal{H}\mathcal{K}\right\}}_{★}\prec 0.\end{array}$$

(16)

Similarly, if

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\mathcal{G}& \rho \mathcal{H}-{\mathcal{K}}^T\mathcal{J}\\ \oplus & \rho \left({\left\{\mathcal{J}\right\}}_{★}\right)\end{array}\right]⪰0,\end{array}$$

(17)

then

$$\begin{array}{c}\hfill \mathcal{G}+{\left\{\mathcal{H}\mathcal{K}\right\}}_{★}⪰0.\end{array}$$

(18)

Assumption A1. 

f satisfies the weak QL condition (7) for $\{R_1,U_1,V_1\}\subset {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$ and the QL condition (6) for $\{R_2,U_2,V_2\}\subset {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$. For each k, $R_k$ is a decision-variable matrix, while $U_k$ and $V_k$ are linearly dependent on $R_k$.

3. Main Results

Theorem 1. 

Under Assumption 1, if there exist $\{\mathcal{Z},\mathcal{E}\}\subset {\mathbb{R}}^{j\times \mathsf{ı}}$, $\tilde{\mathrm{\Lambda }}\in {\mathbb{R}}_{\mathcal{D}}^{j\times j}$, and $\{\tilde{\mathcal{X}},\tilde{\mathcal{Y}}\}\subset {\mathbb{R}}^{\mathsf{ı}\times \ell }$ such that

$$\begin{array}{c}\hfill R_k\succ 0,V_k⪰0,\forall k\in {\mathbb{I}}_2,\end{array}$$

(19)

$$\begin{array}{c}\hfill \tilde{\mathrm{\Lambda }}\succ 0,\end{array}$$

(20)

$$\begin{array}{c}\hfill \left[\begin{array}{ccc}{R}_1& 0_{\mathsf{ı}\times \mathsf{ı}}& \mathcal{Z}{\left[p\right]}^T-\mathcal{E}{\left[p\right]}^T\\ \oplus & R_2& -\mathcal{Z}{\left[p\right]}^T+\mathcal{E}{\left[p\right]}^T\\ \oplus & \oplus & \overline{\upsilon }{\left[p\right]}^2\end{array}\right]⪰0,\forall p\in {\mathbb{I}}_j,\end{array}$$

(21)

$$\begin{array}{c}\hfill \tilde{\mathrm{\Omega }}\prec 0,\end{array}$$

(22)

where

$$\begin{array}{ccc}\hfill \tilde{\mathrm{\Omega }}& =& \left[\begin{array}{ccccc}\mathrm{\Omega }(1,1)& -R_{1}B\mathcal{Z}& R_{1}D& 0_{\mathsf{ı}\times \mathsf{ı}}& -R_{1}B\tilde{\mathrm{\Lambda }}+{\mathcal{E}}^T\\ \oplus & \mathrm{\Omega }(2,2)& 0_{\mathsf{ı}\times \mathsf{ı}}& R_{2}D-\tilde{\mathcal{Y}}M& -{\mathcal{E}}^T\\ \oplus & \oplus & \mathrm{\Omega }(3,3)& 0_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \mathrm{\Omega }(4,4)& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \oplus & -2\tilde{\mathrm{\Lambda }}\end{array}\right],\hfill \end{array}$$

in which

$$\begin{array}{ccc}\hfill \mathrm{\Omega }(1,1)& =& {\{R_{1}A+R_{1}B\mathcal{Z}\}}_{★}+2U_1+Q_1,\mathrm{\Omega }(2,2)={\{R_{2}A-\tilde{\mathcal{X}}M\}}_{★}+2U_2+Q_2,\hfill \\ \hfill \mathrm{\Omega }(3,3)& =& -(1-\check{\hslash })Q_1+2V_1,\mathrm{\Omega }(4,4)=-(1-\check{\hslash })Q_2+2V_2,\hfill \end{array}$$

then system (9) is locally asymptotically stable inside the set $\wp (R,1)$, and the estimated DOA is

$$\nabla =\{\mathrm{\Psi }\in \mathcal{C}\left(\left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right],{\mathbb{R}}^{2\mathsf{ı}}\right)|\parallel \mathrm{\Psi }{\parallel }_s\le \alpha \},$$

(23)

where

$$\begin{array}{c}\hfill \alpha ={\displaystyle \frac{1}{\sqrt{\overline{\lambda }\left(R\right)+\overline{\hslash }\overline{\lambda }\left(Q\right)}}},\end{array}$$

and

$$\begin{array}{c}\hfill R=\left[\begin{array}{cc}{R}_1& 0\\ 0& R_2\end{array}\right],Q=\left[\begin{array}{cc}{Q}_1& 0\\ 0& Q_2\end{array}\right].\end{array}$$

(24)

Under these conditions, the OCtrl gains are

$$\begin{array}{c}\hfill \mathcal{X}=R_2^{-1}\tilde{\mathcal{X}},\mathcal{Y}=R_2^{-1}\tilde{\mathcal{Y}},and\mathcal{Z}.\end{array}$$

(25)

Proof. 

Condition (21) implies that

$$\begin{array}{c}\hfill R⪰{\displaystyle \frac{1}{\overline{\upsilon }{\left[p\right]}^2}}\left[\begin{array}{c}\mathcal{Z}{\left[p\right]}^T-\mathcal{E}{\left[p\right]}^T\\ -\mathcal{Z}{\left[p\right]}^T+\mathcal{E}{\left[p\right]}^T\end{array}\right]\left[\begin{array}{cc}\mathcal{Z}\left[p\right]-\mathcal{E}\left[p\right]& -\mathcal{Z}\left[p\right]+\mathcal{E}\left[p\right]\end{array}\right],\end{array}$$

(26)

which leads to

$$\begin{array}{c}\hfill \mu R\mu ⪰{\displaystyle \frac{1}{\overline{\upsilon }{\left[p\right]}^2}}{(\upsilon \left[p\right]-\xi \left[p\right])}^2,\end{array}$$

(27)

where $\xi =\mathcal{E}\hat{\vartheta }$.

For $\mu \in \wp (R,1)$, (27) implies that $\xi \in \mathrm{\Xi }$, where the set $\mathrm{\Xi }$ is defined in (13).

By applying Lemma 1, it follows that

$$\begin{array}{c}\hfill {\beta }^T(\upsilon ,\overline{\upsilon })\mathrm{\Lambda }\left(\mathcal{E}\hat{\vartheta }-\beta (\upsilon ,\overline{\upsilon })\right)\ge 0,\forall \mathrm{\Lambda }\in {\mathbb{R}}_{\mathcal{D}}^{j\times j},\end{array}$$

(28)

where $\mathrm{\Lambda }\succ 0$.

(28) may be expressed as

$$\begin{array}{c}\hfill {\beta }^T(\upsilon ,\overline{\upsilon })\mathrm{\Lambda }\left(\tilde{\mathcal{E}}\mu -\beta (\upsilon ,\overline{\upsilon })\right)\ge 0,\end{array}$$

(29)

where $\tilde{\mathcal{E}}=\left[\begin{array}{cc}\mathcal{E}& -\mathcal{E}\end{array}\right]$.

We now introduce the following LK functional:

$$\begin{array}{c}\hfill \mathrm{\Gamma }\left({\mu }_t\right)={\mu }^{T}R\mu +{\int }_t^{t+\hslash \left(t\right)}{\mu }^T(s-\hslash \left(t\right))Q\mu (s-\hslash \left(t\right))ds,\end{array}$$

(30)

where ${\mu }_t:=\{\mu (t+\delta ),\delta \in \left[\begin{array}{cc}-\overline{\hslash }& 0\end{array}\right]\}$.

Then, we get

$$\begin{array}{c}\hfill \dot{\mathrm{\Gamma }}\left({\mu }_t\right)\le 2{\mu }^{T}R\left(\mathcal{A}\mu +\mathcal{D}\mu +\mathcal{F}\left(\tilde{\vartheta }\right)+\mathcal{B}\beta (\upsilon ,\overline{\upsilon })\right)+{\mu }^{T}Q\mu -(1-\check{\hslash })\mathit{\mu }Q\mathit{\mu },\end{array}$$

(31)

We have

$$\begin{array}{c}\hfill 2{\mu }^{T}R\mathcal{F}\left(\tilde{\vartheta }\right)=2\langle R_{1}f(\vartheta ,\mathit{\vartheta }),\vartheta \rangle +\langle R_2(f(\vartheta ,\mathit{\vartheta })-f(\hat{\vartheta },\hat{\mathit{\vartheta }})),\vartheta -\hat{\vartheta }\rangle .\end{array}$$

(32)

Since f satisfies the weak QL condition (7) for $\{R_1,U_1,V_1\}\subset {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$ and the QL condition (6) for $\{R_2,U_2,V_2\}\subset {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$, from (32), we obtain

$$\begin{array}{c}\hfill 2{\mu }^{T}R\mathcal{F}\left(\tilde{\vartheta }\right)\le {\left[\begin{array}{c}\vartheta \\ \mathit{\vartheta }\end{array}\right]}^T\left[\begin{array}{cc}2U_1& 0\\ 0& 2V_1\end{array}\right]\left[\begin{array}{c}\vartheta \\ \mathit{\vartheta }\end{array}\right]+{\left[\begin{array}{c}\vartheta -\hat{\vartheta }\\ \mathit{\vartheta }-\hat{\mathit{\vartheta }}\end{array}\right]}^T\left[\begin{array}{cc}2U_2& 0\\ 0& 2V_2\end{array}\right]\left[\begin{array}{c}\vartheta -\hat{\vartheta }\\ \mathit{\vartheta }-\hat{\mathit{\vartheta }}\end{array}\right].\end{array}$$

(33)

Let

$$\begin{array}{ccc}\hfill \mathcal{J}& =& 2{\mu }^{T}R\left(\mathcal{A}\mu +\mathcal{D}\mathit{\mu }+\mathcal{F}\left(\tilde{\vartheta }\right)+\mathcal{B}\beta (\upsilon ,\overline{\upsilon })\right)+{\mu }^{T}Q\mu -(1-\check{\hslash })\mathit{\mu }Q\mathit{\mu }\hfill \\ & & +2{\beta }^T(\upsilon ,\overline{\upsilon })\mathrm{\Lambda }\left(\tilde{\mathcal{E}}\mu -\beta (\upsilon ,\overline{\upsilon })\right),\hfill \end{array}$$

(34)

From (33), we obtain

$$\begin{array}{ccc}\hfill \mathcal{J}& \le & {\left[\begin{array}{c}\mu \\ \mathit{\mu }\\ \beta (\upsilon ,\overline{\upsilon })\end{array}\right]}^T\mathrm{\Omega }\left[\begin{array}{c}\mu \\ \mathit{\mu }\\ \beta (\upsilon ,\overline{\upsilon })\end{array}\right],\hfill \end{array}$$

(35)

where

$$\begin{array}{ccc}\hfill \mathrm{\Omega }& =& \left[\begin{array}{ccccc}\mathrm{\Omega }(1,1)& -R_{1}B\mathcal{Z}& R_{1}D& 0_{\mathsf{ı}\times \mathsf{ı}}& -R_{1}B+{\mathcal{E}}^T\mathrm{\Lambda }\\ \oplus & \mathrm{\Omega }(2,2)& 0_{\mathsf{ı}\times \mathsf{ı}}& R_{2}D-\tilde{\mathcal{Y}}M& -{\mathcal{E}}^T\mathrm{\Lambda }\\ \oplus & \oplus & \mathrm{\Omega }(3,3)& 0_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \mathrm{\Omega }(4,4)& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \oplus & -2\mathrm{\Lambda }\end{array}\right].\hfill \end{array}$$

By pre- and post-multiplying $\mathrm{\Omega }$ by $diag(I_{4\mathsf{ı}\times 4\mathsf{ı}},\tilde{\mathrm{\Lambda }})$, where $\tilde{\mathrm{\Lambda }}={\mathrm{\Lambda }}^{-1}$, we get $\tilde{\mathrm{\Omega }}$. Subsequently, from (22), we deduce that $\mathcal{J}<0$, which, according to (29), implies that $\dot{\mathrm{\Gamma }}\left({\mu }_t\right)<0$.

Thus, $\mathrm{\Gamma }\left({\mu }_t\right)\le \mathrm{\Gamma }\left(\mathrm{\Psi }\right)$, and hence

$$\begin{array}{c}\hfill {\mu }^{T}R\mu \le \mathrm{\Gamma }\left({\mu }_t\right)\le \mathrm{\Gamma }\left(\mathrm{\Psi }\right)\le \left(\overline{\lambda }\left(R\right)+\overline{\hslash }\overline{\lambda }\left(Q\right)\right){\parallel \mathrm{\Psi }\parallel }_s^2.\end{array}$$

(36)

Clearly, all state trajectories originating from the initial conditions within the estimated DOA (23) remain confined inside the ellipsoidal region $\wp (R,1)$. □

Remark 2. 

Condition (22) in Theorem 1 is not an LMI due to the presence of $R_1$. It becomes an LMI only if $R_1$ is fixed rather than treated as a decision variable. Since f satisfies the weak QL condition (7) for $\{R_1,U_1,V_1\}\subset {\mathbb{R}}^{\mathsf{ı}\times \mathsf{ı}}$, $U_1$ and $V_1$ therefore depend on $R_1$. Consequently, they are also fixed and cannot be treated as decision variables. Fixing $\{R_1,U_1,V_1\}$ or treating them as tuning variables will inevitably introduce conservatism. Next, we propose a new result in which the conditions are expressed as LMIs and $\{R_1,U_1,V_1\}$ are treated as decision variables.

Theorem 2. 

Under Assumption A1, for a given $\rho \in \mathbb{R}$ and $\tilde{\mathrm{\Lambda }}\in {\mathbb{R}}_{\mathcal{D}}^{j\times j}$ with $\tilde{\mathrm{\Lambda }}\succ 0$, if there exist $\{\tilde{\mathcal{X}},\tilde{\mathcal{Y}}\}\subset {\mathbb{R}}^{\mathsf{ı}\times \ell }$, $\{\tilde{\mathcal{Z}},\mathcal{E}\}\subset {\mathbb{R}}^{j\times \mathsf{ı}}$, and $\tilde{\mathcal{J}}\in {\mathbb{R}}^{j\times j}$ such that (19) and the following LMIs are satisfied:

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\mathcal{G}}_{1p}& {\tilde{\mathcal{H}}}_{1p}\\ \oplus & {\left\{\tilde{\mathcal{J}}\right\}}_{★}\end{array}\right]⪰0,\forall p\in {\mathbb{I}}_j,\end{array}$$

(37)

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\overline{\mathrm{\Omega }}& {\overline{\mathcal{H}}}_2\\ \oplus & -{\left\{\tilde{\mathcal{J}}\right\}}_{★}\end{array}\right]\prec 0,\end{array}$$

(38)

where

$$\begin{array}{c}\hfill {\mathcal{G}}_{1p}=\left[\begin{array}{ccc}{R}_1& 0_{\mathsf{ı}\times \mathsf{ı}}& -\mathcal{E}{\left[p\right]}^T\\ \oplus & R_2& \mathcal{E}{\left[p\right]}^T\\ \oplus & \oplus & \overline{\upsilon }{\left[p\right]}^2\end{array}\right],{\tilde{\mathcal{H}}}_{1p}=\left[\begin{array}{c}-\rho {\tilde{\mathcal{Z}}}^T\\ \rho {\tilde{\mathcal{Z}}}^T\\ e_p^T\end{array}\right],\end{array}$$

$$\begin{array}{ccc}\hfill \overline{\mathrm{\Omega }}& =& \left[\begin{array}{ccccc}\overline{\mathrm{\Omega }}(1,1)& -B\tilde{\mathcal{Z}}& R_{1}D& 0_{\mathsf{ı}\times \mathsf{ı}}& -R_{1}B\tilde{\mathrm{\Lambda }}+{\mathcal{E}}^T\\ \oplus & \mathrm{\Omega }(2,2)& 0_{\mathsf{ı}\times \mathsf{ı}}& R_{2}D-\tilde{\mathcal{Y}}M& -{\mathcal{E}}^T\\ \oplus & \oplus & \mathrm{\Omega }(3,3)& 0_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \mathrm{\Omega }(4,4)& 0_{\mathsf{ı}\times j}\\ \oplus & \oplus & \oplus & \oplus & -2\tilde{\mathrm{\Lambda }}\end{array}\right],{\overline{\mathcal{H}}}_2=\left[\begin{array}{c}{\overline{\mathcal{H}}}_2\left(1\right)\\ -{\tilde{\mathcal{Z}}}^T\\ 0_{\mathsf{ı}\times j}\\ 0_{\mathsf{ı}\times j}\\ 0_{j\times j}\end{array}\right],\hfill \end{array}$$

in which $e_p\in {\mathbb{R}}^j$ is the p-th canonical basis vector of ${\mathbb{R}}^j$, i.e., $e_p$ has a 1 in the p-th position and 0 elsewhere, $e_p={\left[\begin{array}{c}0,\dots ,0,1,0,\dots ,0\end{array}\right]}^T$,

$$\begin{array}{ccc}\hfill \overline{\mathrm{\Omega }}(1,1)& =& {\{R_1\mathrm{A}+\mathrm{B}\tilde{\mathcal{Z}}\}}_{★}+2U_1+Q_1,{\overline{\mathcal{H}}}_2\left(1\right)=\rho R_{1}B-B\tilde{\mathcal{J}}+{\tilde{\mathcal{Z}}}^T.\hfill \end{array}$$

Then, all requirements in Theorem 1 are met for

$$\begin{array}{ccc}\hfill \mathcal{Z}& =& {\mathcal{J}}^{-1}\tilde{\mathcal{Z}},\hfill \end{array}$$

(39)

with

$$\begin{array}{c}\hfill \mathcal{J}={\displaystyle \frac{1}{\rho }}\tilde{\mathcal{J}}.\end{array}$$

(40)

Hence, system (9) is locally asymptotically stable inside the set $\wp (R,1)$, with the estimated DOA given in (23), and the OCtrl gains are

$$\begin{array}{ccc}\hfill \mathcal{Z}& =& {\mathcal{J}}^{-1}\tilde{\mathcal{Z}},\mathcal{X}=R_2^{-1}\tilde{\mathcal{X}},\mathcal{Y}=R_2^{-1}\tilde{\mathcal{Y}}.\hfill \end{array}$$

(41)

Proof. 

In the first part, we prove that (37) leads to (21).

${\tilde{\mathcal{H}}}_{1p}$ can be formulated as

$$\begin{array}{c}\hfill {\tilde{\mathcal{H}}}_{1p}=\rho {\mathcal{H}}_1-{\mathcal{K}}_{1p}^T\mathcal{J},\end{array}$$

(42)

where

$$\begin{array}{ccc}\hfill {\mathcal{H}}_1=\left[\begin{array}{c}-{\tilde{\mathcal{Z}}}^T\\ {\tilde{\mathcal{Z}}}^T\\ 0_{1\times j}\end{array}\right],{\mathcal{K}}_{1p}& =& -{\mathcal{J}}^{-T}\left[\begin{array}{ccc}{0}_{j\times \mathsf{ı}}& 0_{j\times \mathsf{ı}}& e_p\end{array}\right].\hfill \end{array}$$

Then, (37) is equivalently expressed as

$$\begin{array}{c}\hfill \left[\begin{array}{cc}{\mathcal{G}}_{1p}& \rho {\mathcal{H}}_1-{\mathcal{K}}_{1p}^T\mathcal{J}\\ \oplus & \rho \left({\left\{\mathcal{J}\right\}}_{★}\right)\end{array}\right]⪰0,\forall p\in {\mathbb{I}}_j.\end{array}$$

(43)

By Lemma 2, we have

$$\begin{array}{c}\hfill {\mathcal{G}}_{1p}+{\left\{{\mathcal{H}}_1{\mathcal{K}}_{1p}\right\}}_{★}⪰0,\forall p\in {\mathbb{I}}_j.\end{array}$$

(44)

From (39), we obtain

$$\begin{array}{c}\hfill {\left\{{\mathcal{H}}_1{\mathcal{K}}_{1p}\right\}}_{★}=\left[\begin{array}{ccc}{0}_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times \mathsf{ı}}& {\mathcal{Z}}^T{e}_p\\ 0_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times \mathsf{ı}}& -{\mathcal{Z}}^T{e}_p\\ 0_{\mathsf{ı}\times \mathsf{ı}}& 0_{\mathsf{ı}\times \mathsf{ı}}& 0\end{array}\right].\end{array}$$

(45)

As ${\mathcal{Z}}^T{e}_p=\mathcal{Z}{\left[p\right]}^T$, (44) takes the form of (21).

Next, we demonstrate that (38) leads to (22).

${\overline{\mathcal{H}}}_2$ can be written in the form

$$\begin{array}{c}\hfill {\overline{\mathcal{H}}}_2=\rho {\mathcal{H}}_2+{\mathcal{K}}_2^T{\mathcal{J}}^T,\end{array}$$

(46)

where

$$\begin{array}{ccc}\hfill {\mathcal{H}}_2=\left[\begin{array}{c}{\mathcal{H}}_2\left(1\right)\\ 0_{\mathsf{ı}\times j}\\ 0_{\mathsf{ı}\times j}\\ 0_{\mathsf{ı}\times j}\\ 0_{j\times j}\end{array}\right],{\mathcal{K}}_2& =& {\mathcal{J}}^{-1}\left[\begin{array}{ccccc}\tilde{\mathcal{Z}}& -\tilde{\mathcal{Z}}& 0_{j\times \mathsf{ı}}& 0_{j\times \mathsf{ı}}& 0_{j\times j}\end{array}\right],\hfill \end{array}$$

(47)

in which ${\mathcal{H}}_2\left(1\right)=R_{1}B-B\mathcal{J}$.

Hence, (38) takes the equivalent form

$$\begin{array}{c}\hfill \left[\begin{array}{cc}\overline{\mathrm{\Omega }}& \rho {\mathcal{H}}_2+{\mathcal{K}}_2^T{\mathcal{J}}^T\\ \oplus & -\rho \left({\left\{\mathcal{J}\right\}}_{★}\right)\end{array}\right]\prec 0,\end{array}$$

(48)

which, by Lemma 2, implies that

$$\begin{array}{c}\hfill \overline{\mathrm{\Omega }}+{\left\{{\mathcal{H}}_2{\mathcal{K}}_2\right\}}_{★}\prec 0.\end{array}$$

(49)

□

The following optimization problem aims to maximize the estimated DOA:

Minimize ${\lambda }_1+\overline{\hslash }{\lambda }_2$ subject to (19–37,38) and

$$\begin{array}{c}\hfill {\lambda }_{1}I-R⪰0,{\lambda }_{2}I-Q⪰0.\end{array}$$

(50)

Remark 3. 

The work in [23] addresses the OC of time-delay QL (TQL) systems with AS. However, it does not consider the estimation of the DOA. Estimating this domain is particularly important, as it defines the set of initial conditions from which the system is guaranteed to converge to zero despite actuator limitations. In our work, we explicitly estimate the DOA, providing a more complete characterization of system stability.

Remark 4. 

In [21,22,23], the decoupling technique relies on defining new decision variables, ${\tilde{U}}_1=R_1^{-1}{U}_1{R}_1^{-1}$ and ${\tilde{V}}_1=R_1^{-1}{V}_1{R}_1^{-1}$. However, this approach is not applicable in our case, since $U_1$ and $V_1$ depend linearly on $R_1$. To overcome this difficulty, we employ an alternative decoupling technique that avoids the need for such pre- and post-multiplications.

Remark 5. 

Theorem 2 presents a novel condition for designing an OCtrl for TQL systems under AS. In this design, the conditions depend on the tuning parameters, which include a diagonal matrix $\tilde{\mathrm{\Lambda }}\succ 0$ and a scalar ρ, potentially introducing some conservatism. However, when $U_1$ depends linearly on $R_1$, the new conditions provided are particularly effective in handling this case.

Remark 6. 

To determine $\tilde{\mathrm{\Lambda }}\succ 0$ and ρ that provide feasible solutions to the minimization problem (50), a method was proposed in [27]. It is based on solving a collection of LMI conditions with $\{\eta ,\rho \}\subset \{{10}^{-3},\dots ,{10}^3\}$, where $\tilde{\mathrm{\Lambda }}$ is defined as $\tilde{\mathrm{\Lambda }}=\eta I$. The logarithmically spaced search has been widely tested in previous studies [16].

For each feasible solution, a corresponding minimum value of ${\lambda }_1+\overline{\hslash }{\lambda }_2$ is obtained. The suboptimal solution is the one that achieves the smallest ${\lambda }_1+\overline{\hslash }{\lambda }_2$ among them.

Remark 7. 

Although the present work offers notable advantages, particularly by estimating the DOA and using a decoupling technique to reduce conservatism, further improvements are possible. Specifically, using an improved LK functional with a double integral term instead of a simple integral term could lead to delay-dependent results and further reduce conservatism. However, this approach may introduce additional challenges, and it remains to be investigated whether the decoupling technique can still be effectively applied. This is an open problem for future research.

4. Numerical Example

Consider the system described by (1), where

$$\begin{array}{ccc}\hfill A& =& \left[\begin{array}{ccc}0.1& 0& 0\\ 0& -1& 0\\ 0& 0.5& 1\end{array}\right],D=\left[\begin{array}{ccc}0.1& 0& 0.2\\ 0& 0.1& 0.2\\ 0& 0& 0.1\end{array}\right],B=\left[\begin{array}{c}1\\ 0\\ 0\end{array}\right],M=\left[\begin{array}{ccc}1& 0& 0\\ 0& 1& 0\end{array}\right],\hfill \\ \hfill f(\vartheta ,\mathit{\vartheta })& =& \left[\begin{array}{c}0\\ 0\\ 0.25sin\left(\vartheta \right[3\left]\right)+025sin\left(\mathit{\vartheta }\right[3\left]\right)\end{array}\right],\hfill \end{array}$$

Figure 1 illustrates the dynamic evolution of $\vartheta \left[1\right]$ for $\hslash \left(t\right)=0.5+0.05sin\left(t\right)$, ($\overline{\hslash }=0.55$ and $\check{\hslash }=0.05$), $\phi =\left[\begin{array}{ccc}-1& 2& 1\end{array}\right]$, and $\upsilon =0$, revealing that the system exhibits divergent behavior, which indicates open-loop instability.

Let $R=\left[\begin{array}{ccc}{r}_{11}& r_{12}& r_{13}\\ \oplus & r_{22}& r_{23}\\ \oplus & \oplus & r_{33}\end{array}\right]$. Then, we get

$$\begin{array}{ccc}\hfill & & \langle R(f(\vartheta ,\mathit{\vartheta })-f(\hat{\vartheta },\hat{\mathit{\vartheta }})),\vartheta -\hat{\vartheta }\rangle ={[\vartheta -\hat{\vartheta }]}^{T}R[f(\vartheta ,\mathit{\vartheta })-f(\hat{\vartheta },\hat{\mathit{\vartheta }})]\hfill \\ & & \le {\left[\begin{array}{c}\vartheta -\hat{\vartheta }\\ \mathit{\vartheta }-\hat{\mathit{\vartheta }}\end{array}\right]}^T\left[\begin{array}{cc}U& 0\\ 0& V\end{array}\right]\left[\begin{array}{c}\vartheta -\hat{\vartheta }\\ \mathit{\vartheta }-\mathit{\vartheta }\end{array}\right],\hfill \end{array}$$

(51)

where

$$\begin{array}{c}\hfill U=\left[\begin{array}{ccc}0.125r_{13}& 0& 0.125r_{13}\\ \oplus & 0.125r_{23}& 0.125r_{23}\\ \oplus & \oplus & 0.375r_{33}\end{array}\right],V=\left[\begin{array}{ccc}0& 0& 0\\ \oplus & 0& 0\\ \oplus & \oplus & V_{33}\end{array}\right],\end{array}$$

(52)

in which $V_{33}=0.125r_{13}+0.125r_{23}+0.125r_{33}$.

Let $\overline{\upsilon }=5$. For $k\in {\mathbb{I}}_2$, the matrices $R_k$, $U_k$, and $V_k$ have the same form as R, U, and V, respectively. Considering $\tilde{\mathrm{\Lambda }}=0.1$ and $\rho =10$, a feasible solution to the minimization problem (50) is found as follows:

$$\begin{array}{c}\hfill \mathcal{Z}=\left[\begin{array}{ccc}-0.2705& -0.0125& -0.0146\end{array}\right],\mathcal{X}={10}^3\left[\begin{array}{cc}2.5508& -0.0072\\ -0.0127& 2.8181\\ 0.0197& 0.0377\end{array}\right],\mathcal{Y}=\left[\begin{array}{cc}0.1& 0\\ 0& 0.1\\ 0& 0\end{array}\right]\end{array}$$

(53)

An estimation of the DOA is given by the set in (23), where $\alpha =3.8347$.

We select ${\phi }_1=\left[\begin{array}{ccc}-1& 2& 1\end{array}\right]$ and $\psi =\left[\begin{array}{ccc}0& 0& 0\end{array}\right]$, which lie inside the DOA. Figure 2, Figure 3 and Figure 4 show $\vartheta \left[1\right]$, $\vartheta \left[2\right]$, $\vartheta \left[3\right]$, and their estimates, respectively. These figures illustrate that the closed-loop system exhibits asymptotic stability. Furthermore, the observer accurately estimates the states. Figure 5 depicts the control input.

Figure 6 illustrates the time evolution of $\vartheta \left[1\right]$ in the open-loop case for the initial conditions ${\phi }_1$, ${\phi }_2=\left[\begin{array}{ccc}-1& 1& 2\end{array}\right]$, and ${\phi }_3=\left[\begin{array}{ccc}1& 1& -2\end{array}\right]$. These initial conditions all belong to the set ${\nabla }_{\alpha }$ defined in (12), and their trajectories indicate the instability of the system.

Now, Figure 7 shows the time evolution of the closed-loop states ${\phi }_1$, ${\phi }_2$, and ${\phi }_3$ in the phase plane. All states originating from the selected initial conditions converge to the equilibrium at zero, demonstrating the system’s stability under the designed control.

5. Conclusions

This work focuses on the development of a novel OC strategy for TQL systems with AS. We considered the case where the matrices used to verify the QL condition were treated as decision variables in the design. Criteria guaranteeing the asymptotic stability of the augmented system were first derived using an LK functional. Furthermore, an estimate of the DOA was provided. The resulting conditions were not LMIs, as the matrices associated with the QL condition are incorporated as decision variables in the design. An appropriate decoupling technique, which accounts for the linear dependence of some decision variables on others, was adopted to derive the LMI conditions. To validate the proposed scheme, a numerical example was presented. It should be noted that the proposed design is formulated with two tuning variables, which may increase conservatism. Exploring other decoupling methods to reduce this conservatism represents an important direction for future research.