Controller Design for Continuous-Time Linear Control Systems with Time-Varying Delay

Abstract

This paper addresses the controller design problem for linear systems with time-varying delays. By constructing a novel Lyapunov–Krasovskii functional incorporating delay-partitioning techniques, we establish delay-dependent stability criteria for the solvability of the robust stabilization problem. The derived conditions are formulated as linear matrix inequalities (LMIs) that become affine upon fixing a single scalar parameter, thereby facilitating efficient numerical computation. Furthermore, these criteria guarantee that the reachable set of the closed-loop system remains bounded within a prescribed ellipsoid under zero initial conditions. The effectiveness and superiority of the proposed approach are demonstrated through two comparative numerical examples, including a benchmark problem with varying delay.

1. Introduction

Time delays are ubiquitous in practical engineering systems, including aircraft flight control design [1], chemical process regulation [2], and long-distance pipeline networks [3]. The design of robust controllers for systems with time-varying delays constitutes a challenging yet critical research area in modern control theory, as such delays can severely degrade system stability and dynamic performance. Consequently, developing control strategies with guaranteed robustness against delay uncertainties has become imperative for ensuring reliable operation of safety-critical systems [4,5,6,7,8,9,10].

In the domain of time-delay control systems, both linear and nonlinear, diverse methodologies have been proposed for robust controller synthesis. For continuous-time systems with time-varying delays, Lyapunov–Krasovskii functional (LKF)-based approaches remain prominent. Specifically, a tailored LKF for continuous-time Takagi–Sugeno fuzzy systems with time-varying delays has been introduced [11], which reduces the computational complexity of stability analysis by minimizing the required linear matrix inequality (LMI) count. This enables the derivation of larger permissible delay bounds, thereby enhancing system robustness. Complementarily, reinforcement learning (RL) techniques have emerged as a powerful alternative for robust controller design, particularly for continuous-time uncertain nonlinear systems with input constraints [12]. By reformulating the robust control problem as a constrained optimal control task, RL-based methods generate approximate optimal control policies that maintain stability despite parametric uncertainties and time-varying delays. Another significant advancement involves Lyapunov-based composite nonlinear feedback (CNF) controller design for systems with time-varying delays and input saturation [13]. This approach formulates the controller parameterization as an LMI optimization problem, ensuring robust reference tracking and stability under external disturbances and delay variations. Collectively, these methodologies underscore the critical role of advanced mathematical tools—including Lyapunov theory, RL optimization, and LMI-based synthesis—in developing high-performance controllers for time-delay systems. By integrating these techniques, researchers can systematically address the inherent challenges of delay-induced instability, enabling more reliable and efficient control system implementations.

This paper addresses the robust controller design problem for linear time-delay systems with time-varying delays, focusing on reachable set estimation and confinement within prescribed safety bounds. The coexistence of exogenous disturbances and time-varying delays often leads to performance degradation and potential instability, necessitating precise characterization of the system’s reachable state space to ensure safe operation. While existing literature has extensively studied stability analysis for time-delay systems, the synthesis of controllers with explicit reachability set constraints remains an open challenge—a critical gap since operational safety requires strict confinement of system states within predefined safe regions. For reachable set estimation of time-delay control systems, the predominant approaches in current research employ Lyapunov–Krasovskii functionals combined with linear matrix inequality (LMI) formulations [14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34]. Building upon our recently developed reachability analysis framework for continuous-time linear time-delay systems [35], this paper proposes a novel controller synthesis methodology that guarantees the closed-loop system’s reachable set remains strictly contained within the estimated safe region. The key innovation lies in the integration of delay-dependent reachability bounds into the controller design process, enabling simultaneous stabilization and safety verification through convex optimization.

The principal contributions of this paper are summarized as follows:

• Delay-Dependent Reachable Set Characterization: We derive a sufficient linear matrix inequality (LMI) condition for determining the minimal admissible bounding ellipsoid of the reachable set for linear time-delay systems. This condition explicitly accounts for time-varying delays through delay-dependent Lyapunov–Krasovskii functionals, enabling tighter state space confinement compared to delay-independent approaches.
• Optimal Ellipsoidal-Bounded Controller Synthesis: We propose a state feedback controller design methodology that simultaneously (1) guarantees the reachable set of the closed-loop system is contained within a prescribed ellipsoid and (2) minimizes the volume of this bounding ellipsoid through convex optimization, which means that the resulting controller ensures robust stability while optimizing operational safety margins under worst-case delay scenarios.

Throughout this paper, we adopt standard mathematical notation consistent with the control theory literature. Specifically, ${\mathbb{R}}^n$ is the vector of real numbers, ${\mathbb{R}}^{n\times m}$ is the $n\times m$ real matrix, I is the identity matrix, 0 is the zero matrix, and $A^T$ presents the transpose of A. For a matrix P, $P>0$ denotes P is a symmetric positive definite matrix, $x_t\left(\theta \right)=x(t+\theta ),\theta \in [-h,0]$, and $(\star )$ in a matrix represents the symmetric part.

2. Problem Statement and Preliminary

Consider the following linear time-delay control system with disturbances:

$$\left\{\begin{array}{c}\dot{x}\left(t\right)=Ax\left(t\right)+Dx(t-d\left(t\right))+Bu\left(t\right)+Ew\left(t\right),\hfill \\ x\left(t\right)\equiv 0,t\in [-h,0],\hfill \end{array}\right.$$

(1)

where $x\left(t\right)\in {\mathbb{R}}^n$ is the state vector, $u\left(t\right)$ is the control vector, $A\in {\mathbb{R}}^{n\times n}$, $D\in {\mathbb{R}}^{n\times n}$, $B\in {\mathbb{R}}^{n\times n}$ and $E\in {\mathbb{R}}^{n\times n}$ are constant matrices. The disturbance $w\left(t\right)$ satisfies

$$w^T\left(t\right)w\left(t\right)\le w_m^2$$

(2)

and the time-varying delay $d\left(t\right)$ meets the following conditions:

$$0\le d\left(t\right)\le h,\left|\dot{d}\left(t\right)\right|\le \mu \le 1,$$

(3)

where h, $\mu$ and $w_m$ are constants.

The reachable set $R_x$ of the time-delay linear control system (1) with bounded disturbances (2) is defined in the following form in which

$${\mathcal{R}}_x=\left\{x\left(t\right)\in {\mathbb{R}}^n|x\left(t\right)\mathrm{and}w\left(t\right)\mathrm{meet}\mathrm{conditions}\left(1\right)-\left(3\right),t\ge 0\right\}.$$

For a positive definite matrix, we define an ellipsoid in the following $\mathcal{E}\left(P\right)$ in which

$$\mathcal{E}\left(P\right)=\{x\in {\mathbb{R}}^n:x^T\left(t\right)Px\left(t\right)\le 1,P>0\}.$$

(4)

In this paper, we intend to design the following state feedback control law with time delay

$$u\left(t\right)=Kx\left(t\right)+Gx(t-d(t\left)\right).$$

Considering the control law $u\left(t\right)$, the reachable set of the resulting closed-loop system (5)

$$\dot{x}\left(t\right)=(A+BK)x\left(t\right)+Ew\left(t\right)+(D+BG)x(t-d\left(t\right))$$

(5)

is bounded by a given ellipsoid $\mathcal{E}\left(P\right)$, where K, G is the controller gain to be determined.

Lemma 1 

([36]). The following relationship

$${\displaystyle \frac{d}{dt}}{\int }_{b\left(t\right)}^{a\left(t\right)}f\left(t,s\right)ds=\dot{a}\left(t\right)f[t,a\left(t\right)]-\dot{b}\left(t\right)f[t,b\left(t\right)]+{\int }_{b\left(t\right)}^{a\left(t\right)}{\displaystyle \frac{\delta }{\delta t}}f(t,s)ds$$

is known as Leibniz’s rule.

Lemma 2 

([37]). For a matrix $M\in {\mathbb{R}}^{m\times n}$, $M>0$ and parameters $b>a>0$, as well as the vector function $f(·):[a,b]\to {\mathbb{R}}^n$, the following inequality holds:

$$\begin{array}{cc}\hfill -\left(b-a\right){\int }_a^b{f}^T\left(s\right)Mf\left(s\right)ds& \le -{\left[{\int }_a^{b}f\left(s\right)ds\right]}^{T}M\left[{\int }_a^{b}f\left(s\right)ds,\right]\hfill \\ \hfill -{\displaystyle \frac{b^2-a^2}{2}}{\int }_{-b}^{-a}{\int }_{t+\theta }^t{f}^T\left(s\right)Mf\left(s\right)dsd\theta & \le -{\left({\int }_{-b}^{-a}{\int }_{t+\theta }^{t}f\left(s\right)dsd\theta \right)}^{T}M\left({\int }_{-b}^{-a}{\int }_{t+\theta }^{t}f\left(s\right)dsd\theta \right).\hfill \end{array}$$

Lemma 3 

([38]). S is a symmetric positively definite matrix, and the sufficient and necessary conditions for $S=\left[\begin{array}{cc}{S}_{11}& S_{12}\\ S_{12}^T& S_{22}\end{array}\right]<0$ is

$$S<0;$$

$$S_{11}<0,S_{22}-S_{12}^T{S}_{11}^{-1}{S}_{12}<0;$$

$$S_{22}<0,S_{11}-S_{12}{S}_{22}^{-1}{S}_{12}^T<0;$$

Lemma 4 

([39]). Let $V\left(t\right)$ be a positively definite functional and $V\left(x\left(0\right)\right)=0,w^T\left(t\right)w\left(t\right)\le w_m^2,$ if there exists a parameter $\alpha >0$ such that

$${\displaystyle \frac{d}{dt}}V\left(t\right)+\alpha V\left(t\right)-{\displaystyle \frac{\alpha }{w_m^2}}{w}^T\left(t\right)w\left(t\right)\le 0,$$

then

$$V\left(x_t\right)\le 1,\forall t>0.$$

Proof. 

Multiply both sides of the given equation by $e^{\alpha t}$; then

$$\begin{array}{cc}\hfill e^{\alpha t}{\displaystyle \frac{d}{dt}}V\left(t\right)+\alpha e^{\alpha t}V\left(t\right)& \le e^{\alpha t}{\displaystyle \frac{\alpha }{w_m^2}}{w}^{\top }\left(t\right)w\left(t\right),\hfill \\ \hfill e^{\alpha t}V\left(t\right)& \le e^{\alpha t}{\displaystyle \frac{\alpha }{w_m^2}}{w}^{\top }\left(t\right)w\left(t\right),\hfill \end{array}$$

integrate the inequality given above from 0 to t

$$\begin{array}{cc}\hfill e^{\alpha t}V\left(t\right)& \le {\int }_0^t{\displaystyle \frac{\alpha }{w_m^2}}{e}^{\alpha t}{w}^T\left(t\right)w\left(t\right)dt\le {\int }_0^t\alpha e^{\alpha t}dt=e^{\alpha t}-1.\hfill \end{array}$$

Then, we have $V\left(x_t\right)\le 1$, $\forall t\ge 0.$ □

3. Controller Design for Continuous-Time Linear Control Systems with Time-Varying Delay

Theorem 1. 

For a given scalar h, $\mu >0$, if there exists a parameter $\alpha >0$, matrices $\overline{P}>0$, $\overline{R}>0$, $\overline{Q}>0$, $\overline{W}>0$ and any matrix of appropriate dimensions L, H, $\overline{M}$, $\overline{Z}$, $\overline{F}$, $\overline{U}$, $R_1$, $U_1$, $U_2$ satisfying the following LMIs:

$$\left[\begin{array}{cc}-\overline{P}& \overline{M}\\ \overline{M}& \overline{P_r}\end{array}\right]\le 0,$$

(6)

$$\left[\begin{array}{cc}\overline{M}-\overline{P}& \overline{F}\\ \star & \overline{U}+{\displaystyle \frac{1}{h}}{e}^{-\alpha h}\overline{Q}\end{array}\right]\ge 0,$$

(7)

$$\left[\begin{array}{ccccc}{\overline{\Phi }}_{11}& D\overline{M}+BH-\overline{F}& \overline{M}{A}^T+L^T{B}^T+U_1+\alpha \overline{M}& E& \overline{F}\\ \star & -(1-\mu )e^{-\alpha h}\overline{Q}+{\mu }^2\overline{W}& \overline{M}{D}^T+H^T{B}^T-U_1& 0& 0\\ \star & \star & -{\displaystyle \frac{1}{h}}{e}^{-\alpha h}{R}_1+\alpha U_2& E& U_1^T\\ \star & \star & \star & -{\displaystyle \frac{\alpha }{w_m^2}}& 0\\ \star & \star & \star & \star & -\overline{W}\end{array}\right]\le 0,$$

(8)

where ${\overline{\Phi }}_{11}=A\overline{M}+BL+\overline{M}{A}^T+L^T{B}^T+\alpha \overline{M}+\overline{F}+{\overline{F}}^T+\overline{Q}+h\overline{R}$. Then the reachable set

$$R_x$$

of the system (5) with constraints (2) and (3) is bounded by an ellipsoid $\mathcal{E}\left(P\right)$, and the feedback control is given by $K=L{\overline{M}}^{-1}.$

Proof. 

Construct the following Lyapunov–Krasovskii functional:

$$V\left(x_t\right)=V_1\left(x_t\right)+V_2\left(x_t\right)+V_3\left(x_t\right),$$

(9)

where

$$\begin{array}{cc}\hfill V_1\left(x_t\right)& =x^T\left(t\right)Px\left(t\right),\hfill \\ \hfill V_2\left(x_t\right)& ={\int }_{t-d\left(t\right)}^t{e}^{\alpha (s-t)}[x^T\left(s\right)Qx\left(s\right)+(h-t+s)x^T\left(s\right)Rx\left(s\right)]ds,\hfill \\ \hfill V_3\left(x_t\right)& =\left[\begin{array}{c}{x}^T\left(t\right){\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}Z& F\\ \star & U\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ \eta \left(t\right)\end{array}\right],\hfill \\ \hfill \eta \left(t\right)& ={\int }_{t-d\left(t\right)}^{t}x\left(s\right)ds.\hfill \end{array}$$

□

Because of

$$t-d\left(t\right)\le s\le t,0\le d\left(t\right)\le h,e^{-h}\le e^{-d\left(t\right)}\le e^{s-t}\le 1,0\le h-d\left(t\right)\le h-t+s\le h,$$

then

$$\begin{array}{cc}\hfill V_2\left(x_t\right)& \ge e^{-\alpha h}{\int }_{t-d\left(t\right)}^t{x}^T\left(s\right)Qx\left(s\right)ds\hfill \\ \hfill & \ge {\displaystyle \frac{1}{h}}{e}^{-\alpha h}{\int }_{t-d\left(t\right)}^t{x}^T\left(s\right)dsQ{\int }_{t-d\left(t\right)}^{t}x\left(s\right)ds\hfill \\ \hfill & ={\displaystyle \frac{1}{h}}{e}^{-\alpha h}{\eta }^T\left(t\right)Q\eta \left(t\right),\hfill \end{array}$$

and we have

$$\begin{array}{cc}\hfill V_2\left(x_t\right)+V_3\left(x_t\right)& \ge {\displaystyle \frac{1}{h}}{e}^{-\alpha h}{\eta }^T\left(t\right)Q\eta \left(t\right)+\left[\begin{array}{cc}{x}^T\left(t\right)& {\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}Z& F\\ \star & U\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ \eta \left(t\right)\end{array}\right]\hfill \\ \hfill & \ge \left[\begin{array}{cc}{x}^T\left(t\right)& {\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}Z& F\\ \star & U+{\displaystyle \frac{1}{h}}{e}^{-\alpha h}Q\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ \eta \left(t\right)\end{array}\right]\ge 0.\hfill \end{array}$$

(10)

By calculating the derivative along the system trajectory, and according to Lemma 1 and Lemma 2, we can obtain

$$\begin{array}{cc}\hfill {\dot{V}}_1\left(x_t\right)& =2x^T\left(t\right)P\left[(A+BK)x\left(t\right)+(D+BG)x(t-d(t\left)\right)+Ew\left(t\right)\right],\hfill \end{array}$$

(11)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_2\left(x_t\right)& =x^T\left(t\right)\left(Q+hR\right)x\left(t\right)-(1-\dot{d}\left(t\right))e^{-\alpha d\left(t\right)}{x}^T(t-d\left(t\right))Qx(t-d\left(t\right))\hfill \\ \hfill & -(1-\dot{d}\left(t\right))e^{-\alpha d\left(t\right)}(h-d\left(t\right))x^T(t-d\left(t\right))Rx(t-d\left(t\right))\hfill \\ \hfill & -{\int }_{t-d\left(t\right)}^t{e}^{-\alpha (s-t)}{x}^T\left(s\right)Rx\left(s\right)ds-\alpha V_2\left(x_t\right)\hfill \\ \hfill & \le x^T\left(t\right)\left(Q+hR\right)x\left(t\right)-(1-\mu )e^{-\alpha h}{x}^T(t-d\left(t\right))Qx(t-d\left(t\right))\hfill \\ \hfill & -{\displaystyle \frac{1}{h}}{e}^{-\alpha h}{\eta }^T\left(t\right)R\eta \left(t\right)-\alpha V_2\left(x_t\right),\hfill \end{array}\end{array}$$

(12)

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill {\dot{V}}_3\left(x_t\right)& =2\left[\begin{array}{c}{x}^T\left(t\right){\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}Z& F\\ \star & U\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ x\left(t\right)-x(t-d(t\left)\right)\end{array}\right]+2\dot{d}\left[\begin{array}{c}{x}^T\left(t\right){\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{c}F\\ U\end{array}\right]x(t-d\left(t\right))\hfill \\ \hfill & \le 2\left[\begin{array}{c}{x}^T\left(t\right){\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{cc}Z& F\\ \star & U\end{array}\right]\left[\begin{array}{c}(A+BK)x\left(t\right)+(D+BG)x(t-d(t\left)\right)+Ew\left(t\right)\\ x\left(t\right)-x(t-d(t\left)\right)\end{array}\right]\hfill \\ \hfill & +\left[\begin{array}{c}{x}^T\left(t\right){\eta }^T\left(t\right)\end{array}\right]\left[\begin{array}{c}F\\ U\end{array}\right]W^{-1}\left[\begin{array}{c}{F}^T{U}^T\end{array}\right]\left[\begin{array}{c}x\left(t\right)\\ \eta \left(t\right)\end{array}\right]\hfill \\ \hfill & +{\mu }^2{x}^T(t-d\left(t\right))Wx(t-d\left(t\right)).\hfill \end{array}\end{array}$$

(13)

Based on the inequality $2a^{T}b\le a^T{W}^{-1}a+b^{T}Wb,W>0$ and constraints (2) and (3), Equation (13) can be obtained.

Let ${\xi }_t^T=\left[x^T\left(t\right)x^T(t-d\left(t\right)){\eta }^T\left(t\right)w^T\left(t\right)\right]$, and from Lemma 3 and (11)–(13), we have

$$\begin{array}{cc}\hfill \dot{V}\left(x_t\right)+\alpha V\left(x_t\right)-\beta w^T\left(t\right)w\left(t\right)& \le {\xi }^T\left(t\right)\left(\Omega +\left[\begin{array}{c}{F}^T\\ 0\\ U^T\\ 0\end{array}\right]W^{-1}\left[\begin{array}{cccc}{F}^T& 0& U^T& 0\end{array}\right]\right)\xi \left(t\right)\hfill \\ \hfill & :={\xi }^T\left(t\right)\Psi \xi \left(t\right),\hfill \end{array}$$

(14)

where

$$\begin{array}{c}\hfill \Omega =\left[\begin{array}{cccc}{\varphi }_{11}& (P+Z)(D+BG)-F& {(A+BK)}^{T}F+U+\alpha F& (P+Z)E\\ \star & -(1-\mu )e^{-\alpha h}Q+{\mu }^{2}W& {(D+BG)}^{T}F-U& 0\\ \star & \star & -{\displaystyle \frac{1}{h}}{e}^{-\alpha h}R+\alpha U& F^{T}E\\ \star & \star & \star & -{\displaystyle \frac{\alpha }{w_m^2}}\end{array}\right],\end{array}$$

${\varphi }_{11}=(P+Z)(A+BK)+{(A+BK)}^T(P+Z)+\alpha (P+Z)+F+F^T+Q+hR.$

If $\Psi \le 0$ holds, through Lemma 4, it can be concluded that

$$\Psi =\left[\begin{array}{ccccc}{\Phi }_{11}& M(D+BG)-F& {(A+BK)}^{T}F+U+\alpha Y& ME& F\\ \star & -(1-\mu )e^{-\alpha h}Q+{\mu }^{2}W& {(D+BG)}^{T}F-U& 0& 0\\ \star & \star & -{\displaystyle \frac{1}{h}}{e}^{-\alpha h}R+\alpha U& F^{T}E& U\\ \star & \star & \star & -{\displaystyle \frac{\alpha }{w_m^2}}& 0\\ \star & \star & \star & \star & -W\end{array}\right]\le 0,$$

(15)

where $M=P+Z,{\Phi }_{11}=M(A+BK)+{(A+BK)}^{T}M+\alpha M+F+F^T+Q+hR.$

Let $N_1=diag(M^{-1},M^{-1},F^{-1},I,M^{-1}),\overline{M}=M^{-1},H=G\overline{M},L=K\overline{M},\overline{Z}=\overline{M}Z\overline{M},\overline{F}=\overline{M}F\overline{M},U=\overline{M}U\overline{M},P=\overline{M}P\overline{M},R=\overline{M}R\overline{M},Q=\overline{M}Q\overline{M},W=\overline{M}W\overline{M},R_1=F^{-1}R{F}^{-1},U_1=\overline{M}U{F}^{-1},U_2=F^{-1}R{F}^{-1}.$

Multiplying (15) on the left and right, respectively, we can obtain its equivalent condition:

$$\Psi =\left[\begin{array}{ccccc}{\overline{\Phi }}_{11}& D\overline{M}+BH-\overline{F}& \overline{M}{A}^T+L^T{B}^T+U_1+\alpha \overline{M}& E& \overline{F}\\ \star & -(1-\mu )e^{-\alpha h}\overline{Q}+{\mu }^2\overline{W}& \overline{M}{D}^T+H^T{B}^T-U_1& 0& 0\\ \star & \star & -{\displaystyle \frac{1}{h}}{e}^{-\alpha h}{R}_1+\alpha U_2& E& U_1^T\\ \star & \star & \star & -{\displaystyle \frac{\alpha }{w_m^2}}& 0\\ \star & \star & \star & \star & -\overline{W}\end{array}\right]\le 0.$$

(16)

Similarly, condition (10) can be transformed into

$$\left[\begin{array}{cc}M-P& F\\ \star & U+{\displaystyle \frac{1}{h}}{e}^{-\alpha h}Q\end{array}\right]\ge 0.$$

(17)

Let $N_2=diag(A^{-1},M^{-1})$, and multiplying (17) on the left and on the right, respectively, we can obtain its equivalent condition:

$$\left[\begin{array}{cc}\overline{M}-\overline{P}& \overline{F}\\ \star & \overline{U}+{\displaystyle \frac{1}{h}}{e}^{-\alpha h}\overline{Q}\end{array}\right]\ge 0.$$

(18)

From (16), we obtain

$$\dot{V}\left(x_t\right)+\alpha V\left(x_t\right)-{\displaystyle \frac{\alpha }{w_m^2}}{w}^T\left(t\right)w\left(t\right)\le 0,$$

(19)

which means that, through Lemma 4, we can obtain

$$V\left(x_t\right)=V_1\left(x_t\right)+V_2\left(x_t\right)+V_3\left(x_t\right)\le 1,$$

(20)

and from (10), we can obtain

$$V\left(x_t\right)=V_1\left(x_t\right)+V_2\left(x_t\right)+V_3\left(x_t\right)\ge V_1\left(x_t\right),$$

(21)

and then we have

$$V_1\left(x_t\right)=x^T\left(t\right)Px\left(t\right)\le 1,$$

(22)

Remark 1. 

To ensure that the reachable set of the closed-loop system is bounded by a given ellipsoid, we also need the inequality

$$0<P_r\le P={\overline{M}}^{-1}\overline{P}\overline{M},$$

(23)

Let $N_2=diag(M^{-1},M^{-1})$, multiplying (23) by $N_2$ on the left and $N_2^T$ on the right, respectively, we can obtain its equivalent condition

$$\left[\begin{array}{cc}-\overline{P}& \overline{M}\\ \overline{M}& {\overline{P}}_r\end{array}\right]\le 0.$$

(24)

Then, the reachable set of the system (5) with constraints (2) and (3) is bounded by an ellipsoid $\mathcal{E}\left(P_r\right)$, and the feedback control is given by $K=L{\overline{M}}^{-1}$,$G=H{\overline{M}}^{-1}$

Remark 2. 

The solution to equations (6)–(8) can be obtained using the “feasp” command in the LMI Toolbox. Unlike the reachability set estimation problems discussed in the paper [35], where we aimed for the smallest possible ellipsoidal boundary region, this paper focuses on confining the system state within a given region through appropriate control. Therefore, the problem studied in this chapter is one of existence. As a result, for the parameters α in Theorem 1, we can search for a feasible solution α by fixing the step length from 0 to 2.

Remark 3. 

Theorem 1 is a sufficient condition, only when the feasible set is nonempty, the inequality system is consistent.

4. Numerical Examples

Example 1. 

Consider a time-varying delay linear control system with the following parameters:

$A=\left[\begin{array}{cc}-2& 0\\ 0& 0.7\end{array}\right]$, $D=\left[\begin{array}{cc}-1& 0\\ -1& -0.9\end{array}\right]$, $A=\left[\begin{array}{c}-1\\ 1\end{array}\right]$, $E=\left[\begin{array}{c}-0.5\\ 1\end{array}\right]$, $w_m=1.$ When $h=0.7$, $\mu =0.6$ and $\alpha =0.5$, for a given matrix $P_r=\left[\begin{array}{cc}2& 0\\ 0& 2\end{array}\right]$, from Theorem 1, we can obtain $P={\overline{M}}^{-1}\overline{P}\overline{M}=\left[\begin{array}{cc}2.2840& -0.1705\\ -0.1705& 2.4452\end{array}\right]$, controller gain matrices are $K=[20.1407-24.6733]$ and $G=\left[0.28290.3763\right]$.

Example 2. 

Consider a time-varying delay linear control system with the following parameters: $A=\left[\begin{array}{cc}0& 0\\ 0& 1\end{array}\right]$, $D=\left[\begin{array}{cc}-1& -1\\ 0& -0.9\end{array}\right]$, $A=\left[\begin{array}{c}-1\\ 1\end{array}\right]$, $E=\left[\begin{array}{c}-1\\ -0.5\end{array}\right]$, $w_m=1.$ For an open loop system, the reachable set is unbounded for the reason that it is unstable. When $h=0.7$, $\mu =0.1$ and $\alpha =0.5$, for the given matrix $P_r=\left[\begin{array}{cc}1& 0\\ 0& 1\end{array}\right]$, from Theorem 1, we can obtain controller gain matrices $K=[64.9130-66.5529]$ and $G=[-2.51081.9269]$, and the bounded Ellipsoid can also be obtained $P={\overline{M}}^{-1}\overline{P}{\overline{M}}^{-1}=\left[\begin{array}{cc}1.2072& -0.0832\\ -0.0832& 1.2362\end{array}\right]$.

Example 1 shows that our method is effective for open-loop stable systems, and by solving the inequality systems, we can obtain the ellipsoidal boundary under the control law. Figure 1 depicts the system states $x\left(t\right)$ and reachable sets $P_r$ and P of two closed-loop systems under the control law $K,G$. Figure 1 shows that under the control law obtained from Theorem 1, the ellipsoidal bound P (the inside red solid ellipsoidal line) of the reachable set is obviously bounded by the given $P_r$ (the outside blue solid ellipsoidal line).

By utilizing Theorem 1, the reachable set of the closed-loop system is bounded by an ellipsoid. Figure 2 illustrates the reachable sets of the closed-loop system and two ellipsoids, demonstrating the effectiveness of the method proposed in this paper.

Example 2 shows that our method is effective for open-loop unstable systems, and by solving the inequality systems, we can obtain the ellipsoidal boundary under the control law. Figure 2 depicts the system states $x\left(t\right)$ and reachable sets $P_r$ and P of two closed-loop systems under the control law $K,G$. Figure 1 shows that under the control law obtained from Theorem 1, the ellipsoidal bound P (the inside red solid ellipsoidal line) of the reachable set is obviously bounded by the given $P_r$ (the outside blue solid ellipsoidal line).

5. Conclusions

This paper addresses the robust controller synthesis problem for linear time-delay systems with time-varying delays, focusing on reachable set confinement within safety-critical bounds. The key contributions are threefold:

Theoretical Advancements: We developed novel delay-dependent solvability conditions using tailored Lyapunov–Krasovskii functionals, expressed as parameterized linear matrix inequalities (LMIs). These conditions improve upon existing delay-independent approaches by explicitly incorporating delay rate information, enabling tighter state space characterization.

Methodological Innovation: The proposed framework simultaneously computes the following: minimal-volume bounding ellipsoids for system reachable sets; optimal state feedback controllers guaranteeing ellipsoidal confinement; the LMI formulation becomes convex after fixing a single scalar parameter, facilitating efficient numerical implementation via semidefinite programming.

Practical Validation: Two numerical examples demonstrate robust stability preservation under worst-case delay variations and effective handling of both constant and time-varying delay scenarios.

The results provide a systematic approach for safety-critical control of time-delay systems with applications in networked control, robotics, and power systems. Future work will extend the framework to nonlinear systems and distributed delay architectures.