A Matrix-Separation Approach for Stability of Discrete-Time Systems with Time-Varying Delay

Introduction

Discrete-time systems (DTS) form the backbone of modern digital control and arise across a wide range of engineering applications, including networked control, cyber-physical systems, and power systems [1, 2, 3]. In such settings, delays due to communication latency, computational overhead, or actuator response times can degrade performance or cause instability [4]. This motivates the study of DTS with time-varying delays (TVD), where a central problem is determining the largest delay range over which stability is preserved, and this has attracted sustained attention [5, 6, 7, 8, 9, 10, 11, 12].

The Lyapunov–Krasovskii functional (LKF) method, combined with linear matrix inequality (LMI) techniques, remains the dominant framework for analysing the stability of systems with delays [13]. Reducing conservatism requires both a sufficiently rich LKF and tight inequality to bound the quadratic summation terms in its forward difference. Enriching the functional with finite differences of the state captures additional delay-state interactions [14]. Specifically, introducing a double-summation term involving the state, its first-order difference, and its second-order difference significantly enhances this capability. Regarding the bounding aspect, Jensen inequality [15] were progressively tightened by Wirtinger-based inequality [16], auxiliary-function-based inequality [17], and other extensions, complemented by

reciprocally convex lemmas [18, 19] for handling delay-partitioned terms. This work focuses on the key challenge of bounding the quadratic summation that arises from the forward difference of the LKF.

Two main strategies exist for bounding this term. The first applies Jensen- or Wirtinger-type inequalities treating the state and its first-order difference as a single augmented vector [20, 21, 22]. The second uses matrix-separation-based inequalities (MSBI), which partition the weighting matrix to separately exploit the state and its first-order difference [23, 9, 24]. In [9] parameter-dependent matrices tighten the bound. However, handling the resulting reciprocally convex terms through the zero-equation method increases the number of decision variables while yielding only limited reductions in conservatism. Reference [24] reduces this count by combining MSBI with the reciprocally convex combination lemma, at the cost of a slight increase in conservatism. A common limitation of both strategies is the omission of the second-order state difference from the bounding procedure, leaving potentially useful information unexploited.

Based on the above observations, this letter makes the following contributions:

1. An improved MSBI (IMSBI, Lemma 3) is derived that simultaneously accounts for the state, its first-order difference, and its second-order difference.

2. An augmented LKF is constructed (Section III) that incorporates first-order difference vectors absent in [14], enabling the forward difference to exploit richer delay-state information.

3. A delay-variation-independent stability criterion (Theorem 1) is obtained, verified to be less conservative than state-of-the-art methods, including those that explicitly use delay-variation-rate information, while requiring $187n^2 + 16n$ decision variables — best among state-of-the-art methods.

Notation. $\mathbb{R}^n$, $\mathbb{R}^{m \times n}$, and $\mathbb{S}^n(\mathbb{S}^{n}_{+})$ denote the sets of $n$-dimensional real vectors, $m \times n$ real matrices, and $n \times n$ symmetric (positive definite) matrices, respectively. For any square matrix $A$: $\operatorname{sym}\{A\} = A + A^{\mathsf{T}}$, $\hat{A} = \operatorname{diag}\{A,A\}$, and $\Pi_i(A) = \operatorname{diag}\{A,3A,\ldots,(2i+1)A\}$ ($i+1$ blocks). The shorthand $q_i(s) = s+i$, $i = 1,2$, is used throughout. For a vector-valued sequence $x(\kappa)$, define the first and second-order differences $\zeta(\kappa) = x(\kappa+1)-x(\kappa)$ and $\Delta\zeta(\kappa) = \zeta(\kappa+1)-\zeta(\kappa)$. For integers $a > b \ge0$, the single and double accumulation operators are: $v_1(a,b) = \sum_{i=\kappa-a}^{\kappa-b} x(i)$, $v_2(a,b) = \sum_{i=\kappa-a}^{\kappa-b}\sum_{j=i}^{\kappa-b} x(j)$, $v_1(a,b) = \sum_{i=\kappa-a}^{\kappa-b} \zeta(i)$, $v_2(a,b) = \sum_{i=\kappa-a}^{\kappa-b}\sum_{j=i}^{\kappa-b} \zeta(j)$.

Preliminaries and Problem Formulation

Consider the following DTS, which includes a TVD

Here, $x(\kappa)\in\mathbb{R}^{n}$ represents the state vector, with the initial condition denoted as $\varphi(\kappa)$. The matrices $B$ and $B_d$ are real constant matrices with compatible dimensions. The TVD $d(\kappa):\mathbb{Z}_{+}\to\mathbb{Z}_{+}$ is an integer-valued function satisfying

with known positive integer bounds $d_l$ and $d_u$. The objective of this letter is to establish a less conservative delay-dependent stability criterion for system (1) under the constraint (2).

Before proceeding, we present some lemmas, as they are crucial for proving the theoretical contributions of the letter.

Lemma 1 ([17]): For a matrix $R\in\mathbb{S}_{+}^{n}$, positive integers $\ell_2>\ell_1$, and vector-valued function $x(\kappa)$, The subsequent inequalities are valid:

where $\ell=\ell_2-\ell_1$,

Lemma 2 ([19]): For any scalar $\alpha\in(0,1)$, matrices $R_1,R_2\in\mathbb{S}_{+}^{n}$, and arbitrary matrices $X_1,X_2\in\mathbb{R}^{n\times n}$, the following matrix inequality holds:

where $r_1=R_1-X_2R_2^{-1}X_2^{\mathsf{T}}$ and $r_2=R_2-X_1^{\mathsf{T}}R_1^{-1}X_1$.

Main Results

An Improved Matrix-Separation-Based Inequality

This subsection introduces an improved matrix-separation-based inequality (IMSBI) aimed at enhancing the bounding accuracy of the summation term

where $\eta(i)=\operatorname{col}\{x(i),\zeta(i),\Delta\zeta(i)\}$, and $R\in\mathbb{S}_{+}^{3n}$.

Lemma 3

Let $J(x)$ be defined as in (7), where $R$ admits the decomposition:

with $R_{12},R_{13},R_{23}\in\mathbb{R}^{n\times n}$ and $R_{11},R_{22},R_{33},T,Z\in\mathbb{S}^{n}$. If $R_a\succ0$ and $Z\succ0$, then:

where

Additional notations, including $\chi(\kappa)$ and $\bar e_i$, are introduced and defined in Lemma 1.

Proof. Using the separation matrix (8) of $R$, $J(x)$ can be expressed as the sum of three components:

From Lemma 1, it follows that $R_a>0$ implies:

For $R_b$ and $T\in\mathbb{S}^{n}$, direct computation gives:

With $Z\succ0$ and via (4),(5), we bound $J_c$:

Combining (10), (11), (12), (13) with the relation $E_1=\sigma F_1$ establishes (9). Hence, the proof of Lemma 3 is concluded.

Remark 1: Lemma 3 generalizes the MSBI in [9]: omitting $\Delta\zeta(i)$ from $\eta(i)$ and substituting $T=T_1+\ell T_2$, $Z=Z_1+\ell Z_2$, and further considering the zero-equation

yields its $m=1$ case. However, handling the reciprocally convex terms in (9) via this technique, see Lemma 6 in [14], requires free matrices $M_1,M_2\in\mathbb{R}^{8n\times6n}$, introducing $96n^2$ decision variables per application of the IMSBI. By contrast, combining Lemma 2 with (9) uses only $X_1,X_2\in\mathbb{R}^{6n\times6n}$, reducing this count to $72n^2$, a 25% saving at the cost of a small increase in conservatism. Since the free-matrix dimensions in [9] grow with the order parameter $m$, any selection $m>1$ incurs substantially more decision variables than the proposed approach.

Remark 2: Existing methods estimate $J(\kappa)$ with limited inter-component exploitation: [20], [21], [22] use Jensen/Wirtinger inequalities on $\operatorname{col}\{x(i),\zeta(i)\}$ without capturing component interactions with $\Delta\zeta$; the MSBI in [9] partially addresses this via matrix separation but remains restricted to $\{x(i),\zeta(i)\}$; and [14] employs the full vector $\eta(i)=\operatorname{col}\{x(i),\zeta(i),\Delta\zeta(i)\}$ yet treats it monolithically. The proposed IMSBI overcomes these limitations by explicitly separating $T$ and $Z$ from $R$, thereby incorporating the pairwise relationships among $x(i)$, $\zeta(i)$, and $\Delta\zeta(i)$. In particular, (12) yields a zero-error bound for the $T$-dependent subterm, while the boundary terms $\zeta(\kappa-\ell_1)$, $\zeta(\kappa-\ell_2)$ and the summation terms $v_1(\ell_2,\ell_1)$, $v_2(\ell_2,\ell_1)$ in (9) supply additional delay-dependent information, jointly producing a tighter estimate of $J(\kappa)$.

Delay-Dependent Stability Criterion

This subsection presents a new sufficient condition to ensure the asymptotic stability of DTS (1) in the presence of a TVD satisfying (2). For brevity, the following notations are adopted: $d_{21}=d_u-d_l$, $d_{\kappa1}=d(\kappa)-d_l$, $d_{\kappa2}=d_u-d(\kappa)$.

$\mathcal{T}_i=\operatorname{diag}\{T_i,T_i,-T_i,-T_i\},\ i=1,2,3,$

$c_{0i}=\operatorname{col}\{e_{1+i},e_{11+i},e_{2+i},e_{12+i}\},\ i=1,2,$

$c_{03}=\operatorname{col}\{e_1,e_s-e_1,e_2,e_{12}\},\quad c_1=\operatorname{col}\{e_1,e_s-e_1\},$

$c_i=\operatorname{col}\{e_i,e_{10+i}\},\ i=2,3,4,\quad e_s=Be_1+B_de_3,$

$c_5=\operatorname{col}\{e_{11},e_1-e_{11},e_s-2e_1+e_{11}\},$

$e_g=\begin{bmatrix}e_2^{\mathsf T},e_3^{\mathsf T},e_4^{\mathsf T},e_6^{\mathsf T},e_7^{\mathsf T},e_{12}^{\mathsf T},e_{13}^{\mathsf T},e_{15}^{\mathsf T},e_{16}^{\mathsf T},e_{17}^{\mathsf T},e_{18}^{\mathsf T}\end{bmatrix}^{\mathsf T},$

$e_i=\begin{bmatrix}0_{n,(i-1)n},I_n,0_{n,(18-i)n}\end{bmatrix}^{\mathsf T},\ i=1,2,\ldots,18.$

Proof: To prove asymptotic stability, we construct a candidate LKF of the form,

where

$\eta_{11}(\kappa)=\operatorname{col}\{x(\kappa-1),x(\kappa),x(\kappa-d_l),x(\kappa-d(\kappa)),x(\kappa-d_u),u_1(d_l,1),u_1(d_u,d_l+1),u_2(d_l,1),u_2(d_u,d_l+1)\},$

$\eta_{12}(\kappa)=\operatorname{col}\{\zeta(\kappa-1),v_1(d_l,1),v_1(d_u,d_l+1),v_2(d_l,1),v_2(d_u,d_l+1)\},$

$\eta_2(i)=\operatorname{col}\{x(i),\zeta(i)\},\quad\eta_3(i)=\operatorname{col}\{x(i),\zeta(i),\Delta\zeta(i)\}.$

We define the augmented state vector $\xi(\kappa)$ as follows:

$\xi(\kappa)=\operatorname{col}\{x(\kappa),x(\kappa-d_l),x(\kappa-d(\kappa)),x(\kappa-d_u),\frac{u_1(d_l,0)}{q_1(d_l)},\frac{u_1(d(\kappa),d_l)}{q_1(d_{\kappa1})},\frac{u_1(d_u,d(\kappa))}{q_1(d_{\kappa2})},\frac{u_2(d_l,0)}{q_1(d_l)q_2(d_l)},\frac{u_2(d(\kappa),d_l)}{q_1(d_{\kappa1})q_2(d_{\kappa1})},\frac{u_2(d_u,d(\kappa))}{q_1(d_{\kappa2})q_2(d_{\kappa2})},x(\kappa-1),\zeta(\kappa-d_l),\zeta(\kappa-d(\kappa)),\zeta(\kappa-d_u),\frac{v_1(d(\kappa),d_l)}{q_1(d_{\kappa1})},\frac{v_1(d_u,d(\kappa))}{q_1(d_{\kappa2})},\frac{v_2(d(\kappa),d_l)}{q_1(d_{\kappa1})q_2(d_{\kappa1})},\frac{v_2(d_u,d(\kappa))}{q_1(d_{\kappa2})q_2(d_{\kappa2})}\}.$

It can be verified that the LKF vectors satisfy the relations:

$\eta_{11}(\kappa)=\Gamma_1(d(\kappa))\xi(\kappa),\quad\eta_{11}(\kappa+1)=\Upsilon_1(d(\kappa))\xi(\kappa),$

$\eta_{12}(\kappa)=\Gamma_2(d(\kappa))\xi(\kappa),\quad\eta_{12}(\kappa+1)=\Upsilon_2(d(\kappa))\xi(\kappa).$

Since $P_1\in\mathbb{S}^{9n}_+$, $P_2\in\mathbb{S}^{5n}_+$, $Q_i\in\mathbb{S}^{2n}_+$, and $R,S\in\mathbb{S}^{3n}_+$, it follows directly that $V(\kappa)>0$ for all $\xi(\kappa)\ne0$. Next, by computing the forward difference of the functional $V(\kappa)$ along the solutions of system (1), we derive:

where

Using Lemma 3 in conjunction with (14), the terms $J_i$ for $i=1,2,3$ are estimated as follows:

where $\Psi_{11}(d(\kappa)),\Psi_{12}(d(\kappa)),\Psi_{21},\Psi_{22},\Psi_3,F_{11}$ and $F_{12}$ are defined in Theorem 1.

Using (16) and (17)–(22), we arrive at:

For any matrices $X_1$ and $X_2$ with appropriate dimensions, by applying Lemma 2 with $\alpha=\frac{d_{\kappa1}}{d_{21}^2}$ and utilizing (14), we find:

In addition, the definition of $\xi(\kappa)$ implies that several vectors contained within it satisfy the following relations:

These relations result in the following zero-value terms:

where $G_1$, $G_2$, $G_3$, and $G_4$ are any matrices. Adding (25) and (26) to (24), we obtain that

where $\Xi(d(\kappa))$ is defined in Theorem 1. Applying the Schur complement to (15) shows that the right-hand side of (27) is negative definite. Since $\Xi(d(\kappa))$ is affine in $d(\kappa)$, convexity ensures that (15) holding at the two vertices $d(\kappa)\in\{d_l,d_u\}$ implies $\Delta V(\kappa)<0$ for all nonzero $\xi(\kappa)$ and all $d(\kappa)\in[d_l,d_u]$. Combined with $V(\kappa)>0$, Lyapunov stability theory guarantees asymptotic stability of (1). This completes the proof.

Remark 3: The reduced conservatism of Theorem 1 relative to [6], [7], [8], [9], [10], [11], [12], [5] stands on three pillars. First, the augmented LKF (16) enriches the standard state information with $x(\kappa-1)$, the boundary terms $\zeta(\kappa-d_l)$, $\zeta(\kappa-d(\kappa))$, $\zeta(\kappa-d_u)$, and the weighted summations $v_1(d(\kappa),d_l)$, $v_1(d_u,d(\kappa))$, $v_2(d(\kappa),d_l)$, $v_2(d_u,d(\kappa))$, capturing richer delay-dependent and higher-order difference information. Second, the IMSBI (Lemma 3) yields tighter bounds on $J_1$, $J_2$, and $J_3$ by explicitly exploiting the coupling among $x(i)$, $\zeta(i)$, and $\Delta\zeta(i)$. Third, the zero-equations (25)–(26) and free matrices $G_i$, $i=1,\ldots,4$, inject additional cross-terms that expand the feasibility region of the LMIs.

Numerical Examples

This section validates Theorem 1 via two numerical examples. The MATLAB code is available at Github¹. We assess conservatism via the maximal admissible upper bounds (MAUBs) and computational complexity via the number of decision variables (NDVs).

Example 1: Consider the DTS (1) with matrices given by:

Assuming $\Delta d(\kappa)\coloneqq d(\kappa+1)-d(\kappa)\in[-\mu,\mu]$, we compute the MAUBs $d_u$ for various $d_l$ and $\mu$. Results from Theorem 1 (Table I) are compared with several recent methods. The proposed approach yields larger $d_u$ values than those in [6], [9], [10], [11], [12], [5], indicating reduced conservatism due to the augmented LKF, the IMSBI, and the zero equations (25)–(26). Although the NDVs here exceed those in [6], [9], [5], the computational cost remains lower than in [10], [11], [12]. To further validate the stability criterion proposed in Theorem 1, a numerical simulation is conducted for Example 1. The DTS described by (1) is examined under a TVD randomly selected within the interval $[25,207]$, with initial conditions $\varphi(\kappa)=[0.8,-0.5]^{\mathsf{T}}$ for $\kappa\in[-207,0]$. As shown in Figure 1, the state trajectories converge asymptotically to zero, thereby confirming the system’s stability and supporting the theoretical findings.

Criteria	$\mu$	$d_l=15$	$d_l=25$	$d_l=35$	$d_l=45$	$d_l=50$	NDVs
Th.1 [5]	–	120	135	146	156	160	$39.5n^2+6.5n$
Th.1 [9]	–	135	143	154	166	171	$75.5n^2+8.5n$
Th.3.1 [6]	1	123	145	157	167	169	$40.5n^2+7.5n$
	5	120	136	146	159	163
Th.1 [10]	1	138	159	163	166	167	$372n^2+7n$
	5	128	142	151	158	162
Th.2 [10]	1	141	162	166	168	169	$372n^2+7n$
	5	138	154	162	167	168
Th.2 [9]	1	153	165	170	173	175	$82n^2+12n$
	5	139	149	157	167	170
Th.1.C1 [11]	1	136	156	168	173	177	$267.5n^2+13.5n$
	5	128	149	161	169	171
Th.1.C2 [11]	1	150	165	172	176	178	$267.5n^2+13.5n$
	5	147	162	170	173	176
Th.1 [12]	1	154	166	171	174	175	$420n^2+12n$
	5	136	152	160	167	169
Theorem 1	–	180	207	249	282	290	$187n^2+16n$

Example 2: Consider the DTS (1) with matrices given by:

Table II compares the MAUBs $d_u$ (for various $d_l$) and the corresponding NDVs obtained from Theorem 1 and several existing methods [6], [7], [8], [9], [10], [11], [12], [5]. The comparison assumes $\Delta d(\kappa)\in[-\mu,\mu]$ with $\mu=1$ for [6], [8], [9], [10], [11], [12]. Theorem 1 consistently yields larger MAUBs while requiring relatively few NDVs, indicating reduced conservatism and improved computational efficiency. Example 2 numerically verifies Theorem 1 for system (1) using stochastically generated delays in [6] [38] and initial states $\varphi(\kappa) = [1.5, -2.0]^{\mathsf{T}}$ for $\kappa\in[-38, 0]$. As shown in Figure 2, both states converge asymptotically to the origin, confirming the theoretical result.

Criteria	$d_l$ = 2	6	10	15	20	25	NDVs
Th.1 [9]	22	22	23	25	29	32	$75.5n^2 + 8.5n$
Th.1 [5]	22	22	23	26	29	32	$39.5n^2 + 6.5n$
Cor.8 [7]	22	23	23	26	29	32	$1280.5n^2 + 13.5n$
Th.3.1 [6]	22	23	23	26	29	32	$40.5n^2 + 7.5n$
Th.2 [9]	22	23	24	26	29	33	$82n^2 + 12n$
Prop.3 [8]	23	24	25	27	30	32	$341.5n^2 + 8.5n$
Th.1 [12]	23	25	26	28	31	34	$420n^2 + 12n$
Th.1 [10]	23	25	26	28	31	34	$372n^2 + 7n$
Th.2 [10]	24	25	28	31	34	36	$372n^2 + 7n$
Th.1.C1 [11]	23	25	27	30	33	37	$267.5n^2 + 13.5n$
Th.1.C2 [11]	23	25	27	30	34	38	$267.5n^2 + 13.5n$
Theorem 1	37	38	39	41	43	47	$187n^2 + 16n$

Conclusion

This letter proposed a less conservative framework for stability analysis of discrete-time systems with time-varying delays. The main contributions are an improved matrix-separation-based inequality for tighter estimation of augmented summation terms and an augmented LKF that captures state–delay interactions. Embedding these elements into the forward-difference estimate yields new delay-dependent stability conditions cast as tractable LMIs. Two numerical examples confirm that the approach consistently produces less conservative bounds while retaining computational efficiency.