Robust stability analysis of uncertain discrete-time delayed systems with finite wordlength nonlinearities via delay partitioning approach

Abstract

This paper aims to investigate the problem of robust stability of discrete-time systems with finite wordlength nonlinearities and time-varying delay. Two new delay-dependent stability criteria for such systems are proposed. The first criterion is based on estimating the forward difference of Lyapunov–Krasovskii functional using a summation inequality along with the Wirtinger-based inequality. The second one employs the method of delay partitioning where the delay range is nonuniformly divided into subintervals. The proposed criteria are distinctive and may be considered less stringent than previous criteria. Examples are presented to verify the effectiveness of the criteria.

Keywords

Delay partitioning approach delayed system finite wordlength nonlinearity Lyapunov–Krasovskii method robust stability

Introduction

Since most of the existing systems in practice are nonlinear, significant work has been paid by several researchers in achieving constructive results while handling nonlinear systems. Increasing attention has been focused on study of stability properties of such systems (for instance, process control and networked control system) and their implementation under discrete-time/digital environment. When fixed-point discrete-time systems (DTSs) are practically implemented on digital hardware that is of finite wordlength, nonlinearities (quantization and overflow) may occur, which lead the system toward unstable behavior (Butterweck et al., 1988; Chang, 1981; Classen et al., 1976; Liberzon, 2003). In fixed-point arithmetic, each number is represented by a sign bit and a magnitude. Thus, the magnitude of any number is represented by a string of bits of fixed length b. When multiplying two b-bit numbers, a 2b-bit number is produced. A quantization nonlinearity is generated when the 2b-bit number is reduced in length to b bits. Quantization affects the least significant bits only. Magnitude truncation (MT), value truncation (VT), and round-off (RO) are usually considered for quantization. When the sum of two numbers falls outside the representable range and the result is modified so that it falls back within the representable range, an overflow nonlinearity is introduced. The overflow nonlinearity usually affects the most significant bits as well as the least significant bits of a fixed-point number. To deal with the overflow correction, the nonlinearities such as saturation, zeroing, 2’s complement, and triangular are used. Researchers have drawn the attention toward the stability of a practical system under the combined influence of quantization and overflow (Agarwal and Kar, 2014; Kandanvli and Kar, 2009, 2011; Kar, 2011; Mahmoud, 2013; Singh et al., 2021; Tadepalli et al., 2018; Tadepalli and Kandanvli, 2016).

Other instability sources of the DTS include parameter perturbations and delays. Discrepancy in system parameters or modeling errors usually occurs in the practical system which leads to parameter uncertainties and result in system instability. The effects of parameter uncertainties on the stability of DTS have been studied extensively (Bakule et al., 2006; Chen et al., 2003; Guan et al., 1999; Kandanvli and Kar, 2009, 2011; Singh et al., 2021; Tadepalli et al., 2018). A multi-dimensional parametric model with uncertain non-pharmacological policies has been proposed in Tutsoy et al. (2020) for predicting COVID-19 pandemic casualties. Time delay is commonly inherent in many practical systems owing to transmission, measurement, and computational delays. Delays in such systems may degrade the performance and result in unstable behavior of the system. The problem of stability in delay DTSs with saturation has been investigated in Gupta et al. (2023). The global asymptotic stability problem of DTS with saturation arithmetic and time delay is discussed in Kanithi et al. (2024). The study in Pandey et al. (2024) considers the stability analysis of DTS with saturation, variable time lags, and interference. The moving horizon estimation problem for delayed systems with the Round-Robin protocol scheduling has been investigated in Zou et al. (2019). The stability of uncertain DTS with time-varying delay is essential and has attained great consideration (Chen et al., 2004; Li et al., 2021; Xu et al., 2001). One of the important parameters to measure the effectiveness of a stability criterion for time-varying delayed systems is maximum allowable upper bound delay. For a particular system, larger maximum allowable upper bound delay implies less conservative stability result.

Without considering the effects of parameter uncertainties and delays, several results have been accounted for DTSs with finite wordlength nonlinearities (see, for instance, Agarwal and Kar, 2014; Kar, 2011; Mahmoud, 2013). However, it is a significant and challenging task to analyze the stability of DTSs when quantization, overflow, parameter uncertainties and time-varying delay are all present at the same time. As yet, some attention has been paid for the exploration of this problem (Tadepalli et al., 2018; Tadepalli and Kandanvli, 2016). There are several methods for estimating the sum terms that appear in the forward difference of the Lyapunov–Krasovskii functional (LKF) and such methods (e.g. free weighting matrix (He et al., 2007), Jensen’s inequality (JI) method (Jiang et al., 2005), reciprocally convex inequality (Liu and Zhang, 2012; Park et al., 2011), Wirtinger-based inequality (WBI) method (Nam et al., 2015; Seuret et al., 2015), and improved summation inequality (Zhang et al., 2016a) play an important role in determining the conservatism of the results.

To enhance the stability region, the delay partitioning method was explored (Meng et al., 2010; Tadepalli and Kandanvli, 2016; Xue et al., 2016, 2018; Zhang et al., 2016b). However, the development of existing methods is still partial, and further relaxation of these methods poses an important and demanding problem.

Inspired by the above discussion, the global asymptotic stability problem for uncertain DTSs with time-varying delay and concatenations of overflow and quantization is considered in this paper. Our main aim is to derive less stringent stability criteria by utilizing an improved summation inequality (Zhang et al., 2016a), WBI and delay partitioning method. The major contributions are as follows:

A new stability criterion (Theorem 1) for uncertain DTSs with finite wordlength nonlinearities is derived by utilizing an LKF and an improved summation inequality (Zhang et al., 2016a) together with WBI.

Another stability result (Theorem 2) is obtained by employing a delay partitioning strategy in which the delay interval is partitioned into nonuniform subintervals. Pertaining to the DTSs in absence of nonlinearities and uncertainties, a new stability result (Corollary 1) is also brought out.

The proposed approach generally leads to more relaxed stability results than several existing results. The stability criteria are presented in the linear matrix inequality (LMI) framework.

Numerous examples are provided to highlight the applicability and efficacy of the proposed results.

The leftover part of the paper is ordered as follows. The problem under study is formulated in section “Problem formulation and preliminaries” section. Utilizing the delay partitioning method, new robust stability criteria of DTSs with finite wordlength nonlinearities and time-varying delay are discussed in “Main Results” section. With the help of examples, the merit of the obtained results is exemplified in “Examples” section. The final section concludes the work.

Problem formulation and preliminaries

The following notations and terminologies are used in this paper:

$R^{a}$ Set of $a \times 1$ real vectors

$R^{a \times b}$ Set of $a \times b$ real matrices

0 Null matrix or null vector

$0_{a \times b}$ $a \times b$ null matrix

$I$ Identity matrix

$I_{a}$ $a \times a$ identity matrix

$H^{T}$ Transpose of the matrix $H$

diag (·) Diagonal matrix

H >0 (<0) H is a positive definite (negative definite) symmetric matrix

* Symmetric elements in a symmetric matrix

sym( H ) $H + H^{T}$

$| λ |$ Modulus of a scalar λ

⌊ $λ$ ⌉ Nearest integer to λ

∥·∥ Any vector norm

$f (\cdot)$ Concatenation of overflow and quantization

O (·) Overflow

Q (·) Quantization

DTS Discrete-time system

LKF Lyapunov–Krasovskii functional

MT Magnitude truncation

RO Round-off

VT Value truncation

LMI Linear matrix inequality

WBI Wirtinger-based inequality

The state-space representation of a class of uncertain DTSs influenced by overflow, quantization, and time-varying delay under study is

\begin{matrix} x (c + 1) = O {Q (y (c))} = f (y (c)) \\ = [f_{1} (y_{1} (c)) f_{2} (y_{2} (c)) \dots f_{r} (y_{r} (c))]^{T} \end{matrix}

(1a)

\begin{matrix} y (c) = (A_{11} + Δ A_{11}) x (c) + (A_{1 d} + Δ A_{1 d}) x (c - d (c)) \\ = [y_{1} (c) y_{2} (c) \dots y_{r} (c)]^{T} \end{matrix}

(1b)

x (c) = Θ (c), c \in [- h_{M}, 0]

(1c)

where at time c, $x (c) \in R^{r}$ and $Θ (c) \in R^{r}$ denote the state vector and the initial state vector, respectively, $Q (\cdot)$ and $O (\cdot)$ indicate the quantization and overflow, respectively, $f (\cdot)$ characterizes the combination of quantization and overflow, $A_{11}, A_{1 d} \in R^{r \times r}$ are the known state matrices, $Δ A_{11}, Δ A_{1 d} \in R^{r \times r}$ are the unknown matrices signifying uncertainties in the known state matrices, and d(c) is the time-varying delay satisfying $1 \leq h_{m} \leq d (c) \leq h_{M},$ where $h_{m}$ and $h_{M}$ are the known positive integers.

Note that, for the given system (1), $f_{i} (y_{i} (c))$ occurs due to various combinations of quantization and overflow. The $f_{i} (y_{i} (c))$ is constrained to the sector $[k_{o}, k_{q}]$ , that is

f_{i} (0) = 0, k_{o} y_{i}^{2} (c) \leq f_{i} (y_{i} (c)) y_{i} (c) \leq k_{q} y_{i}^{2} (c), i = 1, 2, . . ., r

(2a)

k_{q} = {\begin{matrix} 1, for MT \\ 2, for RO \end{matrix}

(2b)

k_{o} = {\begin{matrix} 0, for zeroing or saturation \\ - 1 / 3, for triangular \\ - 1, for 2' s complement \end{matrix}

(2c)

where $k_{o}$ and $k_{q}$ represents the lower and upper slopes of the sector, respectively. The nonlinearities described by equation (2) represent the finite wordlength nonlinearities that commonly occur in practical DTSs during implementation (Erickson and Michel, 1985). However, using the sector description in equation (2), it is not possible to characterize the VT quantization due to its peculiar characteristics. The parameter uncertainties are expressed as

Δ A_{11} = H_{0} F_{0} E_{0}

(3a)

Δ A_{1 d} = H_{1} F_{1} E_{1}

(3b)

where $H_{i} \in R^{r \times a_{i}}$ and $E_{i} \in R^{b_{i} \times r} (i = 0, 1)$ are the known matrices, and $F_{i}$ is an unknown matrix satisfying

F_{i}^{T} F_{i} \leq I, i = 0, 1

(3c)

The uncertainty model (3) has been widely used in robust control and filtering (Bakule et al., 2006; Chen et al., 2003; Guan et al., 1999; Kandanvli and Kar, 2009, 2011; Singh et al., 2021; Tadepalli et al., 2018).

A matrix block diagram of nominal DTS (1)–(3) is shown in Figure 1.

Equations (1)–(3) represent a class of DTSs with parameter uncertainties, delay, and different concatenation of quantization and overflow. Distinctive examples of DTS (1)–(3) include adaptive system, microgrid system, networked control system, and neural network (Mary and Rangarajan, 2016; Zou et al., 2019). Packet dropout and network induced delay are the most important factors in the analysis of networked control system. When realizing the networked control system using special-purpose hardware, finite wordlength nonlinearities are introduced in the system. During information transmission over open communication networks, microgrid system generally produces delays.

The goal of this study is to derive new criteria for the global asymptotic stability of systems (1)–(3).

Now, we recall the following definition and lemmas.

Definition 1. The equilibrium of the uncertain discrete-delayed systems (1)–(3) is globally asymptotically stable for all admissible uncertainties if (i) it is stable in terms of Lyapunov, that is, for every $ƛ > 0$ there is a $℘ = ℘ (ƛ)$ satisfying $∥ x (c) ∥ < ƛ$ for all $c = 0, 1, 2, . . .,$ whenever $∥ x (c_{0}) ∥ < ℘,$ $c_{0} = - h_{M}, - h_{M} + 1, . . ., 0$ and (ii) $x (c) \to 0$ as $c \to \infty$ starting from any initial condition given by equation (1c) (Liu and Michel, 1992; Xu et al., 2001).

Lemma 1. Given a symmetric positive definite matrix P and non-negative integers $μ_{1}$ , $μ_{2},$ c, fulfilling $μ_{1} \leq μ_{2} \leq c,$ if (Nam et al., 2015)

\begin{matrix} χ (c, μ_{1}, μ_{2}) = \\ {\begin{matrix} \frac{1}{μ_{2} - μ_{1}} [(2 \sum_{s = c - μ_{2}}^{c - μ_{1} - 1} x (s)) + x (c - μ_{1}) - x (c - μ_{2})], & μ_{1} < μ_{2} \\ 2 x (c - μ_{1}), & μ_{1} = μ_{2} \end{matrix} \end{matrix}

(4)

Then

- (μ_{2} - μ_{1}) \sum_{s = c - μ_{2}}^{c - μ_{1} - 1} Λ^{T} (s) P Λ (s) \leq - {[\begin{matrix} T_{0} \\ T_{1} \end{matrix}]}^{T} [\begin{matrix} P & 0 \\ 0 & 3 P \end{matrix}] [\begin{matrix} T_{0} \\ T_{1} \end{matrix}]

(5)

where $T_{0} = x (c - μ_{1}) - x (c - μ_{2}), T_{1} = x (c - μ_{1}) + x (c - μ_{2}) - χ (c, μ_{1}, μ_{2})$ , $Λ (s) = x (s + 1) - x (s) .$ Over the delay range $[μ_{1}, μ_{2}]$ , $T_{0}$ signifies the average value attained by $x (c)$ and $T_{1}$ is the proportional to the difference of mean value of $x (c)$ and its average value.

Lemma 2. For any symmetric positive definite matrix $P$ and any matrix X , the following relation holds (Zhang et al., 2016a)

\begin{matrix} - (h_{M} - h_{m}) [\sum_{s = c - d (c)}^{c - h_{m} - 1} Λ^{T} (s) P Λ (s) + \sum_{s = c - h_{M}}^{c - d (c) - 1} Λ^{T} (s) P Λ (s)] \\ \leq - γ^{T} (c) {[\begin{matrix} {\tilde{T}}_{0} \\ {\tilde{T}}_{1} \end{matrix}]}^{T} ([\begin{matrix} \tilde{P} & X \\ * & \tilde{P} \end{matrix}] + [\begin{matrix} \frac{h_{M} - d (c)}{h_{M} - h_{m}} M_{1} & 0 \\ * & \frac{d (c) - h_{m}}{h_{M} - h_{m}} M_{2} \end{matrix}]) [\begin{matrix} {\tilde{T}}_{0} \\ {\tilde{T}}_{1} \end{matrix}] γ (c) \end{matrix}

(6)

where $Λ (s) = x (s + 1) - x (s), \tilde{P} = diag {P, 3 P}, M_{1} = \tilde{P} - X {\tilde{P}}^{- 1} X^{T}$

\begin{matrix} M_{2} = \tilde{P} - X^{T} {\tilde{P}}^{- 1} X, {\tilde{T}}_{0} = [\begin{matrix} ρ_{3} - ρ_{2} \\ ρ_{2} + ρ_{3} - ρ_{6} \end{matrix}], {\tilde{T}}_{1} = [\begin{matrix} ρ_{2} - ρ_{4} \\ ρ_{2} + ρ_{4} - ρ_{7} \end{matrix}] \\ ρ_{i} = [\begin{matrix} 0_{r \times (i - 1) r} & I_{r} & 0_{r \times (8 - i) r} \end{matrix}], i = 1, 2, \dots, 8 \\ γ^{T} (c) = [\begin{matrix} x^{T} (c) & x^{T} (c - d (c)) & x^{T} (c - h_{m}) & x^{T} (c - h_{M}) & χ^{T} (c, 0, h_{m}) & χ^{T} (c, h_{m}, d (c)) & χ^{T} (c, d (c), h_{M}) & f^{T} (y (c))] \end{matrix} \end{matrix}

Lemma 3. Let $ϒ, Z, Π$ and $\sum$ are real matrices of proper dimensions and $\sum^{=} \sum^{T} (Boyd et al ., 1994) .$ Then

\sum + ϒ Z Π + Π^{T} Z^{T} ϒ^{T} < 0

(7)

where $Z^{T} Z \leq I,$ if and only if there exists a positive scalar ∈ satisfying

\sum + \in^{- 1} ϒ ϒ^{T} + \in Π^{T} Π < 0

(8)

Main results

New stability criteria for the systems (1)–(3) are presented in this section. The uncertain discrete-delayed systems (1)–(3) are globally asymptotically stable for all admissible uncertainties, if there exists a continuous function $V : R^{r} \to R$ which is radially unbounded, positive definite, and along solutions of equation (1) satisfies

Δ V (x (c)) = V (x (c + 1)) - V (x (c)) < 0

for all $x (c) \in R^{r}$ . The function $V$ said to be a Lyapunov function.

Now, our first result may be stated as follows.

Theorem 1. For the given integers $h_{m}$ and $h_{M},$ the systems (1)–(3) are globally asymptotically stable if there exist matrices

$C = C^{T} = [\begin{matrix} C_{1} & C_{2} & C_{3} \\ * & C_{4} & C_{5} \\ * & * & C_{6} \end{matrix}] > 0, Q_{i} = Q_{i}^{T} > 0 (i = 1, 2, 3), P_{i} = P_{i}^{T} > 0 (i = 1, 2), Y = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}], G = diag (g_{1}, g_{2}, . . ., g_{r}) > 0$ and positive scalars e₀, e₁ such that

W (d (c)) |_{d (c) = h_{m}} = [\begin{matrix} Γ_{1} (d (c)) |_{d (c) = h_{m}} - ϒ_{5, 1} & F_{11}^{T} Y \\ * & - {\tilde{P}}_{2} \end{matrix}] < 0

(9a)

W (d (c)) |_{d (c) = h_{M}} = [\begin{matrix} Γ_{1} (d (c)) |_{d (c) = h_{M}} - ϒ_{5, 2} & F_{22}^{T} Y^{T} \\ * & - {\tilde{P}}_{2} \end{matrix}] < 0

(9b)

where

\begin{matrix} ϒ_{5, 1} = {[\begin{matrix} F_{11} \\ F_{22} \end{matrix}]}^{T} [\begin{matrix} 2 {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} \end{matrix}] [\begin{matrix} F_{11} \\ F_{22} \end{matrix}], ϒ_{5, 2} = {[\begin{matrix} F_{11} \\ F_{22} \end{matrix}]}^{T} [\begin{matrix} {\tilde{P}}_{2} & Y \\ * & 2 {\tilde{P}}_{2} \end{matrix}] [\begin{matrix} F_{11} \\ F_{22} \end{matrix}] \\ F_{11} = [\begin{matrix} τ_{3} - τ_{2} \\ τ_{2} + τ_{3} - τ_{6} \end{matrix}], F_{22} = [\begin{matrix} τ_{2} - τ_{4} \\ τ_{2} + τ_{4} - τ_{7} \end{matrix}] \end{matrix}

\begin{matrix} Γ_{1} (d (c)) = \sum_{i = 1, i \neq 6, 7}^{11} τ_{i}^{T} κ_{i, i} τ_{i} + sym (\sum_{i = 3}^{9} τ_{1}^{T} κ_{1, i} τ_{i} + \sum_{i = 8}^{9} τ_{2}^{T} κ_{2, i} τ_{i} + \sum_{i = 5}^{8} τ_{3}^{T} κ_{3, i} τ_{i} + \sum_{i = 5}^{8} τ_{4}^{T} κ_{4, i} τ_{i} \\ + \sum_{i = 5}^{7} τ_{i}^{T} κ_{i, 8} τ_{8} + \sum_{i = 9}^{11} τ_{8}^{T} κ_{8, i} τ_{i} + \sum_{i = 10}^{11} τ_{9}^{T} κ_{9, i} τ_{i}) \end{matrix}

\begin{matrix} {\tilde{P}}_{2} = diag (P_{2}, 3 P_{2}), τ_{i} = [\begin{matrix} 0_{r \times (i - 1) r} I_{r} 0_{r \times (8 - i + 1) r + a_{0} + a_{1}} \end{matrix}], i = 1, 2, \dots, 9 \\ τ_{10} = [\begin{matrix} 0_{a_{0} \times 9 r} I_{a_{0}} 0_{a_{0} \times a_{1}} \end{matrix}], τ_{11} = [\begin{matrix} {0_{a_{1} \times (9 r +}}_{a_{0})} I_{a_{1}} \end{matrix}], \partial_{1} = - (h_{m}^{2} P_{1} + (h_{M} - h_{m})^{2} P_{2}) \\ κ_{1, 1} = - C_{1} + ((C_{2} + C_{2}^{T}) / 2) - 4 P_{1} + \sum_{i = 1}^{3} Q_{i} + (h_{M} - h_{m}) Q_{3} - \partial_{1} + e_{0} E_{0}^{T} E_{0}, κ_{1, 3} = ((C_{3} - C_{2}) / 2) - 2 P_{1} \end{matrix}

\begin{matrix} κ_{1, 4} = - C_{3} / 2, κ_{1, 5} = h_{m} (C_{4} - C_{2}) / 2 + 3 P_{1}, κ_{1, 6} = (d (c) - h_{m}) (C_{5} - C_{3}) / 2, κ_{1, 7} = (h_{M} - d (c)) (C_{5} - C_{3}) / 2 \\ κ_{1, 9} = - k_{q} \sqrt{- 2 k_{o}} A_{11}^{T} G, κ_{1, 8} = \partial_{1} + C_{2}^{T} / 2 + k_{q} A_{11}^{T} G, κ_{2, 2} = - Q_{3} + e_{1} E_{1}^{T} E_{1}, κ_{2, 8} = k_{q} A_{1 d}^{T} G \\ κ_{3, 3} = - Q_{1} - 4 P_{1}, κ_{2, 9} = - k_{q} \sqrt{- 2 k_{o}} A_{1 d}^{T} G, κ_{3, 5} = 3 P_{1} + h_{m} (C_{5}^{T} - C_{4}) / 2, κ_{3, 6} = (d (c) - h_{m}) (C_{6} - C_{5}) / 2 \\ κ_{3, 7} = (h_{M} - d (c)) (C_{6} - C_{5}) / 2, κ_{3, 8} = (C_{3}^{T} - C_{2}^{T}) / 2, κ_{4, 4} = - Q_{2}, κ_{4, 5} = - h_{m} C_{5}^{T} / 2 \\ κ_{4, 6} = - (d (c) - h_{m}) C_{6} / 2, κ_{4, 7} = - (h_{M} - d (c)) C_{6} / 2, κ_{4, 8} = - C_{3}^{T} / 2, κ_{5, 5} = - 3 P_{1}, κ_{5, 8} = h_{m} C_{2}^{T} / 2 \\ κ_{6, 8} = (d (c) - h_{m}) C_{3}^{T} / 2, κ_{7, 8} = (h_{M} - d (c)) C_{3}^{T} / 2, κ_{8, 8} = C_{1} - \partial_{1} + [(k_{o} / 2 k_{q}) - 2] G \\ κ_{8, 9} = \sqrt{- k_{o} / 2} G, κ_{8, 10} = k_{q} G H_{0}, κ_{8, 11} = k_{q} G H_{1}, κ_{9, 9} = - k_{q} G, κ_{9, 10} = - k_{q} \sqrt{- 2 k_{o}} G H_{0} \\ κ_{9, 11} = - k_{q} \sqrt{- 2 k_{o}} G H_{1}, κ_{10, 10} = - e_{0} I, κ_{11, 11} = - e_{1} I \end{matrix}

Appendix A contains the proof of Theorem 1.

Remark 1. As compared to Kandanvli and Kar (2011), Singh et al. (2021), Tadepalli and Kandanvli (2016) and Tadepalli et al. (2018), an improved summation inequality (Zhang et al., 2016a) along with WBI is employed in this paper to estimate the forward difference of LKF.

Theorem 1 presents the global asymptotic stability criterion for the system without considering delay partition. In our next theorem, the delay partitioning strategy is utilized where the delay interval is nonuniformly divided into $L (\geq 2)$ subintervals (see Figure 2).

Figure 2 shows the delay range $[1, h_{M}]$ which is nonuniformly divided into the delay subintervals $h_{m}, h_{1}, \dots, h_{M} .$ The $n_{i} (i = 1, 2, \dots, L)$ denotes the size of subintervals where $n_{1} = h_{m} - 1, n_{2} = h_{1} - h_{m},$ and so on. These subintervals may be of different sizes.

Now, we present the following theorem for the systems (1)–(3) by partitioning the delay interval into two nonuniform subintervals (i.e. $L = 2$ ).

Theorem 2. For the given integers $h_{m}$ and $h_{M},$ systems (1)–(3) are globally asymptotically stable if there exist matrices

\hat{C} = {\hat{C}}^{T} = [\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ * & C_{5} & C_{6} & C_{7} \\ * & * & C_{8} & C_{9} \\ * & * & * & C_{10} \end{matrix}] > 0, Y = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}], Z = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}], Q_{i} = Q_{i}^{T} > 0

$P_{i} = P_{i}^{T} > 0 (i = 1, 2, 3), G = diag (g_{1}, g_{2}, . . ., g_{r}) > 0$ and positive scalars $e_{0}, e_{1}$ such that

{\tilde{W}}_{1} (d (c)) < 0, \forall d (c) \in {h_{m}, h_{1}}

(10a)

{\tilde{W}}_{2} (d (c)) < 0, \forall d (c) \in {h_{1}, h_{M}}

(10b)

where

{\tilde{W}}_{1} (d (c)) = [\begin{matrix} {\tilde{Γ}}_{1} (d (c)) - {\tilde{ϒ}}_{5, 1} & {\tilde{F}}_{1}^{T} Y ℑ_{1} + {\tilde{F}}_{2}^{T} Y^{T} (1 - ℑ_{1}) \\ * & - {\tilde{P}}_{2} \end{matrix}], ℑ_{1} = \frac{h_{1} - d (c)}{h_{1} - h_{m}}

{\tilde{W}}_{2} (d (c)) = [\begin{matrix} {\tilde{Γ}}_{2} (d (c)) - {\tilde{ϒ}}_{5, 2} & {\tilde{F}}_{3}^{T} Z ℑ_{2} + {\tilde{F}}_{4}^{T} Z^{T} (1 - ℑ_{2}) \\ * & - {\tilde{P}}_{3} \end{matrix}], ℑ_{2} = \frac{h_{M} - d (c)}{h_{M} - h_{1}}

\begin{matrix} {\tilde{ϒ}}_{5, 1} = {[\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}]}^{T} [\begin{matrix} {\tilde{P}}_{2} + ℑ_{1} {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} + (1 - ℑ_{1}) {\tilde{P}}_{2} \end{matrix}] [\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}], {\tilde{ϒ}}_{5, 2} = {[\begin{matrix} {\tilde{F}}_{3} \\ {\tilde{F}}_{4} \end{matrix}]}^{T} [\begin{matrix} {\tilde{P}}_{3} + ℑ_{2} {\tilde{P}}_{3} & Z \\ * & {\tilde{P}}_{3} + (1 - ℑ_{2}) {\tilde{P}}_{3} \end{matrix}] [\begin{matrix} {\tilde{F}}_{3} \\ {\tilde{F}}_{4} \end{matrix}], {\tilde{F}}_{1} = [\begin{matrix} β_{3} - β_{2} \\ β_{3} + β_{2} - β_{7} \end{matrix}], \\ {\tilde{F}}_{2} = [\begin{matrix} β_{2} - β_{4} \\ β_{2} + β_{4} - β_{8} \end{matrix}], {\tilde{F}}_{3} = [\begin{matrix} β_{4} - β_{2} \\ β_{4} + β_{2} - β_{8} \end{matrix}], {\tilde{F}}_{4} = [\begin{matrix} β_{2} - β_{5} \\ β_{2} + β_{5} - β_{9} \end{matrix}] \end{matrix}

\begin{matrix} {\tilde{Γ}}_{1} (d (c)) = \sum_{i = 1, i \neq 7, 8}^{13} β_{i}^{T} α_{i, i} β_{i} + sym (\sum_{i = 3}^{11} β_{1}^{T} α_{1, i} β_{i} + \sum_{i = 10}^{11} β_{2}^{T} α_{2, i} β_{i} + \sum_{i = 6}^{10} β_{3}^{T} α_{3, i} β_{i} + \sum_{i = 5}^{10} β_{4}^{T} α_{4, i} β_{i} \\ + \sum_{i = 6}^{10} β_{5}^{T} α_{5, i} β_{i} + \sum_{i = 6}^{9} β_{i}^{T} α_{i, 10} β_{10} + \sum_{i = 11}^{13} β_{10}^{T} α_{10, i} β_{i} + \sum_{i = 12}^{13} β_{11}^{T} α_{11, i} β_{i}) \end{matrix}

\begin{matrix} {\tilde{Γ}}_{2} (d (c)) = \sum_{i = 1, i \neq 8, 9}^{13} β_{i}^{T} ν_{i, i} β_{i} + sym (\sum_{i = 3}^{11} β_{1}^{T} ν_{1, i} β_{i} + \sum_{i = 10}^{11} β_{2}^{T} ν_{2, i} β_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 5 \end{matrix}}^{10} β_{3}^{T} ν_{3, i} β_{i} + \sum_{i = 6}^{10} β_{4}^{T} ν_{4, i} β_{i} \\ + \sum_{i = 6}^{10} β_{5}^{T} ν_{5, i} β_{i} + \sum_{i = 6}^{9} β_{i}^{T} ν_{i, 10} β_{10} + \sum_{i = 11}^{13} β_{10}^{T} ν_{10, i} β_{i} + \sum_{i = 12}^{13} β_{11}^{T} ν_{11, i} β_{i}) \end{matrix}

\begin{array}{l} {\tilde{P}}_{i} = d i a g (P_{i}, 3 P_{i}), i = 2, 3, β_{i} = [\begin{matrix} 0_{r \times (i - 1) r} & I_{r} & 0_{r \times (10 - i + 1) r + a_{0} + a_{1}} \end{matrix}], i = 1, 2, \dots, 11 \\ β_{12} = [\begin{matrix} 0_{a_{0} \times 11 r} & I_{a_{0}} & 0_{a_{0} \times a_{1}} \end{matrix}], β_{13} = [\begin{matrix} 0_{a_{1} \times (11 r +} a_{0}) & I_{a_{1}} \end{matrix}], \partial_{2} = - (h_{m}^{2} P_{1} + {(h_{1} - h_{m})}^{2} P_{2} + {(h_{M} - h_{1})}^{2} P_{3}) \\ α_{1, 1} = ν_{1, 1} = - C_{1} + ((C_{2} + C_{2}^{T}) / 2) - 4 P_{1} + \sum_{i = 1}^{3} Q_{i} - \partial_{2} + e_{0} E_{0}^{T} E_{0}, α_{1, 3} = ν_{1, 3} = (C_{3} - C_{2}) / 2 - 2 P_{1} \\ α_{1, 4} = ν_{1, 4} = (C_{4} - C_{3}) / 2, α_{1, 5} = ν_{1, 5} = - C_{4} / 2, α_{1, 6} = ν_{1, 6} = 3 P_{1} + h_{m} ((C_{5} - C_{2}) / 2) \\ α_{1, 7} = (d (c) - h_{m}) (C_{6} - C_{3}) / 2, α_{1, 8} = (h_{1} - d (c)) (C_{6} - C_{3}) / 2, α_{1, 9} = (h_{M} - h_{1}) (C_{7} - C_{4}) / 2 \\ α_{1, 10} = ν_{1, 10} = \partial_{2} + C_{2}^{T} / 2 + k_{q} A_{11}^{T} G, α_{1, 11} = ν_{1, 11} = - k_{q} \sqrt{- 2 k_{o}} A_{11}^{T} G, α_{2, 2} = ν_{2, 2} = e_{1} E_{1}^{T} E_{1} \\ α_{2, 10} = ν_{2, 10} = k_{q} A_{1 d}^{T} G, α_{2, 11} = ν_{2, 11} = - k_{q} \sqrt{- 2 k_{o}} A_{1 d}^{T} G, α_{3, 3} = - Q_{1} - 4 P_{1} \\ α_{3, 6} = ν_{3, 6} = h_{m} (C_{6}^{T} - C_{5}) / 2 + 3 P_{1}, α_{3, 7} = ((d (c) - h_{m}) (C_{8} - C_{6}) / 2), α_{3, 8} = ((h_{1} - d (c)) (C_{8} - C_{6}) / 2) \\ α_{3, 9} = (h_{M} - h_{1}) (C_{9} - C_{7}) / 2, α_{3, 10} = ν_{3, 10} = (C_{3}^{T} - C_{2}^{T}) / 2, α_{4, 4} = - Q_{3} - 4 P_{3}, α_{4, 5} = - 2 P_{3} \\ α_{4, 6} = ν_{4, 6} = h_{m} (C_{7}^{T} - C_{6}^{T}) / 2, α_{4, 7} = (d (c) - h_{1}) (C_{9}^{T} - C_{8}) / 2, α_{4, 8} = ((h_{1} - d (c)) (C_{9}^{T} - C_{8}) / 2) \\ α_{4, 9} = 3 P_{3} + (h_{M} - h_{1}) ((C_{10} - C_{9}) / 2), α_{4, 10} = ν_{4, 10} = (C_{4}^{T} - C_{3}^{T}) / 2, α_{5, 5} = - Q_{2} - 4 P_{3} \\ α_{5, 6} = ν_{5, 6} = - C_{7}^{T} h_{m} / 2, α_{5, 7} = - C_{9}^{T} (d (c) - h_{m}) / 2, α_{5, 8} = - C_{9}^{T} (h_{1} - d (c)) / 2, α_{5, 10} = ν_{5, 10} = - C_{4}^{T} / 2 \\ α_{5, 9} = 3 P_{3} - C_{10} (h_{M} - h_{1}) / 2, α_{6, 6} = ν_{6, 6} = - 3 P_{1}, α_{6, 10} = ν_{6, 10} = C_{2}^{T} h_{m} / 2, α_{7, 10} = C_{3}^{T} (d (c) - h_{m}) / 2 \\ α_{8, 10} = C_{3}^{T} (h_{1} - d (c)) / 2, α_{9, 10} = C_{4}^{T} (h_{M} - h_{1}) / 2, α_{9, 9} = - 3 P_{3}, α_{10, 11} = ν_{10, 11} = \sqrt{- k_{o} / 2} G \\ α_{10, 10} = ν_{10, 10} = C_{1} - \partial_{2} + [(k_{o} / 2 k_{q}) - 2] G, α_{10, 12} = ν_{10, 12} = k_{q} G H_{0}, α_{10, 13} = ν_{10, 13} = k_{q} G H_{1} \\ α_{11, 11} = ν_{11, 11} = - k_{q} G, α_{11, 12} = ν_{11, 12} = - k_{q} \sqrt{- 2 k_{o}} G H_{0}, α_{11, 13} = ν_{11, 13} = - k_{q} \sqrt{- 2 k_{o}} G H_{1} \\ α_{12, 12} = ν_{12, 12} = - e_{0} I, α_{13, 13} = ν_{13, 13} - e_{1} I, ν_{1, 7} = (h_{1} - h_{m}) (C_{6} - C_{3}) / 2, ν_{3, 3} = - Q_{1} - 4 P_{1} - 4 P_{2} \\ ν_{1, 8} = (d (c) - h_{1}) (C_{7} - C_{4}) / 2, ν_{1, 9} = (h_{M} - d (c)) (C_{7} - C_{4}) / 2, ν_{3, 4} = - 2 P_{2} \\ ν_{3, 7} = (h_{1} - h_{m}) (C_{8} - C_{6}) / 2 + 3 P_{2}, ν_{3, 8} = (d (c) - h_{1}) (C_{9} - C_{7}) / 2, ν_{3, 9} = (h_{M} - d (c)) (C_{9} - C_{7}) / 2 \\ ν_{4, 4} = - Q_{3} - 4 P_{2}, ν_{4, 7} = 3 P_{2} + (h_{1} - h_{m}) (C_{9}^{T} - C_{8}) / 2, ν_{4, 8} = (d (c) - h_{1}) (C_{10} - C_{9}) / 2 \\ ν_{4, 9} = (h_{M} - d (c)) (C_{10} - C_{9}) / 2, ν_{5, 5} = - Q_{2}, ν_{5, 7} = - C_{9}^{T} (h_{1} - h_{m}) / 2, ν_{5, 8} = - C_{10} (d (c) - h_{1}) / 2 \\ ν_{5, 9} = - C_{10} (h_{M} - d (c)) / 2, ν_{7, 7} = - 3 P_{2}, ν_{7, 10} = C_{3}^{T} (h_{1} - h_{m}) / 2, ν_{8, 10} = C_{4}^{T} (d (c) - h_{1}) / 2 \\ ν_{9, 10} = C_{4}^{T} (h_{M} - d (c)) / 2 \end{array}

Appendix B discusses the proof of Theorem 2.

Remark 2. In general, Theorem 2 provides less conservative stability results than Theorem 1. It may be noted in Theorem 2 that $[h_{m}, h_{M}]$ = $[h_{m}, h_{1}] \cup [h_{1}, h_{M}]$ . A procedure for estimating the optimal values of $h_{1}$ and $h_{M}$ is provided below.

Step 1. For a given $h_{m}$ , choose positive integers $h_{M}$ and $h_{1}$ such that $h_{M} > h_{1} > h_{m}$ .

Step 2. Test the global asymptotic stability of the system (1) via Theorem 2 by checking (10) for $h_{1}$ varying in between $h_{m}$ and $h_{M} .$

If there exists an integer $h_{1} \in [h_{m}, h_{M}]$ such that equation (10) holds, then go to Step 3.

Else, stop the procedure and the maximum allowable delay is $h_{M} - 1 .$

Step 3. Set $h_{M} : = h_{M} + 1$ and go to Step 2.

Remark 3. In this paper, the reduced conservatism is achieved from the following two aspects: (1) nonuniform delay partitioning method is employed which provides more liberty in the selection of partition sizes as compared to uniform delay partitioning (Singh et al., 2021; Xue et al., 2016) and (2) the advantage of an improved summation inequality which helps to bound the sum term $- (h_{1} - h_{m}) \sum_{i = c - h_{1}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i)$ by the following term (see equation (29))

- γ_{1}^{T} (c) {[\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}]}^{T} ([\begin{matrix} {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} \end{matrix}] + [\begin{matrix} \frac{h_{1} - d (c)}{h_{1} - h_{m}} {\tilde{T}}_{1} & 0 \\ * & \frac{d (c) - h_{m}}{h_{1} - h_{m}} {\tilde{T}}_{2} \end{matrix}]) [\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}] γ_{1} (c)

As a special case, when $Y = 0$ , $d (c) = h_{1}$ , equation (29) reduces to

\begin{matrix} - (h_{1} - h_{m}) \sum_{i = c - h_{1}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) \\ \leq - {[\begin{matrix} x (c - h_{m}) - x (c - h_{1}) \\ x (c - h_{m}) + x (c - h_{1}) - χ (c, h_{m}, h_{1}) \end{matrix}]}^{T} {\tilde{P}}_{2} [\begin{matrix} x (c - h_{m}) - x (c - h_{1}) \\ x (c - h_{m}) + x (c - h_{1}) - χ (c, h_{m}, h_{1}) \end{matrix}] \end{matrix}

(11)

which can also be achieved by WBI approach.

Furthermore, as stated in Remark 1 of Seuret et al. (2015), the inequality (11) is less restrictive than the JI (Jiang et al., 2005). Therefore, the improved summation inequality (6) together with WBI employed in the present approach provides relaxed stability results than the WBI and JI approaches.

Remark 4. Following the proofs of Theorems 1 and 2, one can observe that these theorems are also applicable to all nonlinearities confined to the sector $[k_{o}, k_{q}]$ . Using Theorems 1 and 2, one can determine the set of values of $k_{o}$ and $k_{q}$ for which the given system is stable.

Remark 5. The number of decision variables in Theorems 1 and 2 decreases by specifically selecting the matrices ( Y in Theorem 1, Y and Z in Theorem 2) in diagonal form.

Remark 6. Observe that, in the derivation of Theorem 2, we have taken L (i.e. the number of subintervals) as 2. By extending the presented approach, the stability criteria for a general L can also be easily worked out.

Based on Theorem 2, we have the following corollary.

Corollary 1. Consider the system (1) with $Δ A_{11} = Δ A_{1 d} = 0$ and $f (y (c)) = y (c)$ , that is, the system

x (c + 1) = A_{11} x (c) + A_{1 d} x (c - d (c))

(12)

For given positive integers $h_{m}$ and $h_{M},$ the system (12) is globally asymptotically stable if there exist

$\hat{C} = {\hat{C}}^{T} = [\begin{matrix} C_{1} & C_{2} & C_{3} & C_{4} \\ * & C_{5} & C_{6} & C_{7} \\ * & * & C_{8} & C_{9} \\ * & * & * & C_{10} \end{matrix}] > 0, Y = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}], Z = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}]$ and $Q_{i} = Q_{i}^{T} > 0, P_{i} = P_{i}^{T} > 0 (i = 1, 2, 3)$ such that

{\hat{W}}_{1} (d (c)) < 0, \forall d (c) \in {h_{m}, h_{1}}

(13a)

{\hat{W}}_{2} (d (c)) < 0, \forall d (c) \in {h_{1}, h_{M}}

(13b)

where

{\hat{W}}_{1} (d (c)) = [\begin{matrix} {\bar{Γ}}_{1} (d (c)) - {\hat{ϒ}}_{5, 1} & {\hat{F}}_{1}^{T} Y ℑ_{1} + {\hat{F}}_{2}^{T} Y^{T} (1 - ℑ_{1}) \\ * & - {\tilde{P}}_{2} \end{matrix}], ℑ_{1} = \frac{h_{1} - d (c)}{h_{1} - h_{m}}

{\hat{W}}_{2} (d (c)) = [\begin{matrix} {\bar{Γ}}_{2} (d (c)) - {\hat{ϒ}}_{5, 2} & {\hat{F}}_{3}^{T} Z ℑ_{2} + {\hat{F}}_{4}^{T} Z^{T} (1 - ℑ_{2}) \\ * & - {\tilde{P}}_{3} \end{matrix}], ℑ_{2} = \frac{h_{M} - d (c)}{h_{M} - h_{1}}

\begin{matrix} {\hat{ϒ}}_{5, 1} = {[\begin{matrix} {\hat{F}}_{1} \\ {\hat{F}}_{2} \end{matrix}]}^{T} [\begin{matrix} {\tilde{P}}_{2} + ℑ_{1} {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} + (1 - ℑ_{1}) {\tilde{P}}_{2} \end{matrix}] [\begin{matrix} {\hat{F}}_{1} \\ {\hat{F}}_{2} \end{matrix}], {\hat{ϒ}}_{5, 2} = {[\begin{matrix} {\hat{F}}_{3} \\ {\hat{F}}_{4} \end{matrix}]}^{T} [\begin{matrix} {\tilde{P}}_{3} + ℑ_{2} {\tilde{P}}_{3} & Z \\ * & {\tilde{P}}_{3} + (1 - ℑ_{2}) {\tilde{P}}_{3} \end{matrix}] [\begin{matrix} {\hat{F}}_{3} \\ {\hat{F}}_{4} \end{matrix}] \\ {\hat{F}}_{1} = [\begin{matrix} {\tilde{β}}_{3} - {\tilde{β}}_{2} \\ {\tilde{β}}_{3} + {\tilde{β}}_{2} - {\tilde{β}}_{7} \end{matrix}], {\hat{F}}_{2} = [\begin{matrix} {\tilde{β}}_{2} - {\tilde{β}}_{4} \\ {\tilde{β}}_{2} + {\tilde{β}}_{4} - {\tilde{β}}_{8} \end{matrix}], {\hat{F}}_{3} = [\begin{matrix} {\tilde{β}}_{4} - {\tilde{β}}_{2} \\ {\tilde{β}}_{4} + {\tilde{β}}_{2} - {\tilde{β}}_{8} \end{matrix}], {\hat{F}}_{4} = [\begin{matrix} {\tilde{β}}_{2} - {\tilde{β}}_{5} \\ {\tilde{β}}_{2} + {\tilde{β}}_{5} - {\tilde{β}}_{9} \end{matrix}] \end{matrix}

{\bar{Γ}}_{1} (d (c)) = \sum_{i = 1}^{2} {\tilde{β}}_{i}^{T} {\tilde{α}}_{i, i} {\tilde{β}}_{i} + \sum_{i = 3, i \neq 7, 8}^{9} {\tilde{β}}_{i}^{T} α_{i, i} {\tilde{β}}_{i} + sym (\sum_{i = 2}^{9} {\tilde{β}}_{1}^{T} {\tilde{α}}_{1, i} {\tilde{β}}_{i} + \sum_{i = 3}^{9} {\tilde{β}}_{2}^{T} {\tilde{α}}_{2, i} {\tilde{β}}_{i} + \sum_{i = 6}^{9} {\tilde{β}}_{3}^{T} α_{3, i} {\tilde{β}}_{i} + \sum_{i = 5}^{9} {\tilde{β}}_{4}^{T} α_{4, i} {\tilde{β}}_{i} + \sum_{i = 6}^{9} {\tilde{β}}_{5}^{T} α_{5, i} {\tilde{β}}_{i})

{\bar{Γ}}_{2} (d (c)) = \sum_{i = 1}^{2} {\tilde{β}}_{i}^{T} {\tilde{ν}}_{i, i} {\tilde{β}}_{i} + \sum_{i = 3}^{7} {\tilde{β}}_{i}^{T} ν_{i, i} {\tilde{β}}_{i} + sym (\sum_{i = 2}^{9} {\tilde{β}}_{1}^{T} {\tilde{ν}}_{1, i} {\tilde{β}}_{i} + \sum_{i = 3}^{9} {\tilde{β}}_{2}^{T} {\tilde{ν}}_{2, i} {\tilde{β}}_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 5 \end{matrix}}^{9} {\tilde{β}}_{3}^{T} ν_{3, i} {\tilde{β}}_{i} + \sum_{i = 6}^{9} {\tilde{β}}_{4}^{T} ν_{4, i} {\tilde{β}}_{i} + \sum_{i = 6}^{9} {\tilde{β}}_{5}^{T} ν_{5, i} {\tilde{β}}_{i})

{\tilde{P}}_{i} = diag (P_{i}, 3 P_{i}), i = 2, 3

\begin{matrix} {\tilde{β}}_{i} = [\begin{matrix} 0_{r \times (i - 1) r} I_{r} 0_{r \times (8 - i + 1) r} \end{matrix}], i = 1, 2, \dots, 9 \\ \partial_{3} = - (h_{m}^{2} (A_{11} - I)^{T} P_{1} (A_{11} - I) + (h_{1} - h_{m})^{2} (A_{11} - I)^{T} P_{2} (A_{11} - I) + (h_{M} - h_{1})^{2} (A_{11} - I)^{T} P_{3} (A_{11} - I)) \end{matrix}

\begin{matrix} {\tilde{α}}_{1, 1} = {\tilde{ν}}_{1, 1} = - C_{1} + ((C_{2} + C_{2}^{T}) / 2) - 4 P_{1} + \sum_{i = 1}^{3} Q_{i} - \partial_{3} + A_{11}^{T} C_{1} A_{11} + (C_{2}^{T} / 2) A_{11} + A_{11}^{T} C_{2} / 2 \\ {\tilde{α}}_{1, 2} = {\tilde{ν}}_{1, 2} = A_{11}^{T} C_{1} A_{1 d} + (C_{2}^{T} / 2) A_{1 d} + h_{m}^{2} (A_{11} - I)^{T} P_{1} A_{1 d} + (h_{1} - h_{m})^{2} (A_{11} - I)^{T} P_{2} A_{1 d} + (h_{M} - h_{1})^{2} (A_{11} - I)^{T} P_{3} A_{1 d} \\ {\tilde{α}}_{1, 3} = {\tilde{ν}}_{1, 3} = ((A_{11}^{T} + I) (C_{3} - C_{2}) / 2) - 2 P_{1}, {\tilde{α}}_{1, 4} = {\tilde{ν}}_{1, 4} = (A_{11}^{T} + I) (C_{4} - C_{3}) / 2 \\ {\tilde{α}}_{1, 5} = {\tilde{ν}}_{1, 5} = - (A_{11}^{T} + I) C_{4} / 2, {\tilde{α}}_{1, 6} = {\tilde{ν}}_{1, 6} = 3 P_{1} + h_{m} ((C_{5} - C_{2}) / 2) + A_{11}^{T} C_{2} (h_{m} / 2) \\ {\tilde{α}}_{2, 2} = {\tilde{ν}}_{2, 2} = A_{1 d}^{T} C_{1} A_{1 d} + h_{m}^{2} A_{1 d}^{T} P_{1} A_{1 d} + (h_{1} - h_{m})^{2} A_{1 d}^{T} P_{2} A_{1 d} + (h_{M} - h_{1})^{2} A_{1 d}^{T} P_{3} A_{1 d} \\ {\tilde{α}}_{1, 7} = (d (c) - h_{m}) (C_{6} - C_{3}) / 2 + A_{11}^{T} C_{3} ((d (c) - h_{m}) / 2), {\tilde{α}}_{2, 3} = {\tilde{v}}_{2, 3} = A_{1 d}^{T} (C_{3} - C_{2}) / 2 \\ {\tilde{α}}_{1, 8} = (h_{1} - d (c)) (C_{6} - C_{3}) / 2 + A_{11}^{T} C_{3} ((h_{1} - d (c)) / 2), {\tilde{α}}_{2, 4} = {\tilde{ν}}_{2, 4} = A_{1 d}^{T} (C_{4} - C_{3}) / 2 \\ {\tilde{α}}_{1, 9} = (h_{M} - h_{1}) (C_{7} - C_{4}) / 2 + A_{11}^{T} C_{4} ((h_{M} - h_{1}) / 2), {\tilde{α}}_{2, 5} = {\tilde{ν}}_{2, 5} = - A_{1 d}^{T} C_{4} / 2 \\ {\tilde{α}}_{2, 6} = {\tilde{ν}}_{2, 6} = A_{1 d}^{T} C_{2} h_{m} / 2, {\tilde{α}}_{2, 7} = A_{1 d}^{T} C_{3} ((d (c) - h_{m}) / 2), {\tilde{α}}_{2, 8} = A_{1 d}^{T} C_{3} (h_{1} - d (c)) / 2 \\ {\tilde{α}}_{2, 9} = A_{1 d}^{T} C_{4} (h_{M} - h_{1}) / 2, {\tilde{ν}}_{1, 7} = (h_{1} - h_{m}) (C_{6} - C_{3}) / 2 + A_{11}^{T} C_{3} ((h_{1} - h_{m}) / 2), {\tilde{v}}_{2, 7} = A_{1 d}^{T} C_{3} (h_{1} - h_{m}) / 2 \\ {\tilde{ν}}_{1, 8} = (d (c) - h_{1}) (C_{7} - C_{4}) / 2 + A_{11}^{T} C_{4} ((d (c) - h_{1}) / 2), {\tilde{ν}}_{2, 9} = A_{1 d}^{T} C_{4} (h_{M} - d (c)) / 2 \\ {\tilde{ν}}_{1, 9} = (h_{M} - d (c)) (C_{7} - C_{4}) / 2 + A_{11}^{T} C_{4} ((h_{M} - d (c)) / 2), {\tilde{ν}}_{2, 8} = A_{1 d}^{T} C_{4} (d (c) - h_{1}) / 2 \end{matrix}

Remark 7. Recently, the global asymptotic stability of system (12) has been studied in Zhang et al. (2016b) using nonuniform delay partitioning. However, as shown in Example 2, Corollary 1 generally yields less stringent stability conditions than the approach in Zhang et al. (2016b).

Remark 8. The proposed stability conditions are in the form of LMIs and can be easily tested by MATLAB with YALMIP 3.0 (Boyd et al., 1994; Gahinet et al., 1995; Löfberg, 2004). For example, to check the feasibility via Theorem 1, one can solve the constraint (9) through the command “solvesdp (Constraint)” of the YALMIP solver, which gives a feasible result (if exists). To determine the solution S for which (9) is feasible, the command “double( S )” can be used.

Remark 9. The problem of global asymptotic stability of DTSs (1)–(3) has been investigated in Tadepalli et al. (2018) using WBI along with reciprocally convex inequality. The concept of uniform delay partitioning has been utilized in Singh et al. (2021) to deal with the stability of the DTS (1)–(3) using WBI along with reciprocally convex inequality. On the contrary, the approach presented in this paper utilizes an improved summation inequality (Zhang et al., 2016a) together with WBI and a nonuniform delay partitioning method. The nonuniform delay partitioning method yields less stringent stability results than uniform delay partitioning method. Table 1 shows that Theorem 2 is less conservative than several existing results (Kandanvli and Kar, 2011; Singh et al., 2021; Tadepalli et al., 2018; Tadepalli and Kandanvli, 2016).

Remark 10. Observe that with the increase in the number of subintervals (i.e. $L \geq 2$ ), one may achieve less conservative stability results at the cost of an increase in computational complexity. A larger L implies that the feasible results can be exploited in an enhanced stability domain. Since the number of decision variables is directly proportional to L, an increase in L leads to heavier computational complexity. Consequently, for the cases where the computational aspect is not very significant, the presented method may exhibit better stability results than the methods in Kandanvli and Kar (2011), Singh et al. (2021), Tadepalli and Kandanvli (2016), Tadepalli et al. (2018), and Zhang et al. (2016a, 2016b).

Table 1.

Comparison of maximum allowable upper bound delay h_M for a given h_m and computational complexity (Example 1).

$h_{m}$	2	6	10	12	Computational complexity
Theorem 1 (Kandanvli and Kar, 2011)	12	14	16	18	$(26 r + 2 a + 2) (13 r^{2} + 6 r + 2)^{3} = 16, 099, 776$
Theorem 1 (Tadepalli and Kandanvli, 2016)^a	12 $(n = 2,$ $τ = 1)$	16 $(n = 3,$ $τ = 2)$	19 $(n = 5,$ $τ = 2)$	21 $(n = 6,$ $τ = 2)$	$(13 r + 2 nr + 2 a + 2)$ $[(n^{2} + 7) r^{2} + (n + 7) r + 4]^{3} / 8$ $= 1, 365, 606$
Theorem 1 (Tadepalli et al., 2018)	15	17	20	21	$(31 r + 4 a + 2) (11 r^{2} + 5 r + 2)^{3} = 11, 941, 888$
Theorem 1 (L = 2) (Singh et al., 2021)	17	18	21	23	$(37 r + 4 a + 2) (19 r^{2} + 6 r + 2)^{3} = 58, 320, 000$
Theorem 1	16	18	20	22	$(31 r + 4 a + 2) (11 r^{2} + 5 r + 2)^{3} = 11, 941, 888$
Theorem 2 (L = 2)	17 $(h_{1} = 12)$	19 $(h_{1} = 13)$	22 $(h_{1} = 18)$	24 $(h_{1} = 20)$	$(63 r + 8 a + 2) (19 r^{2} + 6 r + 2)^{3} = 99, 144, 000$

In Tadepalli and Kandanvli (2016), $h_{m}$ is divided into n number of partitions such that $h_{m} = τ n$ and $τ$ is the partition size which is an integer.

Examples

Two numerical examples for highlighting the utility of the presented results and checking their conservatism by estimating the maximum allowable upper bound delay are discussed in this section.

Example 1. Consider the systems (1)–(3) with

A_{11} = [\begin{matrix} 0.7 & 0.05 \\ 0.01 & 0.4 \end{matrix}], A_{1 d} = [\begin{matrix} - 0.1 & 0 \\ - 0.1 & - 0.1 \end{matrix}]

H_{0} = H_{1} = [\begin{matrix} 0 & 0.1 \end{matrix}]^{T}, E_{0} = [\begin{matrix} 0.01 & 0 \end{matrix}], E_{1} = [\begin{matrix} 0 & 0.01 \end{matrix}]

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and nonlinearities fall in the sector $[k_{o}, k_{q}] = [0, 1] .$ Therefore, the nonlinearities under consideration cover zeroing, MT, saturation, combination of zeroing and MT, and so on. This system was also considered in Singh et al. (2021). Our aim is to determine $h_{M}$ for a known $h_{m}$ by iteratively solving the conditions in Theorems 1 and 2 with respect to $h_{M} .$ With the help of YALMIP 3.0 (Löfberg, 2004) and MATLAB (Boyd et al., 1994; Gahinet et al., 1995), the maximum allowable upper bound delay (h_M) for different $h_{m}$ obtained by Theorems 1 and 2 together with the previous results is listed in Table 1.

From Table 1, it can be observed that Theorem 2 shows less conservatism than Kandanvli and Kar (2011), Singh et al. (2021), Tadepalli and Kandanvli (2016), and Tadepalli et al. (2018). In Theorems 1 and 2, the numbers of decision variables are $N_{1} = 11 r^{2} + 5 r + 2$ and $N_{2} = 19 r^{2} + 6 r + 2,$ respectively, and the total row sizes of the LMIs are $O_{1} = (31 r + 4 a + 2)$ and $O_{2} = (63 r + 8 a + 2)$ , respectively. The computational complexities of Theorems 1 and 2 are proportional to $O_{1} N_{1}^{3}$ and $O_{2} N_{2}^{3}$ , respectively. Clearly, the computational burden of Theorem 1 is less as compared to Theorem 2. Theorem 2 usually provides larger stability region than Theorem 1.

The state trajectory of the considered system with $F_{0} (c) = \sin (180 (c - 1) / π),$ $F_{1} (c) = \cos (180 (c - 1) / π)$ and $d (c) = 2 + ⌊ | 15 \sin (180 (c - 1) / π) | ⌉$ is shown in Figure 3. Figure 4 depicts the plots of $F_{0} (c)$ and $F_{1} (c)$ . The plot of $d (c)$ is shown in Figure 5. It is clear from Figure 3 that the state trajectories reach to origin as $c \to \infty .$ This is in consistent with the fact, established via Theorem 2, that the DTS is globally asymptotically stable for $2 \leq d (c) \leq 17 .$ Figures 6 and 7 show the impact of various delay ranges on the system state trajectories $x_{1} (c)$ and $x_{2} (c)$ , respectively.

Example 2. Consider the system (12) with

A_{11} = [\begin{matrix} 0.8 & 0 \\ 0.05 & 0.9 \end{matrix}], A_{1 d} = [\begin{matrix} - 0.1 & 0 \\ - 0.2 & - 0.1 \end{matrix}]

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The stability behavior of this system was studied in Zhang et al. (2016a, 2016b). For a given $h_{m}$ , a comparison of maximum allowable upper bound delay ( $h_{M}$ ) arrived at via Corollary 1, and the results in Zhang et al. (2016a, 2016b) for the global asymptotic stability of the current example are given in Table 2. From Table 2, one can see that Corollary 1 leads to less conservative stability results than Zhang et al. (2016a, 2016b).

Table 2.

The maximum allowable upper bound delay $h_{M}$ for various $h_{m}$ (Example 2).

$h_{m}$	2	4	6	10	15	20
Theorem 1 (Zhang et al., 2016b)	19	19	20	21	24	27
Theorem 1 (Zhang et al., 2016a)	21	22	22	23	25	28
Corollary 1	23	23	23	24	26	29

Conclusion

This paper developed two new delay-dependent results (Theorems 1 and 2) to verify the stability of uncertain DTSs with time-varying delay and finite wordlength nonlinearities. The improved summation inequality (Zhang et al., 2016a) along with WBI is used to tackle the sum terms come in the forward difference of LKF. The concept of delay partitioning has been utilized to establish Theorem 2. A stability criterion (Corollary 1) for the DTSs in absence of nonlinearities and uncertainties is also presented. The presented approach is quite distinct and generally provides more relaxed stability results than several existing approaches.

The possible utilization of the presented ideas to analyze the stability of networked time-delay systems (Zou et al., 2019) and externally interfered discrete-delayed systems (Singh et al., 2022) is an important problem for future research. Although we focused on norm-bounded uncertainties in this paper, an extension of the proposed method to systems with randomly occurring uncertainties (Hu et al., 2011; Tutsoy et al., 2020; Wu et al., 2012) is an important issue that requires further study. The proposed work can be extended to 2D delayed DTSs with external disturbances which seems to be a challenging problem for future investigation. The results presented in this work provide only sufficient conditions. Finding the necessary and sufficient conditions for global asymptotic stability appears to be a complex issue that will require more research.

Footnotes

Appendix A

Proof of Theorem 1. Consider an LKF (Tadepalli et al., 2018)

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\begin{matrix} V (x (c)) = Π^{T} (c) C Π (c) + \sum_{i = c - h_{m}}^{c - 1} x^{T} (i) Q_{1} x (i) + \sum_{i = c - h_{M}}^{c - 1} x^{T} (i) Q_{2} x (i) + \sum_{j = - h_{M}}^{- h_{m}} \sum_{i = c + j}^{c - 1} x^{T} (i) Q_{3} x (i) \\ + \sum_{θ = - h_{m} + 1}^{0} \sum_{j = c - 1 + θ}^{c - 1} h_{m} Λ^{T} (j) P_{1} Λ (j) + \sum_{θ = - h_{M} + 1}^{- h_{m}} \sum_{j = c - 1 + θ}^{c - 1} (h_{M} - h_{m}) Λ^{T} (j) P_{2} Λ (j) \end{matrix}

where $Π^{T} (c) = [\begin{matrix} x^{T} (c) \sum_{s = c - h_{m}}^{c - 1} x^{T} (s) \sum_{s = c - h_{M}}^{c - h_{m} - 1} x^{T} (s) \end{matrix}]$ and $Λ (c) = x (c + 1) - x (c) = f (y (c)) - x (c) .$ Along the trajectories of equation (1), one has

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\begin{matrix} Δ V (x (c)) = V (x (c + 1)) - V (x (c)) \\ = γ^{T} (c) ξ (d (c)) γ (c) + x^{T} (c) Q_{1} x (c) - x^{T} (c - h_{m}) Q_{1} x (c - h_{m}) + x^{T} (c) Q_{2} x (c) \\ - x^{T} (c - h_{M}) Q_{2} x (c - h_{M}) + x^{T} (c) Q_{3} x (c) - (h_{M} - h_{m}) x^{T} (c) Q_{3} x (c) \\ - \sum_{c - h_{M}}^{c - h_{m}} x^{T} (i) Q_{3} x (i) + h_{m}^{2} Λ^{T} (c) P_{1} Λ (c) - h_{m} \sum_{i = c - h_{m}}^{c - 1} Λ^{T} (i) P_{1} Λ (i) \\ + (h_{M} - h_{m})^{2} Λ^{T} (c) P_{2} Λ (c) - (h_{M} - h_{m}) \sum_{i = c - h_{M}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) \end{matrix}

where

\begin{matrix} γ^{T} (c) = [\begin{matrix} x^{T} (c) x^{T} (c - d (c)) x^{T} (c - h_{m}) x^{T} (c - h_{M}) χ^{T} (c, 0, h_{m}) χ^{T} (c, h_{m}, d (c)) \end{matrix} \\ \begin{matrix} χ^{T} (c, d (c), h_{M}) f^{T} (y (c)) \end{matrix}] \end{matrix}

\begin{matrix} \begin{matrix} ξ (d (c)) = ρ_{1}^{T} (- C_{1} + ((C_{2} + C_{2}^{T}) / 2)) ρ_{1} + sym \\ (\sum_{i = 3}^{8} ρ_{1}^{T} ƛ_{1, i} ρ_{i} + ρ_{3}^{T} ƛ_{3, 5} ρ_{5} + \sum_{i = 6}^{8} ρ_{3}^{T} κ_{3, i} ρ_{i} + \sum_{i = 5}^{8} ρ_{4}^{T} κ_{4, i} ρ_{i} + ρ_{5}^{T} κ_{5, 8} ρ_{8} + ρ_{8}^{T} ƛ_{8, 8} ρ_{8} + \sum_{i = 6}^{7} ρ_{i}^{T} κ_{i, 8} ρ_{8}) \end{matrix} \\ ρ_{i} = [\begin{matrix} 0_{r \times (i - 1) r} I_{r} 0_{r \times (8 - i) r} \end{matrix}], i = 1, 2, \dots, 8 \end{matrix}

\begin{matrix} ƛ_{1, 3} = (C_{3} - C_{2}) / 2, ƛ_{1, 5} = h_{m} (C_{4} - C_{2}) / 2, ƛ_{1, 6} = (d (c) - h_{m}) (C_{5} - C_{3}) / 2, ƛ_{1, 8} = C_{2}^{T} / 2, ƛ_{8, 8} = C_{1} \\ ƛ_{1, 7} = (h_{M} - d (c)) (C_{5} - C_{3}) / 2, ƛ_{3, 5} = h_{m} (C_{5}^{T} - C_{4}) / 2, ƛ_{1, 4} = - C_{3} / 2 \end{matrix}

Note that

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- \sum_{c - h_{M}}^{c - h_{m}} x^{T} (i) Q_{3} x (i) \leq - x^{T} (c - d (c)) Q_{3} x (c - d (c))

Next, in view of Lemma 1, we have

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\begin{matrix} - h_{m} \sum_{i = c - h_{m}}^{c - 1} Λ^{T} (i) P_{1} Λ (i) \\ \leq - {[\begin{matrix} x (c) - x (c - h_{m}) \\ x (c) + x (c - h_{m}) - χ (c, 0, h_{m}) \end{matrix}]}^{T} [\begin{matrix} P_{1} & 0 \\ * & 3 P_{1} \end{matrix}] [\begin{matrix} x (c) - x (c - h_{m}) \\ x (c) + x (c - h_{m}) - χ (c, 0, h_{m}) \end{matrix}] \end{matrix}

By utilizing Lemma 2, the last sum term in $Δ V (x (c))$ can be expressed as

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\begin{matrix} - (h_{M} - h_{m}) \sum_{i = c - h_{M}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) = - (h_{M} - h_{m}) [\sum_{i = c - d (c)}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) + \sum_{i = c - h_{M}}^{c - d (c) - 1} Λ^{T} (i) P_{2} Λ (i)] \\ \leq γ^{T} (c) ϒ (d (c)) γ (c) \end{matrix}

where $ϒ (d (c)) = - {[\begin{matrix} F_{11} \\ F_{22} \end{matrix}]}^{T} ([\begin{matrix} {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} \end{matrix}] + [\begin{matrix} \frac{h_{M} - d (c)}{h_{M} - h_{m}} T_{1} & 0 \\ * & \frac{d (c) - h_{m}}{h_{M} - h_{m}} T_{2} \end{matrix}]) [\begin{matrix} F_{11} \\ F_{22} \end{matrix}]$

$T_{1} = {\tilde{P}}_{2} - {Y \tilde{P}}_{2}^{- 1} Y^{T}$ and $T_{2} = {\tilde{P}}_{2} - Y^{T} {\tilde{P}}_{2}^{- 1} Y .$ Now, using equations (17)–(20)

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Δ V (x (c)) \leq γ^{T} (c) Ψ (d (c)) γ (c) - ϖ

where

\begin{matrix} \begin{matrix} Ψ (d (c)) = ϒ (d (c)) + ρ_{1}^{T} (ε_{1, 1} - 2 k_{q} k_{o} {\bar{A}}_{11}^{T} G {\bar{A}}_{11}) ρ_{1} + ρ_{2}^{T} (- Q_{3} - 2 k_{q} k_{o} {\bar{A}}_{1 d}^{T} G {\bar{A}}_{1 d}) ρ_{2} + ρ_{8}^{T} ε_{8, 8} ρ_{8} \\ + sym (ρ_{1}^{T} ε_{1, 8} ρ_{8} + ρ_{2}^{T} ε_{2, 8} ρ_{8} + \sum_{i = 2}^{7} ρ_{1}^{T} κ_{1, i} ρ_{i} + \sum_{i = 5}^{8} ρ_{3}^{T} κ_{3, i} ρ_{i} + \sum_{i = 5}^{8} ρ_{4}^{T} κ_{4, i} ρ_{i} + \sum_{i = 5}^{7} ρ_{i}^{T} κ_{i, 8} ρ_{8}) + \sum_{i = 3}^{5} ρ_{i}^{T} κ_{i, i} ρ_{i} \end{matrix} \\ {\bar{A}}_{11} = A_{11} + Δ A_{11}, {\bar{A}}_{1 d} = A_{1 d} + Δ A_{1 d} \end{matrix}

ε_{1, 1} = - C_{1} + ((C_{2} + C_{2}^{T}) / 2) - 4 P_{1} + \sum_{i = 1}^{3} Q_{i} + (h_{M} - h_{m}) Q_{3} - \partial_{1}, κ_{1, 2} = - 2 k_{q} k_{o} {\bar{A}}_{11}^{T} G {\bar{A}}_{1 d}

\begin{matrix} ε_{1, 8} = \partial_{1} + (C_{2}^{T} / 2) + (k_{q} + k_{o}) {\bar{A}}_{11}^{T} G, ε_{2, 8} = (k_{q} + k_{o}) {\bar{A}}_{1 d}^{T} G, ε_{8, 8} = C_{1} - \partial_{1} - 2 G \\ ϖ = 2 [k_{q} y (c) - f (y (c))]^{T} G [f (y (c)) - k_{o} y (c)] \end{matrix}

In view of equation (2), the quantity $ϖ$ is non-negative (Kandanvli and Kar, 2009). Obviously, if $Ψ (d (c)) < 0 \forall d (c) \in [h_{m}, h_{M}]$ , then $Δ V (x (c)) < 0 .$ Thus, the condition $Ψ (d (c)) < 0$ guarantees the global asymptotic stability of the DTS.

By Schur’s complement, $Ψ (d (c)) < 0$ if and only if

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[\begin{matrix} M & N \\ * & κ_{9, 9} \end{matrix}] < 0

where

\begin{matrix} M = ϒ (d (c)) + ρ_{1}^{T} ε_{1, 1} ρ_{1} + ρ_{2}^{T} κ_{2, 2} ρ_{2} + \sum_{i = 3, i \neq 6, 7}^{8} ρ_{i}^{T} κ_{i, i} ρ_{i} + sym \\ (\sum_{i = 3}^{7} ρ_{1}^{T} κ_{1, i} ρ_{i} + \sum_{i = 1}^{2} ρ_{i}^{T} Φ_{i, 8} ρ_{8} + \sum_{i = 5}^{8} ρ_{3}^{T} κ_{3, i} ρ_{i} + \sum_{i = 5}^{8} ρ_{4}^{T} κ_{4, i} ρ_{i} + \sum_{i = 5}^{7} ρ_{i}^{T} κ_{i, 8} ρ_{8}) \end{matrix}

N = {[\begin{matrix} Φ_{1, 9}^{T} & Φ_{2, 9}^{T} & 0 & 0 & 0 & 0 & 0 & κ_{8, 9}^{T} \end{matrix}]}^{T}, Φ_{1, 8} = \partial_{17} + (C_{2}^{T} / 2) + k_{q} {\bar{A}}_{11}^{T} G

Φ_{1, 9} = - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{11}^{T} G, Φ_{2, 8} = k_{q} {\bar{A}}_{1 d}^{T} G, Φ_{2, 9} = - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{1 d}^{T} G, κ_{9, 9} = - k_{q} G

Using equation (3a), equation (22) can be reorganized as

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Ψ_{0} (d (c)) + {\bar{H}}_{0} F_{0} {\bar{E}}_{0} + {\bar{E}}_{0}^{T} F_{0}^{T} {\bar{H}}_{0}^{T} < 0

where ${\bar{H}}_{0}^{T} = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 & 0 & k_{q} H_{0}^{T} G & - k_{q} \sqrt{- 2 k_{o}} H_{0}^{T} G \end{matrix}]$

{\bar{E}}_{0} = [\begin{matrix} E_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}], Ψ_{0} (d (c)) = [\begin{matrix} \tilde{M} & \tilde{N} \\ * & κ_{9, 9} \end{matrix}]

\begin{matrix} \tilde{M} = ϒ (d (c)) + ρ_{1}^{T} ε_{1, 1} ρ_{1} + ρ_{2}^{T} κ_{2, 2} ρ_{2} + \sum_{i = 3}^{8} ρ_{i}^{T} κ_{i, i} ρ_{i} + sym \\ (\sum_{i = 3}^{7} ρ_{1}^{T} κ_{1, i} ρ_{i} + ρ_{1}^{T} {\tilde{Φ}}_{1, 8} ρ_{8} + ρ_{2}^{T} Φ_{2, 8} ρ_{8} + \sum_{i = 5}^{8} ρ_{3}^{T} κ_{3, i} ρ_{i} + \sum_{i = 5}^{8} ρ_{4}^{T} κ_{4, i} ρ_{i} + \sum_{i = 5}^{7} ρ_{i}^{T} κ_{i, 8} ρ_{8}) \end{matrix}

\tilde{N} = [\begin{matrix} {\tilde{Φ}}_{1, 9}^{T} & Φ_{2, 9}^{T} & 0 & 0 & 0 & 0 & 0 & κ_{8, 9}^{T} \end{matrix}]^{T}, {\tilde{Φ}}_{1, 8} = \partial_{1} + (C_{2}^{T} / 2) + k_{q} A_{11}^{T} G, {\tilde{Φ}}_{1, 9} = - k_{q} \sqrt{- 2 k_{o}} A_{11}^{T} G

By employing Lemma 3, equation (23) can be expressed as

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Ψ_{0} (d (c)) + e_{0}^{- 1} {\bar{H}}_{0} {\bar{H}}_{0}^{T} + e_{0} {\bar{E}}_{0}^{T} {\bar{E}}_{0} < 0, e_{0} > 0

By Schur’s complement, equation (24) is equivalent to

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[\begin{matrix} Ψ_{0} (d (c)) & {\bar{H}}_{0} \\ * & κ_{10, 10} \end{matrix}] < 0

Following the similar steps as in equations (23)–(25), one can easily demonstrate that equation (25) is equivalent to $W (d (c)) < 0$ . Observe that, $W (d (c))$ is an affine matrix function over $d (c) \in [h_{m}, h_{M}] .$ Based on the property of affinity, if $W (d (c)) |_{d (c) = h_{m}} < 0$ and $W (d (c)) |_{d (c) = h_{M}} < 0$ hold, then $W (d (c)) < 0 \forall d (c) \in [h_{m}, h_{M}] .$ This completes the proof.

Appendix B

Proof of Theorem 2. Consider the following two delay ranges.

Delay range I: $h_{m} < d (c) < h_{1}$

Consider the LKF as follows (Singh et al., 2021)

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\begin{matrix} V_{1} (x (c)) = Π_{1}^{T} (c) \hat{C} Π_{1} (c) + \sum_{i = c - h_{m}}^{c - 1} x^{T} (i) Q_{1} x (i) + \sum_{i = c - h_{M}}^{c - 1} x^{T} (i) Q_{2} x (i) + \sum_{i = c - h_{1}}^{c - 1} x^{T} (i) Q_{3} x (i) + \sum_{θ = - h_{m} + 1}^{0} \sum_{j = c - 1 + θ}^{c - 1} h_{m} Λ^{T} (j) P_{1} Λ (j) \\ + \sum_{θ = - h_{1} + 1}^{- h_{m}} \sum_{j = c - 1 + θ}^{c - 1} (h_{1} - h_{m}) Λ^{T} (j) P_{2} Λ (j) + \sum_{θ = - h_{M} + 1}^{- h_{1}} \sum_{j = c - 1 + θ}^{c - 1} (h_{M} - h_{1}) Λ^{T} (j) P_{3} Λ (j) \end{matrix}

where $Π_{1}^{T} (c) = [x^{T} (c) \sum_{s = c - h_{m}}^{c - 1} x^{T} (s) \sum_{s = c - h_{1}}^{c - h_{m} - 1} x^{T} (s) \sum_{s = c - h_{M}}^{c - h_{1} - 1} x^{T} (s)] .$ The forward difference of equation (26) along the trajectories of equation (1) is

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\begin{matrix} Δ V_{1} (x (c)) = V_{1} (x (c + 1)) - V_{1} (x (c)) \\ = γ_{1}^{T} (c) ξ_{1} (d (c)) γ_{1} (c) + x^{T} (c) Q_{1} x (c) - x^{T} (c - h_{m}) Q_{1} x (c - h_{m}) + x^{T} (c) Q_{2} x (c) \\ - x^{T} (c - h_{M}) Q_{2} x (c - h_{M}) + x^{T} (c) Q_{3} x (c) - x^{T} (c - h_{1}) Q_{3} x (c - h_{1}) + h_{m}^{2} Λ^{T} (c) P_{1} Λ (c) \\ - h_{m} \sum_{i = c - h_{m}}^{c - 1} Λ^{T} (i) P_{1} Λ (i) + (h_{1} - h_{m})^{2} Λ^{T} (c) P_{2} Λ (c) - (h_{1} - h_{m}) \sum_{i = c - h_{1}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) \\ + (h_{M} - h_{1})^{2} Λ^{T} (c) P_{3} Λ (c) - (h_{M} - h_{1}) \sum_{i = c - h_{M}}^{c - h_{1} - 1} Λ^{T} (i) P_{3} Λ (i) \end{matrix}

where

\begin{matrix} γ_{1}^{T} (c) = [\begin{matrix} x^{T} (c) x^{T} (c - d (c)) x^{T} (c - h_{m}) x^{T} (c - h_{1}) x^{T} (c - h_{M}) χ^{T} (c, 0, h_{m}) \end{matrix} \\ \begin{matrix} χ^{T} (c, h_{m}, d (c)) χ^{T} (c, d (c), h_{1}) χ^{T} (c, h_{1}, h_{M}) f^{T} (y (c)) \end{matrix}] \end{matrix}

\begin{matrix} ξ_{1} (d (c)) = {\tilde{ρ}}_{1}^{T} (- C_{1} + ((C_{2} + C_{2}^{T}) / 2)) {\tilde{ρ}}_{1} + {\tilde{ρ}}_{10}^{T} C_{1} {\tilde{ρ}}_{10} + sym \\ (\sum_{i = 3}^{10} {\tilde{ρ}}_{1}^{T} {\bar{ƛ}}_{1, i} {\tilde{ρ}}_{i} + \sum_{i = 7}^{9} {\tilde{ρ}}_{3}^{T} α_{3, i} {\tilde{ρ}}_{i} + {\tilde{ρ}}_{3}^{T} {\bar{ƛ}}_{3, 6} {\tilde{ρ}}_{6} + {\tilde{ρ}}_{3}^{T} {\bar{ƛ}}_{3, 10} {\tilde{ρ}}_{10} + \sum_{i = 6}^{10} {\tilde{ρ}}_{4}^{T} ƛ_{4, i} {\tilde{ρ}}_{i} + \sum_{i = 6}^{10} {\tilde{ρ}}_{5}^{T} ƛ_{5, i} {\tilde{ρ}}_{i} + \sum_{i = 6}^{9} {\tilde{ρ}}_{i}^{T} ƛ_{i, 10} {\tilde{ρ}}_{10}) \end{matrix}

\begin{matrix} {\bar{ƛ}}_{1, 3} = (C_{3} - C_{2}) / 2, {\bar{ƛ}}_{1, 4} = (C_{4} - C_{3}) / 2, {\bar{ƛ}}_{1, 5} = - C_{4} / 2, {\bar{ƛ}}_{1, 6} = h_{m} (C_{5} - C_{2}) / 2, {\bar{ƛ}}_{1, 10} = C_{2}^{T} / 2 \\ {\bar{ƛ}}_{1, 7} = (d (c) - h_{m}) (C_{6} - C_{3}) / 2, {\bar{ƛ}}_{1, 8} = (h_{1} - d (c)) (C_{6} - C_{3}) / 2, {\bar{ƛ}}_{1, 9} = (h_{M} - h_{1}) (C_{7} - C_{4}) / 2 \\ {\bar{ƛ}}_{3, 6} = h_{m} (C_{6}^{T} - C_{5}) / 2, {\bar{ƛ}}_{3, 10} = (C_{3}^{T} - C_{2}^{T}) / 2, {\bar{ƛ}}_{4, 6} = h_{m} (C_{7}^{T} - C_{6}^{T}) / 2, {\bar{ƛ}}_{4, 7} = (d (c) - h_{1}) (C_{9}^{T} - C_{8}) / 2 \\ {\bar{ƛ}}_{4, 8} = ((h_{1} - d (c)) (C_{9}^{T} - C_{8}) / 2), {\bar{ƛ}}_{4, 9} = (h_{M} - h_{1}) (C_{10} - C_{9}) / 2, {\bar{ƛ}}_{4, 10} = (C_{4}^{T} - C_{3}^{T}) / 2 \\ {\bar{ƛ}}_{5, 10} = - C_{4}^{T} / 2, {\bar{ƛ}}_{5, 6} = - C_{7}^{T} h_{m} / 2, {\bar{ƛ}}_{5, 7} = - C_{9}^{T} (d (c) - h_{m}) / 2, {\bar{ƛ}}_{5, 8} = - C_{9}^{T} (h_{1} - d (c)) / 2 \\ {\bar{ƛ}}_{5, 9} = - C_{10} (h_{M} - h_{1}) / 2, {\bar{ƛ}}_{6, 10} = C_{2}^{T} h_{m} / 2, {\bar{ƛ}}_{7, 10} = C_{3}^{T} (d (c) - h_{m}) / 2, {\bar{ƛ}}_{8, 10} = C_{3}^{T} (h_{1} - d (c)) / 2 \end{matrix}

{\bar{ƛ}}_{9, 10} = C_{4}^{T} (h_{M} - h_{1}) / 2, {\tilde{ρ}}_{i} = [\begin{matrix} 0_{r \times (i - 1) r} I_{r} 0_{r \times (9 - i + 1) r} \end{matrix}], i = 1, 2, \dots, 10

In view of Lemma 1, we have

(28)

- h_{m} \sum_{i = c - h_{m}}^{c - 1} Λ^{T} (i) P_{1} Λ (i) \leq - {[\begin{matrix} x (c) - x (c - h_{m}) \\ x (c) + x (c - h_{m}) - χ (c, 0, h_{m}) \end{matrix}]}^{T} [\begin{matrix} P_{1} & 0 \\ * & 3 P_{1} \end{matrix}] [\begin{matrix} x (c) - x (c - h_{m}) \\ x (c) + x (c - h_{m}) - χ (c, 0, h_{m}) \end{matrix}]

By Lemmas 1 and 2, the sum terms in $Δ V_{1} (x (c))$ can be expressed as

(29)

\begin{matrix} - (h_{1} - h_{m}) \sum_{i = c - h_{1}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) = - (h_{1} - d_{m}) [\sum_{i = c - h_{1}}^{c - d (c) - 1} Λ^{T} (i) P_{2} Λ (i) + \sum_{i = c - d (c)}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i)] \\ \leq - γ_{1}^{T} (c) {[\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}]}^{T} ([\begin{matrix} {\tilde{P}}_{2} & Y \\ * & {\tilde{P}}_{2} \end{matrix}] + [\begin{matrix} ℑ_{1} {\tilde{T}}_{1} & 0 \\ * & (1 - ℑ_{1}) {\tilde{T}}_{2} \end{matrix}]) [\begin{matrix} {\tilde{F}}_{1} \\ {\tilde{F}}_{2} \end{matrix}] γ_{1} (c) \end{matrix}

where ${\tilde{T}}_{1} = {\tilde{P}}_{2} - {Y \tilde{P}}_{2}^{- 1} Y^{T}$ , ${\tilde{T}}_{2} = {\tilde{P}}_{2} - Y^{T} {\tilde{P}}_{2}^{- 1} Y$ and

(30)

\begin{matrix} - (h_{M} - h_{1}) \sum_{i = k - h_{M}}^{k - h_{1} - 1} Λ^{T} (i) P_{3} Λ (i) \\ \leq - {[\begin{matrix} x (c - h_{1}) - x (c - h_{M}) \\ x (c - h_{1}) + x (c - h_{M}) - χ (c, h_{1}, h_{M}) \end{matrix}]}^{T} [\begin{matrix} P_{3} & 0 \\ * & 3 P_{3} \end{matrix}] [\begin{matrix} x (c - h_{1}) - x (c - h_{M}) \\ x (c - h_{1}) + x (c - h_{M}) - χ (c, h_{1}, h_{M}) \end{matrix}] \end{matrix}

Next, by following the similar steps as shown in equations (21)–(25), one obtains ${\tilde{W}}_{1} (d (c)) < 0, \forall d (c) \in [h_{m}, h_{1}] .$

Delay range II: $h_{1} < d (c) < h_{M}$

Consider $Δ V_{1} (x (c))$ similar to Delay range I except some necessary modifications such as

\begin{matrix} γ_{1}^{T} (c) = [\begin{matrix} x^{T} (c) x^{T} (c - d (c)) x^{T} (c - h_{m}) x^{T} (c - h_{1}) x^{T} (c - h_{M}) χ^{T} (c, 0, h_{m}) \end{matrix} \\ \begin{matrix} χ^{T} (c, h_{m}, h_{1}) χ^{T} (c, h_{1}, d (c)) χ^{T} (c, d (c), h_{M}) f^{T} (y (c)) \end{matrix}] \\ \begin{matrix} ξ_{1} (d (c)) = {\tilde{ρ}}_{1}^{T} (- C_{1} + ((C_{2} + C_{2}^{T}) / 2)) {\tilde{ρ}}_{1} + {\tilde{ρ}}_{10}^{T} C_{1} {\tilde{ρ}}_{10} + sym \\ (\sum_{i = 3, i \neq 7, 8, 9}^{10} {\tilde{ρ}}_{1}^{T} ƛ_{1, i} {\tilde{ρ}}_{i} + \sum_{i = 7}^{9} {\tilde{ρ}}_{1}^{T} {\hat{ƛ}}_{1, i} {\tilde{ρ}}_{i} + \sum_{i = 7}^{9} {\tilde{ρ}}_{3}^{T} {\hat{ƛ}}_{3, i} {\tilde{ρ}}_{i} + \sum_{i = 3}^{5} {\tilde{ρ}}_{i}^{T} ƛ_{i, 6} {\tilde{ρ}}_{6} + \sum_{i = 3}^{6} {\tilde{ρ}}_{i}^{T} ƛ_{i, 10} {\tilde{ρ}}_{10} + \sum_{i = 7}^{9} {\tilde{ρ}}_{4}^{T} {\hat{ƛ}}_{4, i} {\tilde{ρ}}_{i} + \sum_{i = 7}^{9} {\tilde{ρ}}_{5}^{T} {\hat{ƛ}}_{5, i} {\tilde{ρ}}_{i} + \sum_{i = 7}^{9} {\tilde{ρ}}_{i}^{T} {\hat{ƛ}}_{i, 10} {\tilde{ρ}}_{10}) \end{matrix} \end{matrix}

where

\begin{matrix} {\hat{ƛ}}_{1, 7} = (h_{1} - h_{m}) (C_{6} - C_{3}) / 2, {\hat{ƛ}}_{1, 8} = (d (c) - h_{1}) (C_{7} - C_{4}) / 2, {\hat{ƛ}}_{1, 9} = (h_{M} - d (c)) (C_{7} - C_{4}) / 2 \\ {\hat{ƛ}}_{3, 7} = (h_{1} - h_{m}) (C_{8} - C_{6}) / 2, {\hat{ƛ}}_{3, 8} = (d (c) - h_{1}) (C_{9} - C_{7}) / 2, {\hat{ƛ}}_{3, 9} = (h_{M} - d (c)) (C_{9} - C_{7}) / 2 \\ {\hat{ƛ}}_{4, 7} = (h_{1} - h_{m}) (C_{9}^{T} - C_{8}) / 2, {\hat{ƛ}}_{4, 8} = (d (c) - h_{1}) (C_{10} - C_{9}) / 2, {\hat{ƛ}}_{4, 9} = (h_{M} - d (c)) (C_{10} - C_{9}) / 2 \\ {\hat{ƛ}}_{5, 7} = - C_{9}^{T} (h_{1} - h_{m}) / 2, {\hat{ƛ}}_{5, 8} = - C_{10} (d (c) - h_{1}) / 2, {\hat{ƛ}}_{5, 9} =, C_{10} (h_{M} - d (c)) / 2 \\ {\hat{ƛ}}_{7, 10} = C_{3}^{T} (h_{1} - h_{m}) / 2, {\hat{ƛ}}_{8, 10} = C_{4}^{T} (d (c) - h_{1}) / 2, {\hat{ƛ}}_{9, 10} = C_{4}^{T} (h_{M} - d (c)) / 2 \end{matrix}

In this case, in view of Lemmas 1 and 2, we have

(31)

\begin{matrix} - (h_{1} - h_{m}) \sum_{i = c - h_{1}}^{c - h_{m} - 1} Λ^{T} (i) P_{2} Λ (i) \\ \leq - {[\begin{matrix} x (c - h_{m}) - x (c - h_{1}) \\ x (c - h_{m}) + x (c - h_{1}) - χ (c, h_{m}, h_{1}) \end{matrix}]}^{T} [\begin{matrix} P_{2} & 0 \\ * & 3 P_{2} \end{matrix}] [\begin{matrix} x (c - h_{m}) - x (c - h_{1}) \\ x (c - h_{m}) + x (c - h_{1}) - χ (c, h_{m}, h_{1}) \end{matrix}] \end{matrix}

(32)

- (h_{M} - h_{1}) \sum_{i = c - h_{M}}^{c - h_{1} - 1} Λ^{T} (i) P_{3} Λ (i) \leq - γ_{1}^{T} (c) {[\begin{matrix} {\tilde{F}}_{3} \\ {\tilde{F}}_{4} \end{matrix}]}^{T} ([\begin{matrix} {\tilde{P}}_{3} & Z \\ * & {\tilde{P}}_{3} \end{matrix}] + [\begin{matrix} ℑ_{2} {\bar{T}}_{1} & 0 \\ * & (1 - ℑ_{2}) {\bar{T}}_{2} \end{matrix}]) [\begin{matrix} {\tilde{F}}_{3} \\ {\tilde{F}}_{4} \end{matrix}] γ_{1} (c)

where ${\bar{T}}_{1} = {\tilde{P}}_{3} - {Z \tilde{P}}_{3}^{- 1} Z^{T}$ , ${\bar{T}}_{2} = {\tilde{P}}_{3} - Z^{T} {\tilde{P}}_{3}^{- 1} Z .$

Next, following the steps similar to equations (21)–(25), one obtains ${\tilde{W}}_{2} (d (c)) < 0, \forall d (c) \in [h_{1}, h_{M}]$ which together with ${\tilde{W}}_{1} (d (c)) < 0, \forall d (c) \in [h_{m}, h_{1}]$ implies $Δ V_{1} (x (c)) < 0$ . Hence, the system is globally asymptotically stable if equation (10) holds. This completes the proof.

Acknowledgements

The authors thank the editors and the anonymous reviewers for their constructive comments and suggestions.

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) received no financial support for the research, authorship, and/or publication of this article.

ORCID iDs

Kalpana Singh

V. Krishna Rao Kandanvli

Haranath Kar

Data availability statement

Data sharing not applicable to this article as no data sets were generated or analyzed during this study.

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