Delay partitioning approach to the robust stability of discrete-time systems with finite wordlength nonlinearities and time-varying delays

Abstract

Keywords

Introduction

System description

Main results

Examples

Conclusions

Footnotes

References

Appendix I

Appendix II

Appendix III

Acknowledgements

Declaration of conflicting interests

Funding

ORCID iDs

This paper is associated with the problem of robust stability of discrete-time systems with time-varying delays and finite wordlength nonlinearities. The main contribution of the paper is two-fold. First, this paper presents a new Lyapunov function based on the idea of partitioning the delay interval into subintervals. The approach may be considered as an advancement over the several existing approaches where only the lower delay bound is partitioned. The second is that reciprocally convex inequality (RCI) and Wirtinger-based inequality (WBI) are used to estimate the sum terms involved in the forward difference of Lyapunov function. The intermediate delay is also included in the Lyapunov function to deal with the delay information more effectively. Finally, several examples are provided to illustrate the less conservatism of the proposed approach as compared to several existing results.

Delay partitioning approach discrete-time system finite wordlength nonlinearity reciprocally convex inequality time-varying delay Wirtinger-based inequality

Over the past few decades, many researchers have paid attention to the stability analysis and design of the discrete-time systems with time delay due to their extensive applications in various fields such as networked control system, neural networks, microgrid system and so on (Mary and Rangarajan, 2016; Naghshtabrizi et al., 2010; Peng et al., 2011; Yu et al., 2013). During the implementation of discrete-time systems on the finite wordlength digital processor, the quantization (roundoff, magnitude truncation and value truncation) and overflow (saturation, zeroing, 2’s complement and triangular) nonlinearities are frequently introduced. These nonlinearities may cause the oscillations typically known as the limit cycles and push the system towards instability (Butterweck et al., 1988; Chang, 1981; Erickson and Michel, 1985). It is essential to find the range of the system parameters for limit cycle-free implementation of the system. Several criteria for the stability of discrete-time systems with quantization nonlinearities have been reported in Bose (1994), Chang (1981) and Lepschy et al. (1988) by ignoring the overflow effects, whereas others (Ahn and Kar, 2015a, 2015b; Chen, 2009; Kandanvli and Kar, 2008, 2009a; Kar, 2007; Kokil et al., 2020; Singh, 2010) have studied the overflow effects in discrete-time systems without considering the quantization effects. The effects of the combination of overflow and quantization are investigated in Kandanvli and Kar (2009b, 2011), Mahmoud (2013), Tadepalli et al. (2015, 2018) and Tadepalli and Kandanvli (2016, 2017).

Apart from above instability sources, the existence of time delays and parameter uncertainties in the discrete-time systems also lead to system instability. Parameter uncertainties occur in the system due to the variation of system parameters and modelling errors. Time delays are inevitable which are encountered if there is transmission of data from one part of the system to another. These delays deeply manipulate the stability of the system and offer deprived performance of the system (Feng et al., 2015; Kwon et al., 2013; Lee and Tsai, 2013; Li et al., 2018; Liu and Zhang, 2012; Mary and Rangarajan, 2016; Meng et al., 2010; Naghshtabrizi et al., 2010; Nam et al., 2015; Seuret et al., 2015; Shao and Han, 2011; Xu et al., 2014; Xue et al., 2016; Yu et al., 2013). Several researchers have explored the stability of uncertain discrete-time systems with time delay (see, Chen et al., 2004; Gao and Chen, 2007; Hua et al., 2014; Huang and Feng, 2010; Xu et al., 2001).

During the last decade, several papers dealing with the construction of suitable Lyapunov functions for analyzing the stability of discrete systems with time-varying delays have appeared (Feng et al., 2015; Gao and Chen, 2007; Kwon et al., 2013; Meng et al., 2010; Nam et al., 2015; Xu et al., 2014; Zhang et al., 2008). During the estimation of forward difference of Lyapunov functions, several techniques (e.g., free weighting matrix (FWM)-based method, inequality-based method etc.) are employed to handle the sum terms. FWM-based method uses the bounding inequalities for cross product between two vectors to reduce the conservatism of the stability conditions. However, this method introduces free weighting matrices, and hence, computationally demanding. In inequality-based method, the sum term is estimated by utilizing the well-known inequalities. Examples of inequality-based methods reported in the literature are reciprocally convex inequality (RCI) method (Liu and Zhang, 2012; Park et al., 2011), Jensen-based inequality (JBI) method (Huang and Feng, 2010; Liu and Zhang, 2012; Tadepalli and Kandanvli, 2016, 2017; Tadepalli et al., 2015; Xu et al., 2014) and Wirtinger-based inequality (WBI) method (Chen et al., 2016; Nam et al., 2015; Seuret et al., 2015; Tadepalli et al., 2018). The RCI is bounded inequality lemma for a linear combination of positive functions where the coefficients are inverses of convex parameters. This approach is usually less conservative than convex combination approach and also requires less decision variables. JBI approach approximates the difference of delay bounds due to the inversely weighted nature of coefficients to improve stability of the system. Criteria derived by WBI method generally provide less conservatism results than those obtained via JBI method (Seuret et al., 2015; Tadepalli et al., 2018). However, all these methods still convey the conservatism to some extent.

From the designer’s point of view, it is important to know the admissible maximum delay bound such that the system with delay less than this bound remains stable. One major concern to find the maximum admissible upper bound is to reduce the conservatism of criteria. In Kandanvli and Kar (2008, 2009a), delay-independent conditions are developed for the stability of discrete systems with constant delays, overflow nonlinearities and uncertainties. A delay-dependent criterion has been derived in Chen (2009) for systems with saturation nonlinearities and time-varying delays. By employing FWM method, Kandanvli and Kar (2011) established a delay-dependent stability criterion for systems with time-varying delays and composite nonlinearities. The stability criteria in Tadepalli and Kandanvli (2016) utilize the concepts of JBI and delay partitioning approach. The criteria in Tadepalli et al. (2018) are derived by adopting the WBI approach which provide less stringent results as compared to Tadepalli and Kandanvli (2016).

To further reduce the conservatism, delay partitioning technique may be adopted which can provide better stability results for the system. This technique is firstly used in Meng et al. (2010) to tackle with stability of the system. The approaches in Meng et al. (2010) and Tadepalli and Kandanvli (2016) are based on partitioning the lower bound of the interval-like time-varying delay only. It makes such approaches less effective if the lower delay bound is 0 or 1 and also the delay information is not exploited adequately. Thus, the derivation of less conservative stability criteria is still an important and challenging research problem.

In order to overcome the limitations in Meng et al. (2010) and Tadepalli and Kandanvli (2016) and utilize the delay information more effectively, in this paper, we have partitioned the delay interval into $N$ (an integer) subintervals which may lead to improved stability results. In particular, our main concern is to partition the delay interval into subintervals and to construct a suitable Lyapunov function including the sum terms whose forward difference involves the cross terms of every subinterval with a view to utilize the subinterval information more effectively.

The aim of this paper is to establish delay-dependent stability criteria for discrete-time systems with parameter uncertainties, delays, quantization and overflow nonlinearities. The contribution of the paper is as follows. Firstly, the system is studied by partitioning the delay interval into $N$ uniform subintervals. Secondly, intermediate delay is introduced to utilize the delay information more effectively while constructing the Lyapunov function. WBI method along with RCI are used to estimate the sum terms of Lyapunov function. The proposed method is shown to be superior than the previous methods (Kandanvli and Kar, 2011; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2018). Moreover, without considering the effects of uncertainties and nonlinearities in discrete-time systems, a new stability criterion is also presented and compared with several existing criteria (Chen et al., 2017; Feng et al., 2015; Liu and Zhang, 2012).

The paper is planned as follows. The next section gives the description of the system in state variable form. Based on delay partitioning approach, new stability criteria for uncertain discrete-time systems involving time-varying delays and finite wordlength nonlinearities are proposed in ‘Main Results’ section. A global asymptotic stability result for the considered system in absence of uncertainties and nonlinearities is also presented in this section. The advantages of the proposed results are illustrated with the help of numerical examples along with simulation results in ‘Examples’ section. The final section concludes the paper.

Notations: In this paper, $R^{m}$ denotes the m-dimensional Euclidean space, $R^{m \times n}$ is the set of $m \times n$ real matrices, $I$ is the identity matrix, $I_{n}$ is the $n \times n$ identity matrix, $0$ represents the null matrix or null vector, $0_{m \times n}$ refers to the $m \times n$ null matrix, $Ξ^{T}$ is the transpose of the matrix $Ξ,$ $Ξ > 0 (\geq 0)$ indicates that $Ξ$ is a positive definite (positive semidefinite) symmetric matrix, $Ξ < 0 (\leq 0)$ represents that $Ξ$ is a negative definite (negative semidefinite) symmetric matrix, diag (·) refers to a diagonal matrix, * denotes symmetric elements in a symmetric matrix and sym( $Ξ$ ) is defined as $Ξ$ + $Ξ^{T} .$ The notation $⌈ λ ⌉$ stands for the smallest integer that is greater than or equal to $λ,$ $| λ |$ refers to the absolute value of a real number $λ$ , $⌊ λ ⌉$ denotes the nearest integer to $λ$ .

This section presents the description of the system under consideration and the problem statement. Some useful lemmas needed for deriving our main results, are also provided.

Consider a class of discrete-time systems influenced by a combination of quantization and overflow nonlinearities, parameter uncertainties and time-varying delays. In particular, the system under consideration is given by

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

(1a)

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

(1b)

x (r) = Θ (r), r \in [- d_{u}, 0],

(1c)

where $x (r) \in R^{m}$ is a state vector, $Q (\cdot)$ denotes quantization nonlinearities, $O (\cdot)$ indicates the overflow nonlinearities, $A_{1}, A_{d 1} \in R^{m \times m}$ are the known constant system matrices, $Δ A_{1}, Δ A_{d 1} \in R^{m \times m}$ are the unknown matrices denoting parametric uncertainties in the state matrices, $Θ (r) \in R^{m}$ indicates the initial state vector at time r, and $f (\cdot)$ characterizes the composite nonlinearities.

The d(r) is a time-varying delay such that

1 \leq d_{l} \leq d (r) \leq d_{u},

(2)

where $d_{l}$ and $d_{u}$ are known positive integers representing the lower and the upper delay bounds, respectively. The delay interval is uniformly divided into $N$ subintervals such that

\begin{matrix} [d_{l}, d_{u}] = [d_{l}, d h_{1}] \cup [{dh}_{1} + 1, d h_{2}] \cup [d h_{2} + 1, d h_{3}] \\ \dots \cup [d h_{i - 1} + 1, d h_{i}] \cup [{dh}_{i} + 1, {dh}_{i + 1}] \dots \cup [d h_{N - 1} + 1, d_{u}] . \end{matrix}

(3a)

Here, the intermediate delay ${dh}_{i}$ $(i = 1, 2, \dots, N - 1)$ is given by

d h_{i} = {\begin{matrix} d_{l} + ⌈ \frac{d_{u} - d_{l}}{N} ⌉, & i = 1, \\ d h_{i - 1} + ⌈ \frac{d_{u} - d h_{i - 1}}{N - i + 1} ⌉, & i = 2, 3, \dots N - 1, \end{matrix}

(3b)

where the notation $⌈ λ ⌉$ signifies the smallest integer that is greater than or equal to $λ .$ Note that $N = 1$ implies no delay partition. For $N = 2,$ the delay interval is subdivided into two intervals (Lee and Tsai, 2013), that is

[d_{l}, d_{u}] = [d_{l}, d h_{1}] \cup [d h_{1} + 1, d_{u}], d h_{1} = d_{l} + ⌈ \frac{d_{u} - d_{l}}{2} ⌉ .

(4)

Similarly, for the case where $N = 3,$

[d_{l}, d_{u}] = [d_{l}, d h_{1}] \cup [d h_{1} + 1, d h_{2}] \cup [d h_{2} + 1, d_{u}],

(5a)

d h_{1} = d_{l} + ⌈ \frac{d_{u} - d_{l}}{3} ⌉, d h_{2} = d h_{1} + ⌈ \frac{d_{u} - d h_{1}}{2} ⌉ .

(5b)

The uncertainties are included as $Δ A_{1}$ and $Δ A_{d 1}$ in the state matrices and supposed to be of the form

Δ A_{1} = H_{0} F_{0} (r) E_{0}, Δ A_{d 1} = H_{1} F_{1} (r) E_{1},

(6a)

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

F_{i}^{T} (r) F_{i} (r) \leq I, i = 0, 1 .

(6b)

A class of discrete-time delayed systems consisting of parameter uncertainties, quantization and overflow nonlinearities can be described by the equations (1), (2) and (6). These systems cover digital control systems with quantization/overflow (Mahmoud, 2013), metal cutting process, material rolling process, wireless sensor networks (Chakrabarty et al., 2002) and so forth. Distinctive examples where the time delays, represented by (2), can occur in various communication networks during the transmission of signals like networked control system (NCS), microgrid system (MGS) and so on (Mary and Rangarajan, 2016; Peng et al., 2011). In the analysis of NCS, network induced delay and packet dropout are most important factors. MGS controller transmits the information in open communication networks that introduces some communication delay in the system.

The aim of this paper is to analyse the stability of the discrete-time system (1) and to establish less conservative stability condition by adopting an effective delay-partitioning approach.

The following lemmas are helpful for establishing the main results of the paper.

Lemma 1: (Nam et al., 2015) For a given matrix R > 0 and integers α > 0, β > 0, r > 0, fulfilling α≤β≤r, if

\begin{matrix} χ (r, α, β) = \\ \frac{1}{β - α} [(2 \sum_{s = r - β}^{r - α - 1} x (s)) + x (r - α) - x (r - β)], α < β \\ = 2 x (r - α), α = β \end{matrix}

(7)

then

- (β - α) \sum_{s = r - β}^{r - α - 1} Λ^{T} (s) R Λ (s) \leq - {[\begin{matrix} T_{0} \\ T_{1} \end{matrix}]}^{T} [\begin{matrix} R & 0 \\ 0 & 3 R \end{matrix}] [\begin{matrix} T_{0} \\ T_{1} \end{matrix}],

(8)

where $T_{0} = x (r - α) - x (r - β),$ $T_{1} = x (r - α) + x (r - β) - χ (r, α, β),$ $Λ (s) = x (s + 1) - x (s) .$ Further in this context, $T_{0}$ denotes the average value achieved by the variable $x (r)$ over the range of the delay $[α, β]$ and $T_{1}$ is proportional to the difference of mean value of $x (r)$ and its average value over the delay range.

Lemma 2: (Liu and Zhang, 2012) For any vectors $γ_{1}, γ_{2}$ , matrices R , S and real numbers $β_{1} \geq 0$ and $β_{2} \geq 0$ satisfying

[\begin{matrix} R & S \\ * & R \end{matrix}] \geq 0, β_{1} + β_{2} = 1,

(9a)

γ_{i} = 0, if β_{i} = 0 (i = 1, 2)

(9b)

then

- \frac{1}{β_{1}} γ_{1}^{T} R γ_{1} - \frac{1}{β_{2}} γ_{2}^{T} R γ_{2} \leq - {[\begin{matrix} γ_{1} \\ γ_{2} \end{matrix}]}^{T} [\begin{matrix} R & S \\ * & R \end{matrix}] [\begin{matrix} γ_{1} \\ γ_{2} \end{matrix}] .

(9c)

Lemma 3: (Boyd et al., 1994) Let $Σ,$ $F,$ $Π$ and $ϒ$ are real matrices with $ϒ$ satisfying $ϒ = ϒ^{T} .$ Then

ϒ + Σ F Π + Π^{T} F^{T} Σ^{T} < 0,

(10)

where $F^{T} F \leq I,$ if and only if there exists a scalar $\in > 0$ such that

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

(11)

This section presents the main results of the paper.

For the given system (1) which is operating in presence of quantization and overflow, $f (\cdot)$ is restricted to the sector $[k_{o}, k_{q}]$ . In other words

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

(12)

where

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

The delay partitioning is one of the very promising methods in reducing the conservativeness of the derived stability criteria. To present our approach in a lucid manner, we proceed with $N = 2$ , i.e., the delay interval is partitioned into two parts. Then, we extend the result to a case with three subintervals ( $N = 3$ ).

Pertaining to $N = 2,$ we have the following result.

Theorem 1: The system represented by (1), (2), (6) and (12) is globally asymptotically stable for given positive integers $d_{l}$ and $d_{u}$ satisfying (4) if there exist matrices

\begin{matrix} P = P^{T} = [\begin{matrix} P_{1} & P_{2} & P_{3} & P_{4} \\ * & P_{5} & P_{6} & P_{7} \\ * & * & P_{8} & P_{9} \\ * & * & * & P_{10} \end{matrix}] > 0, \\ Q_{i} = Q_{i}^{T} > 0, R_{i} = R_{i}^{T} > 0 (i = 1, 2, 3), Δ_{1} = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}], \end{matrix}

$Δ_{2} = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}],$ $G = diag (g_{1}, g_{2}, . . ., g_{m}) > 0$ and positive scalars $\in_{0}, \in_{1}$ such that

{\hat{\nabla}}_{i} = [\begin{matrix} \nabla_{i + 1} & Δ_{i} \\ * & \nabla_{i + 1} \end{matrix}] \geq 0, i = 1, 2,

(13)

Γ_{1} (d (r) = d_{l}) < 0, Γ_{2} (d (r) = d_{u}) < 0,

(14)

where

\begin{matrix} Γ_{1} (d (r)) = \sum_{i = 1}^{13} ϒ_{i}^{T} κ_{i, i} ϒ_{i} + sym (\sum_{i = 3}^{11} ϒ_{1}^{T} κ_{1, i} ϒ_{i} + \sum_{\begin{matrix} i = 3, \\ i \neq 5, 6, 9 \end{matrix}}^{11} ϒ_{2}^{T} κ_{2, i} ϒ_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 5 \end{matrix}}^{10} ϒ_{3}^{T} κ_{3, i} ϒ_{i} + \sum_{i = 5}^{10} ϒ_{4}^{T} κ_{4, i} ϒ_{i} \\ + \sum_{i = 6}^{10} ϒ_{5}^{T} κ_{5, i} ϒ_{i} + \sum_{i = 6}^{9} ϒ_{i}^{T} κ_{i, 10} ϒ_{10} + ϒ_{7}^{T} κ_{7, 8} ϒ_{8} + \sum_{i = 11}^{13} ϒ_{10}^{T} κ_{10, i} ϒ_{i} + \sum_{i = 12}^{13} ϒ_{11}^{T} κ_{11, i} ϒ_{i}), \end{matrix}

\begin{matrix} Γ_{2} (d (r)) = \sum_{i = 1}^{13} ϒ_{i}^{T} ν_{i, i} ϒ_{i} + sym (\sum_{i = 3}^{11} ϒ_{1}^{T} ν_{1, i} ϒ_{i} + \sum_{\binom{i = 4,}{i \neq 6, 7}}^{11} ϒ_{2}^{T} ν_{2, i} ϒ_{i} + \sum_{\binom{i = 4,}{i \neq 5}}^{10} ϒ_{3}^{T} ν_{3, i} ϒ_{i} + \sum_{i = 5}^{10} ϒ_{4}^{T} ν_{4, i} ϒ_{i} \\ + \sum_{i = 6}^{10} ϒ_{5}^{T} ν_{5, i} ϒ_{i} + \sum_{i = 6}^{9} ϒ_{i}^{T} ν_{i, 10} ϒ_{10} + ϒ_{8}^{T} ν_{8, 9} ϒ_{9} + \sum_{i = 11}^{13} ϒ_{10}^{T} ν_{10, i} ϒ_{i} + \sum_{i = 12}^{13} ϒ_{11}^{T} ν_{11, i} ϒ_{i}), \end{matrix}

$\nabla_{i} = diag (R_{i}, 3 R_{i}),$ $i = 2, 3,$ $ϒ_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (10 - i + 1) m + p_{0} + p_{1}} \end{matrix}],$ $i = 1, 2, \dots, 11,$ $ϒ_{12} = [\begin{matrix} 0_{p_{0} \times 11 m} & I_{p_{0}} & 0_{p_{0} \times p_{1}} \end{matrix}],$ $ϒ_{13} = [0_{p_{1} \times (11 m + p_{0})} I_{p_{1}}],$ $\partial_{17} = - (d_{l}^{2} R_{1} + (d h_{1} - d_{l})^{2} R_{2} + (d_{u} - d h_{1})^{2} R_{3}),$

$κ_{1, 1} = ν_{1, 1} = - P_{1} + ((P_{2} + P_{2}^{T}) / 2) - 4 R_{1} + \sum_{i = 1}^{3} Q_{i} - \partial_{17} + \in_{0} E_{0}^{T} E_{0},$ $κ_{6, 6} = ν_{6, 6} = - 3 R_{1},$

$κ_{2, 2} = - 8 R_{2} + Y_{11}^{T} + Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T} + Y_{11} + Y_{12} - Y_{21} - Y_{22} + \in_{1} E_{1}^{T} E_{1},$ $κ_{5, 5} = ν_{5, 5} = - Q_{2} - 4 R_{3},$

$κ_{3, 3} = ν_{3, 3} = - Q_{1} - 4 R_{1} - 4 R_{2},$ $κ_{4, 4} = ν_{4, 4} = - Q_{3} - 4 R_{2} - 4 R_{3},$ $κ_{7, 7} = ν_{7, 7} = κ_{8, 8} = - 3 R_{2},$

$κ_{10, 10} = ν_{10, 10} = P_{1} - \partial_{17} + [(k_{o} / 2 k_{q}) - 2] G,$ $κ_{9, 9} = ν_{8, 8} = ν_{9, 9} = - 3 R_{3},$ $κ_{11, 11} = ν_{11, 11} = - k_{q} G,$

$κ_{12, 12} = ν_{12, 12} = - \in_{0} I,$ $κ_{13, 13} = ν_{13, 13} = - \in_{1} I,$ $κ_{1, 3} = ν_{1, 3} = ((P_{3} - P_{2}) / 2) - 2 R_{1},$ $κ_{1, 4} = ν_{1, 4} = (P_{4} - P_{3}) / 2,$

$κ_{1, 5} = ν_{1, 5} = - P_{4} / 2,$ $κ_{1, 6} = ν_{1, 6} = 3 R_{1} + (d_{l} (P_{5} - P_{2}) / 2),$ $κ_{1, 7} = (d (r) - d_{l}) (P_{6} - P_{3}) / 2,$

$κ_{1, 8} = (d h_{1} - d (r)) (P_{6} - P_{3}) / 2,$ $κ_{1, 9} = (d_{u} - d h_{1}) (P_{7} - P_{4}) / 2,$ $κ_{1, 11} = ν_{1, 11} = - k_{q} \sqrt{- 2 k_{o}} {A_{1}}^{T} G,$ $κ_{1, 10} = ν_{1, 10} = \partial_{17} + P_{2}^{T} / 2 + k_{q} {A_{1}}^{T} G,$ $κ_{2, 3} = - 2 R_{2} - Y_{11}^{T} - Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T},$ $κ_{2, 4} = - 2 R_{2} - Y_{11} + Y_{12} + Y_{21} - Y_{22},$ $κ_{2, 7} = 3 R_{2} + Y_{21}^{T} + Y_{22}^{T},$ $κ_{2, 8} = 3 R_{2} - Y_{12} + Y_{22},$ $κ_{2, 10} = ν_{2, 10} = k_{q} A_{d 1}^{T} G,$ $κ_{2, 11} = ν_{2, 11} = - k_{q} \sqrt{- 2 k_{o}} {A_{d 1}}^{T} G,$ $κ_{3, 4} = Y_{11} - Y_{12} + Y_{21} - Y_{22},$ $κ_{3, 6} = ν_{3, 6} = (d_{l} (P_{6}^{T} - P_{5}) / 2) + 3 R_{1},$ $κ_{3, 7} = ((d (r) - d_{l}) (P_{8} - P_{6}) / 2) + 3 R_{2},$ $κ_{3, 8} = ((d h_{1} - d (r)) (P_{8} - P_{6}) / 2) + Y_{12} + Y_{22},$ $κ_{3, 9} = (d_{u} - d h_{1}) (P_{9} - P_{7}) / 2,$ $κ_{3, 10} = ν_{3, 10} = (P_{3}^{T} - P_{2}^{T}) / 2,$ $κ_{4, 5} = - 2 R_{3},$ $κ_{4, 6} = ν_{4, 6} = d_{l} (P_{7}^{T} - P_{6}^{T}) / 2,$ $κ_{4, 7} = Y_{22}^{T} - Y_{21}^{T} + ((d (r) - d_{l}) (P_{9}^{T} - P_{8}) / 2),$ $κ_{4, 8} = 3 R_{2} + ((d h_{1} - d (r)) (P_{9}^{T} - P_{8}) / 2),$ $κ_{4, 9} = 3 R_{3} + ((d_{u} - d h_{1}) (P_{10} - P_{9}) / 2),$

$κ_{4, 10} = ν_{4, 10} = (P_{4}^{T} - P_{3}^{T}) / 2,$ $κ_{5, 6} = ν_{5, 6} = - P_{7}^{T} d_{l} / 2,$ $κ_{5, 7} = - P_{9}^{T} (d (r) - d_{l}) / 2,$ $κ_{5, 8} = - P_{9}^{T} (d h_{1} - d (r)) / 2,$ $κ_{5, 9} = 3 R_{3} - (P_{10} (d_{u} - d h_{1}) / 2),$ $κ_{5, 10} = ν_{5, 10} = - P_{4}^{T} / 2,$ $κ_{6, 10} = ν_{6, 10} = P_{2}^{T} d_{l} / 2,$ $κ_{7, 10} = P_{3}^{T} (d (r) - d_{l}) / 2,$ $κ_{8, 10} = P_{3}^{T} (d h_{1} - d (r)) / 2,$ $κ_{9, 10} = P_{4}^{T} (d_{u} - d h_{1}) / 2,$ $κ_{7, 8} = - Y_{22},$ $κ_{10, 11} = ν_{10, 11} = \sqrt{- k_{o} / 2} G,$ $κ_{10, 12} = ν_{10, 12} = k_{q} G H_{0},$

$κ_{10, 13} = ν_{10, 13} = k_{q} G H_{1},$ $κ_{11, 13} = ν_{11, 13} = - k_{q} \sqrt{- 2 k_{0}} G H_{1},$ $κ_{11, 12} = ν_{11, 12} = - k_{q} \sqrt{- 2 k_{0}} G H_{0},$

$ν_{2, 2} = - 8 R_{3} + Z_{11}^{T} + Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T} + Z_{11} + Z_{12} - Z_{21} - Z_{22} + \in_{1} E_{1}^{T} E_{1},$

$ν_{1, 7} = (d h_{1} - d_{l}) (P_{6} - P_{3}) / 2,$ $ν_{1, 8} = (d (r) - d h_{1}) (P_{7} - P_{4}) / 2,$ $ν_{1, 9} = (d_{u} - d (r)) (P_{7} - P_{4}) / 2,$

$ν_{2, 4} = - 2 R_{3} - Z_{11}^{T} - Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T},$ $ν_{2, 5} = - 2 R_{3} - Z_{11} + Z_{12} + Z_{21} - Z_{22},$ $ν_{2, 8} = 3 R_{3} + Z_{21}^{T} + Z_{22}^{T},$

$ν_{2, 9} = 3 R_{3} - Z_{12} + Z_{22},$ $ν_{3, 7} = (d h_{1} - d_{l}) (P_{8} - P_{6}) / 2 + 3 R_{2},$ $ν_{3, 8} = (d (r) - d h_{1}) (P_{9} - P_{7}) / 2,$

$ν_{3, 9} = (d_{u} - d (r)) (P_{9} - P_{7}) / 2,$ $ν_{3, 4} = - 2 R_{2},$ $ν_{4, 5} = Z_{11} - Z_{12} + Z_{21} - Z_{22},$ $ν_{4, 7} = 3 R_{2} + (d h_{1} - d_{l}) (P_{9}^{T} - P_{8}) / 2,$

$ν_{4, 8} = 3 R_{3} + (d (r) - d h_{1}) (P_{10} - P_{9}) / 2,$ $ν_{4, 9} = Z_{12} + Z_{22} + (d_{u} - d (r)) (P_{10} - P_{9}) / 2,$

$ν_{5, 7} = - P_{9}^{T} (d h_{1} - d_{l}) / 2,$ $ν_{5, 8} = - P_{10} (d (r) - d h_{1}) / 2 - Z_{21}^{T} + Z_{22}^{T},$ $ν_{5, 9} = 3 R_{3} - P_{10} (d_{u} - d (r)) / 2,$

$ν_{7, 10} = P_{3}^{T} (d h_{1} - d_{l}) / 2,$ $ν_{8, 10} = P_{4}^{T} (d (r) - d h_{1}) / 2,$ $ν_{9, 10} = P_{4}^{T} (d_{u} - d (r)) / 2,$ $ν_{8, 9} = - Z_{22}$ .

The proof of Theorem 1 is given in Appendix I.

Remark 1: The reduced conservatism achieved in this paper is obtained from the three directions as follows. (i) The use of delay partitioning which captures the delay information more effectively. (ii) The use of WBI along with RCI approach, which bound the sum term $- (d h_{1} - d_{l}) \sum_{i = r - d h_{1}}^{r - d_{l} - 1} Λ^{T} (i) R_{2} Λ (i)$ with $d_{l} < d (r) < d h_{1}$ by the following term (see (A.7))

\begin{matrix} - {[\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \\ x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}]}^{T} \\ [\begin{matrix} \nabla_{2} & Δ_{1} \\ * & \nabla_{2} \end{matrix}] [\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \\ x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}], \end{matrix}

with $[\begin{matrix} \nabla_{2} & Δ_{1} \\ * & \nabla_{2} \end{matrix}] \geq 0$ . As a special case, where $Δ_{1} = 0$ and $d (r) = d h_{1}$ , (A.7) reduces to the following form obtained via WBI approach

\begin{matrix} - (d h_{1} - d_{l}) \sum_{i = r - d h_{1}}^{r - d_{l} - 1} Λ^{T} (i) R_{2} Λ (i) \\ \leq - {[\begin{matrix} x (r - d_{l}) - x (r - d h_{1}) \\ x (r - d_{l}) + x (r - d h_{1}) - χ (r, d_{l}, d h_{1}) \end{matrix}]}^{T} \\ \nabla_{2} [\begin{matrix} x (r - d_{l}) - x (r - d h_{1}) \\ x (r - d_{l}) + x (r - d h_{1}) - χ (r, d_{l}, d h_{1}) \end{matrix}] . \end{matrix}

(15)

From Remark 1 of Seuret et al. (2015) and Remark 6 of Tadepalli et al. (2018), it may be noted that (15) is less restrictive than that obtained via JBI. (iii) Unlike estimating the simple difference between upper and lower delay bounds to use a convex combination approach, the RCI approach treats with the inversely weighted convex combination of quadratic sum terms (Feng et al., 2015).

Pertaining to the situation where the delay interval is partitioned into three parts (i.e., $N = 3$ ), an extension of the presented approach leads to (see Appendix II):

Theorem 2: For given positive integers $d_{l}$ and $d_{u}$ satisfying (5), the system (1), (2), (6) and (12) is globally asymptotically stable if there are matrices

\begin{matrix} \hat{P} = {\hat{P}}^{T} = [\begin{matrix} P_{1} & P_{2} & P_{3} & P_{4} & P_{5} \\ * & P_{6} & P_{7} & P_{8} & P_{9} \\ * & * & P_{10} & P_{11} & P_{12} \\ * & * & * & P_{13} & P_{14} \\ * & * & * & * & P_{15} \end{matrix}] > 0, \\ Q_{i} = Q_{i}^{T} > 0, R_{i} = R_{i}^{T} > 0 (i = 1, 2, 3, 4), \end{matrix}

$Δ_{1} = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}], Δ_{2} = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}], Δ_{3} = [\begin{matrix} X_{11} & X_{12} \\ X_{21} & X_{22} \end{matrix}], G = diag (g_{1}, g_{2}, . . ., g_{m}) > 0$ and positive scalars $\in_{0}, \in_{1}$ such that

[\begin{matrix} \nabla_{i + 1} & Δ_{i} \\ * & \nabla_{i + 1} \end{matrix}] \geq 0, i = 1, 2, 3

(16)

\begin{matrix} Γ_{3} (d (r) = d_{l}) < 0, Γ_{4} (d (r) = (d h_{2} - d h_{1}) / 2) < 0, \\ Γ_{5} (d (r) = d_{u}) < 0, \end{matrix}

(17)

where

\begin{matrix} Γ_{3} (d (r)) = \sum_{i = 1}^{15} T_{i}^{T} θ_{i, i} T_{i} + sym (\sum_{i = 3}^{13} T_{1}^{T} θ_{1, i} T_{i} + \sum_{\binom{i = 3,}{i \neq 5, 6, 7, 10, 11}}^{13} T_{2}^{T} θ_{2, i} T_{i} + \sum_{\binom{i = 4,}{i \neq 5, 6}}^{12} T_{3}^{T} θ_{3, i} T_{i} + \sum_{\binom{i = 5,}{i \neq 6}}^{12} T_{4}^{T} θ_{4, i} T_{i} \\ + \sum_{i = 6}^{12} T_{5}^{T} θ_{5, i} T_{i} + \sum_{i = 7}^{12} T_{6}^{T} θ_{6, i} T_{i} + \sum_{i = 7}^{11} T_{i}^{T} θ_{i, 12} T_{12} + T_{8}^{T} θ_{8, 9} T_{9} + \sum_{i = 13}^{15} T_{12}^{T} θ_{12, i} T_{i} + \sum_{i = 14}^{15} T_{13}^{T} θ_{13, i} T_{i}), \end{matrix}

\begin{array}{l} Γ_{4} (d (r)) = \sum_{i = 1}^{15} T_{i}^{T} τ_{i, i} T_{i} + sym \\ (\sum_{i = 3}^{13} T_{1}^{T} τ_{1, i} T_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 6, 7, 8, 11 \end{matrix}}^{13} T_{2}^{T} τ_{2, i} T_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 5, 6 \end{matrix}}^{12} T_{3}^{T} τ_{3, i} T_{i} + \sum_{\begin{matrix} i = 5, \\ i \neq 6 \end{matrix}}^{12} T_{4}^{T} τ_{4, i} T_{i} + \sum_{i = 6}^{12} T_{5}^{T} τ_{5, i} T_{i} \\ + \sum_{i = 7}^{12} T_{6}^{T} τ_{6, i} T_{i} + \sum_{i = 7}^{11} T_{i}^{T} τ_{i, 12} T_{12} + T_{9}^{T} τ_{9, 10} T_{10} + \sum_{i = 13}^{15} T_{12}^{T} τ_{12, i} T_{i} + \sum_{i = 14}^{15} T_{13}^{T} τ_{13, i} T_{i}), \end{array}

\begin{array}{l} Γ_{5} (d (r)) = \sum_{i = 1}^{15} T_{i}^{T} χ_{i, i} T_{i} + sym (\sum_{i = 3}^{13} T_{1}^{T} χ_{1, i} T_{i} + \sum_{\begin{array}{l} i = 5, \\ i \neq 7, 8, 9 \end{array}}^{13} T_{2}^{T} χ_{2, i} T_{i} + \sum_{\begin{array}{l} i = 4, \\ i \neq 5, 6 \end{array}}^{12} T_{3}^{T} χ_{3, i} T_{i} + \sum_{\begin{array}{l} i = 5, \\ i \neq 6 \end{array}}^{12} T_{4}^{T} χ_{4, i} T_{i} + \sum_{i = 6}^{12} T_{5}^{T} χ_{5, i} T_{i} . \\ + \sum_{i = 7}^{12} T_{6}^{T} χ_{6, i} T_{i} + \sum_{i = 7}^{11} T_{i}^{T} χ_{i, 12} T_{12} + T_{10}^{T} χ_{10, 11} T_{11} + \sum_{i = 13}^{15} T_{12}^{T} χ_{12, i} T_{i} + \sum_{\begin{array}{l} i = 14 \end{array}}^{15} T_{13}^{T} χ_{13, i} T_{i}), \end{array}

$\nabla_{i} = diag (R_{i}, 3 R_{i}),$ i = 2, 3, 4, $T_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (12 - i + 1) m + p_{0} + p_{1}} \end{matrix}], i = 1, 2, \dots, 13,$

$T_{14} = [\begin{matrix} 0_{p_{0} \times 13 m} & I_{p_{0}} & 0_{p_{0} \times p_{1}} \end{matrix}],$ $T_{15} = [\begin{matrix} 0_{p_{1} \times (13 m + p_{0})} & I_{p_{1}} \end{matrix}],$ $χ_{10, 12} = P_{5}^{T} (d (r) - d h_{2}) / 2,$

$\partial_{18} = - (d_{l}^{2} R_{1} + (d h_{1} - d_{l})^{2} R_{2} + (d h_{2} - d h_{1})^{2} R_{3} + (d_{u} - d h_{2})^{2} R_{4}),$ $χ_{11, 12} = P_{5}^{T} (d_{u} - d (r)) / 2,$

$τ_{1, 1} = θ_{1, 1} = χ_{1, 1} = - P_{1} + ((P_{2} + P_{2}^{T}) / 2) - 4 R_{1} + \sum_{i = 1}^{4} Q_{i} - \partial_{18} + \in_{0} E_{0}^{T} E_{0},$

$θ_{2, 2} = - 8 R_{2} + Y_{11}^{T} + Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T} + Y_{11} + Y_{12} - Y_{21} - Y_{22} + \in_{1} E_{1}^{T} E_{1},$

$θ_{3, 3} = τ_{3, 3} = χ_{3, 3} = - Q_{1} - 4 R_{1} - 4 R_{2},$ $θ_{4, 4} = τ_{4, 4} = χ_{4, 4} = - Q_{3} - 4 R_{2} - 4 R_{3},$

$θ_{5, 5} = τ_{5, 5} = χ_{5, 5} = - Q_{4} - 4 (R_{3} + R_{4}),$ $θ_{6, 6} = τ_{6, 6} = - Q_{2} - 4 R_{4},$ $θ_{7, 7} = τ_{7, 7} = χ_{7, 7} = - 3 R_{1},$

$θ_{8, 8} = θ_{9, 9} = τ_{8, 8} = χ_{8, 8} = - 3 R_{2},$ $θ_{10, 10} = τ_{9, 9} = τ_{10, 10} = χ_{11, 11} = χ_{9, 9} = - 3 R_{3},$

$θ_{11, 11} = τ_{11, 11} = χ_{10, 10} = - 3 R_{4},$ $θ_{12, 12} = τ_{12, 12} = χ_{12, 12} = P_{1} - \partial_{18} + [(k_{o} / 2 k_{q}) - 2] G,$

$χ_{13, 13} = τ_{13, 13} = θ_{13, 13} = - k_{q} G,$ $χ_{14, 14} = τ_{14, 14} = θ_{14, 14} = - \in_{0} I,$ $χ_{15, 15} = τ_{15, 15} = θ_{15, 15} = - \in_{1} I,$

$θ_{1, 3} = τ_{1, 3} = χ_{1, 3} = ((P_{3} - P_{2}) / 2) - 2 R_{1},$ $θ_{1, 4} = τ_{1, 4} = χ_{1, 4} = (P_{4} - P_{3}) / 2,$ $θ_{1, 5} = τ_{1, 5} = χ_{1, 5} = (P_{5} - P_{4}) / 2,$

$θ_{1, 6} = τ_{1, 6} = χ_{1, 6} = - P_{5} / 2,$ $θ_{1, 7} = τ_{1, 7} = χ_{1, 7} = (d_{l} (P_{6} - P_{2}) / 2) + 3 R_{1},$ $θ_{1, 8} = (d (r) - d_{l}) (P_{7} - P_{3}) / 2,$

$θ_{1, 9} = (d h_{1} - d (r)) (P_{7} - P_{3}) / 2,$ $θ_{1, 10} = (d h_{2} - d h_{1}) (P_{8} - P_{4}) / 2,$ $θ_{1, 11} = τ_{1, 11} = (d_{u} - d h_{2}) (P_{9} - P_{5}) / 2,$

$θ_{1, 12} = τ_{1, 12} = χ_{1, 12} = \partial_{18} + (P_{2}^{T} / 2) + k_{q} A_{1}^{T} G,$ $θ_{1, 13} = τ_{1, 13} = χ_{1, 13} = - k_{q} \sqrt{- 2 k_{o}} A_{1}^{T} G,$

$θ_{2, 3} = - 2 R_{2} - Y_{11}^{T} - Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T},$ $θ_{2, 4} = - 2 R_{2} - Y_{11} + Y_{12} + Y_{21} - Y_{22}, θ_{2, 8} = 3 R_{2} + Y_{21}^{T} + Y_{22}^{T},$

$θ_{2, 9} = 3 R_{2} - Y_{12} + Y_{22},$ $θ_{2, 12} = τ_{2, 12} = χ_{2, 12} = k_{q} A_{d 1}^{T} G,$ $θ_{2, 13} = τ_{2, 13} = χ_{2, 13} = - k_{q} \sqrt{- 2 k_{o}} A_{d 1}^{T} G,$

$θ_{3, 4} = Y_{11} - Y_{12} + Y_{21} - Y_{22},$ $θ_{3, 7} = τ_{3, 7} = χ_{3, 7} = (d_{l} (P_{7}^{T} - P_{6}) / 2) + 3 R_{1},$ $θ_{3, 8} = ((d (r) - d_{l}) (P_{10} - P_{7}) / 2) + 3 R_{2},$

$θ_{3, 9} = ((d h_{1} - d (r)) (P_{10} - P_{7}) / 2) + Y_{12} + Y_{22},$ $θ_{3, 10} = (d h_{2} - d h_{1}) (P_{11} - P_{8}) / 2,$

$θ_{3, 11} = τ_{3, 11} = (d_{u} - d h_{2}) (P_{12} - P_{9}) / 2,$ $θ_{3, 12} = τ_{3, 12} = χ_{3, 12} = (P_{3}^{T} - P_{2}^{T}) / 2,$ $θ_{4, 5} = - 2 R_{3},$

$θ_{4, 7} = τ_{4, 7} = χ_{4, 7} = d_{l} (P_{8}^{T} - P_{7}^{T}) / 2,$ $θ_{4, 8} = Y_{22}^{T} - Y_{21}^{T} + ((d (r) - d_{l}) (P_{11}^{T} - P_{10}) / 2),$

$θ_{4, 9} = 3 R_{2} + ((d h_{1} - d (r)) (P_{11}^{T} - P_{10}) / 2),$ $θ_{4, 10} = 3 R_{3} + ((d h_{2} - d h_{1}) (P_{13} - P_{11}) / 2),$

$θ_{4, 11} = τ_{4, 11} = (d_{u} - d h_{2}) (P_{14} - P_{12}) / 2, θ_{5, 6} = τ_{5, 6} = - 2 R_{4},$ $θ_{4, 12} = τ_{4, 12} = χ_{4, 12} = (P_{4}^{T} - P_{3}^{T}) / 2,$

$θ_{5, 7} = τ_{5, 7} = χ_{5, 7} = d_{l} (P_{9}^{T} - P_{8}^{T}) / 2,$ $θ_{5, 8} = (d (r) - d_{l}) (P_{12}^{T} - P_{11}^{T}) / 2,$ $θ_{5, 9} = (d h_{1} - d (r)) (P_{12}^{T} - P_{11}^{T}) / 2,$

$θ_{5, 10} = 3 R_{3} + ((d h_{2} - d h_{1}) (P_{14}^{T} - P_{13}) / 2),$ $θ_{5, 11} = τ_{5, 11} = 3 R_{4} + ((d_{u} - d h_{2}) (P_{15} - P_{14}) / 2),$

$θ_{5, 12} = τ_{5, 12} = χ_{5, 12} = (P_{5}^{T} - P_{4}^{T}) / 2,$ $θ_{6, 7} = τ_{6, 7} = χ_{6, 7} = - P_{9}^{T} d_{l} / 2,$ $θ_{6, 8} = - P_{12}^{T} (d (r) - d_{l}) / 2,$

$θ_{6, 9} = - P_{12}^{T} (d h_{1} - d (r)) / 2,$ $θ_{6, 10} = - P_{14}^{T} (d h_{2} - d h_{1}) / 2,$ $θ_{6, 11} = τ_{6, 11} = 3 R_{4} - (P_{15} (d_{u} - d h_{2}) / 2),$

$θ_{6, 12} = τ_{6, 12} = χ_{6, 12} = - P_{5}^{T} / 2,$ $θ_{7, 12} = τ_{7, 12} = χ_{7, 12} = P_{2}^{T} d_{l} / 2$ $θ_{8, 9} = - Y_{22},$ $θ_{8, 12} = P_{3}^{T} (d (r) - d_{l}) / 2,$

$θ_{9, 12} = P_{3}^{T} (d h_{1} - d (r)) / 2,$ $θ_{10, 12} = P_{4}^{T} (d h_{2} - d h_{1}) / 2,$ $θ_{11, 12} = τ_{11, 12} = P_{5}^{T} (d_{u} - d h_{2}) / 2,$

$θ_{12, 13} = τ_{12, 13} = χ_{12, 13} = \sqrt{- k_{o} / 2} G,$ $θ_{12, 14} = τ_{12, 14} = χ_{12, 14} = k_{q} {GH}_{0},$ $θ_{12, 15} = τ_{12, 15} = χ_{12, 15} = k_{q} {GH}_{1},$

$θ_{13, 14} = τ_{13, 14} = χ_{13, 14} = - k_{q} \sqrt{- 2 k_{0}} {GH}_{0},$ $θ_{13, 15} = τ_{13, 15} = χ_{13, 15} = - k_{q} \sqrt{- 2 k_{0}} {GH}_{1},$

$τ_{2, 2} = - 8 R_{3} + Z_{11}^{T} + Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T} + Z_{11} + Z_{12} - Z_{21} - Z_{22} + \in_{1} E_{1}^{T} E_{1},$

$τ_{1, 8} = χ_{1, 8} = (d h_{1} - d_{l}) (P_{7} - P_{3}) / 2,$ $τ_{1, 9} = (d (r) - d h_{1}) (P_{8} - P_{4}) / 2,$ $τ_{1, 10} = (d h_{2} - d (r)) (P_{8} - P_{4}) / 2,$

$τ_{2, 5} = - 2 R_{3} - Z_{11} + Z_{12} + Z_{21} - Z_{22},$ $τ_{2, 4} = - 2 R_{3} - Z_{11}^{T} - Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T},$ $τ_{2, 9} = 3 R_{3} + Z_{21}^{T} + Z_{22}^{T},$

$τ_{2, 10} = 3 R_{3} - Z_{12} + Z_{22},$ $τ_{3, 8} = χ_{3, 8} = ((d h_{1} - d_{l}) (P_{10} - P_{7}) / 2) + 3 R_{2},$ $τ_{3, 9} = (d (r) - d h_{1}) (P_{11} - P_{8}) / 2,$

$τ_{3, 10} = (d h_{2} - d (r)) (P_{11} - P_{8}) / 2,$ $τ_{4, 5} = Z_{11} - Z_{12} + Z_{21} - Z_{22},$ $τ_{3, 4} = χ_{3, 4} - 2 R_{2},$

$τ_{4, 8} = χ_{4, 8} = ((d h_{1} - d_{l}) (P_{11}^{T} - P_{10}) / 2) + 3 R_{2},$ $τ_{4, 9} = ((d (r) - d h_{1}) (P_{13} - P_{11}) / 2) + 3 R_{3},$

$τ_{4, 10} = ((d h_{2} - d (r)) (P_{13} - P_{11}) / 2) + Z_{12} + Z_{22},$ $τ_{5, 8} = χ_{5, 8} = (d h_{1} - d_{l}) (P_{12}^{T} - P_{11}^{T}) / 2,$

$τ_{5, 9} = - Z_{21}^{T} + Z_{22}^{T} + ((d (r) - d h_{1}) ({P_{14}}^{T} - P_{13}) / 2),$ $τ_{5, 10} = 3 R_{3} + ((d h_{2} - d (r)) (P_{14}^{T} - P_{13}) / 2),$

$τ_{6, 8} = χ_{6, 8} = - P_{12}^{T} (d h_{1} - d_{l}) / 2,$ $τ_{6, 9} = - P_{14}^{T} (d (r) - d h_{1}) / 2,$ $τ_{6, 10} = - P_{14}^{T} (d h_{2} - d (r)) / 2,$

$τ_{10, 12} = P_{4}^{T} (d h_{2} - d (r)) / 2,$ $τ_{8, 12} = χ_{8, 12} = P_{3}^{T} (d h_{1} - d_{l}) / 2,$ $τ_{9, 10} = - Z_{22},$ $τ_{9, 12} = P_{4}^{T} (d (r) - d h_{1}) / 2,$

$χ_{5, 6} = X_{11} - X_{12} + X_{21} - X_{22},$ $χ_{2, 2} = - 8 R_{4} + X_{11}^{T} + X_{12}^{T} - X_{21}^{T} - X_{22}^{T} + X_{11} + X_{12} - X_{21} - X_{22} + \in_{1} E_{1}^{T} E_{1},$

$χ_{6, 6} = - Q_{2} - 4 R_{3},$ $χ_{1, 9} = (d h_{2} - d h_{1}) (P_{8} - P_{4}) / 2,$ $χ_{1, 10} = (d (r) - d h_{2}) (P_{9} - P_{5}) / 2,$

$χ_{1, 11} = (d_{u} - d (r)) (P_{9} - P_{5}) / 2,$ $χ_{2, 5} = - 2 R_{4} - X_{11}^{T} - X_{12}^{T} - X_{21}^{T} - X_{22}^{T},$ $χ_{2, 6} = - 2 R_{3} - X_{11} + X_{12} + X_{21} - X_{22},$

$χ_{2, 10} = 3 R_{4} + X_{21}^{T} + X_{22}^{T},$ $χ_{2, 11} = 3 R_{3} - X_{12} + X_{22},$ $χ_{3, 9} = (d h_{2} - d h_{1}) (P_{11} - P_{8}) / 2,$ $χ_{4, 5} = - 2 R_{3},$

$χ_{3, 10} = (d (r) - d h_{2}) (P_{12} - P_{9}) / 2,$ $χ_{3, 11} = (d_{u} - d (r)) (P_{12} - P_{9}) / 2,$ $χ_{4, 9} = ((d h_{2} - d h_{1}) (P_{13} - P_{11}) / 2) + 3 R_{3},$

$χ_{4, 10} = (d (r) - d h_{2}) (P_{14} - P_{12}) / 2,$ $χ_{4, 11} = (d_{u} - d (r)) (P_{14} - P_{12}) / 2,$

$χ_{5, 9} = 3 R_{3} + ((d h_{2} - d h_{1}) ({P_{14}}^{T} - P_{13}) / 2),$ $χ_{5, 10} = 3 R_{4} + ((d (r) - d h_{2}) (P_{15} - P_{14}) / 2),$

$χ_{5, 11} = X_{12} + X_{22} + ((d_{u} - d (r)) (P_{15} - P_{14}) / 2),$ $χ_{6, 9} = - P_{14}^{T} (d h_{2} - d h_{1}) / 2,$ $χ_{9, 12} = P_{4}^{T} (d h_{2} - d h_{1}) / 2,$

$χ_{6, 10} = - P_{15} ((d (r) - d h_{2}) / 2) - X_{21}^{T} + X_{22}^{T},$ $χ_{6, 11} = - P_{15} ((d_{u} - d (r)) / 2) + 3 R_{3},$ $χ_{10, 11} = - X_{22} .$

Remark 2: In case of no delay partition (i.e., $N = 1$ ), the time-varying delay $d (r)$ is considered to be in single interval $[d_{l}, d_{u}]$ . This may lead to conservatism stability result due to ineffective utilization of delay information whereas the delay partitioning method can avoid such problems.

Remark 3: The idea of delay partitioning to tackle the stability problem of discrete-time delayed systems has been utilized in Meng et al. (2010), Tadepalli and Kandanvli (2016) and Xue et al. (2016). In Meng et al. (2010) and Tadepalli and Kandanvli (2016), the delay partitioning method is employed in which the lower delay bound ( $d_{l}$ ) is divided into a number of partitions. However, such methods (Meng et al., 2010; Tadepalli and Kandanvli, 2016) lose importance when $d_{l}$ is equal to 0 or 1. In Xue et al. (2016), a delay-range-partition approach is exploited in which the intermediate delays depend on $d_{l}$ , $d_{u}$ and $(d_{u} - d_{l})$ . By contrast, the intermediate delay ( ${dh}_{i}$ ) considered in this paper depends not only on $d_{l}$ , $d_{u}$ and $(d_{u} - d_{l})$ but also on its preceding intermediate delay ( ${dh}_{i - 1}$ ). Hence, the present approach utilizes the delay information more effectively as compared with previous approaches (Meng et al., 2010; Tadepalli and Kandanvli, 2016; Xue et al., 2016).

Remark 4: In the present approach, the number of scalar decision variables increases with the increase of subintervals, which may increase the computational complexity. However, a specific selection of matrices ( $Δ_{1}$ and $Δ_{2}$ in Theorem 1 and $Δ_{1},$ $Δ_{2}$ and $Δ_{3}$ in Theorem 2) as the diagonal one, helps in reducing the number of scalar decision variables.

Remark 5: The conditions (14) and (17) of Theorem 1 and Theorem 2, respectively, rely on the values of sector bounds $k_{o}$ and $k_{q}$ . The given system characterized by (1), (2), (6) and (12) may happen to be stable for some set of values of $k_{o}$ and $k_{q},$ where as it may be unstable for other set of values $k_{o}$ and $k_{q}$ . The proposed theorems can be used to test the global asymptotic stability of the uncertain systems for various concatenations of quantization and overflow.

Remark 6: Parameter uncertainties may be in the form of norm bounded (Gao and Chen, 2007; Kandanvli and Kar, 2009a, 2009b, 2011; Tadepalli and Kandanvli, 2016, 2017; Tadepalli et al., 2018) or polytopic (He et al., 2004). The norm bounded parameter uncertainties are considered in this paper for modelling the uncertainties. The possible extension of the presented approach for delayed systems with polytopic uncertainties requires further investigation.

Remark 7: The stability conditions presented in this paper are specified in terms of linear matrix inequalities (LMIs) and, one can use MATLAB environment (Boyd et al., 1994; Gahinet, 1995) along with YALMIP (Yet Another LMI Parser) 3.0 (Lofberg, 2004) to find the feasible solutions. For instance, to test the feasibility alone via Theorem 1, the constraints (13) and (14) can be solved by using ‘solvesdp (Constraints)’ command of YALMIP solver which returns a feasible solution (if exists). The command ‘double(X)’ can be used to extract the solution X for which LMIs (13) and (14) are feasible.

As a direct consequence of Theorem 1, we have the following corollary for the system (1) and (2) in absence of parameter uncertainties and finite wordlength nonlinearities.

Corollary 1: Consider the system (1) and (2) with no uncertainties $(Δ A_{1} = Δ A_{d 1} = 0)$ and nonlinearities $(f (y (r)) = y (r)),$ that is, the system becomes

x (r + 1) = A_{1} x (r) + A_{d 1} x (r - d (r)) .

(18)

Then the system represented by (18) and (2) is globally asymptotically stable for given positive integers $d_{l}$ and $d_{u}$ satisfying (4) if there exist matrices

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

{\hat{\nabla}}_{i} = [\begin{matrix} \nabla_{i + 1} & Δ_{i} \\ * & \nabla_{i + 1} \end{matrix}] \geq 0, i = 1, 2,

(19)

{\tilde{Γ}}_{1} (d (r) = d_{l}) < 0, {\tilde{Γ}}_{2} (d (r) = d_{u}) < 0,

(20)

where

\begin{matrix} {\tilde{Γ}}_{1} (d (r)) = \sum_{i = 1}^{2} {\tilde{ϒ}}_{i}^{T} {\tilde{κ}}_{i, i} {\tilde{ϒ}}_{i} + \sum_{i = 3}^{9} {\tilde{ϒ}}_{i}^{T} κ_{i, i} {\tilde{ϒ}}_{i} + sym (\sum_{i = 2}^{9} {\tilde{ϒ}}_{1}^{T} {\tilde{κ}}_{1, i} {\tilde{ϒ}}_{i} + \sum_{i = 3}^{9} {\tilde{ϒ}}_{2}^{T} {\tilde{κ}}_{2, i} {\tilde{ϒ}}_{i} + \sum_{\binom{i = 4,}{i \neq 5}}^{9} {\tilde{ϒ}}_{3}^{T} κ_{3, i} {\tilde{ϒ}}_{i} \\ + \sum_{i = 5}^{9} {\tilde{ϒ}}_{4}^{T} κ_{4, i} {\tilde{ϒ}}_{i} + \sum_{i = 6}^{9} {\tilde{ϒ}}_{5}^{T} κ_{5, i} {\tilde{ϒ}}_{i} + {\tilde{ϒ}}_{7}^{T} κ_{7, 8} {\tilde{ϒ}}_{8}), \end{matrix}

\begin{matrix} {\tilde{Γ}}_{2} (d (r)) = \sum_{i = 1}^{2} {\tilde{ϒ}}_{i}^{T} {\tilde{ν}}_{i, i} {\tilde{ϒ}}_{i} + \sum_{i = 3}^{9} {\tilde{ϒ}}_{i}^{T} ν_{i, i} {\tilde{ϒ}}_{i} + sym (\sum_{i = 2}^{9} {\tilde{ϒ}}_{1}^{T} {\tilde{ν}}_{1, i} {\tilde{ϒ}}_{i} + \sum_{i = 3}^{9} {\tilde{ϒ}}_{2}^{T} {\tilde{ν}}_{2, i} {\tilde{ϒ}}_{i} + \sum_{\begin{matrix} i = 4, \\ i \neq 5 \end{matrix}}^{9} {\tilde{ϒ}}_{3}^{T} ν_{3, i} {\tilde{ϒ}}_{i} + \sum_{i = 5}^{9} {\tilde{ϒ}}_{4}^{T} ν_{4, i} {\tilde{ϒ}}_{i} \\ + \sum_{i = 6}^{9} {\tilde{ϒ}}_{5}^{T} ν_{5, i} {\tilde{ϒ}}_{i} + {\tilde{ϒ}}_{8}^{T} ν_{8, 9} {\tilde{ϒ}}_{9}), \end{matrix}

$\nabla_{i} = diag (R_{i}, 3 R_{i}),$ $i = 2, 3,$ ${\tilde{ϒ}}_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (8 - i + 1) m} \end{matrix}],$ $i = 1, 2, \dots, 9,$

${\tilde{\partial}}_{17} = - (d_{l}^{2} (A_{1} - I)^{T} R_{1} (A_{1} - I) + (d h_{1} - d_{l})^{2} (A_{1} - I)^{T} R_{2} (A_{1} - I) + (d_{u} - d h_{1})^{2} (A_{1} - I)^{T} R_{3} (A_{1} - I)),$

${\tilde{κ}}_{1, 1} = {\tilde{ν}}_{1, 1} = - P_{1} + ((P_{2} + P_{2}^{T}) / 2) + \sum_{i = 1}^{3} Q_{i} + A_{1}^{T} P_{1} A_{1} + (P_{2}^{T} / 2) A_{1} + A_{1}^{T} P_{2} / 2 - 4 R_{1} - {\tilde{\partial}}_{17},$

${\tilde{κ}}_{2, 2} = - 8 R_{2} + Y_{11}^{T} + Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T} + Y_{11} + Y_{12} - Y_{21} - Y_{22} + A_{d 1}^{T} P_{1} A_{d 1} + d_{l}^{2} A_{d 1}^{T} R_{1} A_{d 1},$ ${\tilde{κ}}_{1, 2} = {\tilde{ν}}_{1, 2} = A_{1}^{T} P_{1} A_{d 1} + (P_{2}^{T} / 2) A_{d 1} + d_{l}^{2} (A_{1} - I)^{T} R_{1} A_{d 1},$ ${\tilde{κ}}_{1, 3} = {\tilde{ν}}_{1, 3} = (A_{1}^{T} + I) ((P_{3} - P_{2}) / 2) - 2 R_{1},$

${\tilde{κ}}_{1, 4} = {\tilde{ν}}_{1, 4} = (A_{1}^{T} + I) (P_{4} - P_{3}) / 2,$ ${\tilde{κ}}_{1, 5} = {\tilde{ν}}_{1, 5} = - (A_{1}^{T} + I) P_{4} / 2,$ ${\tilde{κ}}_{1, 6} = {\tilde{ν}}_{1, 6} = 3 R_{1} + (d_{l} (P_{5} - P_{2}) / 2) + A_{1}^{T} P_{2} d_{l} / 2,$ ${\tilde{κ}}_{1, 7} = (d (r) - d_{l}) (P_{6} - P_{3}) / 2 + A_{1}^{T} P_{3} (d (r) - d_{l}) / 2,$

${\tilde{κ}}_{1, 8} = (d h_{1} - d (r)) (P_{6} - P_{3}) / 2 + A_{1}^{T} P_{3} (d h_{1} - d (r)) / 2,$ ${\tilde{κ}}_{1, 9} = (d_{u} - d h_{1}) (P_{7} - P_{4}) / 2 + A_{1}^{T} P_{4} (d_{u} - {dh}_{1}) / 2,$

${\tilde{κ}}_{2, 6} = {\tilde{v}}_{2, 6} = A_{d 1}^{T} P_{2} d_{l} / 2,$ ${\tilde{κ}}_{2, 5} = - A_{d 1}^{T} P_{4} / 2,$ ${\tilde{κ}}_{2, 3} = - 2 R_{2} - Y_{11}^{T} - Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T} + A_{d 1}^{T} (P_{3} - P_{2}) / 2,$

${\tilde{κ}}_{2, 4} = - 2 R_{2} - Y_{11} + Y_{12} + Y_{21} - Y_{22} + A_{d 1}^{T} (P_{4} - P_{3}) / 2,$ ${\tilde{κ}}_{2, 7} = 3 R_{2} + Y_{21}^{T} + Y_{22}^{T} + A_{d 1}^{T} P_{3} (d (r) - d_{l}) / 2,$ ${\tilde{κ}}_{2, 8} = 3 R_{2} - Y_{12} + Y_{22} + A_{d 1}^{T} P_{3} (d h_{1} - d (r)) / 2,$ ${\tilde{κ}}_{2, 9} = A_{d 1}^{T} P_{4} (d_{u} - d h_{1}) / 2,$ ${\tilde{v}}_{2, 7} = A_{d 1}^{T} P_{3} (d h_{1} - d_{l}) / 2,$

${\tilde{ν}}_{2, 2} = - 8 R_{3} + Z_{11}^{T} + Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T} + Z_{11} + Z_{12} - Z_{21} - Z_{22} + A_{d 1}^{T} P_{1} A_{d 1} + d_{l}^{2} A_{d 1}^{T} R_{1} A_{d 1},$

${\tilde{ν}}_{1, 7} = (d h_{1} - d_{l}) (P_{6} - P_{3}) / 2 + A_{1}^{T} P_{3} (d h_{1} - d_{l}) / 2,$ ${\tilde{ν}}_{1, 8} = (d (r) - d h_{1}) (P_{7} - P_{4}) / 2 + A_{1}^{T} P_{4} (d (r) - d h_{1}) / 2,$

${\tilde{ν}}_{1, 9} = (d_{u} - d (r)) (P_{7} - P_{4}) / 2 + A_{1}^{T} P_{4} (d_{u} - d (r)) / 2,$ ${\tilde{ν}}_{2, 3} = A_{d 1}^{T} (P_{3} - P_{2}) / 2,$

${\tilde{ν}}_{2, 4} = - 2 R_{3} - Z_{11}^{T} - Z_{12}^{T} - Z_{21}^{T} - Z_{22}^{T} + A_{d 1}^{T} (P_{4} - P_{3}) / 2,$ ${\tilde{ν}}_{2, 5} = - 2 R_{3} - Z_{11} + Z_{12} + Z_{21} - Z_{22} - A_{d 1}^{T} P_{4} / 2,$

${\tilde{ν}}_{2, 8} = 3 R_{3} + Z_{21}^{T} + Z_{22}^{T} + A_{d 1}^{T} P_{4} (d (r) - d h_{1}) / 2,$ ${\tilde{ν}}_{2, 9} = 3 R_{3} - Z_{12} + Z_{22} + A_{d 1}^{T} P_{4} (d_{u} - d (r)) / 2 .$

The proof of Corollary 1 is given in Appendix III.

Remark 8: Several global asymptotic stability criteria for the system (18) and (2) have been reported in Chen et al. (2017), Feng et al. (2015) and Liu and Zhang (2012). By utilizing RCI and a scale inequality, respectively, two novel delay-dependent stability criteria have been proposed in Liu and Zhang (2012) for discrete-time systems with time-varying delay. Utilizing a novel summation inequality, a delay-dependent stability criterion has been reported in Chen et al. (2017). The stability problems for continuous-time/discrete-time delay systems have been studied in Feng et al. (2015) by employing the idea of nonuniform delay partitioning, RCI and JBI. However, it will be illustrated in next section (see, Example 3) that Corollary 1 may yield improved stability results not covered by Chen et al. (2017), Feng et al. (2015) and Liu and Zhang (2012).

This section provides the numerical examples for illustrating the utility of the proposed criteria.

Example 1: Consider the system given by (1), (2), (6) and (12) with

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

(21)

H_{0} = H_{1} = [\begin{matrix} 0 \\ 0.1 \end{matrix}], E_{0} = [\begin{matrix} 0.01 & 0 \end{matrix}], E_{1} = [\begin{matrix} 0 & 0 \end{matrix} . 01] .

(22)

Let the nonlinearities in the system are confined to $[k_{o}, k_{q}] = [0, 1]$ , which cover magnitude truncation, zeroing, saturation, concatenation of magnitude truncation and zeroing, concatenation of magnitude truncation and saturation and so forth. This example was also considered in Kandanvli and Kar (2011), Tadepalli and Kandanvli (2016) and Tadepalli et al. (2018). Using the YALMIP 3.0 parser (Lofberg, 2004) along with MATLAB (Boyd et al., 1994; Gahinet, 1995), our main objective is to find $d_{u}$ for a given $d_{l}$ by solving the LMI conditions in Theorems 1 and 2 with respect to $d_{u}$ iteratively. For comparisons, the different values of $d_{u}$ for a specific $d_{l}$ obtained via various approaches for which the system is guaranteed to be globally asymptotically stable are shown in Table 1.

Table 1.

The maximum allowed $d_{u}$ for various $d_{l}$ for the system in Example 1.

Methods/ $d_{l}$	2	6	8	12
Kandanvli and Kar, 2011, Theorem 1	8	8	10	13
Tadepalli and Kandanvli, 2016, Theorem 1¹	8 ( $n = 2, τ = 1$ )	9 ( $n = 3, τ = 2$ )	10 ( $n = 4, τ = 2$ )	13 ( $n = 6, τ = 2$ )
Tadepalli et al., 2018, Theorem 1	9	10	11	14
Theorem 1 (N=2)	9	10	11	14
Theorem 2 (N=3)	11	11	12	15

¹In Tadepalli and Kandanvli (2016), lower delay bound $d_{l}$ is divided into $n$ number of partitions such that $d_{l} = τ n$ and $τ$ is the partition size which is an integer.

From Table 1, it is clear that Theorem 2 yields fewer conservative results than Theorem 1 in Kandanvli and Kar (2011), Theorem 1 in Tadepalli and Kandanvli (2016) and Theorem 1 in Tadepalli et al. (2018) for the present example. Theorem 1 provides the same upper bound $d_{u}$ as in Theorem 1 (Tadepalli et al., 2018).

Figure 1 shows a plot of state trajectories of the system in Example 1 with an arbitrary initial condition, $F_{0} (r) = \sin (180 (r - 1) / π),$ $F_{1} (r) = \cos (180 (r - 1) / π)$ and $d (r) = 2 + ⌊ | 9 \sin (180 (r - 1) / π) | ⌉ .$ The plots of $F_{0} (r),$ $F_{1} (r)$ are shown in Figure 2. Figure 3 depicts the plot of time-varying delay d(r). It may be observed in Figure 1 that the state trajectories approach to origin as $r \to \infty .$ Therefore, Figure 1 supports the result (obtained via Theorem 2) that the system considered in this example is globally asymptotically stable.

Example 2: Consider the system (1), (2), (6) and (12) with

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

(23)

\begin{matrix} H_{0} = H_{1} = [\begin{matrix} 0 \\ 0.1 \end{matrix}], \\ E_{0} = [\begin{matrix} 0.01 & 0 \end{matrix}], E_{1} = [\begin{matrix} 0 & 0 \end{matrix} . 01], [k_{o}, k_{q}] = [0, 1] . \end{matrix}

(24)

The admissible maximum delay bounds for given lower delay bounds obtained via various criteria for the global asymptotic stability of present system are listed in Table 2. From Table 2, one can infer that the proposed criteria (Theorems 1 and 2) provide reduced conservatism than the existing ones (Kandanvli and Kar, 2011; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2018) for this example.

Table 2.

The maximum allowed $d_{u}$ for various $d_{l}$ for the system in Example 2.

Methods/ $d_{l}$	2	5	7	9
Kandanvli and Kar, 2011, Theorem 1	12	14	16	18
Tadepalli and Kandanvli, 2016, Theorem 1	12 ( $n = 2, τ = 1$ )	14 ( $n = 5, τ = 1$ )	16 ( $n = 7, τ = 1$ )	18 ( $n = 3, τ = 3$ )
Tadepalli et al., 2018, Theorem 1	15	17	18	19
Theorem 1 (N=2)	17	18	20	21
Theorem 2 (N=3)	18	20	21	23

The state trajectories for the present system with an arbitrary initial condition, $F_{0} (r) = \sin (180 (r - 1) / π),$ $F_{1} (r) = \cos (180 (r - 1) / π)$ and $d (r) = 2 + ⌊ | 16 \sin (180 (r - 1) / π) | ⌉$ are depicted in Figure 4. Figure 5 shows the plot of time-varying delay d(r) used in the simulation. It may be noted in Figure 4 that the state trajectories tend to origin as $r \to \infty .$ This supports the result, obtained via Theorem 2, that the system is globally asymptotically stable.

Example 3: Consider the system given by (18) with the following parameters

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

(25)

The global asymptotic stability of this example was investigated in Chen et al. (2017), Feng et al. (2015) and Liu and Zhang (2012). For given $d_{l}$ , a comparison of maximum allowed $d_{u}$ obtained via Corollary 1 and the criteria in Chen et al. (2017), Feng et al. (2015) and Liu and Zhang (2012) for the global asymptotic stable behavior of the present system is shown in Table 3. It is clear from Table 3 that Corollary 1 yields more relaxed results as compared to Chen et al. (2017), Feng et al. (2015) and Liu and Zhang (2012) for the present example.

Table 3.

The maximum allowed $d_{u}$ for various $d_{l}$ for the system in Example 3.

As the number of subintervals ‘N’ becomes larger (≥2), the conservatism of the delay-dependent stability results is reduced while the computational burden increases. This is obvious from Theorems 1 and 2 as the number of decision variables D is directly related to the number of subintervals N. In other words, a larger N implies that the feasibility solution can be explored in a larger set which is confirmed by the results obtained in Examples 1 and 2 (see Tables 1 and 2 also). On the other hand, the increase of number of decision variables results in heavier computational burden. The total number of decision variables for the case N=2 (Theorem 1) is $19 m^{2} + 6 m + 2$ where as for N=3 (Theorem 2) is $28.5 m^{2} + 7.5 m + 2$ .

Pertaining to the system (1), (2), (6) and (12), in the general case, the relation between the total number of decision variables D and the number of subintervals N in the approach adopted in this paper is obtained as $D = \frac{N}{2} (N m^{2} + 14 m^{2} + 3 m) + 3 m^{2} + 3 m + 2$ . Consequently, it is noted that for the situations where the computational burden is non crucial, the presented approach may provide a better platform for achieving larger stability region than previous approaches (Feng et al., 2015; Kandanvli and Kar, 2011; Liu and Zhang, 2012; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2018).

This paper has discussed the problem of delay-dependent stability of uncertain discrete-time systems with time-varying delays and under the influence of quantization and overflow nonlinearities. Two LMI-based delay-dependent criteria (Theorems 1 and 2) have been proposed. Suitable Lyapunov functions are constructed that involve intermediate delays that utilize the delay information more effectively in the proposed criteria. To deal with the sum terms, WBI approach together with RCI approach has been utilized. As illustrated in Examples 1 and 2, Theorem 2 can provide less conservative results as compared with Theorem 1. The presented approach yields improved stability results as compared to previous methods (Kandanvli and Kar, 2011; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2018). Though the stability results have been established in this paper for N=2 and 3, the stability criterion corresponding to a general N can easily be worked out. A global asymptotic stability result (Corollary 1) for the underlying system in absence of uncertainties and nonlinearities is also brought out.

Though the stability criteria proposed in this paper deal with systems with single time-varying delay, the approach can be easily extended to establish the global asymptotic stability criterion for systems with multiple time-varying delays. The problem of obtaining improved stability conditions by combining the presented approach and the ideas of nonuniform delay partitioning (Feng et al., 2015) requires further investigation. The possible extension of the presented approach to develop delay-dependent stability criteria for 2-D uncertain systems (Peng et al., 2018; Tadepalli et al., 2015) is an interesting problem for future investigation. The possible deployment of the presented ideas for developing delay-dependent stability criteria for 1-D and 2-D uncertain discrete time-varying delayed systems with finite wordlength nonlinearities and external interferences (Ahn and Kar, 2015a, 2015b; Kokil et al., 2020) appears to be a realistic and challenging problem for further study.

Proof of Theorem 1: Consider a Lyapunov function as

(A.1a)

V (x (r)) = V_{1} (x (r)) + V_{2} (x (r)) + V_{3} (x (r)) + V_{4} (x (r)) + V_{5} (x (r))

with

(A.1b)

V_{1} (x (r)) = Π_{1}^{T} (r) P Π_{1} (r),

(A.1c)

\begin{matrix} V_{2} (x (r)) = \sum_{i = r - d_{l}}^{r - 1} x^{T} (i) Q_{1} x (i) + \sum_{i = r - d_{u}}^{r - 1} x^{T} (i) Q_{2} x (i) \\ + \sum_{i = r - d h_{1}}^{r - 1} x^{T} (i) Q_{3} x (i), \end{matrix}

(A.1d)

V_{3} (x (r)) = \sum_{θ = - d_{l} + 1}^{0} \sum_{j = r - 1 + θ}^{r - 1} d_{l} Λ^{T} (j) R_{1} Λ (j),

(A.1e)

V_{4} (x (r)) = \sum_{θ = - d h_{1} + 1}^{- d_{l}} \sum_{j = r - 1 + θ}^{r - 1} (d h_{1} - d_{l}) Λ^{T} (j) R_{2} Λ (j),

(A.1f)

V_{5} (x (r)) = \sum_{θ = - d_{u} + 1}^{- d h_{1}} \sum_{j = r - 1 + θ}^{r - 1} (d_{u} - d h_{1}) Λ^{T} (j) R_{3} Λ (j),

where

(A.2)

Π_{1}^{T} (r) = [x^{T} (r) \sum_{s = r - d_{l}}^{r - 1} x^{T} (s) \sum_{s = r - d h_{1}}^{r - d_{l} - 1} x^{T} (s) \sum_{s = r - d_{u}}^{r - d h_{1} - 1} x^{T} (s)],

(A.3)

Λ (r) = x (r + 1)) - x (r) = f (y (r)) - x (r) .

The forward difference of Lyapunov function (A.1) along the trajectories of the system (1) is

Δ V (x (r)) = V (x (r + 1)) - V (x (r))

(A.4a)

= Δ V_{1} (x (r)) + Δ V_{2} (x (r)) + Δ V_{3} (x (r)) + Δ V_{4} (x (r)) + Δ V_{5} (x (r)),

where

(A.4b)

Δ V_{1} (x (r)) = γ^{T} (r) ξ (d (r)) γ (r),

(A.4c)

\begin{matrix} Δ V_{2} (x (r)) = x^{T} (r) Q_{1} x (r) - x^{T} (r - d_{l}) Q_{1} x (r - d_{l}) \\ + x^{T} (r) Q_{2} x (r) - x^{T} (r - d_{u}) Q_{2} x (r - d_{u}) \\ + x^{T} (r) Q_{3} x (r) - x^{T} (r - d h_{1}) Q_{3} x (r - d h_{1}), \end{matrix}

(A.4d)

Δ V_{3} (x (r)) = d_{l}^{2} Λ^{T} (r) R_{1} Λ (r) - d_{l} \sum_{i = r - d_{l}}^{r - 1} Λ^{T} (i) R_{1} Λ (i),

(A.4e)

\begin{matrix} Δ V_{4} (x (r)) = (d h_{1} - d_{l})^{2} Λ^{T} (r) R_{2} Λ (r) \\ - (d h_{1} - d_{l}) \sum_{i = r - d h_{1}}^{r - d_{l} - 1} Λ^{T} (i) R_{2} Λ (i), \end{matrix}

(A.4f)

\begin{matrix} Δ V_{5} (x (r)) = (d_{u} - d h_{1})^{2} Λ^{T} (r) R_{3} Λ (r) \\ - (d_{u} - d h_{1}) \sum_{i = r - d_{u}}^{r - d h_{1} - 1} Λ^{T} (i) R_{3} Λ (i) . \end{matrix}

Using Lemma 1, it can be shown that

(A.5)

\begin{matrix} Δ V_{3} (x (r)) \leq d_{l}^{2} Λ^{T} (r) R_{1} Λ (r) \\ - {[\begin{matrix} x (r) - x (r - d_{l}) \\ x (r) + x (r - d_{l}) - χ (r, 0, d_{l}) \end{matrix}]}^{T} [\begin{matrix} R_{1} & 0 \\ 0 & 3 R_{1} \end{matrix}] \\ [\begin{matrix} x (r) - x (r - d_{l}) \\ x (r) + x (r - d_{l}) - χ (r, 0, d_{l}) \end{matrix}] . \end{matrix}

Now, consider the two possible cases:

In this case, the terms in $Δ V_{1} (x (r))$ can be described as

(A.6a)

\begin{matrix} γ^{T} (r) = \\ [\begin{matrix} x^{T} (r) & x^{T} (r - d (r)) & x^{T} (r - d_{l}) & x^{T} (r - d h_{1}) & x^{T} (r - d_{u}) & x^{T} (r, 0, d_{l}) \end{matrix} \\ x^{T} (r, d_{l}, d (r)) x^{T} (r, d (r), d h_{1}) x^{T} (r, d h_{1}, d_{u}) f^{T} (y (r))], \end{matrix}

(A.6b)

\begin{matrix} ξ (d (r)) = ρ_{1}^{T} (- P_{1} + ((P_{2} + P_{2}^{T}) / 2)) ρ_{1} + ρ_{10}^{T} P_{1} ρ_{10} \\ + sym (\sum_{i = 3}^{10} ρ_{1}^{T} ƛ_{1, i} ρ_{i} + \sum_{i = 6}^{10} ρ_{3}^{T} ƛ_{3, i} ρ_{i} \end{matrix}

+ \sum_{i = 6}^{10} ρ_{4}^{T} ƛ_{4, i} ρ_{i} + \sum_{i = 6}^{10} ρ_{5}^{T} ƛ_{5, i} ρ_{i} + \sum_{i = 6}^{9} ρ_{i}^{T} ƛ_{i, 10} ρ_{10}),

where

$ƛ_{1, 3} = (P_{3} - P_{2}) / 2,$ $ƛ_{1, 4} = (P_{4} - P_{3}) / 2, ƛ_{1, 5} = - P_{4} / 2,$ $ƛ_{1, 6} = d_{l} (P_{5} - P_{2}) / 2,$ $ƛ_{1, 7} = (d (r) - d_{l}) (P_{6} - P_{3}) / 2,$

$ƛ_{1, 8} = (d h_{1} - d (r)) (P_{6} - P_{3}) / 2,$ $ƛ_{1, 9} = (d_{u} - d h_{1}) (P_{7} - P_{4}) / 2,$ $ƛ_{1, 10} = P_{2}^{T} / 2,$ $ƛ_{3, 6} = d_{l} (P_{6}^{T} - P_{5}) / 2,$

$ƛ_{3, 7} = (d (r) - d_{l}) (P_{8} - P_{6}) / 2,$ $ƛ_{3, 8} = (d h_{1} - d (r)) (P_{8} - P_{6}) / 2,$ $ƛ_{3, 9} = (d_{u} - d h_{1}) (P_{9} - P_{7}) / 2,$

$ƛ_{3, 10} = (P_{3}^{T} - P_{2}^{T}) / 2,$ $ƛ_{4, 6} = d_{l} (P_{7}^{T} - P_{6}^{T}) / 2,$ $ƛ_{4, 7} = (d (r) - d_{l}) (P_{9}^{T} - P_{8}) / 2,$

$ƛ_{4, 8} = (d h_{1} - d (r)) (P_{9}^{T} - P_{8}) / 2,$ $ƛ_{4, 9} = (d_{u} - d h_{1}) (P_{10} - P_{9}) / 2,$ $ƛ_{4, 10} = (P_{4}^{T} - P_{3}^{T}) / 2,$ $ƛ_{5, 6} = - P_{7}^{T} d_{l} / 2,$

$ƛ_{5, 7} = - P_{9}^{T} (d (r) - d_{l}) / 2,$ $ƛ_{5, 8} = - P_{9}^{T} (d h_{1} - d (r)) / 2,$ $ƛ_{5, 9} = - P_{10} (d_{u} - d h_{1}) / 2,$ $ƛ_{5, 10} = - P_{4}^{T} / 2,$

$ƛ_{6, 10} = P_{2}^{T} d_{l} / 2,$ $ƛ_{7, 10} = P_{3}^{T} (d (r) - d_{l}) / 2,$ $ƛ_{8, 10} = P_{3}^{T} (d h_{1} - d (r)) / 2,$ $ƛ_{9, 10} = P_{4}^{T} (d_{u} - d h_{1}) / 2 .$

By utilizing Lemmas 1 and 2, the sum terms in $Δ V_{4} (x (r))$ and $Δ V_{5} (x (r))$ can be expressed as

\begin{matrix} - (d h_{1} - d_{l}) \sum_{i = r - d h_{1}}^{r - d_{l} - 1} Λ^{T} (i) R_{2} Λ (i) \\ = - (d h_{1} - d_{l}) [\sum_{i = r - d h_{1}}^{r - d (r) - 1} Λ^{T} (i) R_{2} Λ (i) + \sum_{i = r - d (r)}^{r - d_{l} - 1} Λ^{T} (i) R_{2} Λ (i)] \\ \leq - \frac{(d h_{1} - d_{l})}{(d h_{1} - d (r))} {[\begin{matrix} x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}]}^{T} \\ [\begin{matrix} R_{2} & 0 \\ 0 & 3 R_{2} \end{matrix}] [\begin{matrix} x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}] \end{matrix}

(A.7)

\begin{matrix} - \frac{(d h_{1} - d_{l})}{(d (r) - d_{l})} {[\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \end{matrix}]}^{T} \\ [\begin{matrix} R_{2} & 0 \\ 0 & 3 R_{2} \end{matrix}] [\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \end{matrix}] \\ \leq - {[\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \\ x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}]}^{T} \\ [\begin{matrix} R_{2} & 0 & Y_{11} & Y_{12} \\ * & 3 R_{2} & Y_{21} & Y_{22} \\ * & * & R_{2} & 0 \\ * & * & * & 3 R_{2} \end{matrix}] [\begin{matrix} x (r - d_{l}) - x (r - d (r)) \\ x (r - d_{l}) + x (r - d (r)) - χ (r, d_{l}, d (r)) \\ x (r - d (r)) - x (r - d h_{1}) \\ x (r - d (r)) + x (r - d h_{1}) - χ (r, d (r), d h_{1}) \end{matrix}] . \end{matrix}

The validity of (A.7) lies in the fact (see Lemma 2) that if there exists a matrix $Δ_{1} = [\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}]$ such that ${\hat{\nabla}}_{1} \geq 0$ (see (13)).

- (d_{u} - d h_{1}) \sum_{i = r - d_{u}}^{r - d h_{1} - 1} Λ^{T} (i) R_{3} Λ (i)

(A.8)

\begin{matrix} \leq - {[\begin{matrix} x (r - d h_{1}) - x (r - d_{u}) \\ x (r - d h_{1}) + x (r - d_{u}) - χ (r, d h_{1}, d_{u}) \end{matrix}]}^{T} [\begin{matrix} R_{3} & 0 \\ 0 & 3 R_{3} \end{matrix}] \\ [\begin{matrix} x (r - d h_{1}) - x (r - d_{u}) \\ x (r - d h_{1}) + x (r - d_{u}) - χ (r, d h_{1}, d_{u}) \end{matrix}] . \end{matrix}

Next, using (A.4)–(A.8), one obtains

(A.9)

Δ V (x (r)) \leq γ^{T} (r) Ψ (d (r)) γ (r) - 2 a,

where

(A.10)

\begin{matrix} a = \sum_{i = 1}^{m} g_{i} [k_{q} y_{i} (r) - f_{i} (y_{i} (r))] [f_{i} (y_{i} (r)) - k_{o} y_{i} (r)] \\ = [k_{q} y (r) - f (y (r))]^{T} G [f (y (r)) - k_{o} y (r)], \end{matrix}

(A.11)

\begin{matrix} Ψ (d (r)) = ρ_{1}^{T} (ε_{1, 1} - 2 k_{q} k_{o} {\bar{A}}_{1}^{T} G {\bar{A}}_{1}) ρ_{1} + ρ_{2}^{T} (ε_{2, 2} - 2 k_{q} k_{o} {\bar{A}}_{d 1}^{T} G {\bar{A}}_{d 1}) ρ_{2} + ρ_{10}^{T} ε_{10, 10} ρ_{10} + \sum_{i = 3}^{9} ρ_{i}^{T} κ_{i, i} ρ_{i} \\ + sym (ρ_{1}^{T} ε_{1, 10} ρ_{10} + \sum_{i = 2}^{9} ρ_{1}^{T} κ_{1, i} ρ_{i} + \sum_{\binom{i = 3,}{i \neq 5, 6}}^{8} ρ_{2}^{T} κ_{2, i} ρ_{i} + ρ_{2}^{T} ε_{2, 10} ρ_{10} + \sum_{\binom{i = 4,}{i \neq 5}}^{10} ρ_{3}^{T} κ_{3, i} ρ_{i} \\ + \sum_{i = 5}^{10} ρ_{4}^{T} κ_{4, i} ρ_{i} + \sum_{i = 6}^{10} ρ_{5}^{T} κ_{5, i} ρ_{i} + \sum_{i = 6}^{9} ρ_{i}^{T} κ_{i, 10} ρ_{10} + ρ_{7}^{T} κ_{7, 8} ρ_{8}), \end{matrix}

\begin{matrix} ε_{1, 1} = - P_{1} + (P_{2} + P_{2}^{T}) / 2 - 4 R_{1} + \sum_{i = 1}^{3} Q_{i} - \partial_{17}, \\ {\bar{A}}_{1} = A_{1} + Δ A_{1}, {\bar{A}}_{d 1} = A_{d 1} + Δ A_{d 1}, \\ ε_{2, 2} = - 8 R_{2} + Y_{11}^{T} + Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T} + Y_{11} \\ + Y_{12} - Y_{21} - Y_{22}, ε_{10, 10} = P_{1} - \partial_{17} - 2 G, \\ κ_{1, 2} = - 2 k_{q} k_{o} {\bar{A}}_{1}^{T} G {\bar{A}}_{d 1}, ε_{2, 10} = (k_{q} + k_{o}) {\bar{A}}_{d 1}^{T} G, \\ ε_{1, 10} = \partial_{17} + (P_{2}^{T} / 2) + (k_{q} + k_{o}) {\bar{A}}_{1}^{T} G, \end{matrix}

ρ_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (9 - i + 1) m} \end{matrix}], i = 1, 2, \dots, 10 .

Note that, in light of (12), the quantity ‘a’ (see (A.10)) is non-negative (Kandanvli and Kar, 2009b; Kar and Singh, 2001; Tadepalli et al., 2018). Therefore, the condition $Ψ (d (r)) < 0$ for all $d (r) \in [d_{l}, d h_{1}]$ along with (13) implies $Δ V (x (r)) < 0$ , which assures the global asymptotic stability of the system (1), (2), (6) and (12).

By Schur’s complement, the condition $Ψ (d (r)) < 0$ is expressed as

(A.12)

\begin{matrix} {\bar{ρ}}_{1}^{T} ε_{1, 1} {\bar{ρ}}_{1} + {\bar{ρ}}_{2}^{T} ε_{2, 2} {\bar{ρ}}_{2} + \sum_{i = 3}^{11} {\bar{ρ}}_{i}^{T} κ_{i, i} {\bar{ρ}}_{i} + sym \\ (\sum_{i = 2}^{9} {\bar{ρ}}_{1}^{T} κ_{1, i} {\bar{ρ}}_{i} + \sum_{i = 10}^{11} {\bar{ρ}}_{1}^{T} Φ_{1, i} {\bar{ρ}}_{i} + \sum_{\binom{i = 3,}{i \neq 5, 6, 9}}^{8} {\bar{ρ}}_{2}^{T} κ_{2, i} {\bar{ρ}}_{i} + \sum_{i = 10}^{11} {\bar{ρ}}_{2}^{T} Φ_{2, i} {\bar{ρ}}_{i} \\ + \sum_{\binom{i = 4,}{i \neq 5}}^{10} {\bar{ρ}}_{3}^{T} κ_{3, i} {\bar{ρ}}_{i} + \sum_{i = 5}^{10} {\bar{ρ}}_{4}^{T} κ_{4, i} {\bar{ρ}}_{i} + \sum_{i = 6}^{10} {\bar{ρ}}_{5}^{T} κ_{5, i} {\bar{ρ}}_{i} + \sum_{i = 6}^{9} {\bar{ρ}}_{i}^{T} κ_{i, 10} {\bar{ρ}}_{10} + {\bar{ρ}}_{7}^{T} κ_{7, 8} {\bar{ρ}}_{8} + {\bar{ρ}}_{10}^{T} κ_{10, 11} {\bar{ρ}}_{11}) < 0, \end{matrix}

where $Φ_{1, 10} = \partial_{17} + (P_{2}^{T} / 2) + k_{q} {\bar{A}}_{1}^{T} G,$ $Φ_{1, 11} = - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{1}^{T} G,$ $Φ_{2, 10} = k_{q} {\bar{A}}_{d 1}^{T} G,$ $Φ_{2, 11} = - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{d 1}^{T} G$ and ${\bar{ρ}}_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (10 - i + 1) m} \end{matrix}], i = 1, 2, \dots, 11 .$

In view of (6a), (A.12) can be rearranged as

(A.13)

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

where

(A.14)

{\bar{H}}_{0}^{T} = [\begin{matrix} 0 & 0 & 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}],

(A.15)

{\bar{E}}_{0} = [\begin{matrix} E_{0} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}],

(A.16)

\begin{matrix} Ψ_{0} (d (r)) = {\bar{ρ}}_{1}^{T} ε_{1, 1} {\bar{ρ}}_{1} + {\bar{ρ}}_{2}^{T} ε_{2, 2} {\bar{ρ}}_{2} + \sum_{i = 3}^{11} {\bar{ρ}}_{i}^{T} κ_{i, i} {\bar{ρ}}_{i} + sym \\ (\sum_{i = 2}^{9} {\bar{ρ}}_{1}^{T} κ_{1, i} {\bar{ρ}}_{i} + \sum_{i = 10}^{11} {\bar{ρ}}_{1}^{T} {\bar{Φ}}_{1, i} {\bar{ρ}}_{i} + \sum_{\binom{i = 3,}{i \neq 5, 6, 9}}^{8} {\bar{ρ}}_{2}^{T} κ_{2, i} {\bar{ρ}}_{i} + \sum_{i = 10}^{11} {\bar{ρ}}_{2}^{T} Φ_{2, i} {\bar{ρ}}_{i} + \sum_{\binom{i = 4,}{i \neq 5}}^{10} {\bar{ρ}}_{3}^{T} κ_{3, i} {\bar{ρ}}_{i} \\ + \sum_{i = 5}^{10} {\bar{ρ}}_{4}^{T} κ_{4, i} {\bar{ρ}}_{i} + \sum_{i = 6}^{10} {\bar{ρ}}_{5}^{T} κ_{5, i} {\bar{ρ}}_{i} + \sum_{i = 6}^{9} {\bar{ρ}}_{i}^{T} κ_{i, 10} {\bar{ρ}}_{10} + {\bar{ρ}}_{7}^{T} κ_{7, 8} {\bar{ρ}}_{8} + {\bar{ρ}}_{10}^{T} κ_{10, 11} {\bar{ρ}}_{11}), \end{matrix}

{\bar{Φ}}_{1, 10} = \partial_{17} + (P_{2}^{T} / 2) + k_{q} A_{1}^{T} G, {\bar{Φ}}_{1, 11} = - k_{q} \sqrt{- 2 k_{o}} A_{1}^{T} G .

By employing Lemma 3, (A.13) is equivalent to

(A.17)

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

Using Schur’s complement, (A.17) can also be written as

(A.18)

\begin{matrix} {\bar{Δ}}_{2}^{T} ε_{2, 2} {\bar{Δ}}_{2} + \sum_{\binom{i = 1,}{i \neq 2}}^{12} {\bar{Δ}}_{i}^{T} κ_{i, i} {\bar{Δ}}_{i} + sym \\ (\sum_{i = 2}^{9} {\bar{Δ}}_{1}^{T} κ_{1, i} {\bar{Δ}}_{i} + \sum_{i = 10}^{11} {\bar{Δ}}_{1}^{T} {\bar{Φ}}_{1, i} {\bar{Δ}}_{i} + \sum_{\binom{i = 3,}{i \neq 5, 6, 9}}^{8} {\bar{Δ}}_{2}^{T} κ_{2, i} {\bar{Δ}}_{i} + \sum_{i = 10}^{11} {\bar{Δ}}_{2}^{T} Φ_{2, i} {\bar{Δ}}_{i} + \sum_{\binom{i = 4,}{i \neq 5}}^{10} {\bar{Δ}}_{3}^{T} κ_{3, i} {\bar{Δ}}_{i} \\ + \sum_{i = 5}^{10} {\bar{Δ}}_{4}^{T} κ_{4, i} {\bar{Δ}}_{i} + \sum_{i = 6}^{10} {\bar{Δ}}_{5}^{T} κ_{5, i} {\bar{Δ}}_{i} + \sum_{i = 6}^{9} {\bar{Δ}}_{i}^{T} κ_{i, 10} {\bar{Δ}}_{10} + {\bar{Δ}}_{7}^{T} κ_{7, 8} {\bar{Δ}}_{8} + {\bar{Δ}}_{11}^{T} κ_{11, 12} {\bar{Δ}}_{12} + \sum_{i = 11}^{12} {\bar{Δ}}_{10}^{T} κ_{10, i} {\bar{Δ}}_{i}) < 0, \end{matrix}

where ${\bar{Δ}}_{i} = [\begin{matrix} 0_{m \times (i - 1) m} & I_{m} & 0_{m \times (10 - i + 1) m + p_{0}} \end{matrix}], i = 1, 2, \dots, 11$ and ${\bar{Δ}}_{12} = [\begin{matrix} 0_{p_{0} \times 11 m} & I_{p_{0}} \end{matrix}] .$

Following similar mathematical treatment as shown in equations (A.13)–(A.18), it is easy to show that (A.18) is equivalent to the condition $Γ_{1} (d (r)) < 0$ .

In this case, by adopting the similar procedure as considered in Case I for $Δ V (x (r))$ except some necessary changes in $Δ V_{1} (x (r))$ , $Δ V_{4} (x (r))$ and $Δ V_{5} (x (r))$ , one obtains

(A.19)

Δ V_{1} (x (r)) = {\hat{γ}}^{T} (r) \hat{ξ} (d (r)) \hat{γ} (r),

where

${\hat{γ}}^{T} (r) = [\begin{matrix} x^{T} (r) & x^{T} (r - d (r)) & x^{T} (r - d_{l}) & x^{T} (r - d h_{1}) & x^{T} (r - d_{u}) & x^{T} (r, 0, d_{l}) \end{matrix}$

$\begin{matrix} x^{T} (r, d_{l}, d h_{1}) & x^{T} (r, d h_{1}, d (r)) & x^{T} (r, d (r), d_{u}) & f^{T} (y (r)) \end{matrix}],$

$\hat{ξ} (d (r) = ρ_{1}^{T} (- P_{1} + ((P_{2} + P_{2}^{T}) / 2)) ρ_{1} + ρ_{10}^{T} P_{1} ρ_{10} + sym (\sum_{i = 3}^{10} ρ_{1}^{T} {\hat{ƛ}}_{1, i} ρ_{i} + \sum_{i = 6}^{10} ρ_{3}^{T} {\hat{ƛ}}_{3, i} ρ_{i}$

$+ \sum_{i = 6}^{10} ρ_{4}^{T} {\hat{ƛ}}_{4, i} ρ_{i} + \sum_{i = 6}^{10} ρ_{5}^{T} {\hat{ƛ}}_{5, i} ρ_{i} + \sum_{i = 6}^{9} ρ_{i}^{T} {\hat{ƛ}}_{i, 10} ρ_{10}),$

${\hat{ƛ}}_{1, 3} = (P_{3} - P_{2}) / 2, {\hat{ƛ}}_{1, 4} = (P_{4} - P_{3}) / 2, {\hat{ƛ}}_{1, 5} = - P_{4} / 2,$ ${\hat{ƛ}}_{1, 6} = d_{l} (P_{5} - P_{2}) / 2,$ ${\hat{ƛ}}_{1, 7} = (d h_{1} - d_{l}) (P_{6} - P_{3}) / 2,$

${\hat{ƛ}}_{1, 8} = (d (r) - d h_{1}) (P_{7} - P_{4}) / 2,$ ${\hat{ƛ}}_{1, 9} = (d_{u} - d (r)) (P_{7} - P_{4}) / 2,$ ${\hat{ƛ}}_{1, 10} = P_{2}^{T} / 2,$ ${\hat{ƛ}}_{3, 6} = d_{l} (P_{6}^{T} - P_{5}) / 2,$

${\hat{ƛ}}_{3, 7} = (d h_{1} - d_{l}) (P_{8} - P_{6}) / 2,$ ${\hat{ƛ}}_{3, 8} = (d (r) - d h_{1}) (P_{9} - P_{7}) / 2,$ ${\hat{ƛ}}_{3, 9} = (d_{u} - d (r)) (P_{9} - P_{7}) / 2,$

${\hat{ƛ}}_{3, 10} = (P_{3}^{T} - P_{2}^{T}) / 2,$ ${\hat{ƛ}}_{4, 6} = d_{l} (P_{7}^{T} - P_{6}^{T}) / 2,$ ${\hat{ƛ}}_{4, 7} = (d h_{1} - d_{l}) (P_{9}^{T} - P_{8}) / 2,$

${\hat{ƛ}}_{4, 8} = (d (r) - d h_{1}) (P_{10} - P_{9}) / 2,$ ${\hat{ƛ}}_{4, 9} = (d_{u} - d (r)) (P_{10} - P_{9}) / 2,$ ${\hat{ƛ}}_{4, 10} = (P_{4}^{T} - P_{3}^{T}) / 2,$

${\hat{ƛ}}_{5, 6} = - P_{7}^{T} d_{l} / 2,$ ${\hat{ƛ}}_{5, 7} = - P_{9}^{T} (d h_{1} - d_{l}) / 2,$ ${\hat{ƛ}}_{5, 8} = - P_{10} (d (r) - d h_{1}) / 2,$ ${\hat{ƛ}}_{5, 9} = - P_{10} (d_{u} - d (r)) / 2,$

${\hat{ƛ}}_{6, 10} = P_{2}^{T} d_{l} / 2,$ ${\hat{ƛ}}_{7, 10} = P_{3}^{T} (d h_{1} - d_{l}) / 2,$ ${\hat{ƛ}}_{8, 10} = P_{4}^{T} (d (r) - d h_{1}) / 2,$ ${\hat{ƛ}}_{9, 10} = P_{4}^{T} (d_{u} - d (r)) / 2$

and

Δ V_{4} (x (r)) \leq (d h_{1} - d_{l})^{2} Λ^{T} (r) R_{2} Λ (r) - {[\begin{matrix} x (r - d_{l}) - x (r - d h_{1}) \\ x (r - d_{l}) + x (r - d h_{1}) - χ (r, d_{l}, d h_{1}) \end{matrix}]}^{T} [\begin{matrix} R_{2} & 0 \\ 0 & 3 R_{2} \end{matrix}]

(A.20)

[\begin{matrix} x (r - d_{l}) - x (r - d h_{1}) \\ x (r - d_{l}) + x (r - d h_{1}) - χ (r, d_{l}, d h_{1}) \end{matrix}],

Δ V_{5} (x (r)) \leq (d_{u} - d h_{1})^{2} Λ^{T} (r) R_{3} Λ (r) - {[\begin{matrix} x (r - d h_{1}) - x (r - d (r)) \\ x (r - d h_{1}) + x (r - d (r)) - χ (r, d h_{1}, d (r)) \\ x (r - d (r)) - x (r - d_{u}) \\ x (r - d (r)) + x (r - d_{u}) - χ (r, d (r), d_{u}) \end{matrix}]}^{T} [\begin{matrix} R_{3} & 0 & Z_{11} & Z_{12} \\ * & 3 R_{3} & Z_{21} & Z_{22} \\ * & * & R_{3} & 0 \\ * & * & * & 3 R_{3} \end{matrix}]

(A.21)

[\begin{matrix} x (r - d h_{1}) - x (r - d (r)) \\ x (r - d h_{1}) + x (r - d (r)) - χ (r, d h_{1}, d (r)) \\ x (r - d (r)) - x (r - d_{u}) \\ x (r - d (r)) + x (r - d_{u}) - χ (r, d (r), d_{u}) \end{matrix}] .

Note that the existence of the matrix $Δ_{2} = [\begin{matrix} Z_{11} & Z_{12} \\ Z_{21} & Z_{22} \end{matrix}]$ supports the inequality condition (A.21) such that ${\hat{\nabla}}_{2} \geq 0$ . Following the similar lines as developed in Case I, it can be shown that, the condition $Γ_{2} (d (r)) < 0$ implies $Δ V (x (r)) < 0$ in this case.

It is obvious that $Γ_{1} (d (r))$ and $Γ_{2} (d (r))$ are affine matrix functions over the delay $d (r) \in [d_{l}, d h_{1}]$ and $d (r) \in [d h_{1}, d_{u}]$ , respectively. Based on the affinity property, the conditions $Γ_{1} (d (r)) < 0$ and $Γ_{2} (d (r)) < 0$ for all $d (r) \in [d_{l}, d_{u}]$ satisfy if and only if (14) holds. This completes the proof.

One can easily arrive at Theorem 2 by employing the Lyapunov function

(A.22)

V (x (r)) = V_{1} (x (r)) + V_{2} (x (r)) + V_{3} (x (r)) + V_{4} (x (r)) + V_{5} (x (r)) + V_{6} (x (r)),

where $V_{1} (x (r)) = {\hat{Π}}^{T} (r) \hat{P} \hat{Π} (r),$ $V_{2} (x (r)) = \sum_{i = r - d_{l}}^{r - 1} x^{T} (i) Q_{1} x (i) + \sum_{i = r - d_{u}}^{r - 1} x^{T} (i) Q_{2} x (i) + \sum_{i = r - d h_{1}}^{r - 1} x^{T} (i) Q_{3} x (i)$ $+ \sum_{i = r - d h_{2}}^{r - 1} x^{T} (i) Q_{4} x (i),$ $V_{3} (x (r)) = \sum_{θ = - d_{l} + 1}^{0} \sum_{j = r - 1 + θ}^{r - 1} d_{l} Λ^{T} (j) R_{1} Λ (j),$ $V_{4} (x (r)) = \sum_{θ = - d h_{1} + 1}^{- d_{l}} \sum_{j = r - 1 + θ}^{r - 1} (d h_{1} - d_{l}) Λ^{T} (j) R_{2} Λ (j),$ $V_{5} (x (r)) = \sum_{θ = - d h_{2} + 1}^{- d h_{1}} \sum_{j = r - 1 + θ}^{r - 1} (d h_{2} - d h_{1}) Λ^{T} (j) R_{3} Λ (j),$ $V_{6} (x (r)) = \sum_{θ = - d_{u} + 1}^{- d h_{2}} \sum_{j = r - 1 + θ}^{r - 1} (d_{u} - d h_{2}) Λ^{T} (j) R_{4} Λ (j),$

$\hat{Π} (r) = [x^{T} (r) \sum_{s = r - d_{l}}^{r - 1} x^{T} (s) \sum_{s = r - d h_{1}}^{r - d_{l} - 1} x^{T} (s) \sum_{s = r - d h_{2}}^{r - d h_{1} - 1} x^{T} (s) \sum_{s = r - d_{u}}^{r - d h_{2} - 1} x^{T} (s)]$

and following the steps shown in the proof of Theorem 1.

Proof of Corollary 1: With $f (y (r)) = y (r)$ and $Δ A_{1} = Δ A_{d 1} = 0$ (which signify the absence of uncertainties and nonlinearities in the system (1), (2), (6) and (12)) and following the steps of the proof of Theorem 1 discussed in Appendix I, one can easily arrive at Corollary 1. The details of the proof of Corollary 1 are, therefore, omitted.

The authors wish to thank the Editor, the Associate Editor and the anonymous reviewers for their constructive comments and suggestions.

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

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

Kalpana Singh

Haranath Kar

Methods/ $d_{l}$

Liu and Zhang, 2012, Theorem 1

Feng, et al., 2015, Theorem 3

Chen, et al., 2017, Theorem 1

Corollary 1

Ahn

Kar

(2015a) Expected power bound for two-dimensional digital filters in the Fornasini-Marchesini local state-space model. IEEE Signal Processing Letters 22(8): 1065–1069.

Ahn

Kar

(2015b) Passivity and finite-gain performance for two-dimensional digital filters: The FM LSS model case. IEEE Transactions on Circuits and Systems II: Express Briefs 62(9): 871–875.

Bose

(1994) Asymptotic stability of two-dimensional digital filters under quantization. IEEE Transactions on Signal Processing 42(5): 1172−1177.

Boyd

El Ghaoui

Feron

, et al. (1994) Linear Matrix Inequalities in System and Control Theory. Philadelphia, PA: SIAM.

Butterweck

Ritzerfeld

JHF

Werter

(1988) Finite wordlength effects in digital filters: A review. EUT report 88-E-205, Eindhoven University of Technology, Eindhoven, The Netherlands.

Chakrabarty

Iyengar

, et al. (2002) Grid coverage for surveillance and target location in distributed sensor networks. IEEE Transactions on Computers 51(12): 1448−1453.

Chang

(1981) Suppression of limit cycles in digital filters designed with one magnitude-truncation quantizer. IEEE Transactions on Circuits and Systems 28(2): 107−111.

Chen

(2009) Asymptotic stability of discrete-time systems with time-varying delay subject to saturation nonlinearities. Chaos, Solitons & Fractals 42(2): 1251−1257.

Chen

(2016) Summation inequality and its application to stability analysis for time-delay systems. IET Control Theory & Applications 10(4): 391−395.

10.

Chen

Jia

, et al. (2017) Novel summation inequalities and their applications to stability analysis for systems with time-varying delay. IEEE Transactions on Automatic Control 62(5): 2470−2475.

11.

Chen

Guan

(2004) Delay-dependent guaranteed cost control for uncertain discrete-time systems with both state and input delays. Journal of the Franklin Institute 341(5): 419−430.

12.

Erickson

Michel

(1985) Stability analysis of fixed-point digital filters using computer generated Lyapunov functions-Part I: Direct form and coupled form filters. IEEE Transactions on Circuits and Systems 32(2): 113−132.

13.

Feng

Lam

Yang

G-H

(2015) Optimal partitioning method for stability analysis of continuous/discrete delay systems. International Journal of Robust and Nonlinear Control 25(4): 559−574.

14.

Gahinet

Nemirovskii

Laub

, et al. (1995) LMI control toolbox – For use with Matlab. Natick, MA, USA: The MATH Works Inc.

15.

Gao

Chen

(2007) New results on stability of discrete-time systems with time-varying state delay. IEEE Transactions on Automatic Control 52(2): 328−334.

16.

She

, et al. (2004) Parameter-dependent Lyapunov functional for stability of time-delay systems with polytopic-type uncertainties. IEEE Transactions on Automatic Control 49(5): 828−832.

17.

Hua

Zhang

Chen

, et al. (2014) Delay decomposition approach to robust delay-dependent H_∞ filtering of uncertain stochastic systems with time-varying delays. Transactions of the Institute of Measurement and Control, 36(8): 1143−1152.

18.

Huang

Feng

(2010) Improved approach to delay-dependent stability analysis of discrete-time systems with time-varying delay. IET Control Theory & Applications 4(10): 2152−2159.

19.

Kandanvli

VKR

Kar

(2008) Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities. Nonlinear Analysis: Theory, Methods & Applications 69(9): 2780−2787.

20.

Kandanvli

VKR

Kar

(2009a) Robust stability of discrete-time state-delayed systems with saturation nonlinearities: Linear matrix inequality approach. Signal Processing 89(2): 161−173.

21.

Kandanvli

VKR

Kar

(2009b) An LMI condition for robust stability of discrete-time state-delayed systems using quantization/overflow nonlinearities. Signal Processing 89(11): 2092−2102.

22.

Kandanvli

VKR

Kar

(2011) Delay-dependent LMI condition for global asymptotic stability of discrete-time uncertain state-delayed systems using quantization/overflow nonlinearities. International Journal of Robust and Nonlinear Control 21(14): 1611–1622.

23.

Kar

(2007) An LMI based criterion for the nonexistence of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. Digital Signal Processing 17(3): 685−689.

24.

Kar

Singh

(2001) Stability analysis of 1-D and 2-D fixed-point state-space digital filters using any combination of overflow and quantization nonlinearities. IEEE Transactions on Signal Processing 49 (5): 1097–1105.

25.

Kokil

Jogi

Ahn

, et al. (2020) An improved local stability criterion for digital filters with interference and overflow nonlinearity. IEEE Transactions on Circuits and Systems II: Express Briefs 67(3): 595−599.

26.

Kwon

Park

, et al. (2013) Improved delay-dependent stability criteria for discrete-time systems with time-varying delays. Circuits, Systems and Signal Processing 32(4): 1949−1962.

27.

Lee

Tsai

(2013) Delay-partitioning approach to H_∞ performance analysis of discrete-time systems with interval time-varying delay. Applied Mechanics and Materials 427–429: 1364−1367.

28.

Lepschy

Mian

Viaro

(1988) Effects of quantization in second-order fixed-point digital filters with two’s complement truncation quantizers. IEEE Transactions on Circuits and Systems 35(4): 461−466.

29.

Wang

Zhao

(2018) Stability of discrete-time systems with time-varying delay based on switching technique. Journal of the Franklin Institute 355(13): 6026−6044.

30.

Liu

Zhang

(2012) Note on stability of discrete-time time-varying delay systems. IET Control Theory and Applications 6(2): 335−339.

31.

Lofberg

(2004) YALMIP: A toolbox for modeling and optimization in MATLAB. In: Proceedings of the CACSD Conference, Taipei, Taiwan, 2–4 September, pp. 284−289. IEEE.

32.

Mahmoud

(2013) Stabilization of interconnected discrete systems with quantization and overflow nonlinearities. Circuits, Systems and Signal Processing 32(2): 905−917.

33.

Mary

Rangarajan

(2016) Delay-dependent stability analysis of microgrid with constant and time-varying communication delays. Electric Power Components and Systems 44(13): 1441−1452.

34.

Meng

Lam

, et al. (2010) A delay-partitioning approach to the stability analysis of discrete-time systems. Automatica 46(3): 610−614.

35.

Naghshtabrizi

Hespanha

Teel

(2010) Stability of delay impulsive systems with application to networked control systems. Transactions of the Institute of Measurement and Control 32(5): 511−528.

36.

Nam

Pathirana

Trinh

(2015) Discrete Wirtinger-based inequality and its application. Journal of the Franklin Institute 352(5): 1893−1905.

37.

Park

Jeong

(2011) Reciprocally convex approach to stability of systems with time-varying delays. Automatica 47(1): 235−238.

38.

Peng

Tian

Yue

(2011) Output feedback control of discrete-time systems in networked environments. IEEE Transactions on Systems, Man and Cybernetics-Part A: Systems and Humans 41(1): 185−190.

39.

Peng

Zhang

Hua

, et al. (2018) A delay-partitioning approach to the stability analysis of 2-D linear discrete-time systems with interval time-varying delays. International Journal of Control, Automation and Systems 16(2): 682−688.

40.

Seuret

Gouaisbaut

Fridman

(2015) Stability of discrete-time systems with time-varying delays via a novel summation inequality. IEEE Transactions on Automatic Control 60(10): 2740−2745.

41.

Shao

Han

(2011) New stability criteria for linear discrete-time systems with interval-like time-varying delays. IEEE Transactions on Automatic Control 56(3): 619−625.

42.

Singh

(2010) LMI approach to stability of a class of discrete-time systems using two’s complement arithmetic. Applied Mathematics and Computation 217(6): 2877−2882.

43.

Tadepalli

Kandanvli

VKR

Kar

(2015) A new delay-dependent stability criterion for uncertain 2-D discrete systems described by Roesser model under the influence of quantization/overflow nonlinearities. Circuits, Systems and Signal Processing 34(8): 2537−2559.

44.

Tadepalli

Kandanvli

VKR

(2016) Improved stability results for uncertain discrete-time state-delayed systems in the presence of nonlinearities. Transaction of the Institute of Measurement and Control 38(1): 33−43.

45.

Tadepalli

Kandanvli

VKR

(2017) Delay-dependent stability of discrete-time systems with multiple delays and nonlinearities. International Journal of Innovative Computing, Information and Control 13(3): 891−904.

46.

Tadepalli

Kandanvli

VKR

Vishwakarma

(2018) Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities. Transactions of the Institute of Measurement and Control 40(9): 2868−2880.

47.

Lam

Yang

(2001) Quadratic stability and stabilization of uncertain linear discrete-time systems with state delay. Systems & Control Letters 43(2): 77−84.

48.

Lam

Zhang

, et al. (2014) A new result on the delay-dependent stability of discrete systems with time-varying delays. International Journal of Robust and Nonlinear Control 24(16): 2512−2521.

49.

Xue

Zhang

Han

, et al. (2016) A delay-range-partition approach to analyse stability of linear systems with time-varying delays. International Journal of Systems Science 47(16): 3970−3977.

50.

Deng

, et al. (2013) Design of multiple controllers for networked control systems with delays and packet losses. Transactions of the Institute of Measurement and Control 35(6): 720−729.

51.

Zhang

Zou

(2008) Improved stability criterion and its applications in delayed controller design for discrete-time systems. Automatica 44(11): 2963−2967.