Criteria for stability of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities

Abstract

This paper considers the problem of global asymptotic stability of a class of uncertain discrete-time systems under the influence of finite wordlength nonlinearities (quantization and/or overflow) and time-varying delays. The parameter uncertainties are assumed to be norm-bounded. Utilizing the concept of a Wirtinger-based inequality and a reciprocally convex method, two delay-dependent stability criteria are presented. The selection of the criteria depends on the type of the nonlinearities, that is, a combination of quantization and overflow or saturation overflow nonlinearities involved in the present systems. The approach presented in this paper yields less conservative results and reduces the computational burden as compared to previously reported criteria. Numerical examples are given to illustrate the effectiveness of the presented approach.

Keywords

Delay-dependent stability finite wordlength effect linear matrix inequality Lyapunov method time-varying delay uncertain system

Introduction

In the implementation of stable and linear discrete-time dynamical systems using finite wordlength processors with fixed-point arithmetic, one commonly encounters several kinds of nonlinearities (e.g. quantization and overflow nonlinearities). These finite wordlength nonlinearities may lead to instability in the designed system. The instability may arise in the form of zero-input limit cycles, which characterize nonlinear systems. While designing a discrete-time system under finite wordlength implementation it is, therefore, important to determine the range of system parameters such that the system is free of limit cycles. The quantization and overflow nonlinearities may interact with each other, motivating a considerable number of studies on their combined effects (Kandanvli and Kar, 2009a, 2011; Kar and Singh, 2001; Sim and Pang, 1985; Singh, 2013; Tadepalli and Kandanvli, 2016). However, if the wordlength is large, then the effects of quantization and overflow may be regarded as non-interacting (Claasen et al., 1976; Kandanvli and Kar, 2009a; Singh, 2008; Tadepalli and Kandanvli, 2016). Under this condition, quantization effects are neglected when studying the effects of overflow (Chen, 2009; Kandanvli and Kar, 2009b, 2013; Singh, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015) and vice versa (Bose, 1994; Dewasurendra and Bauer, 2008). Saturation overflow nonlinearity, owing to its larger stability region, has also been widely studied (Chen, 2009; Kandanvli and Kar, 2009b, 2013; Kar, 2007; Singh, 2006; Song and Wang, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015). Therefore, the stability analysis of discrete-time systems under the influence of finite wordlength nonlinearities is considered to be an important subject of system theoretic study (Chen, 2009; Claasen et al., 1976; Kandanvli and Kar, 2009a,b, 2011, 2013; Kar and Singh, 2001; Sim and Pang, 1985; Singh, 2008, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015).

Apart from the instabilities due to finite wordlength implementation, the presence of parameter uncertainties and delays may also lead to instability in discrete-time systems. It is well known that, while deriving delay-dependent stability criteria for such systems, the issue of computational burden may play a vital role (see the recent works by Nam et al., 2015, and Seuret et al., 2015). Several studies have been performed on the stability of discrete-time systems under the simultaneous influence of finite wordlength nonlinearities, parameter uncertainties and delays (Kandanvli and Kar, 2008, 2009a,b, 2011, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014, 2015). Examples of such systems include digital control systems (Tadepalli et al., 2015), digital filter design (Bose, 1994; Chen, 2009; Kandanvli and Kar, 2009a,b, 2011, 2013; Kar and Singh, 2001; Singh, 2008, 2013; Tadepalli and Kandanvli, 2016; Tadepalli et al., 2014), neural networks defined on hypercubes (Kandanvli and Kar, 2009a), combined marketing and production problems (Kandanvli and Kar, 2008, 2009b; Tadepalli and Kandanvli, 2016), and so on.

In Kandanvli and Kar (2008, 2009b), delay-independent stability criteria have been established for discrete-time systems in the presence of overflow nonlinearities, uncertainties and constant delays. A delay-dependent stability criterion for discrete-time systems with saturation nonlinearities and interval-like time-varying delays has been given in Chen (2009). In Kandanvli and Kar (2009a), a delay-independent stability problem for the global asymptotic stability of uncertain discrete-time delayed systems under the combined influence of quantization and overflow nonlinearities has been studied, whereas Kandanvli and Kar (2011) consider a delay-dependent stability problem employing a free-weighting matrix (FWM) approach (He et al., 2008). Lee et al. (2012) provide a delay-dependent regional asymptotic stability criterion for discrete-time delayed systems with saturation nonlinearity. By utilizing a FWM approach, delay-dependent stability analysis of uncertain discrete-time delayed systems under the influence of saturation nonlinearities has been performed in Kandanvli and Kar (2013). Recently, in Tadepalli and Kandanvli (2016), two delay-dependent stability criteria have been established for discrete-time systems under the influence of saturation nonlinearities and for systems involving composite nonlinearities (i.e. a combination of quantization and overflow nonlinearities), respectively. The criteria developed in Tadepalli and Kandanvli (2016) are based on utilizing both delay-partitioning and reciprocally convex methods. Although some results are available on the delay-dependent stability of discrete-time systems involving finite wordlength nonlinearities, parameter uncertainties and time-varying delays, there still remains great scope for improvement in terms of conservativeness and computational burden.

In this paper, motivated by the preceding discussion, we revisit the problem of stability of discrete-time systems under the influence of finite wordlength nonlinearities, parameter uncertainties and time-varying delays. The larger the stability region in the parameter space of a discrete-time system, the greater would be the flexibility in selecting the system parameters. In other words, the important objective in the design of a discrete-time system is to have an efficient (less conservative and smaller computational burden) criterion for choosing the values of the system parameters in the parameter space so that the designed system is globally asymptotically stable. However, it remains a challenge to derive such a criterion for testing the stability of discrete-time systems, particularly in the simultaneous presence of finite wordlength nonlinearities, parameter uncertainties and time-varying delays.

The approach presented in this paper utilizes a Lyapunov functional to derive the corresponding linear matrix inequality (LMI)-based stability criteria as feasibility tests. However, selection of an appropriate Lyapunov functional is one of the key steps to reduce the conservativeness, and a lot of work (see, for instance, He et al., 2008; Jiang et al., 2005; Liu and Zhang, 2012; Nam et al., 2015; Seuret et al., 2015) has been done regarding this aspect. To establish delay-dependent stability criteria, several techniques have been employed which include a FWM method (He et al., 2008; Kandanvli and Kar, 2008, 2009a,b, 2011) and the bounding techniques (reciprocal convexity, see Liu and Zhang, 2012; Park et al., 2011; discrete Jensen inequality, see Jiang et al., 2005, etc.) of the cross-terms and sum terms in the forward difference of the Lyapunov functional. Though the FWM method (He et al., 2008) helps in reducing the conservatism, introduction of multiple free parameters makes the criteria more complex and computationally demanding. The reciprocal convexity technique was originally derived for continuous-time systems in Park et al. (2011). It is a lower-bound lemma for a linear combination of positive functions with inverses of convex parameters as the coefficients (Park et al., 2011). The application of a reciprocal convexity technique may support obtaining improvement in terms of conservativeness and number of decision variables. Different from existing techniques (He et al., 2008; Liu and Zhang, 2012; Park et al., 2011), a combination of a discrete version of the Wirtinger-based inequality (Nam et al., 2015), which is much tighter than the discrete Jensen inequality, and a reciprocally convex approach is employed in this paper to derive better results. The efficiency of the presented approach is tested with the help of examples.

The objective of this paper is to derive efficient delay-dependent criteria for the global asymptotic stability of the systems under consideration. Following are the main contributions of the paper. (i) The present system encompasses a broader class of discrete-time systems involving parameter uncertainties, finite wordlength nonlinearities and time-varying delays. (ii) To establish the delay-dependent stability criteria for such systems, a discrete version of the Wirtinger-based inequality (Nam et al., 2015; Seuret et al., 2015) along with a reciprocally convex approach (Liu and Zhang, 2012; Park et al., 2011) are used in this paper, which may provide improvement in terms of computational burden and conservatism. (iii) The criteria presented in this paper are in the form of LMI-based conditions and, therefore, computationally tractable. (iv) A comparison of the present criteria with the previously reported criteria (Kandanvli and Kar, 2011, 2013; Tadepalli and Kandanvli, 2016) is made on the basis of conservatism and computational burden.

The paper is organized as follows. In ‘System description’, the description of system under consideration is given. ‘Main results’ provides the main results of the paper. In ‘Examples’, examples are provided to illustrate the effectiveness of the presented results.

Notation

The following notation is used in this paper: $ℝ^{p}$ denotes the p-dimensional Euclidean space; $ℝ^{p \times q}$ is the set of p × q real matrices; 0 represents the null matrix or null vector of appropriate dimensions; I is the identity matrix of appropriate dimensions; B ^T stands for the transpose of the matrix (or vector) B ; B > 0 (≥0) means that B is a positive-definite (semidefinite) symmetric matrix; B < 0 means that B is a negative-definite symmetric matrix; diag(·) is a diagonal matrix with diagonal elements; and the symbol ‘*’ represents the symmetric terms in a symmetric matrix.

System description

The system under consideration is represented by

\begin{matrix} x (k + 1) & = f (y (k)) \\ = {[f_{1} (y_{1} (k)) f_{2} (y_{2} (k)) \dots f_{n} (y_{n} (k))]}^{T} \end{matrix}

(1a)

\begin{matrix} y (k) & = (A + Δ A) x (k) + (A_{d} + Δ A_{d}) x (k - d (k)) \\ = {[y_{1} (k) y_{2} (k) \dots y_{n} (k)]}^{T} \end{matrix}

(1b)

x (k) = φ (k), \forall k \in [- h_{2}, 0]

(1c)

where $x (k) \in ℝ^{n}$ is the state vector; A , $A_{d} \in ℝ^{n \times n}$ are the known constant matrices; Δ A , $Δ A_{d} \in ℝ^{n \times n}$ are the unknown matrices representing norm-bounded parametric uncertainties in the state matrices; $φ (k) \in ℝ^{n}$ is the initial state value at time k; f (·) represents the finite wordlength nonlinearities; and d(k) is a time-varying delay satisfying

1 \leq h_{1} \leq d (k) \leq h_{2}

(2)

where h₁ and h₂ are known positive integers representing the lower and upper delay bounds, respectively. The uncertainties in the state matrices are assumed to be of the form (Bakule et al., 2006; Kandanvli and Kar, 2009a, 2011; Parlakçı, 2015; Tadepalli and Kandanvli, 2016; Xie et al., 1992)

Δ A = H_{0} F_{0} E_{0}

(3a)

Δ A_{d} = H_{1} F_{1} E_{1}

(3b)

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

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

(3c)

Equations (1) to (3) can be used to describe a broad class of uncertain discrete-time delayed systems involving finite wordlength nonlinearities. Examples of such systems include digital control systems with finite wordlength nonlinearities (Mahmoud, 2013), digital filters implemented in finite register lengths (Kandanvli and Kar, 2009a,b, 2011), fixed-point processors used in sensor networks (Dewasurendra and Bauer, 2008), Hopfield neural networks (Kandanvli and Kar, 2009a), cold rolling mills, and many other engineering problems. One typical example of the system represented by (1)–(3) can be found in networked control systems (Tadepalli and Kandanvli, 2016; Wu et al., 2011), where the delays induced by the network transmission (either from sensor to controller or from controller to actuator) are actually time-varying and can be assumed to have minimum and maximum delay bounds without loss of generality. While implementing the networked control systems using computer or special-purpose hardware with fixed-point arithmetic for data processing in the network transmission, the nonlinearities due to finite wordlength are generated.

The following lemmas play an important role in the proof of our main results.

Lemma 1.(Nam et al., 2015) For a given positive-definite matrix R and three non-negative integers a, b, k, satisfying a ≤ b ≤ k, if

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

(4)

then

- (b - a) \sum_{s = k - b}^{k - a - 1} η^{T} (s) R η (s) \leq - {[\begin{matrix} Ω_{0} \\ Ω_{1} \end{matrix}]}^{T} [\begin{matrix} R & 0 \\ 0 & 3 R \end{matrix}] [\begin{matrix} Ω_{0} \\ Ω_{1} \end{matrix}]

(5)

where

Ω_{0} = x (k - a) - x (k - b)

(6a)

Ω_{1} = x (k - a) + x (k - b) - χ (k, a, b)

(6b)

η (s) = x (s + 1) - x (s)

(6c)

The vector Ω ₀represents the average value gained by the discrete-time variable x (k) over the delay range [a, b], whereasΩ₁is proportional to the difference between the mean value of x (k) and its average over the delay range.

Lemma 2.(Liu and Zhang, 2012; Park et al., 2011) For any vectors ξ ₁, ξ ₂, matrices R , S and real numbers α₁ ≥ 0, α₂ ≥ 0 satisfying

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

(7)

ξ_{i} = 0, if α_{i} = 0 (i = 1, 2)

(8)

then

- \frac{1}{α_{1}} ξ 1^{T} R ξ 1 - \frac{1}{α_{2}} ξ 2^{T} R ξ 2 \leq - {[\begin{matrix} ξ 1 \\ ξ 2 \end{matrix}]}^{T} [\begin{matrix} R & S \\ * & R \end{matrix}] [\begin{matrix} ξ 1 \\ ξ 2 \end{matrix}]

(9)

Lemma 3.(Boyd et al., 1994) Let Σ, Γ, F andΛbe real matrices of appropriate dimensions withΛsatisfyingΛ = Λ^T. Then

Λ + Σ F Γ + Γ^{T} F^{T} Σ^{T} < 0

(10)

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

Λ + ϵ^{- 1} Σ Σ^{T} + ϵ Γ^{T} Γ < 0

(11)

Next, we present the main results of the paper. The main results section is divided into two subsections. In the first subsection, the system under consideration (1)–(3) involves, in particular, a combination of quantization and overflow nonlinearities, that is, composite nonlinearities. The second subsection considers the case where the present system (1)–(3) is under the influence of saturation overflow nonlinearities.

Main results

Discrete-time systems under the simultaneous influence of quantization and overflow nonlinearities

Consider the system (1) to (3) under the influence of quantization and overflow nonlinearities (Kandanvli and Kar, 2011; Tadepalli and Kandanvli, 2016):

x (k + 1) = O {Q (y (k))} = f (y (k))

(12)

where Q (·) represents the quantization nonlinearities, O (·) denotes the overflow nonlinearities and f (·) represents the composite nonlinearities. Further, when quantization Q (·) is of magnitude truncation or round-off, f (·) is confined to the sector [k_o, k_q], that is,

f_{i} (0) = 0, k_{o} y_{i}^{2} (k) \leq f_{i} (y_{i} (k)) y_{i} (k) \leq k_{q} y_{i}^{2} (k), i = 1, 2, \dots, n

(13a)

where

k_{q} = {\begin{matrix} 1, & for magnitude truncation \\ 2, & for round - off \end{matrix}

(13b)

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

(13c)

Next, we will provide a delay-dependent criterion for the global asymptotic stability of the system given by (1) to (3), (12) and (13).

Theorem 1.Given positive integers h ₁and h ₂, the system represented by (1) to (3) along with (12) and (13) is globally asymptotically stable if there exist positive-definite symmetric matrices

P = [\begin{matrix} P_{1} & P_{2} & P_{3} \\ * & P_{4} & P_{5} \\ * & * & P_{6} \end{matrix}] \in ℝ^{3 n \times 3 n}

Q _i (i = 1,2,3), R ₁, $R_{2} \in ℝ^{n \times n}$ , a matrix

[\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}] \in ℝ^{2 n \times 2 n}

a positive-definite diagonal matrix G = diag(g₁, g₂,…,g_n) and positive scalars $ϵ_{0}$ , $ϵ_{1}$ such that

[\begin{matrix} R_{2} & 0 & Y_{11} & Y_{12} \\ * & 3 R_{2} & Y_{21} & Y_{22} \\ * & * & R_{2} & 0 \\ * & * & * & 3 R_{2} \end{matrix}] \geq 0

(14)

and the following LMIs hold simultaneously:

Π_{1} (d (k) = h_{1}) < 0

(15a)

Π_{1} (d (k) = h_{2}) < 0

(15b)

where

\begin{matrix} Π_{1} (d (k)) = [\begin{matrix} ϱ 11 + ϵ_{0} E_{0}^{T} E_{0} & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 + ϵ_{1} E_{1}^{T} E_{1} & ϱ 23 & ϱ 24 & 0 & ϱ 26 \\ * & * & - Q_{1} - 4 (R_{1} + R_{2}) & ϱ 34 & ϱ 35 + 3 R_{1} & ϱ 36 + 3 R_{2} \\ * & * & * & - Q_{2} - 4 R_{2} & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & - 3 R_{1} & 0 \\ * & * & * & * & * & - 3 R_{2} \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + k_{q} A^{T} G & ϱ 19 & 0 & 0 \\ ϱ 27 & k_{q} {A_{d}}^{T} G & ϱ 29 & 0 & 0 \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 & 0 & 0 & 0 \\ ϱ 47 & - P_{3}^{T} / 2 & 0 & 0 & 0 \\ 0 & h_{1} P_{2}^{T} / 2 & 0 & 0 & 0 \\ - Y_{22} & (d (k) - h_{1}) P_{3}^{T} / 2 & 0 & 0 & 0 \\ - 3 R_{2} & (h_{2} - d (k)) P_{3}^{T} / 2 & 0 & 0 & 0 \\ * & ϱ 88 & \sqrt{\frac{- k_{o}}{2}} G & k_{q} G H_{0} & k_{q} G H_{1} \\ * & * & - k_{q} G & - k_{q} \sqrt{- 2 k_{o}} G H_{0} & - k_{q} \sqrt{- 2 k_{o}} G H_{1} \\ * & * & * & - ϵ_{0} I & 0 \\ * & * & * & * & - ϵ_{1} I \end{matrix}] \end{matrix}

(16)

ϱ 11 = - P_{1} + (P_{2} + P_{2}^{T}) / 2 - 4 R_{1} + \sum_{i = 1}^{3} Q_{i} + h_{12} Q_{3} - ϱ 18

(17)

ϱ 15 = h_{1} (P_{4} - P_{2}) / 2

(18)

ϱ 16 = (d (k) - h_{1}) (P_{5} - P_{3}) / 2

(19)

ϱ 17 = (h_{2} - d (k)) (P_{5} - P_{3}) / 2

(20)

ϱ 18 = - (h_{1}^{2} R_{1} + h_{12}^{2} R_{2})

(21)

ϱ 19 = - k_{q} \sqrt{- 2 k_{o}} A^{T} G

(22)

\begin{matrix} ϱ 22 = & - Q_{3} - 8 R_{2} + Y_{11} + Y_{11}^{T} + Y_{12} \\ + Y_{12}^{T} - Y_{21} - Y_{21}^{T} - Y_{22} - Y_{22}^{T} \end{matrix}

(23)

ϱ 23 = - 2 R_{2} - Y_{11}^{T} - Y_{12}^{T} - Y_{21}^{T} - Y_{22}^{T}

(24)

ϱ 24 = - 2 R_{2} - Y_{11} + Y_{12} + Y_{21} - Y_{22}

(25)

ϱ 26 = 3 R_{2} + Y_{21}^{T} + Y_{22}^{T}

(26)

ϱ 27 = 3 R_{2} - Y_{12} + Y_{22}

(27)

ϱ 29 = - k_{q} \sqrt{- 2 k_{o}} A_{d}^{T} G

(28)

ϱ 34 = Y_{11} - Y_{12} + Y_{21} - Y_{22}

(29)

ϱ 35 = h_{1} (P_{5}^{T} - P_{4}) / 2

(30)

ϱ 36 = (d (k) - h_{1}) (P_{6} - P_{5}) / 2

(31)

ϱ 37 = (h_{2} - d (k)) (P_{6} - P_{5}) / 2 + Y_{12} + Y_{22}

(32)

ϱ 46 = - (d (k) - h_{1}) P_{6} / 2 - Y_{21}^{T} + Y_{22}^{T}

(33)

ϱ 47 = - (h_{2} - d (k)) P_{6} / 2 + 3 R_{2}

(34)

ϱ 88 = P_{1} - ϱ 18 + [(\frac{k_{o}}{2 k_{q}}) - 2] G

(35)

h_{12} = h_{2} - h_{1}

(36)

Proof. Consider a Lyapunov functional

\begin{matrix} V (x (k)) = Γ^{T} (k) P Γ (k) + \sum_{i = k - h_{1}}^{k - 1} x^{T} (i) Q_{1} x (i) \\ + \sum_{i = k - h_{2}}^{k - 1} x^{T} (i) Q_{2} x (i) \\ + \sum_{j = - h_{2}}^{- h_{1}} \sum_{i = k + j}^{k - 1} x^{T} (i) Q_{3} x (i) \\ + \sum_{θ = - h_{1} + 1}^{0} \sum_{j = k - 1 + θ}^{k - 1} h_{1} η^{T} (j) R_{1} η (j) \\ + \sum_{θ = - h_{2} + 1}^{- h_{1}} \sum_{j = k - 1 + θ}^{k - 1} h_{12} η^{T} (j) R_{2} η (j) \end{matrix}

(37)

where

Γ^{T} (k) = [x^{T} (k) \sum_{s = k - h_{1}}^{k - 1} x^{T} (s) \sum_{s = k - h_{2}}^{k - h_{1} - 1} x^{T} (s)]

(38)

η (k) = x (k + 1) - x (k) = f (y (k)) - x (k)

(39)

Taking the forward difference of the Lyapunov functional V( x (k)) along the trajectories of the system (1), we obtain

\begin{matrix} Δ V (x (k)) = V (x (k + 1)) - V x (k) \\ = ξ^{T} (k) Φ (d (k)) ξ (k) + x^{T} (k) Q_{1} x (k) \\ - x^{T} (k - h_{1}) Q_{1} x (k - h_{1}) \\ + x^{T} (k) Q_{2} x (k) - x^{T} (k - h_{2}) Q_{2} x (k - h_{2}) \\ + x^{T} (k) Q_{3} x (k) + h_{12} x^{T} (k) Q_{3} x (k) \\ - \sum_{i = k - h_{2}}^{k - h_{1}} x^{T} (i) Q_{3} x (i) + h_{1}^{2} η^{T} (k) R_{1} η (k) \\ - h_{1} \sum_{i = k - h_{1}}^{k - 1} η^{T} (i) R_{1} η (i) \\ + h_{12}^{2} η^{T} (k) R_{2} η (k) - h_{12} \sum_{i = k - h_{2}}^{k - h_{1} - 1} η^{T} (i) R_{2} η (i) \end{matrix}

(40)

where

\begin{matrix} ξ^{T} (k) & = [x^{T} (k) x^{T} (k - d (k)) x^{T} (k - h_{1}) x^{T} (k - h_{2}) χ^{T} (k, 0, h_{1}) χ^{T} (k, h_{1}, d (k)) χ^{T} (k, d (k), h_{2}) f^{T} (y (k))] \end{matrix}

(41)

\begin{matrix} Φ (d (k)) = [\begin{matrix} - P_{1} + (P_{2} + P_{2}^{T}) / 2 & 0 & (P_{3} - P_{2}) / 2 & - P_{3} / 2 & ϱ 15 & ϱ 16 \\ * & 0 & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & - (d (k) - h_{1}) P_{6} / 2 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & 0 \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & P_{2}^{T} / 2 \\ 0 & 0 \\ (h_{2} - d (k)) (P_{6} - P_{5}) / 2 & (P_{3}^{T} - P_{2}^{T}) / 2 \\ - (h_{2} - d (k)) P_{6} / 2 & - P_{3}^{T} / 2 \\ 0 & h_{1} P_{2}^{T} / 2 \\ 0 & (d (k) - h_{1}) P_{3}^{T} / 2 \\ 0 & (h_{2} - d (k)) P_{3}^{T} / 2 \\ * & P_{1} \end{matrix}] \end{matrix}

(42)

and χ (k, 0, h₁), χ (k, h₁, d(k)), χ (k, d(k), h₂) are defined by (4).

Note that

- \sum_{i = k - h_{2}}^{k - h_{1}} x^{T} (i) Q_{3} x (i) \leq - x^{T} (k - d (k)) Q_{3} x (k - d (k))

(43)

Next, by employing Lemma 1, that is, the discrete version of the Wirtinger-based inequality, the 10th and the last terms of (40) are bounded by (5) and are as follows:

\begin{matrix} - h_{1} \sum_{i = k - h_{1}}^{k - 1} η T (i) R_{1} η (i) \\ \leq - {[\begin{matrix} x (k) - x (k - h_{1}) \\ x (k) + x (k - h_{1}) - χ (k, 0, h_{1}) \end{matrix}]}^{T} [\begin{matrix} R_{1} & 0 \\ 0 & 3 R_{1} \end{matrix}] \\ [\begin{matrix} x (k) - x (k - h_{1}) \\ x (k) + x (k - h_{1}) - χ (k, 0, h_{1}) \end{matrix}] \end{matrix}

(44)

\begin{matrix} - h_{12} \sum_{i = k - h_{2}}^{k - h_{1} - 1} η^{T} (i) R_{2} η (i) \\ = - h_{12} [\sum_{i = k - h_{2}}^{k - d (k) - 1} η^{T} (i) R_{2} η (i) + \sum_{i = k - d (k)}^{k - h_{1} - 1} η^{T} (i) R_{2} η (i)] \\ \leq - \frac{(h_{2} - h_{1})}{(h_{2} - d (k))} {[\begin{matrix} x (k - d (k)) - x (k - h_{2}) \\ x (k - d (k)) + x (k - h_{2}) - χ (k, d (k), h_{2}) \end{matrix}]}^{T} \\ \times [\begin{matrix} R_{2} & 0 \\ 0 & 3 R_{2} \end{matrix}] [\begin{matrix} x (k - d (k)) - x (k - h_{2}) \\ x (k - d (k)) + x (k - h_{2}) - χ (k, d (k), h_{2}) \end{matrix}] \\ - \frac{(h_{2} - h_{1})}{(d (k) - h_{1})} {[\begin{matrix} x (k - h_{1}) - x (k - d (k)) \\ x (k - h_{1}) + x (k - d (k)) - χ (k, h_{1}, d (k)) \end{matrix}]}^{T} \\ \times [\begin{matrix} R_{2} & 0 \\ 0 & 3 R_{2} \end{matrix}] [\begin{matrix} x (k - h_{1}) - x (k - d (k)) \\ x (k - h_{1}) + x (k - d (k)) - χ (k, h_{1}, d (k)) \end{matrix}] \end{matrix}

(45)

The reciprocal convexity method of Lemma 2 ensures that if there exists a matrix $[\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}] \in ℝ^{2 n \times 2 n}$ such that (14) holds true, then the upper bound of (45) can be written as

\begin{matrix} - h_{12} \sum_{i = k - h_{2}}^{k - h_{1} - 1} η^{T} (i) R_{2} η (i) \\ \leq - {[\begin{matrix} x (k - h_{1}) - x (k - d (k)) \\ x (k - h_{1}) + x (k - d (k)) - χ (k, h_{1}, d (k)) \\ x (k - d (k)) - x (k - h_{2}) \\ x (k - d (k)) + x (k - h_{2}) - χ (k, d (k), h_{2}) \end{matrix}]}^{T} \\ \times [\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 (k - h_{1}) - x (k - d (k)) \\ x (k - h_{1}) + x (k - d (k)) - χ (k, h_{1}, d (k)) \\ x (k - d (k)) - x (k - h_{2}) \\ x (k - d (k)) + x (k - h_{2}) - χ (k, d (k), h_{2}) \end{matrix}] \end{matrix}

(46)

Employing (40) to (44) and (46), we have the following inequality

Δ V (x (k)) \leq ξ^{T} (k) ξ (d (k)) ξ (k) - 2 β

(47)

where

\begin{matrix} β & = \sum_{i = 1}^{n} g_{i} [k_{q} y_{i} (k) - f_{i} (y_{i} (k))] [f_{i} (y_{i} (k)) - k_{o} y_{i} (k)] \\ = [k_{q} y (k) - f (y (k {))]}^{T} G [f (y (k)) - k_{o} y (k)] \end{matrix}

(48)

\begin{matrix} ξ (d (k)) = [\begin{matrix} ϱ 11 - 2 k_{q} k_{o} {\bar{A}}^{T} G \bar{A} & - 2 k_{q} k_{o} {\bar{A}}^{T} G {\bar{A}}_{d} & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 - 2 k_{q} k_{o} {\bar{A}}_{d}^{T} G {\bar{A}}_{d} & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & 0 \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + (k_{q} + k_{o}) {\bar{A}}^{T} G \\ 0 & (k_{q} + k_{o}) {\bar{A}}_{d}^{T} G \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 \\ ϱ 47 & - P_{3}^{T} / 2 \\ 0 & h_{1} P_{2}^{T} / 2 \\ 0 & (d (k) - h_{1}) P_{3}^{T} / 2 \\ 0 & (h_{2} - d (k)) P_{3}^{T} / 2 \\ * & P_{1} - ϱ 18 - 2 G \end{matrix}] \end{matrix}

(49)

\bar{A} = A + Δ A, {\bar{A}}_{d} = A_{d} + Δ A_{d}

(50)

In view of (13), (48) is non-negative. From (47) and (48) it follows that ΔV ( x (k)) < 0 if Ψ(d(k)) < 0 for all d(k) ∈ [h₁, h₂]. Thus, Ψ(d(k)) < 0 along with (14) are sufficient conditions for the global asymptotic stability of the system (1) to (3), (12) and (13). The condition Ψ(d(k)) < 0 can also be expressed as

\begin{matrix} [\begin{matrix} ϱ 11 & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 & ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + k_{q} {\bar{A}}^{T} G \\ * & ϱ 22 & 0 & 0 & 0 & 0 & 0 & k_{q} {\bar{A}}_{d}^{T} G \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 & ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & ϱ 46 & ϱ 47 & - P_{3}^{T} / 2 \\ * & * & * & * & 0 & 0 & 0 & h_{1} P_{2}^{T} / 2 \\ * & * & * & * & * & 0 & 0 & (d (k) - h_{1}) P_{3}^{T} / 2 \\ * & * & * & * & * & * & 0 & (h_{2} - d (k)) P_{3}^{T} / 2 \\ * & * & * & * & * & * & * & ϱ 88 \end{matrix}] \\ - [\begin{matrix} - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}^{T} G \\ - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{d}^{T} G \\ 0 \\ 0 \\ 0 \\ 0 \\ 0 \\ \sqrt{\frac{- k_{o}}{2}} G \end{matrix}] \\ \times [- k_{q} G]^{- 1} \times [\begin{matrix} - k_{q} \sqrt{- 2 k_{o}} G \bar{A} & - k_{q} \sqrt{- 2 k_{o}} G {\bar{A}}_{d} & 0 & 0 & 0 & 0 & 0 \sqrt{\frac{- k_{o}}{2}} G \end{matrix}] < 0 \end{matrix}

(51)

Using the well-known Schur’s complement, (51) is equivalent to

\begin{matrix} [\begin{matrix} ϱ 11 & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & 0 \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + k_{q} {\bar{A}}^{T} G & - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}^{T} G \\ 0 & k_{q} {\bar{A}}_{d}^{T} G & - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{d}^{T} G \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 & 0 \\ ϱ 47 & - P_{3}^{T} / 2 & 0 \\ 0 & h_{1} P_{2}^{T} / 2 & 0 \\ 0 & (d (k) - h_{1}) P_{3}^{T} / 2 & 0 \\ 0 & (h_{2} - d (k)) P_{3}^{T} / 2 & 0 \\ * & ϱ 88 & \sqrt{\frac{- k_{o}}{2}} G \\ * & * & - k_{q} G \end{matrix}] < 0 \end{matrix}

(52)

Further, using (3a), the condition (52) can be rewritten as

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

(53)

where

{\bar{H}}_{0}^{T} = [0 0 0 0 0 0 0 k_{q} H_{0}^{T} G - k_{q} \sqrt{- 2 k_{o}} H_{0}^{T} G]

(54)

{\bar{E}}_{0} = [E_{0} 0 0 0 0 0 0 0 0]

(55)

and

\begin{matrix} ξ_{0} (d (k)) = [\begin{matrix} ϱ 11 & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & 0 \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + k_{q} A^{T} G & - k_{q} \sqrt{- 2 k_{o}} A^{T} G \\ 0 & k_{q} {\bar{A}}_{d}^{T} G & - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{d}^{T} G \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 & 0 \\ ϱ 47 & - P_{3}^{T} / 2 & 0 \\ 0 & h_{1} P_{2}^{T} / 2 & 0 \\ 0 & (d (k) - h_{1}) P_{3}^{T} / 2 & 0 \\ 0 & (h_{2} - d (k)) P_{3}^{T} / 2 & 0 \\ * & ϱ 88 & \sqrt{\frac{- k_{o}}{2}} G \\ * & * & - k_{q} G \end{matrix}] < 0 \end{matrix}

(56)

By Lemma 3, (53) to (56) is equivalent to

ξ_{0} (d (k)) + ϵ_{0}^{- 1} {\bar{H}}_{0} {\bar{H}}_{0}^{T} + ϵ_{0} {\bar{E}}_{0}^{T} {\bar{E}}_{0} < 0

(57)

where $ϵ_{0} > 0$ . Using Schur’s complement, (57) can also be expressed as

\begin{matrix} [\begin{matrix} ϱ 11 & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 & 0 & 0 & 0 & 0 \\ * & * & 0 & 0 & ϱ 35 & ϱ 36 \\ * & * & * & 0 & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & 0 & 0 \\ * & * & * & * & * & 0 \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + k_{q} A^{T} G & - k_{q} \sqrt{- 2 k_{o}} A^{T} G & 0 \\ 0 & k_{q} {\bar{A}}_{d}^{T} G & - k_{q} \sqrt{- 2 k_{o}} {\bar{A}}_{d}^{T} G & 0 \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 & 0 & 0 \\ ϱ 47 & - P_{3}^{T} / 2 & 0 & 0 \\ 0 & h_{1} P_{2}^{T} / 2 & 0 & 0 \\ 0 & (d (k) - h_{1}) P_{3}^{T} / 2 & 0 & 0 \\ 0 & (h_{2} - d (k)) P_{3}^{T} / 2 & 0 & 0 \\ * & ϱ 88 & \sqrt{\frac{- k_{o}}{2}} G & k_{q} G H_{0} \\ * & * & - k_{q} G & - k_{q} \sqrt{- 2 k_{o}} G H_{0} \\ * & * & * & - ϵ_{0} I \end{matrix}] < 0 \end{matrix}

(58)

Following steps similar to (53) to (58), it is easy to show that (58) is equivalent to the condition Π₁(d(k)) < 0. It is apparent that Π₁(d(k)) is an affine matrix function with respect to the delay d(k). According to the property of an affine function, Π₁(d(k)) < 0 holds if and only if (15a) and (15b) hold simultaneously. This completes the proof of Theorem 1.

Remark 1. The number of decision variables of Theorem 1 is 11n² + 5n + 2. Similarly, for Theorem 1 by Kandanvli and Kar (2011) and Theorem 1 by Tadepalli and Kandanvli (2016), the number of decision variables can be expressed as 13n² + 6n + 2 and [(m² + 7)n² + (m + 7)n + 4]/2, respectively. Here, ‘n’ represents the order of the system while ‘m’ represents the number of partitions of the lower delay bound h₁ (h₁ = τm, where τ is the partition size). It may be noted that in the delay-partitioning approach considered in Tadepalli and Kandanvli (2016), by increasing the number of partitions ‘m’, one may expect better conservative results at the cost of number of decision variables. Thus, keeping this fact in view, and from Table 1, the proposed Theorem 1 may be considered a better choice in terms of number of decision variables than Theorem 1 by Kandanvli and Kar (2011) and Theorem 1 by Tadepalli and Kandanvli (2016).

Table 1.

Comparison of number of decision variables.

Methods	Number of decision variables for n = 2
Theorem 1 (Kandanvli and Kar, 2011)	66

Theorem 1 (Tadepalli and Kandanvli, 2016)	59 (when m = 4)
Theorem 1 (Tadepalli and Kandanvli, 2016)	78 (when m = 5)
	128 (when m = 7)

Theorem 1 (Proposed)	56

Remark 2. Table 2 shows (see also Example 1) that the proposed Theorem 1 may yield less conservative results than Theorem 1 (Kandanvli and Kar, 2011) and Theorem 1 (Tadepalli and Kandanvli, 2016).

Table 2.

Upper delay bound h₂ for various h₁ for the system described in Example 1.

Methods/h₁	2	6	8	12
Theorem 1 (Kandanvli and Kar, 2011)	8	8	10	13
Theorem 1 (Tadepalli and Kandanvli, 2016)	8 (m = 2, τ = 1)	9 (m = 3, τ = 2)	10 (m = 4, τ = 2)	13 (m = 6, τ = 2)
Theorem 1 (Proposed)	9	10	11	14

In general, the efficiency of a particular criterion depends on how much improvement, in terms of conservativeness and number of decision variables, it will provide. Given this, and from Tables 1 and 2, it is clear that Theorem 1 may provide better results than Theorem 1 (Kandanvli and Kar, 2011) and Theorem 1 (Tadepalli and Kandanvli, 2016).

Discrete-time systems in the presence of saturation nonlinearities

Pertaining to the situation where the system (1) to (3) involves saturation overflow nonlinearities f (·) given by

\begin{matrix} f_{i} (y_{i} (k)) & = y_{i} (k), & | y_{i} (k) | \leq 1 \\ f_{i} (y_{i} (k)) & = 1, & y_{i} (k) > 1 \\ f_{i} (y_{i} (k)) & = - 1, & y_{i} (k) < - 1 \end{matrix}}, i = 1, 2, \dots, n

(59)

we have the following result.

Theorem 2.Given positive integers h ₁and h ₂, the system represented by (1) to (3) and (59) is globally asymptotically stable if there exist positive-definite symmetric matrices

P = [\begin{matrix} P_{1} & P_{2} & P_{3} \\ * & P_{4} & P_{5} \\ * & * & P_{6} \end{matrix}] \in ℝ^{3 n \times 3 n}

Q _i (i = 1,2,3), R ₁, $R_{2} \in ℝ^{n \times n}$ , a matrix

[\begin{matrix} Y_{11} & Y_{12} \\ Y_{21} & Y_{22} \end{matrix}] \in ℝ^{2 n \times 2 n}

and positive scalars $ϵ_{0}$ , $ϵ_{1}$ , α_ij, β_ij (i,j = 1,2,…,n;i≠j) such that (14) and (60) hold true:

Π_{2} (d (k) = h_{1}) < 0 and Π_{2} (d (k) = h_{2}) < 0

(60)

where

\begin{matrix} Π_{2} (d (k)) = [\begin{matrix} ϱ 11 + ϵ_{0} E_{0}^{T} E_{0} & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 + ϵ_{1} E_{1}^{T} E_{1} & ϱ 23 & ϱ 24 & 0 & ϱ 26 \\ * & * & - Q_{1} - 4 (R_{1} + R_{2}) & ϱ 34 & ϱ 35 + 3 R_{1} & ϱ 36 + 3 R_{2} \\ * & * & * & - Q_{2} - 4 R_{2} & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & - 3 R_{1} & 0 \\ * & * & * & * & * & - 3 R_{2} \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + A^{T} C & 0 & 0 \\ ϱ 27 & {A_{d}}^{T} C & 0 & 0 \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 & 0 & 0 \\ ϱ 47 & - P_{3}^{T} / 2 & 0 & 0 \\ 0 & h_{1} P_{2}^{T} / 2 & 0 & 0 \\ - Y_{22} & (d (k) - h_{1}) P_{3}^{T} / 2 & 0 & 0 \\ - 3 R_{2} & (h_{2} - d (k)) P_{3}^{T} / 2 & 0 & 0 \\ * & P_{1} - ϱ 18 - (C + C^{T}) & C^{T} H_{0} & C^{T} H_{1} \\ * & * & - ϵ_{0} I & 0 \\ * & * & * & - ϵ_{1} I \end{matrix}] \end{matrix}

(61)

and $C = [c_{ij}] \in ℝ^{n \times n}$ denotes a matrix (Kandanvli and Kar, 2013; Tadepalli and Kandanvli, 2016) defined by

c_{ii} = \sum_{j = 1, j \neq i}^{n} (α_{ij} + β_{ij}), i = 1, 2, \dots, n

(62a)

c_{ij} = α_{ij} - β_{ij}, i, j = 1, 2, \dots, n (i \neq j)

(62b)

α_{ij} > 0, β_{ij} > 0, i, j = 1, 2, \dots, n (i \neq j)

(62c)

Proof. By choosing the Lyapunov functional (37) and employing (40) to (44) and (46), we obtain

Δ V (x (k)) \leq ξ^{T} (k) Ξ (d (k)) ξ (k) - δ

(63)

where

\begin{matrix} δ = \sum_{i = 1}^{n} 2 [y_{i} (k) - f_{i} (y_{i} (k))] \\ [\sum_{j = 1, j \neq i}^{n} {(α_{ij} + β_{ij}) f_{i} (y_{i} (k)) + (α_{ij} - β_{ij}) f_{j} (y_{j} (k))}] \\ = y^{T} (k) Cf (y (k)) + f^{T} (y (k)) C^{T} y (k) - f^{T} (y (k)) (C + C^{T}) f (y (k)) \end{matrix}

(64)

\begin{matrix} Ξ (d (k)) = [\begin{matrix} ϱ 11 & 0 & (P_{3} - P_{2}) / 2 - 2 R_{1} & - P_{3} / 2 & ϱ 15 + 3 R_{1} & ϱ 16 \\ * & ϱ 22 & ϱ 23 & ϱ 24 & 0 & ϱ 26 \\ * & * & - Q_{1} - 4 (R_{1} + R_{2}) & ϱ 34 & ϱ 35 + 3 R_{1} & ϱ 36 + 3 R_{2} \\ * & * & * & - Q_{2} - 4 R_{2} & - h_{1} P_{5}^{T} / 2 & ϱ 46 \\ * & * & * & * & - 3 R_{1} & 0 \\ * & * & * & * & * & - 3 R_{2} \\ * & * & * & * & * & * \\ * & * & * & * & * & * \end{matrix} \\ \begin{matrix} ϱ 17 & ϱ 18 + P_{2}^{T} / 2 + {\bar{A}}^{T} C \\ ϱ 27 & {\bar{A}}_{d}^{T} C \\ ϱ 37 & (P_{3}^{T} - P_{2}^{T}) / 2 \\ ϱ 47 & - P_{3}^{T} / 2 \\ 0 & h_{1} P_{2}^{T} / 2 \\ - Y_{22} & (d (k) - h_{1}) P_{3}^{T} / 2 \\ - 3 R_{2} & (h_{2} - d (k)) P_{3}^{T} / 2 \\ * & P_{1} - ϱ 18 - (C + C^{T}) \end{matrix}] \end{matrix}

(65)

Observe that, for the saturation nonlinearities given by (59) along with (62c), the quantity δ (see (64)) is non-negative (Kandanvli and Kar, 2013; Kar, 2007; Singh, 1990; Tadepalli and Kandanvli, 2016).

From (63), it is clear that ΔV( x (k)) < 0, if Ξ(d(k)) < 0 for all d(k) ∈ [h₁, h₂]. Thus, Ξ(d(k)) < 0 together with (14) is a sufficient condition for the global asymptotic stability of the system (1) to (3) and (59).

Next, by following the approach presented in the proof of Theorem 1, the condition Ξ(d(k)) < 0 leads to Π₂(d(k)) < 0. Based on the properties of affine matrix functions, the condition Π₂(d(k)) < 0 if and only if Π₂(d(k) = h₁) < 0 and Π₂(d(k) = h₂) < 0 which concludes the proof of Theorem 2.

Remark 3.Similar to Remark 1, the number of decision variables for Theorem 2 is 13n² + 2n + 2, whereas for Theorem 3.1 (Kandanvli and Kar, 2013) and Theorem 2 (Tadepalli and Kandanvli, 2016) the number of decision variables can be expressed as 15n² + 3n + 2 and [(m² + 11)n² + (m + 1)n + 4]/2, respectively. From Table 3, one can observe that the proposed Theorem 2 requires fewer decision variables than Theorem 3.1 (Kandanvli and Kar, 2013) and Theorem 2 (Tadepalli and Kandanvli, 2016).

Table 3.

Comparison of number of decision variables.

Methods	Number of decision variables for n = 2
Theorem 3.1 (Kandanvli and Kar, 2013)	68

Theorem 2 (Tadepalli and Kandanvli, 2016)	61 (when m = 4)
	80 (when m = 5)
	130 (when m = 7)
	376 (when m = 13)

Theorem 2 (Proposed)	58

Remark 4.From Table 4, it is easy to demonstrate the effectiveness of Theorem 2. It can be inferred (see Example 2) that Theorem 2 may provide less conservative results compared to the previously reported criteria (Kandanvli and Kar, 2013; Tadepalli and Kandanvli, 2016).

Table 4.

Upper delay bound h₂ for various h₁ for the system described in Example 2.

Methods/ h₁	2	4	8	12
Theorem 3.1 (Kandanvli and Kar, 2013)	8	8	10	13
Theorem 2 (Tadepalli and Kandanvli, 2016)	8 (m = 2, τ = 1)	8 (m = 2, τ = 2)	10 (m = 4, τ = 2)	13 (m = 6, τ = 2)
Theorem 2 (Proposed)	9	9	11	14

Remark 5.From (13), when quantization is of magnitude truncation or round-off, the function f (·) considered in ‘Discrete-time systems under the simultaneous influence of quantization and overflow nonlinearities’ is confined to the sector [k_o, k_q], that is

f_{i} (0) = 0, k_{o} \leq \frac{f_{i} (y_{i} (k))}{y_{i} (k)} \leq k_{q}, i = 1, 2, \dots, n

(66a)

which satisfies the following sector-bounded condition

{[f (y (k)) - k_{o} y (k)]}^{T} [f (y (k)) - k_{q} y (k)] \leq 0

(66b)

Note that by choosing G = I , and from (13), (48) implies (66b). For technical ease, the nonlinear function f ( y (k)) is decomposed into linear and nonlinear parts as $f (y (k)) = k_{o} y (k) + \hat{f} (y (k))$ . Then it follows from (66b) that

{\hat{f}}^{T} (y (k)) [\hat{f} (y (k)) - (k_{q} - k_{o}) y (k)] \leq 0

(66c)

The nonlinear description in (66b) is more general and includes the usual Lipschitz conditions as a special case. On the other hand, for the case where the system (1)–(3) is under the influence of only saturation overflow nonlinearities, the function f (·) considered in ‘Discrete-time systems in the presence of saturation nonlinearities’ (see (59) and (62)) can be characterized by (64). Note that saturation overflow nonlinearities are confined to the sector [0, 1] (Kandanvli and Kar, 2009b) and also belong to a class of odd and 1-Lipschitz nonlinearities (Chu and Glover, 1999; Kandanvli and Kar, 2008).

Remark 6. As mentioned in Remark 1 of Seuret et al. (2015), it may be noted that the inequality (5) shown in Lemma 1 yields

(b - a) \sum_{s = k - b}^{k - a - 1} η^{T} (s) R η (s) \geq Ω_{0}^{T} R Ω_{0}

(67)

which is identical to the discrete Jensen inequality (Jiang et al., 2005). Hence, it may be concluded that Lemma 1 is less conservative than the discrete Jensen inequality (Jiang et al., 2005) since a positive term is added to the right-hand side of the inequality of Lemma 1.

Remark 7.The characterization of nonlinearities employed in Theorem 2 is quite distinct from that in Theorem 1. Pertaining to saturation overflow nonlinearities (61), a substantially improved characterization of the nonlinearities δ(δ > 0) given by (65) which exploits the structural properties of the multiple saturation nonlinearities in greater detail (i.e. the so-called passivity property; Singh, 1990) than merely the sector information is considered in Theorem 2. By contrast, the non-negative quantity β (see (50)) used in Theorem 1 covers a broader class of nonlinearities belonging to the sector [k_o, k_q] which includes both quantization and overflow. The results obtained via Theorem 2 may not be applicable to systems involving a combination of quantization and saturation nonlinearities. However, situations may also arise where Theorem 1 will fail to establish global asymptotic stability in the presence of various combinations of quantization and overflow nonlinearities, whereas Theorem 2 will succeed in establishing global asymptotic stability in the presence of saturation overflow nonlinearities only.

Examples

In this section, we will give examples to show the significance of the presented results.

Example 1. Consider the system (1) to (3), (12) and (13) with

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

(68)

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

(69)

and the nonlinearities belonging to the sector [k_o, k_q] = [0, 1], which includes saturation, zeroing, magnitude truncation, a combination of saturation and magnitude truncation, a combination of zeroing and magnitude truncation, and so on. This example has been considered in Kandanvli and Kar (2011, 2013) and Tadepalli and Kandanvli (2016). Now, using the SeDuMi 1.21 solver (Sturm, 1999) and the YALMIP 3.0 parser (Löfberg, 2004), our objective is to obtain an upper delay bound h₂ for a given lower delay bound h₁ by iteratively solving the linear matrix inequalities with respect to h₂. The various values of h₂ obtained are summarized in Table 2. From Tables 1 and 2, one can observe that Theorem 1 yields less conservative results along with a reduced computational burden compared to Theorem 1 in Kandanvli and Kar (2011) and Theorem 1 in Tadepalli and Kandanvli (2016).

By choosing an arbitrary initial condition, a plot of the state trajectories of the present system with F ₀ = F ₁ = 1 and 2 ≤ d(k) ≤ 9 is shown in Figure 1. It may be observed that Figure 1 supports the fact (which has been arrived at via Theorem 1) that the present system is globally asymptotically stable.

Figure 1.

State response of the system in Example 1.

Example 2. Consider the system described by (1) to (3) and (59) with (68) and (69), which represents a class of discrete-time systems under the influence of saturation nonlinearities, uncertainties and delays. The upper delay bounds h₂ which guarantee the global asymptotic stability of the system under consideration for various values of the lower delay bound h₁ are listed in Table 4. It can also be seen that the results of Theorem 2 are better than those in Theorem 3.1 of Kandanvli and Kar (2013) and Theorem 2 of Tadepalli and Kandanvli (2016). For the number of decision variables listed in Table 3, the proposed Theorem 2 has much fewer decision variables than the corresponding theorems in Kandanvli and Kar (2013) and Tadepalli and Kandanvli (2016). The state response in Figure 2 verifies the findings of Theorem 2 for the parameters considered in Example 2 along with F ₀ = F ₁ = 1 and 2 ≤ d(k) ≤ 9.

Figure 2.

State response of the system in Example 2.

Summarizing the above, the criteria presented in this paper may provide better stability conditions (i.e. less conservative and computationally less demanding) which are outside the scope of previously reported criteria. In other words, in the design of a class of uncertain discrete-time systems employing finite wordlength nonlinearities and time-varying delays, the present criteria could help to obtain a larger stability region than that obtainable via previously reported criteria. This is important, since for efficient design it is necessary to have as large a stability region as possible.

Conclusion

In this paper, with the help of a discrete version of the Wirtinger-based inequality and a reciprocally convex approach, two delay-dependent stability criteria for a class of uncertain discrete-time systems with time-varying delays and finite wordlength nonlinearities have been presented. Relative to previous criteria, the presented approach involves fewer decision variables and, hence, the computational burden has reduced greatly. From the examples it has also been shown that the presented results are less conservative than the previous ones.

Footnotes

Acknowledgements

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

Funding

This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.

References

Bakule

Rodellar

Rossell

(2006) Robust overlapping guaranteed cost control of uncertain state-delay discrete-time systems. IEEE Transactions on Automatic Control 51: 1943–1950.

Bose

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

Boyd

El Ghaoui

Feron

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

Chen

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

Chu

Glover

(1999) Bounds of the induced norm and model reduction errors for systems with repeated scalar nonlinearities. IEEE Transactions on Automatic Control 44: 471–483.

Claasen

TACM

Mecklenbräuker

WFG

Peek

JBH

(1976) Effects of quantization and overflow in recursive digital filters. IEEE Transactions on Acoustics, Speech, and Signal Processing 24: 517–529.

Dewasurendra

Bauer

(2008) A novel approach to grid sensor networks. In: Proceedings of 15th IEEE international conference on electronics, circuits and systems, 31 August–3 September, St. Julian’s, Malta: IEEE pp. 1191–1194.

Liu

et al . (2008) Output feedback stabilization for a discrete-time system with a time-varying delay. IEEE Transactions on Automatic Control 53: 2372–2377.

Jiang

Han

Q-L

(2005) Stability criteria for linear discrete-time systems with interval-like time-varying delay. In: Proceedings of the American control conference, June 8–10, 2005, Portland, OR: IEEE, pp. 2817–2822.

10.

Kandanvli

VKR

Kar

(2008) Robust stability of discrete-time state-delayed systems employing generalized overflow nonlinearities. Nonlinear Analysis 69: 2780–2787.

11.

Kandanvli

VKR

Kar

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

12.

Kandanvli

VKR

Kar

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

13.

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: 1611–1622.

14.

Kandanvli

VKR

Kar

(2013) Delay-dependent stability criterion for discrete-time uncertain state-delayed systems employing saturation nonlinearities. Arabian Journal for Science and Engineering 38: 2911–2920.

15.

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: 685–689.

16.

Kar

Singh

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

17.

Lee

Kwon

Park

(2012) Regional asymptotic stability analysis for discrete delayed systems with saturation nonlinearity. Nonlinear Dynamics 67: 885–892.

18.

Liu

Zhang

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

19.

Löfberg

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

20.

Mahmoud

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

21.

Nam

Pathirana

Trinh

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

22.

Park

Jeong

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

23.

Parlakçi

MNA

(2015) A robust H-infinity controller design approach for neutral systems with time-varying delay and plant and controller uncertainties. Transactions of the Institute of Measurement and Control 37: 645–651.

24.

Seuret

Gouaisbaut

Fridman

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

25.

Sim

Pang

(1985) Design criterion for zero-input asymptotic overflow-stability of recursive digital filters in the presence of quantization. Circuits Systems and Signal Processing 4: 485–502.

26.

Singh

(1990) Elimination of overflow oscillations in fixed-point state-space digital filters using saturation arithmetic. IEEE Transactions on Circuits and Systems 37: 814–818.

27.

Singh

(2006) Modified form of Liu-Michel’s criterion for global asymptotic stability of fixed-point state-space digital filters using saturation arithmetic. IEEE Transactions on Circuits and Systems II: Express Briefs 53: 1423–1425.

28.

Singh

(2008) Stability analysis of 2-D discrete systems described by the Fornasini-Marchesini second model with state saturation. IEEE Transactions on Circuits and Systems II: Express Briefs 55: 793–796.

29.

Singh

(2013) A novel LMI-based criterion for the stability of direct-form digital filters utilizing a single two’s complement nonlinearity. Nonlinear Analysis: Real World Applications 14: 684–689.

30.

Song

Wang

(2013) A delay partitioning approach to output feedback control for uncertain discrete time-delay systems with actuator saturation. Nonlinear Dynamics 74: 189–202.

31.

Sturm

(1999) Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones. Optimizations Method Software 11–12: 625–653.

32.

Tadepalli

Kandanvli

VKR

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

33.

Tadepalli

Kandanvli

VKR

Kar

(2014) Stability criteria for uncertain discrete-time systems under the influence of saturation nonlinearities and time-varying delay. ISRN Applied Mathematics: 861–759.

34.

Tadepalli

Kandanvli

VKR

Kar

(2015) Comment on ‘Stability analysis and controller synthesis for discrete linear time-delay systems with state saturation nonlinearities’. International Journal of Systems Science 46: 2818–2819.

35.

Lam

Yao

et al . (2011) Robust guaranteed cost control of discrete-time networked control systems. Optimal Control Applications and Methods 32: 95–112.

36.

Xie

de Souza

(1992) H_∞ control and quadratic stabilization of systems with parameter uncertainty via output feedback. IEEE Transactions on Automatic Control 37: 1253–1256.