An anti-windup control strategy for discrete time-delay system based on triple Lyapunov functional

Abstract

This paper addresses the design of a dynamic output feedback-based anti-windup compensator to mitigate the windup phenomenon for a discrete time-varying delayed system with input saturation to establish the asymptotic stability. Linear matrix inequalities–based stability conditions are derived locally and globally. The controller for the closed-loop system is designed to reduce the effect of external bounded disturbances by utilizing the linear matrix inequality approach. The novel triple Lyapunov–Krasovskii functional along with reciprocal convex inequality is used to solve the expressions contained in the forward difference of the functional and to maximize the basin of attraction. Finally, industrial examples are simulated to prove the effectiveness of the proposed criterion.

Keywords

Discrete-time system actuator saturation anti-windup asymptotic stability

Introduction

Time delay is a natural phenomenon which is inevitably present in all practical systems due to finite computation capability of real-time system and transportation lag in transmitted signals (Huang et al., 2022; Singh et al., 2021; Wang et al., 2020). It is a major cause of instability, poor performance, and oscillations (Chen et al., 2019; Kwon et al., 2013). Therefore, a lot of researchers have addressed the issues related to stabilization of time-delay systems by choosing the delay-dependent approach (Qian et al., 2015; Xu et al., 2012; Zhao et al., 2019). Furthermore, in this direction, new less-conservative stability results are derived by introducing different lemmas such as reciprocal convex inequality, Wirtinger inequality, triple Lyapunov–Krasovskii functional (LKF), bivariate quadratic function negative-determination lemma, and so on. (Kwon et al., 2013; Li et al., 2009; Liu and Zhang, 2012; Park et al., 2011; Singh et al., 2021; Sun et al., 2010). Nowadays, the use of microcontroller and fast, precise, and accurate digital processors has transformed the overall world into new domain of digitalization. Therefore, the research in the field of controller design related to discrete system is growing day by day (Chen et al., 2019; Singh et al., 2021).

The controller sends control signal to final control elements like actuator and valve for getting the desired response from the industrial systems (Valenzuela, 2021). Due to various physical limitations in terms of finite voltage and current handling capacity, saturation appears in different practical systems, causing deterioration in their performance (Wang et al., 2020; Xu et al., 2012). A controller with actuator constraints in unstable mode may exhibit windup problems as discussed in the study by Doyle et al. (1987). Control constraints nonlinearity may cause poor performance of the system. They have been troubleshooted in many ways, with the anti-windup being the earliest one (Da Silva and Tarbouriech, 2006). The windup phenomenon comes into the picture when there is a divergence between the plant input and the controller output. Its occurrence causes the feedback loop to break, so an additional increment in the control signal does not assist the rapid response of the system (Da Silva et al., 2018). The anti-windup concept is to develop an additional element in the control system to monitor the control signal (Obaiah and Subudhi, 2020). If saturation occurs, the control system is modified to improve during and after saturation (Huang et al., 2016). The complexity of the system vertically increases by considering the effect of delay and actuator saturation (Xu et al., 2012). In the literature, less attention has been paid to anti-windup problems for discrete-time systems (Pal et al., 2020). In continuous time systems, the proposed designs do not explicitly address the issue of enlarging the domain of the closed-loop system stability. Hence, a discrete-time system with anti-windup is an essential topic for the researchers (Valenzuela, 2021).

Concerning available results related to this work, Negi et al. (2012) have designed an anti-windup compensator for linear time-varying delay system with saturation but not considered either triple Lyapunov or disturbance. Although stability analysis of discrete delayed system has been analyzed by Pal and Negi (2017) and Pal et al. (2020), the approach to tackle saturation via anti-windup method and for mitigating different disturbances (like step) is not considered in the above papers. To bridge this gap, the present work is motivated by aforesaid papers to take up the problem of a discretized delayed system with anti-windup and different types of disturbances such as exponential decaying and step-load disturbance for stability analysis of the system.

The novelty of this work is that the triple Lyapunov-based LKF is used for designing anti-windup compensator to tackle the complex issues of discrete time-delay systems subjected to actuator saturation and exponential decay as well as step disturbance using the Linear Matrix Inequality (LMI) technique.

Inspired by the aforementioned observations, this paper aims to design an anti-windup controller for discrete-time system with saturation, disturbance, reciprocal convexity, and triple Lyapunov method to establish the asymptotic stability of the system.

The prime contribution of this script is as follows:

A novel dynamic output feedback $H_{\infty}$ stabilization controller with an anti-windup compensator is devised for the discrete time-delay saturated system with disturbance using triple Lyapunov and reciprocal convex inequality to mitigate the impact of windup phenomenon and disturbance.

Anti-windup gain matrix $F_{va}$ and basin of attraction are evaluated for various delay ranges using optimization techniques.

The delay range has increased in comparison with the actual result in the previous literature (Negi et al., 2012; Pal and Negi, 2018). The result obtained in the presented work is less conservative due to the application of triple Lyapunov and reciprocal convex approach than the available results.

Numerical instances prove the effectiveness of the obtained outcomes.

The script is structured along these lines. Section “Notations” shows the system considered. Section “Problem formulation” portrays the global asymptotic stability of the discrete time-delay system in the context of LMI-based conditions employing saturation and $H_{\infty}$ controller to stabilize the system. In Section “Main results,” numerical instances prove the advantage of the obtained results.

Notations

The notations used in this paper are as follows:

$ℜ^{q \times n}$ is the set of $q \times n$ real matrices, $ℜ^{q}$ denotes set of $q \times 1$ real matrices, $P$ $> 0 (\geq 0)$ denotes that $P$ is real symmetric and positive-definite (positive semidefinite) matrix, $0$ is a null matrix or null vector, $I$ is an identity matrix with appropriate dimension, $λ_{max} (\bar{A})$ denotes maximum eigenvalue of any given matrix $\bar{A}$ , symbol ‘*’ represents symmetric terms in symmetric matrix, ||.|| represents norm of matrix or vector, $diag {a_{1}, a_{2} . . . a_{n}}$ denotes block diagonal matrix with diagonal elements $a_{1}, a_{2} . . . a_{n}$ , and $‖ w ‖_{2} = \sqrt{\sum_{i = 0}^{\infty} {‖ w (i) ‖}^{2}}$ is $l_{2}$ norm of signal $w (i) \in l_{2} {(0, \infty)}$ , if $‖ w ‖_{2} < \infty$ .

Problem formulation

Consider a linear discrete time-varying delay system

x (k + 1) = Ax (k) + A_{d} x (k - g (k)) + D_{Θ} w (k) + Bu (k)

(1a)

y (k) = Cx (k)

(1b)

Z (k) = C_{p} x (k) + D_{p} w (k)

(1c)

where $x (k) \in ℜ^{n}$ , $u (k) \in ℜ^{m}$ , $y (k) \in ℜ^{p}$ , and $Z (k) \in ℜ^{m}$ are state, input, measured, and controlled output vectors, respectively. The external interference in system is denoted as $w (k) \in ℜ^{q}$ . Matrices $A$ , $A_{d}$ , $B$ , $C$ , $D_{Θ}$ , $C_{p}$ , $D_{p}$ are constant matrices of appropriate dimensions, and $g (k)$ is the time-varying delay satisfying

g_{1} \leq g (k) \leq g_{2}

(2)

where $g_{1}$ and $g_{2}$ are constant non-negative integers representing the lower and upper bounds, respectively.

The controller to stabilize the output of system (1) is considered as

x_{va} (k + 1) = A_{va} x_{va} (k) + B_{va} y (k)

(3a)

v_{va} (k) = C_{va} x_{va} (k) + D_{va} Cx (k)

(3b)

where $x_{va} (k) \in ℜ^{n_{c}}$ and $v_{va} (k) \in ℜ^{n_{c}}$ denote the controller state and output vectors, respectively. Matrices $A_{va}$ , $B_{va}$ , $C_{va}$ , $D_{va}$ are constant controller matrices. To stabilize system (1) without control saturation, a controller (3) is required to be designed. The architecture of anti-windup scheme is shown in Figure 1.

The amplitude constraint is applied to the input vector $u (k)$ as under

- u_{0 (i)} \leq u_{(i)} \leq u_{0 (i)}

(4)

where $u_{0 (i)} > 0$ , $i = 1, . . ., m$ , is the control amplitude bounds. Therefore, the actual control signal injected within the plant is given by

u (k) = sat (v_{va} (k)) = sat (C_{va} x_{va} (k) + D_{va} Cx (k))

(5)

The saturation nonlinearities are defined as

sat {(v_{va} (k))}_{(i)} = {\begin{matrix} - u_{0 (i)} & if & v_{va (i)} < - u_{0 (i)} \\ v_{va (i)} & if & - u_{0 (i)} \leq v_{va (i)} \leq u_{0 (i)}, \\ u_{0 (i)} & if & v_{va (i)} > u_{0 (i)} \end{matrix} i = 1, 2, . . . m

(6)

Substituting equation (5) in equation (1), we get

\begin{matrix} x (k + 1) = Ax (k) + A_{d} x (k - g (k)) + D_{Θ} w (k) \\ + B (C_{va} x_{va} (k) + D_{va} Cx (k)) \\ - B Ψ_{l} (C_{va} x_{va} (k) + D_{va} Cx (k)) \end{matrix}

(7)

where

\hat{Ψ} (v) = v - sat (v)

(8)

In this case, $\hat{Ψ} (v)$ corresponds to decentralized deadzone nonlinearity.

Introducing an anti-windup term $F_{va} (sat (v_{va} (k)) - v_{va} (k)$ to the controller to mitigate the undesirable effects of the windup caused by input saturation, equation (3a) can be modified as

\begin{matrix} x_{va} (k + 1) = A_{va} x_{va} (k) + B_{va} C x (k) - F_{va} \hat{Ψ} (C_{va} x_{va} (k) + D_{va} Cx (k)) \end{matrix}

(9)

where $F_{va}$ is anti-windup gain matrix.

An extended state vector can be expressed as

Ψ (k + 1) = [\begin{matrix} x (k + 1) \\ x_{va} (k + 1) \end{matrix}] \in ℜ^{n + n_{va}},

(10)

and the matrices

\begin{matrix} \bar{A} = [\begin{matrix} A + B D_{va} C & B C_{va} \\ B_{va} C & A_{va} \end{matrix}], {\bar{A}}_{d} = [\begin{matrix} A_{d} & 0 \\ 0 & 0 \end{matrix}], \\ \bar{B} = [\begin{matrix} B \\ 0 \end{matrix}], Ξ = [\begin{matrix} 0 \\ I_{n_{va}} \end{matrix}], \bar{K} = [\begin{matrix} D_{va} C & C_{va} \end{matrix}], \\ {\bar{D}}_{Θ} = [\begin{matrix} D_{Θ} & 0 \end{matrix}], W (ϑ) = [\begin{matrix} w (ϑ) \\ 0 \end{matrix}] \in ℜ^{q + 1} . \end{matrix}

(11)

Utilizing equations (1)–(11), the closed-loop system can be represented as

\begin{matrix} Ψ (k + 1) = \bar{A} Ψ (k) + {\bar{A}}_{d} Ψ (k - g (k)) - (\bar{B} + Ξ F_{va}) \\ \hat{Ψ} (\bar{K} Ψ (k)) + {\bar{D}}_{Θ} W (k) \end{matrix}

(12)

The initial conditions are defined as follows

Ψ_{0} = Ç_{Ψ} (k), k = - g_{2}, - g_{2} + 1, . . . ., 0 be \oint (k, Ψ_{0})

The basin of attraction of the origin of system (12) is defined as

ℶ \overset{Δ}{=} {Ç_{Ψ} (k), k = - g_{2}, - g_{2} + 1, . . . ., 0 : \underset{k \to \infty}{lim \oint (k, Ψ_{0}) = 0}}

(13)

An estimate of attraction basin is given by $X_{μ} \subset ℶ$ where $μ$ is the largest possible scalar such that asymptotic stability of the closed-loop system (12) is ensured for all time-varying delays satisfying equation (2)

X_{μ} \overset{Δ}{=} {Ç_{Ψ} (k), k = - g_{2}, - g_{2} + 1, . . . ., 0 : \underset{k \to \infty}{max | Ç_{Ψ} (k) | \leq μ}}

(14)

The saturation nonlinearity is tackled using sector conditions. Consider a matrix $G \in ℜ^{m \times (n + n_{c})}$ and define the following polyhedral set as

ℓ \overset{Δ}{=} {Ψ \in ℜ^{(η + η_{c})}; - u_{0 (i)} \leq ({\bar{K}}_{(i)} - G_{(i)} Ψ (k)) \leq u_{0 (i)}, i = 1, 2, . . . . m}

(15)

Referring to Da Silva and Tarbouriech (2006), if $Ψ (k) \in ℓ$ , then

Ω = {\hat{Ψ}}^{T} (\bar{K} Ψ (k)) {\hat{Ψ} (\bar{K} Ψ (k)) - G Ψ (k)} \leq 0

(16)

where $Đ \in ℜ^{m \times m}$ is a positive definite diagonal matrix.

The main results of the paper are established in view of the following Lemmas.

Lemma 1. (Kwon et al., 2013; Sun et al., 2010; Wang et al., 2013) For any constant matrix $W \in ℜ^{m \times m}$ with $W = W^{T} > 0$ , positive integers $l_{1}$ and $l_{2}$ satisfying $l_{1} < l_{2}$ , vector function $ω : {l_{1}, l_{1} + 1, . . ., l_{2}} \to ℜ^{m}$ , following inequalities hold

\begin{matrix} (l_{2} - l_{1} + 1) \sum_{i = l_{1}}^{l_{2}} ω^{T} (i) W ω (i) \geq {(\sum_{i = l_{1}}^{l_{2}} ω (i))}^{T} W (\sum_{i = l_{1}}^{l_{2}} ω (i)) \\ \frac{(l_{2}^{2} - l_{1}^{2})}{2} \sum_{i = - l_{2}}^{- l_{1} - 1} \sum_{j = k + i}^{k - 1} ω^{T} (j) W ω (j) \geq {(\sum_{i = - l_{2}}^{- l_{1} - 1} \sum_{j = k + i}^{k - 1} ω (j))}^{T} \end{matrix}

(17)

Lemma 2. (Park et al., 2011) For any vectors $σ_{1}, σ_{2}$ matrices $T, S$ and real numbers $α_{1} \geq 0$ , $α_{2} \geq 0$ satisfying

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

(18a)

σ_{k} = 0 if α_{k} = 0, (k = 1, 2)

(18b)

then

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

(18c)

Lemma 3. (Li et al., 2009) For constant matrices $Ω$ , $Ξ_{1}$ , and $Ξ_{2}$ of appropriate dimensions, $α \in [0, 1]$ , then $Ω + [α Ξ_{1} + (1 - α) Ξ_{2}] < 0$ holds, if the inequalities $Ω + Ξ_{1} < 0$ and $Ω + Ξ_{2} < 0$ hold simultaneously.

Lemma 4. (Liu and Zhang, 2012) If there exist symmetric matrices $R > 0$ and $Ξ$ of appropriate dimension, constant matrix $Γ$ , then the following statements are equivalent: (1) $Ξ - Γ^{T} R Γ < 0$ (2) and if there exists a matrix of appropriate dimension $Λ$ , the following is true

[\begin{matrix} Ξ + Γ^{T} Λ + Λ^{T} Γ & Λ^{T} \\ * & - R \end{matrix}] < 0

(19)

Main results

The main results are stated as follows.

Delay-dependent stability analysis in the absenceof disturbance

Theorem 1. For given positive integers $g_{1}$ and $g_{2}$ satisfying $0 < g_{1} < g_{2}$ , if there exist $14 (n_{c} + n) \times (n_{c} + n)$ matrices $K_{i} (i = 1, 2, 3, 4)$ , positive definite symmetric matrices $0 < P \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < P_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < Q \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < S \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < T \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < X_{k} (k = 1, \dots, 6) \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , a diagonal positive definite matrix $L \in ℜ^{m \times m}$ , $G \in ℜ^{m \times (n + n_{va})}$ , $H \in ℜ^{n_{va} \times m}$ , matrices $S_{k} (k = 1, 2, 3)$ of appropriate dimensions satisfying the set of LMIs (20)–(22)

\oint_{k} = [\begin{matrix} Z_{k} \end{matrix}] \geq 0, k = 1, 2, 3

(20)

[\begin{matrix} P & K_{i}^{T} - G_{i}^{T} \\ K_{i} - G_{i} & u_{0 (i)}^{2} \end{matrix}] \geq 0

(21)

[\begin{matrix} £ 1 + F_{i} K^{T} + K F_{i}^{T} & K_{i} & K_{j} \\ * & - Z_{1} & 0 \\ * & * & - Z_{2} \end{matrix}] < 0, i = 1, 2; j = 3, 4

(22)

where $Z_{k} = [\begin{matrix} R_{k} & S_{k} \\ S_{k}^{T} & R_{k} \end{matrix}], k = 1, 2, 3$ with

£ 1 = [\begin{matrix} {\bar{Λ}}_{11} & 0 & T & 0 & g_{1} Q & 0 & 0 & G^{T} & {\bar{A}}^{T} & \frac{g_{1}^{2}}{2} {\hat{A}}^{T} & g_{1} {\hat{A}}^{T} & \frac{(g_{2} - g_{1})}{2} {\hat{A}}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\hat{A}}^{T} & (g_{2} - g_{1}) {\hat{A}}^{T} \\ * & {\bar{Λ}}_{22} & R_{3} - S_{3} & R_{3} - S_{3} & 0 & 0 & R & 0 & {\bar{A}}_{d}^{T} & \frac{g_{1}^{2}}{2} {\bar{A}}_{d}^{T} & g_{1} {\bar{A}}_{d}^{T} & \frac{(g_{2} - g_{1})}{2} {\bar{A}}_{d}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\bar{A}}_{d}^{T} & (g_{2} - g_{1}) {\bar{A}}_{d}^{T} \\ * & * & Λ_{33} & S_{3} & 0 & R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - P_{2} - R_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & - Q - S & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - 2 L & - {\hat{B}}^{T} & - \frac{g_{1}^{2}}{2} {\hat{B}}^{T} & - g_{1} {\hat{B}}^{T} & - \frac{(g_{2} - g_{1})}{2} {\hat{B}}^{T} & - \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\hat{B}}^{T} & - (g_{2} - g_{1}) {\hat{B}}^{T} \\ * & * & * & * & * & * & * & * & £ 1_{9.9} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & £ 1_{10.10} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & £ 1_{11.11} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & * & £ 1_{12.12} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & * & * & £ 1_{13.13} & 0 \\ * & * & * & * & * & * & * & * & * & * & * & * & * & £ 1_{14.14} \end{matrix}] < 0

(23)

where

\begin{matrix} K = [\begin{matrix} K_{1} & K_{2} & K_{3} & K_{4} \end{matrix}], {\bar{Λ}}_{11} = - P + P_{1} + P_{3} \\ + (g_{2} - g_{1}) P_{3} - g_{1}^{2} Q + g_{1}^{2} S + (g_{2} - g_{1})^{2} R_{2} - T, \\ {\bar{Λ}}_{22} = - P_{3} \\ - R - 2 R_{3} + S_{3} + S_{3}^{T}, Λ_{33} = - P_{1} - P_{2} - R - T - R_{3}, \\ £ 1_{9.9} = - 2 X_{1} + X_{1} P X_{1}, £ 1_{10, 10} = - 2 X_{2} + X_{2} Q X_{2}, \\ £ 1_{11, 11} = - 2 X_{3} + X_{3} T X_{3}, £ 1_{12, 12} = - 2 X_{4} + X_{4} R X_{4}, \\ £ 1_{13, 13} = - 2 X_{5} + X_{5} R_{1} X_{5}, £ 1_{14, 14} = - 2 X_{6} + X_{6} R_{3} X_{6}, \\ (\bar{A} - I) = \hat{A}, (\bar{B} \bar{L} + Ξ H) = \hat{B}, \end{matrix}

{\bar{F}}_{1}^{T} = [\begin{matrix} I_{n} 0_{n . 4 n} - I_{n} 0_{n . 8 n} \\ 0_{n . 5 n} 0_{n . 9 n} \\ 0_{n . 5 n} - I_{n} 0_{n . 8 n} \\ 0_{n . 5 n} 0_{n . 9 n} \end{matrix}], {\bar{F}}_{2}^{T} = [\begin{matrix} 0_{n . 5 n} 0_{n . 9 n} \\ I_{n} 0_{n . 5 n} - I_{n} 0_{n . 7 n} \\ 0_{n . 5 n} 0_{n . 9 n} \\ 0_{n . 6 n} - I_{n} 0_{n . 7 n} \end{matrix}]

(24)

then for the gain matrix $F_{va} = H L^{- 1}$ , the closed-loop system (12) is asymptotically stable. An estimated attraction basin for equation (12)is given by $ℶ_{μ} \leq 1$ where $ℶ_{μ} = μ^{2} H$

\begin{matrix} H = λ_{max} (P) + g_{1} λ_{max} (P_{1}) + (g_{2} - g_{1}) λ_{max} (P_{2}) \\ + (g (k) + (g_{2} - g_{1}) \frac{(g_{2} + g_{1} - 1)}{2}) λ_{max} (P_{3}) \\ + \frac{g_{1}^{2}}{2} (g_{1} + 1) λ_{max} (S) + \frac{g_{1}^{3}}{12} (g_{1} + 1) (g_{1} + 2) λ_{max} (Q) \\ + \frac{(g_{2} - g_{1}) (g_{2} - g_{1} + 1) (g_{2} - g_{1} + 2)}{12} \\ \times λ_{max} (R) + \frac{g_{1}^{2}}{2} (g_{1} + 1) λ_{max} (T) + \frac{(g_{2}^{2} - g_{1}^{2})}{12} \\ (g_{2} (g_{2} + 1) (g_{2} + 2) - g_{1} (g_{1} + 1) (g_{1} + 2)) λ_{max} \\ \times (R_{1}) + \frac{{(g_{2} - g_{1})}^{2}}{2} (g_{2} + g_{1} + 1) λ_{max} (R_{2}) \\ + \frac{{(g_{2} - g_{1})}^{2}}{2} (g_{2} + g_{1} + 1) λ_{max} (R_{3}) \end{matrix}

(25)

Proof. Define

∐ (k) = Ψ (k + 1) - Ψ (k)

(26a)

= (A - I) Ψ (k) + {\bar{A}}_{d} Ψ (k - g (k)) - (\bar{B} - Ξ F_{va}) \hat{Ψ} (\bar{K} Ψ (k))

(26b)

Consider an LKF given by

V (Ψ (k)) = \sum_{a = 1}^{4} V_{a} (Ψ (k)),

(27)

where

V_{1} (Ψ (k)) = Ψ^{T} (k) P Ψ (k)

(28a)

\begin{matrix} V_{2} (Ψ (k)) = \sum_{a = k - g_{1}}^{k - 1} Ψ^{T} (a) P_{1} Ψ (a) + \sum_{a = k - g_{2}}^{k - g_{1} - 1} Ψ^{T} (a) P_{2} Ψ (a) \\ + \sum_{a = k - g (k)}^{k - 1} Ψ^{T} (a) P_{3} Ψ (a) \\ + \sum_{a = k - g_{2} + 1}^{k - g_{1}} \sum_{b = a}^{k - 1} Ψ^{T} (a) P_{3} Ψ (a) \end{matrix}

(28b)

\begin{matrix} V_{3} (Ψ (k)) = \frac{g_{1}^{2}}{2} \sum_{a = - g_{1}}^{- 1} \sum_{b = a}^{0} \sum_{c = k + b}^{k - 1} ∐^{T} (c) Q ∐ (c) \\ + \frac{1}{2} \sum_{a = k - g_{2}}^{k - g_{1} - 1} \sum_{b = a}^{k - g_{1} - 1} \sum_{c = b}^{k - g_{1} - 1} ∐^{T} (c) R ∐ (c) \\ \frac{g_{2}^{2} - g_{1}^{2}}{2} \sum_{a = - g_{2}}^{- g_{1} - 1} \sum_{b = a}^{0} \sum_{c = k + b}^{k - 1} ∐^{T} (c) R_{1} ∐ (c) \end{matrix}

(29)

\begin{matrix} V_{4} (Ψ (k)) = g_{1} \sum_{a = - g_{1}}^{- 1} \sum_{b = k + a}^{k - 1} Ψ^{T} (b) Ψ S (b) \\ g_{1} \sum_{a = - g_{1}}^{- 1} \sum_{b = k + a}^{k - 1} ∐^{T} (b) T ∐ (b) + (g_{2} - g_{1}) \\ \times \sum_{a = - g_{2}}^{- g_{1} - 1} \sum_{b = k + a}^{k - 1} Ψ^{T} (b) R_{2} (b) Ψ (b) \\ + (g_{2} - g_{1}) \sum_{a = - g_{2}}^{- g_{1} - 1} \sum_{b = k + a}^{k - 1} ∐^{T} (b) R_{3} ∐ (b) \end{matrix}

(30)

The forward difference of equation (27) along the trajectories of the system (12) and employing Lemma 1 and Lemma 2 and equation (20) on $Δ V_{3} (Ψ (k))$ and $Δ V_{4} (Ψ (k))$ , $Δ V (Ψ (k))$ is given as

\begin{matrix} Δ V (Ψ (k)) \leq φ^{T} (k) Π φ (k) - þ^{T} (k) [\begin{matrix} σ (k) I & ε (k) I \\ - σ (k) I & 0 \\ 0 & - ε (k) I \end{matrix}] \\ [\begin{matrix} R_{1} & S_{1} \\ * & R_{1} \end{matrix}] {[\begin{matrix} σ (k) I & ε (k) I \\ - σ (k) I & 0 \\ 0 & - ε (k) I \end{matrix}]}^{T} \end{matrix}

(31)

where

\begin{matrix} þ^{T} (k) = [\begin{matrix} Ψ^{T} (k) \frac{1}{(g (k) - g_{1})} [\sum_{a = k - g (k)}^{k - g_{1} - 1} Ψ^{T} (a)] \frac{1}{(g_{2} - g (k))} [\sum_{a = k - g_{2}}^{k - g (k) - 1} Ψ^{T} (a)] \end{matrix}], \end{matrix}

$σ (k) = (g (k) - g_{1})$ , $ε (k) = (g_{2} - g (k))$ , $Ω$ is given by equation (16) and

\begin{matrix} Π^{T} (k) = [\begin{matrix} Ψ^{T} (k) Ψ^{T} (k - g (k)) Ψ^{T} (k - g_{1}) Ψ^{T} (k - g_{2}) \sum_{a = k - g_{1}}^{k - 1} Ψ^{T} (a) \end{matrix} \\ \begin{matrix} \frac{1}{(g (k) - g_{1})} \sum_{a = k - g (k)}^{k - g_{1} - 1} Ψ^{T} (a) \frac{1}{(g_{2} - g (k))} \sum_{a = k - g_{2}}^{k - g (k) - 1} Ψ^{T} (a) {\hat{Ψ}}^{T} (\bar{K} Ψ (k)) \end{matrix}] \end{matrix}

(32)

Π = [\begin{matrix} Λ_{11} & Λ_{12} & T & 0 & g_{1} Q & 0 & 0 & Λ_{18} \\ * & Λ_{22} & R_{3} - S_{3} & R_{3} - S_{3} & 0 & 0 & R & Λ_{28} \\ * & * & Λ_{33} & S_{3} & 0 & R & 0 & 0 \\ * & * & * & - P_{2} - R_{3} & 0 & 0 & 0 & 0 \\ * & * & * & * & - Q - S & 0 & 0 & 0 \\ * & * & * & * & * & - R & 0 & 0 \\ * & * & * & * & * & * & - R & - 2 Đ \\ * & * & * & * & * & * & * & Λ_{88} \end{matrix}]

(33)

\begin{matrix} Λ_{11} = {\bar{Λ}}_{11} + {\bar{A}}^{T} P \bar{A} + \frac{g_{1}^{4}}{4} {\hat{A}}^{T} Q \hat{A} + \frac{{(g_{2} - g_{1})}^{2}}{4} {\hat{A}}^{T} R \hat{A} \\ + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} {\hat{A}}^{T} R_{1} {\hat{A}}^{T} + g_{1}^{2} {\hat{A}}^{T} T \hat{A} + {(g_{2} - g_{1})}^{2} {\hat{A}}^{T} R_{3} \hat{A} \\ Λ_{12} = {\bar{A}}^{T} P {\bar{A}}_{d} + \frac{g_{1}^{4}}{4} {\hat{A}}^{T} Q {\bar{A}}_{d} + \frac{{(g_{2} - g_{1})}^{2}}{4} {\hat{A}}^{T} R {\bar{A}}_{d} \\ + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} {\hat{A}}^{T} R_{1} {\bar{A}}_{d} + g_{1}^{2} {\hat{A}}^{T} T {\bar{A}}_{d} + {(g_{2} - g_{1})}^{2} {\hat{A}}^{T} R_{3} {\bar{A}}_{d} \\ Λ_{22} = {\bar{Λ}}_{22} + \frac{g_{1}^{2}}{4} ({\bar{A}}_{d}^{T} Q {\bar{A}}_{d}) + \frac{{(g_{2} - g_{1})}^{2}}{4} {\bar{A}}_{d}^{T} R {\bar{A}}_{d} + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} \\ \times {\bar{A}}_{d}^{T} R_{1} {\bar{A}}_{d} + g_{1}^{2} {\bar{A}}_{d}^{T} T {\bar{A}}_{d} + {\bar{A}}_{d}^{T} P {\bar{A}}_{d} + {(g_{2} - g_{1})}^{2} {\bar{A}}_{d}^{T} R_{3} {\bar{A}}_{d} \\ Λ_{18} = - {\bar{A}}^{T} P (\bar{B} + Ξ F_{va}) - {\hat{A}}^{T} \\ [\frac{g_{1}^{4}}{4} Q + \frac{{(g_{2} - g_{1})}^{2}}{4} R + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} R_{1} + g_{1}^{2} T + {(g_{2} - g_{1})}^{2} R_{3}] \\ (\bar{B} + Ξ F_{va}) + G^{T} Đ, \\ Λ_{28} = {\bar{A}}_{d}^{T} P (\bar{B} + Ξ F_{va}) - {\bar{A}}_{d}^{T} \\ [\frac{g_{1}^{4}}{4} Q + \frac{{(g_{2} - g_{1})}^{2}}{4} R + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} R_{1} + g_{1}^{2} T + {(g_{2} - g_{1})}^{2} R_{3}] \\ \times (\bar{B} + Ξ F_{va}) \\ Λ_{88} = {(\bar{B} + Ξ F_{va})}^{T} P (\bar{B} + Ξ F_{va}) + {(\bar{B} + Ξ F_{va})}^{T} \\ [\frac{g_{1}^{4}}{4} Q + \frac{{(g_{2} - g_{1})}^{2}}{4} R + \frac{{(g_{2}^{2} - g_{1}^{2})}^{2}}{4} R_{1} + g_{1}^{2} T + {(g_{2} - g_{1})}^{2} R_{3}] \\ \times (\bar{B} + Ξ F_{va}) - 2 Đ \end{matrix}

Using Schur’s complement and then pre and post multiplying equation (33) by

$d i a g {\begin{matrix} I & I & I & I & I & I & I & Ð^{- 1} & I & I & I & I & I & I \end{matrix}}$ , we obtain equation (34)

M 1 = [\begin{matrix} {\bar{Λ}}_{11} & 0 & T & 0 & g_{1} Q & 0 & 0 & G^{T} & {\bar{A}}^{T} & \frac{g_{1}^{2}}{2} {\hat{A}}^{T} & g_{1} {\hat{A}}^{T} & \frac{(g_{2} - g_{1})}{2} {\hat{A}}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\hat{A}}^{T} & (g_{2} - g_{1}) {\hat{A}}^{T} \\ * & {\bar{Λ}}_{22} & R_{3} - S_{3} & R_{3} - S_{3} & 0 & 0 & R & 0 & {\bar{A}}_{d}^{T} & \frac{g_{1}^{2}}{2} {\bar{A}}_{d}^{T} & g_{1} {\bar{A}}_{d}^{T} & \frac{(g_{2} - g_{1})}{2} {\bar{A}}_{d}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\bar{A}}_{d}^{T} & (g_{2} - g_{1}) {\bar{A}}_{d}^{T} \\ * & * & Λ_{33} & S_{3} & 0 & R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - P_{2} - R_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & - Q - S & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - 2 L & Y & \frac{g_{1}^{2}}{2} Y & g_{1} Y & \frac{(g_{2} - g_{1})}{2} Y & \frac{(g_{2}^{2} - g_{1}^{2})}{2} Y & (g_{2} - g_{1}) Y \\ * & * & * & * & * & * & * & * & - P_{1}^{- 1} & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & - Q^{- 1} & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & - T^{- 1} & 0 & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & * & - R^{- 1} & 0 & 0 \\ * & * & * & * & * & * & * & * & * & * & * & * & - R_{1}^{- 1} & 0 \\ * & * & * & * & * & * & * & * & * & * & * & * & * & - R_{3}^{- 1} \end{matrix}] < 0

(34)

where $(\bar{B} \bar{L} + Ξ H) = \hat{B}$ , $(\bar{A} - I) = \hat{A}$ , $Đ^{- 1} = L$ , and $F_{va} = H L^{- 1}$ .

For the system given in equation (12) to be asymptotically stable, the constraint $Δ V (Ψ (k)) < 0$ for $φ (k) \neq 0$ is retrieved from equation (31). Thus, it can be written as

Δ V (Ψ (k)) < Ψ^{T} (k) [M 1 - F (k) [\begin{matrix} Z_{1} & 0 \\ 0 & Z_{2} \end{matrix}] F^{T} (k)] Ψ (k)

(35)

where $M 1$ is presented in equation (34) and

F^{T} (k) = [\begin{matrix} σ (k) I & 0_{n . 4 n} & - σ (k) I & 0 & 0_{n . 7 n} \\ ε (k) I & 0_{n . 4 n} & 0 & - ε (k) I & 0_{n . 7 n} \\ 0 & 0_{n . 4 n} & - σ (k) I & 0 & 0_{n . 7 n} \\ 0 & 0_{n . 4 n} & 0 & - ε (k) I & 0_{n . 7 n} \end{matrix}]

(36a)

From LMI (22), it is implied that

M 1 + F_{i} K^{T} + K F_{i}^{T} < 0 for i = 1, 2

(36b)

Employing Lemma 3 in equation (36b), it is ensured that

M 1 + F (k) K^{T} + K F^{T} (k) < 0

(37)

Hence, LMI (22) makes the following inequality true

\begin{matrix} [\begin{matrix} M 1 + F (k) K^{T} + K F^{T} (k) & K_{i} & K_{j} \\ * & - Z_{1} & 0 \\ * & * & - Z_{2} \end{matrix}] < 0, \\ i = 1, 2; j = 3, 4 \end{matrix}

(38)

Thus, in the light of Lemma 4, it follows from equation (38) that

M 1 + F (k) [\begin{matrix} Z_{1} & 0 \\ 0 & Z_{2} \end{matrix}] F^{T} (k) < 0

(39)

It can be seen that $Δ V (Ψ (k)) < 0$ for $φ (k) \neq 0$ and equations (20) and (21) offer constraints for the asymptotic stability of the system (12).

The satisfaction of constraints stated in equation (21) indicates that set $\partial (P) = {Ψ \in ℜ^{n}, Ψ^{T} P Ψ \leq 1}$ is included in the polyhedral set $ℓ (H)$ as defined in equation (15). It can be proved that $\partial (P) = {Ψ \in ℜ^{n}, Ψ^{T} P Ψ \leq 1}$ is equivalent to (Boyd et al., 1994)

P - (K_{i}^{T} - G_{i}^{T}) {(K_{i} - G)}_{i} u_{0 (i)}^{- 2} > 0, i = 1, 2, . . . . m

(40)

Pre and post multiplication of equation (40) by $Ψ^{T}$ and $Ψ$ , respectively, follows that $Ψ \in ℓ (K - G)$ for all $Ψ \in \partial (P)$ . The inequality (21) is obtained by implementing Schur’s complement in equation (40).

The relation (40) shows that the set $\partial (P, 1) \overset{Δ}{=} {Ψ \in ℜ^{n + n_{c}}; Ψ^{T} P Ψ \leq 1}$ is included in the polyhedral set $ℓ$ defined in equation (15). Thus, $\forall Ψ (k) \in (P, 1)$ ; thereby, $\hat{Ψ} (\bar{K} Ψ (k)) = \bar{K} Ψ (k) - sat (\bar{K} Ψ (k))$ satisfies the sector condition (16).

Remark 1. Primary guess for a positive definite symmetric matrix $X_{1}$ could be taken as $X_{1} = α_{c} (A^{T} A + I)^{- 1}$ .

Remark 2. Although the complexities of space and time have increased, the conservativeness of stability is reduced in terms of discrete delay system, as seen in previous papers. There is a tradeoff between space and time complexities and conservativeness. The dimension/size of the undertaken system is 14×14, as seen in equation (23). It is a large order system; it occupies sufficient RAM/system memories. Since the simulation is performed on 2.26 GHz, exploring the optimum solution of the defined problem takes significant time. To optimize space and time complexities for the defined problem (12), a high-end computation platform is required.

As a direct consequence of Theorem 1, we have the following result.

Corollary 1. For given positive integers $g_{1}$ and $g_{2}$ satisfying $0 < g_{1} < g_{2}$ , if there exist $14 (n_{c} + n) \times (n_{c} + n)$ matrices $K_{i} (i = 1, 2, 3, 4)$ , positive definite symmetric matrices $0 < P \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < P_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < Q \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < S \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < T \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < X_{k} (k = 1, \dots, 6) \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , a diagonal positive definite matrix $L \in ℜ^{m \times m}$ , $G \in ℜ^{m \times (n + n_{va})}$ , $H \in ℜ^{n_{va} \times m}$ , matrices $S_{k} (k = 1, 2, 3)$ satisfying LMIs (20)–(22), the closed-loop system (12) is globally asymptotically stable for the gain matrix $F_{va} = H L^{- 1}$ .

Proof. Choosing $G = \bar{K}$ , one can see that equation (15) is automatically met for all $Ψ (k) \in ℜ^{n + n_{c}}$ . Substituting $G = \bar{K}$ into equation (23), we obtain global asymptotic stability condition (41) defined as $M 2$ , where $M 2 = M 1$ when all elements of $M 2$ are equal to $M 1$ except $M 1_{1, 8}$ , where

M 2_{1, 8} = {\bar{K}}^{T}

(41)

This completes the proof. The global asymptotic stability is valid only when the open loop system is asymptotically stable (Da Silva and Tarbouriech, 2006).

Delay-dependent stability analysis in the presenceof disturbance

Theorem 2. For given positive integers $g_{1}$ and $g_{2}$ satisfying $0 < g_{1} < g_{2}$ , if there exists $16 (n_{c} + n) \times (n_{c} + n)$ matrices $K_{i} (i = 1, 2, 3, 4)$ , positive definite symmetric matrices $0 < P \in ℜ^{(n + n_{v a}) \times (n + n_{v a})}$ , $0 < P_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < Q \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < R_{k} (k = 1, 2, 3) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < S \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < T \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , $0 < X_{k} (k = 1, . . ., 6) \in ℜ^{(n + n_{va}) \times (n + n_{va})}$ , a diagonal positive definite matrix $L \in ℜ^{m \times m}$ , $G \in ℜ^{m \times (n + n_{va})}$ , $H \in ℜ^{n_{va} \times m}$ , matrices $S_{k} (k = 1, 2, 3)$ of proper dimensions convincing the following set of LMIs (42)–(44)

\oint_{k} = [Z_{k}] \geq 0, k = 1, 2, 3

(42)

[\begin{matrix} P & K_{i}^{T} - G_{i}^{T} \\ K_{i} - G_{i} & {hu}_{0 (i)}^{2} \end{matrix}] \geq 0

(43)

[\begin{matrix} W 1 + F_{i} K^{T} + K F_{i}^{T} & K_{i} & K_{j} \\ * & - Z_{1} & 0 \\ * & * & - Z_{2} \end{matrix}] < 0, i = 1, 2; j = 3, 4

(44)

with $h = 1 / (1 + ℘^{2} {\hat{σ}}^{2})$

W 1 = [\begin{matrix} {\bar{Λ}}_{11} & 0 & T & 0 & g_{1} Q & 0 & 0 & G^{T} & 0 & {\bar{A}}^{T} & \frac{g_{1}^{2}}{2} {\hat{A}}^{T} & g_{1} {\hat{A}}^{T} & \frac{(g_{2} - g_{1})}{2} {\hat{A}}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\hat{A}}^{T} & (g_{2} - g_{1}) {\hat{A}}^{T} & C_{p}^{T} \\ * & {\bar{Λ}}_{22} & R_{3} - S_{3} & R_{3} - S_{3} & 0 & 0 & R & 0 & 0 & {\bar{A}}_{d}^{T} & \frac{g_{1}^{2}}{2} {\bar{A}}_{d}^{T} & g_{1} {\bar{A}}_{d}^{T} & \frac{(g_{2} - g_{1})}{2} {\bar{A}}_{d}^{T} & \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\bar{A}}_{d}^{T} & (g_{2} - g_{1}) {\bar{A}}_{d}^{T} & 0 \\ * & * & Λ_{33} & S_{3} & 0 & R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & - P_{2} - R_{3} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & - Q - S & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & - R & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ * & * & * & * & * & * & * & - 2 L & 0 & - {\hat{B}}^{T} & - \frac{g_{1}^{2}}{2} {\hat{B}}^{T} & - g_{1} {\hat{B}}^{T} & - \frac{(g_{2} - g_{1})}{2} {\hat{B}}^{T} & - \frac{(g_{2}^{2} - g_{1}^{2})}{2} {\hat{B}}^{T} & - (g_{2} - g_{1}) {\hat{B}}^{T} & 0 \\ * & * & * & * & * & * & * & * & - ℘^{2} I & {\bar{D}}_{Θ}^{T} & \frac{g_{1}^{2}}{2} {\bar{D}}_{Θ}^{T} & g_{1} {\bar{D}}_{Θ}^{T} & \frac{(g_{2} - g_{1})}{2} {\bar{D}}_{Θ}^{T} & \frac{(g_{2}^{2} - g_{1}^{2}) {\bar{D}}_{Θ}^{T}}{2} & - (g_{2} - g_{1}) {\bar{D}}_{Θ}^{T} & D_{p}^{T} \\ * & * & * & * & * & * & * & * & * & £ 1_{9.9} & 0 & 0 & 0 & 0 & 0 & * \\ * & * & * & * & * & * & * & * & * & * & £ 1_{10.10} & 0 & 0 & 0 & 0 & * \\ * & * & * & * & * & * & * & * & * & * & * & £ 1_{11.11} & 0 & 0 & 0 & * \\ * & * & * & * & * & * & * & * & * & * & * & * & £ 1_{12.12} & 0 & 0 & * \\ * & * & * & * & * & * & * & * & * & * & * & * & * & £ 1_{13.13} & 0 & * \\ * & * & * & * & * & * & * & * & * & * & * & * & * & * & £ 1_{14.14} & * \\ * & * & * & * & * & * & * & * & * & * & * & * & * & * & * & - I \end{matrix}] < 0,

(45)

\begin{matrix} K = [\begin{matrix} K_{1} & K_{2} & K_{3} & K_{4} \end{matrix}], \\ {\bar{Λ}}_{11} = - P + P_{1} + P_{3} + (g_{2} - g_{1}) P_{3} - g_{1}^{2} Q + g_{1}^{2} S \\ + (g_{2} - g_{1})^{2} R_{2} - T, \\ {\bar{Λ}}_{22} = - P_{3} - R - 2 R_{3} + S_{3} + S_{3}^{T}, Λ_{33} = - P_{1} - P_{2} \\ - R - T - R_{3}, \end{matrix}

{\bar{F}}_{1}^{T} = [\begin{matrix} I_{n} 0_{n . 4 n} - I_{n} 0_{n . 10 n} \\ 0_{n . 5 n} 0_{n . 11 n} \\ 0_{n . 5 n} - I_{n} 0_{n . 10 n} \\ 0_{n . 5 n} 0_{n . 11 n} \end{matrix}], {\bar{F}}_{2}^{T} = [\begin{matrix} 0_{n . 5 n} 0_{n . 11 n} \\ I_{n} 0_{n . 5 n} - I_{n} 0_{n . 9 n} \\ 0_{n . 5 n} - I_{n} 0_{n . 11 n} \\ 0_{n . 6 n} I_{n} 0_{n . 9 n} \end{matrix}]

(46)

then for the gain matrix $F_{va} = H L^{- 1}$ , the closed-loop system given by equation (12) has a stipulated $H_{\infty}$ interference attenuation level ℘ for all primary conditions satisfying $ℶ_{μ} \leq 1$ and the region of asymptotic stability is defined by an ellipsoid

\partial (P, 1 + ℘^{2} {\hat{σ}}^{2}) = {Ψ \in ℜ^{n}; Ψ^{T} P Ψ \leq 1 + ℘^{2} {\hat{σ}}^{2}}

(47)

An estimated attraction basin for equation (12) is given by $ℶ_{μ} \leq 1$ .

Proof. When disturbance is present in a system (12), disturbance attenuation can be intended in sense of $H_{\infty}$ by considering equation (48)

β (k) = Δ v (x (k)) + ℨ^{T} (k) ℨ (k) - ℘^{2} w^{T} (k) w (k)

(48)

On the same lines as Theorem 1 and including disturbance, the proof of Theorem 2 can be carried out.

LMI constraints (42) and (43) and $β (k) < 0$ are sufficient conditions for asymptotic stability of system (12). The asymptotic condition $β (k) < 0$ can be characterized as

\overset{⌣}{G} - F (k) [\begin{matrix} Z_{1} & 0 \\ 0 & Z_{2} \end{matrix}] F^{T} (k) < 0

(49)

Where

{\bar{F}}^{T} = [\begin{matrix} σ (k) I & 0_{n . 4 n} & - σ (k) I & 0 & 0_{n . 9 n} \\ ε (k) I & 0_{n . 4 n} & 0 & - ε (k) I & 0_{n . 9 n} \\ 0 & 0_{n . 4 n} & - σ (k) I & 0 & 0_{n . 9 n} \\ 0 & 0_{n . 4 n} & 0 & - ε (k) I & 0_{n . 9 n} \end{matrix}]

(50)

Analogous to the proof of Theorem 1 and $M 1 + F (k) K^{T} + K F^{T} (k) < 0$ from equation (37), it is possible to demonstrate that $β (k) < 0$ converges to equation (44). Furthermore, for zero initial condition, it can be indicated that

‖ X (k) ‖_{2}^{2} < ℘ ‖ w (k) ‖_{2}^{2}

(51)

The satisfaction of LMI (43) for all initial states $ℶ_{μ} < 1$ follows that $Ψ^{T} (k) P Ψ (k) < 1 + ℘^{2} {\hat{σ}}^{2}$ . So the system trajectories beginning from $ℶ_{μ} < 1$ will stay within the ellipsoid given as $\partial (P, 1 + ℘^{2} {\hat{σ}}^{2})$ .

Maximization of basin of attraction

An anti-windup gain $F_{va} = H L^{- 1}$ provides a maximized estimate of the attraction basin yielded by $μ_{max} = 1 / \sqrt{H} .$

Remark 3. The arithmetic complexity of Theorems and Corollaries described in the paper might be diminished by putting a couple of $n \times n$ matrices in $K = [\begin{matrix} K_{1} & K_{2} & K_{3} & K_{4} \end{matrix}]$ to zero (Wang et al., 2013).

Numerical examples

This section provides examples to validate the applicability of the significant findings.

Example 1. Consider the system and controller (1)–(13) with the parameters

\begin{matrix} A = [\begin{matrix} 0.215 & 0.08 \\ 0.2 & - 0.04 \end{matrix}], A_{d} = [\begin{matrix} - 0.1 & - 0.1 \\ 0 & - 0.1 \end{matrix}], B = [\begin{matrix} 0.8 \\ 0 \end{matrix}], \\ C = [\begin{matrix} 1 & 0 \\ 0 & 0.1 \end{matrix}], D_{Θ} = [\begin{matrix} 0.01 & 0 \\ 0.007 & 0.008 \end{matrix}], C_{p} = [\begin{matrix} 0.01 & 0 \\ 0.006 & 0.002 \end{matrix}], \\ D_{p} = [\begin{matrix} 0.006 & 0.005 \\ 0.002 & 0.006 \end{matrix}], w (k) = [\begin{matrix} - 0.01 e^{- 2 k} \cos k \\ 0.01 e^{- 0.5 k} \sin k \end{matrix}], \\ A_{va} = [\begin{matrix} - 0.051 & 0 \\ 0 & - 0.501 \end{matrix}], B_{va} = [\begin{matrix} 0.5 & 0 \\ 0 & 0.6 \end{matrix}], \\ C_{va} = [- 0.4958 0], \\ D_{va} = [\begin{matrix} - 0.011 & - 0.527 \end{matrix}], and \hat{σ} = 0.01 \end{matrix}

The saturated control signal (6) is given into the plant where $- 1 \leq u_{0 (i)} \leq 1 .$ By utilizing the LMI workbench (Gahinet et al., 1995), the LMI constraints (42)–(44) stated in Theorem 2 are encountered feasible for $℘ = 0.8$ , and $F_{va} = H L^{- 1} = {[\begin{matrix} - 0.0100 & - 0.0758 \end{matrix}]}^{T}$ .

The trajectories of plant states and controller states are revealed in Figure 2(a) and (b) for initial condition $x_{0} = [\begin{matrix} 300 & 20 \end{matrix} 0]^{T}$ . The plot of $u (k)$ and $v_{va} (k)$ is displayed in Figure 2(c) where the effect of actuator saturation nonlinearity is observed. It is noticed that the designed $H_{\infty}$ controller with anti-windup gain stabilizes the system in the presence of the above nonlinearities. The plant states are asymptotically stable as it approaches the origin as $k \to \infty$ . The maximized basin of attraction is calculated as $μ_{max} = 0.0120$ (Table 1).

Table 1.

Computation of anti-windup gain for Example 1.

$g_{1} \leq g (k) \leq g_{2}$	$μ_{max}$	$F_{va} = H L^{- 1}$
$1 \leq g (k) \leq 3$	$0.0155$	${[\begin{matrix} - 0.0727 & - 0.0424 \end{matrix}]}^{T}$
$1 \leq g (k) \leq 4$	$0.0134$	${[\begin{matrix} - 0.0481 & - 0.459 \end{matrix}]}^{T}$
$1 \leq g (k) \leq 5$	$0.0128$	${[\begin{matrix} - 0.0331 & - 0.0570 \end{matrix}]}^{T}$
$1 \leq g (k) \leq 6$	$0.0120$	${[\begin{matrix} - 0.0100 & - 0.0758 \end{matrix}]}^{T}$
$1 \leq g (k) \leq 7$	$Infeasible$	$Infeasible$

For exponential disturbance, the plots are as given in Figure 2.

For step load disturbance, the plots are as given in Figure 3.

Note 1. The trajectory of plant states in Figures 2(a) and 3(a) with reference to given exponentially decaying disturbance and step load disturbance at time k = 4; as seen from the state trajectories, the system are asymptotically stable. From the observation of control effort $u (k)$ , the effect may be seen from the controller part. It may be observed, controller state $x_{c 1}$ has increased toward the value 5 in Figure 3(b). So it is a type of overshoot, that is, due to step disturbance. It is concluded from the proposed approach that it is possible to mitigate the effect of step disturbance also.

In future, we may extend this method for other type of disturbances also. From Figures 2 and 3, it can be inferred that the designed controller is capable against the exponential disturbance and step disturbance also.

Note 2. The control effort has been shown in graph of Figure 2b. After the system becomes stable, it tends to zero. In this way controller is working in optimized manner; as a result, cost of controller is also minimized. Table 2 shows the comparison of present work with available results. We can see the effect of disturbance has been vanished. We have employed anti-windup technique, delay range has increased, and less conservative results are obtained. In this way, the proposed method is superior to existing one.

The novelty of this work is that the triple LKF is used for designing of anti-windup compensator to tackle the complex issue of discrete time-delay systems subjected to actuator saturation and exponentially decaying as well as step disturbance using LMI technique.

Table 2.

Delay bounds.

Method	Delay range $(g_{1} \leq g (k) \leq g_{2})$	Nonlinearities
Theorem 2 (Negi et al., 2012)	$1 \leq g (k) \leq 5$	When the system is subjected to actuator saturation and time-varying delay
Theorem 1 (Xu et al., 2012)	$1.4 \leq g (k) \leq 3.8$	When system subjected to actuator saturation, time-varying delay and interference
Theorem 1 (Qian et al., 2015)	$1 \leq g (k) \leq 4$	When system is subjected to actuator saturation, time-varying delay, and interference with attenuation level $℘ = 1.3424$
Theorem 2 (Qian et al., 2015)	$1 \leq g (k) \leq 3$	When system is subjected to actuator saturation, time-varying delay and interference with attenuation level $℘ = 1.4683$
Theorem 1 (Chen et al., 2019)	$1 \leq g (k) \leq 5$	When system is subjected to actuator saturation, time-varying delay and uncertainty
Theorem 2 (proposed work)	$1 \leq g (k) \leq 6$	When the system is subjected to actuator saturation, stabilizing controller, time-varying delay and interference with attenuation level $℘ = 0.8$

The comparison of the outcomes is depicted in Table 2.

Note 3. It is seen from Table 2 that the upper limit of the delay range has increased in comparison to existing results (Chen et al., 2018; Negi et al., 2012; Qian et al., 2015; Xu et al., 2012). The interference attenuation level is also reduced.

Example 2. Consider a fluid flows through two tanks (Blanchini et al., 2009; Huang et al., 2016).

The effect of delay is also considered for the two-tank system in Figure 4. The parameters are as follows

\begin{matrix} A = [\begin{matrix} - 0.1 & 0.1 \\ 0.1 & - 0.1 \end{matrix}], A_{d} = [\begin{matrix} - 0.1 & - 0.1 \\ 0 & - 0.1 \end{matrix}], B = [\begin{matrix} 0.9 \\ 1.1 \end{matrix}], \\ C = [\begin{matrix} 0.12 & 0.1 \\ 0 & 0 \end{matrix}], D_{Θ} = [\begin{matrix} 0.3 & 0 \\ 0.2 & 0 \end{matrix}], C_{p} = [\begin{matrix} 0.3 & 0.2 \\ 0.2 & 0.3 \end{matrix}], \\ D_{p} = [\begin{matrix} 0.1 \\ 0.2 \end{matrix}], w (k) = [\begin{matrix} - 0.01 e^{- 2 k} \cos k \\ 0.01 e^{- 0.5 k} \sin k \end{matrix}], A_{va} = [\begin{matrix} 0.2 & 0.1 \\ 0.2 & 0.4 \end{matrix}], \\ B_{va} = [\begin{matrix} 0.3 & 0 \\ 0.4 & 0 \end{matrix}], C_{va} = [\begin{matrix} - 0.2 & 0.1 \end{matrix}], D_{va} = 3.5 . \end{matrix}

The saturated control signal (6) is given into the plant where $- 1 \leq u_{0 (i)} \leq 1$ . By utilizing the LMI workbench (Gahinet et al., 1995), the LMI constraints (42)–(44) stated in Theorem 2 are encountered feasible for $1 \leq g (k) \leq 5$ , $℘ = 1$ , $\hat{σ} = 1$ and $F_{va} = H L^{- 1} = [\begin{matrix} 0.2998 & 0.0713 \end{matrix}]^{T}$ .

The trajectories of plant states and controller states are displayed in Figure 5(a) and (b) for initial states $x_{0} = [\begin{matrix} 10 & - 10 \end{matrix}]^{T}$ . It approaches to origin as $k \to \infty$ . The plot of the control effort $u (k)$ and $v_{va} (k)$ is shown in Figure 5(c) where the effect of actuator saturation nonlinearity is observed. It is clear from the graph that control effort ( $u (k) = sat (v_{va} (k))$ ) is windup to the given saturation limit (i.e. $- 1 \leq u_{0 (i)} \leq 1$ ) due to the anti-windup control strategy. It is inferred that the designed $H_{\infty}$ controller with anti-windup gain stabilizes the system in the presence of the above nonlinearities and $μ_{max} = 0.0126$ . In this example, delay and disturbance are both considered, and the system is found to be stable for the range $1 \leq g (k) \leq 5$ as compared to the previous result where the delay was not considered. The comparison is shown in Table 3.

Table 3.

Computation of delay range and anti-windup gain.

Method	Delay range $(g_{1} \leq g (k) \leq g_{2})$	$F_{va} = H L^{- 1}$
Theorem 4(Huang et al., 2016)	No delay	${[\begin{matrix} 0.5279 - 0.3415 \end{matrix}]}^{T}$
Theorem 2(proposed work)	$1 \leq g (k) \leq 5$	${[\begin{matrix} 0.2998 0.0713 \end{matrix}]}^{T}$

From Table 3, it is clear that the two tanks’ fluid flow systems are stable in the presence of delay and disturbances $w (k)$ .

Example 3. (Nise, 2010; Pal and Negi, 2018) The missile control system can be marked by equation (12) with parameters as follows

\begin{matrix} A = [\begin{matrix} - 0.2428 & - 0.6238 \\ 1 & 0 \end{matrix}], A_{d} = [\begin{matrix} - 0.016 & - 0.012 \\ 0 & - 0.013 \end{matrix}], \\ B = [\begin{matrix} 1 \\ 0 \end{matrix}], C = [\begin{matrix} 1.31 & 0.5566 \end{matrix}], \end{matrix}

\begin{matrix} D_{Θ} = [\begin{matrix} 0.00004 & 0.0006 \\ 0.00079 & 0.0008 \end{matrix}], C_{p} = [\begin{matrix} - 0.014 & 0.253 \\ 0 & 0.01 \end{matrix}], \\ D_{p} = [\begin{matrix} 0.109 & 0 \\ 0 & 0.012 \end{matrix}], \hat{σ} = 0.01, \end{matrix}

\begin{matrix} w (k) = [\begin{matrix} - 0.01 e^{- 2 k} \cos k \\ 0.01 e^{- 0.5 k} \sin k \end{matrix}], A_{va} = [\begin{matrix} - 0.00051 & 0 \\ 0 & - 0.501 \end{matrix}], \\ B_{va} = [\begin{matrix} - 0.0213 & 0.0001 \\ - 0.0621 & 0.008 \end{matrix}], C_{va} = [\begin{matrix} - 0.014958 & 0 \end{matrix}], \\ D_{va} = [- 0.011 - 0.527] . \end{matrix}

The saturated control signal (6) is given into the plant where $- 1 \leq u_{0 (i)} \leq 1 .$ By utilizing the LMI workbench (Gahinet et al., 1995), the LMI constraints (42)–(44) stated in Theorem 2 are encountered feasible for $1 \leq g (k) \leq 97$ , $℘ = 1.8$ , $\hat{σ} = 0.01$ , and $F_{va} = H L^{- 1} = [\begin{matrix} - 0.5602 & - 1607 \end{matrix}]^{T}$ .

The trajectories of plant states and controller states are displayed in Figure 6(a) and (b). It approaches to origin as $k \to \infty$ . The plot of the control effort $u (k)$ and $v_{va} (k)$ is shown in Figure 6(c) where the effect of actuator saturation nonlinearity is observed. It is clear from the graph that control effort $(u (k) = sat (v_{va} (k)))$ is bounded within the saturation limit (i.e. $- 1 \leq u_{0 (i)} \leq 1$ ) due to the anti-windup control strategy. It is inferred that the designed $H_{\infty}$ controller with anti-windup gain stabilizes the system in the presence of the above nonlinearities.

Conclusion

In this manuscript, a novel $H_{\infty}$ controller is designed to stabilize a discrete time-delay system with two different disturbances (exponential decaying and step). The proposed criterion employs triple summation LKF and an anti-windup controller for actuator saturation, delay, and external disturbances. The efficacy of the proposed criteria has been shown in simulation results for the practical systems. The effect of delay is analyzed for a two-tank model (Blanchini et al., 2009; Huang et al., 2016) of process control industrial system, which is generally used for batch processing industry (like the pharmaceutical industry, chemical industry. Another practical example for the missile control system has also been given (Nise, 2010; Pal and Negi, 2018). The extension of the proposed work may be seen in the field of telecommunication along with probability density distribution communication delay by event-triggered mechanism (Gu et al., 2022a) and for network fault detection in interval type 2 adaptive memory-event-triggered mechanism (Gu et al., 2022b).

Footnotes

Declaration of conflicting interests

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

Funding

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

ORCID iDs

Komal Agrawal

Vipin Chandra Pal

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