Sliding mode controller design via delay-dependent $H_{\infty }$ stabilization criterion for load frequency regulation

Abstract

This work presents a control approach based on sliding-mode-control (SMC) to design robust $H_{\infty }$ state feedback controllers for load frequency regulation of delayed interconnected power system (IPS) with parametric uncertainties. Considering both state feedback control strategy and delayed feedback control strategy, two SMC laws are proposed. The proposed control laws are designed to improve the stability and disturbance rejection performance of delayed IPS, while stabilization criteria in the form of linear matrix inequality are derived by choosing a Lyapunov–Krasovskii functional. An artificial time-delay is incorporated in the control law design of the delayed feedback control structure to enhance the controller performance. A numerical example is considered to study the control performance of the proposed controllers and simulation results are provided to observe the dynamic response of the IPS.

1 Introduction

Stable power system operation needs stability in frequency because frequency reflects the status of the real power balance between power supply and load demand [1, 2]. Alternators and other power devices deviate from stable operating condition due to fluctuation in frequency, which can lead to unstable and unsafe operation of the entire power system [3, 4]. Disturbance in load demand is the main cause of originating frequency and voltage oscillations in power system [5, 6]. Therefore, load frequency control (LFC) is an essential need of power system for its smooth operation [7, 8]. Using modern technologies, simple power system areas are interconnected through tie-lines to create an interconnected power system (IPS) [9, 10]. The main objective of a well designed IPS is to regulate frequency variation within tolerable limits during abnormal condition of IPS operation, i.e., load variation or sudden change in load demand [6, 11, 12].

In the last few decades, many traditional control techniques have been proposed to solve LFC problem in power system. Proportional-Integral (PI) control method [13] is first proposed to solve LFC problem, while decentralized robust PI controller based on Kharitonov’s theorem is designed for LFC of a multi-area IPS in [14]. Various control schemes are designed for solving LFC problem of uncertain IPS, such as robust control scheme based on Riccati-equation [15] and adaptive control scheme with system parameter variation [16]. An active disturbance rejection control approach is proposed to develop a robust decentralized LFC algorithm for a three-area IPS in [17], while in [18], pole-placement method based variable structure controller is proposed for LFC of IPS.

A finite time is always needed to measure control signals, compute control action and actuate the plant. This finite time requirement in every closed-loop system is termed as time-delay. Stability of closed-loop systems is affected by this delay because of its destabilizing nature [8]. Time-delay appears in IPS due to two main reasons, i.e.: (i) time needed for frequency and tie-line power measurement; and (ii) time taken in signal transmission from the remote terminal unit to the control centre and from the control centre to the generating unit [19]. Presence of time-delay in IPS sometimes leads to system instability and performance degradation [20, 21]. Therefore, time-delay should be included properly in LFC design of IPS to solve frequency deviation problem. Different control schemes have been used to design controllers for solving the LFC problem of time-delayed IPS. A decentralized robust PI controller is designed for LFC of power system with delay in [22], while state feedback control strategy based on linear matrix inequality (LMI) is adopted for LFC of IPS with delay in [23]. An iterative LMI based robust LFC algorithm is developed in [24] for time-delay IPS, while Lyapunov theory based delay-dependent stability criterion is developed in [25] for LFC by considering constant delay and time-varying delay in IPS. In [26], a delay-dependent robust PID-type controller is designed to regulate frequency of a three-area IPS with communication delay. Considering the existence of time-delay in the local PI-type LFC signals of an IPS, delay-dependent stability criteria are developed for the delayed LFC scheme in [27]. Although the above mentioned literatures based on delayed LFC design have well considered the effect of time-delay in LFC scheme, no specific method has been implemented to improve the performance of LFC scheme which is degraded due to the existence of time-delay.

Stabilizing nature of time-delay in system dynamics is an interesting aspect studied in [28, 29]. A finite valued known time-delay is introduced intentionally in the control law design to improve the stability of a delayed system in [8]. A few research results are available in literatures where time-delay is introduced in control design for improving the control performance of multi-area delayed IPS. In [19], a known finite delay is used to design $H_{\infty }$ two-term frequency controller for enhancing the control performance of a two-area IPS considering communication delays, while tolerable delay margin of IPS is enhanced using time-delay in a robust $H_{\infty }$ frequency controller design in [21]. The time-delay used purposefully in controller design is termed as artificial delay. Besides the above discussed control approaches, sliding mode control (SMC) approach is very effective to solve the LFC problem by designing robust state feedback controller for time-delayed multi-area IPS. SMC approach is popular for its insensitivity to parameters variation, excellent control performance and finite-time convergence [6, 30]. In [31], a non-linear sliding mode controller is designed for load frequency regulation of multi-area power system with time-varying delay. SMC scheme for LFC is considered for wind-integrated delayed IPS in [32], and SMC scheme is proposed to design a robust state feedback controller for LFC of multi-area power system with time-delay in [6, 33]. In [34], a decentralised robust load frequency controller is proposed for interconnected time-delay power system using sliding mode technique, while in [35], an event-triggered SMC design is proposed for LFC of power systems with sensor faults and communication delay. In [36], an adaptive delay-dependent sliding mode fault-tolerant LFC design is proposed for nonlinear power system with unknown time-varying state and input delays, whereas a dynamic integral SMC based LFC scheme is proposed for an interconnected delayed power system in [37].

Based on the above study, it is clearly observed that $H_{\infty }$ control and SMC approaches are mostly used for the LFC design of IPS with time-delay and parametric uncertainty. The reason behind for the application of SMC approach on LFC design of delayed IPS is that SMC based controllers are very insensitive to parametric variation [37]. Similarly, the popularity of $H_{\infty }$ control is widespread for its disturbance rejection performance and robustness against uncertainties [2, 19]. As time-delay and parametric uncertainty are two inevitable phenomena in LFC of IPS and they have negative impact on system stability, the effects of these phenomena on LFC should be accounted properly during LFC design for IPS. To neutralize the destabilizing effect of the existence of time-delay in a system, purposeful use of a known finite time-delay may be considered while designing the control method [38]. In view of the above discussions, to obtain the benefit of both SMC and $H_{\infty }$ control methods, sliding mode controller design based on delay-dependent $H_{\infty }$ stabilization criterion is proposed for load frequency regulation of IPS considering the existence of time-delay and parametric uncertainty in this paper. The main contributions of this paper are described as follows:

• Two new SMC laws are proposed considering two different controller structures, including state feedback controller and delayed feedback controller designing strategies, for load frequency regulation of delayed IPS with parametric uncertainty. Chattering is a major issue in SMC approach because it degrades control performance [33]. This paper takes care of the chattering issue of SMC approach to ensure that the controller output signals are free from chattering.

• A novel sliding surface with the incorporation of artificial delay is defined to obtain the proposed SMC law based on delayed feedback controller designing strategy. An important feature of the proposed SMC law based on delayed feedback controller designing strategy is that a finite known delay is introduced judiciously in the proposed sliding mode LFC structure to improve closed-loop IPS performance which is degraded due to the existence of time-delay in uncontrolled IPS.

• New delay-dependent $H_{\infty }$ stabilization criteria in LMI framework for the sliding mode dynamics of IPS are derived by using a simple Lyapunov–Krasovskii (LK) functional approach. These stabilization criteria satisfy parametric uncertainty and time-delay existence of the closed-loop IPS, while the gains of the proposed sliding mode controllers are computed by solving these stabilization criteria. The inclusion of $H_{\infty }$ criterion into the stabilization condition improves disturbance rejection performance.

• A well-known numerical example of a two-area IPS with time-delay and parametric uncertainty is considered to study the performance of the proposed load frequency regulation method. The performance of the proposed frequency regulation method is compared with some existing methods [19, 21] to show its superiority.

Notation: Throughout this paper, the superscript ‘T’ stands for matrix transposition, the superscript ‘$-1$’ stands for matrix inverse. For any arbitrary matrix B and two symmetric matrices A and C, $\left[ \begin{matrix} A &{} B \\ * &{} C \\ \end{matrix} \right]$ denotes a symmetric matrix, where $*$ represents $B^{T}$.

2 Modeling of delayed interconnected power system (IPS)

A two-area IPS with time-delay in each control area is considered in this paper to analyse the LFC problem. The LFC model of the two-area IPS with time-delay in each control area is presented in Fig. 1 [19].

Fig. 1

LFC model of two-area IPS with time-delay

From Fig. 1, the dynamic model of the two-area IPS with time-delay can be described as [21]:

$$\begin{aligned} \Delta {{\dot{P}}_{mi}}(t)= & {} \frac{{1}}{{{T}_{chi}}}({\Delta {{P}_{vi}}(t)}-{\Delta {{P}_{mi}}(t)}), \end{aligned}$$

(1)

$$\begin{aligned} \Delta {{\dot{E}}_{i}}(t)= & {} {{k}_{i}}(\Delta {{P}_{12}}(t)+{{B}_{i}}\Delta {{f}_{i}}(t)),\end{aligned}$$

(2)

$$\begin{aligned} \Delta {{\dot{P}}_{12}}(t)= & {} 2\pi {{T}_{1}}(\Delta {{f}_{1}}(t)-\Delta {{f}_{2}}(t)),\end{aligned}$$

(3)

$$\begin{aligned} \Delta {{\dot{P}}_{vi}}(t)= & {} -\frac{{1}}{{{T}_{gi}}}(\frac{\Delta {{f}_{i}}(t)}{{{R}_{i}}}+{\Delta {{P}_{vi}}(t)}+{\Delta {{E}_{i}}(t-{{\tau }_{i}})}-{{{u}_{i}}(t)}),\end{aligned}$$

(4)

$$\begin{aligned} \Delta {{\dot{f}}_{i}}(t)= & {} -\frac{{{k}_{pi}}}{{{T}_{pi}}}\left( \Delta {{P}_{di}}(t)+\Delta {{P}_{ij}}(t)-\Delta {{P}_{mi}}(t)\right) -\frac{{\Delta {{f}_{i}}(t)}}{{{T}_{pi}}}, \end{aligned}$$

(5)

with $\Delta {{P}_{12}}=-\Delta {{P}_{21}}$, $i,j=1,2,\,i\ne j$.

The above dynamic model can be represented in state-space form as [2]:

$$\begin{aligned} \dot{x}(t)= & {} Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})+{{A}_{d2}}x(t-{{\tau }_{2}})+Bu(t)+Dw(t),\end{aligned}$$

(6)

$$\begin{aligned} y(t)= & {} Cx(t), \end{aligned}$$

(7)

where the state vector

$$\begin{aligned} x(t)= & {} {{\left[ \begin{matrix} \Delta Ae1(t) &{} \Delta P_{12}(t) &{} \Delta Ae2(t) \\ \end{matrix} \right] }^{T}},\\ \Delta Ae1(t)= & {} \left[ \begin{matrix} \Delta f_{1}(t) &{} \Delta P_{m1}(t) &{} \Delta P_{v1}(t) &{} \Delta E_{1}(t) \\ \end{matrix} \right] ,\\ \Delta Ae2(t)= & {} \left[ \begin{matrix} \Delta f_{2}(t) &{} \Delta P_{m2}(t) &{} \Delta P_{v2}(t) &{} \Delta E_{2}(t) \\ \end{matrix} \right] , \end{aligned}$$

and load disturbance vector

$w(t)={{\left[ \begin{matrix} \Delta P_{d1}(t) &{} \Delta P_{d2}(t) \\ \end{matrix} \right] }^{T}}$.

Here, $\Delta P_{d1}(t)$ and $\Delta P_{d2}(t)$ are norm bounded, and satisfy $\left\| \Delta {{P}_{d1}}(t) \right\| <{{b}_{1}}$ and $\left\| \Delta {{P}_{d2}}(t) \right\| <{{b}_{2}}$. ${{b}_{1}}$ and ${{b}_{2}}$ are positive constants, while ${{\tau }_{1}}$ and ${{\tau }_{2}}$ are time delays of area 1 and area 2, respectively. The system parameters are given for $e=1,2$ as:

$$\begin{aligned} A= & {} \left[ \begin{array}{lll} {{A}_{1}} &{} {{A}_{3}} &{} {{0}_{4\times 4}} \\ {{A}_{5}} &{} {{0}_{1\times 1}} &{} -{{A}_{5}} \\ {{0}_{4\times 4}} &{} {{A}_{4}} &{} {{A}_{2}} \\ \end{array} \right] , \,\, {{A}_{e}}=\left[ \begin{array}{llll} -\frac{1}{{{T}_{pe}}} &{} \frac{{{k}_{pe}}}{{{T}_{pe}}} &{} 0 &{} 0 \\ 0 &{} -\frac{1}{{{T}_{che}}} &{} \frac{1}{{{T}_{che}}} &{} 0 \\ -\frac{1}{{{R}_{e}}{{T}_{ge}}} &{} 0 &{} -\frac{1}{{{T}_{ge}}} &{} 0 \\ {{k}_{e}}{{B}_{e}} &{} 0 &{} 0 &{} 0 \\ \end{array} \right] ,\\ \,\, {{A}_{3}}= & {} {{\left[ \begin{array}{llll} -\frac{k_{p1}}{T_{p1}} &{} 0 &{} 0 &{} k_{1} \\ \end{array} \right] }^{T}}, \,\, {{A}_{4}}={{\left[ \begin{array}{llll} -\frac{k_{p2}}{T_{p2}} &{} 0 &{} 0 &{} k_{2} \\ \end{array} \right] }^{T}}, \,\, {{A}_{5}}=\left[ \begin{array}{llll} 2\pi {{T}_{12}} &{} 0 &{} 0 &{} 0 \\ \end{array} \right] ,\\ \,\, {{A}_{d1}}= & {} \left[ \begin{array}{lll} {{0}_{2\times 3}} &{} {{0}_{2\times 1}} &{} {{0}_{2\times 5}} \\ {{0}_{1\times 3}} &{} -\frac{1}{{{T}_{g1}}} &{} {{0}_{1\times 5}} \\ {{0}_{6\times 3}} &{} {{0}_{6\times 1}} &{} {{0}_{6\times 5}} \\ \end{array} \right] , \,\, {{A}_{d2}}=\left[ \begin{array}{ll} {{0}_{7\times 8}} &{} {{0}_{7\times 1}} \\ {{0}_{1\times 8}} &{} -\frac{1}{{{T}_{g2}}} \\ {{0}_{1\times 8}} &{} {{0}_{1\times 1}} \\ \end{array} \right] ,\\ B= & {} {{\left[ \begin{array}{lllllllll} 0 &{} 0 &{} -\frac{1}{{{T}_{g1}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{1}{{{T}_{g2}}} &{} 0 \\ \end{array} \right] }^{T}},\\ C= & {} \left[ \begin{array}{lllllllll} 1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 1 &{} 0 &{} 0 &{} 0 \\ \end{array} \right] , \,\, D={{\left[ \begin{array}{llllllllll} \frac{{{k}_{p1}}}{{{T}_{p1}}} &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} \frac{{{k}_{p2}}}{{{T}_{p2}}} &{} 0 &{} 0 &{} 0 \\ \end{array} \right] }^{T}}. \end{aligned}$$

Considering parametric uncertainties in system matrices of (6), one may write:

$$\begin{aligned} \dot{x}(t) & = {\hat{A}}(t)x(t)+{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}}) \\ & \quad +{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}})+Bu(t)+Dw(t), \end{aligned}$$

(8)

where $\hat{A}(t)$, ${{\hat{A}}_{d1}}(t)$ and ${{\hat{A}}_{d2}}(t)$ are uncertain state matrices, i.e., $\hat{A}(t)=A+\Delta A(t)$, ${{\hat{A}}_{d1}}(t)={{A}_{d1}}+\Delta {{A}_{d1}}(t)$ and ${{\hat{A}}_{d2}}(t)={{A}_{d2}}+\Delta {{A}_{d2}}(t)$. $\Delta A(t)$, $\Delta {{A}_{d1}}(t)$ and $\Delta {{A}_{d2}}(t)$ are uncertainties with the system matrices A, ${{A}_{d1}}$ and ${{A}_{d2}}$, respectively. The uncertainty in system matrices can be considered to be bounded with norm, and can be described as:

$$\begin{aligned} \left[ \begin{matrix} \Delta A(t) &{} \Delta {{A}_{d1}}(t) &{} \Delta {{A}_{d2}}(t) \\ \end{matrix} \right] =\left[ \begin{matrix} HF(t){{E}_{1}} &{} HF(t){{E}_{2}} &{} HF(t){{E}_{3}} \\ \end{matrix} \right] , \end{aligned}$$

(9)

where H, ${{E}_{1}}$, ${{E}_{2}}$, and ${{E}_{3}}$ are constant matrices with appropriate dimensions. F(t) is a time-varying matrix, which satisfies ${{F}^{T}}(t)F(t)\le I$.

3 Controller designing strategy

This paper aims to design load frequency controllers based on sliding mode control for delayed IPS, whereas they satisfy the $H_{\infty }$ criterion, defined as [2]:

$$\begin{aligned} {{\left\| {{T}_{wy}} \right\| }_{\infty }}=\frac{{{\left\| y \right\| }_{2}}}{{{\left\| w \right\| }_{2}}}=\frac{\sqrt{\int \limits _{0}^{\infty }{{{y}^{T}}(t)y(t)dt}}}{\sqrt{\int \limits _{0}^{\infty }{{{w}^{T}}(t)w(t)dt}}}\le \gamma , \end{aligned}$$

(10)

where $\gamma$ is the $H_{\infty }$ performance indicator, which indicates rejection of load disturbance. Minimization of $\gamma$ is required to obtain the minimal effect of load variation in the IPS performance.

Fig. 2

Controller designing strategy for load frequency regulation of two-area IPS with time-delay

The controller designing strategy proposed for load frequency regulation of the two-area IPS with time-delay is shown in Fig. 2. As shown, state vector (x(t)) of the two-area delayed IPS acts as input to the proposed sliding mode controller, and the controller outputs ($u_{1}$ and $u_{2}$) act as control inputs to the two-area delayed IPS, provided that the controller gains are based on $H_{\infty }$ performance condition (10). Some steps are required to compute $H_{\infty }$ performance-based gains of the sliding mode controller, which are described as follows.

• Step-1 Selection of switching surface, which is a function of the state vector of delayed IPS.

• Step-2 Differentiate the selected switching surface and obtain an equivalent controller by equating this differentiation to zero.

• Step-3 Obtain the closed-loop sliding mode dynamics of delayed IPS by using the equivalent controller in the state-space form of delayed IPS.

• Step-4 Develop delay-dependent ${{H}_{\infty }}$ stabilization criterion in LMI framework for the closed-loop sliding mode dynamics of delayed IPS.

• Step-5 Obtain $H_{\infty }$ performance-based gains of sliding mode controller from the solution of delay-dependent ${{H}_{\infty }}$ stabilization criterion.

After obtaining the $H_{\infty }$ performance-based controller gains, sliding mode control law is designed by using these gains and system state vector for load frequency regulation of IPS with time-delay.

Subsequently, it proceeds to design the proposed ${{H}_{\infty }}$ performance-based sliding mode controller by using both state feedback controller designing strategy and delayed feedback controller designing strategy.

3.1 Robust ${{H}_{\infty }}$ state feedback sliding mode controller designing strategy

3.1.1 Selection of switching surface

A switching surface can be selected considering delay in IPS as [6]:

$$\begin{aligned} \sigma (t)= & {} \,\, Jx(t)-\int \limits _{0}^{t}{J\hat{A}(t)x(\tau )d\tau }+\int \limits _{0}^{t}{JBKx(\tau )d\tau }\nonumber \\{} & {} -\int \limits _{0}^{t-{{\tau }_{1}}}{J{{\hat{A}}_{d1}}(t)x(\tau )d\tau } -\int \limits _{0}^{t-{{\tau }_{2}}}{J{{\hat{A}}_{d2}}(t)x(\tau )d\tau }, \end{aligned}$$

(11)

where J and K are two constant matrices, and J is selected such that the matrix JB becomes nonsingular. Time derivative of (11) is given as:

$$\begin{aligned} {\dot{\sigma }}(t)=\,\,J\dot{x}(t)-J\hat{A}(t)x(t)+JBKx(t) -J{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}})-J{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}}). \end{aligned}$$

(12)

When delayed IPS trajectory reaches its sliding mode, the conditions of ${\sigma }(t)=0$ and ${\dot{\sigma }}(t)=0$ are satisfied by the switching function. Substituting (8) in (12), one can obtain:

$$\begin{aligned} {\dot{\sigma }}(t)=JBKx(t)+JBu(t)+JDw(t). \end{aligned}$$

(13)

The following equivalent controller can be obtained from (13) by considering ${\dot{\sigma }}(t)=0$:

$$\begin{aligned} u{{(t)}_{equ}}=-Kx(t)-{{(JB)}^{-1}}JDw(t). \end{aligned}$$

(14)

The closed-loop system of delayed IPS is obtained by substituting (14) in (8) as:

$$\begin{aligned} \dot{x}(t)=\,\,\hat{A}(t)x(t)+{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}}) +{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}}) -BKx(t)+\hat{D}w(t), \end{aligned}$$

(15)

where matrix $\hat{D}=D-B{{(JB)}^{-1}}JD$. System (15) without parametric uncertainty can be written as:

$$\begin{aligned} \dot{x}(t)=Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})+{{A}_{d2}}x(t-{{\tau }_{2}}) -BKx(t)+\hat{D}w(t). \end{aligned}$$

(16)

Remark 1

The equivalent controller structure (14) contains an inverse matrix of ${{(JB)}^{-1}}$. It is well known that a singular matrix can not be inverted, and thus, matrix JB must be a nonsingular matrix. Therefore, care must be taken while selecting matrix J for the construction of the switching surface (11).

3.1.2 Robust ${{H}_{\infty }}$ stabilization criterion

Theorem 1

Closed-loop IPS (15) satisfies the $H_{\infty }$ criterion $\left\| {{T}_{wy}} \right\| \le \gamma$ and $\gamma >0$, if there exist positive definite matrices $\hat{P}$, ${{\hat{Q}}_{1}}$, ${{\hat{Q}}_{2}}$, ${{\hat{R}}_{{{\tau }_{1}}}}$, ${{\hat{R}}_{{{\tau }_{2}}}}$ and matrix G, such that an LMI holds as follows:

$$\begin{aligned} \left[ \begin{matrix} {\bar{\Lambda }} &{} {{{\bar{\Gamma }} }_{1}} &{} {{{\bar{\Gamma }} }_{2}} &{} {{{\bar{\Gamma }} }_{3}} &{} {{{\bar{\Gamma }} }_{4}} &{} {{{\bar{\Gamma }} }_{5}} \\ * &{} -{{\varepsilon }_{1}}I &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} -{{\varepsilon }_{2}}I &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} -{{\varepsilon }_{3}}I &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} -{{\gamma }^{2}} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} -I \\ \end{matrix} \right] <0, \end{aligned}$$

(17)

where

$$\begin{aligned} {\bar{\Lambda }}=\left[ \begin{matrix} {{{\bar{\Lambda }} }_{11}} &{} {{{\bar{\Lambda }} }_{12}} &{} {{{\bar{\Lambda }} }_{13}} &{} {{{\bar{\Lambda }} }_{14}} \\ * &{} {{{\bar{\Lambda }} }_{22}} &{} {{{\bar{\Lambda }} }_{23}} &{} {{{\bar{\Lambda }} }_{24}} \\ * &{} * &{} {{{\bar{\Lambda }} }_{33}} &{} {{{\bar{\Lambda }} }_{34}} \\ * &{} * &{} * &{} {{{\bar{\Lambda }} }_{44}} \\ \end{matrix} \right] , \end{aligned}$$

$$\begin{aligned} {{{\bar{\Lambda }} }_{11}}= & {} A{{Y}^{T}}+Y{{A}^{T}}-BG-{{G}^{T}}{{B}^{T}}+{{\hat{Q}}_{1}}+{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{1}}}} -{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{12}}= & {} {{A}_{d1}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{1}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{13}}= & {} {{A}_{d2}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{14}}= & {} -{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+\hat{P}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{22}}= & {} {{A}_{d1}}{{Y}^{T}}+YA_{d1}^{T}-{{\hat{Q}}_{1}}-{{\hat{R}}_{{{\tau }_{1}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{23}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d1}^{T}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{24}}= & {} -{{Y}^{T}}+YA_{d1}^{T}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{33}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d2}^{T}-{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{34}}= & {} -{{Y}^{T}}+YA_{d2}^{T}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Lambda }} }_{44}}= & {} -{{Y}^{T}}-Y+\tau _{1}^{2}{{\hat{R}}_{{{\tau }_{1}}}}+\tau _{2}^{2}{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{i=1}^{3}{{{\varepsilon }_{i}}H{{H}^{T}}},\\ {{{\bar{\Gamma }} }_{1}}= & {} {{\left[ \begin{matrix} {{E}_{1}}{{Y}^{T}} &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}, {{{\bar{\Gamma }} }_{2}}={{\left[ \begin{matrix} {{0}_{1n\times 1n}} &{} {{E}_{2}}{{Y}^{T}} &{} {{0}_{1n\times 2n}} \\ \end{matrix} \right] }^{T}},\\ {{{\bar{\Gamma }} }_{3}}= & {} {{\left[ \begin{matrix} {{0}_{1n\times 2n}} &{} {{E}_{3}}{{Y}^{T}} &{} {{0}_{1n\times 1n}} \\ \end{matrix} \right] }^{T}}, {{{\bar{\Gamma }} }_{4}}={{\left[ \begin{matrix} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} \\ \end{matrix} \right] }^{T}},\\ {{{\bar{\Gamma }} }_{5}}= & {} {{\left[ \begin{matrix} C{{Y}^{T}} &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}, \end{aligned}$$

and n is the dimension of x(t).

The corresponding robust ${{H}_{\infty }}$ controller gain matrix is obtained as $K=G{{\left( {{Y}^{T}} \right) }^{-1}}$.

Proof

For investigating the stability of (15), the LK functional [39] is considered as:

$$\begin{aligned} V(t)={{x}^{T}}(t)Px(t)+\sum \limits _{k=1}^{2}{{{V}_{k}}(t)}, \end{aligned}$$

(18)

where

$$\begin{aligned} {{V}_{k}}(t)=\int \limits _{t-{{\tau }_{k}}}^{t}{{{x}^{T}}(s){{Q}_{k}}x(s)ds}+{{\tau }_{k}}\int \limits _{t-{{\tau }_{k}}}^{t}{\int \limits _{s }^{t}{{{{\dot{x}}}^{T}}(\phi ){{R}_{{{\tau }_{k}}}}\dot{x}(\phi )d\phi ds }}. \end{aligned}$$

Differentiating (18) yields:

$$\begin{aligned} \dot{V}(t)= & {} 2{{x}^{T}}(t)P\dot{x}(t)+{{\dot{x}}^{T}}(t)\left[ \sum \limits _{k=1}^{2}{\tau _{k}^{2}{{R}_{{{\tau }_{k}}}}} \right] \dot{x}(t) \nonumber \\{} & {} +\sum \limits _{k=1}^{2}\left[ {{{x}^{T}}(t){{Q}_{k}}x(t)}-{{{x}^{T}}(t-{{\tau }_{k}}){{Q}_{k}}x(t-{{\tau }_{k}})} \right. \nonumber \\{} & {} \left. -{{{\tau }_{k}}\int \limits _{t-{{\tau }_{k}}}^{t}{{{{\dot{x}}}^{T}}(s ){{R}_{{{\tau }_{k}}}}\dot{x}(s)ds }}\right] . \end{aligned}$$

(19)

Referring to Lemma 1 of [40], the integral terms in (19) may be replaced with inequalities as:

$$\begin{aligned} \dot{V}(t)\le & {} 2{{x}^{T}}(t)P\dot{x}(t)+{{\dot{x}}^{T}}(t)\left[ \sum \limits _{k=1}^{2}{\tau _{k}^{2}{{R}_{{{\tau }_{k}}}}} \right] \dot{x}(t) \nonumber \\{} & {} +\sum \limits _{k=1}^{2}\left[ {{{x}^{T}}(t){{Q}_{k}}x(t)-{{x}^{T}}(t-{{\tau }_{k}}){{Q}_{k}}x(t-{{\tau }_{k}})} \right. \nonumber \\{} & {} \left. +{{{\left[ \begin{matrix} x(t) \\ x(t-{{\tau }_{k}}) \\ \end{matrix} \right] }^{T}}\left[ \begin{matrix} -{{R}_{{{\tau }_{k}}}} &{} {{R}_{{{\tau }_{k}}}} \\ * &{} -{{R}_{{{\tau }_{k}}}} \\ \end{matrix} \right] \left[ \begin{matrix} x(t) \\ x(t-{{\tau }_{k}}) \\ \end{matrix} \right] }\right] . \end{aligned}$$

(20)

Stabilization condition requires information regarding system dynamics. Therefore, the zero-valued quadratic formulation of IPS dynamics (15) is considered to incorporate IPS dynamics in the stabilization criterion instead of replacing $\dot{x}(t)$ in (20) directly by using (15), i.e.:

$$\begin{aligned} 2{{\xi }^{T}}\left( t \right) S\left[ -\dot{x}(t)+\hat{A}(t)x(t)+{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}})\right. \nonumber \\ \left. +{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}})-BKx(t)+\hat{D}w(t) \right] =0, \end{aligned}$$

(21)

where $\xi \left( t \right) ={{\left[ \begin{matrix} {{x}^{T}}(t) &{} {{x}^{T}}(t-{{\tau }_{1}}) &{} {{x}^{T}}(t-{{\tau }_{2}}) &{} {{{\dot{x}}}^{T}}(t) \\ \end{matrix} \right] }^{T}}$, and $S={{\left[ \begin{matrix} S_{1}^{T} &{} S_{2}^{T} &{} S_{3}^{T} &{} S_{4}^{T} \\ \end{matrix} \right] }^{T}}$. $S_1$, $S_2$, $S_3$ and $S_4$ are proper dimensional arbitrary matrices. The above zero term (21) is able to fulfill the requirement of involving system states in stability condition. One can rewrite (21) by splitting certain and uncertain terms as:

$$\begin{aligned} 2{{\xi }^{T}}(t)S\left\{ -\dot{x}(t)+Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})+{{A}_{d2}}x(t-{{\tau }_{2}})\right. \left. -BKx(t)+{\hat{D}}w(t) \right\} \nonumber \\ +2{{\xi }^{T}}(t)S\left\{ \Delta Ax(t)\right. \left. +\Delta {{A}_{d1}}x(t-{{\tau }_{1}})+\Delta {{A}_{d2}}x(t-{{\tau }_{2}}) \right\} =0. \end{aligned}$$

(22)

The certain terms in (22) can be written as:

$$\begin{aligned} 2{{\xi }^{T}}(t)S\left\{ -\dot{x}(t)+Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})+{{A}_{d2}}x(t-{{\tau }_{2}})\right. \nonumber \\ \left. -BKx(t)+{\hat{D}}w(t) \right\} ={{\xi }^{T}}(t){\bar{\Theta }} \xi (t)+2{{\xi }^{T}}(t){\bar{\theta }} w(t), \end{aligned}$$

(23)

where ${\bar{\Theta }}={{\left[ {{{\bar{\Theta }} }_{lm}} \right] }_{l,m=1,2,3,4}}$,

$$\begin{aligned} {{{\bar{\Theta }} }_{11}}= & {} {{S}_{1}}A+{{A}^{T}}S_{1}^{T}-{{S}_{1}}BK-{{K}^{T}}{{B}^{T}}S_{1}^{T}, \\ {{{\bar{\Theta }} }_{12}}= & {} {{S}_{1}}{{A}_{d1}}+{{A}^{T}}S_{2}^{T}-{{K}^{T}}{{B}^{T}}S_{2}^{T}, \\ {{{\bar{\Theta }} }_{13}}= & {} {{S}_{1}}{{A}_{d2}}+{{A}^{T}}S_{3}^{T}-{{K}^{T}}{{B}^{T}}S_{3}^{T}, \\ {{{\bar{\Theta }} }_{14}}= & {} -{{S}_{1}}+{{A}^{T}}S_{4}^{T}-{{K}^{T}}{{B}^{T}}S_{4}^{T},\\ {{{\bar{\Theta }} }_{22}}= & {} {{S}_{2}}{{A}_{d1}}+A_{d1}^{T}S_{2}^{T}, {{{\bar{\Theta }} }_{23}}={{S}_{2}}{{A}_{d2}}+A_{d1}^{T}S_{3}^{T}, \\ {{{\bar{\Theta }} }_{24}}= & {} -{{S}_{2}}+A_{d1}^{T}S_{4}^{T}, {{{\bar{\Theta }} }_{33}}={{S}_{3}}{{A}_{d2}}+A_{d2}^{T}S_{3}^{T},\\ {{{\bar{\Theta }} }_{34}}= & {} -{{S}_{3}}+A_{d2}^{T}S_{4}^{T}, {{{\bar{\Theta }} }_{44}}=-{{S}_{4}}-S_{4}^{T},\\ {\bar{\theta }}= & {} {{\left[ \begin{matrix} {{\hat{D}}^{T}}S_{1}^{T} &{} {{\hat{D}}^{T}}S_{2}^{T} &{} {{\hat{D}}^{T}}S_{3}^{T} &{} {{\hat{D}}^{T}}S_{4}^{T} \\ \end{matrix} \right] }^{T}}. \end{aligned}$$

The second term in the right hand side (RHS) of (23) can be expressed as [41]:

$$\begin{aligned} 2{{\xi }^{T}}(t){\bar{\theta }} w(t)\le {{\gamma }^{-2}}{{\xi }^{T}}(t){\bar{\theta }} {{{\bar{\theta }} }^{T}}\xi (t)+{{\gamma }^{2}}{{w}^{T}}(t)w(t). \end{aligned}$$

(24)

Substituting (24) into (23) yields:

$$\begin{aligned} 2{{\xi }^{T}}(t)S\left\{ -\dot{x}(t)+Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})\right. \left. +{{A}_{d2}}x(t-{{\tau }_{2}})-BKx(t)+\hat{D}w(t) \right\} \nonumber \\ \le {{\xi }^{T}}(t){\bar{\Theta }} \xi (t) +{{\gamma }^{-2}}{{\xi }^{T}}(t){\bar{\theta }} {{{\bar{\theta }} }^{T}}\xi (t)+{{\gamma }^{2}}{{w}^{T}}(t)w(t). \end{aligned}$$

(25)

By following Lemma 2 of [40], the uncertain terms of (22) may be represented as:

$$\begin{aligned} 2{{\xi }^{T}}(t)S\left\{ \left( HF(t){{E}_{1}} \right) x(t)\right. \left. +\left( HF(t){{E}_{2}} \right) x(t-{{\tau }_{1}})+\left( HF(t){{E}_{3}} \right) x(t-{{\tau }_{2}}) \right\} \nonumber \\ \le \sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{\xi }^{T}}(t){\bar{\varphi }} \xi (t)}+{{\xi }^{T}}\left( t \right) \sum \limits _{j=1}^{3}{{{\Xi }_{j}}}\xi \left( t \right) , \end{aligned}$$

(26)

where

$$\begin{aligned} {\bar{\varphi }}= & {} SH{{H}^{T}}{{S}^{T}}, {{\Xi }_{1}} = \left[ \begin{matrix} \varepsilon _{1}^{-1}E_{1}^{T}{{E}_{1}} &{} {{0}_{1n\times 3n}} \\ {{0}_{3n\times 1n}} &{} {{0}_{3n\times 3n}} \\ \end{matrix} \right] ,\\ {{\Xi }_{2}}= & {} \left[ \begin{matrix} {{0}_{1n\times 1n}} &{} {{0}_{1n\times 1n}} &{} {{0}_{1n\times 2n}} \\ {{0}_{1n\times 1n}} &{} \varepsilon _{2}^{-1}E_{2}^{T}{{E}_{2}} &{} {{0}_{1n\times 2n}} \\ {{0}_{2n\times 1n}} &{} {{0}_{2n\times 1n}} &{} {{0}_{2n\times 2n}} \\ \end{matrix} \right] , {{\Xi }_{3}} = \left[ \begin{matrix} {{0}_{2n\times 2n}} &{} {{0}_{2n\times 1n}} &{} {{0}_{2n\times 1n}} \\ {{0}_{1n\times 2n}} &{} \varepsilon _{3}^{-1}E_{3}^{T}{{E}_{3}} &{} {{0}_{1n\times 1n}} \\ {{0}_{1n\times 2n}} &{} {{0}_{1n\times 1n}} &{} {{0}_{1n\times 1n}} \\ \end{matrix} \right] . \end{aligned}$$

Substituting (25) and (26) into (22) yields:

$$\begin{aligned} {{\xi }^{T}}(t){\bar{\Theta }} \xi (t)+{{\gamma }^{-2}}{{\xi }^{T}}(t){\bar{\theta }} {{{\bar{\theta }} }^{T}}\xi (t)+{{\gamma }^{2}}{{w}^{T}}(t)w(t)\nonumber \\ +\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{\xi }^{T}}(t){\bar{\varphi }} \xi (t)}+{{\xi }^{T}}\left( t \right) \sum \limits _{j=1}^{3}{{{\Xi }_{j}}}\xi \left( t \right) \ge 0. \end{aligned}$$

(27)

Addition of (20) and (27) gives:

$$\begin{aligned} \dot{V}(t)\le {{\xi }^{T}}\left( t \right) \left\{ {\bar{\Psi }} +\sum \limits _{j=1}^{3}{{{\Xi }_{j}}}+{{\gamma }^{-2}}{\bar{\theta }} {{{\bar{\theta }} }^{T}} \right\} \xi \left( t \right) +{{\gamma }^{2}}{{w}^{T}}(t)w(t), \end{aligned}$$

(28)

where ${\bar{\Psi }} ={{\left[ {{{\bar{\Psi }} }_{lm}} \right] }_{l,m=1,2,3,4}},$

$$\begin{aligned} {{{\bar{\Psi }} }_{11}}= & {} {{{\bar{\Theta }} }_{11}}+{{Q}_{1}}+{{Q}_{2}}-{{R}_{{{\tau }_{1}}}}-{{R}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{1}}H{{H}^{T}}S_{1}^{T}}, \\ {{{\bar{\Psi }} }_{12}}= & {} {{{\bar{\Theta }} }_{12}}+{{R}_{{{\tau }_{1}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{1}}H{{H}^{T}}S_{2}^{T}},\\ {{{\bar{\Psi }} }_{13}}= & {} {{{\bar{\Theta }} }_{13}}+{{R}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{1}}H{{H}^{T}}S_{3}^{T}}, \\ {{{\bar{\Psi }} }_{14}}= & {} {{{\bar{\Theta }} }_{14}}+P+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{1}}H{{H}^{T}}S_{4}^{T}},\\ {{{\bar{\Psi }} }_{22}}= & {} {{{\bar{\Theta }} }_{22}}-{{Q}_{1}}-{{R}_{{{\tau }_{1}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{2}}H{{H}^{T}}S_{2}^{T}}, \\ {{{\bar{\Psi }} }_{23}}= & {} {{{\bar{\Theta }} }_{23}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{2}}H{{H}^{T}}S_{3}^{T}},\\ {{{\bar{\Psi }} }_{24}}= & {} {{{\bar{\Theta }} }_{24}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{2}}H{{H}^{T}}S_{4}^{T}}, \\ {{{\bar{\Psi }} }_{33}}= & {} {{{\bar{\Theta }} }_{33}}-{{Q}_{2}}-{{R}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{3}}H{{H}^{T}}S_{3}^{T}},\\ {{{\bar{\Psi }} }_{34}}= & {} {{{\bar{\Theta }} }_{34}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{3}}H{{H}^{T}}S_{4}^{T}}, \\ {{{\bar{\Psi }} }_{44}}= & {} {{{\bar{\Theta }} }_{44}}+\tau _{1}^{2}{{R}_{{{\tau }_{1}}}}+\tau _{2}^{2}{{R}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}{{S}_{4}}H{{H}^{T}}S_{4}^{T}}. \end{aligned}$$

A cost function is obtained from (10) for the investigation of ${{H}_{\infty }}$ criterion, as:

$$\begin{aligned} {{J}_{yw}}=\int \limits _{0}^{\infty }{\left[ {{y}^{T}}(t)y(t)-{{\gamma }^{2}}{{w}^{T}}(t)w(t) \right] dt}. \end{aligned}$$

(29)

Closed-loop IPS (15) satisfies the criterion (10) only when ${{J}_{yw}}\le 0$. For initial condition $V(0)=0$ and since $V(\infty )\ge 0$, one can write:

$$\begin{aligned} {{J}_{yw}}\le \int \limits _{0}^{\infty }{\left[ {{y}^{T}}(t)y(t)-{{\gamma }^{2}}{{w}^{T}}(t)w(t)+\dot{V}(t) \right] dt} \end{aligned}$$

(30)

Substituting (28) into (30), one obtains:

$$\begin{aligned} {{J}_{yw}}\le \int \limits _{0}^{\infty }{{{\xi }^{T}}(t){\bar{\Omega }} \xi (t)dt}, \end{aligned}$$

(31)

where ${\bar{\Omega }} ={\bar{\Psi }} +\sum \limits _{j=1}^{3}{{{\Xi }_{j}}}+{{\gamma }^{-2}}{\bar{\theta }} {{{\bar{\theta }} }^{T}}+\hat{C}{{\hat{C}}^{T}}$ and $\hat{C}={{\left[ \begin{matrix} C &{} 0 &{} 0 &{} 0 \\ \end{matrix} \right] }^{T}}$.

The condition ${{J}_{yw}}\le 0$ is satisfied, if there is:

$$\begin{aligned} {\bar{\Omega }} <0 \end{aligned}$$

(32)

Now, using Schur complement [41] in inequality (32), we can write:

$$\begin{aligned} \left[ \begin{matrix} {\bar{\Psi }} &{} {{{\bar{\Phi }} }_{1}} &{} {{{\bar{\Phi }} }_{2}} &{} {{{\bar{\Phi }} }_{3}} &{} {{{\bar{\Phi }} }_{4}} &{} {{{\bar{\Phi }} }_{5}} \\ * &{} -{{\varepsilon }_{1}}I &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} -{{\varepsilon }_{2}}I &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} -{{\varepsilon }_{3}}I &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} -{{\gamma }^{2}}I &{} 0 \\ * &{} * &{} * &{} * &{} * &{} -I \\ \end{matrix} \right] <0, \end{aligned}$$

(33)

where ${{{\bar{\Phi }} }_{1}}={{\left[ \begin{matrix} {{E}_{1}} &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}$, ${{{\bar{\Phi }} }_{2}}={{\left[ \begin{matrix} {{0}_{1n\times 1n}} &{} {{E}_{2}} &{} {{0}_{1n\times 2n}} \\ \end{matrix} \right] }^{T}}$, ${{{\bar{\Phi }} }_{3}}={{\left[ \begin{matrix} {{0}_{1n\times 2n}} &{} {{E}_{3}} &{} {{0}_{1n\times 1n}} \\ \end{matrix} \right] }^{T}}$, ${{{\bar{\Phi }} }_{4}}={{\left[ \begin{matrix} {{\hat{D}}^{T}}S_{1}^{T} &{} {{\hat{D}}^{T}}S_{2}^{T} &{} {{\hat{D}}^{T}}S_{3}^{T} &{} {{\hat{D}}^{T}}S_{4}^{T} \\ \end{matrix} \right] }^{T}}$, ${{{\bar{\Phi }} }_{5}}={{\left[ \begin{matrix} C &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}}$.

Considering ${{S}_{1}}={{S}_{2}}={{S}_{3}}={{S}_{4}}=S$, pre- and post-multiplying with $diag\left\{ \begin{matrix} {{S}^{-1}} &{} {{S}^{-1}} &{} {{S}^{-1}} &{} {{S}^{-1}} &{} I &{} I &{} I &{} I &{} I \\ \end{matrix} \right\}$ and its transpose in (33), and finally adopting variables change as ${{S}^{-1}}=Y$, $K{{\left( {{S}^{-1}} \right) }^{T}}=K{{Y}^{T}}=G$, ${{S}^{-1}}{{Q}_{1}}{{\left( {{S}^{-1}} \right) }^{T}}={{\hat{Q}}_{1}}$, ${{S}^{-1}}{{Q}_{2}}{{\left( {{S}^{-1}} \right) }^{T}}={{\hat{Q}}_{2}}$, ${{S}^{-1}}{{R}_{{{\tau }_{1}}}}{{\left( {{S}^{-1}} \right) }^{T}}={{\hat{R}}_{{{\tau }_{1}}}}$, ${{S}^{-1}}{{R}_{{{\tau }_{2}}}}{{\left( {{S}^{-1}} \right) }^{T}}={{\hat{R}}_{{{\tau }_{2}}}}$,

${{S}^{-1}}P{{\left( {{S}^{-1}} \right) }^{T}}=\hat{P}$, one obtains (17). Hence, proof of Theorem 1 is completed. $\square$

Remark 2

For a specific value of $\gamma$, ${{H}_{\infty }}$ controller gain matrix K may be obtained from the feasible solution of LMI (17). As per (10), the minimum value of $\gamma$ should be utilized for computing controller gains to have minimal disturbance effect on system response. The minimum $\gamma$ value can be obtained by minimizing ${{\gamma }^{2}}$ of LMI (17) and K can then be easily obtained from the solution of LMI (17) by using this $\gamma$ value. But, this process results in high value of controller gains [19]. From Theorem 1, it is known that variables G and Y of LMI (17) are involved in the calculation of K. So, minimization of $\left\| G \right\|$ and $\left\| {{Y}^{-1}} \right\|$ can result in the values of K within limits. Based on the analysis, following LMI optimization problem is designed, and solution of this optimization problem gives minimum $\gamma$ as well as stabilizing gain K.

LMI Optimization Problem 1:

Min ${\gamma }^{2}+{\bar{g}}+{\bar{y}}$ Subject to (17), $\left[ \begin{matrix} {\bar{g}}I &{} G \\ * &{} I \\ \end{matrix} \right] >0$ and $\left[ \begin{matrix} Y &{} I \\ * &{} {\bar{y}}I \\ \end{matrix} \right] >0.$

where ${\bar{y}}$ and ${\bar{g}}$ denote the matrix norms $\left\| {{Y}^{-1}} \right\|$ and $\left\| G \right\|$, respectively.

Remark 3

The LK functional defined in (18) has double integral terms whereas the LK functionals chosen in [6, 21] have single integral terms. Use of multiple integral terms in LK functional reduces conservatism of delay-dependent stabilization criterion. On the other hand, it increases the computational burden. To reduce computational burden as well as conservatism of stabilization criterion, LK functional defined in this paper is restricted to double integral terms.

The following corollary presents ${{H}_{\infty }}$ performance based criterion for stabilization of delayed IPS (16).

Corollary 1

Closed-loop IPS (16) satisfies $H_{\infty }$ criterion $\left\| {{T}_{wy}} \right\| \le \gamma$ and $\gamma >0$, for positive definite matrices $\hat{P}$, ${{\hat{Q}}_{1}}$, ${{\hat{Q}}_{2}}$, ${{\hat{R}}_{{{\tau }_{1}}}}$, ${{\hat{R}}_{{{\tau }_{2}}}}$ and arbitrary matrix G, such that an LMI holds as follows:

$$\begin{aligned} \left[ \begin{matrix} {\tilde{\Lambda }} &{} {\tilde{D}} &{} {\tilde{C}}\\ * &{} -{{\gamma }^{2}} &{} 0\\ * &{} * &{} -I\\ \end{matrix}\right] <0, \end{aligned}$$

(34)

where${\tilde{\Lambda }}=\left[ \begin{matrix} {{{\tilde{\Lambda }} }_{11}} &{} {{{\tilde{\Lambda }} }_{12}} &{} {{{\tilde{\Lambda }} }_{13}} &{} {{{\tilde{\Lambda }} }_{14}} \\ * &{} {{{\tilde{\Lambda }} }_{22}} &{} {{{\tilde{\Lambda }} }_{23}} &{} {{{\tilde{\Lambda }} }_{24}} \\ * &{} * &{} {{{\tilde{\Lambda }} }_{33}} &{} {{{\tilde{\Lambda }} }_{34}} \\ * &{} * &{} * &{} {{{\tilde{\Lambda }} }_{44}} \\ \end{matrix} \right]$,

$$\begin{aligned} {{{\tilde{\Lambda }} }_{11}}= & {} A{{Y}^{T}}+Y{{A}^{T}}-BG-{{G}^{T}}{{B}^{T}}+{{\hat{Q}}_{1}}+{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{1}}}}-{{\hat{R}}_{{{\tau }_{2}}}},\\ {{{\tilde{\Lambda }} }_{12}}= & {} {{A}_{d1}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{1}}}},\\ {{{\tilde{\Lambda }} }_{13}}= & {} {{A}_{d2}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{2}}}},\\ {{{\tilde{\Lambda }} }_{14}}= & {} -{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+\hat{P},\\ {{{\tilde{\Lambda }} }_{22}}= & {} {{A}_{d1}}{{Y}^{T}}+YA_{d1}^{T}-{{\hat{Q}}_{1}}-{{\hat{R}}_{{{\tau }_{1}}}},\\ {{{\tilde{\Lambda }} }_{23}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d1}^{T}, {{{\tilde{\Lambda }} }_{24}}=-{{Y}^{T}}+YA_{d1}^{T},\\ {{{\tilde{\Lambda }} }_{33}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d2}^{T}-{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{2}}}}, {{{\tilde{\Lambda }} }_{34}}=-{{Y}^{T}}+YA_{d2}^{T},\\ {{{\tilde{\Lambda }} }_{44}}= & {} -{{Y}^{T}}-Y+\tau _{1}^{2}{{\hat{R}}_{{{\tau }_{1}}}}+\tau _{2}^{2}{{\hat{R}}_{{{\tau }_{2}}}},\\ {\tilde{D}}= & {} {{\left[ \begin{matrix}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}}\\ \end{matrix}\right] }^{T}}, {\tilde{C}}={{\left[ \begin{matrix}C{{Y}^{T}}&{}{{0}_{1n\times 3n}}\\ \end{matrix}\right] }^{T}}. \end{aligned}$$

The corresponding ${{H}_{\infty }}$ controller gain matrix is obtained as $K=G{{\left( {{Y}^{T}} \right) }^{-1}}$.

Proof

One can prove Corollary 1 by adopting similar procedures as in proof of Theorem 1 without considering uncertainties of IPS parameters. $\square$

3.1.3 Sliding mode control law based on state feedback control strategy

Theorem 2

Switching control law satisfying reaching condition ${{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0$ is given by:

$$\begin{aligned} {{u}_{i}}(t)=-{{K}_{i}}x(t)-{{b}_{i}}{{({{J}_{i}}{{B}_{i}})}^{-1}}\left\| {{J}_{i}}{{D}_{i}} \right\| \left( \frac{{{\sigma }_{i}}(t)}{\left\| {{\sigma }_{i}}(t) \right\| +\zeta } \right) , \end{aligned}$$

(35)

where $\zeta$ is a constant having positive value close to zero.

Proof

Proof of Theorem 2 is given in Appendix. $\square$

Remark 4

Signum function of switching surface is generally used in sliding mode controller structure to drive system trajectory into the predefined sliding surface [6, 31, 32]. However, such controller has chattering issue, and therefore, use of signum function is avoided in design of sliding mode controller (35) to reduce the chattering effect.

3.2 Robust ${{H}_{\infty }}$ delayed feedback sliding mode controller designing strategy

3.2.1 Selection of switching surface

The switching surface is selected by including artificial delay to improve the performance of closed-loop IPS, as:

$$\begin{aligned} \sigma (t)= & {} Jx(t)-\int \limits _{0}^{t}{J\hat{A}(t)x(\tau )d\tau }+\int \limits _{0}^{t}{JBKx(\tau )d\tau }+ \int \limits _{0}^{t-h}{JB{{K}_{h}}x(\tau )d\tau }\nonumber \\{} & {} -\int \limits _{0}^{t-{{\tau }_{1}}}{J{{\hat{A}}_{d1}}(t)x(\tau )d\tau } -\int \limits _{0}^{t-{{\tau }_{2}}}{J{{\hat{A}}_{d2}}(t)x(\tau )d\tau }, \end{aligned}$$

(36)

where ${K}_{h}$ is a constant matrix, and h is the artificial delay chosen by the control designer. Derivative of (36) gives:

$$\begin{aligned} {\dot{\sigma }}(t)= & {} J\dot{x}(t)-J\hat{A}(t)x(t)+JBKx(t)+JB{{K}_{h}}x(t-h)\nonumber \\{} & {} -J{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}})-J{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}}). \end{aligned}$$

(37)

Substituting (8) into (37) yields:

$$\begin{aligned} {\dot{\sigma }}(t)=JBKx(t)+JB{{K}_{h}}x(t-h)+JBu(t)+JDw(t). \end{aligned}$$

(38)

Considering ${\dot{\sigma }}(t)=0$, the equivalent controller is derived from (38) as:

$$\begin{aligned} u{{(t)}_{equ}}=-Kx(t)-{{K}_{h}}x(t-h)-{{(JB)}^{-1}}JDw(t). \end{aligned}$$

(39)

The sliding mode closed-loop delayed IPS is obtained by substituting (39) into (8) as:

$$\begin{aligned} \dot{x}(t)= & \, \, {} \hat{A}(t)x(t)+{{\hat{A}}_{d1}}(t)x(t-{{\tau }_{1}})+{{\hat{A}}_{d2}}(t)x(t-{{\tau }_{2}})\nonumber \\{} & {} -BKx(t)-B{{K}_{h}}x(t-h)+\hat{D}w(t), \end{aligned}$$

(40)

where matrix $\hat{D}=D-B{{(JB)}^{-1}}JD$. System (40) without parametric uncertainty can be written as:

$$\begin{aligned} \dot{x}(t)= & {} Ax(t)+{{A}_{d1}}x(t-{{\tau }_{1}})+{{A}_{d2}}x(t-{{\tau }_{2}})\nonumber \\{} & {} -BKx(t)-B{{K}_{h}}x(t-h)+\hat{D}w(t). \end{aligned}$$

(41)

3.2.2 Robust ${{H}_{\infty }}$ stabilization criterion

Theorem 3

Closed-loop IPS (40) satisfies $H_{\infty }$ criterion $\left\| {{T}_{wy}} \right\| \le \gamma$ and $\gamma >0$, for positive definite matrices $\hat{P}$, ${{\hat{Q}}_{1}}$, ${{\hat{Q}}_{2}}$, ${{\hat{Q}}_{h}}$, ${{\hat{R}}_{{{\tau }_{1}}}}$, ${{\hat{R}}_{{{\tau }_{2}}}}$, ${{\hat{R}}_{h}}$ and arbitrary matrices G and V, such that an LMI holds as follows:

$$\begin{aligned} \left[ \begin{matrix} {\hat{\Omega }} &{} {{{\hat{\Theta }} }_{1}} &{} {{{\hat{\Theta }} }_{2}} &{} {{{\hat{\Theta }} }_{3}} &{} {{{\hat{\Theta }} }_{4}} &{} {{{\hat{\Theta }} }_{5}} \\ * &{} -{{\varepsilon }_{1}}I &{} 0 &{} 0 &{} 0 &{} 0 \\ * &{} * &{} -{{\varepsilon }_{2}}I &{} 0 &{} 0 &{} 0 \\ * &{} * &{} * &{} -{{\varepsilon }_{3}}I &{} 0 &{} 0 \\ * &{} * &{} * &{} * &{} -{{\gamma }^{2}} &{} 0 \\ * &{} * &{} * &{} * &{} * &{} -I \\ \end{matrix} \right] <0, \end{aligned}$$

(42)

where ${\hat{\Omega }} =\left[ \begin{matrix} {{{\hat{\Omega }} }_{11}} &{} {{{\hat{\Omega }} }_{12}} &{} {{{\hat{\Omega }} }_{13}} &{} {{{\hat{\Omega }} }_{14}} &{} {{{\hat{\Omega }} }_{15}} \\ * &{} {{{\hat{\Omega }} }_{22}} &{} {{{\hat{\Omega }} }_{23}} &{} {{{\hat{\Omega }} }_{24}} &{} {{{\hat{\Omega }} }_{25}} \\ * &{} * &{} {{{\hat{\Omega }} }_{33}} &{} {{{\hat{\Omega }} }_{34}} &{} {{{\hat{\Omega }} }_{35}} \\ * &{} * &{} * &{} {{{\hat{\Omega }} }_{44}} &{} {{{\hat{\Omega }} }_{45}} \\ * &{} * &{} * &{} * &{} {{{\hat{\Omega }} }_{55}} \\ \end{matrix} \right] ,$

$$\begin{aligned} {{{\hat{\Omega }} }_{11}}= & {} A{{Y}^{T}}+Y{{A}^{T}}-BG-{{G}^{T}}{{B}^{T}}+{{\hat{Q}}_{1}}+{{\hat{Q}}_{2}}+{{\hat{Q}}_{h}} -{{\hat{R}}_{h}}-{{\hat{R}}_{{{\tau }_{1}}}}-{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{12}}= & {} {{A}_{d1}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{1}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{13}}= & {} {{A}_{d2}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{14}}= & {} Y{{A}^{T}}-BV-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{h}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{15}}= & {} -{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+\hat{P}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{22}}= & {} {{A}_{d1}}{{Y}^{T}}+YA_{d1}^{T}-{{\hat{Q}}_{1}}-{{\hat{R}}_{{{\tau }_{1}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{23}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d1}^{T}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{24}}= & {} -BV+YA_{d1}^{T}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{25}}= & {} -{{Y}^{T}}+YA_{d1}^{T}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }}}_{33}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d2}^{T}-{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{2}}}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{34}}= & {} -BV+YA_{d2}^{T}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{35}}= & {} -{{Y}^{T}}+YA_{d2}^{T}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{44}}= & {} -BV-{{V}^{T}}{{B}^{T}}-{{\hat{Q}}_{h}}-{{\hat{R}}_{h}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{45}}= & {} -{{Y}^{T}}-{{V}^{T}}{{B}^{T}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ {{{\hat{\Omega }} }_{55}}= & {} -{{Y}^{T}}-Y+\tau _{1}^{2}{{\hat{R}}_{{{\tau }_{1}}}}+\tau _{2}^{2}{{\hat{R}}_{{{\tau }_{2}}}}+{{h}^{2}}{{\hat{R}}_{h}}+\sum \limits _{j=1}^{3}{{{\varepsilon }_{j}}H{{H}^{T}}},\\ \end{aligned}$$

$$\begin{aligned} {{{\hat{\Theta }} }_{1}}= & {} {{\left[ \begin{matrix} {{E}_{1}}{{Y}^{T}} &{} {{0}_{1n\times 4n}} \\ \end{matrix} \right] }^{T}}, {{{\hat{\Theta }} }_{2}}={{\left[ \begin{matrix} {{0}_{1n\times 1n}} &{} {{E}_{2}}{{Y}^{T}} &{} {{0}_{1n\times 3n}} \\ \end{matrix} \right] }^{T}},\\ {{{\hat{\Theta }} }_{3}}= & {} {{\left[ \begin{matrix} {{0}_{1n\times 2n}} &{} {{E}_{3}}{{Y}^{T}} &{} {{0}_{1n\times 2n}} \\ \end{matrix} \right] }^{T}}, \\ {{{\hat{\Theta }} }_{4}}= & {} {{\left[ \begin{matrix} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} &{} {{\hat{D}}^{T}} \\ \end{matrix} \right] }^{T}},\\ {{{\hat{\Theta }} }_{5}}= & {} {{\left[ \begin{matrix} C{{Y}^{T}} &{} {{0}_{1n\times 4n}} \\ \end{matrix} \right] }^{T}} \end{aligned}$$

The corresponding robust ${{H}_{\infty }}$ controller gain matrices are obtained as $K=G{{\left( {{Y}^{T}} \right) }^{-1}}$ and ${{K}_{h}}=V{{\left( {{Y}^{T}} \right) }^{-1}}$.

Proof

To investigate the stability of (40), the LK functional is considered as:

$$\begin{aligned} V(t)={{x}^{T}}(t)Px(t)+{{V}_{h}}(t)+\sum \limits _{j=1}^{2}{{{V}_{j}}(t)}, \end{aligned}$$

(43)

where

$$\begin{aligned} {{V}_{h}}(t)=\int \limits _{t-h}^{t}{{{x}^{T}}(\theta ){{Q}_{h}}x(\theta )d\theta }+h\int \limits _{t-h}^{t}{\int \limits _{\theta }^{t}{{{{\dot{x}}}^{T}}(\phi ){{R}_{h}}\dot{x}(\phi )d\phi d\theta }}, \end{aligned}$$

and

$$\begin{aligned} {{V}_{j}}(t)=\int \limits _{t-{{\tau }_{j}}}^{t}{{{x}^{T}}(\theta ){{Q}_{j}}x(\theta )d\theta }+{{\tau }_{1}}\int \limits _{t-{{\tau }_{j}}}^{t}{\int \limits _{\theta }^{t}{{{{\dot{x}}}^{T}}(\phi ){{R}_{{{\tau }_{j}}}}\dot{x}(\phi )d\phi d\theta }}. \end{aligned}$$

Next, by implementing similar steps as in proof of Theorem 1, one obtains (42). This completes the proof of Theorem 3. $\square$

LMI Optimization Problem 2:

Min ${\gamma }^{2}+{\bar{g}}+{\bar{v}}+{\bar{y}}$

Subject to (42), $\left[ \begin{matrix} {\bar{g}}I &{} G \\ * &{} I \\ \end{matrix} \right] >0$, $\left[ \begin{matrix} {\bar{v}}I &{} V \\ * &{} I \\ \end{matrix} \right] >0$ and $\left[ \begin{matrix} Y &{} I \\ * &{} {\bar{y}}I \\ \end{matrix} \right] >0.$

where ${\bar{v}}$ represents the norm of matrix $\left\| V \right\|$. Stabilizing gains K and $K_h$ can be calculated by solving the above LMI optimization problem.

${{H}_{\infty }}$ performance based stabilization criterion for (41) can be deduced from Theorem 3. The following corollary presents the criterion for (41).

Corollary 2

Closed-loop IPS (41) satisfies $H_{\infty }$ criterion $\left\| {{T}_{wy}} \right\| \le \gamma$ and $\gamma >0$, for positive definite matrices $\hat{P}$, ${{\hat{Q}}_{1}}$, ${{\hat{Q}}_{2}}$, ${{\hat{Q}}_{h}}$, ${{\hat{R}}_{{{\tau }_{1}}}}$, ${{\hat{R}}_{{{\tau }_{2}}}}$, ${{\hat{R}}_{h}}$ and arbitrary matrices G and V, such that an LMI holds as follows:

$$\begin{aligned} \left[ \begin{matrix} {\tilde{\varphi }} &{} {\overset{\frown }{D}} &{} {\overset{\frown }{C}}\\ * &{} -{{\gamma }^{2}} &{} 0\\ * &{} * &{} -I\\ \end{matrix}\right] <0, \end{aligned}$$

(44)

where ${\tilde{\varphi }} =\left[ \begin{matrix} {{{\tilde{\varphi }} }_{11}} &{} {{{\tilde{\varphi }} }_{12}} &{} {{{\tilde{\varphi }} }_{13}} &{} {{{\tilde{\varphi }} }_{14}} &{} {{{\tilde{\varphi }} }_{15}} \\ * &{} {{{\tilde{\varphi }} }_{22}} &{} {{{\tilde{\varphi }} }_{23}} &{} {{{\tilde{\varphi }} }_{24}} &{} {{{\tilde{\varphi }} }_{25}} \\ * &{} * &{} {{{\tilde{\varphi }} }_{33}} &{} {{{\tilde{\varphi }} }_{34}} &{} {{{\tilde{\varphi }} }_{35}} \\ * &{} * &{} * &{} {{{\tilde{\varphi }} }_{44}} &{} {{{\tilde{\varphi }} }_{45}} \\ * &{} * &{} * &{} * &{} {{{\tilde{\varphi }} }_{55}} \\ \end{matrix} \right]$,

$$\begin{aligned} {{{\tilde{\varphi }} }_{11}}= & {} A{{Y}^{T}}+Y{{A}^{T}}-BG-{{G}^{T}}{{B}^{T}}+{{\hat{Q}}_{1}}+{{\hat{Q}}_{2}}+{{\hat{Q}}_{h}} -{{\hat{R}}_{h}}-{{\hat{R}}_{{{\tau }_{1}}}}-{{\hat{R}}_{{{\tau }_{2}}}},\\ {{{\tilde{\varphi }} }_{12}}= & {} {{A}_{d1}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{1}}}},\\ {{{\tilde{\varphi }} }_{13}}= & {} {{A}_{d2}}{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{{{\tau }_{2}}}},\\ {{{\tilde{\varphi }} }_{14}}= & {} Y{{A}^{T}}-BV-{{G}^{T}}{{B}^{T}}+{{\hat{R}}_{h}},\\ {{{\tilde{\varphi }} }_{15}}= & {} -{{Y}^{T}}+Y{{A}^{T}}-{{G}^{T}}{{B}^{T}}+\hat{P},\\ {{{\tilde{\varphi }} }_{22}}= & {} {{A}_{d1}}{{Y}^{T}}+YA_{d1}^{T}-{{\hat{Q}}_{1}}-{{\hat{R}}_{{{\tau }_{1}}}},\\ {{{\tilde{\varphi }} }_{23}}= & {} {{A}_{d2}}{{Y}^{T}}+YA_{d1}^{T}, {{{\tilde{\varphi }} }_{24}}=-BV+YA_{d1}^{T},\\ {{{\tilde{\varphi }} }_{25}}= & {} -{{Y}^{T}}+YA_{d1}^{T}, {{{\tilde{\varphi }} }_{33}}={{A}_{d2}}{{Y}^{T}}+YA_{d2}^{T}-{{\hat{Q}}_{2}}-{{\hat{R}}_{{{\tau }_{2}}}},\\ {{{\tilde{\varphi }} }_{34}}= & {} -BV+YA_{d2}^{T}, {{{\tilde{\varphi }} }_{35}}=-{{Y}^{T}}+YA_{d2}^{T},\\ {{{\tilde{\varphi }} }_{44}}= & {} -BV-{{V}^{T}}{{B}^{T}}-{{\hat{Q}}_{h}}-{{\hat{R}}_{h}}, {{{\tilde{\varphi }} }_{45}}=-{{Y}^{T}}-{{V}^{T}}{{B}^{T}},\\ {{{\tilde{\varphi }} }_{55}}= & {} -{{Y}^{T}}-Y+\tau _{1}^{2}{{\hat{R}}_{{{\tau }_{1}}}}+\tau _{2}^{2}{{\hat{R}}_{{{\tau }_{2}}}}+{{h}^{2}}{{\hat{R}}_{h}},\\ {\overset{\frown }{D}}= & {} {{\left[ \begin{matrix}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}} &{}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}}&{}{{{\hat{D}}}^{T}}\\ \end{matrix}\right] }^{T}}, {\overset{\frown }{C}}={{\left[ \begin{matrix}C{{Y}^{T}}&{}{{0}_{1n\times 4n}}\\ \end{matrix}\right] }^{T}}. \end{aligned}$$

The corresponding ${{H}_{\infty }}$ controller gain matrices are obtained as $K=G{{\left( {{Y}^{T}} \right) }^{-1}}$ and ${{K}_{h}}=V{{\left( {{Y}^{T}} \right) }^{-1}}$.

Proof

One may follow Theorem 3 to prove Corollary 2. $\square$

3.2.3 Sliding mode control law based on delayed feedback control strategy

Theorem 4

Switching control law satisfying reaching condition ${{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0$ is given by:

$$\begin{aligned} {{u}_{i}}(t)=-{{K}_{i}}x(t)-{{K}_{hi}}x(t-h) -{{b}_{i}}{{({{J}_{i}}{{B}_{i}})}^{-1}}\left\| {{J}_{i}}{{D}_{i}} \right\| \left( \frac{{{\sigma }_{i}}(t)}{\left\| {{\sigma }_{i}}(t) \right\| +\zeta } \right) , \end{aligned}$$

(45)

Proof

Proof of Theorem 4 is similar to that of Theorem 2, and thus no further description is given here. $\square$

Remark 5

The performance of the proposed SMC scheme may be improved by selecting a suitable h value which can improve the transient stability of a delayed system. It can also be used to improve the dynamic response of a system with time-delay.

4 Results analysis

4.1 Numerical example

A well-known numerical example of an IPS with two areas is considered for analyzing the performance of the proposed SMC schemes. Parameters of the IPS are given in Table 1 [19, 21].

Table 1 Parameters of the two-area IPS

The uncertainties in IPS parameters are assumed as $E_{1}=E_{2}=E_{3}=H=0.001I$ as described in (9). Time-delays of area-1 and area-2 are considered to be fixed as ${\tau }_{1}=0.1s$ and ${\tau }_{2}=0.2s$, respectively.

For the ${{H}_{\infty }}$ state feedback sliding mode controller (35) and ${{H}_{\infty }}$ delayed feedback sliding mode controller (45), the design parameters $\zeta$ and $b_{i}$ are set as 0.001 and 1, respectively, whereas matrix J is selected as:

$$\begin{aligned} J=\left[ \begin{matrix} 10 &{} 0 &{} -0.1 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 0 \\ 0 &{} 0 &{} 0 &{} 0 &{} 0 &{} 12 &{} 0 &{} 0.4 &{} 0 \\ \end{matrix} \right] . \end{aligned}$$

Besides $\zeta$, $b_{i}$ and J, controller gain matrix K is required to design controller (35), and an artificial delay (h) and controller gains (K and $K_{h}$) are needed to design controller (45). h is tuned at 0.86s, while LMI Optimization Problem 1 and Problem 2 are solved by using mincx solver of LMI control Toolbox in MATLAB to obtain ${{H}_{\infty }}$ performance index $\gamma$ and gains of controllers (35) and (45), respectively. $\gamma$ for controller (35) is computed as 1.4942 and the corresponding gain matrix K is presented in (46). Similarly, $\gamma$ for controller (45) is computed as 2.6686 and the corresponding gain matrices K and $K_{h}$ are presented in (47) and (48). Replacing (17) with (34) in LMI Optimization Problem 1 and (42) with (44) in LMI Optimization Problem 2, ${{H}_{\infty }}$ performance index and gains of controllers (35) and (45) are computed neglecting the uncertainties of IPS parameters. At this condition, $\gamma$ is obtained as 1.4938 and K is computed as (49) for controller (35). Similarly, for controller (45), $\gamma$ is obtained as 2.6679, and K and $K_{h}$ are computed as (50) and (51).

Table 2 A comparative study on the two-area closed-loop IPS without considering parametric uncertainty

$$\begin{aligned} \small {K}= & {} \left[ \begin{matrix} -1.4381 &{} -0.0708 &{} -0.0256 &{} -0.0892 &{} 0.0860 &{} -13.3958 &{} -0.1344 &{} -0.0709 &{} -0.7348 \\ -18.7202 &{} -0.2593 &{} -0.0540 &{} -0.6518 &{} -0.7116 &{} 40.1205 &{} 0.4610 &{} 0.4992 &{} 0.8281 \\ \end{matrix} \right] \end{aligned}$$

(46)

$$\begin{aligned} \small {K}= & {} \left[ \begin{matrix} 12.9569 &{} 0.1235 &{} 0.0090 &{} 0.9976 &{} 0.3836 &{} -46.6727 &{} -0.4983 &{} -0.3601 &{} -1.7016 \\ -39.3775 &{} -0.6914 &{} -0.1782 &{} -1.8792 &{} -1.2455 &{} 81.3836 &{} 0.9720 &{} 1.1957 &{} 1.7138 \\ \end{matrix} \right] \end{aligned}$$

(47)

$$\begin{aligned} \small {{K}_{h}}= & {} \left[ \begin{matrix} 3.3359 &{} 0.0373 &{} 0.0049 &{} 0.2395 &{} 0.0957 &{} -10.8436 &{} -0.1175 &{} -0.0914 &{} -0.3812 \\ -7.6956 &{} -0.1540 &{} -0.0439 &{} -0.3067 &{} -0.2435 &{} 11.8959 &{} 0.1536 &{} 0.2358 &{} 0.1204 \\ \end{matrix} \right] \end{aligned}$$

(48)

$$\begin{aligned} \small {K}= & {} \left[ \begin{matrix} -1.2050 &{} -0.0611 &{} -0.0221 &{} -0.0587 &{} 0.0883 &{} -13.4954 &{} -0.1348 &{} -0.0773 &{} -0.7272 \\ -18.9126 &{} -0.2665 &{} -0.0566 &{} -0.6697 &{} -0.7214 &{} 40.2924 &{} 0.4640 &{} 0.5123 &{} 0.8353 \\ \end{matrix} \right] \end{aligned}$$

(49)

$$\begin{aligned} \small {K}= & {} \left[ \begin{matrix} 13.0885 &{} 0.1440 &{} 0.0111 &{} 1.0001 &{} 0.3854 &{} -47.5581 &{} -0.5056 &{} -0.3468 &{} -1.6900 \\ -40.3655 &{} -0.7169 &{} -0.1827 &{} -1.8897 &{} -1.2745 &{} 80.9309 &{} 1.0033 &{} 1.2113 &{} 1.7092 \\ \end{matrix} \right] \end{aligned}$$

(50)

$$\begin{aligned} \small {{K}_{h}}= & {} \left[ \begin{matrix} 3.3394 &{} 0.0380 &{} 0.0103 &{} 0.2510 &{} 0.1151 &{} -11.0578 &{} -0.1207 &{} -0.1124 &{} -0.3707 \\ -7.9684 &{} -0.1498 &{} -0.0492 &{} -0.3119 &{} -0.2342 &{} 12.1195 &{} 0.1651 &{} 0.2224 &{} 0.1192 \\ \end{matrix} \right] \end{aligned}$$

(51)

Fig. 3

${{H}_{\infty }}$ state feedback SMC model for LFC of the two-area IPS

Fig. 4

${{H}_{\infty }}$ delayed feedback SMC model for LFC of the two-area IPS

In Table 2, a comparative study of h and $\gamma$ values is presented. Larger h and smaller $\gamma$ are obtained by using the proposed control approach in comparison to the other existing values [19, 21]. A larger value of $\sigma$ indicates better delay tolerability of the closed-loop IPS [2], while a smaller value of $\gamma$ indicates a better reduction of disturbance effect on the system performance [8]. As the existence of time-delay in LFC scheme of IPS is an inevitable phenomenon, improvement in delay tolerability improves the stability of the IPS. As load disturbance is the main cause of load frequency deviation in IPS, the reduction of disturbance effect on system performance results in better stability of IPS.

Next, simulation studies of the proposed LFC schemes are performed in MATLAB/Simulink using the above mentioned parameters of the two-area delayed IPS and the ${{H}_{\infty }}$ performance-based sliding mode controllers. The ${{H}_{\infty }}$ state feedback sliding mode LFC model is shown in Fig. 3. As seen, the LFC model consists of the two-area delayed IPS and the proposed ${{H}_{\infty }}$ state feedback sliding mode controller. In Fig. 3, $w_{1}$ and $w_{2}$ are the disturbances of area-1 and area-2, respectively, while ${\sigma _{1}}(t)$ and ${\sigma _{2}}(t)$ are the sliding mode switching surface functions for area-1 and area-2, respectively. $u_{1}$ and $u_{2}$ are the respective ${{H}_{\infty }}$ state feedback SMC terms to area-1 and area-2. $B_1$ and $D_1$ are the first columns of the matrices B and D, respectively, while $B_2$ and $D_2$ are the second columns of the matrices B and D, respectively. $J_1$ and $J_2$ are the first and second rows of the matrix J, while $K_1$ and $K_2$ are the first and second rows of the controller gain matrix K, respectively.

Similarly, the ${{H}_{\infty }}$ delayed feedback sliding mode LFC model is shown in Fig. 4. As shown, the LFC model consists of the two-area delayed IPS and the proposed ${{H}_{\infty }}$ delayed feedback sliding mode controller. In Fig. 4, $w_{1}$ and $w_{2}$ are the disturbances of area-1 and area-2, respectively, ${\sigma _{1}}(t)$ and ${\sigma _{2}}(t)$ are the sliding mode switching surface functions for area-1 and area-2, respectively, and $u_{1}$ and $u_{2}$ are the ${{H}_{\infty }}$ delayed feedback SMC terms to area-1 and area-2, respectively. $B_1$ and $D_1$ are the first columns of the matrices B and D, respectively, while $B_2$ and $D_2$ are the second columns of the matrices B and D, respectively. $J_1$ and $J_2$ are the first and second rows of the matrix J. $K_1$ and ${K}_{h1}$ are the first rows of the controller gain matrices K and $K_{h}$, respectively, while $K_2$ and $K_{h2}$ are the second rows of the controller gain matrices K and $K_{h}$, respectively.

Fig. 5

Frequency deviations for step load change in the closed-loop IPS

Fig. 6

Control inputs to the closed-loop IPS with step load change

Fig. 7

Frequency deviations for step load change in the uncertain closed-loop IPS

Fig. 8

Control inputs to the uncertain closed-loop IPS with step load change

Fig. 9

Random load change in the two-area IPS

Fig. 10

Frequency deviations for random load change in the closed-loop IPS

Fig. 11

Control inputs to the closed-loop IPS with random load change

Fig. 12

Frequency deviations for random load change in the uncertain closed-loop IPS

Fig. 13

Control inputs to the uncertain closed-loop IPS with random load change

4.2 Simulation results

The two-area closed-loop delayed power system without parametric uncertainty [i.e., state-space model of two-area IPS (6) with SMC laws (35) and (45)], and the two-area closed-loop delayed power system with parametric uncertainty [i.e., state-space model of two-area IPS (8) with SMC laws (35) and (45)], are simulated in MATLAB/Simulink platform. Simulation studies on LFC schemes (i.e., state feedback ${{H}_{\infty }}$ SMC and delayed feedback ${{H}_{\infty }}$ SMC) are analyzed by considering step load disturbances and random load disturbances in both areas of the IPS.

In [19], a two-term ${{H}_{\infty }}$ LFC scheme based on delayed feedback controller designing strategy is proposed for a two-area IPS with time-delay. Moreover, in [21], delayed feedback ${{H}_{\infty }}$ LFC scheme is proposed for a two-area IPS with time-delay and parametric uncertainty. By using the two-area IPS parameters shown in Table 1, simulation studies of the schemes in [19] and [21] are performed in MATLAB/Simulink for both step load disturbances and random load disturbances in the two areas of the IPS. The simulation results of the proposed LFC schemes and the LFC schemes of [19] and [21] are then compared.

4.2.1 Step-load disturbance

Step-load changes in control area-1 and control area-2 are set as $\Delta P_{d1}=0.1$ p.u. and $\Delta P_{d2}=0.2$ p.u., respectively. Simulation results on frequency deviations and control signals of the closed-loop IPS without parametric uncertainty are shown in Figs. 5 and 6, respectively. In comparison, the frequency deviations and control signals of the closed-loop IPS with parametric uncertainty are shown in Figs. 7 and 8, respectively.

It is observed from Figs. 5 and 7 that the frequency deviations of the two-area delayed IPS with and without parametric uncertainty are converging to zero within 15s by adopting the proposed delayed feedback ${{H}_{\infty }}$ SMC law (45). In comparison, by adopting the proposed state feedback ${{H}_{\infty }}$ SMC law (35) or the control approaches of [19, 21] (i.e., delayed feedback ${{H}_{\infty }}$ control approaches), the frequency deviations of the two-area delayed IPS with and without parametric uncertainty take almost 30s for converging to zero. Furthermore, the observation of these frequency response curves in between 25 to 30 s is interesting, during which the frequency deviations of the two-area delayed IPS with and without parametric uncertainty are almost zero by using the proposed delayed feedback ${{H}_{\infty }}$ SMC law (45). The damping of frequency deviation of the two-area delayed IPS without parametric uncertainty by using the proposed state feedback ${{H}_{\infty }}$ SMC law (35) is not much better than those using the control approaches of [19, 21] (see Fig. 5), whereas, the damping of frequency deviation of the two-area delayed IPS with parametric uncertainty by using the proposed state feedback ${{H}_{\infty }}$ SMC law (35) is better than that of using the control approaches of [19, 21] (see Fig. 7). Therefore, by comparing the frequency response curves of Figs. 5 and 7 from 25s to 30s, one can observe that the proposed control approaches perform better than the existing control approaches for the consideration of parametric uncertainty in delayed IPS. There is not much difference in the respective frequency response curves of delayed IPS with and without parametric uncertainty by using the proposed approaches. This study shows that the proposed ${{H}_{\infty }}$ performance-based sliding mode controllers are not sensitive to parametric variation.

One can observe from Fig. 6 that the convergence of control signals generated by the proposed delayed feedback ${{H}_{\infty }}$ SMC law (45) for the two-area delayed IPS without parametric uncertainty is faster than the control signals generated by the proposed state feedback ${{H}_{\infty }}$ SMC law (35) or the control approaches of [19, 21]. Also, From Fig. 8, one can observe that with consideration of parametric uncertainty in the delayed IPS, the convergence of control signals generated by both the proposed control approaches is faster than those generated by the control approaches of [19, 21]. Moreover, it can be observed from Figs. 6 and 8 that the control signals generated by the proposed methods are free from chattering. There are significant differences in the convergence values of control signals produced by the proposed ${{H}_{\infty }}$ SMC laws (35) and (45), due to the difference in the structure of ${{H}_{\infty }}$ SMC laws (35) and (45). The ${{H}_{\infty }}$ SMC law (45) contains a delayed state feedback term, which is not present in the structure of the ${{H}_{\infty }}$ SMC law (35).

The above analysis of the simulation results indicates that the performance of the proposed ${{H}_{\infty }}$ SMC approaches is better than the performance of the existing ${{H}_{\infty }}$ control approaches [19, 21] in terms of robustness against parametric uncertainty, fast convergence and disturbance rejection performance. Furthermore, better load frequency regulation of delayed IPS is obtained with the proposed delayed feedback ${{H}_{\infty }}$ SMC approach than the proposed state feedback ${{H}_{\infty }}$ SMC approach. This means that artificial delay incorporation in SMC structure improves stability of delayed power system.

4.2.2 Random-load disturbance

Random-load disturbances applied to control area-1 and control area-2 are shown in Fig. 9. Frequency deviations and control signals of the closed-loop delayed IPS without parametric uncertainty are depicted in Figs. 10 and 11, respectively. Considering uncertainty in the closed-loop delayed IPS parameters, frequency deviations and control signals are depicted in Figs. 12 and 13, respectively.

It is observed from Figs. 10 and 12, the proposed control schemes reduce frequency deviations of the delayed IPS satisfactorily for random-load changes. Damping of frequency deviations by using the delayed feedback $H_{\infty }$ SMC strategy (45) is better than that of state feedback $H_{\infty }$ SMC strategy (35). Moreover, from Figs. 11 and 13, one can observe that the control signals generated by the delayed feedback $H_{\infty }$ SMC strategy (45) have faster convergence rate than those by the state feedback $H_{\infty }$ SMC strategy (35). This observation shows that incorporation of artificial delay in SMC structure improves the stability of IPS with time-delay. The control signals [i.e., $u_{1}(t)$ and $u_{2}(t)$] generated by the proposed $H_{\infty }$ SMC approaches for the delayed IPS areas are chattering free. Hence, $H_{\infty }$ criterion based sliding mode controllers (35) and (45) give chattering free control signals. Furthermore, one can observe from these simulation results (i.e., Figs. 10,  11,  12 and 13) that the performance of the proposed control approaches is better than the existing control approaches [19, 21] in terms of frequency regulation and control signal convergence of the closed-loop delayed IPS with random-load change in each control area.

The objective of any control design for load frequency regulation of IPS is to maintain any load frequency oscillation within acceptable range, which in power system is defined by two main levels such as statutory limit and operational limit [4]. Frequency oscillation of $\pm 0.5$ Hz and $\pm 0.2$ Hz are acceptable in statutory and operational limits, respectively [8]. From the frequency response curves shown in Figs. 5, 7, 10, and 12, it is clearly observed that frequency oscillations in each area of the closed-loop delayed IPS with and without parametric uncertainty are within the range of $\pm 0.2$ Hz. This observation indicates that the load frequency oscillation of delayed IPS with and without parametric uncertainty is damped satisfactorily by using the proposed $H_{\infty }$ performance-based sliding mode controllers. Hence, application of the proposed sliding mode $H_{\infty }$ LFC designs based on the proposed state feedback control and delayed feedback control strategies can make IPS operation safe, secure and stable.

$H_{\infty }$ performance-based state feedback sliding mode controller (35) and $H_{\infty }$ performance-based delayed feedback sliding mode controller (45) take state variables of the state-space model (6) as their inputs and generate control signals for the two-area delayed IPS. Performance of controller (35) depends on the selection of suitable values of controller parameters ${b}_{1}$, ${{b}_{2}}$, J and K. Similarly, performance of controller (45) depends on the selection of suitable values of controller parameters ${b}_{1}$, ${{b}_{2}}$, h, J, K and $K_{h}$. The values of ${b}_{1}$ and ${b}_{2}$ are selected according to the disturbances of area-1 and area-2, respectively, while the value of h is tuned by considering the values of time-delays appeared in area-1 and area-2. Selection of matrix J depends on the input matrix B of the state-space model (6), while computation of K and $K_{h}$ needs the values of $\tau _{1}$, $\tau _{2}$, h, and matrices A, $A_{d1}$, $A_{d2}$, B, C, D of the state-space model (6) and (2). The matrices A, $A_{d1}$, $A_{d2}$, B, C and D are obtained by using the values of the two-area IPS parameters given in Table 1. In real situation, the number of control areas of an IPS is not limited to two, and there may be a variety of generation technologies in each control area. For such situation, the number of state variables of the IPS (i.e., number of inputs to controller) and number of control inputs to IPS (i.e., number of controller outputs) may increase, which increase the dimensions of matrices A, $A_{d1}$, $A_{d2}$, B, C, D and J. Hence, dimensions of controller gain matrices K and $K_{h}$ are increased and new values of K and $K_{h}$ are computed. The above discussed changes in power system and controller parameters may change the control performance in real situations.

Conventional control design for load frequency regulation of IPS consists of an integral controller or a PI controller in each control area of the IPS to control the frequency oscillation of the respective control area [19]. Integral controllers and PI controllers are preferred to design conventional LFC schemes due to their simple structure and availability of various tuning methods [21], and when used in each control area of IPS for load frequency regulation, they are known as local controllers. For high and random load demand change as shown in Fig. 9, the local controllers may take a long time (up to several minutes) to bring the frequency oscillation to its tolerable limit. To handle such situation, the proposed sliding mode $H_{\infty }$ LFC designs based on state feedback or delayed feedback control strategies can be considered, since in the proposed LFC designs, local controllers are considered as integral parts of the IPS while an $H_{\infty }$ sliding mode controller is used to solve LFC problem of such IPS which already has local controllers. By observing the results analyzed in this paper, it is evident that the application of the proposed $H_{\infty }$ performance-based sliding mode LFC designs can bring the frequency oscillation of IPS to its tolerable limit within a minute.

5 Conclusions

Two types of sliding mode controller designs are proposed in this paper by considering two different strategies, such as state feedback control strategy and delayed feedback control strategy, for power frequency regulation of delayed IPS with parametric uncertainty. In the delayed feedback sliding mode controller design, an artificial delay is incorporated intentionally to obtain better dynamic performance of the closed-loop delayed IPS. To obtain minimal disturbance effect on the performance of closed-loop systems (delayed IPS with state feedback sliding mode controller and with delayed feedback sliding mode controller), two delay-dependent $H_{\infty }$ criteria (Theorems 1 and 3) in LMI framework are developed separately using two separate LK functionals for the stabilization of the two closed-loop IPSs with time-delay and parametric uncertainty. Stabilizing controller gains of these two types of sliding mode controllers are computed from the solution of these developed $H_{\infty }$ criteria. These proposed LFC methods are tested for a numerical example of a two-area power system. Results of the test system show that these control methods can damp frequency deviations adequately for both step and random load changes in each control area. The findings of this paper are summarized as follows:

• The proposed ${{H}_{\infty }}$ performance-based sliding mode controller designs for load frequency regulation of delayed IPS with parametric uncertainty are not affected by chattering. So, its control performance is improved.

• Damping of frequency oscillation is significantly improved by using the delayed feedback ${{H}_{\infty }}$ SMC approach in comparison to the state feedback ${{H}_{\infty }}$ SMC approach, which indicates that incorporation of an artificial delay in SMC design improves the stability of IPS with time-delay.

• For the high and random load changes in each area of the delayed IPS (see Fig. 9), the frequency oscillations are within the operational limit ($\pm 0.2$ Hz) by adopting the proposed ${{H}_{\infty }}$ SMC approaches (see Figs. 10 and 12). This observation shows the applicability of the proposed control approaches for making the operation of IPS stable, secure and safe.

• The performance of the proposed delayed feedback ${{H}_{\infty }}$ sliding mode LFC approach is superior to those of the existing delayed feedback ${{H}_{\infty }}$ LFC approaches [19, 21] in terms of robustness against parametric variation, fast convergence and disturbance attenuation performance.

The study may be extended to analyzing the load frequency regulation of IPS considering a variety of generation technologies in each control area. Controller design for load frequency regulation of time-delay IPS with parametric uncertainty may also be proposed by using fractional order sliding mode controller in place of the integral sliding mode controller.

Appendix A Proof of Theorem 2

Consider the Lyapunov function as:

$$\begin{aligned} V(t)=\frac{1}{2}{{\sigma }^{T}}(t)\sigma (t). \end{aligned}$$

(52)

Derivative of (52) is given by:

$$\begin{aligned} \dot{V}(t)=\frac{1}{2}{{{\dot{\sigma }}}^{T}}(t)\sigma (t)+\frac{1}{2}{{\sigma }^{T}}(t){\dot{\sigma }}(t)={{\sigma }^{T}}(t){\dot{\sigma }}(t). \end{aligned}$$

(53)

Using (13), one can write (53) as:

$$\begin{aligned} \dot{V}(t)=\sum \limits _{i=1}^{2}{{{\sigma }_{i}}(t)}\{{{J}_{i}}{{B}_{i}}{{K}_{i}}x(t)+{{J}_{i}}{{B}_{i}}{{u}_{i}}(t)+{{J}_{i}}{{D}_{i}}{{w}_{i}}(t)\}. \end{aligned}$$

(54)

Substituting (35) in (54), one obtains:

$$\begin{aligned} \dot{V}(t)=\sum \limits _{i=1}^{2}{{{\sigma }_{i}}(t)\left[ -{{b}_{i}}\left\| {{J}_{i}}{{D}_{i}} \right\| \left( \frac{{{\sigma }_{i}}(t)}{\left\| {{\sigma }_{i}}(t) \right\| +\zeta } \right) +{{J}_{i}}{{D}_{i}}{{w}_{i}}(t)\right] }. \end{aligned}$$

(55)

Equation (55) can be expressed as:

$$\begin{aligned} \dot{V}(t)\le \sum \limits _{i=1}^{2}{\left[ -{{b}_{i}}\left\| {{J}_{i}}{{D}_{i}} \right\| \left\| {{\sigma }_{i}}(t) \right\| +\left\| {{\sigma }_{i}}(t) \right\| \left\| {{J}_{i}}{{D}_{i}} \right\| \left\| {{w}_{i}}(t) \right\| \right] }. \end{aligned}$$

(56)

The above equation may be further solved as:

$$\begin{aligned} \dot{V}(t)\le \sum \limits _{i=1}^{2}{\left\| {{\sigma }_{i}}(t) \right\| \left\| {{J}_{i}}{{D}_{i}} \right\| }\left[ -{{b}_{i}}+\left\| {{w}_{i}}(t) \right\| \right] <0. \end{aligned}$$

(57)

Thus, control law (35) ensures ${{\sigma }_{i}}(t){{{\dot{\sigma }}}_{i}}(t)<0$.