Robust discrete-time sliding mode control for systems with time-varying state delay and uncertainties on state and control input

Abstract

This paper proposes a discrete-time sliding mode control (DT-SMC) algorithm for uncertain linear systems with time-varying delay on the state. The parameter uncertainties and the external disturbances are assumed to be unknown but norm-bounded. The sliding mode existence depends on that of the stable linear sliding surface, which is ensured by a sufficient condition that depends on the delay bounds and is derived using LMI. A numerical example as well as an application to the Two-Area Four-Machine power system has been presented to illustrate and confirm the usefulness and the effectiveness of the proposed control strategy.

Keywords

Discrete sliding mode control time-varying delay uncertainty LMI

Introduction

Time-delays are frequently encountered in various physical processes such as network control systems, chemical systems, long transmission line systems, hydraulic systems, nuclear reactors and so on (see e.g. Brayton (1968), Fridman (2014), Halanay and Rasvan (1997), Musmade et al. (2015), and the references therein). These delays may result from many sources, in particular sensors and actuators. They are considered to be sources of instability and oscillation, and may dwindle the system performance. Similarly, the parameter uncertainties as well as the external disturbances may lead to instability and cause more degradation in performance. Therefore, the problem of stability-analysis and control of uncertain time-delay systems have been largely investigated. Recently, considerable attention has been devoted to the problem of delay-dependent stability analysis and controller-synthesis for uncertain systems with time-delay, and several analysis and control design approaches have been reported in Gu et al. (2003). A less-conservative delay-dependent criterion for robust stability of continuous-time systems with time-varying delay is proposed using the Leibniz–Newton formula and Linear Matrix Inequality (LMI) technique in Wu et al. (2004). In Gao and Chen (2007), promising results on stability are derived by defining a new Lyapuov function. When uncertainties are present, the robustness performance is sought through a control strategy qualified as a robust control technique. In this way, the sliding mode control (SMC) is well-known as a robust control method that copes with variable structure systems. SMC is often adopted due to its inherent advantages of easy realization, fast response and good transient performance, as well as its insensitivity to parameter uncertainties and external disturbances (Boukhobza et al., 1996; Edwards and Spurgeon, 1996; Fu and Xiet, 2005; Musmade et al., 2013; Seungrohk and Hassan, 1995; Castillo-Toledo et al., 2008; Ullah et al., 2015).

In recent years, the widespread use of digital computers and Digital Signal Processing (DSP) chips to implement a control algorithm requires a discrete-time control design. To do so, we have to extend the results obtained for continuous SMC to the discrete case. Pai (2008) studied a discrete-time output feedback SMC to stabilize a class of linear uncertain systems. While the study of continuous SMC in the presence of state delay has been ongoing (Lin et al., 2011; Pai, 2008, 2010, 2013; Xia, Zhu et al., 2010; Yan and Shi, 2008), results in the discrete-time case are rare in the literature (Bandyopadhyay et al., 2010; Hu et al., 2013, 2014; Ignaciuk and Bartoszewicz, 2011; Kumar et al., 2013; Pai, 2012; Sun et al., 2013; Xia, Liu et al., 2007; Xia, Fu et al., 2010), which motivates the current study. In Yan and Shi (2008), a sufficient condition for the existence of stable sliding surfaces depending on the delay bounds for a class of linear uncertain systems has been addressed. In Pai (2010), a DT-SMC method for the robust stabilization of linear uncertain multi-input discrete-time systems with constant state and input delays has been presented. However, it is worth mentioning that most of the results are concerned with the constant-delay case and research on the stability-analysis of discrete-time systems with time-varying delay on the state have seldom been reported in existing literature. Moreover, in practice, the delays we often encounter are assumed to be time-varying and bounded, which motivates the current study. This paper deals with the synthesis of a DT-SMC for systems with time-varying delay on the state and where the uncertainties are on the state and on the control input as well.

The paper is organized as follows. The second section states briefly the problem formulation and assumptions. The third section is structured in two phases: the first is concerned with the switching surface design and the stability-analysis while the second is concerned with the existence of the quasi-sliding mode (QSM) and the synthesis of the control law. In the fourth section we present some results from numerical simulations to validate the proposed approach. Finally, and in order to give more insight into the usefulness and the effectiveness of the proposed control strategy we illustrate, in the fifth section, results provided from the application of this strategy to a benchmark of the Two-Area Four-Machine (TAFM) system (Kundur et al., 1994).

System description and problem formulation

Consider the uncertain linear discrete-time system with time-varying state delay represented by

{\begin{matrix} \begin{matrix} x (k + 1) = (A + Δ A) x (k) + (A_{d} + Δ A_{d}) x (k - r (k)) \\ + (B + Δ B) u (k) + f (x (k), k) \end{matrix} \\ x (k) = ψ (k) \forall k = - r_{M}, - r_{M} + 1, \dots, 0 \end{matrix}

(1)

where $x \in ℝ^{n}$ is the state vector, $u \in ℝ^{m}$ the input, A, B and $A_{d}$ are real constant matrices of dimensions $ℝ^{n \times n}$ , $ℝ^{n \times m}$ and $ℝ^{n \times n}$ respectively, $ψ (k)$ is the initial state, $f (x (k), k) \in ℝ^{n}$ represents the external state disturbance, which is assumed to be unknown but norm-bounded, $Δ A$ , $Δ A_{d}$ and $Δ B$ denote the uncertainties on matrices A, $A_{d}$ and B respectively and $r (k)$ is the time-varying state delay assumed to be bounded too as $r_{m} \leq r (k) \leq r_{M}$ .

Equation (1) can be written

{\begin{matrix} \begin{matrix} x (k + 1) = (A + Δ A) x (k) + (A_{d} + Δ A_{d}) x (k - r (k)) \\ + B (I + Δ B_{1}) u (k) + B Δ f (x (k), k) \end{matrix} \\ x (k) = ψ (k) \forall k = - r_{M}, - r_{M} + 1, \dots, 0 \end{matrix}

(2)

and

Δ B = B Δ B_{1} and B Δ f (x (k), k) = f (x (k), k)

(3)

where $Δ B_{1} \in ℝ^{m \times m}$ and $Δ f (x (k), k) \in ℝ^{m}$ .

Let us define E, $O_{1}$ and $O_{2}$ as known real constant matrices of appropriate dimensions that represent the structure of uncertainties, and $D (k)$ as an unknown matrix function satisfying

D^{T} (k) D (k) \leq I

(4)

where I is the identity matrix of appropriate dimensions, such that

Δ A (k) = E D (k) O_{1} and Δ A_{d} (k) = E D (k) O_{2}

(5)

The pair $(A, B)$ in system (2) is assumed to be controllable. The uncertainties $Δ A$ , $Δ A_{d}$ , $Δ B$ and $Δ f$ are unknown and bounded.

Let us define the matrix $Θ = {[\begin{matrix} Θ_{2} & Θ_{1} \end{matrix}]}^{T}$ where $Θ_{1} \in ℝ^{n \times m}$ and $Θ_{2} \in ℝ^{n \times (n - m)}$ are the sub-blocks of the unitary matrix resulting from the singular value decomposition of B

B = [\begin{matrix} Θ_{2} & Θ_{1} \end{matrix}] [\begin{matrix} Λ \\ 0_{(n - m) \times m} \end{matrix}] V^{T}

(6)

where $Λ \in ℝ^{m \times m}$ is a diagonal positive-definite matrix and $0_{i \times j}$ is the matrix of dimensions i and j with all its components equal to zero. To obtain a regular form of the system (2), the matrix $Θ$ satisfies

Θ B = [\begin{matrix} 0_{(n - m) \times m} \\ B_{2} \end{matrix}]

(7)

where $B_{2} = Λ V^{T}$ is non singular matrix $\in ℝ^{m \times m}$ . The matrix $Θ$ will be used as a transformation matrix to define a new state vector $z (k) = Θ x (k)$ so that the state vector will be partitioned in two blocks and the control signal appears in only one block

{\begin{matrix} \begin{matrix} z (k + 1) = (\bar{A} + Δ \bar{A}) z (k) + ({\bar{A}}_{d} + Δ {\bar{A}}_{d}) z (k - r (k)) \\ + [\begin{matrix} 0_{(n - m) \times m} \\ B_{2} \end{matrix}] (I + Δ B_{1}) u (k) \\ + [\begin{matrix} 0_{(n - m) \times m} \\ B_{2} \end{matrix}] Δ f' (z (k), k) \end{matrix} \\ z (k) = \bar{ψ} (k) \forall k = - r_{M}, - r_{M} + 1, \dots, 0 \end{matrix}

(8)

where $\bar{A} = Θ A Θ^{- 1} = [\begin{matrix} {\bar{A}}_{11} & {\bar{A}}_{12} \\ {\bar{A}}_{21} & {\bar{A}}_{22} \end{matrix}]$ , $Δ \bar{A} = Θ Δ A Θ^{- 1} = [\begin{matrix} Δ {\bar{A}}_{11} & Δ {\bar{A}}_{12} \\ Δ {\bar{A}}_{21} & Δ {\bar{A}}_{22} \end{matrix}]$ , ${\bar{A}}_{d} = Θ A_{d} Θ^{- 1} = [\begin{matrix} {\bar{A}}_{d 11} & {\bar{A}}_{d 12} \\ {\bar{A}}_{d 21} & {\bar{A}}_{d 22} \end{matrix}]$ and $Δ {\bar{A}}_{d} = Θ Δ A_{d} Θ^{- 1} = [\begin{matrix} Δ {\bar{A}}_{d 11} & Δ {\bar{A}}_{d 12} \\ Δ {\bar{A}}_{d 21} & Δ {\bar{A}}_{d 22} \end{matrix}]$

with ${\bar{A}}_{11}$ , $Δ {\bar{A}}_{11}$ , ${\bar{A}}_{d 11}$ and $Δ {\bar{A}}_{d 11}$ $\in ℝ^{(n - m) \times (n - m)}$ , ${\bar{A}}_{12}, Δ {\bar{A}}_{12}, {\bar{A}}_{d 12} and Δ {\bar{A}}_{d 12}$ $\in ℝ^{(n - m) \times m}$ , ${\bar{A}}_{21}, Δ {\bar{A}}_{21}, {\bar{A}}_{d 21} and Δ {\bar{A}}_{d 21} \in ℝ^{m \times (n - m)}$ , ${\bar{A}}_{22}, Δ {\bar{A}}_{22}$ , ${\bar{A}}_{d 22}$ and $Δ {\bar{A}}_{d 22}$ $\in ℝ^{m \times m}$ and $\bar{ψ} (k) = Θ ψ (k) = [\begin{matrix} {\bar{ψ}}_{1} (k) \\ {\bar{ψ}}_{2} (k) \end{matrix}]$ with ${\bar{ψ}}_{1} \in ℝ^{(n - m)}$ and ${\bar{ψ}}_{2} \in ℝ^{m}$ .

Equation (8) can be expressed in the following regular form

{\begin{array}{l} z_{1} (k + 1) = ({\bar{A}}_{11} + Δ {\bar{A}}_{11}) z_{1} (k) + ({\bar{A}}_{d 11} + Δ {\bar{A}}_{d 11}) z_{1} (k - r (k)) \\ + ({\bar{A}}_{12} + Δ {\bar{A}}_{12}) z_{2} (k) + ({\bar{A}}_{d 12} + Δ {\bar{A}}_{d 12}) z_{2} (k - r (k)) \\ z_{2} (k + 1) = ({\bar{A}}_{21} + Δ {\bar{A}}_{21}) z_{1} (k) + ({\bar{A}}_{d 21} + Δ {\bar{A}}_{d 21}) z_{1} (k - r (k)) \\ + ({\bar{A}}_{22} + Δ {\bar{A}}_{22}) z_{2} (k) + ({\bar{A}}_{d 22} + Δ {\bar{A}}_{d 22}) z_{2} (k - r (k)) \\ + B_{2} [(I + Δ B_{1}) u (k) + Δ f^{'} (z (k), k)] \\ z_{1} (k) = {\bar{ψ}}_{1} (k) \forall k = - r_{M}, - r_{M} + 1, \dots,0 \\ z_{2} (k) = {\bar{ψ}}_{2} (k) \forall k = - r_{M}, - r_{M} + 1, \dots,0 \end{array}

(9)

The main objective of this paper is to design a discrete-time sliding surface $S (k)$ and a robust DT-SMC $u (k)$ so that the states of the uncertain discrete-time-delay system (2) will asymptotically approach zero in spite of the presence of time-varying unmatched uncertainties, unknown disturbances and time-varying state delay.

Design of robust DT-SMC

For discrete-time uncertain systems with time-varying state delay, the linear sliding surface is chosen as

\begin{matrix} S (k) & = \bar{C} z (k) = [\begin{matrix} C & I_{m} \end{matrix}] z (k) = [\begin{matrix} C & I_{m} \end{matrix}] [\begin{matrix} z_{1} (k) \\ z_{2} (k) \end{matrix}] \\ = C z_{1} (k) + z_{2} (k) \end{matrix}

(10)

where $\bar{C} \in ℝ^{m \times n}$ and $C \in ℝ^{m \times (n - m)}$ is a real matrix to be designed. The sliding surface should be zero $(S (k) = 0)$ , therefore, $z_{2} (k) = - C z_{1} (k)$ and system (9) yields

{\begin{array}{l} z_{1} (k + 1) = ({\bar{A}}_{11} + Δ {\bar{A}}_{11}) z_{1} (k) + ({\bar{A}}_{d 11} + Δ {\bar{A}}_{d 11}) z_{1} (k - r (k)) \\ - ({\bar{A}}_{12} + Δ {\bar{A}}_{12}) C z_{1} (k) + ({\bar{A}}_{d 12} + Δ {\bar{A}}_{d 12}) C z_{1} (k - r (k)) \\ = ({\bar{A}}_{11} + Δ {\bar{A}}_{11} - {\bar{A}}_{12} C - Δ {\bar{A}}_{12} C) z_{1} (k) \\ + ({\bar{A}}_{d 11} + Δ {\bar{A}}_{d 11} - {\bar{A}}_{d 12} C - Δ {\bar{A}}_{d 12} C) z_{1} (k - r (k)) \\ z_{2} (k + 1) = - C z_{1} (k + 1) \\ z_{1} (k) = {\bar{ψ}}_{1} (k), k = - r_{M}, - r_{M} + 1, \dots,0 \end{array}

(11)

Here ${\bar{ψ}}_{1} (k) \in ℝ^{n - m}$ stands for initial values of $z_{1} (k)$ .

The design of the sliding surface requires the determination of the matrix C, which is a difficult task as this matrix appears on the state and on its delay as well. This matrix should satisfy some constraints resulting from asymptotic stability of system (11). The following theorem formulates such constraints as an LMI problem.

Theorem 1. (Yan and Shi, 2008) The system in equation (11) is robustly asymptotically stable if there exist matrices $L > 0$ , $P > 0$ , $W > 0$ , $Z > 0$ , $X > 0$ , R, $Q > 0$ $\in ℝ^{(n - m) \times (n - m)}$ and matrix $C \in ℝ^{m \times (n - m)}$ satisfying the following LMIs

[\begin{matrix} X & R \\ R^{T} & Z \end{matrix}] \geq 0

(12)

and

[\begin{matrix} - L & 0 & ({\bar{A}}_{11} - {\bar{A}}_{12} C) & ({\bar{A}}_{d 11} - {\bar{A}}_{d 12} C) & Θ_{2}^{T} E & 0 \\ * & - r_{M}^{- 1} W & ({\bar{A}}_{11} - {\bar{A}}_{12} C - I) & ({\bar{A}}_{d 11} - {\bar{A}}_{d 12} C) & Θ_{2}^{T} E & 0 \\ * & * & ϒ & - R & 0 & {[O_{1} (Θ_{2} - Θ_{1} C)]}^{T} \\ * & * & * & - Q & 0 & {[O_{2} (Θ_{2} - Θ_{1} C)]}^{T} \\ * & * & * & * & - I & 0 \\ * & * & * & * & * & - I \end{matrix}] < 0

(13)

subject to:

PL = I, ZW = I

(14)

where

ϒ = - P + r_{M} X + R + R^{T} + (r_{M} - r_{m} + 1) Q

Design of the control law

In Yan and Shi (2008) the designed DT-SMC assumes that uncertainties on control are neglected. In this section we propose the appropriate discrete SMC law taking into account the uncertainties on the control signal. The design of a discrete sliding mode controller for uncertain systems with time-varying bounded state delay must guarantee that, once the system’s trajectory crosses the switching plane the first time, it will cross the plane again in every successive sampling period (Bartoszewicz, 1996). The design parameter $ε T$ is chosen adequately such that system (8) converges in finite time into a QSM band. Besides, the reaching condition should be satisfied in spite of the presence of uncertainties and/or time lag. Let us consider the reaching condition proposed by Gao (Gao et al., 1995)

{\begin{array}{l} Δ S (k) = S (k + 1) - S (k) \\ \leq - ε T sgn (S (k)) - q T S (k) i f S (k) > 0 \\ Δ S (k) = S (k + 1) - S (k) \\ \geq - ε T sgn (S (k)) - q T S (k) i f S (k) < 0 \end{array}

(15)

where $q > 0$ , $ε > 0$ and T is the sampling period such that $(1 - qT) > 0$ . From equations (8) and (10) we obtain

\begin{matrix} Δ S (k) = & \bar{C} z (k + 1) - \bar{C} z (k) \\ = & \bar{C} (\bar{A} + Δ \bar{A}) z (k) + \bar{C} ({\bar{A}}_{d} + Δ {\bar{A}}_{d}) z (k - r (k)) \\ + B_{2} u (k) + B_{2} Δ B_{1} u (k) + B_{2} Δ f' (z (k), k) - \bar{C} z (k) \end{matrix}

(16)

Let $p (z (k), u (k), k) = Δ B_{1} u (k) + Δ f' (z (k), k)$

The designed control law is formulated by the proposed Theorem 2.

Theorem 2. When the state delay is time-varying and bounded as $r_{m} \leq r (k) \leq r_{M}$ , matrices L, P, X, W, Z, R, Q satisfy equations (13) and (14), assuming that the linear sliding surface is given by equation (10), then the trajectory of the closed-loop system (8) can be driven into a QSM band in finite time and will remain inside a specified band due to the following control law

u (k) = - B_{2}^{- 1} [\begin{matrix} \bar{C} \bar{A} z (k) - (1 - Tq) S (k) + ε T sgn (S (k)) \\ + {\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0} + (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) sgn (S (k)) \end{matrix}]

(17)

where $T > 0$ and q satisfy

\begin{matrix} {\underline{α}}_{0} = \frac{{\underline{α}}_{\min} + {\underline{α}}_{\max}}{2}, δ_{\underline{α}} = \frac{{\underline{α}}_{\max} - {\underline{α}}_{\min}}{2} \\ {\underline{β}}_{0} = \frac{{\underline{β}}_{\min} + {\underline{β}}_{\max}}{2}, δ_{\underline{β}} = \frac{{\underline{β}}_{\max} - {\underline{β}}_{\min}}{2} \\ {\underline{γ}}_{0} = \frac{{\underline{γ}}_{\min} + {\underline{γ}}_{\max}}{2}, δ_{\underline{γ}} = \frac{{\underline{γ}}_{\max} - {\underline{γ}}_{\min}}{2} \end{matrix}

(18)

with ${{\underline{α}}_{\min}; {\underline{α}}_{\max}}$ ; ${{\underline{β}}_{\min}; {\underline{β}}_{\max}}$ and ${{\underline{γ}}_{\min}; {\underline{γ}}_{\max}}$ are respectively the lower and the upper bounds of $\bar{C} Δ \bar{A} z (k)$ ; $\bar{C} ({\bar{A}}_{d} + Δ {\bar{A}}_{d}) z (k - r (k))$ and $B_{2} (Δ B_{1} u (k) + Δ f' (z (k), k))$ .

Proof. Using the improved Razumikhin theorem (Liu and Marquez, 2007), for any solution $z (k - r (k))$ of equation (10), there exists a constant $σ > 1$ such that

‖ z (k - r (k)) ‖ < σ ‖ z (k) ‖ \forall k and r_{m} \leq r (k) \leq r_{M}

and assuming that $Δ A$ , $Δ A_{d}$ , $Δ B$ and $Δ f (x (k), k)$ are norm-bounded, then $p (z (k), u (k), k)$ is bounded too. Let us define $\underline{α} (k) = \bar{C} Δ \bar{A} z (k)$ , $\underline{β} (k) = \bar{C} ({\bar{A}}_{d} + Δ {\bar{A}}_{d}) z (k - r (k))$ and $\underline{γ} (k) = B_{2} p (z (k), u (k), k)$ . These expressions are bounded as follows

\begin{matrix} {\underline{α}}_{\min} \leq \underline{α} (k) \leq {\underline{α}}_{\max} \\ {\underline{β}}_{\min} \leq \underline{β} (k) \leq {\underline{β}}_{\max} \\ {\underline{γ}}_{\min} \leq \underline{γ} (k) \leq {\underline{γ}}_{\max} \end{matrix}

(19)

Substituting equation (17) into equation (16) we get the following expression

\begin{matrix} Δ S (k) = \underline{α} (k) + \underline{β} (k) + \underline{γ} (k) - T q S (k) - ε T s i g n (S (k)) \\ - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0}) - (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) s i g n (S (k)) \end{matrix}

(20)

From equations (18) and (19) we have

\begin{matrix} \underline{α} (k) \leq {\underline{α}}_{0} + δ_{\underline{α}} sign (S (k)) if S (k) > 0 \\ \underline{α} (k) \geq {\underline{α}}_{0} + δ_{\underline{α}} sign (S (k)) if S (k) < 0 \\ \underline{β} (k) \leq {\underline{β}}_{0} + δ_{\underline{β}} sign (S (k)) if S (k) > 0 \\ \underline{β} (k) \geq {\underline{β}}_{0} + δ_{\underline{β}} sign (S (k)) if S (k) < 0 \\ \underline{γ} (k) \leq {\underline{γ}}_{0} + δ_{\underline{γ}} sign (S (k)) if S (k) > 0 \\ \underline{γ} (k) \geq {\underline{γ}}_{0} + δ_{\underline{γ}} sign (S (k)) if S (k) < 0 \end{matrix}

(21)

From equations (20) and (21) it can be seen that the proposed control law (17) satisfies the reaching condition (15). Thus, the increment $Δ S (k)$ and $S (k)$ have opposite sign regardless of the values that can be taken by uncertainties $\underline{α} (k)$ , $\underline{β} (k)$ and $\underline{γ} (k)$

Using equation (20) we obtain

\begin{array}{l} S (k + 1) = (1 - T q) S (k) - ε T sign (S (k)) + \underline{α} (k) + \underline{β} (k) + \underline{γ} (k) \\ - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0}) - (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) sign (S (k)) \\ = \underline{α} (k) + \underline{β} (k) + \underline{γ} (k) - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0}) - (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) ρ \\ + λ S (k) - μ ρ \end{array}

(22)

where $λ = 1 - Tq$ , $μ = ε T$ and $ρ = sign (S (k))$ .

To determine the QSM band width $2 θ = 4 (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) + 2 μ$ , and with respect to the so-called ‘reaching condition’, in the QSM we need to ensure the following assumption

sign (S (k + 2)) = - sign (S (k + 1)) = sign (S (k)) = ρ

(23)

Using equations (22) and (23) we have

\begin{matrix} S (k + 2) = & λ S (k + 1) - μ sign (S (k + 1)) \\ + \underline{α} (k + 1) + \underline{β} (k + 1) + \underline{γ} (k + 1) \\ - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0}) - (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) sign (S (k + 1)) \\ = ρ [λ^{2} | S (k) | + Tq μ + Tq (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}})] \\ + λ {\underline{α} (k) + \underline{β} (k) + \underline{γ} (k) - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0})} \\ + {\underline{α} (k + 1) + \underline{β} (k + 1) + \underline{γ} (k + 1) - ({\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0})} \end{matrix}

(24)

From equation (24) it is obvious that the sign of the first term on the right-hand side is $ρ$ , while the sign of the second and the third term on the right-hand side may differ from the sign of $S (k)$ . To ensure that $sign (S (k + 2)) = sign (S (k))$ , and according to relation $(30)$ in reference Bartoszewicz (1996) the inequality $δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}} < \frac{qT μ}{2 (1 - Tq)}$ should be guaranteed, this leads to the following constraint

\begin{matrix} qT [(δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) + μ] > (1 + λ) (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) \end{matrix}

(25)

From equation (25), the QSM band width satisfies

2 θ = 4 (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) + 2 μ < 4 \frac{qT μ}{2 λ} + 2 μ

Hence

2 θ < 2 \frac{ε T}{1 - Tq}

(26)

Remark 1. The relation in equation (26) clearly shows that, in the steady-state, the magnitude of oscillations of $S (k)$ around zero depends on $ε$ , T and q: the smaller $ε$ , Tandqare, the smaller the QSM band width is.

Theorem 3. When the state delay is constant, $r_{m} = r = r_{M}$ , and matrices L, P, X, W, Z, R, Q satisfy equations (13) and (14), assuming that the linear sliding surface is given by equation (10), then the trajectory of the closed-loop system in equation (8) can be driven into a QSM band in finite time and remain in a residual set of the origin with the control

u (k) = - B_{2}^{- 1} [\begin{array}{l} \bar{C} \bar{A} z (k) + \bar{C} {\bar{A}}_{d} z (k - r) - (1 - T q) S (k) + ε T sgn (S (k)) \\ + {\underline{α}}_{0} + {\underline{β}}_{0} + {\underline{γ}}_{0} + (δ_{\underline{α}} + δ_{\underline{β}} + δ_{\underline{γ}}) sgn (S (k)) \end{array}]

(27)

It is important to point out that $\underline{α} (k)$ and $\underline{γ} (k)$ have the same expressions as in the case of time-varying state delay, while $\underline{β} (k)$ has the following expression

\underline{β} (k) = \bar{C} Δ {\bar{A}}_{d} z (k - r) and {\underline{β}}_{\min} \leq \underline{β} (k) \leq {\underline{β}}_{\max}

Proof. To prove Theorem 3 we need to follow the same methodology as in the Proof of Theorem 2.

Illustrative example

To illustrate the efficiency of the proposed control algorithm, we present simulation results obtained for a numerical example of a Multiple Input Multiple Output (MIMO) system described by the state equation (8). A comparative study between the proposed control algorithm and that developed in Yan and Shi (2008) has been presented to raise the sufficiency and the supremacy of our proposition in terms of precision, robustness and time convergence.

Let us consider a time-varying state delay function chosen as a random integer variable and illustrated in Figure 1.

To have a complete description of the handled system, let us take

\begin{matrix} A = [\begin{matrix} 0.03 & - 1.4 & 0.1 \\ 0.4 & - 0.2 & 0.2 \\ 0.14 & 0.5 & 0 \end{matrix}]; A_{d} = [\begin{matrix} - 0.2 & 0 & 0 \\ 0 & - 0.1 & 0 \\ 0 & 0 & - 0.2 \end{matrix}]; \\ B = [\begin{matrix} B_{11} & B_{12} \\ B_{21} & B_{22} \\ B_{31} & B_{32} \end{matrix}] = [\begin{matrix} 17 & 3 \\ 0 & 0.9 \\ 0.3 & 0.1 \end{matrix}] \\ Θ = [\begin{matrix} - 0.0176 & - 0.0522 & 0.9985 \\ - 0.9998 & - 0.0091 & - 0.0181 \\ 0.0100 & - 0.9986 & - 0.0520 \end{matrix}]; B_{2} = [\begin{matrix} - 17.0019 & - 3.0094 \\ 0.1547 & - 0.8739 \end{matrix}] \\ Δ B (k) = [\begin{matrix} Δ B_{11} & Δ B_{12} \\ Δ B_{21} & Δ B_{22} \\ Δ B_{31} & Δ B_{32} \end{matrix}] = [\begin{matrix} 0.7 \cos (‖ z (k) ‖) & 0.1 \sin (‖ z (k) ‖) \\ \cos (‖ z (k) ‖) & 0 \\ 0.2 \sin (z_{1} (k)) & 0.1 \end{matrix}] \\ Δ B_{1} (k) = [\begin{matrix} Δ b_{11} & Δ b_{12} \\ Δ b_{21} & Δ b_{22} \end{matrix}] = [\begin{matrix} \frac{Δ B_{11} - B_{12} Δ b_{21}}{B_{11}} & \frac{Δ B_{12} - B_{12} Δ b_{22}}{B_{11}} \\ \frac{Δ B_{21} - Δ B_{11} B_{21}}{B_{11} B_{22} - B_{21} B_{12}} & \frac{Δ B_{22} - Δ B_{12} B_{21}}{B_{11} B_{22} - B_{21} B_{12}} \end{matrix}] \\ f (z (k), k) = [\begin{matrix} f_{1} \\ f_{2} \\ f_{3} \end{matrix}] = [\begin{matrix} 0.06 \cos (z_{1} (k)) \\ 0.3 \sin (z_{2} (k)) \\ rand (1) \end{matrix}]; \\ Δ f' (z (k), k) = [\begin{matrix} Δ f'_{1} \\ Δ f'_{2} \end{matrix}] = [\begin{matrix} \frac{f_{1} - B_{12} Δ f'_{2}}{B_{11}} \\ \frac{f 2 - f 1 B_{21}}{B_{11} B_{22} - B_{21} B_{12}} \end{matrix}] \\ O_{1} = [\begin{matrix} 5 & 0.12 & 1.4 \end{matrix}]; O_{2} = [\begin{matrix} 0.05 & 0.01 & 0.02 \end{matrix}]; \\ E = {[\begin{matrix} 0.31 & 0.01 & 0.02 \end{matrix}]}^{T} \\ D (k) = \cos (3 ‖ z (k) ‖); r_{m} = 1; r_{M} = 4; z (0) = {[\begin{matrix} - 0.1 & 0.2 & 0 \end{matrix}]}^{T} \end{matrix}

(28)

Using CVX Toolbox we obtain the following feasible solutions $L = 0.6258$ , $P = 1.5980$ , $W = 1.6029$ , $Z = 0.6239$ , $X = 0.0102$ , $R = - 0.0798$ , $Q = 0.1766$ and $C = [\begin{matrix} - 0.3029 \\ 1.4686 \end{matrix}]$ , therefore $\bar{C} = [\begin{matrix} - 0.3029 & 1 & 0 \\ 1.4686 & 0 & 1 \end{matrix}]$ , set a sampling period $T = 0.1$ , $ε = 0.5$ and $q = 5$

Under the same simulation framework, we illustrate the results of the Monte Carlo simulations that were performed over 100 runs using different sequences of delay. In each of the figures below we present simultaneously simulation results corresponding to the proposed controller as well as those corresponding to the control strategy given in Yan and Shi (2008). In Figure 2 we draw system states. Sliding surfaces are illustrated in Figure 3, while the control inputs are depicted in Figure 4.

Figure 2 shows that the system states are asymptotically stable, and $x_{1}$ reaches the origin of state space within $1.8 s$ while $x_{2}$ and $x_{3}$ require $2 s$ and $2.5 s$ respectively. It is obvious that our proposed algorithm provides a faster response and better transient behaviour than that developed by Yan and Shi (2008). In fact, $x_{1 YAN}$ converges to zero during $3 s$ , while $x_{2 YAN}$ and $x_{3 YAN}$ take $3.7 s$ and $4 s$ respectively.

Moreover, the amplitude of oscillations of state components concerning the method in Yan and Shi (2008) are greater. Thus, in the transient response, the maximum absolute value obtained by $x_{1}$ is at about $25.73$ whereas it is $61.4$ for $x_{1 YAN}$ , similarly for $x_{2 \max} = 35.73$ , $x_{2 Yan max} = 88.08$ , $x_{3} = 1.869$ and $x_{3 Yan max} = 4.905$ . Furthermore, although oscillations can occur in the transient response due to the presence of delay and both state and input uncertainties, we can nevertheless affirm that time-varying delay and uncertainties have no effects when the sliding mode is reached, which attests to the robustness of the proposed control strategy.

Through Figures 3 and 4 and comparing amplitudes of the sliding surfaces as well as the control input signals corresponding to both methods, we can observe that the amplitudes corresponding to the designed control strategy are smaller than those of the method given in Yan and Shi (2008). Besides, our algorithm is advantageous since it takes into account uncertainties on control inputs.

Applying the Monte Carlo simulation to 100 sequences of delay, and respecting the same system structure defined previously in equation (28), we present in Figure 5, simultaneously for both control strategies, the time taken by $x_{1}$ to converge to the origin of state space. Similar draws can be done for $x_{2}$ and $x_{3}$ .

As can be seen from Figure 5, when Yan and Shi (2008) is applied, the convergence time increases until the $71 st$ sequence of delay. However, when the proposed controller is used, the convergence time keeps the same profile for every 11 sequences of delay. Note that convergence time values corresponding to the designed method are close to each other; for example, the difference in the convergence time between the first and the $11 th$ sequence is $3.1 s$ , while this difference is $6 s$ between the first and the seventy first sequence in case of the Yan and Shi (2008) algorithm.

This confirms that the proposed controller is faster, less influenced by the delay variations, and has a better transient behaviour than that of Yan and Shi (2008). In Figure 6, we illustrate simultaneously the steady-state errors for $x_{1}$ , referring to both control methods, with respect to each delay sequence. We note that the steady-state errors corresponding to Yan and Shi (2008) had reached greater values than those provided by the proposed controller strategy.

In Table 1 we give the values of time convergence, steady-state errors and maximum amplitude obtained as the mean values of the Monte Carlo simulations that were performed using 100 sequences of delay. The results summarized in this table refer to both approaches and are obtained under a similar simulation framework. In fact, different values shown in Table 1 had been procured via the use of a computing loop which had been repeated until the satisfaction of a convergence test. This latter depends on the fact that the absolute value of the difference between each state component at the $j^{th}$ and the $(j + 1) th$ iterations becomes less than a fixed threshold $ζ = 10^{- 3}$ . According to Table 1, all variables corresponding to the proposed algorithm were observed to converge faster and they have small amplitude oscillations than those of the controller (Yan and Shi, 2008), furthermore, the steady-state error for whole variables obtained when both methods are applied shows clearly that the suggested algorithm is more precise and offers faster convergence and better robustness than that of Yan and Shi (2008). Thus, this table definitely asserts not merely the efficiency as well as the quality of the designed controller but also its supremacy compared with the control algorithm developed in Yan and Shi (2008).

Table 1.

Performance of the proposed algorithm compared with Yan and Shi (2008).

Criterion	Method	Variables
Criterion	Method	$x_{1}$	$x_{2}$	$x_{3}$	$S_{1}$	$S_{2}$	$u_{1}$	$u_{2}$
Mean time convergence (s)	Proposed controller	4.9455	6.3556	6.0588	5.4700	5.4700	6.0750	6.0667
Mean time convergence (s)	Controller (Yan and Shi, 2008)	6.4535	8.7308	8.5481	9.1294	9.1294	9.0778	9.3500
Mean steady error	Proposed controller	0.0839	0.0737	0.0332	0.0933	0.0375	0.0065	0.0181
Mean steady error	Controller (Yan and Shi, 2008)	0.0941	0.1005	0.1005	0.0944	0.0413	0.0121	0.0189
Mean maximum amplitude	Proposed controller	29.8814	39.6119	2.0392	25.8675	35.7625	8.4004	54.3655
Mean maximum amplitude	Controller (Yan and Shi, 2008)	62.9419	88.3020	4.9576	64.1474	91.2999	20.8599	120.3017

Application to the Two-Area Four-Machine power system

To provide valuable insight into the effectiveness of the proposed control strategy, simulation results are provided from the application of such a control approach to the TAFM power system shown in Figure 7, subject to norm-bounded disturbances and uncertainties as well as bounded time-varying state delay.

Process description

The TAFM power system consists of two similar areas connected by a weak tie. Each area is composed of two coupled units with 900 MVA and 20 kV for each unit. Further details and a complete description of different parts of the TAFM power system (including the exciter, the power system stabilizer, the generator parameters and initial values, transmission line data and power flow) can be found in Kundur et al. (1994).

The nonlinear model of the TAFM power system is linearised at a nominal operating point (Kundur et al., 1994). In Li et al. (2015), the authors presented a linearised power system model of the following compact form

{\begin{matrix} \overset{\cdot}{x} (t) = \underline{A} x (t) + \sum_{i = 1}^{4} {\underline{A}}_{d_{i}} x (t - r_{i} (t)) + \underline{B} u (t) + B_{ω} ω (t) \\ l (t) = Gx (t) \end{matrix}

(29)

where the state vector is

\begin{array}{l} x (t) = [Δ δ_{1}, Δ ω_{1}, Δ δ_{2}, Δ ω_{2}, Δ E_{q 2}^{'}, Δ E_{d 2}^{'}, Δ E_{f 2}, Δ y_{2}, Δ z_{2}, Δ δ_{3}, \\ Δ ω_{3}, Δ δ_{4}, Δ ω_{4}, Δ E_{q 4}^{'}, Δ E_{d 4}^{'}, Δ E_{f 4}, Δ y_{4}, Δ z_{4}]^{T} \end{array}

The control input is

u (t) = {[Δ P_{m_{1}}, Δ P_{m_{2}}, Δ V_{2}, Δ P_{m_{3}}, Δ P_{m_{4}}, Δ V_{m_{4}}]}^{T}

Where $ω (t)$ is added to the model to represent the external disturbances, and $l (t)$ stands for the system output vector. We have to mention that the outputs are taken to be the deviation of the rotor angle $(Δ δ_{i} = δ_{i} - δ_{0})$ and the deviation of the rotor speed $(Δ ω_{i} = ω_{i} - ω_{0}); i = (1, \dots, 4)$

The TAFM power system model given by equation (29) is modified so that all time-delays are equal, i.e. $r_{i} (t) = r (t); i = (1, \dots, 4)$ . Moreover, $Δ \underline{A}$ , $Δ {\underline{A}}_{d_{i}}$ and $Δ \underline{B}$ are added to the model (29) as uncertainties on matrices $\underline{A}$ , ${\underline{A}}_{d_{i}}$ and $\underline{B}$ respectively. With respect to these new assumptions, the TAFM power system model (29) is henceforth written as

{\begin{matrix} \overset{\cdot}{x} (t) = & (\underline{A} + Δ \underline{A} (t)) x (t) + \sum_{i = 1}^{4} ({\underline{A}}_{d_{i}} + Δ {\underline{A}}_{d_{i}} (t)) x (t - r (t)) \\ + (\underline{B} + Δ \underline{B} (t)) u (t) + B_{ω} ω (x (t), t) \\ l (t) = & Gx (t) \end{matrix}

(30)

Using the Euler method, the discretised TAFM power system is therefore given by

{\begin{matrix} \begin{array}{l} \frac{x (k + 1) - x (k)}{T} = (\underline{A} + Δ \underline{A} (k)) x (k) + \sum_{i = 1}^{4} ({\underline{A}}_{d_{i}} + Δ {\underline{A}}_{d_{i}} (k)) x (k - r (k)) \\ + (\underline{B} + Δ \underline{B} (k)) u (k) + B_{ω} ω (x (k), k) \end{array} \\ l (k) = G x (k) \end{matrix}

(31)

where T is the sampling period. The state representation (31) can be also written as

{\begin{matrix} \begin{array}{l} x (k + 1) = (A + Δ A (k)) x (k) + (A_{d} + Δ A_{d} (k)) x (k - r (k)) \\ + (B + Δ B (k)) u (k) + f (x (k), k) \end{array} \\ l (k) = G x (k) \end{matrix}

(32)

where $A = (T \underline{A} + I_{n})$ , $A_{d} = T \sum_{i = 1}^{4} {\underline{A}}_{d_{i}}$ , $B = T \underline{B}$ , $f = T B_{ω} ω$ , $Δ A = T Δ \underline{A}$ , $Δ A_{d} = T \sum_{i = 1}^{4} Δ {\underline{A}}_{d_{i}}$ and $Δ B = T Δ \underline{B}$ .

Validation

This section aims to sustain the effectiveness of the proposed control strategy. To do so, an application to the TAFM system described in equation (32) has been presented. A comparative study between our approach and Yan and Shi (2008) has been provided to corroborate the sufficiency and the supremacy of our proposition in terms of precision, robustness and time convergence.

Let us consider the time-varying state delay function as a bounded random integer variable, depicted in Figure 8. System parameters, an LMI solution as well as matrices concerning the simulation tests on the TAFM power system are listed in the Appendix.

With respect to a common simulation framework, we illustrate the results of the Monte Carlo simulations that were achieved over 100 runs using different sequences of delay. In Figures 9 –14 we depict simultaneously results generated by the proposed control algorithm and the control approach (Yan and Shi, 2008). In Figure 9 we draw the deviation of rotor angle for all generators, in Figure 10 we depict the deviation of rotor speed angle for each generator, sliding surfaces are illustrated in Figure 11 while the control inputs are shown in Figure 12.

From these simulation results it can be clearly seen that the system is asymptotically stable. The convergence times and the maximum amplitudes of the rotor angle deviations and rotor speed angle are summarized in Table 2 for the case of our proposed approach and that of Yan and Shi (2008). From Table 2 we notice that the proposed control algorithm provides faster response and better transient behaviour compared with Yan and Shi (2008).

Table 2.

Convergence times and maximum amplitudes for both approaches.

Criterion	Method	Variables
Criterion	Method	$Δ δ_{1}$	$Δ δ_{2}$	$Δ δ_{3}$	$Δ δ_{4}$	$Δ ω_{1}$	$Δ ω_{2}$	$Δ ω_{3}$	$Δ ω_{4}$
Convergence time (s)	Proposed controller	3.60	1.08	2.28	0.96	1.08	0.96	2.46	0.54
Convergence time (s)	Controller (Yan and Shi, 2008)	4.02	5.34	4.50	4.68	5.28	1.14	4.62	5.22
Maximum amplitude	Proposed controller	1.44	0.36	1.44	0.35	0.13	0.07	1.44	0.35
Maximum amplitude	Controller (Yan and Shi, 2008)	0.59	0.39	0.49	0.60	0.49	36	0.48	0.49

Moreover, oscillations are damped faster and very well once the proposed control algorithm is used. As illustrated in Table 2, there is a huge difference between the maximum amplitude of oscillations concerning the deviation of rotor speed when comparing both control strategies. Furthermore, although the maximum oscillation amplitudes for $(Δ δ_{1})$ and $(Δ δ_{3})$ are greater than $(Δ δ_{1 YAN})$ and $(Δ δ_{3 YAN})$ respectively, these oscillations are lessened rapidly, while those yielded by Yan and Shi (2008) take more time to approach zero. Therefore, despite the presence of time-varying state delay, uncertainties and disturbances, the proposed control strategy has manifested its robustness.

From Figures 11 and 12 a comparison between both algorithms in terms of the oscillation amplitude and the convergence time of sliding surfaces and control inputs raises the supremacy of our proposition.

In Figure 13 we illustrate for both control methods the steady-state error for $Δ ω_{1}$ for all delay sequences. We notice that this error never exceeds $0.01988$ for our proposition, however it can reach $0.08422$ in the case of the algorithm from Yan and Shi (2008). The same remark is valid for $Δ δ_{i}$ and $Δ ω_{i}; i = (1, \dots, 4)$ .

In Figure 14 we draw for both control strategies the convergence time for $Δ ω_{2}$ . The averaged convergence time is smaller for our proposition.

In Table 3 we give the values of time convergence, steady-state errors and maximum amplitude obtained from the Monte Carlo simulations averaged over 100 runs of the program. We present in this table values corresponding to deviation of rotor angle and deviation of rotor speed. Similar tasks are done in Tables 4 and 5 for sliding surfaces and control inputs. The results illustrated in Tables 3, 4 and 5 are related to both approaches and are obtained under a similar simulation framework. To avoid redundancy we have fixed a threshold $ξ = 10^{- 3}$ . The values given in Tables 3, 4 and 5 confirm that the proposed control algorithm is more precise, as it provides faster convergence and manifests better robustness than Yan and Shi (2008).

Table 3.

Performance of the proposed algorithm compared with Yan and Shi (2008) corresponding to different TAFM power system outputs.

Criterion	Method	Variables
Criterion	Method	$Δ δ_{1}$	$Δ δ_{2}$	$Δ δ_{3}$	$Δ δ_{4}$	$Δ ω_{1}$	$Δ ω_{2}$	$Δ ω_{3}$	$Δ ω_{4}$
Mean time convergence (s)	Proposed controller	3.7003	3.2439	3.6750	3.2491	3.2756	3.6408	3.6400	3.2290
Mean time convergence (s)	Controller (Yan and Shi, 2008)	4.1271	4.2717	3.4895	3.9813	4.5984	4.4700	3.7762	4.0550
Mean steady error	Proposed controller	0.0208	0.0004	0.0205	0.0003	0.0007	0.0027	0.0109	0.0006
Mean steady error	Controller (Yan and Shi, 2008)	0.0154	0.0237	0.0013	0.0148	0.0123	0.1449	0.0014	0.0086
Mean maximum amplitude	Proposed controller	0.0768	0.2584	0.0768	0.3598	0.2588	0.0703	0.1018	0.3598
Mean maximum amplitude	Controller (Yan and Shi, 2008)	0.4908	0.2365	0.3911	0.4908	0.2866	35.9983	0.3885	0.4901

Table 4.

Performance of both algorithms concerning sliding surface variables of the TAFM power system.

Criterion	Method	Sliding surface variables
Criterion	Method	$S_{1}$	$S_{2}$	$S_{3}$	$S_{4}$	$S_{5}$	$S_{6}$
Mean time convergence (s)	Proposed controller	1.1486	1.6850	1.8626	1.500	2.2800	0.7800
Mean time convergence (s)	Controller (Yan and Shi, 2008)	5.6100	3.0600	5.5800	5.400	4.8000	5.3400
Mean steady error	Proposed controller	0.0129	0.0815	0.0041	0.0173	0.0165	0.0027
Mean steady error	Controller (Yan and Shi, 2008)	0.0320	0.01299	0.0843	0.0785	0.0084	0.0403
Mean maximum amplitude	Proposed controller	47.3500	6.0701	6.0703	38.7200	33.0900	5.9490
Mean maximum amplitude	Controller (Yan and Shi, 2008)	600	6.5480	6.548	600	8.1481	8.151

Table 5.

Performance of both algorithms corresponding to control inputs of the TAFM power system.

Criterion	Method	Control inputs
Criterion	Method	$u_{1}$	$u_{2}$	$u_{3}$	$u_{4}$	$u_{5}$	$u_{6}$
Mean time convergence (s)	Proposed controller	1.4400	3.3050	0.1200	3.0601	1.3121	0.1200
Mean time convergence (s)	Controller (Yan and Shi, 2008)	5.2800	5.1662	0.1200	5.4000	5.2276	0.1200
Mean steady error	Proposed controller	0.0828	0.0296	0.0400	0.0176	0.0828	0.0001
Mean steady error	Controller (Yan and Shi, 2008)	0.1053	0.0494	0.0901	0.0687	0.0835	0.0013
Mean maximum amplitude	Proposed controller	62.2734	65.1955	1.0000	5.7710	73.2961	1.0000
Mean maximum amplitude	Controller (Yan and Shi, 2008)	279.8324	337.6121	1.0000	7.6410	398.9679	1.0000

Conclusion

The DT-SMC algorithm for uncertain linear systems with time-varying state delay with uncertainties on the state and input control has been discussed. Based on linear matrix inequalities, a sufficient condition for the existence of a stable sliding surface, depending on delay bounds, has been derived. An SMC law has been designed. Finally, a comparative study performed on a numerical example as well as on the TAFM power system has been presented to assess the effectiveness and the supremacy of the proposed control methodology compared with that developed by Yan and Shi (2008).

Footnotes

Appendix

\begin{matrix} A = [\begin{matrix} {\underline{A}}_{1} & {\underline{A}}_{2} & 0_{3 \times 3} & 0_{3 \times 6} \\ 0_{6 \times 3} & {\underline{A}}_{3} & 0_{6 \times 3} & 0_{6 \times 6} \\ 0_{3 \times 3} & 0_{3 \times 6} & {\underline{A}}_{1} & {\underline{A}}_{2} \\ 0_{6 \times 3} & 0_{6 \times 6} & 0_{6 \times 3} & {\underline{A}}_{4} \end{matrix}] with {\underline{A}}_{1} = [\begin{matrix} 1 & 376.991 T & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end{matrix}]; {\underline{A}}_{2} = [\begin{matrix} 0 & 0_{1 \times 5} \\ 0 & 0_{1 \times 5} \\ 376.991 T & 0_{1 \times 5} \end{matrix}] \end{matrix}

{\underline{A}}_{3} = [\begin{matrix} 1 & - . 093 T & 0.041 T & 0 & 0 & 0 \\ 0 & - 0.629 T + 1 & - 0.022 T & - 0.125 T & 0 & 0 \\ 0 & 0.334 T & - 7.142 T + 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & - 50 T + 1 & 1019.8 T & 29846000 T \\ 0 & 0 & 0 & 0 & 1 & T \\ 0 & 0 & 0 & 0 & - 2.243 T & - 27.001 T + 1 \end{matrix}]

{\underline{A}}_{4} = [\begin{matrix} 1 & - . 078 T & 0.036 T & 0 & 0 & 0 \\ 0 & - 0.629 T + 1 & - 0.022 T & - 0.125 T & 0 & 0 \\ 0 & 0.334 T & - 7.142 T & 0 & 0 & 0 \\ 0 & 0 & 0 & - 50 T + 1 & 1019.8 T & 29846000 T \\ 0 & 0 & 0 & 0 & 1 & T \\ 0 & 0 & 0 & 0 & - 2.2432 T & - 27.0012 T + 1 \end{matrix}]

A_{d} = T \sum_{i = 1}^{4} {\underline{A}}_{di} where {\underline{A}}_{d 1} = {\underline{A}}_{d 3} = 0_{18 \times 18}; {\underline{A}}_{d 2} = [\begin{matrix} 0_{6 \times 3} & 0_{6 \times 1} & 0_{6 \times 14} \\ 0_{3 \times 3} & {\underline{A}}_{d 1} & 0_{3 \times 14} \\ 0_{9 \times 3} & 0_{9 \times 1} & 0_{9 \times 14} \end{matrix}]

{\underline{A}}_{d 4} = [\begin{matrix} 0_{15 \times 12} & 0_{15 \times 1} & 0_{15 \times 5} \\ 0_{3 \times 12} & {\underline{A}}_{d 1} & 0_{3 \times 5} \end{matrix}] with {\underline{A}}_{d 1} = [\begin{matrix} - 1019.8 \\ 0 \\ 2.2432 \end{matrix}]

B = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ 0.0093 T & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0.0093 T & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 10000 T & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0.0079 T & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0.0079 T & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 10000 T \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]; f = [\begin{matrix} 0 \\ T \cos (‖ z (k) ‖ \\ 0 \\ T \cos (‖ z (k) ‖ \\ 0 \\ 0 \\ 0.04 \cos (0.8 k π) \\ 0 \\ 0 \\ 0 \\ 0.01 \cos (0.5 k π) \\ 0 \\ 0.01 \cos (0.5 k π) \\ 0 \\ 0 \\ 0.04 \cos (0.8 k π) \\ 0 \\ 0 \end{matrix}]

$Δ f' (z (k), k) = {[\begin{matrix} \frac{f (2)}{B (2, 1)} & \frac{f (4)}{B (4, 2)} & \frac{f (7)}{B (7, 3)} & \frac{f (11)}{B (11, 4)} & \frac{f (13)}{B (13, 5)} & \frac{f (16)}{B (16, 6)} \end{matrix}]}^{T}$ where $f (i)$ stands for the component i of the vector f, and $B (i, j)$ represents the component $(i, j)$ of the matrix B.

Δ B = [\begin{matrix} 0 & 0 & 0 & 0 & 0 & 0 \\ δ b_{1} & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & δ b_{1} & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & δ b_{2} & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & δ b_{3} & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & δ b_{3} & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & δ b_{2} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}] Δ B_{1} = [\begin{matrix} \frac{δ b_{1}}{B (2, 1)} & 0 & 0 & 0 & 0 & 0 \\ 0 & \frac{δ b_{1}}{B (4, 2)} & 0 & 0 & 0 & 0 \\ 0 & 0 & \frac{δ b_{2}}{B (7, 3)} & 0 & 0 & 0 \\ 0 & 0 & 0 & \frac{δ b_{3}}{B (11, 4)} & 0 & 0 \\ 0 & 0 & 0 & 0 & \frac{δ b_{3}}{B (13, 5)} & 0 \\ 0 & 0 & 0 & 0 & 0 & \frac{δ b_{2}}{B (16, 6)} \end{matrix}]

where $δ b_{1} = 0.01 \cos (0.2 Tk π); δ b_{2} = 0.01 \sin (0.2 Tk π); δ b_{3} = 0.02 \cos (0.1 Tk π)$

G = [\begin{matrix} I_{4} & 0_{4 \times 5} & 0_{4 \times 4} & 0_{4 \times 5} \\ 0_{4 \times 4} & 0_{4 \times 5} & I_{4} & 0_{4 \times 5} \end{matrix}]; O 1 = O 2 = [\begin{matrix} I_{12 \times 12} & 0_{12 \times 6} \end{matrix}]

D = [\begin{matrix} D_{11} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} \\ 0_{2 \times 2} & D_{11} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} \\ 0_{2 \times 2} & 0_{2 \times 2} & D_{11} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} \\ 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & D_{11} & 0_{2 \times 2} & 0_{2 \times 2} \\ 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & D_{11} & 0_{2 \times 2} \\ 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & 0_{2 \times 2} & D_{11} \end{matrix}] where D_{11} = [\begin{matrix} 0.1 \sin (Tk π) & 0 \\ 0 & 0.1 \cos (Tk π) \end{matrix}]

Using the CVX toolbox we obtain the following feasible solutions

\begin{matrix} P = [\begin{matrix} P_{1} & P_{2} \end{matrix}]; where \\ P_{1} = [\begin{matrix} 9.2544 & - 0.0487 & - 0.1435 & 0.1547 & - 0.0115 & - 0.0047 \\ - 0.0487 & 10.7733 & 0.0502 & - 0.1596 & - 0.8175 & - 0.3932 \\ - 0.1435 & 0.0502 & 10.9220 & - 0.1909 & 0.0631 & - 0.3723 \\ 0.1547 & - 0.1596 & - 0.1909 & 9.2503 & 0.1706 & - 0.1087 \\ - 0.0115 & - 0.8175 & 0.0631 & 0.1706 & 11.1209 & 0.4221 \\ - 0.0047 & - 0.3932 & - 0.3723 & - 0.1087 & 0.4221 & 11.2543 \\ - 0.0796 & 0.0731 & 0.0262 & - 0.0493 & - 0.0045 & 0.0451 \\ - 0.1415 & - 0.5799 & - 0.1070 & - 0.0808 & 0.0323 & - 0.2411 \\ 0.0558 & - 0.2683 & - 0.2633 & 0.0888 & 0.0804 & 0.2425 \\ - 0.0136 & 0.4776 & - 0.1332 & 0.0146 & - 0.3054 & - 0.3618 \\ - 0.1915 & 0.1991 & 0.1972 & - 0.1345 & - 0.0909 & - 0.2457 \\ 0.0825 & 0.4635 & - 0.8309 & 0.0733 & - 0.0612 & - 0.3251 \end{matrix}] \end{matrix}

P_{2} = [\begin{matrix} - 0.0796 & - 0.1415 & 0.0558 & - 0.0136 & - 0.1915 & 0.0825 \\ 0.0731 & - 0.5799 & - 0.2683 & 0.4776 & 0.1991 & 0.4635 \\ 0.0262 & - 0.1070 & - 0.2633 & - 0.1332 & 0.1972 & - 0.8309 \\ - 0.0493 & - 0.0808 & 0.0888 & 0.0146 & - 0.1345 & 0.0733 \\ - 0.0045 & 0.0323 & 0.0804 & - 0.3054 & - 0.0909 & - 0.0612 \\ 0.0451 & - 0.2411 & 0.2425 & - 0.3618 & - 0.2457 & - 0.3251 \\ 9.3550 & 0.3363 & 0.0271 & - 0.0575 & 0.3244 & 0.0350 \\ 0.3363 & 3.1036 & - 0.3539 & 0.1767 & 0.2548 & 0.4165 \\ 0.0271 & - 0.3539 & 11.2300 & 0.0274 & - 0.2645 & - 0.4947 \\ - 0.0575 & 0.1767 & 0.0274 & 10.3484 & 0.2662 & 1.2825 \\ 0.3244 & 0.2548 & - 0.2645 & 0.2662 & 9.9191 & 0.2134 \\ 0.0350 & 0.4165 & - 0.4947 & 1.2825 & 0.2134 & 13.3466 \end{matrix}]

$L = [\begin{matrix} L_{1} & L_{2} \end{matrix}]$ ; where

L_{1} = [\begin{matrix} 0.1082 & 0.0007 & 0.0013 & - 0.0017 & 0.0002 & 0.0002 \\ 0.0007 & 0.0949 & - 0.0003 & 0.0017 & 0.0066 & 0.0032 \\ 0.0013 & - 0.0003 & 0.0923 & 0.0019 & - 0.0007 & 0.0032 \\ - 0.0017 & 0.0017 & 0.0019 & 0.1083 & - 0.0016 & 0.0013 \\ 0.0002 & 0.0066 & - 0.0007 & - 0.0016 & 0.0906 & - 0.0031 \\ 0.0002 & 0.0032 & 0.0032 & 0.0013 & - 0.0031 & 0.0896 \\ 0.0007 & - 0.0014 & - 0.0003 & 0.0004 & 0.0000 & - 0.0008 \\ 0.0049 & 0.0192 & 0.0031 & 0.0031 & - 0.0002 & 0.0071 \\ - 0.0003 & 0.0026 & 0.0024 & - 0.0007 & - 0.0004 & - 0.0014 \\ 0.0001 & - 0.0040 & 0.0006 & - 0.0002 & 0.0023 & 0.0026 \\ 0.0019 & - 0.0019 & - 0.0018 & 0.0013 & 0.0006 & 0.0018 \\ - 0.0008 & - 0.0033 & 0.0058 & - 0.0006 & - 0.0002 & 0.0017 \end{matrix}]

L_{2} = [\begin{matrix} 0.0007 & 0.0049 & - 0.0003 & 0.0001 & 0.0019 & - 0.0008 \\ - 0.0014 & 0.0192 & 0.0026 & - 0.0040 & - 0.0019 & - 0.0033 \\ - 0.0003 & 0.0031 & 0.0024 & 0.0006 & - 0.0018 & 0.0058 \\ 0.0004 & 0.0031 & - 0.0007 & - 0.0002 & 0.0013 & - 0.0006 \\ 0.0000 & - 0.0002 & - 0.0004 & 0.0023 & 0.0006 & - 0.0002 \\ - 0.0008 & 0.0071 & - 0.0014 & 0.0026 & 0.0018 & 0.0017 \\ 0.1075 & - 0.0118 & - 0.0007 & 0.0009 & - 0.0032 & 0.0000 \\ - 0.0118 & 0.3314 & 0.0102 & - 0.0050 & - 0.0077 & - 0.0097 \\ - 0.0007 & 0.0102 & 0.0897 & - 0.0010 & 0.0020 & 0.0031 \\ 0.0009 & - 0.0050 & - 0.0010 & 0.0983 & - 0.0022 & - 0.0090 \\ - 0.0032 & - 0.0077 & 0.0020 & - 0.0022 & 0.1014 & - 0.0011 \\ 0.0000 & - 0.0097 & 0.0031 & - 0.0090 & - 0.0011 & 0.0768 \end{matrix}]

$[W = W_{1} W_{2}]$ ; where

W_{1} = 10^{3} [\begin{matrix} 10.2555 & - 0.0390 & - 0.0131 & - 0.5713 & 0.0477 & - 0.0417 \\ - 0.0390 & 0.0129 & 0.0025 & 0.1081 & - 0.0076 & 0.0092 \\ - 0.0131 & 0.0025 & 0.0035 & 0.0341 & - 0.0030 & 0.0031 \\ - 0.5713 & 0.1081 & 0.0341 & 1.4476 & - 0.1192 & 0.1121 \\ 0.0477 & - 0.0076 & - 0.0030 & - 0.1192 & 0.0118 & - 0.0094 \\ - 0.0417 & 0.0092 & 0.0031 & 0.1121 & - 0.0094 & 0.0112 \\ 0.0442 & - 0.0103 & - 0.0029 & - 0.1039 & 0.0082 & - 0.0083 \\ - 0.0390 & 0.0100 & 0.0025 & 0.1084 & - 0.0086 & 0.0092 \\ - 0.0390 & 0.0097 & 0.0027 & 0.1083 & - 0.0087 & 0.0087 \\ 0.0669 & - 0.0164 & - 0.0035 & - 0.1846 & 0.0150 & - 0.0149 \\ - 0.1728 & 0.0416 & 0.0098 & 0.4950 & - 0.0403 & 0.0421 \\ - 0.0436 & 0.0088 & 0.0033 & 0.1099 & - 0.0090 & 0.0087 \end{matrix}]

W_{2} = 10^{3} [\begin{matrix} 0.0442 & - 0.0390 & - 0.0390 & 0.0669 & - 0.1728 & - 0.0436 \\ - 0.0103 & 0.0100 & 0.0097 & - 0.0164 & 0.0416 & 0.0088 \\ - 0.0029 & 0.0025 & 0.0027 & - 0.0035 & 0.0098 & 0.0033 \\ - 0.1039 & 0.1084 & 0.1083 & - 0.1846 & 0.4950 & 0.1099 \\ 0.0082 & - 0.0086 & - 0.0087 & 0.0150 & - 0.0403 & - 0.0090 \\ - 0.0083 & 0.0092 & 0.0087 & - 0.0149 & 0.0421 & 0.0087 \\ 0.0232 & - 0.0101 & - 0.0096 & 0.0145 & - 0.0342 & - 0.0096 \\ - 0.0101 & 0.0109 & 0.0093 & - 0.0157 & 0.0421 & 0.0084 \\ - 0.0096 & 0.0093 & 0.0116 & - 0.0156 & 0.0420 & 0.0095 \\ 0.0145 & - 0.0157 & - 0.0156 & 0.0300 & - 0.0742 & - 0.0142 \\ - 0.0342 & 0.0421 & 0.0420 & - 0.0742 & 0.2122 & 0.0386 \\ - 0.0096 & 0.0084 & 0.0095 & - 0.0142 & 0.0386 & 0.0114 \end{matrix}]

$Z = [Z_{1} Z_{2}]$ ; where

Z_{1} = [\begin{matrix} 0.0666 & - 0.0106 & - 0.0361 & 0.0386 & - 0.0057 & - 0.0039 \\ - 0.0106 & 0.4322 & - 0.0042 & - 0.0233 & - 0.2518 & - 0.1016 \\ - 0.0361 & - 0.0042 & 0.4605 & - 0.0240 & 0.0292 & - 0.1080 \\ 0.0386 & - 0.0233 & - 0.0240 & 0.0336 & 0.0511 & - 0.0127 \\ - 0.0057 & - 0.2518 & 0.0292 & 0.0511 & 0.6778 & 0.1295 \\ - 0.0039 & - 0.1016 & - 0.1080 & - 0.0127 & 0.1295 & 0.5510 \\ - 0.0208 & 0.0222 & 0.0076 & - 0.0129 & - 0.0052 & 0.0061 \\ - 0.0390 & - 0.1262 & - 0.0026 & - 0.0325 & 0.0065 & - 0.0605 \\ - 0.0323 & - 0.0767 & - 0.0053 & - 0.0168 & 0.0326 & 0.0606 \\ 0.0007 & 0.1286 & - 0.0473 & 0.0066 & - 0.0927 & - 0.0874 \\ - 0.0265 & 0.0267 & 0.0394 & - 0.0182 & - 0.0156 & - 0.0647 \\ 0.0281 & - 0.0243 & - 0.1104 & - 0.0040 & 0.0107 & 0.0095 \end{matrix}]

Z_{2} = [\begin{matrix} - 0.0208 & - 0.0390 & - 0.0323 & 0.0007 & - 0.0265 & 0.0281 \\ 0.0222 & - 0.1262 & - 0.0767 & 0.1286 & 0.0267 & - 0.0243 \\ 0.0076 & - 0.0026 & - 0.0053 & - 0.0473 & 0.0394 & - 0.1104 \\ - 0.0129 & - 0.0325 & - 0.0168 & 0.0066 & - 0.0182 & - 0.0040 \\ - 0.0052 & 0.0065 & 0.0326 & - 0.0927 & - 0.0156 & 0.0107 \\ 0.0061 & - 0.0605 & 0.0606 & - 0.0874 & - 0.0647 & 0.0095 \\ 0.0935 & 0.0885 & 0.0289 & - 0.0167 & - 0.0133 & 0.0300 \\ 0.0885 & 0.6815 & - 0.0405 & 0.0308 & - 0.0316 & 0.0650 \\ 0.0289 & - 0.0405 & 0.4785 & 0.0093 & - 0.0264 & - 0.1649 \\ - 0.0167 & 0.0308 & 0.0093 & 0.3534 & 0.0798 & - 0.0269 \\ - 0.0133 & - 0.0316 & - 0.0264 & 0.0798 & 0.0702 & - 0.0244 \\ 0.0300 & 0.0650 & - 0.1649 & - 0.0269 & - 0.0244 & 0.4502 \end{matrix}]

$X = [X_{1} X_{2}]$ ; where

X_{1} = [\begin{matrix} 0.0059 & - 0.0029 & - 0.0030 & 0.0038 & - 0.0002 & - 0.0008 \\ - 0.0029 & 0.0608 & 0.0097 & - 0.0180 & - 0.0242 & - 0.0044 \\ - 0.0030 & 0.0097 & 0.0609 & - 0.0157 & 0.0031 & - 0.0057 \\ 0.0038 & - 0.0180 & - 0.0157 & 0.0256 & 0.0068 & - 0.0135 \\ - 0.0002 & - 0.0242 & 0.0031 & 0.0068 & 0.0572 & 0.0035 \\ - 0.0008 & - 0.0044 & - 0.0057 & - 0.0135 & 0.0035 & 0.0736 \\ - 0.0018 & 0.0037 & 0.0030 & - 0.0015 & - 0.0011 & - 0.0001 \\ 0.0001 & - 0.0045 & - 0.0063 & 0.0026 & 0.0003 & - 0.0027 \\ 0.0000 & - 0.0007 & 0.0001 & 0.0003 & - 0.0006 & 0.0008 \\ 0.0004 & 0.0074 & - 0.0055 & 0.0011 & - 0.0053 & - 0.0025 \\ - 0.0004 & 0.0010 & 0.0004 & - 0.0008 & 0.0000 & - 0.0014 \\ - 0.0000 & - 0.0024 & - 0.0047 & 0.0029 & 0.0066 & - 0.0029 \end{matrix}]

X_{2} = [\begin{matrix} - 0.0018 & 0.0001 & 0.0000 & 0.0004 & - 0.0004 & - 0.0000 \\ 0.0037 & - 0.0045 & - 0.0007 & 0.0074 & 0.0010 & - 0.0024 \\ 0.0030 & - 0.0063 & 0.0001 & - 0.0055 & 0.0004 & - 0.0047 \\ - 0.0015 & 0.0026 & 0.0003 & 0.0011 & - 0.0008 & 0.0029 \\ - 0.0011 & 0.0003 & - 0.0006 & - 0.0053 & 0.0000 & 0.0066 \\ - 0.0001 & - 0.0027 & 0.0008 & - 0.0025 & - 0.0014 & - 0.0029 \\ 0.0071 & 0.0003 & 0.0001 & - 0.0007 & 0.0001 & - 0.0004 \\ 0.0003 & 0.0416 & - 0.0038 & 0.0009 & 0.0030 & 0.0040 \\ 0.0001 & - 0.0038 & 0.0021 & 0.0000 & - 0.0005 & - 0.0007 \\ - 0.0007 & 0.0009 & 0.0000 & 0.0227 & 0.0009 & - 0.0006 \\ 0.0001 & 0.0030 & - 0.0005 & 0.0009 & 0.0017 & 0.0005 \\ - 0.0004 & 0.0040 & - 0.0007 & - 0.0006 & 0.0005 & 0.0049 \end{matrix}]

$R = [R_{1} R_{2}]$ ; where

R_{1} = [\begin{matrix} 0.0166 & - 0.0006 & - 0.0050 & 0.0093 & - 0.0023 & - 0.0068 \\ - 0.0026 & 0.1093 & - 0.0076 & - 0.0075 & - 0.0646 & - 0.0073 \\ - 0.0041 & - 0.0158 & 0.1058 & - 0.0075 & 0.0143 & - 0.0252 \\ 0.0067 & 0.0009 & 0.0089 & 0.0084 & 0.0114 & - 0.0122 \\ - 0.0030 & - 0.0688 & 0.0191 & 0.0148 & 0.1861 & 0.0247 \\ - 0.0053 & - 0.0151 & - 0.0367 & - 0.0107 & 0.0252 & 0.1545 \\ - 0.0047 & 0.0060 & 0.0011 & - 0.0027 & - 0.0010 & 0.0028 \\ 0.0021 & 0.0067 & 0.0062 & 0.0034 & 0.0035 & 0.0027 \\ - 0.0004 & 0.0039 & 0.0003 & - 0.0008 & - 0.0036 & - 0.0036 \\ 0.0035 & 0.0300 & - 0.0175 & 0.0049 & - 0.0226 & - 0.0125 \\ 0.0011 & - 0.0016 & - 0.0017 & 0.0011 & 0.0013 & 0.0024 \\ - 0.0005 & - 0.0099 & 0.0118 & 0.0029 & 0.0189 & 0.0070 \end{matrix}]

R_{2} = [\begin{matrix} - 0.0069 & - 0.0186 & - 0.0064 & 0.0062 & - 0.0003 & 0.0008 \\ 0.0134 & 0.0059 & 0.0239 & 0.0189 & - 0.0119 & - 0.0150 \\ 0.0127 & 0.0432 & 0.0081 & - 0.0208 & - 0.0053 & 0.0507 \\ - 0.0059 & - 0.0344 & - 0.0019 & 0.0097 & 0.0049 & - 0.0241 \\ - 0.0042 & 0.0021 & 0.0077 & - 0.0184 & 0.0012 & - 0.0220 \\ - 0.0002 & - 0.0229 & - 0.0137 & - 0.0027 & 0.0009 & 0.0362 \\ 0.0250 & 0.0215 & 0.0069 & - 0.0054 & - 0.0045 & 0.0079 \\ - 0.0023 & - 0.0413 & 0.0043 & - 0.0059 & - 0.0027 & - 0.0056 \\ 0.0009 & 0.0039 & - 0.0031 & 0.0014 & 0.0006 & 0.0042 \\ - 0.0029 & 0.0158 & - 0.0013 & 0.0818 & 0.0098 & - 0.0066 \\ - 0.0005 & - 0.0027 & 0.0009 & - 0.0014 & - 0.0020 & - 0.0005 \\ - 0.0021 & - 0.0018 & 0.0025 & - 0.0063 & - 0.0011 & - 0.0296 \end{matrix}]

$Q = [Q_{1} Q_{2}]$ ; where

Q_{1} = [\begin{matrix} 1.4924 & - 0.0021 & 0.0011 & - 0.0030 & 0.0006 & 0.0036 \\ - 0.0021 & 1.4717 & 0.0112 & - 0.0080 & 0.0250 & - 0.0089 \\ 0.0011 & 0.0112 & 1.4724 & - 0.0145 & - 0.0076 & 0.0229 \\ - 0.0030 & - 0.0080 & - 0.0145 & 1.5086 & - 0.0049 & 0.0017 \\ 0.0006 & 0.0250 & - 0.0076 & - 0.0049 & 1.4159 & - 0.0081 \\ 0.0036 & - 0.0089 & 0.0229 & 0.0017 & - 0.0081 & 1.4455 \\ 0.0025 & - 0.0029 & - 0.0009 & 0.0015 & 0.0015 & - 0.0014 \\ 0.0091 & - 0.0068 & - 0.0267 & 0.0185 & - 0.0013 & 0.0092 \\ 0.0032 & - 0.0132 & - 0.0031 & 0.0009 & - 0.0028 & 0.0082 \\ - 0.0017 & - 0.0096 & 0.0076 & - 0.0035 & 0.0083 & 0.0042 \\ - 0.0002 & 0.0068 & 0.0027 & - 0.0029 & - 0.0011 & - 0.0018 \\ - 0.0002 & 0.0085 & - 0.0253 & 0.0126 & 0.0077 & - 0.0205 \end{matrix}]

Q_{2} = [\begin{matrix} 0.0025 & 0.0091 & 0.0032 & - 0.0017 & - 0.0002 & - 0.0002 \\ - 0.0029 & - 0.0068 & - 0.0132 & - 0.0096 & 0.0068 & 0.0085 \\ - 0.0009 & - 0.0267 & - 0.0031 & 0.0076 & 0.0027 & - 0.0253 \\ 0.0015 & 0.0185 & 0.0009 & - 0.0035 & - 0.0029 & 0.0126 \\ 0.0015 & - 0.0013 & - 0.0028 & 0.0083 & - 0.0011 & 0.0077 \\ - 0.0014 & 0.0092 & 0.0082 & 0.0042 & - 0.0018 & - 0.0205 \\ 1.4876 & - 0.0102 & - 0.0034 & 0.0022 & 0.0023 & - 0.0043 \\ - 0.0102 & 0.0505 & - 0.0042 & - 0.0066 & 0.0033 & 0.0048 \\ - 0.0034 & - 0.0042 & 0.0033 & 0.0004 & - 0.0007 & - 0.0007 \\ 0.0022 & - 0.0066 & 0.0004 & 1.4577 & - 0.0041 & 0.0045 \\ 0.0023 & 0.0033 & - 0.0007 & - 0.0041 & 0.0022 & 0.0002 \\ - 0.0043 & 0.0048 & - 0.0007 & 0.0045 & 0.0002 & 0.0059 \end{matrix}]

$C = [C_{1} C_{2}]$ ; where

C_{1} = [\begin{matrix} - 0.0001 & - 0.0434 & - 0.0092 & - 0.0024 & - 0.0136 & - 0.0036 \\ 11.1814 & - 180.1642 & - 18.5191 & 9.1468 & - 19.8215 & 15.1742 \\ 0.0007 & 0.0025 & - 0.0003 & - 0.0017 & 0.1266 & - 0.0048 \\ 0.1532 & - 0.0011 & - 0.0020 & - 0.0007 & 0.0014 & 0 \\ 0.0005 & 0.0003 & 0.0010 & - 0.1545 & 0.0011 & - 0.0010 \\ - 0.0008 & 0.0013 & - 0.0010 & - 0.0012 & 0.0049 & - 0.1330 \end{matrix}]

\begin{matrix} C_{2} = [\begin{matrix} 0.0456 & 0.0699 & - 0.0087 & 0.0016 & 0.0020 & - 0.0127 \\ 544.2000 & 263.6614 & - 23.6124 & 3.0506 & 9.8088 & - 23.4539 \\ - 0.0066 & - 0.0036 & 0.0001 & 0 & - 0.0001 & 0.0021 \\ 0.0035 & 0.0020 & 0.0002 & 0.0006 & 0.0003 & 0.0035 \\ - 0.0008 & - 0.0005 & - 0.0003 & 0.0007 & - 0.0001 & - 0.0016 \\ - 0.0008 & - 0.0007 & - 0.0035 & 0.0007 & 0.0008 & 0.0112 \end{matrix}] \\ T = 0.06 s, q = 0.5 and ε = 0.5 \end{matrix}

Funding

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

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