2. State Key Laboratory of Synthetical Automation for Process Industries, Shenyang 110819, China;
3. College of Electrical Engineering, North China University of Water Resources and Electric Power, Zhengzhou 450011, China
Time-delay is inevitably encountered in practical systems including inferred grinding models, automatic control systems and so on [1]-[7]. It may degrade the system performance or destabilize the systems under consideration. Therefore, the stability analysis of time-delay systems has received an increasing interest [8]-[13].
Lyapunov stability theory is a favorable tool to study the stability of time-delay systems. Not only the choice of the Lyapunov-Krasovskii functional (LKF) but also the estimation of the derivative of LKF shows their comprehensive influence on the stability criteria [14]-[18]. Very recently, many powerful inequalities have been proposed, and their applications to stability [19]-[22] have shown an impressive improvement, such as the Wirtinger-based integral inequality [23], free-matrix-based integral inequality [24], refined Jensen-based inequality [25], and Bessel-Legendre inequality [26]. It is noteworthy that almost all of the mentioned inequalities deal with only a single integral term
It is well-known that the choice of the LKF plays a crucial role in deriving less conservative stability criteria. Up to now, two classes of LKFs are usually considered to study the stability analysis of time-delay systems: the augmented LKF and the delay-partitioning-based LKF. A common feature of the two classes of LKFs is that they include several double or triple integral terms. Looking at the literatures on the subject [27]-[29], one can find that the stability criteria derived by choosing triple integral forms of LKF are universally superior to those derived by choosing double integral forms of LKF. This phenomenon seems to reveal an interesting law: increasing the multiplicity of the integral terms in LKF helps to yield less conservative stability results. In fact, this law has been shown in [30]-[33] where the multiple integral
In this paper, we are further concerned with the stability analysis of time-delay systems via the multiple integral approach. Firstly, a novel multiple integral inequality, named refined Jensen-based multiple integral inequality (RJMII), is proposed. We show that the proposed inequality encompasses some existing ones. To proceed with, the inequality is applied to estimate the derivative of LKF with multiple integral terms, a new delay-dependent sufficient condition is then formulated to warrant that the considered time-delay system is globally asymptotically stable. Finally, our result is verified by two comparison examples.
Throughout this paper,
In this section, we present the RJMII based on the refined Jensen-based inequality in [25].
Lemma 1 (Refined Jensen-Based Inequality [25]): For any positive definite symmetric matrix
$ (b-a)\int_{a}^{b}x^T(s)Mx(s)ds\geq\xi_1^TM\xi_1+3\xi_2^TM\xi_2+5\xi_3^TM\xi_3 $ | (1) |
where
For continuous vector function
$ x^{[r]}=\int_{a}^{b}\int_{\theta_1}^{b}\int_{\theta_2}^{b}\cdots\int_{\theta_r}^{b}x(s)dsd\theta_r\cdots d\theta_2d\theta_1. $ |
Based on Lemma 1, we can obtain the following lemma.
Lemma 2 (RJMII): For any positive definite symmetric matrix
$ \frac{(b-a)^{r+1}}{(r+1)!}\int_{a}^{b}\int_{\theta_1}^{b}\int_{\theta_2}^{b}\cdots\int_{\theta_r}^{b}x^T(s)Mx(s)dsd\theta_r\cdots d\theta_2d\theta_1 \nonumber\\ \geq x^{[r]T}Mx^{[r]}+\frac{r+3}{r+1}\chi_a^TM\chi_a +\frac{r+5}{r+1}\chi_b^TM\chi_b $ | (2) |
where
$ \chi_a=x^{[r]}-\frac{r+2}{b-a}x^{[r+1]}\\ \chi_b=x^{[r]}-\frac{2(r+3)}{b-a}x^{[r+1]}+\frac{(r+3)(r+4)}{(b-a)^2}x^{[r+2]}. $ |
Proof: Based on Lemma 1 with the facts
$ \begin{align}\label{RJMI-p1} &(b-a)\int_{a}^{b}x(s)ds\nonumber\\ =&\int_{a}^{b}\int_{a}^{s}x(u)duds%\nonumber\\ +\int_{a}^{b}\int_{s}^{b}x(u)duds \end{align} $ | (3) |
$ \begin{align}\label{RJMI-p2} &\int_{a}^{b}\int_{a}^{s}\int_{a}^{u}x(\rho)d\rho duds-\int_{a}^{b}\int_{s}^{b}\int_{u}^{b}x(\rho)d\rho duds\nonumber\\ =&\frac{(b-a)^2}{2}\int_{a}^{b}x(s)ds-(b-a)\int_{a}^{b}\int_{s}^{b}x(s)duds \end{align} $ | (4) |
the following inequality holds for any
$ \int_{\theta_r}^{b}x^T(s)Mx(s)ds\geq\varpi^T(\theta_r)\Omega(\theta_r)\varpi(\theta_r) $ | (5) |
where
$ \begin{align*} &\Omega(\theta_r)=\begin{bmatrix}\frac{9}{b-\theta_r}&-\frac{36}{(b-\theta_r)^2}&\frac{60}{(b-\theta_r)^3}\\[2mm] *&\frac{192}{(b-\theta_r)^3}&-\frac{360}{(b-\theta_r)^4}\\[2mm] *&*&\frac{720}{(b-\theta_r)^5}\end{bmatrix}\otimes M\\[2mm] &\varpi^T(\theta_r)=\Big[ \begin{matrix}\int_{\theta_r}^{b}x^T(s)ds & \int_{\theta_r}^{b}\int_{s}^{b}x^T(\nu)d\nu ds \end{matrix} \\&\ \ \ \ \ \ \ \ \ \ \ \ \ \ \begin{matrix} \int_{\theta_r}^{b}\int_{s}^{b}\int_{\nu}^{b}x^T(\rho)d\rho d\nu ds\end{matrix}\Big]. \end{align*} $ |
We can verify that
$ \begin{align}\label{RJMI-p4} \begin{bmatrix}\int_{\theta_r}^{b}x^T(s)Mx(s)ds&\varpi^T(\theta_r)\\ *&\widetilde{\Omega}(\theta_r)\end{bmatrix}\geq0 \end{align} $ | (6) |
where
$ \begin{align} \widetilde{\Omega}(\theta_r)&=\Omega^{-1}(\theta_r)\\ &= \begin{bmatrix}b-\theta_r&\frac{(b-\theta_r)^2}{2}&\frac{(b-\theta_r)^3}{6}\\ *&\frac{(b-\theta_r)^3}{3}&\frac{(b-\theta_r)^4}{8}\\ *&*&\frac{(b-\theta_r)^5}{20}\end{bmatrix}\otimes M^{-1}>0. \end{align} $ |
We can obtain from (6) that
$ \begin{align}\label{RJMI-p5} &\left[\begin{matrix}\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\int_{\theta_r}^{b}x^T(s)Mx(s)dsd\theta_r\cdots d\theta_2d\theta_1\\ *\end{matrix}\right.\nonumber\\ & \ \ \ \ \ \ \ \ \ \ \left.\begin{matrix} \int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\varpi^T(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\\ \int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\widetilde{\Omega}(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\end{matrix}\right]\geq0 \end{align} $ | (7) |
for any
$ \begin{align*} &\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\widetilde{\Omega}(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\\ =&\frac{(b - a)^{r+1}}{(r+1)!} \begin{bmatrix}1&\frac{b-a}{r+2}&\frac{(b-a)^2}{(r+3)(r+2)}\\[2mm] *&\frac{2(b-a)^2}{(r+3)(r+2)}&\frac{3(b-a)^3}{(r+4)(r+3)(r+2)}\\[2mm] *&*&\frac{6(b-a)^4}{(r+5)(r+4)(r+3)(r+2)}\end{bmatrix} \otimes M^{-1}. \end{align*} $ |
By Schur complement [34], one has from (7) that
$ \begin{align}\label{RJMI-p6} &\int_{a}^{b}\int_{\theta_1}^{b}\int_{\theta_2}^{b}\cdots\int_{\theta_r}^{b}x^T(s)Mx(s)dsd\theta_r\cdots d\theta_2d\theta_1\nonumber\\ \geq&\left(\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\varpi(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\right)^T \nonumber\\ &\times \left(\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\widetilde{\Omega}(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\right)^{-1}\nonumber\\ &\times\left(\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\varpi(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\right) \end{align} $ | (8) |
where
$ \begin{align*} &\left(\int_{a}^{b}\int_{\theta_1}^{b}\cdots\int_{\theta_{r-1}}^{b}\widetilde{\Omega}(\theta_r)d\theta_r\cdots d\theta_2d\theta_1\right)^{-1}\nonumber\\ =&\frac{(r+3)(r+1)!}{(r+1)(b-a)^{r+1}}\nonumber\\ &\times\begin{bmatrix}3&-\frac{3(r+4)}{b-a}&\frac{(r+4)(r+5)}{(b-a)^2}\\ *&\frac{(r+4)(5r+16)}{(b-a)^2}&\frac{-2(r+3)(r+4)(r+5)}{(b-a)^3}\\ *&*&\frac{(r+3)(r+4)^2(r+5)}{(b-a)^4}\end{bmatrix}\otimes M. \end{align*} $ |
Rearranging (8) yields (2).
Remark 1: By the refined Jensen-based inequality (i.e., Lemma 1), several stability criteria for linear time-delay systems have been developed in [25]. Since the refined Jensen-based inequality is superior to the Jensen-and Writinger-based inequalities, the approach in [25] leads to improved conditions in comparison to some existing results, such as the results in [23], [24], [27]. However, this inequality is merely applied to the LKF with double integral terms. In Lemma 2, the refined Jensen-based inequality has been extended to the multiple integral version which can be utilized to estimate the derivative of LKF with multiple integral terms. Particularly, we can verify that the RJMII (2) includes the refined Jensen-based inequality (1) as a special case with
Remark 2: If the last two terms in (2) are removed, Lemma 2 is reduced to Lemma 1 in [30], i.e., the Jensen-based multiple integral inequality. Clearly, Lemma 2 offers a more precise lower bound for the integral term on the left of inequality (2) than the Lemma 1 in [30] since
It is well-known that the LKF usually takes into account the integral quadratic terms of variable
Lemma 3: For any positive definite symmetric matrix
$ \begin{align}\label{L-WM2} &\frac{(b-a)^{r+1}}{(r+1)!}\int_{a}^{b}\int_{\theta_1}^{b}\int_{\theta_2}^{b}\cdots\int_{\theta_r}^{b}\dot{x}^T(s)M\dot{x}(s)dsd\theta_r\cdots d\theta_2d\theta_1 \nonumber\\ \geq& \left[\frac{(b-a)^r}{r!}x(b)-x^{[r-1]}\right]^TM\left[\frac{(b-a)^r}{r!}x(b)-x^{[r-1]}\right]\nonumber\\ &+\frac{r+3}{r+1}\hat{\Omega}_1^TM\hat{\Omega}_1 +\frac{r+5}{r+1}\hat{\Omega}_2^TM\hat{\Omega}_2 \end{align} $ | (9) |
where
$ \quad \quad \hat{\Omega}_1=\frac{r+2}{b-a}x^{[r]}-x^{[r-1]}-\frac{(b-a)^r}{(r+1)!}x(b) \\ \hat{\Omega}_2 = \frac{2(b - a)^r}{(r+2)!}x(b) - x^{[r - 1]} + \frac{2(r + 3)}{b-a}x^{[r]} - \frac{(r + 3)(r + 4)}{(b-a)^2}x^{[r + 1]}. $ |
Applying the RJMII, this section presents a novel delay-dependent stability criterion for the following linear time-delay systems:
$ \begin{align}\label{eq-1} \left\{ \begin{array}{l} \dot{x}(t) = Ax(t) + A_dx(t - \tau) + A_D\int^{t}_{t - \tau}x(s)ds\ \ \forall\ t \geq 0\\ x(t) = \phi(t)\ \ \forall\ t\in[-\tau, 0] \end{array} \right. \end{align} $ | (10) |
where
Theorem 1: For a given positive integer
$ \Gamma=P_m+\frac{1}{\tau}\Theta_{m-1}+\frac{1}{\tau}\sum\limits_{r=0}^{m-2}\Theta_r>0 $ | (11) |
$ \Xi=\Xi_1+\Xi_2+\Xi_3<0 $ | (12) |
where
$ \begin{align*} \Theta_s=&(s+1)\epsilon_{s+2}Q_r\epsilon_{s+2}^T +(s+3)\Sigma_{1s}Q_s\Sigma_{1s}^T\\ &+(s+5)\Sigma_{2s}Q_s\Sigma_{2s}^T, \ \ \ \ \ \ \ s=0, 1, \ldots, m-2\\ \Theta_{m-1}&=m\epsilon_{m+1}Q_{m-1}\epsilon_{m+1}^T+(m+2)\Sigma_{1, m-1} Q_{m-1}\Sigma_{1, m-1}^T \end{align*} $ |
$ \begin{align*} &\, \Xi_1=\Pi_1P_m\Pi_2^T+\Pi_1^TP_m\Pi_2\\ &\Xi_2=e_1Q_0e_1^T-e_2Q_0e_2^T \\ &~~~~~~+\sum\limits_{r=1}^{m-1}\Big[(\frac{\tau^r}{r!})^2e_1Q_re_1^T-e_{r+2}Q_re_{r+2}^T\\ &~~~~~~-\frac{r+2}{r}\Delta_{1r}Q_r\Delta_{1r}^T -\frac{r+4}{r}\Delta_{2r}Q_r\Delta_{2r}^T\Big]\\ &\Xi_3=\sum\limits_{r=1}^{m}\Big[(\frac{\tau^r}{r!})^2\Psi R_r\Psi^T -\Lambda_{1r}R_r\Lambda_{1r}^T\\ &~~~~~~-\frac{r+2}{r}\Lambda_{2r}R_r\Lambda_{2r}^T -\frac{r+4}{r}\Lambda_{3r}R_r\Lambda_{3r}^T\Big]\\ &\Pi_1=\begin{bmatrix}e_1&e_3&e_4&\cdots&e_{m+3} \end{bmatrix}\\ &\Pi_2=\begin{bmatrix}\Psi&e_1-e_2&\tau &e_1-e_3&\cdots&\frac{\tau^m}{m!}e_1-e_{m+2}\end{bmatrix}\\ &\Sigma_{1p}=\epsilon_{p+2}-\frac{p+2}{\tau}\epsilon_{p+3}\\ &\Sigma_{2p}=\epsilon_{p+2}-\frac{2(p+3)}{\tau}\epsilon_{p+3}+\frac{(p+3)(p+4)}{\tau^2}\epsilon_{p+4}\\ &\Delta_{1p}=e_{p+2}-\frac{p+1}{\tau}e_{p+3}\\ &\Delta_{2p}=e_{p+2}-\frac{2(p+2)}{\tau}e_{p+3}+\frac{(p+2)(p+3)}{\tau^2}e_{p+4}\\ &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ p=0, 1, \ldots, m-1\\ &\Lambda_{1q}=\frac{\tau^{q-1}}{(q-1)!}e_1-e_{q+1}\\ &\Lambda_{2q}=\frac{\tau^{q-1}}{q!}e_1+e_{q+1}-\frac{q+1}{\tau}e_{q+2}\\ &\Lambda_{3q}=\frac{2\tau^{q-1}}{(q+1)!}e_1-e_{q+1} +\frac{2(q+2)}{\tau}e_{q+2}\\ &~~~~~~~-\frac{(q+2)(q+3)}{\tau^2}e_{q+3}, \ \ q=1, 2, \ldots, m\\ &\Psi=e_1A^T+e_2A_d^T+e_3A_D^T\\ &\epsilon_s^T=\begin{bmatrix}0_{n\times(s-1)n}&I_n&0_{n\times(m+2-s)n} \end{bmatrix}, s=1, 2, \ldots, m+2\\ &e_i^T=\begin{bmatrix}0_{n\times(i-1)n}&I_n&0_{n\times(m+3-i)n} \end{bmatrix}, i=1, 2, \ldots, m+3. \end{align*} $ |
Proof: For simplifying the expression of the proof, we first define the following variables:
$ \begin{align*} &\omega_0(t)=\int_{t-\tau}^{t}x(s)ds\\[3mm] &\omega_r(t)=\int_{t-\tau}^{t}\int_{\theta_1}^{t}\cdots\int_{\theta_{r-1}}^{t} \int_{\theta_r}^{t}x(s)dsd\theta_{r}d\theta_{r-1}\cdots d\theta_{1}\\[3mm] &\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ r=1, 2, \ldots, m\\[2mm] &\omega(t)=\begin{bmatrix}\omega_0^T(t)&\omega_1^T(t)&\ldots&\omega_m^T(t)\end{bmatrix}^T\\[2mm] &\eta(t)=\begin{bmatrix}x^T(t)&x^T(t-\tau)&\omega^T(t)\end{bmatrix}^T. \end{align*} $ |
Let us consider the following LKF candidate with multiple integral terms:
$ V(t)=V_1(t)+V_2(t)+V_3(t) $ | (13) |
where
$ \begin{align*} &V_1(t)=X^T(t)P_mX(t)\\ &V_2(t)= \int_{t-\tau}^{t}x^T(s)Q_0x(s)ds+\sum\limits_{r=1}^{m-1}\frac{\tau^r}{r!}Y_r(t)\\ &V_3(t)=\sum\limits_{r=1}^{m}\frac{\tau^r}{r!}Z_r(t) \end{align*} $ |
where
$ \begin{align*} &X(t)=\begin{bmatrix}x^T(t)&\omega^T(t)\end{bmatrix}^T\\ &Y_r(t) = \int_{t - \tau}^{t}\int_{\theta_1}^{t}\cdots\int_{\theta_{r - 1}}^{t} \int_{\theta_r}^{t}x^T(s)Q_rx(s)dsd\theta_{r}d\theta_{r - 1}\cdots d\theta_{1}\\ &Z_r(t) = \int_{t - \tau}^{t}\int_{\theta_1}^{t}\cdots\int_{\theta_{r - 1}}^{t} \int_{\theta_r}^{t}\dot{x}^T(s)R_r\dot{x}(s)dsd\theta_{r}d\theta_{r - 1}\cdots d\theta_{1}. \end{align*} $ |
Because
$ \begin{align} \frac{\tau^r}{r!}Y_r(t)\geq& \frac{1}{\tau}\Bigg[(r+1)\omega_r^TQ_r\omega_r+(r+3)\Big(\omega_r-\frac{r+2}{\tau}\omega_{r+1}\Big)^T\nonumber\\ &\times Q_r\big(\omega_r-\frac{r+2}{\tau}\omega_{r+1}\big)+(r+5)\nonumber\\ &\times\Big(\omega_r - \frac{2(r + 3)}{\tau}\omega_{r+1} + \frac{(r + 3)(r + 4)}{\tau^2}\omega_{r + 2}\Big)^TQ_r\nonumber\\ &\times \Big(\omega_r-\frac{2(r+3)}{\tau}\omega_{r+1}+\frac{(r+3)(r+4)}{\tau^2}\omega_{r+2}\Big)\Bigg]\nonumber\\ &~~~~~~~~~~~~~~r=0, 1, \ldots, m-2\label{eq-Wa}\end{align} $ | (14) |
$ \begin{align}\label{eq-Wb} \frac{\tau^{m-1}}{(m-1)!}Y_{m-1}(t)\geq& \frac{1}{\tau}\Bigg[m\omega_{m-1}^TQ_{m-1}\omega_{m-1}\nonumber\\ &+(m+2)\Big(\omega_{m-1}-\frac{m+1}{\tau}\omega_m\Big)^T\nonumber\\ &\times Q_{m-1}\Big(\omega_{m-1}-\frac{m+1}{\tau}\omega_m\Big)\Bigg]. \end{align} $ | (15) |
According to (14) and (15), it can be verified that
Calculating the derivative of
$ \dot{V}_1(t)=2X^T(t)P_m\dot{X}(t) =\eta^T(t)\Xi_1\eta(t). $ | (16) |
Based on Lemma 2 and Lemma 3, the estimations of the derivative of
Obviously, we have from (16)-(18) ((17) and (18) are shown at the bottom of this page) that
$ \begin{align} \dot{V}_2(t)=\, &x^T(t)Q_0x(t)-x^T(t-\tau)Q_0x(t-\tau)+\sum\limits_{r=1}^{m-1}\Bigg[(\frac{\tau^r}{r!})^2x^T(t)Q_rx(t)\nonumber\\[-3mm] &-\frac{\tau^r}{r!} \int_{t-\tau}^{t}\int_{\theta_1}^{t}\cdots\int_{\theta_{r-1}}^{t} x^T(s)Q_rx(s)dsd\theta_{r-1}\cdots d\theta_{1}\Bigg]\nonumber\\[-1mm] \leq\, &x^T(t)Q_0x(t)-x^T(t-\tau)Q_0x(t-\tau)%\nonumber\\ +\sum\limits_{r=1}^{m-1}\Bigg[(\frac{\tau^r}{r!})^2x^T(t)Q_rx(t) -\omega_{r-1}^T(t)Q_r\omega_{r-1}(t)\nonumber\\[-1mm] &-\frac{r+2}{r}\Big(\omega_{r-1}(t)-\frac{r+1}{\tau}\omega_r(t)\Big)^TQ_r\Big(\omega_{r-1}(t) -\frac{r+1}{\tau}\omega_r(t)\Big)\nonumber\\[-1mm] &-\frac{r+4}{r}\Big(\omega_{r-1}(t)-\frac{2(r+2)}{\tau}\omega_r(t)+\frac{(r+2)(r+3)}{\tau^2}\omega_{r+1}(t)\Big)^T\nonumber\\[-1mm] &\times Q_r\Big(\omega_{r-1}(t)-\frac{2(r+2)}{\tau}\omega_r(t)+\frac{(r+2)(r+3)}{\tau^2}\omega_{r+1}(t)\Big)\Bigg]\nonumber\\[-1mm] =\, &\eta^T(t)\Xi_2\eta(t) \label{eq-W3}\end{align} $ | (17) |
$ \begin{align} \dot{V}_3(t)=&\sum\limits_{r=1}^{m}\Bigg[(\frac{\tau^r}{r!})^2\dot{x}^T(t)R_r\dot{x}(t)%\nonumber\\ -\frac{\tau^r}{r!}\int_{t-\tau}^{t}\int_{\theta_1}^{t}\cdots\int_{\theta_{r-1}}^{t} \dot{x}^T(s)R_r\dot{x}(s)dsd\theta_{r-1}\cdots d\theta_{1}\Bigg]\nonumber\\[-1mm] \leq&\sum\limits_{r=1}^{m}\Bigg[(\frac{\tau^r}{r!})^2\dot{x}^T(t)R_r\dot{x}(t)%\nonumber\\ -\Big(\frac{\tau^{r-1}}{(r-1)!}x(t)-\omega_{r-2}(t)\Big)^T R_r\Big(\frac{\tau^{r-1}}{(r-1)!}x(t)-\omega_{r-2}(t)\Big)\nonumber\\[-1mm] &-\frac{r+2}{r}\Big(\frac{\tau^{r-1}}{r!}x(t)+\omega_{r-2}(t)-\frac{r+1}{\tau}\omega_{r-1}(t)\Big)^T\\&\times R_r\Big(\frac{\tau^{r-1}}{r!}x(t)+\omega_{r-2}(t)-\frac{r+1}{\tau}\omega_{r-1}(t)\Big)\nonumber\\[-1mm] &-\frac{r+4}{r}\Big(\frac{2\tau^{r-1}}{(r+1)!}x(t)-\omega_{r-2}(t) +\frac{2(r+2)}{\tau}\omega_{r-1}(t)-\frac{(r+2)(r+3)}{\tau^2}\omega_r(t)\Big)^T\nonumber\\[-1mm] &\times R_r\Big(\frac{2\tau^{r-1}}{(r+1)!}x(t)-\omega_{r-2}(t)%\nonumber\\ +\frac{2(r+2)}{\tau}\omega_{r-1}(t)-\frac{(r+2)(r+3)}{\tau^2}\omega_r(t)\Big) \Bigg]\nonumber\\[-1mm] =\, &\eta^T(t)\Xi_3\eta(t) \label{eq-W4} \end{align} $ | (18) |
Remark 3: By using the RJMII to estimate the derivative of LKF (13), a novel stability criterion has been shown in Theorem 1. An inevitable problem is that a larger
Remark 4: By constructing an augmented LKF with multiple integral terms and establishing a multiple integral inequality to estimate the derivative of LKF, a stability criterion is derived in [30]. When the same
1) The preferable inequalities (2) and (9) are employed to bound the multiple integral terms in the derivative of LKF, which helps us to obtain less conservative result than the one in [30].
2) The Lyapunov matrix
3) The number of decision variables of the stability conditions in Theorem 1 is also less than that in [30] (see the example in next section for details).
Remark 5: When
$ \begin{align*} V(t)=\, &\tilde{x}^T(t)P\tilde{x}(t)+\int_{t-\tau}^{t}x^T(s)Q_0x(s)ds\\ &+\tau\int_{t-\tau}^{t}\int_{\theta_1}^{t}\dot{x}^T(s)R_1\dot{x}(s)dsd\theta_1 \end{align*} $ |
where
Remark 6: The multiple integral approach and delay partitioning approach are two completely different ways on the stability analysis of delayed systems. The distance between them is much large:
1) The partitioning approach divides the integral interval
2) The delay partitioning approach reduces the conservatism of stability criteria by increasing the partitions of interval
3) When using the delay partitioning approach, one usually chooses the LKF with double or triple integral terms. But the multiple integral approach focuses on the LKF with arbitrary multiple integral terms.
In Theorem 1, the Lyapunov matrix
Corollary 1: For a given positive integer
In this section, we check our results by providing two time-delay systems (10) which are listed in Table Ⅰ. By verifying Theorem 1, we obtain Table Ⅱ which displays the maximal allowable delays or delay ranges for the systems listed in Table Ⅰ. From Table Ⅱ, we can clearly see that Theorem 1 is superior to most of the existing results in terms of conservatism. For example, our results are superior to the result in [12] where the delay
On the other hand, we can see from Table Ⅱ that the results in the cases of
Based on the refined Jensen integral inequality in [25], this brief has established a multiple integral inequality which was named RJMII. It is shown that the proposed RJMII improves some existing results, such as the Jensen-and Wirtinger-based multiple integral inequalities. The RJMII has been applied to the stability analysis of linear time-delay systems, and the associated stability criterion has been presented. By employing two typical numerical experiments, the effectiveness of our theoretical results has been fully demonstrated.
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