Global stabilization by state feedback
This section presents the delay-dependent stabilization conditions obtained by means of the LMI method. The state feedback controller is given by
$$ u=K(\varepsilon)x , $$
(5)
where \(K(\varepsilon)= [\frac{k_{1}}{\varepsilon^{n}},\ldots, \frac{k_{n}}{\varepsilon}]\) and \(K= [k_{1},\ldots,k_{n}]\) such that \(A_{K}:=A+BK\) is Hurwitz.
Theorem 6
Suppose that Assumption
1
is satisfied. Then there exist symmetric positive definite matrices
S, Q, Z
and there exists a positive constant
ε
such that the following LMIs hold:
$$\begin{aligned}& \frac{1}{\varepsilon}\Psi+a(\varepsilon)I< 0 , \end{aligned}$$
(6)
$$\begin{aligned}& \frac{-1}{\varepsilon}Q+b(\varepsilon)I< 0 , \end{aligned}$$
(7)
where
$$\begin{gathered} \Psi=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} A^{T}_{K}S + SA_{K}+Q& S &\bar{ \tau} A_{K}^{T}Z & \bar{\tau} A_{K}^{T}Z \\ S & -I & 0 & 0 \\ \bar{\tau}ZA_{K} & 0 & -\bar{\tau} I & 0 \\ \bar{\tau} ZA_{K} & 0& 0 & -\bar{\tau} Z \end{array}\displaystyle \right ], \\a(\varepsilon)= \varepsilon n^{2} \bigl( \bar{\tau}\bigl(\|Z\|+1\bigr)+1 \bigr) \gamma_{1}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon ) \bigr), \\ b(\varepsilon)= \varepsilon n^{2} \bigl(\bar{\tau}\bigl(\|Z\|+1\bigr)+1 \bigr) \gamma_{2}(\varepsilon) \bigl(\gamma_{1}(\varepsilon)+ \gamma_{2}(\varepsilon) \bigr). \end{gathered} $$
Then the closed loop time-delay system (3)-(5) is asymptotically stable for any time delay
τ
satisfying
\(0\leq\tau \leq\bar{\tau}\).
Proof
The closed loop system is given by
$$\dot{x}= \bigl(A+BK(\varepsilon) \bigr)x+f \bigl(x,x^{\tau},u \bigr). $$
For \(\varepsilon>0\), let \(D(\varepsilon)=\operatorname{diag}[1,\varepsilon,\ldots ,\varepsilon^{n-1}]\) and \(\chi=D(\varepsilon)x\).
Using the fact that \(A+BK(\varepsilon)= \frac{1}{\varepsilon}D(\varepsilon)^{-1}A_{K}D(\varepsilon)\), we get
$$\dot{\chi}=\frac{1}{\varepsilon}A_{K}\chi+D(\varepsilon)f \bigl(x,x^{\tau},u \bigr). $$
Let us choose a Lyapunov-Krasovskii functional candidate as follows:
$$ W(\chi_{t}) = W_{1}(\chi_{t})+W_{2}( \chi _{t})+W_{3}(\chi_{t}), $$
(8)
where
$$\begin{gathered}W_{1}(\chi_{t})= \chi^{T} S\chi, \\ W_{2}(\chi_{t})=\varepsilon \int_{-\tau}^{0} \int_{t+\beta}^{t}\dot {\chi}^{T}(s)Z\dot{ \chi}(s) \,ds \,d\beta, \\ W_{3}(\chi_{t})=\frac{1}{\varepsilon} \int_{t-\tau}^{t}\chi^{T}(s)Q\chi (s)\,ds. \end{gathered} $$
Since S is symmetric positive definite, for all \(\chi\in\mathbb{R}^{n}\),
$$\lambda_{\min}(S)\|\chi\|^{2}\leq\chi^{T}S\chi\leq \lambda_{\max}(S)\| \chi\|^{2}. $$
This implies that, on the one hand,
$$W(\chi_{t})\geq\lambda_{\min}(S)\|\chi\|^{2}, $$
and on the other hand,
$$\begin{aligned} W(\chi_{t})&= W_{1}( \chi_{t})+W_{2}(\chi_{t})+ W_{3}( \chi_{t}) \\ &= \chi^{T} S \chi+\varepsilon \int_{-\tau}^{0} \int_{t+\beta}^{t} \dot{\chi}^{T}(s) Z \dot{ \chi}(s) \,ds \,d\beta + \frac{1}{\varepsilon} \int_{t-\tau}^{t} \chi^{T}(s) Q \chi(s)\,ds \\ & = \chi^{T} S \chi+ \varepsilon \int_{-\tau}^{0} \int_{\beta}^{0} \dot {\chi}^{T}(s+t) Z \dot{ \chi}(s+t) \,ds \,d\beta + \frac{1}{\varepsilon} \int_{-\tau}^{0} \chi^{T}(s+t) Q \chi(s+t)\, ds \\ & \leq \lambda_{\mathrm{max}}(S) \|{\chi}\|^{2} + \varepsilon \int_{-\tau}^{0} \lambda_{\max}(Z) \int_{\beta}^{0} \big\| \dot{\chi_{t}}(s) \big\| ^{2} \,ds \, d\beta + \frac{1}{\varepsilon}\lambda_{\max}(Q) \int_{-\tau}^{0} \big\| \chi _{t}(s) \big\| ^{2}\,ds \\ & \leq \lambda_{\mathrm{max}}(S) \|{\chi}\|^{2} - \varepsilon\tau \lambda_{\max }(Z) \int_{0}^{\beta} \big\| \dot{\chi_{t}}(s) \big\| ^{2} \,ds + \frac{1}{\varepsilon}\lambda_{\max}(Q) \int_{-\tau}^{0} \|\chi_{t}\| _{\infty}^{2}\,ds \\ & \leq \biggl(\lambda_{\mathrm{max}}(S)+ \frac{\tau}{\varepsilon}\lambda_{\max}(Q) \biggr)\| \chi_{t}\|_{\infty}^{2}. \end{aligned} $$
The time derivative of \(W_{1}\) is
$$\begin{gathered}\dot{W}_{1}(\chi_{t})= \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{K}S + SA_{K} \bigr)\chi +2\chi^{T} SD(\varepsilon)f \bigl(x,x^{\tau},u \bigr). \end{gathered} $$
So by Assumption 1 we get
$$\begin{aligned} \big\| D(\varepsilon)f \bigl(x,x^{\tau},u \bigr) \big\| &\leq \sum_{i=1}^{n} \varepsilon^{i-1}\big|f_{i} \bigl(x,x^{\tau},u \bigr)\big| \\ &\leq \gamma_{1}(\varepsilon) \sum_{i=1}^{n} \varepsilon ^{i-1}|x_{i}|+\gamma_{2}(\varepsilon) \sum_{i=1}^{n}\varepsilon^{i-1}\big|x_{i}^{\tau}\big| \\ &\leq n\gamma_{1}(\varepsilon) \big\| D(\varepsilon)x\big\| +n \gamma_{2}(\varepsilon) \big\| D(\varepsilon)x^{\tau} \big\| , \end{aligned} $$
which implies that
$$ \big\| D(\varepsilon)f \bigl(x,x^{\tau},u \bigr)\big\| \leq n \gamma_{1}(\varepsilon) \|\chi\|+n\gamma _{2}( \varepsilon) \big\| \chi^{\tau}\big\| . $$
(9)
Using Lemma 5 we deduce that
$$\begin{aligned} {\dot{W}}_{1}(\chi_{t})\leq{}& \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{K}S + SA_{K} \bigr)\chi+ \frac{1}{\varepsilon}\chi^{T} S S\chi+ \varepsilon\big\| D(\varepsilon )f \bigl(x,x^{\tau},u \bigr) \big\| ^{2} \\ \leq {}& \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{K}S + SA_{K} \bigr)\chi+ \frac{1}{\varepsilon}\chi^{T} S S\chi+ \varepsilon \bigl(n\gamma _{1}(\varepsilon) \|\chi\|+n \gamma_{2}(\varepsilon) \big\| \chi^{\tau}\big\| \bigr)^{2} \\ \leq{}& \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{K}S + SA_{K} \bigr)\chi+ \frac{1}{\varepsilon}\chi^{T} S S\chi+ \varepsilon n^{2}\gamma ^{2}_{1}(\varepsilon) \| \chi\|^{2} +\varepsilon n^{2} \gamma^{2}_{2}( \varepsilon) \big\| \chi^{\tau}\big\| ^{2} \\ & +\varepsilon n^{2}\gamma_{1}(\varepsilon) \gamma_{2}(\varepsilon) \bigl( \|\chi\|^{2}+ \big\| \chi^{\tau}\big\| ^{2} \bigr) \\ \leq{}& \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{K}S + SA_{K} \bigr)\chi+ \frac{1}{\varepsilon}\chi^{\tau} S S\chi+ \varepsilon n^{2}\gamma _{1}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \|\chi \|^{2} \\ & +\varepsilon n^{2} \gamma_{2}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
Using Lemma 5 and (9), the time derivative of \(W_{2}\) is
$$\begin{aligned}\dot{W}_{2}(\chi_{t})={}& \varepsilon\biggl( \int_{-\tau }^{0} \bigl(\dot{\chi}^{T}(t)Z\dot{ \chi}(t)- \dot{\chi}^{T}(t+\beta)Z\dot{\chi}(t+\beta) \bigr)\,d\beta\biggr) \\ ={}& \varepsilon\tau\dot{\chi}^{T}(t)Z\dot{\chi}(t)-\varepsilon \int _{t-\tau}^{t}\dot{\chi}^{T}(s)Z\dot{ \chi}(s) \,ds \\ \leq{}& \varepsilon\bar{\tau} \biggl[\frac{1}{\varepsilon}A_{K}\chi +D( \varepsilon)f \bigl(x,x^{\tau},u \bigr) \biggr]^{T} Z \biggl[ \frac{1}{\varepsilon}A_{K}\chi +D(\varepsilon)f \bigl(x,x^{\tau},u \bigr) \biggr] \\ \leq{}& \frac{\bar{\tau}}{\varepsilon}\chi^{T} \bigl(A_{K}^{T}ZA_{K} \bigr)\chi+ 2\bar{\tau}\chi^{T}A_{K}^{T}ZD( \varepsilon)f \bigl(x,x^{\tau},u \bigr) \\ & + \varepsilon\bar{\tau}\|Z\|\big\| D(\varepsilon)f \bigl(x,x^{\tau },u \bigr)\big\| ^{2} \\ \leq{}& \frac{\bar{\tau}}{\varepsilon}\chi^{T} \bigl(A_{K}^{T}ZA_{K} \bigr)\chi+ \frac{\bar{\tau}}{\varepsilon}\chi^{T}A_{K}^{T}ZZA_{K} \chi +\varepsilon\bar{\tau}\big\| D(\varepsilon)f \bigl(x,x^{\tau},u \bigr)\big\| ^{2} \\ & + \varepsilon\bar{\tau}\|Z\|\big\| D(\varepsilon)f \bigl(x,x^{\tau },u \bigr)\big\| ^{2} \\ \leq {}&\frac{\bar{\tau}}{\varepsilon}\chi^{T} \bigl(A_{K}^{T}ZA_{K}+ A_{K}^{T}ZZA_{K} \bigr)\chi +\varepsilon\bar{ \tau}\bigl(\|Z\|+1\bigr)n^{2}\gamma_{1}(\varepsilon) \bigl( \gamma _{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \| \chi \|^{2} \\ & +\varepsilon\bar{\tau}\bigl(\|Z\|+1\bigr)n^{2}\gamma_{2}( \varepsilon) \bigl( \gamma _{1}(\varepsilon)+\gamma_{2}( \varepsilon) \bigr)\big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
The time derivative of \(W_{3}\) is
$$\dot{W}_{3}(\chi_{t})=\frac{1}{\varepsilon}\chi^{T}Q \chi^{T}-\frac {1}{\varepsilon} \bigl(\chi^{\tau} \bigr)^{T}Q \chi^{\tau}. $$
Hence, we have
$$ \begin{aligned}\dot{W}(\chi_{t})\leq{}& \frac{1}{\varepsilon} \chi^{T} \bigl\{ \bigl( A^{T}_{K}S + SA_{K}+SS \bigr)+ \bar{\tau} \bigl(A_{K}^{T}ZA_{K}+ A_{K}^{T}ZZA_{K} \bigr)+Q \bigr\} \chi \\ & -\frac{1}{\varepsilon} \bigl(\chi^{\tau} \bigr)^{T}Q \chi^{\tau}+a(\varepsilon)\| \chi\|^{2} +b(\varepsilon)\big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
(10)
Then, using the Lyapunov-Krasovskii stability Theorem 2 and the Schur complement Lemma 4, we conclude that the closed loop time-delay system (3)-(5) is asymptotically stable if (6) and (7) hold. □
Global stabilization by output feedback
In [6], under Assumption 1, if the conditions does not depend on the delay τ, global exponential stability by the dynamic output feedback control is achieved. In this subsection, we study the problem of global asymptotic stability by output feedback control under Assumption 1 and delay-dependent conditions. The following system is proposed:
$$ \dot{\hat{x}}(t) = A\hat{x} + Bu(t) - L(\varepsilon) (y - C\hat{x}), $$
(11)
where \(L(\varepsilon)= [\frac{l_{1}}{\varepsilon},\ldots, \frac{l_{n}}{\varepsilon^{n}}]^{T}\) and \(L= [l_{1},\ldots,l_{n}]^{T}\) such that \(A_{L}:=A+LC\) is Hurwitz. The output feedback controller is given by
$$ u=K(\varepsilon)\hat{x}. $$
(12)
Theorem 7
Suppose that Assumption
1
is satisfied. Then there exist symmetric positive definite matrices
P, M, N
and there exists a positive constant
ε
such that the following LMIs hold:
$$\begin{aligned}& \Phi=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c@{\quad}c} A^{T}_{L}P + PA_{L}+N & P &\bar{ \tau} A_{L}^{T}M & \bar{\tau} A_{L}^{T}M \\ P & -I & 0 & 0 \\ \bar{\tau}MA_{L} & 0 & -\bar{\tau} I & 0 \\ \bar{\tau} MA_{L} & 0& 0 & -\bar{\tau} M \end{array}\displaystyle \right ]< 0, \end{aligned}$$
(13)
$$\begin{aligned}& \frac{1}{\varepsilon}\Psi+ \bigl(a(\varepsilon)+c(\varepsilon ) \bigr)I< 0, \end{aligned}$$
(14)
$$\begin{aligned}& \frac{-1}{\varepsilon}Q+ \bigl(b(\varepsilon)+d(\varepsilon ) \bigr)I< 0, \end{aligned}$$
(15)
where
$$\begin{gathered}c(\varepsilon)= \varepsilon n^{2} \bigl( \bar{\tau}\bigl(\|M\|+1\bigr)+1 \bigr) \gamma_{1}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon ) \bigr), \\ d(\varepsilon)= \varepsilon n^{2} \bigl(\bar{\tau}\bigl(\|M\|+1\bigr)+1 \bigr) \gamma_{2}(\varepsilon) \bigl(\gamma_{1}(\varepsilon)+ \gamma_{2}(\varepsilon) \bigr). \end{gathered} $$
Then the closed loop time-delay system (3)-(12) is asymptotically stable for any time delay
τ
satisfying
\(0\leq\tau \leq\bar{\tau}\).
Proof
Define \(e=x-\hat{x}\). We have
$$ \dot{e}= \bigl(A+L(\varepsilon)C \bigr)e+f \bigl(x,x^{\tau},u \bigr). $$
(16)
For \(\varepsilon>0\), let \(D(\varepsilon)=\operatorname{diag}[1,\varepsilon,\ldots ,\varepsilon^{n-1}]\) and \(\eta=D(\varepsilon)e\).
Using the fact that \(A+L(\varepsilon)C= \frac{1}{\varepsilon}D(\varepsilon)^{-1}A_{L}D(\varepsilon)\), we get
$$ \dot{\eta}=\frac{1}{\varepsilon}A_{L}\eta +D(\varepsilon)f \bigl(x,x^{\tau},u \bigr). $$
(17)
Let us choose a Lyapunov-Krasovskii functional candidate as follows:
$$ V(\eta_{t}) = V_{1}(\eta_{t})+V_{2}( \eta _{t})+V_{3}(\eta_{t}), $$
(18)
where
$$\begin{gathered}V_{1}(\eta_{t})= \eta^{T}P\eta, \\ V_{2}(\eta_{t})=\varepsilon \int_{-\tau}^{0} \int_{t+\beta}^{t}\dot {\eta}^{T}(s)M\dot{ \eta}(s) \,ds \,d\beta, \\ V_{3}(\eta_{t})=\frac{1}{\varepsilon} \int_{t-\tau}^{t}\eta^{T}(s)N\eta (s)\,ds. \end{gathered} $$
The time derivative of \(V_{1}\) is
$$\begin{aligned}\dot{V}_{1}(\eta_{t})={}& \frac{1}{\varepsilon} \chi^{T} \bigl( A^{T}_{L}P + PA_{L} \bigr)\eta +2\eta^{T} PD(\varepsilon)f \bigl(x,x^{\tau},u \bigr) \\ \leq{} & \frac{1}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L} \bigr)\eta+ \frac{1}{\varepsilon}\eta^{T} P P\eta+ \varepsilon\big\| D(\varepsilon )f \bigl(x,x^{\tau},u \bigr) \big\| ^{2} \\ \leq {}& \frac{1}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L} \bigr)\eta+ \frac{1}{\varepsilon}\eta^{T} P P\eta+ \varepsilon \bigl(n\gamma _{1}(\varepsilon) \|\chi\|+n \gamma_{2}(\varepsilon) \big\| \chi^{\tau}\big\| \bigr)^{2} \\ \leq{} & \frac{1}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L} \bigr)\eta+ \frac{1}{\varepsilon}\eta^{T} P P\eta+ \varepsilon n^{2}\gamma ^{2}_{1}(\varepsilon) \| \chi\|^{2} +\varepsilon n^{2} \gamma^{2}_{2}( \varepsilon) \big\| \chi^{\tau} \big\| ^{2} \\ & +\varepsilon n^{2}\gamma_{1}(\varepsilon) \gamma_{2}(\varepsilon) \bigl( \|\chi\|^{2}+ \big\| \chi^{\tau}\big\| ^{2} \bigr) \\ \leq{}& \frac{1}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L} \bigr)\eta+ \frac{1}{\varepsilon}\eta^{T}P P\eta+ \varepsilon n^{2}\gamma _{1}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \big\| \chi \big\| ^{2} \\ & +\varepsilon n^{2} \gamma_{2}(\varepsilon) \bigl( \gamma_{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
The time derivative of \(V_{2}\) is
$$\begin{aligned}\dot{V}_{2}(\eta_{t})={}& \varepsilon\biggl( \int_{-\tau }^{0} \bigl(\dot{\eta}^{T}(t)M\dot{ \eta}(t)- \dot{\eta}^{T}(t+\beta)M\dot{\eta}(t+\beta) \bigr)\,d\beta\biggr) \\ ={}& \varepsilon\tau\dot{\eta}^{T}(t)M\dot{\eta}(t)-\varepsilon \int _{t-\tau}^{t}\dot{\eta}^{T}(s)M\dot{ \eta}(s) \,ds \\ \leq{}& \varepsilon\bar{\tau} \biggl[\frac{1}{\varepsilon}A_{L}\eta +D( \varepsilon)f \bigl(x,x^{\tau},u \bigr) \biggr]^{T} M \biggl[ \frac{1}{\varepsilon}A_{L}\eta+ D(\varepsilon)f \bigl(x,x^{\tau},u \bigr) \biggr] \\ \leq {}& \frac{\bar{\tau}}{\varepsilon}\eta^{T} \bigl(A_{L}^{T}MA_{L} \bigr)\eta+ 2\bar{\tau}\eta^{T}A_{L}^{T}MD( \varepsilon)f \bigl(x,x^{\tau},u \bigr) \\ & + \varepsilon\bar{\tau}\|M\|\big\| D(\varepsilon)f \bigl(x,x^{\tau },u \bigr)\big\| ^{2} \\ \leq {}& \frac{\bar{\tau}}{\varepsilon}\eta^{T} \bigl(A_{L}^{T}MA_{L} \bigr)\eta+ \frac{\bar{\tau}}{\varepsilon}\eta^{T}A_{L}^{T}MMA_{L} \eta +\varepsilon\bar{\tau}\big\| D(\varepsilon)f \bigl(x,x^{\tau},u \bigr)\big\| ^{2} \\ & +\varepsilon\bar{\tau} \|M\|\big\| D(\varepsilon)f \bigl(x,x^{\tau },u \bigr)\big\| ^{2} \\ \leq {}& \frac{\bar{\tau}}{\varepsilon}\eta^{T} \bigl(A_{L}^{T}MA_{L}+ A_{L}^{T}MMA_{L} \bigr)\eta +\varepsilon\bar{ \tau}\bigl(1+\|M\|\bigr)n^{2}\gamma_{1}(\varepsilon) \bigl( \gamma _{1}(\varepsilon)+\gamma_{2}(\varepsilon) \bigr) \| \chi \|^{2} \\ & +\varepsilon\bar{\tau}\bigl(1+\|M\|\bigr)n^{2}\gamma_{2}( \varepsilon) \bigl( \gamma _{1}(\varepsilon)+\gamma_{2}( \varepsilon) \bigr) \big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
The time derivative of \(V_{3}\) is
$$\dot{V}_{3}(\eta_{t})=\frac{1}{\varepsilon}\eta^{T}N \eta-\frac {1}{\varepsilon} \bigl(\eta^{\tau} \bigr)^{T}N \eta^{\tau}. $$
So we have
$$ \begin{aligned}[b]\dot{V}\leq{}& \frac{1}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L}+PP+ \bar{\tau }A_{L}^{T}MA_{L}+\bar{\tau}A_{L}^{T}MMA_{L}+ N \bigr)\eta \\ &-\frac{1}{\varepsilon} \bigl(\eta^{\tau} \bigr)^{T}N \eta^{\tau}+c(\varepsilon)\| \chi\|^{2} +d(\varepsilon)\big\| \chi^{\tau}\big\| ^{2}. \end{aligned} $$
(19)
Let
$$U( \eta_{t},\chi_{t})=\alpha V(\eta_{t})+ W( \chi_{t}), $$
where W is given by (8). Using (10) and (19), we get
$$\begin{aligned} \dot{U}( \eta_{t},\chi_{t}) \leq{}& \frac{\alpha}{\varepsilon} \eta^{T} \bigl( A^{T}_{L}P + PA_{L}+PP+ \bar{\tau }A_{L}^{T}MA_{L}+ \bar{\tau}A_{L}^{T}MMA_{L}+ N \bigr)\eta \\ & -\frac{\alpha}{\varepsilon} \bigl(\eta^{\tau} \bigr)^{T}N \eta^{\tau}+ \bigl\{ \alpha c(\varepsilon)+a(\varepsilon) \bigr\} \|\chi \|^{2} + \bigl\{ \alpha d(\varepsilon)+b(\varepsilon) \bigr\} \big\| \chi^{\tau}\big\| ^{2} \\ & +\frac{1}{\varepsilon}\chi^{T} \bigl\{ \bigl( A^{T}_{K}S + SA_{K}+SS \bigr) +\bar{\tau} \bigl(A_{K}^{T}ZA_{K}+ A_{K}^{T}ZZA_{K} \bigr)+Q \bigr\} \chi \\ & -\frac{1}{\varepsilon} \bigl(\chi^{\tau} \bigr)^{T}Q \chi^{\tau}. \end{aligned} $$
Finally, we select α such that
$$\alpha< \min \biggl(-\frac{1}{\varepsilon}\frac{\lambda_{\min}(\Psi )+\varepsilon a(\varepsilon)}{c(\varepsilon)},\ \frac{1}{\varepsilon} \frac{\lambda_{\max}(Q)-\varepsilon b(\varepsilon )}{d(\varepsilon)} \biggr). $$
□
Remark 8
The nonlinear matrix inequalities which appeared in the criteria are successfully transformed into the LMIs to be solved easily by various effective optimization algorithms [23] or using the MATLAB LMI Control Toolbox [35].
Remark 9
Compared with [11] and [10], our new criteria overcome some of the main sources of conservatism, and contain the criteria in [11] and [10] as a special case of a class of linear delay systems. Furthermore, the new criteria also contain the well-known delay-independent stability condition in [6] and [25].
Remark 10
In [6], state feedback and output controllers for a certain class of nonlinear timing systems cover the class of systems satisfying a linear growth condition [17], using the Lyapunov-Krasovskii functions. Authors derived delay-independent conditions to ensure global exponential stability of the closed-loop systems. In this paper, in order to reduce the conservatism, a new delay-dependent stability criterion is obtained in Theorem 6 and Theorem 7 by constructing a new Lyapunov-Krasovskii functional given by (8) and (18).
Numerical example
To check the effectiveness of the result, consider the following system:
$$ \begin{gathered} \dot{x}_{1}=x_{2}(t)+ \frac{1}{10}x_{2}\sin x_{3}\cos u+\frac {1}{10}x_{2}(t- \tau)\cos u, \\ \dot{x}_{2}=x_{3}(t), \\ \dot{x}_{3}=u. \end{gathered} $$
(20)
Following the notation used throughout the paper, let \(f_{1}(x,x^{\tau},u)=\frac{1}{10}x_{2}\sin x_{3}\cos u+\frac {1}{10}x_{2}(t-\tau)\cos u\), and \(f_{2}(x,x^{\tau},u)=f_{3}(x,x^{\tau},u)=0\). Since \(f_{1}\) depends on \(x_{3}\) and \(x_{2}^{\tau}\), the output feedback scheme in [2, 20] is not applicable. It is easy to check that system (20) satisfies Assumption 1 with \(\gamma_{1}(\varepsilon) =\gamma_{2}(\varepsilon) =\frac {1}{10\varepsilon}\).
Select \(K=[ {-}4\ {-}9\ {-}4 ]\) and \(L=[ {-}2\ {-}4 \ {-}2 ]^{T}\). \(A_{K}\) and \(A_{L}\) are Hurwitz. Applying Theorem 7 and the MATLAB LMI Control Toolbox, we find that conditions (6) and (13) are given, respectively, by
$$\begin{gathered}S=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 7.2525 & 6.2371 & 1.1651\\ 6.2371 & 9.1658 & 1.6812\\ 1.1651 & 1.6812 & 0.9827 \end{array}\displaystyle \right ],\qquad P=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 2.2881 & -0.7102 & -0.0859\\ -0.7102 & 1.0878 & -0.7938\\ -0.0859 & -0.7938 & 1.2502 \end{array}\displaystyle \right ], \\ Z=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0.1768 & 0.1126 & 0.0211\\ 0.1126 & 0.2428 & 0.0259\\ 0.0211 & 0.0259 & 0.0103 \end{array}\displaystyle \right ],\qquad M=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0.0426 & -0.0142 & -0.0090\\ -0.0142 & 0.0226 & -0.0294\\ -0.0090 & -0.0294 & 0.0685 \end{array}\displaystyle \right ], \\ Q=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0.8717 & 1.5881 & 0.5684\\ 1.5881 & 3.7280 & 1.6842\\ 0.5684 & 1.6842 & 1.8074 \end{array}\displaystyle \right ],\qquad N=\left [ \textstyle\begin{array}{c@{\quad}c@{\quad}c} 0.3636 & -0.0875 & -0.0198\\ -0.0875 & 0.1448 & -0.0873\\ -0.0198 & -0.0873 & 0.1389 \end{array}\displaystyle \right ]. \end{gathered} $$
The above system is asymptotically stable for any τ satisfying \(0\leq\tau\leq1.1125\) and \(0\leq\tau\leq0.2594\). So \(\|Z\|=0.3306\) and \(\|M\|=0.0830\). This implies that condition (14) is satisfied for all \(\varepsilon>0.2279\) and condition (15) is satisfied for all \(\varepsilon>0.2189\). For our numerical simulation, we choose the delay \(\tau= 0.2\), and \(\varepsilon=0.4\). Corresponding numerical simulation results are shown in Figures 1-3.