Passivity-Based Sliding Mode Control for Lur’e Singularly Perturbed Time-Delay Systems with Input Nonlinearity

Abstract

This paper addresses the passivity-based sliding mode control (SMC) for Lur’e singularly perturbed time-delay systems with input nonlinearity. First, a novel $\varepsilon$-dependent sliding mode surface is designed such that resulting sliding mode dynamics is still a singularly perturbed system. Based on this, a SMC law is designed to guarantee that the system state can be driven onto the specified sliding mode surface in a finite time and stays there for all future time. Then, delay-dependent and delay-independent sufficient conditions are proposed such that the sliding mode dynamics is passive and asymptotically stable. The criteria presented are both independent of the small parameter and the upper bound for the passivity can be obtained in a workable computation way. Finally, the validity of the developed methods is illustrated by three numerical examples.

Appendix: Proof of Theorem 2

Proof

By substituting (19) into (18), we obtain that the inequality (18) is equivalent to.

$$ \left( {\begin{array}{*{20}c} {\overline{\Omega }_{11} } & {\Omega_{12} } & {\Omega_{13} } & {\overline{B}_{\omega } - (CX)^{\text{T}} } & {d(\overline{A}X)^{\text{T}} } & {dX^{\text{T}} WX} \\ * & { - X^{\text{T}} QX} & {(LC_{d} X)^{\text{T}} } & { - \,(C_{d} X)^{\text{T}} } & {d(A_{d} X)^{\text{T}} } & O \\ * & * & { - \,2\eta^{ - 1} I} & {LD_{\omega } } & {\eta^{ - 1} d\overline{B}^{\text{T}} } & O \\ * & * & * & { - \,(D_{\omega } + D_{\omega }^{\text{T}} )} & {d\overline{B}_{\omega }^{\text{T}} } & O \\ * & * & * & * & { - \,d\Pi^{ - 1} } & O \\ * & * & * & * & * & { - \,d(X^{\text{T}} + X - \Pi^{ - 1} )} \\ \end{array} } \right) < 0, $$

(27)

where $\overline{\Omega }_{11} = X^{\text{T}} \overline{A}^{\text{T}} + \overline{A}X + X^{\text{T}} (W + W^{\text{T}} + Q)X$.

Pre- and post-multiplying the inequality (26) by ${\text{diag}}(I,I,\eta I,\,\,I,I,I)$ and its transpose, respectively. Then the inequality (26) is equivalent to

$$ \left( {\begin{array}{*{20}c} {\overline{\Omega }_{11} } & {\Omega_{12} } & {\overline{B} + \eta (LCX)^{\text{T}} } & {\overline{B}_{\omega } - (CX)^{\text{T}} } & {d(\overline{A}X)^{\text{T}} } & {dX^{\text{T}} WX} \\ * & { - X^{\text{T}} QX} & {\eta (LC_{d} X)^{\text{T}} } & { - \,(C_{d} X)^{\text{T}} } & {d(A_{d} X)^{\text{T}} } & O \\ * & * & { - \,2\eta I} & {\eta LD_{\omega } } & {d\overline{B}^{\text{T}} } & O \\ * & * & * & { - \,(D_{\omega } + D_{\omega }^{\text{T}} )} & {d\overline{B}_{\omega }^{\text{T}} } & O \\ * & * & * & * & { - \,d\Pi^{ - 1} } & O \\ * & * & * & * & * & { - \,d(X^{\text{T}} + X - \Pi^{ - 1} )} \\ \end{array} } \right) < 0. $$

(28)

Notice that $0 \le (X - \Pi^{ - 1} )^{\text{T}} \Pi (X - \Pi^{ - 1} ) = X^{\text{T}} \Pi X - X - X^{\text{T}} + \Pi^{ - 1}$, which implies that

$$ - X^{\text{T}} \Pi X \le - X - X^{\text{T}} + \Pi^{ - 1} . $$

(29)

Pre- and post-multiplying inequality (27) by ${\text{diag}}(X^{ - T} ,X^{ - T} ,I,\,\,I,\Pi ,X^{ - T} )$ and ${\text{diag}}(X^{ - 1} ,\,X^{ - 1} ,I,\,I,\Pi ,X^{ - 1} )$, respectively, let $X = P^{ - 1}$, $Y = KP^{ - 1}$, then it follows from (27) and (28) that the following inequality holds.

$$ \tilde{\Phi } = \left( {\begin{array}{*{20}c} {\tilde{\Omega }_{11} } & {P^{\text{T}} A_{d} - W} & {P^{\text{T}} \overline{B} + \eta (LC)^{\text{T}} } & {P^{\text{T}} \overline{B}_{\omega } - C^{\text{T}} } & {d\overline{A}^{\text{T}} \Pi } & {dW} \\ * & { - \,Q} & {\eta (LC_{d} )^{\text{T}} } & { - \,C_{d}^{\text{T}} } & {dA_{d}^{\text{T}} \Pi } & O \\ * & * & { - \,2\eta I} & {\eta LD_{\omega } } & {d\overline{B}^{\text{T}} \Pi } & O \\ * & * & * & { - \,(D_{\omega } + D_{\omega }^{\text{T}} )} & {d\overline{B}_{\omega }^{\text{T}} \Pi } & O \\ * & * & * & * & { - \,d\Pi } & O \\ * & * & * & * & * & { - \,d\Pi } \\ \end{array} } \right) < 0, $$

(30)

where $\tilde{\Omega }_{11} = \overline{A}^{\text{T}} P + P^{\text{T}} \overline{A} + W + W^{\text{T}} + Q$.

Since $X$ is a lower triangular matrix and satisfies $0 < X_{11} \in R^{n \times n}$ and $0 < X_{22} \in R^{m \times m}$, so does $P$. Let us assume that $P$ has the following form

$$ P = \left( {\begin{array}{*{20}c} {P_{11} } & O \\ {P_{21} } & {P_{22} } \\ \end{array} } \right), $$

(31)

then $P_{11}$ and $P_{22}$ are both symmetric and positive definite. Thus, there exists a scalar $\varepsilon_{11} > 0$ such that $P_{11} - \varepsilon P_{21}^{\text{T}} P_{22}^{ - 1} P_{21} > 0$ for all $\varepsilon \in (0,\varepsilon_{11} ]$. According to Schur’s Complement Lemma, it yields

$$ E_{\varepsilon } P_{\varepsilon } = \left( {\begin{array}{*{20}c} {P_{11} } & {\varepsilon P_{21}^{\text{T}} } \\ {\varepsilon P_{21} } & {\varepsilon P_{22} } \\ \end{array} } \right) > 0, $$

where $P_{\varepsilon } = \left( {\begin{array}{*{20}c} {P_{11} } & {\varepsilon P_{21}^{\text{T}} } \\ {P_{21} } & {P_{22} } \\ \end{array} } \right)$. We choose the storage function candidate as follows:

$$ V(x_{t} ) = \frac{1}{2}\left( {x^{\text{T}} (t)E_{\varepsilon } P_{\varepsilon } \,x(t) + \int_{t - d}^{t} {x^{\text{T}} (\tau )Qx(\tau ){\text{d}}\tau + } \int_{ - d}^{0} {\int_{t + \theta }^{t} {\dot{x}^{\text{T}} (\tau )E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(\tau ){\text{d}}\tau {\text{d}}\theta } } } \right), $$

where $x_{t} = x(t + \theta )$, $\theta \in [ - d,0]$. Then, computing the derivative of $V(x_{t} )$ along trajectories of the sliding mode dynamics yields

$$ \begin{aligned} & \dot{V}(x_{t} ) - \omega^{\text{T}} y \\ & \quad = \frac{1}{2}[2x^{\text{T}} (t)P_{\varepsilon }^{\text{T}} E_{\varepsilon } \,\dot{x}(t) + x^{\text{T}} (t)Qx(t) - x^{\text{T}} (t - d)Qx(t - d) \\ & \quad \quad + d\dot{x}^{\text{T}} (t)E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(t) - \int_{t - d}^{t} {\dot{x}^{\text{T}} (s)E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(s)ds} - 2\omega^{\text{T}} y] \\ & \quad = \frac{1}{2}[2x^{\text{T}} (t)P_{\varepsilon }^{\text{T}} (\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t))\\ &\quad + x^{\text{T}} (t)Qx(t) - x^{\text{T}} (t - d)Qx(t - d) \\ & \quad \quad + d(\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t))^{\text{T}} \Pi (\overline{A}x(t) \\ &\quad + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t)) \\ & \quad \quad - {\kern 1pt} {\kern 1pt} \int_{t - d}^{t} {\dot{x}^{\text{T}} (s)E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(s)ds} - 2\omega^{\text{T}} (Cx(t) + C_{d} x(t - d) + D_{\omega } \omega (t))]. \\ \end{aligned} $$

(32)

Notice that $x(t) - x(t - d) = \int_{t - d}^{t} {\dot{x}(s)ds}$. Then for $W \triangleq \left( {W_{1} ,{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} O} \right)$, one has

$$ x^{\text{T}} (t)W\left( {x(t) - x(t - d) - \int_{t - d}^{t} {\dot{x}(s){\text{d}}s} } \right) = 0,\quad WE_{\varepsilon } = \left( {\begin{array}{*{20}c} {W_{1} } & O \\ \end{array} } \right)\left( {\begin{array}{*{20}c} {I_{n} } & O \\ O & {\varepsilon I_{m} } \\ \end{array} } \right) = W. $$

(33)

By (31)–(32) and noticing the sector condition (5), for any scalar $\eta > 0$, we have

$$ \begin{aligned} & \dot{V}(x_{t} ) - \omega^{\text{T}} y \\ & \quad \le \frac{1}{2}\left[2x^{\text{T}} (t)P_{\varepsilon }^{\text{T}} (\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t)) + x^{\text{T}} (t)Qx(t) - x^{\text{T}} (t - d)Qx(t - d) \right.\\ & \quad \quad + {\kern 1pt} {\kern 1pt} d(\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t))^{\text{T}} \Pi (\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t)) \\ & \quad \quad + {\kern 1pt} {\kern 1pt} 2x^{\text{T}} (t)W(x(t) - x(t - d)) - 2\omega^{\text{T}} (Cx(t) + C_{d} x(t - d) + D_{\omega } \omega (t)) \\ & \left.\quad \quad - 2\eta \varphi^{\text{T}} (t,y)(\varphi (t,y) - Ly)\right] - \frac{1}{2}\left[2x^{\text{T}} (t)W\int_{t - d}^{t} {\dot{x}^{\text{T}} (s){\text{d}}s} + \int_{t - d}^{t} {\dot{x}^{\text{T}} (s)E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(s){\text{d}}s} \right] \\ & \quad = \frac{1}{2}\left[2x^{\text{T}} (t)P_{\varepsilon }^{\text{T}} (\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t)) + x^{\text{T}} (t)Qx(t) - x^{\text{T}} (t - d)Qx(t - d) \right.\\ & \quad \quad + {\kern 1pt} {\kern 1pt} d(\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t))^{\text{T}} \Pi (\overline{A}x(t) + A_{d} x(t - d) + \overline{B}\varphi (t,y) + \overline{B}_{\omega } \omega (t)) \\ & \quad \quad + {\kern 1pt} {\kern 1pt} 2x^{\text{T}} (t)W(x(t) - x(t - d)) - 2\omega^{\text{T}} (Cx(t) + C_{d} x(t - d) + D_{\omega } \omega (t)) + dx^{\text{T}} (t)W\Pi^{ - 1} Wx(t) \\ & \left.\quad \quad - 2\eta \varphi^{\text{T}} (t,y)(\varphi (t,y) - Ly)\right] \\ & \quad \quad - \frac{1}{2}\left[\int_{t - d}^{t} {x^{\text{T}} (t)W\Pi^{ - 1} Wx(t){\text{d}}s} - 2x^{\text{T}} (t)W\int_{t - d}^{t} {\dot{x}^{\text{T}} (s){\text{d}}s} + \int_{t - d}^{t} {\dot{x}^{\text{T}} (s)E_{\varepsilon } \Pi E_{\varepsilon } \dot{x}(s){\text{d}}s} \right] \\ & \quad = \left( {\begin{array}{*{20}c} {x^{\text{T}} (t)} & {x^{\text{T}} (t - d)} & {\varphi^{\text{T}} } & {\omega^{\text{T}} } \\ \end{array} } \right)\frac{1}{2}\Theta_{\varepsilon } \left( {\begin{array}{*{20}c} {x^{\text{T}} (t)} & {x^{\text{T}} (t - d)} & {\varphi^{\text{T}} } & {\omega^{\text{T}} } \\ \end{array} } \right)^{{\text{T}}} \\ & \quad \quad - \frac{1}{2}\int_{t - d}^{t} {(W^{\text{T}} x(t) + \Pi E_{\varepsilon } \dot{x}(s))^{\text{T}} R^{ - 1} (W^{\text{T}} x(t) + \Pi E_{\varepsilon } \dot{x}(s)){\text{d}}s} ]. \\ \end{aligned} $$

(34)

where

$$ \Theta_{\varepsilon } = \left( {\begin{array}{*{20}c} {\Theta_{11} } & {P_{\varepsilon }^{\text{T}} A_{d} - W} & {P_{\varepsilon }^{\text{T}} \overline{B} + \eta (LC)^{\text{T}} } & {P_{\varepsilon }^{\text{T}} \overline{B}_{\omega } - C^{\text{T}} } \\ * & { - \,Q} & {\eta (LC_{d} )^{\text{T}} } & { - \,C_{d}^{\text{T}} } \\ * & * & { - \,2\eta I} & {\eta LD_{\omega } } \\ * & * & * & { - \,(D_{\omega } + D_{\omega }^{\text{T}} )} \\ \end{array} } \right) + d\left( {\begin{array}{*{20}c} {\overline{A}^{\text{T}} } \\ {A_{d}^{\text{T}} } \\ {\overline{B}^{\text{T}} } \\ {\overline{B}_{\omega }^{\text{T}} } \\ \end{array} } \right)\Pi \left( {\begin{array}{*{20}c} {\overline{A}^{\text{T}} } \\ {A_{d}^{\text{T}} } \\ {\overline{B}^{\text{T}} } \\ {\overline{B}_{\omega }^{\text{T}} } \\ \end{array} } \right)^{\text{T}} $$

with $\Theta_{11} = \overline{A}^{\text{T}} P_{\varepsilon } + P_{\varepsilon }^{\text{T}} \overline{A} + W + W^{\text{T}} + Q + dW\Pi^{ - 1} W$. By Schur’s complement lemma, $\Theta_{\varepsilon } < 0$ is equivalent to

$$ \tilde{\Phi } + \varepsilon \tilde{\Phi }_{0} < 0, $$

where

$$ \tilde{\Phi }_{0} = \left( {\begin{array}{*{20}c} {\overline{A}^{\text{T}} P_{0} + P_{0}^{\text{T}} \overline{A}} & {P_{0}^{\text{T}} \overline{A}_{d} } & {P_{0}^{\text{T}} \overline{B}} & {P_{0}^{\text{T}} \overline{B}_{\omega } } & O & O \\ * & O & O & O & O & O \\ * & * & O & O & O & O \\ * & * & * & O & O & O \\ * & * & * & * & O & O \\ * & * & * & * & * & O \\ \end{array} } \right),\quad P_{0} = \left( {\begin{array}{*{20}c} O & {P_{21}^{\text{T}} } \\ O & O \\ \end{array} } \right). $$

It follows from $\tilde{\Phi } < 0$ that there exists a sufficiently small scalar $\varepsilon_{12} > 0$ such that $\tilde{\Phi } + \varepsilon \tilde{\Phi }_{0} < 0$ can be guaranteed for $\varepsilon \in (0,\varepsilon_{12} ]$. Let $\varepsilon^{ * } = \min \{ \varepsilon_{11} ,\,\,\varepsilon_{12} \}$, and then we have $V(x_{t} ) > 0$ and

$$ \dot{V}(x_{t} ) \le \omega^{\text{T}} y $$

for any given $\varepsilon \in (0,\varepsilon^{ * } ]$. According to Definition 1, the sliding mode dynamics (15)–(17) is passive.

We now show that the sliding mode dynamics with $\omega (t) = 0$ is asymptotically stable. Reconsider the inequality (33), we have

$$ \dot{V}(x_{t} ) - \omega^{\text{T}} y \le \left( {\begin{array}{*{20}c} {x^{\text{T}} (t)} & {x^{\text{T}} (t - d)} & {\varphi^{\text{T}} } & {\omega^{\text{T}} } \\ \end{array} } \right)\frac{1}{2}\Theta_{\varepsilon } \left( {\begin{array}{*{20}c} {x^{\text{T}} (t)} & {x^{\text{T}} (t - d)} & {\varphi^{\text{T}} } & {\omega^{\text{T}} } \\ \end{array} } \right)^{{\text{T}}} $$

for any given $\varepsilon \in (0,\varepsilon^{ * } ]$. Let $\lambda_{\varepsilon } = \lambda_{\min } ( - \frac{1}{2}\Theta_{\varepsilon } )$, then $\lambda_{\varepsilon } > 0$ for $\varepsilon \in (0,\varepsilon^{ * } ]$. Furthermore, in case of $\omega (t) = 0$, we obtain that

$$ \dot{V}(x_{t} ) \le - \lambda_{\varepsilon } ||\xi (t)||^{2} , $$

where $\xi (t) = \left( {\begin{array}{*{20}c} {x^{\text{T}} (t)} & {x^{\text{T}} (t - d)} \\ \end{array} } \right)^{\text{T}}$. Therefore, the sliding mode dynamics is asymptotically stable for $\varepsilon \in (0,\varepsilon^{ * } ]$. This completes the proof. □