Velocity-free event-triggered control for a class of uncertain axis-motion systems with prescribed performance

Control system design

In this section, a robust prescribed performance control scheme is proposed to keep the real-time position of the slide block in a safe region $|{x_1}|\; (\(\varepsilon$ is a safe setting boundary satisfying $\varepsilon >{\beta _1}$, where ${\beta _1}$ is given in Assumption 1), while ensuring that tracking error can achieve the specified tracking control accuracy in a fixed time.

The controller design is based on the following coordinates transformation, i.e.,

$${{{e}}_1}=\varpi \cdot \left( {{{{x}}_1} – {{{x}}_d}} \right),$$

(24)

$${{{e}}_2}={{{\hat {x}}}_2} – {{\alpha }}.$$

(25)

In (25), $\alpha$ is the virtual control law. For subsequent analysis, we define a compact set as

$${\Omega _e}=\{ {\kern 1pt} {\kern 1pt} {e_1} \in {\mathbb{R}}|\,\;\;e_{1}^{2}

(26)

It should be noted that the state constraint $|{x_1}|\; can be achieved if \(e_{1}^{2} for \(\forall t \geq 0$ and $\zeta =\varepsilon – {\beta _1}$. This is due to the fact that if $e_{1}^{2} for \(\forall t \geq 0$, then $\varpi \cdot \left| {{{{x}}_1} – {{{x}}_d}} \right|, we can conclude that \(\left| {{x_1}} \right| in view of \(\zeta =\varepsilon – {\beta _1}$ and $0. From the above analysis, it is clear that the state constraints \(|{x_1}|\; is not violated as long as \(e_{1}^{2} (\(\zeta =\varepsilon – {\beta _1}$) for $\forall t \geq 0$. In the following, we will design an efficient control strategy to guarantee $e_{1}^{2} for \(\forall t \geq 0$. The design process of the control scheme is as follows:

Step 1. Differentiating ${e_1}$ with respect to time, we have

$${\dot {e}_1}=\dot {\varpi } \cdot \left( {{x_1} – {x_d}} \right)+\varpi \cdot \left( {{x_2} – {{\dot {x}}_d}} \right).$$

(27)

Considering that ${\hat {x}_2} – {x_2}={\varpi ^{ – 1}}({\hat {z}_2}+\mu {x_1}) – {\varpi ^{ – 1}}({z_2}+\mu {x_1})={\varpi ^{ – 1}}({\hat {z}_2} – {\hat {z}_2})={\varpi ^{ – 1}}{\tilde {z}_2}$, we get form (25) that

$$\begin{aligned} {\dot {e}_1}&=\dot {\varpi } \cdot \left( {{x_1} – {x_d}} \right)+\varpi \cdot \left( {{x_2} – {{\hat {x}}_2}+{{\hat {x}}_2} – {{\dot {x}}_d}} \right)\\ &=\dot {\varpi } \cdot \left( {{x_1} – {x_d}} \right)+\varpi \cdot \left( {{e_2}+\alpha – {{\dot {x}}_d}} \right) – {\tilde {z}_2}.\end{aligned}$$

(28)

Design the virtual control law $\alpha$ as

$$\alpha = – \frac{{\dot {\varpi }}}{\varpi } \cdot \left( {{x_1} – {x_d}} \right)+{\dot {x}_d} – \frac{{{l_2} \cdot {e_1}}}{\varpi } – \frac{{\varpi \cdot {e_1}}}{{{\zeta ^2} – e_{1}^{2}}},$$

(29)

where ${l_2}>0$. Putting (29) into (28), ${{{e_1}{{\dot {e}}_1}} \mathord{\left/ {\vphantom {{{e_1}{{\dot {e}}_1}} {({\zeta ^2} – e_{1}^{2})}}} \right. \kern-0pt} {({\zeta ^2} – e_{1}^{2})}}$ in ${\Omega _e}$ can be written as

$$\frac{{{e_1}{{\dot {e}}_1}}}{{{\zeta ^2} – e_{1}^{2}}}=\frac{{{e_1}}}{{{\zeta ^2} – e_{1}^{2}}} \cdot \left( {\varpi \cdot {e_2} – {{\tilde {z}}_2} – {l_2} \cdot {e_1} – \frac{{{\varpi ^2}}}{{{\zeta ^2} – e_{1}^{2}}} \cdot {e_1}} \right).$$

(30)

Utilizing Young inequality, one has

$$\left\{ \begin{array}{l} \frac{{ – {e_1}{{\tilde {z}}_2}}}{{{\zeta ^2} – e_{1}^{2}}} \leq \frac{{e_{1}^{2}}}{{2{{({\zeta ^2} – e_{1}^{2})}^2}}}+\frac{{\tilde {z}_{2}^{2}}}{2} \hfill \\ \frac{{\varpi \cdot {e_1}{e_2}}}{{{\zeta ^2} – e_{1}^{2}}} \leq \frac{{{\varpi ^2} \cdot e_{1}^{2}}}{{2{{({\zeta ^2} – e_{1}^{2})}^2}}}+\frac{{e_{2}^{2}}}{2} \hfill \\ \end{array} \right.$$

(31)

Based on (31) and ${\varpi ^2} \geq 1$, it follows that

$$\begin{aligned} \frac{{{e_1}{{\dot {e}}_1}}}{{{\zeta ^2} – e_{1}^{2}}} &\leq \frac{{{\varpi ^2} \cdot e_{1}^{2}}}{{2{{({\zeta ^2} – e_{1}^{2})}^2}}}+\frac{{e_{2}^{2}}}{2}+\frac{{e_{1}^{2}}}{{2{{({\zeta ^2} – e_{1}^{2})}^2}}}+\frac{{\tilde {z}_{2}^{2}}}{2}\\ &- \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}} – \frac{{{\varpi ^2} \cdot e_{1}^{2}}}{{{{({\zeta ^2} – e_{1}^{2})}^2}}} \leq \frac{{e_{2}^{2}}}{2}+\frac{{\tilde {z}_{2}^{2}}}{2} – \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}}.\end{aligned}$$

(32)

Step 2. With the help of (10) and (11), the time derivative of ${{{\hat {x}}}_2}$ can be expressed as

$$\begin{aligned} {{\dot {\hat {x}}}_2}&=\frac{1}{\varpi }\left( {{{\dot {\hat {z}}}_2}+\mu {{\dot {x}}_1}+\dot {\mu }{x_1}} \right) – \frac{{\dot {\varpi }}}{{{\varpi ^2}}}\left( {{{\hat {z}}_2}+\mu {x_1}} \right) \hfill \\ & =\frac{1}{\varpi } \cdot \left[ {\left( { – \frac{{{\mu ^2}}}{\varpi }+\frac{{\dot {\varpi }\mu }}{\varpi } – \dot {\mu }} \right){{\hat {z}}_1}+\left( {\frac{{\dot {\varpi }}}{\varpi } – \frac{\mu }{\varpi }} \right){{\hat {z}}_2}+\frac{{\varpi {\nu _c}}}{m} – {L_2}{{\tilde {z}}_1}} \right] \hfill \\ &+\frac{1}{\varpi } \cdot \left[ {\frac{1}{\varpi }\left( {\mu {z_2}+{\mu ^2}{z_1}} \right)+\dot {\mu }{z_1}} \right] – \frac{{\dot {\varpi }}}{{{\varpi ^2}}} \cdot \left( {{{\hat {z}}_2}+\mu \cdot {z_1}} \right)\\ &=\frac{{{\nu _c}}}{m} – \frac{{{L_2}{{\tilde {z}}_1}}}{\varpi }+\frac{1}{\varpi } \cdot \left( {\frac{{\dot {\varpi }\mu }}{\varpi } – \frac{{{\mu ^2}}}{\varpi } – \dot {\mu }} \right) \cdot {\tilde {z}_1} – \frac{{\mu {{\tilde {z}}_2}}}{{{\varpi ^2}}}.\end{aligned}$$

(33)

Since ${L_2}=\frac{{\dot {\varpi }\mu }}{\varpi } – \frac{{{\mu ^2}}}{\varpi } – \dot {\mu } – \frac{1}{\varpi }$, the above equation can be rewritten as

$${\dot {\hat {x}}_2}=\frac{{{\nu _c}}}{m}+\frac{{{{\tilde {z}}_1}}}{{{\varpi ^2}}} – \frac{{\mu {{\tilde {z}}_2}}}{{{\varpi ^2}}}.$$

(34)

Noting that ${{{e}}_2}={{{\hat {x}}}_2} – {{\alpha }}$, we get form (34) that

$$m{{{\dot {e}}}_2}=m\left( {{{{{\dot {\hat {x}}}}}_2} – {{\dot {\alpha }}}} \right)={\nu _c} – m \cdot \left( {\frac{{\mu {{\tilde {z}}_2}}}{{{\varpi ^2}}} – \frac{{{{\tilde {z}}_1}}}{{{\varpi ^2}}}+{{\dot {\alpha }}}} \right).$$

(35)

To mitigate resource wastage associated with unnecessary fixed-period signal sampling and enhance the efficiency of control actions, a robust event-triggered control law incorporating an event-triggered strategy is developed for the nonlinear system (1) as follows:

$${\nu _c}\left( t \right)={{\sigma }}\left( {{t_K}} \right),\,\,\forall t \in \left[ {{t_K},\;{t_{K+1}}} \right),\;\left( {K=0,1,2, \ldots ,n,\;\;{t_0}=0} \right),$$

(36)

$${t_{K+1}}=\inf \left\{ {t>{t_K}\left| {\;\left| {\theta
(37)

in which \(\theta
(38)

Substituting (38) into (35) yields

$$m{{{\dot {e}}}_2}={{\sigma }}\left( t \right)+\beta \cdot \delta \cdot \varpi – m \cdot \left( {\frac{{\mu {{\tilde {z}}_2}}}{{{\varpi ^2}}} – \frac{{{{\tilde {z}}_1}}}{{{\varpi ^2}}}+{{\dot {\alpha }}}} \right).$$

(39)

Now, the signal ${{\sigma }}\left( t \right)$ is designed as follows:

$${{\sigma }}\left( t \right)= – \frac{{\psi \cdot {\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}} \cdot {{{e}}_2},$$

(40)

where $\psi$ is a positive design parameter. From the expression of \(\sigma

Theorem 2

Consider the closed loop system composed of axis-motion model (1), observer system (10), virtual control law (29), and the event-triggered control strategy (36)–(37). If initial condition ${e_1}(0) \in {\Omega _e}$ and design parameter ${l_1}$ is chosen as (13), then the following goals can be achieved.

1)

The movement position of slide block never exceeds the safety setting boundary, i.e., $\left| {{x_1}} \right|, \(\forall t \geq 0;$ in the meantime the control loop signals are bounded.
2)

the tracking error can reach a prespecified error constraint domain $\left| {{{{x}}_1} – {{{x}}_d}} \right| for \(t \geq {T_f}.$
3)

Zeno behavior never occurs as long as the closed loop system is functioning.

Proof

Consider the following Lyaponuv function

$${V_G}=\frac{1}{2} \cdot \log \left( {\frac{{{\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}} \right)+\frac{1}{2}m{{e}}_{2}^{2},$$

(41)

where ${V_G}$ is valid in the set ${\Omega _e}$. By using (32) and (39), the time derivative of ${V_G}$ can be calculated as

$$\begin{aligned} {{\dot {V}}_G}&=\frac{{{e_1}{{\dot {e}}_1}}}{{{\zeta ^2} – e_{1}^{2}}}+m{{{e}}_2}{{{{\dot {e}}}}_2} \hfill \\ & \leq \frac{{e_{2}^{2}}}{2}+\frac{{\tilde {z}_{2}^{2}}}{2} – \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}}+{e_2} \cdot {{\sigma }}
(42)

where $\Phi =\beta \cdot \delta \cdot \varpi – m \cdot ({\varpi ^{ – 2}}\mu {\tilde {z}_2} – {\varpi ^{ – 2}}{\tilde {z}_1}+{{\dot {\alpha }}})$. Note that $\left| \Phi \right|$ is bounded in ${\Omega _e}$, and therefore there exists a constant ${\vartheta ^{{\max} }}$ such that \({\Phi ^2}\,. Using young inequality yields

$${{{e}}_2}\Phi \leq e_{2}^{2} \cdot {\vartheta ^{{\max} }}+\frac{1}{4}.$$

(43)

Substituting (40) into (42) and using (43), it follows that

$$\begin{aligned} {\dot {V}_G} &\leq e_{2}^{2} \cdot \left( {{\vartheta ^{{\max} }}+\frac{{(1+{l_2})}}{2} – \frac{{\psi \cdot {\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}} \right) – \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}} – \frac{{{l_2}e_{2}^{2}}}{2}+\frac{{\tilde {z}_{2}^{2}}}{2}+\frac{1}{4}\\ &\leq e_{2}^{2} \cdot \left( {{\vartheta ^{{\max} }}+\kappa – \frac{{\psi \cdot {\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}} \right) – \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}} – \frac{{{l_2}e_{2}^{2}}}{2}+{\Delta _G}.\end{aligned}$$

(44)

where $\kappa ={{(1+{l_2})} \mathord{\left/ {\vphantom {{(1+{l_2})} 2}} \right. \kern-0pt} 2}$ and ${\Delta _G}={{\tilde {z}_{2}^{2}} \mathord{\left/ {\vphantom {{\tilde {z}_{2}^{2}} 2}} \right. \kern-0pt} 2}+{1 \mathord{\left/ {\vphantom {1 4}} \right. \kern-0pt} 4}$. For the sake of analyzing, an auxiliary function $\chi$ is defined as

$$\chi ={\zeta ^2} \cdot \left( {1 – \frac{\psi }{{{\vartheta ^{{\max} }}+\kappa }}} \right),$$

(45)

and then the following cases need to be addressed in ${\Omega _e}$.

Case 1

(If ${\vartheta ^{{\max} }}+\kappa >\psi$ and $e_{1}^{2}>\chi$)

In this case, we obtain from (45) that \(\chi ={\zeta ^2}\left( {1 – \frac{\psi }{{{\vartheta ^{{\max} }}+\kappa }}} \right) and

$${\zeta ^2} – \chi ={\zeta ^2} – {\zeta ^2} \cdot \left( {1 – \frac{\psi }{{{\vartheta ^{{\max} }}+\kappa }}} \right)=\frac{{{\zeta ^2}\psi }}{{{\vartheta ^{{\max} }}+\kappa }}.$$

(46)

Since $\frac{{\psi \cdot {\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}>\frac{{\psi \cdot {\zeta ^2}}}{{{\zeta ^2} – \chi }}={\vartheta ^{{\max} }}+\kappa$ and $- \frac{{e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}} \leq – \log \frac{{{\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}$, therefore (44) can be reduced to

$${\dot {V}_G} \leq – \frac{{{l_2} \cdot e_{1}^{2}}}{{{\zeta ^2} – e_{1}^{2}}} – \frac{{{l_2} \cdot e_{2}^{2}}}{2}+{\Delta _G} \leq – {l_2}{V_G}+{\Delta _G}.$$

(47)

Solving (47), it follows that

$$\frac{1}{2} \cdot \log \left( {\frac{{{\zeta ^2}}}{{{\zeta ^2} – e_{1}^{2}}}} \right) \leq {V_G}\left( t \right) \leq \frac{{{\Delta _G}}}{{{l_2}}}+\left( {{V_G}\left( 0 \right) – \frac{{{\Delta _G}}}{{{l_2}}}} \right) \cdot {e^{ – {l_2}t}}.$$

(48)

Using exponentials operation on both side of (48) yields

$$e_{1}^{2} \leq {\zeta ^2}\left( {1 – {e^{ – 2\left( {{V_G}(0)+{{{\Delta _G}} \mathord{\left/ {\vphantom {{{\Delta _G}} {{l_2}}}} \right. \kern-0pt} {{l_2}}}} \right)}}} \right)

(49)

From (49), it is seen that $e_{1}^{2}$ is constrained within ${\Omega _e}.$

Case 2

(If ${\vartheta ^{{\max} }}+\kappa >\psi$ and $e_{1}^{2} \leq \chi$)

It is obviously that the relation $e_{1}^{2} \leq {\zeta ^2} \cdot (1 – {\psi \mathord{\left/ {\vphantom {\psi {({\vartheta ^{{\max} }}+\kappa )}}} \right. \kern-0pt} {({\vartheta ^{{\max} }}+\kappa )}}) holds under this case. It means that the variable \({e_1}$ is constrained with the compact set ${\Omega _e}.$

Case 3

(If ${\vartheta ^{{\max} }}+\kappa \leq \psi$)

Notice that ${{{\zeta ^2}} \mathord{\left/ {\vphantom {{{\zeta ^2}} {({\zeta ^2} – e_{1}^{2})}}} \right. \kern-0pt} {({\zeta ^2} – e_{1}^{2})}}>1$, we further deduce ${{\psi {\zeta ^2}} \mathord{\left/ {\vphantom {{\psi {\zeta ^2}} {({\zeta ^2} – e_{1}^{2})}}} \right. \kern-0pt} {({\zeta ^2} – e_{1}^{2})}}>\psi \geq {\vartheta ^{{\max} }}+\kappa$, it follows from (44) that ${\dot {V}_G} \leq – {l_2}{V_G}+{\Delta _G}$, similar to mathematical operation in Case1, we easily obtain

$$e_{1}^{2} \leq {\zeta ^2}\left( {1 – {e^{ – 2\left( {{V_G}(0)+{{{\Delta _G}} \mathord{\left/ {\vphantom {{{\Delta _G}} {{l_2}}}} \right. \kern-0pt} {{l_2}}}} \right)}}} \right)

(50)

which it implies that $e_{1}^{2}$ is restrained in ${\Omega _e}$.

Combining the above analysis, we can conclude that the variable ${e_1}$ will never break the constraint \(e_{1}^{2}
(51)