Event-triggered compound learning tracking control of nonstrict-feedback nonlinear systems in sensor-to-controller channel

Abstract

This paper investigates the event-triggered tracking control of the nonstrict-feedback nonlinear system with time-varying disturbances. While the fuzzy logic systems (FLSs) approximate the unknown dynamics, an event-triggered compound learning algorithm is originally developed to accurately estimate the total uncertainties. By referring to an event-triggered adaptive model, the control laws are derived without provoking the problem of “algebraic loop,” seeing Remark 3. The command filters are employed to generate the continuous substitutes for both the virtual control laws and their derivatives, so as to solve the recently proposed problem of “jumps of virtual control laws” arising in the backstepping-based event-triggered control (ETC). The triggering condition is constructed to guarantee the similarity between the adaptive model and the original system. Estimation of optimal fuzzy weights and compound disturbances follows from the event-triggered update laws. While the satisfactory learning performance is achieved, the proposed control scheme can guarantee the semi-globally uniformly ultimate boundedness (SGUUB) of all the tracking errors. Finally, a numerical experiment verifies the effectiveness of the proposed control scheme.

Appendices

Proof of bounded estimation errors

Select the Lyapunov function as $V_{wi}={\tilde{W}}_i^\mathrm {T}{\tilde{W}}_i$. By invoking (8), (9) and the Young’s inequality, $\varDelta V_{wi}$ is derived as

$$\begin{aligned} \varDelta V_{wi}= & {} -{\tilde{W}}_i^\mathrm {T}{\tilde{W}}_i+(W_i-l_{wi}{\hat{W}}_i \nonumber \\&-\gamma _{wi}\phi _i\phi _i^\mathrm {T}{\tilde{W}}_i-\gamma _{wi}\phi _iD_i)^\mathrm {T} \nonumber \\&\times (W_i-l_{wi}{\hat{W}}_i-\gamma _{wi}\phi _i\phi _i^\mathrm {T}{\tilde{W}}_i -\gamma _{wi}\phi _iD_i) \nonumber \\= & {} (l_{wi}{\tilde{W}}_i^\mathrm {T}+(1-l_{wi})W_i^\mathrm {T} \nonumber \\&-\gamma _{wi}{\tilde{W}}_i^\mathrm {T}\phi _i\phi _i^\mathrm {T} -\gamma _{wi}D_i\phi _i^\mathrm {T})\nonumber \\&\times (l_{wi}{\tilde{W}}_i+(1-l_{wi})W_i^\mathrm {T} -\gamma _{wi}\phi _i\phi _i^\mathrm {T}{\tilde{W}}_i\nonumber \\&-\gamma _{wi}\phi _iD_i) -{\tilde{W}}_i^\mathrm {T}{\tilde{W}}_i \nonumber \\\le & {} 4(1-l_{wi})^2W_i^\mathrm {T}W_i+3l_{wi}^2{\tilde{W}}_i^\mathrm {T} {\tilde{W}}_i \nonumber \\&+4\gamma _{wi}^2D_i^2\phi _i^\mathrm {T}\phi _i \nonumber \\&+3\gamma _{wi}^2{\tilde{W}}_i^\mathrm {T}\phi _i\phi _i^\mathrm {T} \phi _i\phi _i^\mathrm {T}{\tilde{W}}_i-{\tilde{W}}_i^\mathrm {T}{\tilde{W}}_i \nonumber \\\le & {} -(1-3l_{wi}^2-3\gamma _{wi}^2n_{\phi _i}^2){\tilde{W}}_i^\mathrm {T} {\tilde{W}}_i+4\gamma _{wi}^2n_{\phi _i}b_{D_i}\nonumber \\&+4(1-l_{wi})^2W_i^\mathrm {T}W_i \end{aligned}$$

(A-1)

where $\Vert \phi _i\Vert ^2\le n_{\phi _i}$, $D_i^2\le b_{D_i}$ and $\lambda _{max}(\phi _i\phi _i^\mathrm {T}\phi _i\phi _i^\mathrm {T})\le n_{\phi _i}^2$. Because the maximum inter-event time $T_{max}$ is finite and $\Vert \varphi _i\Vert$ and $\sigma _i$ are bounded, it is tenable that the positive constants of $n_{\phi _i}$ and $b_{D_i}$ exist.

Fig. 12

Accumulation of sampling times

Fig. 13

Inter-event time of ET-CL

Select the Lyapunov function as $V_{\sigma i}={\tilde{\sigma }}_i^2$. By invoking (12), (13) and the Young’s inequality, $\varDelta V_{\sigma i}$ is derived as

$$\begin{aligned} \begin{aligned} \varDelta V_{\sigma _i}&=(\sigma _i-l_{\sigma i}{\hat{\sigma }}_i-\gamma _{\sigma i}{\tilde{W}}_i^\mathrm {T}\phi _i-\gamma _{\sigma i}D_i)^2-{\tilde{\sigma }}_i^2\\&=(l_{\sigma i}{\tilde{\sigma }}_i+(1-l_{\sigma i})\sigma _i-\gamma _{\sigma i}{\tilde{W}}_i^\mathrm {T}\phi _i-\gamma _{\sigma i}D_i)^2\\&\quad -{\tilde{\sigma }}_i^2\\&\le 4(1-l_{\sigma i})^2\sigma _i^2+4l_{\sigma i}^2{\tilde{\sigma }}_i^2+4\gamma _{\sigma i}^2{\tilde{W}}_i^\mathrm {T}\phi _i\phi _i^\mathrm {T}{\tilde{W}}_i\\&\quad -{\tilde{\sigma }}_i^2 +4\gamma _{\sigma i}^2D_i^2\\&\le -(1-4l_{\sigma i}^2){\tilde{\sigma }}_i^2+4(1-l_{\sigma i})^2b_{\sigma _i}+4\gamma _{\sigma i}^2b_{D_i}\\&\quad +4\gamma _{\sigma i}^2n_{\phi _i}{\tilde{W}}^\mathrm {T}_i{\tilde{W}}_i \end{aligned} \end{aligned}$$

(A-2)

where $b_{\sigma _i}=(b_{d_i}+b_{\varepsilon _i})^2$. Define $V_i=V_{wi}+V_{\sigma i}$. By adding up (A-1) and (A-2), it renders

$$\begin{aligned} \begin{aligned} \varDelta V_i&\le -(1-3l_{wi}^2-3\gamma _{wi}^2n_{\phi _i}^2-4\gamma _{\sigma i}^2n_{\phi _i}){\tilde{W}}_i^\mathrm {T}{\tilde{W}}_i\\&\quad +4\gamma _{\sigma i}^2b_{D_i}\\&\quad -(1-4l_{\sigma i}^2){\tilde{\sigma }}_i^2+4\gamma _{wi}^2n_{\phi _i} b_{D_i}\\&\quad +4(1-l_{wi})^2W_i^\mathrm {T}W_i\\&\quad +4(1-l_{\sigma i})^2b_{\sigma _i} \end{aligned} \end{aligned}$$

(A-3)

By selecting $l_{\sigma i}<1/2$, $3\gamma _{wi}^2n_{\phi _i}^2+4\gamma _{\sigma i}^2n_{\phi _i}<1$ and $l_{wi}<(1-3\gamma _{wi}^2n_{\phi _i}^2-4\gamma _{\sigma i}^2n_{\phi _i})/\sqrt{3}$, and letting $a_i=\min \{1-3l_{wi}^2-3\gamma _{wi}^2n_{\phi _i}^2-4\gamma _{\sigma i}^2n_{\phi _i},1-4l_{\sigma i}^2\}$ and $b_i=4\gamma _{wi}^2n_{\phi _i}b_{D_i}+4(1-l_{wi})^2W_i^\mathrm {T}W_i+4(1-l_{\sigma i})^2b_{\sigma _i}+4\gamma _{\sigma i}^2b_{D_i}$, (A-3) can be simplified as $\varDelta V_i\le -a_iV_i+b_i$. Because ${\tilde{\sigma }}_i$ is changing during the inter-event time, one cannot infer the boundedness of $|{\tilde{\sigma }}_i|$ from $\varDelta V_i\le -a_iV_i+b_i$. A further processing is needed.

Because $\dot{{\tilde{W}}}_i=0$ in $t\in (t_j,t_{j+1}]$, it is obvious that $\varTheta V_i=\varDelta V_i+{\tilde{\sigma }}_i^2(t_{j+1})-{\tilde{\sigma }}_i^2(t_j^+)$. Assume $|{\tilde{\sigma }}_i|$ grows with the fastest speed during the inter-vent time. According to Assumption 1 and (13), it is tenable to assume $|\dot{{\tilde{\sigma }}}_i|=|{\dot{\sigma }}_i|\le b_{{\dot{\sigma }}_i}$ in $t\in (t_j,t_{j+1}]$, where $b_{{\dot{\sigma }}_i}$ is an unknown constant. Thus, the following inequality holds.

$$\begin{aligned} \begin{aligned}&{\tilde{\sigma }}_i^2(t_{j+1})-{\tilde{\sigma }}_i^2(t_j^+)\\&\quad \le 2b_{{\dot{\sigma }}_i}T_{max}|{\tilde{\sigma }}_i(t_j^+)|+b_{{\dot{\sigma }}_i}^2T_{max}^2\\&\quad \le 2b_{{\dot{\sigma }}}T_{max}\sqrt{V_i+\varDelta V_i}+b_{{\dot{\sigma }}_i}^2T_{max}^2. \end{aligned} \end{aligned}$$

(A-4)

Thus, a new interconnected system is obtained as

$$\begin{aligned} \left\{ \begin{aligned} \varDelta V_i&\le -a_iV_i+b_i\\ \varTheta V_i&\le -a_iV_i+2b_{{\dot{\sigma }}_i}T_{max} \sqrt{V_i+\varDelta V_i}\\&\quad +b_{{\dot{\sigma }}_i}^2T_{max}^2+b_i \end{aligned}\right. . \end{aligned}$$

(A-5)

If $V_i(t_j)>b_i/a_i$, it has $\varDelta V_i<0$ for the first inequality in (A-5). In this case, the second inequality in (A-5) is transformed to $\varTheta V_i\le -a_iV_i+2b_{{\dot{\sigma }}_i}T_{max}\sqrt{V_i}+b_{{\dot{\sigma }}_i}^2T_{max}^2+b_i$. By solving this inequality, it has $\varTheta V_i<0$ for $V_i(t_j)>(b_{{\dot{\sigma }}_i}T_{max}+\sqrt{b_{{\dot{\sigma }}_i}^2T_{max}^2(1+a_i)+a_ib_i})^2/a_i^2$. Thus, it is concluded that $\varTheta V_i<0$ for $V_i(t_j)>\max \{b_i/a_i,(b_{{\dot{\sigma }}_i}T_{max}+\sqrt{b_{{\dot{\sigma }}_i}^2T_{max}^2(1+a_i)+a_ib_i})^2/a_i^2\}$. Because $V_i(t_j^+)<V_i(t)<V_i(t_{j+1})$ in $t\in (t_j,t_{j+1}]$, it is inferred that $V_i(t)$ is bounded all along and is ultimately bounded by $\max \{b_i/a_i,(b_{{\dot{\sigma }}_i}T_{max}{+}\sqrt{b_{{\dot{\sigma }}_i}^2T_{max}^2(1+a_i)+a_ib_i})^2/a_i^2\}$. The proof is completed.

It is obvious that the larger $T_{max}$ (namely by increasing $\mu$) will lead to the smaller $a_i$ and the larger ultimate bound of $V_i$, which suggests the slower learning rate and the worse identification accuracy.

Proof of Lemma 2

For convenience, set $\zeta =1$ in (17). By subtracting $[\alpha _i,0]^\mathrm {T}$ from (17), it renders

$$\begin{aligned} \left[ \begin{matrix}{\dot{q}}_i\\ {\dot{\eta }}_i\end{matrix}\right] =\left[ \begin{matrix}0&{}1\\ omega^2&{}-2\omega \end{matrix}\right] \left[ \begin{matrix}q_i\\ \eta _i\end{matrix}\right] -\left[ \begin{matrix}{\dot{\alpha }}_i\\ 0\end{matrix}\right] . \end{aligned}$$

(B-1)

Define $A=[0,1;-\omega ^2,-2\omega ]$. It is calculated that A has two repeated eigenvalues of $\lambda (A)=-\omega$. Through the Hamilton–Cayley theorem, it can be derived that

$$\begin{aligned} \left\{ \begin{aligned}&a_1(t)\lambda (A)+a_0(t)=e^{\lambda (A)t}\\&a_1(t)=te^{\lambda (A)t} \end{aligned}\right. . \end{aligned}$$

(B-2)

Thus, $a_0(t)=(1+\omega t)e^{-\omega t}$ and $a_1(t)=te^{-\omega t}$ are obtained. Ulteriorly, the state transformation matrix of (B-1) in $t\in (t_j,t_{j+1}]$ is obtained as $\varPhi (t)=a_1(t-t_j)A+a_0(t-t_j)$. Then, the analytical solution of (B-1) is derived as

$$\begin{aligned} \left[ \begin{matrix}q_i(t)\\ \eta _i(t)\end{matrix}\right]&=\varPhi (t)\left[ \begin{matrix}q_i(t_j^+)\\ \eta _i(t_j)\end{matrix}\right] \nonumber \\&\quad + \int _{t_j}^t\varPhi (\tau )\left[ \begin{matrix}{\dot{\alpha }}_i(t-t_j -\tau )\\ 0\end{matrix}\right] ~\mathrm {d}\tau . \end{aligned}$$

(B-3)

Because all the states are defined in a compact set, it is tenable to assume $|{\dot{\alpha }}_i(t)|\le b_{{\dot{\alpha }}_i}$, where $b_{{\dot{\alpha }}_i}$ is an unknown constant. Then, the following inequality can be derived from (B-3).

$$\begin{aligned} \left\{ \begin{aligned} |q_i(t)|&\le |q_i(t_j^+)|(1+\omega (t-t_j))e^{-\omega (t-t_j)}\\&\quad +\left( \frac{2}{\omega }-\frac{2}{\omega }e^{-\omega (t-t_j)} -(t-t_j)e^{-\omega (t-t_j)}\right) b_{{\dot{\alpha }}_i} \\&\quad +|\eta _i(t_j)|(t-t_j)e^{-\omega (t-t_j)}\\ |\eta _i(t)|\le&\,|q_i(t_j^+)|\omega ^2(t-t_j)e^{-\omega (t-t_j)}\\&\quad +|(1-\omega (t-t_j))e^{-\omega (t-t_j)}|\times |\eta _i(t_j)|\\&\quad +(1-e^{-\omega (t-t_j)}-w(t-t_j)e^{-\omega (t-t_j)})b_{{\dot{\alpha }}_i} \end{aligned}\right. . \end{aligned}$$

(B-4)

For (B-4), an assumption is made at first that $|q_i(t_j^+)|\le \mu _{q_i}+b_{q_i}$ and $|\eta _i(t_j)|\le \mu _{\eta _i}+b_{{\dot{\alpha }}_i}$, where $\mu _{q_i}$ and $\mu _{\eta _i}$ can two arbitrarily small positive constants. For the first inequality in (B-4), it is inferred that $|q_i(t)|\rightarrow 0$ with $\omega \rightarrow +\infty$. Thus, there must exist $\omega =\omega _{q_i}$ such that $|q_i(t_j+\delta _t)|\le \mu _{q_i}$ holds, where $\delta _t$ is the minimum inter-event time proved in Section IV. Invoking $|\varDelta q_i|\le b_{q_i}$, $|q_i(t_j^+)|\le \mu _{q_i}+b_{q_i}$ holds. For the second inequality in (B-4), it is inferred that $|\eta _i(t)|\le b_{{\dot{\alpha }}_i}$ with $\omega \rightarrow +\infty$. Thus, there must exist $\omega =\omega _{\eta _i}$ such that $|\eta _i(t_j+\delta _t)|\le \mu _{\eta _i}+b_{{\dot{\alpha }}_i}$ holds. By selecting $\omega =\max \{\omega _{q_i},\omega _{\eta _i}\}$, this assumption holds.

From the first inequality in (B-4), it can be found that $|(1+\omega (t-t_j))e^{-\omega (t-t_j)}|<1$, $|2/\omega -2/\omega e^{-\omega (t-t_j)}-(t-t_j)e^{-\omega (t-t_j)}|<2/\omega$ and $|(t-t_j)e^{-\omega (t-t_j)}|<1/(\omega e)$. By invoking the above assumption, the first inequality in (B-4) can be transformed to

$$\begin{aligned} |q_i(t)|\le \mu _{q_i}+b_{q_i}+\frac{\mu _{\eta _i}+b_{{\dot{\alpha }}_i}}{\omega e}+\frac{2b_{{\dot{\alpha }}_i}}{\omega }. \end{aligned}$$

(B-5)

Let ${\bar{b}}_{q_i}=\mu _{q_i}+b_{q_i}+(\mu _{\eta _i}+b_{{\dot{\alpha }}_i})/(\omega e)+2b_{{\dot{\alpha }}_i}/\omega$. It is obvious that ${\bar{b}}_{q_i}\rightarrow b_{q_i}$ with $\omega \rightarrow +\infty$. The proof is completed.