Dynamic-Gain-Based Adaptive Fuzzy Control for Uncertain Delayed Non-triangular Switched Nonlinear Systems

Abstract

This work studies the adaptive issue of nonlinear switched system with state delays. Different from the reported works, the research is more challenging since the system is in non-triangular form and has multiple uncertainties such as unmodeled dynamic, unknown parameters, and unknown coefficients. By presenting a dynamic-gain-based adaptive fuzzy regulation strategy, and constructing a novel Lyapunov–Krasovskii (L–K) functional, a new robust adaptive controller is successfully designed. The raised strategy is validated using a simulation example.

Appendix

Proof of (21)

According to (19), we get

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}L_{k,\sigma (t)}\nonumber \\= & {} \Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}} \Bigg ({\bar{\theta }}\sum ^m_{j=1}|\zeta _j|^{r_k+\omega }\nonumber \\&+{\bar{\theta }}\sum _{j=1}^{k-1}\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg | \sum ^m_{l=1}|\zeta _l|^{r_j+\omega } \Bigg ). \end{aligned}$$

(33)

Using Lemma 1, we have

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}{\bar{\theta }}\sum ^m_{j=1}|\zeta _j|^{r_k+\omega }\nonumber \\&\le \frac{\kappa _1}{2(n+1)} \sum ^m_{j=1}|\zeta _j|^{2r_n}+\Theta _k\phi _{k1}z_k^{\frac{2r_n}{r_k}}, \end{aligned}$$

(34)

where $\phi _{k1}=m2^{\frac{r_{k+1}}{2r_n-r_{k+1}}}\frac{2r_n-r_{k+1}}{r_n} \Bigg (\frac{r_{k+1}(n+1)}{r_n\kappa _1}\Bigg )^{\frac{r_{k+1}}{2r_n-r_{k+1}}}$. Similarly, we have

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}} {\bar{\theta }}\sum _{j=1}^{k-1}\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg | \sum ^m_{l=1}|\zeta _l|^{r_j+\omega }\nonumber \\&\le \frac{\kappa _1}{2(n+1)}\sum ^m_{l=1}|\zeta _l|^{2r_n} +\Theta _k\phi _{k2}z_k^{\frac{2r_n}{r_k}} +\frac{C_{k1}}{\epsilon _k}, \end{aligned}$$

(35)

where $\phi _{k2}$ is a smooth function, $C_{k1}$ and $\epsilon _k$ are positive constants. Then, using (34) and (35), we obtain

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}L_{k,\sigma (t)}\nonumber \\&\le \Theta _k\varphi _{k2}(\epsilon _k,{\bar{\eta }}_{j-2},{\bar{x}}_k,\bar{\hat{\Theta }}_{k-1})z_k^{\frac{2r_n}{r_k}}\nonumber \\&+\frac{\kappa _1}{n+1}\sum _{j=1}^{m}\zeta ^{2r_n}_j+\frac{C_{k1}}{\epsilon _k}, \end{aligned}$$

(36)

where $\varphi _{k2}=\phi _{k1}+\phi _{k2}$. This completes the proof. $\square$

Proof of (22)

By (19), we have

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}\nonumber \\&+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}\bar{L}_k\nonumber \\&\le \frac{1}{\alpha _k} |z_k|^{\frac{2r_n-r_{k+1}}{r_k}} \Bigg ({\bar{H}}_{k,\sigma (t)}(x) + \Bigg (\frac{1}{\alpha _k\delta _{k-1}}\nonumber \\&+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}\Bigg (\sum _{j=1}^{k-1}\Bigg (\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg | {\bar{H}}_{j,\sigma }(x)\nonumber \\&+\Bigg |\frac{\partial x_{k}^{*}}{\partial \hat{\Theta }_j} \dot{\hat{\Theta }}_j\Bigg | +\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg ||D_{j,\sigma }x_{j+1}|\Bigg )\Bigg ). \end{aligned}$$

(37)

Defining ${{\tilde{H}}}_j(x)=\max _{1\le l\le S}\{{\bar{H}}_{j, l}(x)\}, 1\le j\le k$, by Lemma 2.2 in [19], in a compact set, we can find a fuzzy system $\Phi _j^{\top }S_j$ such that ${{\tilde{H}}}_j(x)=\Phi _j^{\top }S_j+\varsigma _j$, where $|\varsigma _j|\le \iota _j$ is the error. It follows that

$$\begin{aligned} \frac{1}{\alpha _k} |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}{\bar{H}}_{k,\sigma (t)}(x)\le & {} \frac{1}{\alpha _k} |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}|\Phi _k^\top S_k+\varsigma _k| \nonumber \\\le & {} \frac{1}{\epsilon _k}+\frac{1}{\epsilon _k}\iota _k^{\frac{2r_n}{r_{k+1}}} +\Theta _k{{\bar{\varphi }}}_{k1}z_k^{\frac{2r_n}{r_k}}, \end{aligned}$$

(38)

where ${{\bar{\varphi }}}_{k1}=\frac{2r_n-r_{k+1}}{r_n} \big (\frac{r_{k+1}\epsilon _k}{r_n}\big )^{\frac{r_{k+1}}{2r_n-r_{k+1}}}$. Similarly, we have

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}\sum _{j=1}^{k-1}\Bigg |\frac{\partial x_k^*}{\partial x_j} \Bigg |{\bar{H}}_{j,\sigma }(x) \nonumber \\&\le \frac{1}{\epsilon _k}+\frac{1}{\epsilon _k}\sum ^{k-1}_{j=1}\iota _j^{\frac{2r_n}{r_{k+1}}} +\Theta _k{{\bar{\varphi }}}_{k2}z_k^{\frac{2r_n}{r_k}}, \end{aligned}$$

(39)

where ${{\bar{\varphi }}}_{k2}$ denotes a smooth function. Using Lemma 1, it follows that

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}\sum \limits _{j=1}^{k-1}\Bigg |\frac{\partial x_{k}^{*}}{\partial \hat{\Theta }_j}\dot{\hat{\Theta }}_j\Bigg | \nonumber \\&\le \frac{k-1}{\epsilon _k} +\Theta _k{{\bar{\varphi }}}_{k3}z_k^{\frac{2r_n}{r_k}}, \end{aligned}$$

(40)

where ${{\bar{\varphi }}}_{k3}$ denotes a smooth function. Similarly, we obtain

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}\sum _{j=1}^{k-1}\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg | |D_{j,\sigma }x_{j+1}| \nonumber \\&\le \frac{c}{2}\sum _{j=1}^{k-1}z_j^{\frac{2r_n}{r_j}}+\frac{2}{\epsilon _k} +\Theta _k{{\bar{\varphi }}}_{k4} z_k^{\frac{2r_n}{r_k}}, \end{aligned}$$

(41)

where ${{\bar{\varphi }}}_{k4}$ is a smooth function. By (37)–(41), and let $\varphi _{k3}=\sum ^{k}_{j=1}{{\bar{\varphi }}}_{kj}$, we show that (22) holds. $\square$

Proof of (23)

For $1\le j\le k$, defining $\bar{H}_j({\bar{x}}_j(t-d_k))=\max _{1\le l\le S}\{H_{j,l}({\bar{x}}_k(t-d_k))\}$, we get

$$\begin{aligned}&\Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}}{\bar{\theta }}{{\tilde{L}}}_k \nonumber \\ {}\le & {} \Bigg (\frac{1}{\alpha _k\delta _{k-1}}+\frac{1}{\alpha _k\delta _k}\Bigg ) |z_k|^{\frac{2r_n-r_{k+1}}{r_k}} {\bar{\theta }}\Bigg ({\bar{H}}_{k} \sum \limits _{j=1}^{k}|x_j(t-d_j)|^{\frac{r_k+\omega }{r_j}} \nonumber \\&+\sum _{j=1}^{k-1}\Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg |{\bar{H}}_{j} \sum \limits _{l=1}^{j}|x_l(t-d_l)|^{\frac{r_j+\omega }{r_l}}\Bigg ). \end{aligned}$$

(42)

$\square$

By using (2), we obtain

$$\begin{aligned}&{\bar{H}}_{k} \sum \limits _{j=1}^{k}|x_j(t-d_j)|^{\frac{r_k+\omega }{r_j}} \nonumber \\&\le \sum \limits _{j=1}^{k}{{\tilde{H}}}_{j}({\bar{z}}_j(t-d_j))|z_j(t-d_l)|^{\frac{r_j+\omega }{r_j}}\nonumber \\&+\sum \limits _{j=1}^{k-1}{{\tilde{H}}}_{j}({\bar{z}}_j(t-d_{j+1}))|z_j(t-d_{j+1})|^{\frac{r_j+\omega }{r_j}}+{\bar{C}}_{k1}. \end{aligned}$$

(43)

where ${{\tilde{H}}}_{j}$ is a smooth function, ${\bar{C}}_{k1}>0$ is a constant. Similarly, there holds

$$\begin{aligned}&|{\bar{H}}_{j}({\bar{x}}_j(t-d_j)) \sum \limits _{l=1}^{j}|x_l(t-d_l)|^{\frac{r_j+\omega }{r_j}} \nonumber \\&\le \sum \limits _{l=1}^{j}{{\hat{H}}}_{l}({\bar{z}}_l(t-d_j))|z_l(t-d_l)|^{\frac{r_j+\omega }{r_l}}\nonumber \\&+\sum \limits _{l=1}^{j-1}{{\hat{H}}}_{l}({\bar{z}}_l(t-d_{l+1}))|z_l(t-d_{l+1})|^{\frac{r_j+\omega }{r_l}}+{\bar{C}}_{j1}, \end{aligned}$$

(44)

where ${\bar{C}}_{j1}>0$ is a constant, ${{\hat{H}}}_{l}$ denotes a smooth function. Also, we obtain

$$\begin{aligned} \Bigg |\frac{\partial x_k^*}{\partial x_j}\Bigg |\le \Bigg |g_{k-1}\cdots g_j\frac{r_k}{r_j}\Bigg |\Bigg (|z_{k-1}|^{\frac{(k-j)\omega }{r_{k-1}}} +\cdots +|z_j|^{\frac{(k-j)\omega }{r_j}}\Bigg ). \end{aligned}$$

(45)

By (42)–(45) and Lemma 1, we show that (23) holds.