A combined NN and dynamic gain-based approach to further stabilize nonlinear time-delay systems

Abstract

This paper focuses on the state-feedback control problem for a class of high-order nonlinear systems with unknown time delay and control coefficients. Based on a novel dynamic gain-based backstepping technique and radial basis function neural network (RBF NN) approximation approach, the restrictions on high-order and nonlinearities are removed or further relaxed. Under these weaker conditions, a smooth state-feedback controller is skillfully constructed with only one adaptive parameter. In addition, the knowledge of time delay, NN nodes and weights is not necessary to be known a priori. It is proven that the designed controller can render the closed-loop system be semi-globally uniformly ultimately bounded. Finally, both practical and numerical examples are shown to demonstrate the effectiveness of the proposed scheme.

Appendix

Proof of Proposition 1

We prove it by induction. Assume that at step ($i-1$), there exist a series of virtual control laws

$$\begin{aligned} z_1 & = {} \,x_1,\nonumber \\ z_2 & = {} \,x_2-\alpha _2,\ \alpha _2=-z_1\beta _1^{\frac{1}{p_1}}(z_1,\hat{\theta }),\nonumber \\&\vdots&\nonumber \\ z_i & = {}\, x_i-\alpha _i,\ \alpha _i=-z_{i-1}(L_1\cdots L_{i-2})^{\frac{1}{p_{i-1}}} \beta _{i-1}^{\frac{1}{p_{i-1}}}(\bar{z}_{i-1},\bar{L}_{i-3},\hat{\theta }), \end{aligned}$$

(51)

such that $U_{i-1}$ satisfies

$$\begin{aligned} \dot{U}_{i-1}& \le {} \,{-}\sum _{j=1}^{i-1}(\beta _j-Y_j)z_j^{p+3} +\Phi _{i,0}z_{i}^{p+3}+\sum _{j=1}^{i-1}|z_j|^{p-p_j+3}\tilde{\theta }+\Delta _{i-1}\nonumber \\&+\sum _{j=1}^{i-2}\sum _{k=j+1}^{i-1}\frac{1}{L_{j}}\varrho _{kj}z_{j}^{p+3} -\sum _{j=1}^{i-1}\frac{\sigma _j}{(p-p_j+4)L_j}z_j^{p+3}. \end{aligned}$$

(52)

Now, we prove (52) still holds for (30). Firstly, from (4) and (7), one has

$$\begin{aligned} \dot{z}_i=x_{i+1}^{p_i}+\bar{f}_i+\bar{g}_{id}, \end{aligned}$$

(53)

where $\bar{f}_i=f_i-\sum _{j=1}^{i-1}\frac{\partial \alpha _i}{\partial x_j} (x_{j+1}^{p_j}+f_j)\,-\sum _{j=1}^{i-2}\frac{\partial \alpha _i}{\partial L_j}\dot{L}_j$ and $\bar{g}_{id}=g_{id}-\sum _{j=1}^{i-1}\frac{\partial \alpha _i}{\partial x_j}g_{jd}$. Choosing Lyapunov function (30) and using (31), (52)–(53), one can get

$$\begin{aligned} \dot{V}_i&\le {} \,{-}\sum _{j=1}^{i-1}(\beta _j-Y_j)z_j^{p+3}-(\beta _i-1-\Phi _{i,0})z_{i}^{p+3} +{2|z_i|^{p-p_i+3}|x_{i+1}^{p_i}-\alpha _{i+1}^{p_i}|}\nonumber \\&+\,{2|z_i|^{p-p_i+3}|\bar{f}_i|} +\frac{2}{L_1\cdots L_{i-1}}{|z_i|^{p-p_i+3}|\bar{g}_{id}|} +\,\sum _{j=1}^{i-1}|z_j|^{p-p_j+3}\tilde{\theta }+\Delta _{i-1}\nonumber \\&+\,\sum _{j=1}^{i-2}\sum _{k=j+1}^{i-1}\frac{1}{L_{j}}\varrho _{kj}z_{j}^{p+3} -\sum _{j=1}^{i}\frac{\sigma _j}{(p-p_j+4)L_j}z_j^{p+3}. \end{aligned}$$

(54)

In the sequel, we focus on estimating the third–fifth terms on the right-hand side of (54). From (2), one gets $\bar{f}_i(X_i)=W_i^{*^T}S_i(X_i)+\delta _i(X_i)$, $|\delta _i(X_i)|\le \varepsilon _i$, where $\varepsilon _i>0$ is a design constant, $X_i=(\bar{z}_i,\bar{L}_{i-1},\hat{\theta })^T \in S_{X_i}=\{X_i|X_i\in S_x\}$. Then, using $(a+b)^n=\sum _{i=0}^nC_n^ia^{n-i}b^i$, (5)–(7), Lemmas 1 and 2 and $\Vert W_{i}^{*^T}\Vert ^2\Vert S_{i}\Vert ^2\le \Vert W_{i}^{*^T}\Vert ^2N_{i}\le \theta$, there exist positive constants $\xi _i$, $\bar{\gamma }_i=\sum _{j=0}^{\frac{p_i-1}{2}}\gamma _{ij}$, $\gamma _{ij}$, $\tilde{\gamma }_i$, and smooth functions $\lambda _i(\hat{\theta })$, $\Phi _{i}(\hat{\theta })$, $\Phi _{i+1,0}(\bar{z}_i,\bar{L}_{i-1},\hat{\theta })$, $\varrho _{ij}(\bar{z}_{jd})$, $\varrho _{ii}(\bar{z}_{id},\bar{L}_{i-2,d})$ with $j=1,\ldots ,i-1$ such that

$$\begin{aligned} {2|z_i|^{p-p_i+3}|\bar{f}_i|}&\le {} \,\xi _i+\Phi _i(\hat{\theta })z_i^{p+3} +z_i^{p-p_i+3}\tilde{\theta }, \end{aligned}$$

(55)

$$\begin{aligned} {2|z_i|^{p-p_i+3}|x_{i+1}^{p_i}-\alpha _{i+1}^{p_i}|}&\le {} \, \bar{\gamma }_iz_i^{p+3}+\Phi _{i+1,0}(\bar{z}_i,\bar{L}_{i-1},\hat{\theta })z_{i+1}^{p+3}, \end{aligned}$$

(56)

$$\begin{aligned} \frac{2}{L_1\cdots L_{i-1}}{|z_i|^{p-p_i+3}|\bar{g}_{id}|}& \le {} \, \tilde{\gamma }_iz_i^{p+3}+\frac{1}{L_{1d}\cdots L_{i-1,d}} \sum _{j=1}^{i}\bar{\varrho }_{ij}(\bar{x}_{jd})x_{jd}^{p+3}\nonumber \\& \le {} \, \tilde{\gamma }_iz_i^{p+3}+\sum _{j=1}^{i-1}\frac{1}{L_{jd}} \varrho _{ij}(\bar{z}_{jd},\bar{L}_{j-1,d})z_{jd}^{p+3} +\varrho _{ii}(\bar{z}_{id},\bar{L}_{i-2,d})z_{id}^{p+3}. \end{aligned}$$

(57)

Finally, by choosing $U_i$ as (30) and substituting (55)–(57) into (54), one can yield (32). The proof is completed. $\square$