Global adaptive tracking for a class of nonlinear time-delay systems

Abstract

This paper aims to extend the newly developed global adaptive regulation scheme to adaptive tracking for a general class of nonlinear systems with both parameter uncertainties and unknown time delay for all the system states. With a simple yet novel decomposition of some coupling functions derived from the introduction of a reference signal, a backstepping tracking control strategy is proposed which possesses the feature that for each design step, a dynamic gain is introduced to generate an additional term to counteract the system nonlinearities and parameter uncertainties. As a result, a memoryless adaptive state-feedback controller is obtained to achieve the global adaptive tracking.

Keywords:

• Time-delay systems
• dynamic gain
• global tracking
• adaptive control

Disclosure statement

No potential conflict of interest was reported by the authors.

ORCID

Wanli Wang  http://orcid.org/0000-0003-1155-6736

Notes

1 Please see the Appendix for more details of the deduction of (13(13) $\begin{array}{rl}{\dot{V}}_{1LK}& \le -h{V}_{1K}-({\beta }_1-R_1)z_1^2+z_2^2-\frac{{\rho }_1}{2l_1}{z}_1^2+Y_1\\ & +\frac{1}{\delta }\tilde{\vartheta }\left({\tau }_1-\dot{\hat{\vartheta }}\right)+\sigma \tilde{\vartheta }\hat{\vartheta },\end{array}$(13) ) here and (17(17) $\begin{array}{rl}{\dot{V}}_{2LK}& \le -h{V}_{1K}-h{V}_{2K}-({\beta }_1-R_1-2)z_1^2-({\beta }_2-R_2)z_2^2\\ & +z_3^2-\frac{{\rho }_1}{2l_1}{z}_1^2-\frac{{\rho }_2}{2l_2}{z}_2^2+\frac{{\mathrm{e}}^{hd}}{l_1}{\psi }_{21}{z}_1^2+Y_2\\ & +[\frac{1}{\delta }\tilde{\vartheta }+\frac{1}{l_1}\left(1+\frac{1}{l_2}\right)z_2\frac{\mathrm{\partial }{\alpha }_1}{\mathrm{\partial }\hat{\vartheta }}]\left({\tau }_2-\dot{\hat{\vartheta }}\right)+\sigma \tilde{\vartheta }\hat{\vartheta },\end{array}$(17) ), (21(21) $\begin{array}{rl}{\dot{V}}_{iLK}& \le -h\sum _{j=1}^i{V}_{jK}-\sum _{j=1}^i\left({\beta }_j-R_j-2i+2j\right)z_j^2\\ & +z_{i+1}^2-\sum _{j=1}^i\frac{{\rho }_j}{2l_j}{z}_j^2+\sum _{j=1}^{i-1}\sum _{p=j+1}^i\frac{{\mathrm{e}}^{hd}}{l_j}{\psi }_{pj}{z}_j^2+Y_i\\ & +[\frac{1}{\delta }\tilde{\vartheta }+\sum _{j=1}^{i-1}\frac{1}{l_1\cdots l_j}\left(1+\frac{1}{l_{j+1}}\right)z_{j+1}\frac{\mathrm{\partial }{\alpha }_j}{\mathrm{\partial }\hat{\vartheta }}]\left({\tau }_i-\dot{\hat{\vartheta }}\right)\\ & +\sigma \tilde{\vartheta }\hat{\vartheta },\end{array}$(21) ), (25(25) $\begin{array}{rl}\dot{V}& \le -h\sum _{j=1}^n{V}_{jK}-\sum _{j=1}^n({\beta }_j-R_j-2n+2j)z_j^2\\ & -\sum _{j=1}^{n-1}\frac{{\rho }_j}{2l_j}{z}_j^2+\hat{\vartheta }\sum _{j=1}^{n-1}\sum _{p=j+1}^n\frac{1}{l_j}{\psi }_{pj}{z}_j^2+Y_n\\ & +[\frac{1}{\delta }\tilde{\vartheta }+\sum _{j=1}^{n-2}\frac{1}{l_1\cdots l_j}\left(1+\frac{1}{l_{j+1}}\right)z_{j+1}\frac{\mathrm{\partial }{\alpha }_j}{\mathrm{\partial }\hat{\vartheta }}\\ & +\frac{1}{l_1\cdots l_{n-1}}{z}_n\frac{\mathrm{\partial }{\alpha }_{n-1}}{\mathrm{\partial }\hat{\vartheta }}]\\ & \times \left({\tau }_n-\dot{\hat{\vartheta }}\right)+\sigma \tilde{\vartheta }\hat{\vartheta },\end{array}$(25) ) in the subsequent design steps.

Additional information

Funding

This work was supported in part by National Natural Science Foundation of China under Grants 61673038, 61433011 and the National Basic Research Program of China (973 Program) under Grant 2014CB046406.