Semiglobal stabilisation for time-delay nonlinear systems using high-gain observers with an upper bound and without a lower bound on ϵ

ABSTRACT

This paper addresses high-gain observer design for nonlinear systems with time delays. In the previous work of Lei and Khalil, a high-gain observer is designed for a time-delay nonlinear system taking advantage of time delay so that a constraint on lower bound occurs to high-gain-observer parameter ϵ (besides there is an upper bound on ϵ). Unlike the previous work, in this paper the time delays are totally ignored in high-gain-observer construction. It is demonstrated that the output feedback using this high-gain observer will still semiglobally stabilise the system. In return, the ϵ is required to be sufficiently small without restriction on its lower bound (except zero and besides there is also an upper bound on ϵ). Meanwhile, the nominal and the uncertain time delays are required sufficiently small. To pursue the essence of the distinction between these two high-gain observer design schemes, an insight is given into the theoretical analysis. Moreover, simulations are conducted in order to distinguish the specifics of features of these two schemes, which relates to the theoretical reasons.

KEYWORDS:

• Nonlinear systems
• high-gain observers
• time delays
• singular perturbation method

Acknowledgment

The author appreciates Professor Hassan K. Khalil for the discussions and Professor P. Pepe for the comments on this paper.

Disclosure statement

No potential conflict of interest was reported by the author.

Notes

¹ Inequality (A18(A18) $$\begin{array}{ccc}& & \parallel \Phi \left(x_t,x_{t-{\tau }^0-\delta },z_t,D\left(\epsilon \right){\eta }_t,D\left(\epsilon \right){\eta }_{t-{\tau }^0-\delta }\right)\parallel \hfill \\ & & \le l_4{∥{\eta }_t∥}_s+l_5,\hfill \end{array}$$(A18) ) associated with (A8(A8) $$\begin{array}{ccc}& & \Phi \left(x\left(t\right),x(t-{\tau }^0-\delta ),z\left(t\right),D\left(\epsilon \right)\eta \left(t\right),\right.\hfill \\ & & \left.D\left(\epsilon \right)\eta (t-{\tau }^0-\delta )\right)\hfill \\ & & =\phi \left(x\left(t\right),z\left(t\right),\gamma \left(\widehat{x}(t-{\tau }_1^0-{\delta }_1)\right)\right)\hfill \\ & & -{\phi }_0\left(\widehat{x}(t+{\tau }_2^0+{\delta }_2),\gamma \left(\widehat{x}(t+{\tau }_2^0+{\delta }_2)\right)\right).\hfill \end{array}$$(A8) ) needs the boundedness of φ₀ in $\widehat{x}$.

Additional information

Funding

This work was supported in part by the National Science Foundation of China [grant number 61364012], [grant number 11361076].