Predictor Control for Non Forward Complete Nonlinear System With Time-Varying Input Delay

Abstract

We consider <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\dot {\textrm {X}}(\textrm {t})=\textrm {X(t)}^{2}+\textrm {U}(\textrm {t}-\textrm {D}(\textrm {t}))$ </tex-math></inline-formula> , where D(t) is a long time-varying delay. If D(t) = 0, <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textrm {U(t)}=-\textrm {X(t)}^{2}-\textrm {cX(t)},\,\,\textrm {c}&gt;0$ </tex-math></inline-formula> is a simply control, but it just delays finite time escape for this system. We design a predictor control and prove that the attraction region is <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\textrm {X}(0) + \sup _{\theta \in [\varphi (0),\,0]} \int _{\varphi (0)}^{\theta } {\frac {\textrm {U}(\theta)}{\varphi ^{\prime }(\varphi ^{-1}(\theta))}\textrm {d}\theta } &lt; \frac {1}{\sigma (0)}$ </tex-math></inline-formula> , with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\varphi (\theta)=\theta -\textrm {D}(\theta)$ </tex-math></inline-formula> , and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\sigma (\theta)=\varphi ^{-1}(\theta)$ </tex-math></inline-formula> . Further, the predictor control locally exponentially stabilizes this system.