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Constructive Lyapunov stabilization of nonlinear cascade systems. (English) Zbl 0869.93039

Consider the system \[ \dot x= f(x)+h(x,\xi),\quad f(0)=0,\quad h(x,0)\equiv 0,\quad\dot\xi= A\xi\tag{1} \] with \(A\) a Hurwitz matrix. The authors consider a Lyapunov function of the form \(V_0(x,\xi)= W(x)+ \psi(x,\xi)+\xi^*P\xi\), where the cross term \(\psi\) is constructed in such a way that \(V_0(x,\xi)>0\) is radially unbounded and nonincreasing along the trajectories. This is the main step in the global stabilization of systems with input obtained by various augmentations of (1) such as \[ \dot x= f(x)+h(x,\xi)+ g_1(x,\xi)u,\quad\dot\xi= A\xi+bu,\tag{2} \]
\[ \dot x= f(x)+h(x,\xi)+ g_2(x,\xi,y)y,\quad\dot\xi= A\xi+by,\quad\dot y=u\tag{3} \] and \[ \dot z= \varphi(z)+ \kappa(z,x,\xi)+ \rho(z,x,\xi)u,\quad\dot x= f(x)+h(x,\xi)+ g(x,\xi)u,\quad\dot\xi= A\xi+bu.\tag{4.} \]

MSC:

93D15 Stabilization of systems by feedback
93A99 General systems theory
93D30 Lyapunov and storage functions
93C10 Nonlinear systems in control theory
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