Stabilization of nonlinear systems with homogeneous stability (Q2747402)

The authors use the center manifold technique to obtain via linear state feedback \(k\)-th order homogeneous stabilization of nonlinear control systems NEWLINE\[NEWLINE \dot z=Az+ \xi(z,\eta)+g(z,\eta)u,\quad \dot\eta=p(z,\eta),NEWLINE\]NEWLINE where \(g(z,\eta)=B+\delta(z,\eta)\), \(\delta(z,\eta)=O(\|z,\eta\|)\) and \((A,B)\) is completely controllable. An illustrative example is given to show the authors' motivation. The application to the affine nonlinear system NEWLINE\[NEWLINE \dot x=f(x)+\sum_{j=1}^mg_j(x)u_j,\quad y_i=h_i(x),\quad x\in\mathbb R^n,\quad i=1,\dots,m, NEWLINE\]NEWLINE (without the usual relative degree because of singularity) is given.