zbMATH — the first resource for mathematics

Homogeneous feedback controls for homogeneous systems. (English) Zbl 0877.93088
Summary: Let \(X(x)=\sum ^{n}_{j=2}a_{j}(x)\partial /\partial x_{j}\) be a real analytic, n-dimensional vector field with \(X(0)=0\) which is homogeneous of degree \(m\leq 0\) with respect to a dilation \(\delta ^{r}_{\varepsilon}x= (\varepsilon ^{r_{1}}x_{1},\ldots,\varepsilon ^{r_{n}}x_{n})\) and \(Y=\partial /\partial x_{1}\). We show that if the control system \(\dot x=X(x)+uY\) admits a continuous, asymptotically stabilizing, feedback control \(u(x)\), then a one-dimensional dynamic extension of the system admits a homogeneous, asymptotically stabilizing (dynamic) feedback control \(\hat u(\tau,x)\) such that the vector field \[ \hat W(\tau,x)=\alpha \tau^{1-m}\partial /\partial\tau+ \hat u(\tau,x)\partial /\partial x_{1}+\sum ^{n}_{j=2}a_{j}(x)\partial /\partial x_{j} \] describing the extended, controlled, system is homogeneous of degree m with respect to the extended dilation \[ \hat\delta^{r}_{\varepsilon}(\tau,x)= (\varepsilon\tau,\varepsilon^{r_{1}}x_{1},\ldots, \varepsilon^{r_{n}}x_{n}). \]

93D15 Stabilization of systems by feedback
Full Text: DOI
[1] Brockett, R.W, Asymptotic stability and feedback stabilization, (), 181-191 · Zbl 0528.93051
[2] Coron, J.M, A necessary condition for feedback stabilization, Systems control lett., 14, 227-232, (1990) · Zbl 0699.93075
[3] W.P. Dayawansa, C.F. Martin and S. Samelson, Asymptotic stabilization of a generic class of three-dimensional homogeneous quadratic systems, preprint. · Zbl 0877.93090
[4] Hermes, H, Asymptotically stabilizing feedback controls, J. differential equations, 92, 76-89, (1991) · Zbl 0736.93069
[5] Hermes, H, Asymptotically stabilizing feedback controls and the nonlinear regulator problem, SIAM J. control optim., 29, 185-196, (1991) · Zbl 0738.93061
[6] Hermes, H, Nilpotent and high order approximations of vector field systems, SIAM rev., 33, 238-264, (1991) · Zbl 0733.93062
[7] Hermes, H, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, (), 249-260
[8] Kawski, M, Stabilization and nilpotent approximations, (), 1244-1248
[9] Kawski, M, Homogeneous stabilizing feedback laws, (), 497-516
[10] Kurzweil, J; Kurzweil, J; Kurzweil, J, On the reversibility of the theorem of Lyapunov concerning the stability of motion, Czechoslovak math. J., Czechoslovak math. J., Czechoslovak math. J., 6, 474-484, (1956), (English summary) · Zbl 0127.30702
[11] Rosier, L, Homogeneous Lyapunov functions for homogeneous vector fields, Systems control lett., 19, 467-473, (1992) · Zbl 0762.34032
[12] Wilson, F.W, The structure of the level surfaces of a Lyapunov function, J. differential equations, 4, 323-329, (1967) · Zbl 0152.28701
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.