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Adding an integrator for the stabilization problem. (English) Zbl 0747.93072
Summary: We study the relationship between the following two properties: P1: The system \(\dot x=f(x,y)\), \(\dot y=v\) is locally asymptotically stabilizable; and P2: The system \(\dot x=f(x,u)\) is locally asymptotically stabilizable; where \(x\in\mathbb{R}^ n\), \(y\in\mathbb{R}\). Dayawansa, Martin and Knowles have proved that these properties are equivalent if the dimension \(n=1\). Here, using the so-called Control Lyapunov function approach, (a) we propose another more constructive and somewhat simpler proof of Dayawansa, Martin and Knowles’s result; (b) we show that, in general, P1 does not imply P2 for dimensions \(n\) larger than 1; (c) we prove that P2 implies P1 if some extra assumptions are added like homogeneity of the system. By using the latter result recursively, we obtain a sufficient condition for the local asymptotic stabilizability of systems in a triangular form.

93D20 Asymptotic stability in control theory
93C15 Control/observation systems governed by ordinary differential equations
93D30 Lyapunov and storage functions
Full Text: DOI
[1] Andreini, A; Bacciotti, A; Stefani, G, Global stabilizability of homogeneous vector fields of odd degree, Systems control lett., 10, 251-256, (1988) · Zbl 0653.93040
[2] Artstein, Z, Stabilization with relaxed controls, Nonlinear anal. TMA, 7, 1163-1173, (1983) · Zbl 0525.93053
[3] Boothby, W.M; Marino, R, Feedback stabilization of planar nonlinear systems II, (), 1970-1974, FA 4-8:15
[4] Calderon, A.P; Zygmund, A, Local properties of solutions of elliptic partial differential equations, Studia math., 20, 171-225, (1961) · Zbl 0099.30103
[5] Dayawansa, W.P; Martin, C.F, Asymptotic stabilization of two dimensional real analytic systems, Systems control lett., 12, 205-211, (1989) · Zbl 0673.93064
[6] Dayawansa, W.P; Martin, C.F, Two examples of stabilizable second order systems, () · Zbl 0703.93056
[7] Dayawansa, W.P; Martin, C.F, Some sufficient conditions for the asymptotic stabilizability of three dimensional homogeneous polynomial systems, () · Zbl 0738.93065
[8] Dayawansa, W.P; Martin, C.F; Knowles, G, Asymptotic stabilization of a class of smooth two-dimensional systems, SIAM J. control optim., 28, 1321-1349, (Nov. 1990)
[9] Hahn, W, Stability of motion, (1967), Springer-Verlag Berlin · Zbl 0189.38503
[10] Hermes, H, Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, () · Zbl 0711.93069
[11] Krasnoselskii, M.A; Zabreiko, P.P, Geometric methods in nonlinear analysis, (1983), Springer-Verlag Berlin
[12] Kawski, M, Stabilization of nonlinear systems in the plane, Systems control lett., 12, 169-175, (1989) · Zbl 0666.93103
[13] M. Kawski, Homogeneous stabilizing feedback laws, Internal Report, Department of Mathematics, Arizona State University, Tempe. Submitted for publication. · Zbl 0736.93020
[14] Kurzweil, J, On the inversion of Lyapunov’s second theorem on stability of motion, Ann. math. soc. transl. ser. 2, 24, 19-77, (1956)
[15] Praly, L; d’Andréa-Novel, B; Coron, J.M; Praly, L; d’Andréa-Novel, B; Coron, J.M, Lyapunov design of stabilizing controllers for cascaded systems, (), IEEE trans. automat. control, (1991), Also, to appear in · Zbl 0746.93064
[16] Sontag, E.D, A ‘universal’ construction of Artstein’s theorem on nonlinear stabilization, Systems control lett., 13, 117-123, (1989) · Zbl 0684.93063
[17] Sontag, E.D, Feedback stabilization of nonlinear systems, () · Zbl 0758.93013
[18] Sontag, E.D; Sussmann, H.J, Remarks on continuous feedback, Ieee cdc, Vol. 2, 916-921, (1980), Albuquerque, NM
[19] Tsinias, J, Sufficient Lyapunov-like conditions for stabilization, Math. control. signals systems, 2, 343-357, (1989) · Zbl 0688.93048
[20] Whitney, H; Bruhat, F, Quelques propriétés fondamentales des ensembles analytiques réels, Comm. math. helv., 33, 132-160, (1959) · Zbl 0100.08101
[21] Zubov, V.I, Methods of A.M. Lyapunov and their application, (1964), Noordhoff Leiden · Zbl 0115.30204
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