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.
MSC:
| 93D20 | Asymptotic stability in control theory |
| 93C15 | Control/observation systems governed by ordinary differential equations |
| 93D30 | Lyapunov and storage functions |
References:
| [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, (Proc. 28th Conference on Decision and Control. Proc. 28th Conference on Decision and Control, Tampa, FL (1989)), 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, (Conference on Computation and Control (June 1988), Montana State University) · Zbl 0703.93056 |
| [7] | Dayawansa, W. P.; Martin, C. F., Some sufficient conditions for the asymptotic stabilizability of three dimensional homogeneous polynomial systems, (Proc. 28th IEEE Conference on Decision and Control. Proc. 28th IEEE Conference on Decision and Control, Tampa, FL (Dec. 1989)) · 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) · Zbl 0731.93076 |
| [9] | Hahn, W., Stability of Motion (1967), Springer-Verlag: Springer-Verlag Berlin · Zbl 0189.38503 |
| [10] | Hermes, H., Homogeneous coordinates and continuous asymptotically stabilizing feedback controls, (Elaydi, S., Differential Equations: Stability and Control. Differential Equations: Stability and Control, Proceedings of Colorado Spring Conference (1990), Marcel Dekker: Marcel Dekker New York) · Zbl 0711.93069 |
| [11] | Krasnoselskii, M. A.; Zabreiko, P. P., Geometric Methods in Nonlinear Analysis (1983), Springer-Verlag: 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.; 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) · Zbl 0127.30703 |
| [15] | Praly, L.; d’Andréa-Novel, B.; Coron, J. M., 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, (Proc. of the MTNS-89 (1990), Birkhaüser: Birkhaüser Basel-Boston) · 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: Noordhoff Leiden · Zbl 0115.30204 |
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