Semi-global stabilization of nonlinear systems by nonsmooth output feedback
Bo Yang
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA
Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, China
Search for more papers by this authorCorresponding Author
Wei Lin
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA
Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, China
Correspondence to: Wei Lin, Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA.
E-mail: [email protected]
Search for more papers by this authorBo Yang
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA
Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, China
Search for more papers by this authorCorresponding Author
Wei Lin
Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA
Harbin Institute of Technology, Shenzhen Graduate School, Shenzhen, China
Correspondence to: Wei Lin, Department of Electrical Engineering and Computer Science, Case Western Reserve University, Cleveland, OH 44106, USA.
E-mail: [email protected]
Search for more papers by this authorSUMMARY
Semi-global stabilization by output feedback is studied for a class of nonuniformly observable and nonsmoothly stabilizable nonlinear systems. The contribution of this paper is to point out that most of the restrictive growth conditions required in the previous work can be relaxed or removed if a less demanding control objective, namely, semi-global instead of global stabilization is sought. In particular, it is proved that without imposing restrictive conditions, semi-global stabilization by nonsmooth output feedback can be achieved for a chain of odd power integrators perturbed by a smooth triangular vector field, although it is neither smoothly stabilizable nor uniformly observable. Extensions to nonstrictly triangular systems are also discussed in the two-dimensional case. Several examples are provided to illustrate the key features of the proposed semi-global output feedback controllers. Copyright © 2013 John Wiley & Sons, Ltd.
REFERENCES
- 1 Celikovsky S, Aranda-Bricaire E. Constructive non-smooth stabilization of triangular systems. Systems & Control Letters 1999; 36: 21–37.
- 2 Coron JM, Praly L. Adding an integrator for the stabilization problem. Systems & Control Letters 1991; 17: 89–104.
- 3 Qian C, Lin W. A continuous feedback approach to global strong stabilization of nonlinear systems. IEEE Transactions on Automatic Control 2001; 46: 1061–1079.
- 4 Tzamtzi M, Tsinias J. Explicit formulas of feedback stabilizers for a class of triangular systems with uncontrollable linearization. Systems & Control Letters 1999; 38: 115–126.
- 5 Mazenc F, Praly L, Dayawansa WD. Global stabilization by output feedback: examples and counterexamples. Systems and Control Letters 1994; 23: 119–125.
- 6 Teel A, Praly L. Global stabilizability and observability imply semi-global stabilizality by output feedback. Systems & Control Letters 1994; 22: 313–325.
- 7 Gauthier JP, Hammouri H, Othman S. A simple observer for nonlinear systems, applications to bioreactors. IEEE Transactions on Automatic Control 1992; 37: 875–880.
- 8 Teel A, Praly L. Tools for semiglobal stabilization by partial state and output feedback. SIAM Journal on Control and Optimization 1995; 33(1): 1443–1488.
- 9 Isidori A. A tool for semiglobal stabilization of uncertain non-minimum-phase nonlinear systems via output feedback. IEEE Transactions on Automatic Control 2000; 45: 1817–1827.
- 10
Isidori A. Nonlinear Control Systems II. Springer-Verlag: New York, 1999.
10.1007/978-1-4471-0549-7 Google Scholar
- 11 Kawski M. Stabilization of nonlinear systems in the plane. Systems & Control Letters 1989; 12: 169–175.
- 12 Rui C, Reyhanoglu M, Kolmanovsky I, Cho S, McClamroch NH. Non-smooth stabilization of an underactuated unstable two degrees of freedom mechanical system. Proceedings of the 36th IEEE CDC, San Diego, 1997; 3998–4003.
- 13 Qian C, Lin W. Recursive observer design, homogeneous approximation, and nonsmooth output feedback stabilization of nonlinear systems. IEEE Transactions on Automatic Control 2006; 51: 1457–1471.
- 14 Bacciotti A. Local Stabilizability of Nonlinear Control Systems. World Scientific: Singapore, 1992.
- 15 Dayawansa WP. Recent advances in the stabilization problem for low dimensional systems. Proceedings of the 2nd IFAC NOLCOS, Bordeaux, 1992; 1–8.
- 16 Dayawansa WP, Martin CF, Knowles G. Asymptotic stabilization of a class of smooth two dimensional systems. SIAM Journal on Control and Optimization 1990; 28: 1321–1349.
- 17 Dayawansa W. Personal Communication, 2002.
- 18 Kawski M. Homogeneous stabilizing feedback laws. Control Theory and Advanced Technology 1990; 6: 497–516.
- 19 Kawski M. Geometric homogeneity and applications to stabilization. Proceedings of the 3rd IFAC NOLCOS, Lake Tahoe, CA, 1995; 164–169.
- 20 Hermes H. Homogeneous coordinates and continuous asymptotically stabilizing feedback controls. In Differential Equations Stability and Control, Vol. 109, S Elaydi (ed.), Lecture Notes in Applied Mathematics. Marcel Dekker: New York, 1991; 249–260.
- 21 Rosier L. Homogeneous Lyapunov functions for homogeneous continuous vector field. Systems & Control Letters 1992; 19: 467–473.
- 22 Lin W, Lei H. Taking advantage of homogeneity: a unified framework for output feedback control of nonlinear systems. Proceedings of the 7th IFAC NOLCOS, Pretoria, South Africa, 2007; 27–38.
- 23 Yang B, Lin W. Homogeneous observers, iterative design and global stabilization of high order nonlinear systems by smooth output feedback. IEEE Transactions on Automatic Control 2004; 49(7): 1069–1080.
- 24 Yang B, Lin W. Robust output feedback stabilization of uncertain nonlinear systems with uncontrollable and unobservable linearization. IEEE Transactions on Automatic Control 2005; 50(5): 619–630.
- 25 Khalil H, Esfandiari F. Semi-global stabilization of a class of nonlinear systems using output feedback. IEEE Transactions on Automatic Control 1995; 38: 1412–1415.
- 26 Yang B, Lin W. On semi-global stabilizability of MIMO nonlinear systems by output feedback. Automatica 2006; 42(6): 1049–1054.
- 27 Cheng D, Lin W. On p-normal forms of nonlinear systems. IEEE Transactions on Automatic Control 2003; 48: 1242–1248.
- 28
Jakubczyk B,
Respondek W. Feedback equivalence of planar systems and stability. In Robust Control of Linear Systems and Nonlinear Control, MA Kaashoek, JH Schuppen, ACM Ran, et al. (eds). Birkhäuser: Basel, 1990; 447–456.
10.1007/978-1-4612-4484-4_43 Google Scholar
- 29 Khalil HK. High-gain observers in nonlinear feedback control. In New Directions in Nonlinear Observer Design, H Nijmeijer, TI Fossen (eds). Springer-Verlag: Berlin, 1999.
- 30 Krener AJ, Kang W. Backstepping design of nonlinear observers. SIAM Journal on Control and Optimization 2003; 42(1): 155–177.
- 31 Lin W, Pongvuthithum R. Global stabilization of cascade systems by C0 partial state feedback. IEEE Transactions on Automatic Control 2002; 47: 1356–1362.
