1. Y. Lin, E.D. Sontag, and Y. Wang. A smooth converse Lyapunov theorem for robust stability. SIAM J. Control Optim., 34(1):124-160, 1996. [PDF] [doi:http://dx.doi.org/10.1137/S0363012993259981] Keyword(s): input to state stability. Abstract:

   This paper presents a Converse Lyapunov Function Theorem motivated by robust control analysis and design. Our result is based upon, but generalizes, various aspects of well-known classical theorems. In a unified and natural manner, it (1) allows arbitrary bounded time-varying parameters in the system description, (2) deals with global asymptotic stability, (3) results in smooth (infinitely differentiable) Lyapunov functions, and (4) applies to stability with respect to not necessarily compact invariant sets.




2. Y. Lin and E.D. Sontag. Control-Lyapunov universal formulas for restricted inputs. Control Theory Adv. Tech., 10(4, part 5):1981-2004, 1995. [PDF] Keyword(s): control-Lyapunov functions, saturation, bounded inputs. Abstract:

   We deal with the question of obtaining explicit feedback control laws that stabilize a nonlinear system, under the assumption that a "control Lyapunov function" is known. In previous work, the case of unbounded controls was considered. Here we obtain results for bounded and/or positive controls. We also provide some simple preliminary remarks regarding a set stability version of the problem and a version for systems subject to disturbances.




3. Y. Lin, E.D. Sontag, and Y. Wang. Input to state stabilizability for parametrized families of systems. Internat. J. Robust Nonlinear Control, 5(3):187-205, 1995. [PDF] Keyword(s): ISS, stabilization. Abstract:

   This paper studies various stability issues for parameterized families of systems, including problems of stabilization with respect to sets. The study of such families is motivated by robust control applications. A Lyapunov-theoretic necessary and sufficient characterization is obtained for a natural notion of robust uniform set stability; this characterization allows replacing ad hoc conditions found in the literature by more conceptual stability notions. We then use these techniques to establish a result linking state space stability to ``input to state'' (bounded-input bounded-state) stability. In addition, the preservation of stabilizability under certain types of cascade interconnections is analyzed.




4. Y. Lin and E.D. Sontag. A universal formula for stabilization with bounded controls. Systems Control Lett., 16(6):393-397, 1991. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(91)90111-Q] Keyword(s): stabilization, nonlinear systems, saturation, bounded inputs, control-Lyapunov functions, real-analytic functions. Abstract:

   We provide a formula for a stabilizing feedback law using a bounded control, under the assumption that an appropriate control-Lyapunov function is known. Such a feedback, smooth away from the origin and continuous everywhere, is known to exist via Artstein's Theorem. As in the unbounded-control case treated in a previous note, we provide an explicit and ``universal'' formula given by an algebraic function of Lie derivatives. In particular, we extend to the bounded case the result that the feedback can be chosen analytic if the Lyapunov function and the vector fields defining the system are analytic.

1. Y. Lin and E.D. Sontag. On control-Lyapunov functions under input constraints. In Proc. IEEE Conf. Decision and Control, Orlando, Dec. 1994, IEEE Publications, 1994, pages 640-645, 1994. Keyword(s): control-Lyapunov functions.



2. Y. Lin, E.D. Sontag, and Y. Wang. Recent results on Lyapunov-theoretic techniques for nonlinear stability. In Proc. Amer. Automatic Control Conf., Baltimore, June 1994, pages 1771-1775, 1994.



3. Y. Lin, E.D. Sontag, and Y. Wang. Lyapunov-function characterizations of stability and stabilization for parameterized families of systems. In Proc. IEEE Conf. Decision and Control, San Antonio, Dec. 1993, IEEE Publications, 1993, pages 1978-1983, 1993.



4. Y. Lin and E.D. Sontag. Gradient techniques for steering systems with no drift. In Proc. Conf. Inform. Sci. and Systems, Princeton University, March 1992, pages 1003-1008, 1992.



5. E.D. Sontag and Y. Lin. Stabilization with respect to noncompact sets: Lyapunov characterizations and effect of bounded inputs. In Nonlinear Control Systems Design 1992, IFAC Symposia Series, M. Fliess Ed., Pergamon Press, Oxford, 1993, pages 43-49, 1992. Note: Also in Proc. Nonlinear Control Systems Design Symp., Bordeaux, June 1992,(M. Fliess, Ed.), IFAC Publications, pp. 9--14. [PDF] Keyword(s): saturation, bounded inputs.



6. Y. Lin and E.D. Sontag. Further universal formulas for Lyapunov approaches to nonlinear stabilization. In Proc. Conf. Inform. Sci. and Systems, John Hopkins University, March 1991, pages 541-546, 1991.

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