R. H. Middleton, K. Lau and J.H. Braslavsky, Conjectures and Counterexamples on Optimal L2 Disturbance Attenuation in Nonlinear Systems. CIDAC Technical Report EE03005, March 2003. Presented at the 42nd CDC, Maui, Hawaii USA, December 2003.
Abstract: This paper considers the problem of optimal $L_2$ disturbance attenuation with global asymptotic stability for strict feedback nonlinear systems. It is known from previous results that this problem cannot be solved with an arbitrary level of disturbance attenuation (almost disturbance decoupling) if the disturbance input drives unstable zero dynamics of the system. In this case, the problem can only be solved to achieve a level of disturbance attenuation above a nonzero optimal bound. An explicit expression of this lowest optimal bound is known for linear systems, and an approximate bound exists for a special subclass of nonlinear systems with second order zero dynamics. A more general expression for the lowest bound remains unknown. In this paper we provide background to the problem, and discuss the feasibility of obtaining such a general expression by presenting a series of conjectures, examples and counterexamples. We first present a conjecture that might appear as a natural generalisation of the linear expression but that, as we show by means of a counterexample, is generally false. Finally, we present a second conjecture, which holds generally for the linear case, and also for a class of scalar nonlinear systems. A general proof, or a counterexample, to this conjecture are still questions open to further research.
R. Middleton and J.H. Braslavsky, Towards Quantitative Time Domain Design Tradeoffs in Nonlinear Control. Internal Report IACI, Universidad Nacional de Quilmes, September 2001. Presented at the ACC 2001, Anchorage, USA.
Abstract: This paper analyzes the feasibility of quantifying design tradeoffs on the transient step response of a class of nonlinear systems. This feasibility analysis builds on available tools for the characterization of performance limitations in the optimal quadratic response of the class of strict feedback nonlinear systems. We present results that show that, as in linear systems, for certain classes of nonminimum phase systems, the closed loop transient step response must display undershoot. A lower bound on this undershoot can be computed based on the settling time of the system, and this bound increases as the settling time is decreased.
J.H. Braslavsky, J.S. Freudenberg, R. Middleton, Cheap Control Performance of a Class of Non-Right-Invertible Nonlinear Systems. Technical Report EE99007, March 1999. Published in the IEEE Transactions on Automatic Control.
Abstract: For strict-feedback nonlinear systems, this paper shows that it is impossible to reduce to zero the optimal cost in the regulation of more states than the number of control inputs in the system, even using unrestricted control effort. By constructing a near optimal cheap control law, we characterise the infimum value of the optimal regulation cost as the optimal value of a reduced-order regulator problem where the states with lower relative degree drive those with higher relative degree. We illustrate our results with two examples of practical interest: the optimal regulation of the rotational motion of a free rigid body, and the optimal control of a magnetic suspension system.
J.H. Braslavsky, M.M. Seron and P.V. Kokotovic. Near-Optimal Cheap Control of Nonlinear Systems. Technical Report CCEC971117, December 1997. Presented at NOLCOS 98.
Abstract: For strict-feedback nonlinear systems with relative degree r, we show that there exists a coordinate transformation under which input-output feedback linearization with Butterworth pole placement is an O(µ) approximation of the cheap control that minimizes a quadratic cost functional with control weighting µ^{2r}.
M.M. Seron, J.H. Braslavsky and P.V. Kokotovic. Stability Margins of Backstepping Designs. Technical Report CCEC970822, August 1997.
Abstract: We characterize linear input unmodeled dynamics for which nonlinear feedback systems designed with the integrator backstepping technique preserve global asymptotic stability. The comparison of two backstepping designs demonstrates that the ``domination'' of potentially harmful terms in the control law is more robust than their ``cancelation''.
Julio Braslavsky
Last modified: Fri Aug 30 11:39:53 EST 2002