1. A. C. B. de Oliveira, M. Siami, and E. D. Sontag. Regularising numerical extremals along singular arcs: a Lie-theoretic approach. In M.A. Belabbas, editor, Geometry and Topology in Control, Proceedings of BIRS Workshop. American Institute of Mathematical Sciences Press, 2024. Note: To appear.[PDF] Keyword(s): optimal control, nonlinear control, Lie algebras, robotics. Abstract:

   Numerical ``direct'' approaches to time-optimal control often fail to find solutions that are singular in the sense of the Pontryagin Maximum Principle. These approaches behave better when searching for saturated (bang-bang) solutions. In previous work by one of the authors, singular solutions were theoretically shown to exist for the time-optimal problem for two-link manipulators under hard torque constraints. The theoretical results gave explicit formulas, based on Lie theory, for singular segments of trajectories, but the global structure of solutions remains unknown. In this work, we show how to effectively combine these theoretically found formulas with the use of general-purpose optimal control softwares. By using the explicit formula given by theory in the intervals where the numerical solution enters a singular arcs, we not only obtain an algebraic expression for the control in that interval, but we are also able to remove artifacts present in the numerical solution. In this way, the best features of numerical algorithms and theory complement each other and provide a better picture of the global optimal structure. We showcase the technique on a 2 degrees of freedom robotic arm example, and also propose a way of extending the analyzed method to robotic arms with higher degrees of freedom through partial feedback linearization, assuming the desired task can be mostly performed by a few of the degrees of freedom of the robot and imposing some prespecified trajectory on the remaining joints.




2. W. Maass, P. Joshi, and E.D. Sontag. Computational aspects of feedback in neural circuits. PLoS Computational Biology, 3:e165 1-20, 2007. [PDF] Keyword(s): machine learning, neural networks, feedback linearization, computation by cortical microcircuits, fading memory. Abstract:

   It had previously been shown that generic cortical microcircuit models can perform complex real-time computations on continuous input streams, provided that these computations can be carried out with a rapidly fading memory. We investigate in this article the computational capability of such circuits in the more realistic case where not only readout neurons, but in addition a few neurons within the circuit have been trained for specific tasks. This is essentially equivalent to the case where the output of trained readout neurons is fed back into the circuit. We show that this new model overcomes the limitation of a rapidly fading memory. In fact, we prove that in the idealized case without noise it can carry out any conceivable digital or analog computation on time-varying inputs. But even with noise the resulting computational model can perform a large class of biologically relevant real-time computations that require a non-fading memory.




3. A. Arapostathis, B. Jakubczyk, H.-G. Lee, S. I. Marcus, and E.D. Sontag. The effect of sampling on linear equivalence and feedback linearization. Systems Control Lett., 13(5):373-381, 1989. [PDF] [doi:http://dx.doi.org/10.1016/0167-6911(89)90103-5] Keyword(s): discrete-time, sampled-data systems, discrete-time systems, sampling. Abstract:

   We investigate the effect of sampling on linearization for continuous time systems. It is shown that the discretized system is linearizable by state coordinate change for an open set of sampling times if and only if the continuous time system is linearizable by state coordinate change. Also, it is shown that linearizability via digital feedback imposes highly nongeneric constraints on the structure of the plant, even if this is known to be linearizable with continuous-time feedback.

1. B. Jakubczyk and E.D. Sontag. The effect of sampling on feedback linearization. In Proc. IEEE Conf. Decision and Control, Los Angeles, Dec.1987, pages 1374-1379, 1987.

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