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Product Measure
Published in Kenneth Kuttler, Modern Analysis, 2017
In calculus, the actual evaluation of a multiple integral involves iterated integrals. An important manipulation is the change in order of integration. We are asked to believe that iterated integrals in different order are all equal and they all equal the value of the multiple integral. In fact, this is not always true. The following is an example.
An algebraic version of the active disturbance rejection control for second-order flat systems
Published in International Journal of Control, 2021
Carlos Aguilar-Ibanez, Hebertt Sira-Ramirez, José Ángel Acosta, Miguel S. Suarez-Castanon
In this work, we propose an algebraic version of the ADRC approach for unknown uncertain single-input single-output nonlinear second-order plants. We must mention that this approach can be seen as a nonlinear second-order flat system with a matching perturbation, which lumped nonlinearities and external perturbations. The proposed approach uses an algebraic estimator for the unknown disturbances affecting the system. This task is customarily carried out, in ADRC schemes, using extended high-gain observers. In our approach, the required, unmeasured, phase variables associated with the flat output are online recovered using an algebraic estimator. The algebraic estimator comprises a readily usable formula. This formula is based on the computation of an iterated integral, which accepts an analytical solution. The algebraic time derivatives estimator can be, alternatively, substituted by any other time derivative estimation technique, like high-gain observers or sliding mode observers, to mention just two of the available options.
Survey on algebraic numerical differentiation: historical developments, parametrization, examples, and applications
Published in International Journal of Systems Science, 2022
Amine Othmane, Lothar Kiltz, Joachim Rudolph
Recall from Doetsch (1974), Mikusinski (1983) that the differentiation with respect to the operator s corresponds to a multiplication with in the time domain. A product , , where is the operational counterpart of a function u, corresponds to the k-th order iterated integral The Cauchy formula for repeated integration is used in the latter equation to transform the iterated integral into a single one.
On a linear input–output approach for the control of nonlinear flat systems
Published in International Journal of Control, 2018
H. Sira-Ramírez, E. W. Zurita-Bustamante, E. Hernández-Flores, M. A. Aguilar-Orduña
We extend the previous result to include a robustness feature for the proposed dynamical controller. We aim at including, m > 0, additional pure integrations in the controller, keeping the globally exponentially stabilising property of the output tracking error. We accomplish this while conserving the proportional-plus-iterated integral character of the original controller in the previous Lemma. Such a modification makes the control scheme robust with respect to disturbances locally described by time-polynomial disturbances of up to degree m − 1. Such a controller is described by