51 research outputs found
Computing Dynamic Output Feedback Laws
The pole placement problem asks to find laws to feed the output of a plant
governed by a linear system of differential equations back to the input of the
plant so that the resulting closed-loop system has a desired set of
eigenvalues. Converting this problem into a question of enumerative geometry,
efficient numerical homotopy algorithms to solve this problem for general
Multi-Input-Multi-Output (MIMO) systems have been proposed recently. While
dynamic feedback laws offer a wider range of use, the realization of the output
of the numerical homotopies as a machine to control the plant in the time
domain has not been addressed before. In this paper we present symbolic-numeric
algorithms to turn the solution to the question of enumerative geometry into a
useful control feedback machine. We report on numerical experiments with our
publicly available software and illustrate its application on various control
problems from the literature.Comment: 20 pages, 3 figures; the software described in this paper is publicly
available via http://www.math.uic.edu/~jan/download.htm
Complexity of Model Testing for Dynamical Systems with Toric Steady States
In this paper we investigate the complexity of model selection and model
testing for dynamical systems with toric steady states. Such systems frequently
arise in the study of chemical reaction networks. We do this by formulating
these tasks as a constrained optimization problem in Euclidean space. This
optimization problem is known as a Euclidean distance problem; the complexity
of solving this problem is measured by an invariant called the Euclidean
distance (ED) degree. We determine closed-form expressions for the ED degree of
the steady states of several families of chemical reaction networks with toric
steady states and arbitrarily many reactions. To illustrate the utility of this
work we show how the ED degree can be used as a tool for estimating the
computational cost of solving the model testing and model selection problems
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