The sum of squares (SOS) decomposition technique allows numerical methods such as semidefinite programming to be used for proving the positivity of multivariable polynomial functions. It is well known that it is not an easy task to find Lyapunov functions for stability analysis of nonlinear systems. An algorithmic tool is used in this work for solving this problem. This approach is presented as SOS programming and solutions were obtained with a Matlab toolbox. Simple examples of SOS concepts, stability analysis for nonlinear polynomial and rational systems with uncertainties in parameters are presented to show the use of this tool. Besides these approaches, an alternative stability analysis for switched systems using a polynomial approach is also presented.
In this paper we develop and improve a piecewiselinear approximations of nonlinear cellular growth using orthonormal canonical piecewise linear functions [5], and it is tested by a probing control strategy [13] for the feed rate. The work is with the mammalian cells BHK (Baby Hamster Kidney) in bioreactor in batch, fed-batch and continuous mode operation. Preliminary results show that this piecewise linear approximation is well suited for modeling such nonlinear dynamics.
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