Abstract
Hybrid models of genetic regulatory networks allow for a simpler analysis with respect to fully detailed quantitative models, still maintaining the main dynamical features of interest. In this paper we consider a piecewise affine model of a genetic regulatory network, in which the parameters describing the production function are affected by polytopic uncertainties. In the first part of the paper, after recalling how the problem of finding a Lyapunov function is solved in the nominal case, we present the considered polytopic uncertain system and then, after describing how to deal with sliding mode solutions, we prove a result of existence of a parameter dependent Lyapunov function subject to the solution of a feasibility linear matrix inequalities problem. In the second part of the paper, based on the previously described Lyapunov function, we are able to determine a set of domains where the system is guaranteed to converge, with the exception of a zero measure set of times, independently from the uncertainty realization. Finally a three nodes network example shows the validity of the results.
Highlights
In the last few years control theory tools have been extensively used in biology, to both understand natural biological systems or design new ones to perform specific tasks (Blanchini et al 2018; Qian et al 2018)
We provide an linear matrix inequalities (LMIs) framework whose solution describes a parameter dependent piecewise quadratic Lyapunov function (PD-Piecewise Quadratic Lyapunov function (PWQ-LF)) for the system, i.e. a Lyapunov function that depends explicitly on the unknown parameters describing the particular uncertainty realization, which will allow us to describe a convergence set for the system, robust with respect to the uncertainty
It is remarked that even if in literature there are results on parameter dependent Lyapunov functions with more complex and general structures, in this work we consider parameter dependent Lyapunov functions which are only affinely dependent on the system’s uncertain parameters, as these allow to deal in a straightforward manner with sliding mode monotonicity and continuity on the boundary of regulatory domains, as it will be clear in the following
Summary
In the last few years control theory tools have been extensively used in biology, to both understand natural biological systems or design new ones to perform specific tasks (Blanchini et al 2018; Qian et al 2018). In our previous work (Pasquini and Angel 2019) we considered the aforementioned PWA model of a GRN and, using the model structure and the information from the STG, we developed an LMI framework to find a Lyapunov function for the system, in order to assess its convergence properties, even in the presence of cycles in the STG. The first one consists in searching for a Lyapunov function which is common between all the systems obtained by considering the realizations associated with the vertices of the uncertainty polytope (Liberzon 2003; Lin and Antsaklis 2005) This approach, allowing to conclude certain stability and stabilizability properties of the system, is challenging and can lead to a conservative solution, as it does not consider how the extremal behaviors are combined in any given uncertainty realization.
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