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On Qualitative Behaviours of a Class of Piecewise-Linear Control Systems (Part II: A Case Study)
Abstract
Qualitative models of complex systems recently emerged as useful tools for behaviour prediction in case of incomplete system knowledge or as a variant to a difficult analytical analysis. A class of piecewise (PL) differential systems can be abstracted to a logical automaton, within a hybrid control systems (HCS) framework from control engineering, and the resulting discrete evolution provides information about the qualitative behaviours; of the continuous system. The basic general models and related design problems were presented in Part I. Part II presents, within a case study, a solution to the problem of constructing a logical automaton as a discrete event abstraction of a PL differential system. Relevant qualitative evolutions are successfully checked against simulated trajectories and future research directions are proposed.
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Hybrid control systems, that is, systems which contain both continuous dynamics and discrete event dynamics are studied in this paper. First, a model is introduced that describes the continuous plant and discrete event controller along with an interface which connects them. A Discrete Event System (DES) automaton description is employed to describe the plant together with the interface and it is used to analyze the hybrid control system. Controllability is defined for hybrid control systems, enhancing existing DES control concepts. It is then used to obtain a controller design method for hybrid control systems.
A relaxation of Lyapunov's direct method has been proposed elsewhere that allows for an algorithmic construction of Lyapunov functions to prove stability of equilibria in nonlinear systems, but the search is restricted to systems with polynomial vector fields. In the paper, the above technique is extended to include systems with equality, inequality, and integral constraints. This allows certain non-polynomial nonlinearities in the vector field to be handled exactly and the constructed Lyapunov functions to contain non-polynomial terms. It also allows robustness analysis to be performed. Some examples are given to illustrate how this is done.
Specific modelling and simulation tools have recently emerged for the study of complex systems, allowing systematic prediction of their behaviour. In case of incomplete knowledge of the original physical system, qualitative modelling tools arc preferred to quantitative ones. Starting from a hybrid control systems (HCS) framework in control engineering, this paper proposes a qualitative modelling approach for a class of piecewise linear (PL) differential systems, firstly introduced as a genetic network model and having a natural feedback structure. The thresholds limits imposed to the state variables partition the state space into open regions and the resulting qualitative model is a logical abstraction of the families of continuous trajectories mapped to this partition. Part I describes the basic general models and associated problems.
The study of genetic regulatory networks has received a major impetus from the recent development of experimental techniques
allowing the measurement of patterns of gene expression in a massively parallel way. This experimental progress calls for
the development of appropriate computer tools for the modeling and simulation of gene regulation processes. We present a method
for the hybrid modeling and simulation of genetic regulatory networks, based on a class of piecewiselinear (PL) differential
equations that has been well-studied in mathematical biology. Distinguishing characteristics of the method are that it makes
qualitative predictions of the behavior of regulatory systems and that it deals with discontinuities in the right-hand side
of the differential equations. The simulation method has been implemented in Java in the computer tool Genetic Network Analyzer
(GNA). The method and the tool have been used to analyze several networks of biological interest, including the network underlying
the initiation of sporulation in Bacillus subtilis.
We propose a mapping to study the qualitative properties of continuous biochemical control networks which are invariant to the parameters used to describe the networks but depend only on the logical structure of the networks. For the networks, we are able to place a lower limit on the number of steady states and strong restrictions on the phase relations between components on cycles and transients. The logical structure and the dynamical behavior for a number of simple systems of biological interest, the feedback (predator-prey) oscillator, the bistable switch, the phase dependent switch, are discussed. We discuss the possibility that these techniques may be extended to study the dynamics of large many component systems.