Christian Breindl

Maximally Permissive Hierarchical Control of Decentralized Discrete Event Systems

The subject of this paper is the synthesis of natural projections that serve as nonblocking and maximally permissive abstractions for the hierarchical and decentralized control of large-scale discrete event systems. To this end, existing concepts for nonblocking abstractions such as natural observers and marked string accepting (msa)-observers are extended by local control consistency (LCC) as a novel sufficient condition for maximal permissiveness. Furthermore, it is shown that, similar to the natural observer condition and the msa-observer condition, also LCC can be formulated in terms of a quasi-congruence. Based on existing algorithms in the literature, this allows to algorithmically compute natural projections that are either natural observers or msa-observers and that additionally fulfill LCC. The obtained results are illustrated by the synthesis of nonblocking and maximally permissive supervisors for a manufacturing system.

On maximal permissiveness of hierarchical and modular supervisory control approaches for discrete event systems

Recently, several efficient modular and hierarchical approaches for the control of discrete event systems (DES) have been proposed. Although these methods are very suitable for dealing with the state space explosion problem, their common limitation is that either maximal permissiveness is not addressed or unnecessarily restrictive conditions are required in order to ensure maximally permissive control. In this paper we develop a unified framework for the investigation of maximal permissiveness of modular control in multi-level hierarchies.We identify a set of conditions that is met by several approaches, and prove its sufficiency for maximally permissive control.

A framework for state attraction of discrete event systems under partial observation

State attraction for discrete event systems (DES) addresses the problem of reaching a desired subset of the plant state space after a bounded number of event occurrences. The problem of state attraction arises for example in fault-tolerant supervisory control or in the control of reconfigurable manufacturing systems, and is also applicable to systems biological problems such as the control of gene regulatory networks. State attraction is investigated with the assumption of full event observation in the existing literature. This paper extends the concept of state attraction to the case of partial observation. The notion of weak attraction under partial observation (WAPO) is introduced and necessary and sufficient conditions for the existence of a supervisor under partial observation that achieves WAPO are derived. Furthermore, a solution algorithm is proposed that finds such supervisor whenever it exists. It is shown that such supervisor can always be realized as a subautomaton of the observer automaton of the DES plant. An application example from systems biology illustrates the obtained results.

A robustness measure for the stationary behavior of qualitative gene regulation networks

Abstract In this paper the stationary behavior of uncertain and possibly multistable gene regulation networks is considered. We first introduce a modeling framework which is able to represent the qualitative knowledge which is typically available for these systems. Then we turn to the problem of model discrimination: Given several alternative model structures that can all reproduce the experimental observations, is it possible to decide which structure may be the most appropriate description of the real system. To this end, a robustness measure for qualitative multistable gene regulation networks is introduced and also a method for the computation of this measure is presented. The benefit of the developed method is twofold: On the one hand it allows to compare the robustness properties of different model structures, on the other hand also the most fragile interconnections of a network can be detected. Finally, an example network is analyzed with this method.

Verification of multistability in gene regulation networks: A combinatorial approach

Bi- or even multistable behavior is a recurrent phenomenon in gene regulation networks. These networks have the capacity to operate in two or more distinct modes in a stable manner. In this work, we consider gene regulation networks with known interaction structure but unknown reaction kinetics. Additionally, it is assumed that several distinct operation modes were observed experimentally whereby also the measurements of the individual protein concentrations are uncertain. For this setup, the important question of model validation is addressed: Can the given network structure in principle reproduce the observed operation modes? To approach this problem, the work builds on an existing modeling framework which is appropriate to deal with uncertain gene regulation networks. By regarding each operation mode as a forward-invariant set in the state space and with the notion of compatible intervals, it is shown how the validation problem can be translated into a combinatorial one. An algorithm is developed which can efficiently solve the new problem. Finally, the well-known bistable lactose utilization network is analyzed with this new method.

A linear reformulation of Boolean optimization problems and structure identification of gene regulation networks

We consider the problem of estimating Boolean models of gene regulation networks from few and noisy measurements. To this end, we use a representation of Boolean functions as multi-affine polynomials, leading to a reformulation of the estimation problem as mixed integer linear program. We then show that the integer constraints can be omitted which improves existing results and reduces the required computing time drastically. Also certain properties of Boolean functions such as unateness or the canalizing property can be included in the linear formulation. The benefits of this reformulation are demonstrated with the help of a large Boolean model of the network of the segment polarity genes in Drosophila melanogaster.

Structure estimation for unate Boolean models of gene regulation networks

This paper deals with the reconstruction of the interaction structure of a gene regulation network from qualitative data in a Boolean framework. The problem in this setup is to find update functions which are in agreement with the data. As the search space grows exponentially with the system size but data are rare, large uncertainties remain in the reconstructed networks. In order to attenuate this problem, we propose to restrict the search space to the biologically meaningful class of unate functions. Using sign-representations, the problem of exploring this reduced search space is transformed into a linear feasibility problem. The sign-representation furthermore allows to incorporate robustness considerations and gives rise to a new measure which can be used to further reduce the uncertainties. The proposed methodology is demonstrated with a Boolean apoptosis signaling model.

Structural requirements and discrimination of cell differentiation networks

Abstract Mathematical models of stem cell differentiation are commonly based upon the concept of subsequent cell fate decisions, each controlled by a gene regulatory network. These networks exhibit a multistable behavior and cause the system to switch between qualitatively distinct stable steady states. However, the network structure of such a switching module is often uncertain, and there is lack of knowledge about the exact reaction kinetics. In this paper, we therefore perform an elementary study of small networks consisting of three interacting transcriptional regulators responsible for cell differentiation: We investigate which network structures can reproduce a certain multistable behavior, and how robustly this behavior is realized by each network. In order to approach these questions, we use a modeling framework which only uses qualitative information about the network, yet allows model discrimination as well as to evaluate the robustness of the desired multistability properties. We reveal structural network properties which are necessary and sufficient to realize distinct steady state patterns required for cell differentiation. Our results also show that structural and robustness properties of the networks are related to each other.

Approximative classification of regions in parameter spaces of nonlinear ODEs yielding different qualitative behavior

Nonlinear dynamical systems can show a variety of different qualitative behaviors depending on the actual parameter values. As in many situations of practical relevance the parameter values are not precisely known it is crucial to determine the region in parameter space where the system exhibits the desired behavior.