Hybrid Control of a Bioreactor with Quantized Measurements: Extended Version

We consider the problem of global stabilization of an unstable bioreactor model (e.g. for anaerobic digestion), when the measurements are discrete and in finite number ("quantized"), with control of the dilution rate. The model is a differential system with two variables, and the output is the biomass growth. The measurements define regions in the state space, and they can be perfect or uncertain (i.e. without or with overlaps). We show that, under appropriate assumptions, a quantized control may lead to global stabilization: trajectories have to follow some transitions between the regions, until the final region where they converge toward the reference equilibrium. On the boundary between regions, the solutions are defined as a Filippov differential inclusion. If the assumptions are not fulfilled, sliding modes may appear, and the transition graphs are not deterministic

Limit cycles in piecewise-affine gene network models with multiple interaction loops

In this paper we consider piecewise affine differential equations modeling gene networks. We work with arbitrary decay rates, and under a local hypothesis expressed as an alignment condition of successive focal points. The interaction graph of the system may be rather complex (multiple intricate loops of any sign, multiple thresholds...). Our main result is an alternative theorem showing that, if a sequence of region is periodically visited by trajectories, then under our hypotheses, there exists either a unique stable periodic solution, or the origin attracts all trajectories in this sequence of regions. This result extends greatly our previous work on a single negative feedback loop. We give several examples and simulations illustrating different cases

Periodic solutions of piecewise affine gene network models: the case of a negative feedback loop

In this paper the existence and unicity of a stable periodic orbit is proven, for a class of piecewise affine differential equations in dimension 3 or more, provided their interaction structure is a negative feedback loop. It is also shown that the same systems converge toward a unique stable equilibrium point in dimension 2. This extends a theorem of Snoussi, which showed the existence of these orbits only. The considered class of equations is usually studied as a model of gene regulatory networks. It is not assumed that all decay rates are identical, which is biologically irrelevant, but has been done in the vast majority of previous studies. Our work relies on classical results about fixed points of monotone, concave operators acting on positive variables. Moreover, the used techniques are very likely to apply in more general contexts, opening directions for future work

Global Qualitative Behavior of a Class of Nonlinear Biological Systems; Application to the Qualitative Validation of Phytoplankton Growth Models

In this paper we propose a methodology to analyze the global qualitative behavior of a class of nonlinear differential systems with respect to their structure. This class of loop structured systems with monotonous interactions encompasses numerous biological models. We show that, independently of the parameters values or of the analytical formulation of the system, the possible successions with respect to time of some qualitative events that characterize the transients of state variables are strongly related to the signs of the Jacobian matrix (structure of the model). We propose a procedure to derive the transition graph; this graph summarizes the set of possible qualitative features for the state according to the structure of the model. The comparison of the graph with experimental (even noisy) data allows to validate directly this structure. The method is illustrated with a set of models usually used to describe phytoplanctonic growth in the chemostat. The corresponding transition graphis derived and compared with experimental data

Feedback control for nonmonotone competition models in the chemostat

This paper deals with the problem of feedback control of competition between two species with one substrate in the chemostat with nonmonotone growth functions. Without control, the generic behavior is competitive exclusion. The aim of this paper is to find a feedback control of the dilution rate, depending only on the total biomass such that coexistence holds. We obtain a sufficient condition for the global asymptotic stability of an unique equilibrium point in the positive orthant for a three dimensional differential system which arises from this controlled competition model. This paper generalizes the results obtained by De Leenher and Smith in \cite{smith}

Global stability of reversible enzymatic metabolic chains

International audienceWe consider metabolic networks with reversible enzymatic reactions. The model is written as a system of ordinary differential equations, possibly with inputs and outputs. We prove the global stability of the equilibrium (if it exists), using techniques of monotone systems and compartmental matrices. We show that the equilibrium does not always exist. Finally, we consider a metabolic system coupled with a genetic network, and we study the dependence of the metabolic equilibrium (if it exists) with respect to concentrations of enzymes. We give some conclusions concerning the dynamical behavior of coupled genetic/metabolic systems

Exact control of genetic networks in a qualitative framework: the bistable switch example

International audienceA qualitative method to control piecewise affine differential systems is proposed and explored for application to genetic regulatory networks. This study considers systems whose outputs and inputs are of a qualitative form, well suited to experimental devices: the measurements indicate whether the variables are "strongly" or "weakly" expressed, that is, only the region of the state space where trajectories evolve at each instant can be known. The control laws are piecewise constant functions in each region and in time, and are only allowed to take three qualitative values corresponding to no control (u=1u=1), high synthesis rates (View the MathML sourceu=umax) or low synthesis rates (View the MathML sourceu=umin). The problems of controlling the bistable switch to each of its steady states is considered. Exact solutions are given to asymptotically control the system to either of its two stable steady states. Two approximate solutions are suggested to the problem of controlling the system to the unstable steady state: either control to a neighborhood of the state, or in the form of a periodic cycle that passes through the state

Robust control for an uncertain chemostat model

In this paper we consider a control problem for an uncertain chemostat model with a general monotone growth function. This uncertainty affects the model (growth function) as well as the outputs (measurements of substrate). Despite this lack of information, an upper bound and a lower bound for those uncertainties are assumed to be known a priori. We are able to build a family of feedback control laws on the dilution rate, giving a guaranteed estimation on the unmeasured variable (biomass), and stabilizing asymptotically the two variables in a rectangular set, around a reference value of the substrate. This could be implemented in a real chemostat, keeping a high level of substrate, avoiding the washout of the bioreactor

Qualitative stability patterns for Lotka-Volterra systems on rectangles

We present a qualitative analysis of the Lotka-Volterra differential equation within rectangles that are transverse with respect to the flow. In similar way to existing works on affine systems (and positively invariant rectangles), we consider here nonlinear Lotka-Volterra n-dimensional equation, in rectangles with any kind of tranverse patterns. We give necessary and sufficient conditions for the existence of symmetrically transverse rectangles (containing the positive equilibrium), giving notably the method to build such rectangles. We also analyse the stability of the equilibrium thanks to this transverse pattern. We finally propose an analysis of the dynamical behavior inside a rectangle containing the positive equilibrium, based on Lyapunov stability theory. More particularly, we make use of Lyapunov-like functions, built upon vector norms. This work is a first step towards a qualitative abstraction and simulation of Lotka-Volterra systems

Practical and Polytopic Observers for Nonlinear Uncertain Systems

For a class of dynamical systems, with uncertain nonlinear terms considered as «unknown inputs», we give sufficient conditions for observability. We show also that there does not exist any exact observer independent of the unknown inputs. Under the additional assumption that the uncertainty is bounded, we build practical observers whose error converges exponentially towards an arbitrary small neighborhood of the origin. For the general case, when the system might be not observable with unknown inputs, we build polytopic observers providing time-varying bounds (depending on the uncertainty bounds) on the state variables. We illustrate these results on a biological model of a structured population