New approach to the stability and control of reaction networks

A new system-theoretic approach for studying the stability and control of chemical reaction networks (CRNs) is proposed, and analyzed. This has direct application to biological applications where biochemical networks suffer from high uncertainty in the kinetic parameters and exact structure of the rate functions. The proposed approach tackles this issue by presenting "structural" results, i.e. results that extract important qualitative information from the structure alone regardless of the specific form of the kinetics which can be arbitrary monotone kinetics, including Mass-Action. The proposed method is based on introducing a class of Lyapunov functions that we call Piecewise Linear in Rates (PWLR) Lyapunov functions. Several algorithms are proposed for the construction of these functions. Subject to mild technical conditions, the existence of these functions can be used to ensure powerful dynamical and algebraic conditions such as Lyapunov stability, asymptotic stability, global asymptotic stability, persistence, uniqueness of equilibria and exponential contraction. This shows that this class of networks is well-behaved and excludes complicated behaviour such as multi-stability, limit cycles and chaos. The class of PWLR functions is then shown to be a subset of larger class of Robust Lyapunov functions (RLFs), which can be interpreted by shifting the analysis to reaction coordinates. In the new coordinates, the problem transforms into finding a common Lyapunov function for a linear parameter varying system. Consequently, dual forms of the PWLR Lyapunov functions are presented, and the interpretation in terms of the variational dynamics and contraction analysis are given. An other class of Piecewise Quadratic in Rates Lyapunov function is also introduced. Relationship with consensus dynamics are also pointed out. Control laws for the stabilization of the proposed class of networks are provided, and the concept of control Lyapunov function is briefly discussed. Finally, the proposed framework is shown to be widely applicable to biochemical networks.Open Acces

A robust Lyapunov criterion for nonoscillatory behaviors in biological interaction networks

We introduce the notion of nonoscillation, propose a constructive method for its robust verification, and study its application to biological interaction networks (also known as, chemical reaction networks). We begin by revisiting Muldowney’s result on the nonexistence of periodic solutions based on the study of the variational system of the second additive compound of the Jacobian of a nonlinear system. We show that exponential stability of the latter rules out limit cycles, quasi-periodic solutions, and broad classes of oscillatory behavior. We focus then on nonlinear equations arising in biological interaction networks with general kinetics, and we show that the dynamics of the aforementioned variational system can be embedded in a linear differential inclusion. We then propose algorithms for constructing piecewise linear Lyapunov functions to certify global robust nonoscillatory behavior. Finally, we apply our techniques to study several regulated enzymatic cycles, where available methods are not able to provide any information about their qualitative global behavior

Polyhedral Lyapunov functions structurally ensure global asymptotic stability of dynamical networks iff the Jacobian is non-singular

For a vast class of dynamical networks, including chemical reaction networks (CRNs) with monotonic reaction rates, the existence of a polyhedral Lyapunov function (PLF) implies structural (i.e., parameter-free) local stability. Global structural stability is ensured under the additional assumption that each of the variables (chemical species concentrations in CRNs) is subject to a spontaneous infinitesimal dissipation. This paper solves the open problem of global structural stability in the absence of the infinitesimal dissipation, showing that the existence of a PLF structurally ensures global convergence if and only if the system Jacobian passes a structural non-singularity test. It is also shown that, if the Jacobian is structurally non-singular, under positivity assumptions for the system partial derivatives, the existence of an equilibrium is guaranteed. For systems subject to positivity constraints, it is shown that, if the system admits a PLF, under structural non-singularity assumptions, global convergence within the positive orthant is structurally ensured, while the existence of an equilibrium can be proven by means of a linear programming test and the computation of a piecewise-linear-in-rate Lyapunov function.Accepted Author ManuscriptTeam Tamas Keviczk

Distributed Model Predictive Control for Reconfigurable Large-Scale Systems

Large-scale Systems are gaining more importance in the modern world requiring flexible techniques capable of handling interactions. This thesis is concerned with the development of suitable algorithms based on Model Predictive Control (MPC) that guarantee stability, recursive feasibility and constraint satisfaction. In the first part of this thesis, the main properties and control challenges for controlling an Large-Scale System are brought together, and the main distributed approaches for solving these problems are surveyed. Also, two novel Distributed MPC algorithms are presented. A non-centralised approach to the output-feedback variant of tube-based model predictive control of dynamically coupled linear time-invariant systems with shared constraints. A tube-based algorithm capable of handling the interactions–not rejecting them– that replaces the conventional linear disturbance rejection controller with a second MPC controller, as is done in tube-based nonlinear MPC. Following this, a smart-grids application of the developed algorithm is presented to solve the load frequency control for a power network. The approach achieves guaranteed constraint satisfaction, the recursive feasibility of the MPC problems and stability while maintaining on-line complexity similar to conventional MPC. The second part of the thesis covers reconfigurable distributed MPC. Two novel approaches are considered: a nominal MPC methodology that incorporates information of external disturbances, and a coalitional approach for robust distributed MPC. The first approach uses available disturbance predictions within a nominal model predictive control formulation is studied. The main challenge that arises is the loss of recursive feasibility and stability guarantees when a disturbance, which may change from time step to time step, is resent in the model and on the system. We show how standard stabilising terminal conditions may be modified to account for the use of disturbances in the prediction model. Robust stability and feasibility are established under the assumption that the disturbance change across sampling instances is limited. The proposed coalitional approach to robust Distributed MPC aims to tackle the existing trade-off between communication and performance in Large-Scale System by exploiting the different network topologies of system dynamics. The algorithm employs a method to switch between topologies using a multi-rate control approach. The optimal topology selection problem is solved using a consensus approach appropriately constrained to reduce the effects of any combinatorial explosion. The robust control algorithm is capable of recomputing the necessary parameters online to readjust to new partitions. Robust constraint satisfaction, recursive and stability are guaranteed by the proposed algorithm