Feedback tuning of bifurcations

The present paper studies a feedback regulation problem that arises in at least two different biological applications. The feedback regulation problem under consideration may be interpreted as an adaptive control problem for tuning bifurcation parameters, and it has not been studied in the control literature. The goal of the paper is to formulate this problem and to present some preliminary results.     

Certifying spatially uniform behavior in reaction–diffusion PDE and compartmental ODE systems

We present a condition that guarantees spatial uniformity for the asymptotic behavior of the solutions of a reaction–diffusion PDE with Neumann boundary conditions. This condition makes use of the Jacobian matrix of the reaction terms and the second Neumann eigenvalue of the Laplacian operator on the given spatial domain, and eliminates the global Lipschitz assumptions commonly used in mathematical biology literature. We then derive numerical procedures that employ linear matrix inequalities to certify this condition, and illustrate these procedures on models of several biochemical reaction networks. Finally, we present an analog of this PDE result for the synchronization of a network of identical ODE models coupled by diffusion terms. From a systems biology perspective, the main contribution of the paper is to blend analytical and numerical tools from nonlinear systems and control theory to derive a relaxed and verifiable condition for spatial uniformity of biological processes.     

Equilibrium-independent passivity: A new definition and numerical certification

We extend the traditional notion of passivity to a forced system whose equilibrium is dependent on the control input by defining equilibrium-independent passivity, a system property characterized by a dissipation inequality centered at an arbitrary equilibrium point. We provide a necessary input/output condition which can be tested for systems of arbitrary dimension and sufficient conditions to certify this property for scalar systems. An example from network stability analysis is presented which demonstrates the utility of this new definition. We then proceed to numerical certification of equilibrium-independent passivity using sum-of-squares programming. Finally, through numerical examples we show that equilibrium-independent passivity is less restrictive than incremental passivity.     

Global output tracking control of a class of Euler-Lagrange systems with monotonic non-linearities in the velocities

In this paper, we address the problem of output feedback tracking control of a class of Euler-Lagrange systems subject to non-linear dissipative loads. By imposing a monotone damping condition on the non-linearities of the unmeasured states, the common restriction that the non-linearities be globally Lipschitz is removed. The proposed observer-controller scheme renders the origin of the error dynamics uniformly globally asymptotically stable, in the general case. Under certain additional assumptions, the result continue to hold for a simplified control law that is less sensitive to noise and unmodelled phenomena.     

Synchronization of diffusively-coupled limit cycle oscillators

We develop analytical and numerical conditions to determine whether limit cycle oscillations synchronize in diffusively coupled systems. We examine two classes of systems: reaction–diffusion PDEs with Neumann boundary conditions, and compartmental ODEs, where compartments are interconnected through diffusion terms with adjacent compartments. In both cases the uncoupled dynamics are governed by a nonlinear system that admits an asymptotically stable limit cycle. We provide two-time scale averaging methods for certifying stability of spatially homogeneous time-periodic trajectories in the presence of sufficiently small or large diffusion and develop methods using the structured singular value for the case of intermediate diffusion. We highlight cases where diffusion stabilizes or destabilizes such trajectories.     

A sufficient condition for additive -stability and application to reaction–diffusion models

The matrix A is said to be additively D-stable if A−D remains Hurwitz for all non-negative diagonal matrices D. In reaction–diffusion models, additive D-stability of the matrix describing the reaction dynamics guarantees the stability of the homogeneous steady-state, thus ruling out the possibility of diffusion-driven instabilities. We present a new criterion for additive D-stability using the concept of compound matrices. We first give conditions under which the second additive compound matrix has non-negative off-diagonal entries. We then use this Metzler property of the compound matrix to prove additive D-stability with the help of an additional determinant condition. This result is then applied to investigate the stability of cyclic reaction networks in the presence of diffusion. Finally, a reaction network structure that fails to achieve additive D-stability is exhibited.     

Robust global stabilization with ignored input dynamics: aninput-to-state stability (ISS) small-gain approach

This paper presents a re-design for global asymptotic stabilization in the presence of ignored input dynamics, restricted to be minimum-phase and relative degree zero. Using nonlinear small-gain arguments, we design a static feedback control law to achieve global asymptotic stabilization. In the absence of full-state information, an observer-based stabilizing controller is proposed     

A Two-Time-Scale Design for Edge-Based Detection and Rectification of Uncooperative Flows

Existing Internet protocols rely on cooperative behavior of end users. We present a control-theoretic algorithm to counteract uncooperative users which change their congestion control schemes to gain larger bandwidth. This algorithm rectifies uncooperative users; that is, forces them to comply with their fair share, by adjusting the prices fed back to them. It is to be implemented at the edge of the network (e.g., by ISPs), and can be used with any congestion notification policy deployed by the network. Our design achieves a separation of time-scales between the network congestion feedback loop and the price-adjustment loop, thus recovering the fair allocation of bandwidth upon a fast transient phase     

Using electrochemical impedance to determine airflow rates

This paper proposes the use of electrochemical impedance spectroscopy (EIS) to estimate the cathode flow rate in a fuel cell system. Through experimental testing of an eight-cell, hydrogen-fueled polymer electrolyte stack, it shows that the ac impedance measurements are highly sensitive to the airflow rates at varying current densities. The ac impedance magnitude at 0.1 Hz allows the distinction of airflow rates (stoichiometry of 1.5–3.0) at current densities as low as 0.1 A/cm². Using experimental data and regression analysis, a simple algebraic equation that estimates the airflow rate using impedance measurements at a frequency of 0.1 Hz is developed. The derivation of this equation is based on the operating cell voltage equation that accounts for all the irreversibilities.