David Angeli

Detection of multistability, bifurcations, and hysteresis in a large class of biological positive-feedback systems

It is becoming increasingly clear that bistability (or, more generally, multistability) is an important recurring theme in cell signaling. Bistability may be of particular relevance to biological systems that switch between discrete states, generate oscillatory responses, or “remember” transitory stimuli. Standard mathematical methods allow the detection of bistability in some very simple feedback systems (systems with one or two proteins or genes that either activate each other or inhibit each other), but realistic depictions of signal transduction networks are invariably much more complex. Here, we show that for a class of feedback systems of arbitrary order the stability properties of the system can be deduced mathematically from how the system behaves when feedback is blocked. Provided that this open-loop, feedback-blocked system is monotone and possesses a sigmoidal characteristic, the system is guaranteed to be bistable for some range of feedback strengths. We present a simple graphical method for deducing the stability behavior and bifurcation diagrams for such systems and illustrate the method with two examples taken from recent experimental studies of bistable systems: a two-variable Cdc2/Wee1 system and a more complicated five-variable mitogen-activated protein kinase cascade.

On Average Performance and Stability of Economic Model Predictive Control

Control performance and cost optimization can be conflicting goals in the management of industrial processes. Even when optimal or optimization-based control synthesis tools are applied, the economic cost associated with plant operation is often only optimized according to static criteria that pick, among all feasible steady states, those with minimal cost. In mathematical terms, an economic cost functional differs from stage costs commonly adopted in model predictive control (MPC) as it need not be minimal at its best equilibrium. This note collects and illustrates some recent advances in receding horizon optimization of nonlinear systems that allow the control designer to simultaneously and dynamically optimize transient and steady-state economic performance. In particular, we show that average performance of economic MPC is never worse than the optimal steady-state operation. We introduce a dissipation inequality and supply function that extend previous sufficient conditions for asymptotic stability of economic MPC. Dissipativity is also shown to be a sufficient condition for concluding that steady-state operation is optimal. We show how to modify an economic cost function so that steady-state operation is asymptotically stable when that feature is deemed desirable. Finally, for the case when steady-state operation is not optimal, we develop two modified MPC controllers that asymptotically guarantee 1) improved performance compared to optimal periodic control and 2) satisfaction of constraints on average values of states and inputs.

Economic optimization using model predictive control with a terminal cost

Abstract In the standard model predictive control implementation, first a steady-state optimization yields the equilibrium point with minimal economic cost. Then, the deviation from the computed best steady state is chosen as the stage cost for the dynamic regulation problem. The computed best equilibrium point may not be the global minimum of the economic cost, and hence, choosing the economic cost as the stage cost for the dynamic regulation problem, rather than the deviation from the best steady state, offers potential for improving the economic performance of the system. It has been previously shown that the existing framework for MPC stability analysis, which addresses to the standard class of problems with a regulation objective, does not extend to economic MPC. Previous work on economic MPC developed new tools for stability analysis and identified sufficient conditions for asymptotic stability. These tools were developed for the terminal constraint MPC formulation, in which the system is stabilized by forcing the state to the best equilibrium point at the end of the horizon. In this work, we relax this constraint by imposing a region constraint on the terminal state instead of a point constraint, and adding a penalty on the terminal state to the regulator cost. We extend the stability analysis tools, developed for terminal constraint economic MPC, to the proposed formulation and establish that strict dissipativity is sufficient for guaranteeing asymptotic stability of the closed-loop system. We also show that the average closed-loop performance outperforms the best steady-state performance. For implementing the proposed formulation, a rigorous analysis for computing the appropriate terminal penalty and the terminal region is presented. A further extension, in which the terminal constraint is completely removed by modifying the regulator cost function, is also presented along with its stability analysis. Finally, an illustrative example is presented to demonstrate the differences between the terminal constraint and the proposed terminal penalty formulation.

Fundamentals of economic model predictive control

The goal of most current advanced control systems is to guide a process to a target setpoint rapidly and reliably. Model predictive control has become a popular technology in many applications because it can handle large, multivariable systems subject to hard constraints on states and inputs. The optimal steady-state setpoint is usually provided by some other information management system that determines, among all steady states, which is the most profitable. For an increasing number of applications, however, this hierarchical separation of information and purpose is no longer optimal or desirable. A recently proposed alternative to the hierarchical decomposition is to take the economic objective directly as the objective function of the control system. In this approach, known as economic MPC, the controller optimizes directly in real time the economic performance of the process, rather than tracking to a setpoint. The purpose of this tutorial is to explain how to design these kinds of control systems and what kinds of closed-loop properties one can achieve with them. We cover the following issues: asymptotic average performance; closed-loop stability and convergence, strong duality and dissipativity; designing terminal costs, terminal regions, and terminal periodic constraints. Several examples are included to illustrate these results.

A Stochastic Approach to “Dynamic-Demand” Refrigerator Control

Dynamic demand management is a very promising research direction for improving power system resilience. This paper considers the problem of managing power consumption by means of “smart” thermostatic control of domestic refrigerators. In this approach, the operating temperature of these appliances and thus their energy consumption, is modified dynamically, within a safe range, in response to mains frequency fluctuations. Previous research has highlighted the potential of this idea for responding to sudden power plant outages. However, deterministic control schemes have proved inadequate as individual appliances tend to “synchronize” with each other, leading to unacceptable levels of overshoot in energy demand, when they “recover” their steady-state operating cycles. In this paper we design decentralized random controllers that are able to respond to sudden plant outages and which avoid the instability phenomena associated with other feedback strategies. Stochasticity is used to achieve desynchronization of individual refrigerators while keeping overall power consumption tightly regulated.

Economic model predictive control with self-tuning terminal cost ☆

Abstract In this paper, we propose an economic model predictive control (MPC) framework with a self-tuning terminal weight, which builds on a recently proposed MPC algorithm with a generalized terminal state constraint. First, given a general time-varying terminal weight, we derive an upper bound on the closed-loop average performance which depends on the limit value of the predicted terminal state. After that, we derive conditions for a self-tuning terminal weight such that bounds for this limit value can be obtained. Finally, we propose several update rules for the self-tuning terminal weight and analyze their respective properties. We illustrate our findings with several examples.

Graph-theoretic characterizations of monotonicity of chemical networks in reaction coordinates

This paper derives new results for certain classes of chemical reaction networks, linking structural to dynamical properties. In particular, it investigates their monotonicity and convergence under the assumption that the rates of the reactions are monotone functions of the concentrations of their reactants. This is satisfied for, yet not restricted to, the most common choices of the reaction kinetics such as mass action, Michaelis-Menten and Hill kinetics. The key idea is to find an alternative representation under which the resulting system is monotone. As a simple example, the paper shows that a phosphorylation/dephosphorylation process, which is involved in many signaling cascades, has a global stability property. We also provide a global stability result for a more complicated example that describes a regulatory pathway of a prevalent signal transduction module, the MAPK cascade.

Economic and Environmental Benefits of Dynamic Demand in Providing Frequency Regulation

Increase of penetration of intermittent renewable power connected to the system will increase the requirements for frequency regulation services. If these services are met by conventional plant running part-loaded, this will not only reduce the system operational efficiency but will also limit the ability of the system to accommodate renewable generation. This work quantifies the value of Dynamic Demand (DD) concept, which enables domestic refrigeration appliances to contribute to primary frequency regulation through an advanced stochastic control algorithm. The benefits of DD providing frequency response are determined for a wide range of future low-carbon generation systems, using an efficient generation scheduling model which includes scheduling of frequency regulation and reserve services. The analysis also considers the potential impact of wind generation on system inertia and primary frequency regulation. Simulations indicate that the benefits of DD increase considerably in systems with high wind penetration, making DD an attractive option for significantly improving system efficiency.

Receding horizon cost optimization for overly constrained nonlinear plants

A receding horizon control algorithm, originally proposed for tracking best-possible steady-states in the presence of overly stringent state and/or input constraints, is analyzed for the case of nonlinear plant models and possibly non-convex cost functionals. Unlike the linear case (with convex cost functionals), convergence to equilibrium is not always possible and only average performance bounds are guaranteed in general.

Integral Input to State Stable systems in cascade

The Integral Input to State Stability (iISS) property is studied in the context of nonlinear time-invariant systems in cascade. Some sufficient conditions for the preservation of the iISS property under a cascade interconnection are presented. These are first given as growth restrictions on the supply functions of the storage function associated with each subsystem and are then expressed as solutions-based requirements. A Lyapunov-based condition guaranteeing that the cascade composed of an iISS system driven by a Globally Asymptotically Stable (GAS) one remains GAS is also provided. We also show that some of these results extend to cascades composed of more than two subsystems.