Shaping pulses to control bistable systems: Analysis, computation and counterexamples

Abstract

In this paper we study how to shape temporal pulses to switch a bistable system between its stable steady states. Our motivation for pulse-based control comes from applications in synthetic biology, where it is generally difficult to implement real-time feedback control systems due to technical limitations in sensors and actuators. We show that for monotone bistable systems, the estimation of the set of all pulses that switch the system reduces to the computation of one non-increasing curve. We provide an efficient algorithm to compute this curve and illustrate the results with a genetic bistable system commonly used in synthetic biology. We also extend these results to models with parametric uncertainty and provide a number of examples and counterexamples that demonstrate the power and limitations of the current theory. In order to show the full potential of the framework, we consider the problem of inducing oscillations in a monotone biochemical system using a combination of temporal pulses and event-based control. Our results provide an insight into the dynamics of bistable systems under external inputs and open up numerous directions for future investigation.

Pulse-Based Control Using Koopman Operator Under Parametric Uncertainty

“…In [4], [5], it has been proposed to solve the switching problem using temporal pulses of a fixed length τ and a fixed magnitude µ for monotone systems (cf. [6]).…”

“…In the left panel, boundaries of basins of attraction of the toggle switch for different parameter values. In the right panel a trajectory controlled by an event-based procedure in order to keep the flow inside the box[4,10]×[4,10].…”

Optimal control formulation of pulse-based control using Koopman operator

“…The authors derived a computational procedure to estimate the set of control signals allowing the switch. The control strategy proposed by [11] is open loop so that the control signal cannot be adjusted during the experiment. [12] considered a similar setting, but using the Koopman operator framework to estimate the set of all switching pulses and provide estimates for convergence time.…”

“…[12] considered a similar setting, but using the Koopman operator framework to estimate the set of all switching pulses and provide estimates for convergence time. In this paper, we present an optimization program for solving the convergence problem, which is a generalization of the switching problem considered by [11] and [12].…”

“…To summarize, in this paper, we consider a convergence problem similarly to [13], the target set is chosen similarly to [14], while the class of systems and the control signals are restricted as in the works by [11] and [12]. We opted for restricting the class of systems in order to show optimality of our approach in this specific case, which indicates that systems' properties can be used to derive easy-to-implement and interpretable optimal solutions.…”