Convex, monotone systems are optimally operated at steady-state

Abstract

We consider a special class of monotone systems for which the system equations are also convex in both the state and the input. For such systems we study optimal infinite horizon operation with respect to an objective function that is also monotone and convex. The main results state that, under some technical assumptions, these systems are optimally operated at steady state, i.e. there does not exist any timevarying trajectory over an infinite horizon that outperforms stabilizing the system in the optimal equilibrium. We draw a connection to recent results on dissipative systems in the context of Economic Model Predictive Control, where systems that are optimally operated at steady state have already been studied. Finally, we apply the main result to a problem in traffic control, where we are able to disprove the existence of improving periodic trajectories involving the alternation of congestion and free flow for freeway ramp metering.

On Robustness of Equilibria in Dynamical Transportation Networks

“…While other equilibria to the dynamical system (2)-(3) together with the constraints (7)-(8) exist, the equilibrium point x * ∈ F is the desired one, since it achieves the same throughput in the network with less traffic present in the network compared to any other equilibria. Moreover, as shown in [20], it is most efficient to operate the transportation network at this equilibrium point, rather than trying to steer some time-varying trajectories. Therefore this equilibrium will be the focus of the studies in this paper.…”

Convexity and monotonicity in nonlinear optimal control under uncertainty

“…[83] Three remarks are in order. First, system ( 5) is a straightforward generalization of the convex-state-monotone system studied in [76], which in turn generalizes the convex-monotone system studied in [75]. A precise name for system (5) would be convex-nonlinear-state-monotone, since the monotonicity requirement applies only to the states with nonlinear dynamics.…”

“…Many more examples of nonlinear convex and concave functions can be found in §3 of [83]. Systems with nonlinear convex dynamics have arisen in applications ranging from cancer and HIV treatment scheduling [75] to voltage control in power systems [75] to freeway congestion management [76] to energy storage control [84], [85]. The drug treatment model in [75] is a special case of a more general class of bilinear systems that, through a logarithmic transformation, can be cast as convex systems.…”

“…In [75], Rantzer and Bernhardsson observe that in convex-monotone systems, where the dynamics are convex and nondecreasing in the state and control input, the state trajectory is convex in the control input trajectory. In [76], Schmitt et al generalize this result to convex-state-monotone systems, where the dynamics need not be monotone in the control input.…”

“…An immediate consequence of the observations in [75], [76] is that for some nonlinear systems, open-loop optimal control (where the decision variable is a fixed control trajectory, rather than a feedback policy) is a convex optimization problem. This holds for convex-state-monotone systems in particular, provided the cost and constraints are nondecreasing in the states.…”