Closed-loop performance analysis for economic model predictive control of time-varying systems

Abstract

In this paper we study closed loop performance estimates for economic model predictive control (MPC) of timevarying systems. In order to consider general cost functionals we use the concept of overtaking optimality. Under turnpike and continuity assumptions we show that the cost of the MPC closed loop approximates the cost of an infinite horizon optimal trajectory. The theoretical results are illustrated by a numerical example.

Optimal power flow: an introduction to predictive, distributed and stochastic control challenges

“…Therefore, it is fair to ask whether one needs to a add stabilizing constraints / terminal penalties to (15) to enforce convergence and stability in an economic nmpc framework? Or shall one avoid those constraints and rather analyze (15) in a notion of time-varying turnpike properties [39,45]? From a power systems point of view, it would be interesting to analyze how to combine efficient online optimization with approaches of alternating optimization and power-flow solution and how to encode additional application requirements (e.g.…”

Toward economic NMPC for multistage AC optimal power flow

“…§ We remark that, in principle, one could as well apply recent results on time-varying turnpike properties of ocps here. 57,58 However, without prior knowledge of the disturbance sequence d(·), these conditions are hard to check. Thus, we leave this point for future work.…”

Strict Dissipativity Implies Turnpike Behavior for Time-Varying Discrete Time Optimal Control Problems

“…In particular, the turnpike property guarantees that the optimal open-loop trajectories, which are calculated in the MPC iterations, are close to the infinite horizon optimal trajectory for a certain number of time steps. Together with continuity of the optimal value function, this allows for the construction of a Lyapunov function as well as convergence and performance estimates for time-invariant MPC, see [8] and [13], and for performance estimates of the MPC closed-loop solution in the time-varying setting, see [12].…”

“…In this section we provide an example of a time-varying optimal control problem, that was introduced in [12]. It can be interpreted as a very simple room heating/cooling model that has to react to external influences (the weather).…”

“…Since this trajectory is very difficult to compute, compared to the time-invariant setting the situation reverses: instead of using the turnpike property and the knowledge about the optimal equilibrium for the approximation of finite horizon optimal trajectories, now we may use finite horizon optimal trajectories (which can be efficiently computed numerically if the horizon is not too long) and the turnpike property in order to approximate the infinite-horizon optimal trajectory. This can be done directly by numerically computing optimal trajectories on finite horizons with increasing length, or indirectly via a receding horizon or model predictive control (MPC) approach, see Remark 1 and [12]. However, in order to decide whether these methods can be employed, we need to find ways to check whether the given optimal control problem exhibits the turnpike property.…”