On Average Performance and Stability of Economic Model Predictive Control

Abstract

Control performance and cost optimization can be conflicting goals in the management of industrial processes. Even when optimal or optimizationbased 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 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 (i) improved performance compared to optimal periodic control and (ii) satisfaction of constraints on average values of states and inputs.

Decentralised Hierarchical Multi-rate Control of Large-Scale Drinking Water Networks

“…Recently, Sampathirao et al [9] proposed a control framework for large-scale DWNs where pumping actions are computed by minimising a cost index. Such approaches are in the spirit of economic MPC [10], and, despite the fact that are proven to lead to improved closed-loop behaviour, may fail to guarantee the satisfaction of state constraints in closed loop. The proposed methodology allows the operator to command reference signals to the sub-systems of the network according to some cost-optimisation strategy in such a way so as to satisfy the constraints during controlled operation.…”

“…Note that, except for the last consequence of the theorem, no convergence of the system's trajectories to some constant value is assumed or required. This suggests that a purely cost-driven approach can be applied where the system's trajectories move in an oscillatory manner leading to an economically profitable performance determined by the optimisation of a performance criterion in a receding horizon fashion [9,10].…”

Safeness Index-Based Economic Model Predictive Control of Stochastic Nonlinear Systems

“…Motivated by this, recently, a new class of economic model predictive control systems (EMPC), in which the cost function penalizes process economics instead of the distances from the steady-state in a general quadratic form, was utilized to account for process operational safety and economic optimality based on a function called the Safeness Index [2,3]. These new EMPC methods complement previous efforts on economic model predictive control (e.g., [4][5][6][7]), which were not concerned explicitly with process operational safety. Specifically, in [2], a Safeness Index function that indicates the level of safety of a given state was utilized to characterize a safe operating region and used as a constraint in the EMPC design such that the closed-loop state of a nonlinear process is guaranteed to be driven into the safe operating region in finite time in the presence of sufficiently small bounded disturbances and, if the Safeness Index takes a special form related to a Lyapunov function used in the EMPC design, to never again exit that safe operating region while maximizing the economics of the process.…”

On the Relation Between Turnpike Properties for Finite and Infinite Horizon Optimal Control Problems

“…1 Moreover, the turnpike property can also be rigorously established in control systems governed by partial differential equations [11], significantly enlarging the classes of systems for which these methods are applicable. In the context of economic model predictive control, strict dissipativity can be used as an alternative to assuming the turnpike property [2]. However, as [8] shows, these two properties are almost equivalent, in the sense that under a controllability and a reachability condition, strict dissipativity holds if and only if a robust version of the turnpike property holds.…”