Economic MPC

The core feature of EMPC is its cost function, which directly encodes a process's economic performance metric. Unlike traditional MPC that minimizes tracking error (e.g., (y - y_ref)²), EMPC minimizes a function like operating cost or maximizes production profit.

• Example: For a chemical reactor, the cost could be J = cost(feedstock) + cost(utilities) - value(product).
• This shifts the controller's goal from 'stay here' to 'operate most profitably,' often leading to operation at constraint boundaries.

EMPC naturally drives the system to its economically optimal steady-state (EOSS), which is typically not a pre-defined setpoint but is dynamically calculated. The EOSS is the operating point that maximizes profit or minimizes cost given current prices, constraints, and disturbances.

• The controller solves a real-time optimization (RTO) problem implicitly within its horizon.
• It can handle time-varying economics, such as changing electricity prices, by continuously re-computing the optimal target.

EMPC explicitly manages hard and soft constraints (e.g., tank levels, pressure limits, actuator saturation) while pursuing economic goals. This is critical because the most profitable operating point often lies at the intersection of multiple constraints.

• Constraint satisfaction is guaranteed over the prediction horizon.
• The controller can temporarily violate soft constraints (e.g., a storage tank level) if it yields a significant economic benefit, paying a penalty in the cost function.

A key advantage is the seamless integration of dynamic economic signals into the control law. This allows the system to proactively adjust its behavior based on forecasts.

• Primary Example: Demand Response in power systems, where an EMPC controller for a building HVAC or industrial process shifts energy consumption to avoid high electricity price periods.
• The cost function incorporates a forecasted price vector λ(k) over the horizon: J = Σ λ(k) * P_consumed(k).

EMPC is not limited to driving systems to a steady-state. It can optimally manage transients and enforce periodic operation if it is economically beneficial.

• Example: In batch or cyclic processes (e.g., refrigeration cycles, pressure swing adsorption), the most economic mode is a repeating cycle, not a steady state. EMPC can optimize the entire periodic trajectory.
• The controller computes the optimal path between states, considering the trade-off between transition speed (energy cost) and economic gain.

Guaranteeing closed-loop stability is more complex than in tracking MPC because the economic cost does not inherently penalize deviation from a fixed point. Stability is typically enforced using terminal ingredients.

• Common Methods: Adding a terminal cost (often a Lyapunov function) or a terminal constraint that forces the final predicted state into a stabilizing set.
• Average Constraint and Lyapunov-like techniques are also used to ensure that the infinite-horizon economic performance is bounded and satisfactory.

The Optimal Control Problem (OCP) is the core mathematical formulation solved at each step of an MPC controller. It defines the optimization over a future time horizon, comprising:

• A dynamic model (e.g., differential equations) predicting state evolution.
• A cost function (objective function) to be minimized (e.g., tracking error, economic cost).
• Constraints on system states, inputs, and outputs. For Economic MPC, the OCP's cost function directly encodes economic metrics like profit or energy cost, rather than purely technical setpoint tracking.

Real-Time Optimization (RTO) is a higher-level, slower-time-scale process that calculates optimal steady-state operating points or setpoints for a process plant, typically to maximize profit or minimize cost. Economic MPC acts as the dynamic, fast-time-scale layer that drives the system to these RTO-computed targets while respecting dynamic constraints. The integration of Economic MPC with RTO creates a two-layer hierarchy for plant-wide economic optimization.

Lyapunov-Based Economic MPC is a design approach that incorporates a Lyapunov function (an energy-like function) into the controller formulation to provide rigorous guarantees of closed-loop stability and average performance. Since a purely economic cost can lead to undesirable steady-state oscillations, this method adds a strictly convex technical cost (weighted by the Lyapunov function's gradient) to the economic objective. This ensures the system is driven to a economically optimal, stable equilibrium.

Dissipativity Theory provides a fundamental framework for analyzing the stability of Economic MPC without requiring a strictly convex technical cost. A system is dissipative with respect to its supply rate (the economic cost) if there exists a storage function showing that stored energy is bounded by what is supplied. In Economic MPC, if the system is strictly dissipative at the optimal steady state, it guarantees that the closed-loop system converges to this economically optimal point, even with a non-convex economic objective.

Periodic Economic MPC is a variant designed for systems where the economically optimal operation is inherently cyclic or time-varying, not a fixed steady state. Examples include batch processes, energy systems with daily price fluctuations, or seasonal production. The controller's prediction horizon and cost function are designed to optimize over one or more periods of the known cycle, explicitly handling the periodic nature of constraints and economic objectives to find the optimal repeating trajectory.

Stochastic Economic MPC extends the framework to explicitly account for uncertainties in economic parameters, such as fluctuating energy prices, variable raw material costs, or uncertain product demand. Instead of using deterministic forecasts, it optimizes the expected value of the economic cost (or a risk measure like Conditional Value-at-Risk) over a distribution of possible future scenarios. This leads to control policies that are robust to market volatility while maximizing expected profit.