Adaptive Economic Model Predictive Control: Performance Guarantees for Nonlinear Systems

Abstract

We consider the problem of optimizing the economic performance of nonlinear constrained systems subject to uncertain time-varying parameters and bounded disturbances. In particular, we propose an adaptive economic model predictive control (MPC) framework that: (i) directly minimizes transient economic costs, (ii) addresses parametric uncertainty through online model adaptation, (iii) determines optimal setpoints online, and (iv) ensures robustness by using a tube-based approach. The proposed design ensures recursive feasibility, robust constraint satisfaction, and a transient performance bound. In case the disturbances have a finite energy and the parameter variations have a finite path length, the asymptotic average performance is (approximately) not worse than the performance obtained when operating at the best reachable steady-state. We highlight performance benefits in a numerical example involving a chemical reactor with unknown time-invariant and time-varying parameters.

Figures (4)

• Figure 1: We minimize the economic cost and adapt the prediction model during online operation. Utilizing existing robust MPC designs guarantees constraint satisfaction and thanks to artificial references, optimize setpoints online, leveraging the adapted model.
• Figure 2: For each $x\in \mathbb{X}_{k}$, we can predict future states for different disturbances $d\in \mathbb{D}$ and parameters $\theta\in\Theta$. All of these predictions are contained in the robust one-step reachable set $\Phi(\mathbb{X}_{k}, z_{k}, v_{k})$.
• Figure 3: Performance comparison of AE-MPC and E-MPC (no adaptation). First plot: Closed-loop evolution of the state $[x]_2$. Second plot: Averaged transient performance for AE-MPC (blue, solid), E-MPC (no adaptation) with constant parameter estimate $\hat{\theta}_0$ (red, dashed), E-MPC (known parameters) with knowledge of $\theta_k$ (green, dotted), and optimal steady-state cost for $\theta_k$ (grey, dash-dotted). Third and fourth plot: Evolution of the parameter estimates of AE-MPC. The LMS is activated at time step $200$ (vertical grey dashed line).
• Figure 4: Performance comparison of AE-MPC and E-MPC for time-varying parameters $\theta_k$. First plot: Averaged transient performance for AE-MPC (blue, solid), E-MPC (no adaptation) with constant $\hat{\theta}_0$ (red, dashed), and E-MPC (known parameters) with knowledge of $\theta_k$ (green, dotted). Second and third plot: Evolution of the parameter estimates of AE-MPC. The LMS is activated at time step $200$ (vertical grey dashed line).

Theorems & Definitions (8)

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