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Provably Safe and Robust Learning-Based Model Predictive Control
Anil Aswani, Humberto Gonzalez, S. Shankar Sastry, Claire Tomlin
TL;DR
The paper addresses how to improve control performance without sacrificing deterministic safety and robustness. It introduces LBMPC, which optimizes a statistically learned model while checking the same actions against an uncertainty-bounded nominal model. The framework supports constrained control, multiple statistical identification tools, and probabilistic convergence to true-dynamics MPC under sufficient excitation.
Problem
Control design must reconcile robustness from approximate models with performance from accurate models, while statistical learning alone does not guarantee safety or stability.
Method
LBMPC maintains an uncertainty-bounded nominal model for robust safety checks and a statistically updated learned model for cost-based performance optimization.
Results
LBMPC improved real-system performance, including up to 30% energy savings on warm days and up to 70% on cooler days at the BRITE testbed.
Takeaways & Limitations
LBMPC provides deterministic robustness and safety guarantees while accommodating parametric or nonparametric learning tools and richer system models.
Takeaways & Limitations
The convergence condition for complex nonparametric regression is sufficient rather than necessary, so tools failing it may still yield stochastic controller convergence.
Abstract
from arXiv · showhide
Controller design faces a trade-off between robustness and performance, and the reliability of linear controllers has caused many practitioners to focus on the former. However, there is renewed interest in improving system performance to deal with growing energy constraints. This paper describes a learning-based model predictive control (LBMPC) scheme that provides deterministic guarantees on robustness, while statistical identification tools are used to identify richer models of the system in order to improve performance; the benefits of this framework are that it handles state and input constraints, optimizes system performance with respect to a cost function, and can be designed to use a wide variety of parametric or nonparametric statistical tools. The main insight of LBMPC is that safety and performance can be decoupled under reasonable conditions in an optimization framework by maintaining two models of the system. The first is an approximate model with bounds on its uncertainty, and the second model is updated by statistical methods. LBMPC improves performance by choosing inputs that minimize a cost subject to the learned dynamics, and it ensures safety and robustness by checking whether these same inputs keep the approximate model stable when it is subject to uncertainty. Furthermore, we show that if the system is sufficiently excited, then the LBMPC control action probabilistically converges to that of an MPC computed using the true dynamics.
1 Introduction
The paper addresses the robustness–performance trade-off by combining statistical model learning with deterministic safety guarantees in learning-based MPC. LBMPC maintains separate models so learned dynamics improve performance while uncertainty-bounded dynamics support robustness, constraints, and convergence.
- Controller design trades off robustness, which approximate models support, against performance, which requires accurate models.
- The paper targets constrained control that optimizes a cost, learns model uncertainties statistically, and provably converges.
- Statistical methods converge probabilistically, so combining learned uncertainty models with deterministic safety guarantees is a central challenge.
- LBMPC minimizes cost using a statistically updated learned model while robust MPC checks the same inputs against an uncertainty-bounded nominal model.
- LBMPC differs from related adaptive MPC by using statistical model updates, tolerating poor updates, and separating performance learning from robustness certification.
2 Preliminaries
The preliminaries define the system variables, constrained state-space setting, modeling error, and mathematical tools used to analyze LBMPC. The setup assumes bounded modeling error and focuses on fully measured states without measurement noise, while noting extensions.
- The preliminaries introduce Lyapunov stability, Minkowski and Pontryagin set operations, and asymptotic-probability notation for later analysis.
- The paper uses x for state, u for control input, and y for output, with states and inputs constrained by polytopes X and U.
- The true dynamics are represented by a nominal linear model plus unmodeled dynamics g(x, u), treated as modeling error.
- Modeling error is assumed to lie in a bounded polytope W over the constrained state–input domain.
- The analysis focuses on fully measured states and assumes no measurement noise, while allowing set-theoretic estimation and W ⊕ D extensions in principle.
3 Learning-Based MPC
LBMPC decouples performance from robustness by optimizing a learned model while certifying feasibility and safety with an uncertainty-bounded nominal model. Reachability-based invariant sets yield deterministic guarantees, and stability results hold under explicit assumptions on costs, dynamics, and oracle error.
- 3 Learning-Based MPC: LBMPC maintains a nominal linear model with disturbance and a learned model updated by an oracle, separating robustness certification from performance optimization.
- 3 Learning-Based MPC: The terminal set Ω is a maximal output-admissible disturbance-invariant set that preserves state and input constraints under bounded disturbances.
- 3 Learning-Based MPC: Tube MPC guarantees that the true trajectory remains within a feedback-controlled tube around the nominal trajectory.
- 3.2 Stability and Safety of LBMPC: Using the learned model for cost evaluation leaves LBMPC stability and robustness independent of the specific cost terms, thereby decoupling safety from performance.
- 3.2 Stability and Safety of LBMPC: Robust feasibility and constraint satisfaction imply closed-loop stability, and feasible initialization guarantees stability, constraints, and feasibility for all future times.
- 3.2 Stability and Safety of LBMPC: These guarantees remain robust under disturbances when modeling error respects its prescribed bound and the invariant set Ω can be computed.
- 3.2 Stability and Safety of LBMPC: Under continuity, strict convexity, a suitable Lyapunov function, and bounded continuous oracle error, the LBMPC control law is robustly asymptotically stable.
- 3.2.1 Example: Tracking in Linearized Systems: A quadratic tracking cost with positive definite matrices satisfies the Lyapunov requirement when the feedback matrix is Schur stable and P solves the discrete-time Lyapunov equation.
4 The Oracle
The oracle lets LBMPC incorporate statistical models while preserving robustness through boundedness and differentiability requirements. Parametric and nonparametric oracles support model learning, and sufficient conditions yield convergence toward MPC using the true dynamics.
- Oracle framework: Oracles are abstract functions whose continuity or boundedness suffices for proving LBMPC robustness and stability, regardless of their internal construction.They can therefore include arbitrary or worst-case behavior in the preceding theorems.
- Parametric oracles: Parametric oracles represent learned dynamics as χ(x, u; λn), with coefficients estimated from input and trajectory data, although nonlinear estimation can be computationally difficult.Least-squares estimation is generally a nonlinear optimization problem in the general parametric case.
- Parametric oracles: When an oracle is linear in its coefficients, least-squares estimation is convex even with nonlinear basis functions, simplifying computation.This subclass has the form On(x, u) = P_i λn,iχi(x, u).
- Nonparametric oracles: Nonparametric methods avoid prior assumptions about function form, and LBMPC permits their use because robustness depends on computing a bounded uncertainty set.Neural networks are one example and can be used with LBMPC for learned nonlinear dynamics.
- Nonparametric oracles: The modified Nadaraya-Watson estimator L2NW uses regularization to ensure both boundedness and differentiability, supporting robustness and numerical optimization.Theorem 3 establishes these properties when the kernel conditions hold and λ > 0.
- Convergence: Under sufficient excitation, learned-oracle LBMPC converges in probability to MPC with the true model; for L2NW, this requires shrinking bandwidth, λ = O(hn), and Lipschitz dynamics.For broader nonparametric tools, the stated convergence condition is sufficient but not necessary.
5 Experimental and Numerical Results
Experiments and simulations applied LBMPC to HVAC systems, quadrotor flight, and a nonlinear compressor model, demonstrating improved performance alongside robustness and constraint guarantees.
- Energy-efficient Building Automation: LBMPC achieved up to 30% energy savings on warm days and up to 70% on cooler days in the BRITE single-room HVAC testbed.Semiparametric regression estimated exogenous heating loads from thermostat temperature measurements, informing HVAC control adjustments.
- Energy-efficient Building Automation: LBMPC achieved an average of 1.5MWh of energy savings per day in the seven-floor BRITE-S office-building testbed.
- High Performance Quadrotor Helicopter Flight: Quadrotor experiments showed lower overshoot and settling time than linear MPC, while overcoming ground-effect difficulties near the ground.An extended Kalman filter corrected A and B matrix coefficients during learning.
- High Performance Quadrotor Helicopter Flight: LBMPC prevented ground crashes even when the EKF was deliberately made unstable and mis-learned.An integrated experiment also demonstrated learning generalization by enabling the quadrotor to catch ping-pong balls thrown by a human.
- Moore-Greitzer Compressor Model: The compressor example satisfied the theorem assumptions, maintaining robust constraints and feasibility, closed-loop stability, and input-to-state stability with respect to modeling error.The controller used a linearization-based approximate model, bounded modeling error, and invariant-set computation.
- Moore-Greitzer Compressor Model: In the compressor simulation, LBMPC converged faster than linear MPC but required 0.9s per step versus 0.3s for linear MPC.LBMPC performed as well as nonlinear MPC, which required 0.4s per step, without requiring the engineer to model nonlinearities.
6 Conclusion
LBMPC combines bounded-uncertainty linear models with statistical learning to provide deterministic robustness and safety while improving performance. The paper reports simulation and testbed evidence of improvement, while identifying better globally regularized learning methods as an open problem.
- LBMPC constructs invariant sets from a linear model with bounded uncertainty, providing deterministic guarantees on robustness and safety.
- The framework accommodates many statistical identification tools and includes a nonparametric estimator with properties needed for optimization and robustness.
- A simulation improved over linear MPC, and experiments on HVAC and quadrotor testbeds transferred that improvement to real systems.
- Designing globally regularized nonparametric methods with theoretical properties favorable for LBMPC remains an open problem.The paper notes that such methods can outperform the strictly local L2NW estimator.