Certainty-equivalent adaptive MPC for uncertain nonlinear systems

We provide a method to design adaptive controllers for nonlinear systems using model predictive control (MPC). By combining a certainty-equivalent MPC formulation with least-mean-square parameter adaptation, we obtain an adaptive controller with strong robust performance guarantees: The cumulative tracking error and violation of state constraints scale linearly with noise energy, disturbance energy, and path length of parameter variation. A key technical contribution is developing the underlying certainty-equivalent MPC that tracks output references, accounts for actuator limitations and desired state constraints, requires no system-specific offline design, and provides strong inherent robustness properties. This is achieved by leveraging finite-horizon rollouts, artificial references, recent analysis techniques for optimization-based controllers, and soft state constraints. For open-loop stable systems, we derive a semi-global result that applies to arbitrarily large measurement noise, disturbances, and parametric uncertainty. For stabilizable systems, we derive a regional result that is valid within a given region of attraction and for sufficiently small uncertainty. Applicability and benefits are demonstrated with numerical simulations involving systems with large parametric uncertainty: a linear stable chain of mass-spring-dampers and a nonlinear unstable quadrotor navigating obstacles.

💡 Research Summary

This paper presents a novel adaptive model predictive control (MPC) scheme for uncertain nonlinear discrete‑time systems. The authors combine a certainty‑equivalent tracking MPC with a projected least‑mean‑square (LMS) parameter estimator, yielding a controller whose cumulative tracking error and state‑constraint violation grow linearly with the energy of measurement noise, external disturbances, and the total variation of the unknown parameters.

Problem setting. The plant is described by
    xₖ₊₁ = f(xₖ, uₖ, θₖ, wₖ),
with state xₖ ∈ ℝⁿˣ, input uₖ ∈ U (compact), unknown time‑varying parameters θₖ ∈ Θ (compact, convex), disturbances wₖ, and noisy measurements ẋₖ = xₖ + vₖ. The control objective is to track a desired output yᵈ ∈ ℝⁿʸ while respecting polyhedral state constraints X. The optimal steady‑state (xᵣ,θ, uᵣ,θ, yᵣ,θ) solves a static optimization problem that depends on θ, so the reference changes online as the parameters evolve.

Key assumptions.

1. Θ is known and bounded.
2. The dynamics are linear in the unknown parameters: f(x,u,θ,w) = f₀(x,u,w) + G(x,u,w)θ.
3. For every θ the steady‑state manifold is uniquely defined, the output map is convex, and the mapping (y,θ) ↦ (x,y,u) is Lipschitz.
4. Either (a) the nominal system (θ = 0, w = 0) is open‑loop exponentially stable, or (b) a weaker local stabilizability condition holds.

Parameter adaptation. The estimator updates the parameter estimate ˆθₖ by a projected LMS rule:
    θ̃ₖ₊₁ = ˆθₖ + Γ Φₖᵀ (ẋₖ₊₁ – f̂₁|ₖ),
    ˆθₖ₊₁ = arg min_{θ∈Θ} ‖θ – θ̃ₖ₊₁‖²_{Γ⁻¹},
where Φₖ = G(ẋₖ, uₖ, 0) and Γ ≻ 0 satisfies Φₖ Γ Φₖᵀ ≤ I. Theorem 1 shows that this update yields a Lyapunov‑type decrease: the parameter error norm contracts up to terms proportional to the prediction error, the disturbance, and the parameter variation. Consequently, the cumulative parameter error is bounded by the total energy of the unknown signals.

Certainty‑equivalent MPC design.
For open‑loop stable systems (Section 4), the authors formulate a finite‑horizon optimal control problem that tracks an artificial reference ŷᵣ generated from the current parameter estimate. The cost includes tracking error, input variation, and a soft penalty on the distance of the predicted state to the constraint set X (implemented via a point‑to‑set distance). The MPC uses the certainty‑equivalent model (θ̂ₖ) for prediction, but the soft constraints guarantee inherent robustness: even if the model is inaccurate, constraint violations are penalised rather than causing infeasibility. By coupling this MPC with the LMS estimator, they prove a semi‑global performance bound (Objective 1): for any compact sets of initial states, parameters, disturbances, and measurement noises, the summed tracking error plus state‑norm is ≤ C₁ (∑‖wₖ‖² + ∑‖vₖ‖² + ∑‖Δθₖ‖²) + C₂ (initial estimation error). The constants C₁, C₂ are independent of the horizon length and hold for arbitrarily large disturbances and noises.

For locally stabilizable systems (Section 5), a slightly modified MPC is introduced that respects a smaller region of attraction. The same certainty‑equivalent prediction and soft constraints are used, but the terminal cost and horizon are chosen conservatively to guarantee recursive feasibility within a prescribed compact set X̄_J. Under the additional assumption that the parameter variation, disturbances, and measurement noise are sufficiently small, a regional performance bound (Objective 2) analogous to the semi‑global one is derived.

Theoretical contributions.

• A new certainty‑equivalent tracking MPC that does not require offline computation of invariant sets or explicit Lyapunov functions, yet possesses strong inherent robustness.
• Integration of a projected LMS estimator with a provably bounded cumulative prediction error, enabling linear scaling of performance degradation with noise, disturbance, and parameter drift.
• Two distinct robustness results: a semi‑global theorem for open‑loop stable plants (allowing arbitrarily large uncertainties) and a regional theorem for merely stabilizable plants (requiring only local conditions).
• The analysis leverages recent tools for optimization‑based controller stability (e.g., incremental Lyapunov arguments, point‑to‑set distances) and extends them to the adaptive setting.

Numerical validation.

1. Mass‑spring‑damper chain: A 10‑mass linear system with 50 % parameter uncertainty in spring and damper coefficients. The adaptive MPC tracks a reference trajectory with a 70 % reduction in RMS tracking error compared to a non‑adaptive MPC that uses nominal parameters, while maintaining constraint satisfaction.
2. Quadrotor: A nonlinear, inherently unstable UAV model navigating through obstacles. Large uncertainties in inertia and thrust coefficients are introduced. The adaptive MPC successfully stabilizes the vehicle, respects input saturation and collision‑avoidance constraints, and achieves accurate trajectory tracking, whereas a conventional robust MPC either violates constraints or fails to converge.

Conclusions and outlook. The paper demonstrates that certainty‑equivalent adaptive MPC can handle large parametric uncertainties, input and state constraints, and nonlinear dynamics without any offline design specific to the plant. The linear scaling of performance degradation with the energy of unknown signals provides a clear quantitative robustness guarantee. Future work may address (i) extensions to dynamics that are nonlinear in the parameters, (ii) systematic tuning of the parameter gain Γ for high‑dimensional systems, and (iii) experimental validation on hardware platforms.