Certainty-equivalent adaptive MPC for uncertain nonlinear systems

Abstract: 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.

• The paper develops a certainty-equivalent adaptive MPC method that combines projected LMS parameter estimation, artificial setpoints, finite-tail terminal costs, and soft state constraints without requiring offline Lyapunov or invariant-set design.
• The theoretical results bound cumulative tracking error and constraint violations by measurement-noise energy, disturbance energy, parameter-variation path length, and initial estimation error, with semi-global guarantees for stable systems and regional guarantees for locally stabilizable systems.
• Numerical studies show robust performance under ±50% uncertainty and 420 uncertain parameters, while ablation results demonstrate that adaptation and finite-tail terminal costs substantially reduce tracking error, constraint violations, oscillations, and convergence time.

Problem setting and main result

This paper, by Johannes Kohler, develops an adaptive model predictive control (MPC) scheme for nonlinear discrete-time systems with large parametric uncertainty. The plant is described by $x_{k+1}=f(x_k,u_k,\theta_k,w_k)$ with linearly parametrized uncertainty in $\theta_k$, polytopic state constraints $\mathbb{X}$, compact input constraints $\mathbb{U}$, noisy measurements $\hat{x}_k=x_k+v_k$, and a desired output target $y_{\mathrm{d}}$. The optimal setpoint $$(x_{\mathrm{rd},\theta},u_{\mathrm{rd},\theta},y_{\mathrm{rd},\theta})$$ depends on the unknown parameters and must therefore be recomputed online.

The central guarantee is a cumulative performance bound: the sum of squared output tracking errors plus squared state constraint violations over any horizon is bounded by constants times the accumulated noise energy, disturbance energy, and path length of parameter variation, plus a transient term involving the initial parameter error. This is stated as two objectives: a semi-global result for open-loop exponentially stable systems, valid for arbitrarily large noise, disturbances, and parameter sets; and a regional result for merely locally stabilizable (e.g., unstable) systems, valid within a sublevel-set region of attraction for sufficiently small uncertainty. The paper's key claim is that these guarantees are obtained without any system-specific offline design — no terminal CLF, no robust positively invariant sets, no common Lyapunov function over the parameter set.

Parameter adaptation via projected LMS

The adaptation is a projected least-mean-square update: the estimate is corrected by the gain $\Gamma$ times the regressor transpose applied to the one-step prediction error, followed by a weighted projection onto the known compact parameter set $\Theta$. The parameter gain must satisfy $\hat{\Phi}_k\Gamma\hat{\Phi}_k^\top\preceq I$ for all time. Theorem 1 establishes three properties of this update: a Lyapunov-like decrease of the weighted parameter error by the squared prediction error due to parametric mismatch, a bound on the per-step estimate change, and a bound on the prediction error attributable to disturbances and measurement noise. These results extend prior LMS analyses to general gains, noisy measurements, and time-varying parameters. The consequence is that prediction error due to parametric mismatch decays whenever disturbances, noise, and parameter variation decay — the property that later converts MPC robustness into adaptive performance guarantees.

Certainty-equivalent tracking MPC design

The controller solves a single optimization problem combining four ingredients:

• Artificial setpoints: the steady-state triple $\theta_k$0 is a decision variable, with a quadratic offset cost $\theta_k$1, following the MPC-for-tracking literature.
• Finite-tail terminal cost: instead of an offline CLF, the terminal penalty is the stage cost accumulated along an $\theta_k$2-step rollout under the steady-state input (stable case) or a local stabilizing feedback $\theta_k$3 (unstable case).
• Soft state constraints: constraint violations enter the stage cost through quadratic penalties, which avoids recursive feasibility issues inherent to certainty-equivalent schemes.
• No offline design: stability is enforced implicitly by choosing the rollout horizon $\theta_k$4, weight $\theta_k$5, and horizon $\theta_k$6 so that a computable condition holds.

For open-loop stable systems, Propositions 1–3 derive a quadratic bound on the rollout stage cost, an exponential cost controllability bound $\theta_k$7, and an approximate-CLF property of the finite-tail cost with error $\theta_k$8 given in closed form. Theorem 2 then proves a strict decrease condition on the value function with uniform constant $\theta_k$9: notably, once $\mathbb{X}$0 and $\mathbb{X}$1, any horizon $\mathbb{X}$2 yields stability. Corollary 1 combines this with a bound relating the artificial setpoint to the optimal reachable one, establishing exponential stability of $\mathbb{X}$3 in the nominal closed loop.

Inherent robustness and closed-loop guarantees

Theorem 3 proves that the value function contracts geometrically ($\mathbb{X}$4) despite measurement noise, disturbances, and parameter mismatch — including the fact that the MPC policy may be discontinuous, handled via arguments from robustness analysis of sampled/discontinuous feedback. The proof rests on a continuity bound for the optimal cost under perturbations of state and parameters, enabled precisely by the soft state constraints.

Combining this contraction with the LMS properties via a composite Lyapunov function $\mathbb{X}$5 yields Theorem 4: the semi-global objective holds with explicit constants. A corollary shows that with zero noise/disturbance and constant parameters, the cumulative tracking error and constraint violation remain bounded despite arbitrarily large initial parameter error — i.e., exact asymptotic convergence to the optimal feasible setpoint. For unstable systems, Theorems 5–7 mirror this development within the region of attraction $\mathbb{X}$6, requiring sufficiently small mismatch so that the sublevel set remains invariant; enlarging $\mathbb{X}$7 requires a horizon $\mathbb{X}$8 scaling linearly in it.

Discussion and relation to prior work

The paper positions the scheme against two alternatives. Classical nonlinear adaptive control requires offline construction of parametrized CLFs or contraction metrics, and handles constraints poorly. Existing adaptive/robust MPC methods are characterized as robust schemes that use data only to reduce conservatism: they require small uncertainty and typically offline-computed polytopic invariant sets, and mostly address stabilization of a known equilibrium rather than tracking. The proposed method avoids both limitations at the price of allowing transient constraint violations (soft constraints) rather than pointwise-in-time satisfaction — the author notes explicitly that pointwise safety would require reverting to robust MPC designs applicable only under small parametric uncertainty. For linear dynamics, most standing assumptions reduce to standard conditions (Schur stability or stabilizability, a rank condition for unique setpoints), and the online problem becomes a convex QP.

Numerical results

Two examples with large parametric uncertainty support the theory:

System	Uncertainty	Comparison
10-mass spring-damper chain ($\mathbb{X}$9, $\mathbb{U}$0, non-minimum-phase)	±50% deviation, 420 uncertain parameters	Robust adaptive MPC [Lorenzen et al.] could not be implemented even at ±20% because no common Lyapunov function exists
Planar quadrotor ($\mathbb{U}$1, unstable, obstacle avoidance)	Initial estimate off by factor 2 (>50%)	Robust adaptive MPC [Sasfi et al.] became infeasible at 6% uncertainty

In the mass-spring-damper case, normalized cumulative tracking error was 1.00 (proposed), 2.51 (no terminal cost), 1.38 (no adaptation); normalized constraint violation was 1.00, 18.97, and 698.05 respectively. Computation took $\mathbb{U}$2 ms against a 500 ms sampling period. For the quadrotor, the no-adaptation variant diverged; removing the terminal cost slowed convergence roughly fivefold and left oscillations above 10 cm around the target versus approximately 4 mm hovering accuracy for the full scheme, at $\mathbb{U}$3 ms solve time against a 25 ms sampling period. These results indicate that both the finite-tail terminal cost and the adaptation are essential to the observed performance, and that the certainty-equivalent approach tolerates uncertainty levels far beyond the reach of comparable robust adaptive MPC formulations.

Limitations and open questions

Several caveats are conceded directly. State constraints are satisfied only in a soft, cumulative sense; pointwise safety is not guaranteed and would require conservative robust machinery. The regional result restricts uncertainty qualitatively ("sufficiently small") via conservative Lipschitz bounds, without a sharp characterization. The uniform global Lipschitz assumption on the dynamics can be restrictive; relaxing it to local Lipschitz continuity additionally requires sufficiently small parameter variations $\mathbb{U}$4. The quadrotor study does not verify the sufficient conditions of Theorem 7, so its success is empirical rather than certified. Finally, the author notes that robustification of the least-squares update (deadzone, normalization) and experimental validation remain unaddressed.

Conclusion

The paper provides a self-contained adaptive MPC framework whose theoretical guarantees — linear scaling of tracking error and constraint violation with noise, disturbance, and parameter-variation energy — hold for general nonlinear systems under input and soft state constraints, with no offline control design. The combination of artificial setpoints, finite-tail rollouts, soft constraints, and projected LMS adaptation is what makes both the implementation simplicity and the robustness analysis possible. The main open questions concern hard state constraints, quantitative uncertainty margins for the regional result, and experimental reliability.