Robust Model Predictive Control

• Robust Model Predictive Control is an advanced strategy that integrates optimization with robust state estimation to maintain performance under uncertainty.
• It employs techniques like tube-based predictions, min-max formulations, and robust Kalman filtering to ensure constraint satisfaction and stability.
• Tunable parameters such as robustness tolerance balance improved tracking and resilience against model mismatches with increased control action aggressiveness.

Robust Model Predictive Control (MPC) is a class of advanced control strategies that preserve the structure and advantages of nominal Model Predictive Control—particularly constraint handling and multi-variable optimization—while explicitly guaranteeing performance and constraint satisfaction under model uncertainty and external disturbances. The robust MPC framework typically employs techniques to quantify, bound, or compensate for the effect of uncertainties in the plant dynamics or measurement process, merging estimation theory with robust optimal control. Distinct architectures such as robust state estimation, min-max optimization, tube-based predictions, and adaptive or learning-enabled approaches are prevalent, with formal performance and feasibility guarantees provided under various modeling assumptions.

1. Foundational Principles and Architectures

The foundational paradigm in robust MPC is to ensure robust constraint satisfaction and closed-loop stability despite plant-model mismatch, noise, and disturbances. Classical MPC relies on the assumption that the state estimate, usually obtained through a standard Kalman filter, accurately reflects the true system state. When this assumption is violated by uncertainties, state estimation errors may lead to constraint violations and performance degradation. Robust MPC architectures address this challenge in several ways:

• Decoupled Robust State Estimation: The state estimator is designed to be robust against model and noise uncertainty, commonly via minimax or risk-sensitive criteria. This robust estimate is then passed to the MPC optimizer, while the optimization structure itself remains entirely nominal (Zenere et al., 2017, Zenere et al., 2018).
• Modified Estimation-Driven Tube-Based MPC: The robust estimator determines a bound or set within which the real state must reside; the MPC then optimizes using this set by tightening constraints or planning over "tubes" to absorb uncertainty. This preserves constraint satisfaction for all plausible realizations of model and measurement error (Köhler et al., 2021).
• Direct Min-Max and Risk-Sensitive Formulations: The MPC finite-horizon optimization itself is modified, either by taking a worst-case (min-max) cost over the admissible uncertainty set or by incorporating a risk-sensitive cost. This approach directly incorporates robustness into the optimal action computation and may, but need not, leverage robust estimation (Parsi et al., 2019, Zhong et al., 2022).
• System Level Synthesis (SLS) and Learning-Integrated Methods: SLS-based robust MPC and data-driven/learning-augmented methods address high-dimensional or adaptive scenarios by parameterizing closed-loop responses and seeking optimal controllers joint with uncertainty bounds, or by integrating model learning (e.g., Gaussian processes or Koopman linearization) with robust constraint handling (Chen et al., 2021, Dubied et al., 2 Jul 2025, Zhang et al., 2019).

2. Robust State Estimation in MPC: The Robust Kalman Filter

A central theme is the integration of robust state estimators—especially the robust Kalman filter—into the MPC feedback loop. This approach distinctly separates the estimation and control problems, allowing each to be robustified independently:

• Robust estimation is posed as a minimax problem: find a state estimator that minimizes the worst-case mean squared estimation error over all predictive distributions $f̃_t$ within a Kullback–Leibler divergence ball of radius $c$ around the nominal model $f_t$. Explicitly:

$$\hat{x}_{t+1|t} = \arg\min_{g_t \in \mathcal{G}_t} \max_{f̃_t \in \mathcal{S}_t} E_{f̃_t}[ \| x_{t+1} - g_t(y_t) \|^2 | Y_{t-1} ]$$

where $$\mathcal{S}_t = \{ f̃_t \ | \mathcal{D}(f̃_t\,||\,f_t) \leq c \}$$ (Zenere et al., 2017, Zenere et al., 2018).

• The recursion for the robust Kalman filter resembles a standard Kalman filter but with modified gains and covariance,

$L_t = V_t C^\top (C V_t C^\top + D D^\top )^{-1}$

$V_t = (P_t^{-1} - \theta_t I)^{-1}$

with $\theta_t$ found by solving an implicit equation involving the entropy ball radius $c$:

$$-\log\det((I - \theta_t P_t)^{-1}) + \operatorname{tr}\left[(I - \theta_t P_t)^{-1} - I\right] = c$$

As $c$ increases, the estimator is increasingly conservative, and the controller is more robust but applies larger control actions.

• Integration with MPC occurs by simply feeding the robust estimate into the same open-loop quadratic optimization as the nominal controller, maintaining the core MPC structure and computational tractability.

3. Uncertainty Quantification and Tuning

Robust MPC methods provide explicit tuning parameters to mediate the trade-off between robustness and performance:

• The robustness tolerance $c$ directly determines the size of the uncertainty set considered, acting as a knob to interpolate between nominal and robust performance.
• The cost weights $Q_k$, $R_k$ in the MPC stage cost remain essential as in standard MPC for achieving a desired compromise between reference-tracking and control effort.
• In min-max or distributionally robust formulations, analogous tuning parameters include the size of the uncertainty set (e.g., via Wasserstein ball radius), the risk-aversion weight (in risk-sensitive costs), or explicit constraints on worst-case deviation.

A critical insight is that increased robustness (e.g., larger $c$) yields improved tracking and resilience under plant-model mismatch but at the cost of more aggressive, less smooth control and possibly elevated actuation energy (Zenere et al., 2017).

4. Theoretical Guarantees: Feasibility, Stability, and Performance

Robust MPC strategies typically ensure the following properties:

• Recursive Feasibility: The designed controller guarantees that, if the robust MPC problem is feasible at an initial time, it remains feasible at every subsequent step under bounded uncertainties. This requires appropriate design of invariant or contractive terminal sets (robust positive invariance), possibly using adjustable or set-based invariance constructs (Kim et al., 2018, Köhler et al., 2021).
• Robust Stability: By leveraging the separation into robust estimation and control, the closed-loop system is shown to be input-to-state stable (ISS) with respect to the disturbances and uncertainty, with a Lyapunov or dissipation argument often carried through the ISS property of the (possibly adaptive) value function (Chen et al., 2021).
• Constraint Satisfaction: Robust MPC methods ensure that all input and state constraints are satisfied robustly—that is, the actual trajectory remains within the admissible set for all uncertainties within the pre-specified set (e.g., Kullback–Leibler ball) and for all realizations of bounded disturbances.

5. Empirical Performance and Practical Considerations

Empirical results on nonlinear plant control—specifically a servomechanism with nonlinear friction—indicate that robust MPC integrating a robust Kalman filter (with $c = 10^{-1}$ or $c = 1$) achieves substantially improved tracking under model mismatch compared to standard MPC (with a nominal Kalman filter) (Zenere et al., 2017, Zenere et al., 2018):

Scenario	S-MPC (std. Kalman)	R-MPC1 ($c=10^{-1}$)	R-MPC2 ($c=1$)
Nominal Model	Accurate/smooth input	Accurate/less smooth	Accurate/less smooth
Model Mismatch	Persistent tracking error	Improved accuracy	Best accuracy; more energetic/less smooth

• Increasing $c$ yields higher accuracy/tracking under uncertainty but increases the magnitude and variability of control inputs.
• Design is modular: the robust estimator and nominal optimizer are implemented independently with minimal change to system architecture.
• The computational burden of solving for $\theta_t$ (defining the robustified covariance) at each step is negligible compared to the benefit—no need for complex min-max optimization inside the control loop.
• Monte Carlo simulations further validate that robust MPC consistently leads to reduced tracking errors and improved worst-case performance when exposed to plant-model mismatch and nonlinear externalities.

6. Extensions, Limitations, and Research Impact

The decoupled robust estimation plus MPC optimization paradigm exemplified here has been influential in shaping robust and stochastic predictive control. Notable extensions and ongoing research include:

• Development of efficient algorithms for large-scale and distributed systems using similar decoupled or modular robust estimation-control decomposition.
• Alternative robust estimation techniques (e.g., H-infinity filtering, set-based observers) and distributionally robust control approaches (e.g., Wasserstein ball ambiguity sets) adopting related modular architectures.
• Continued analysis of the trade-off between robustness and performance/conservatism, especially in disturbance-rich, highly uncertain settings.
• New methods for adaptive and learning-based robust MPC, employing reinforcement learning and model set adaptation integrated with robust optimization principles.
• Application to domains such as robotics, autonomous vehicles, and process control, especially where model uncertainty and external noise are substantial relative to controller bandwidth.

While robust MPC methods of this form guarantee strong resilience and safety properties under uncertainty, the main limitation is the potential for conservatism and degraded smoothness when the robustness parameter is set excessively high. Thus, careful calibration to domain-specific uncertainty scales remains essential.

7. Summary Table: Robust Kalman Filter Enhanced MPC Workflow

Component	Robustification	Parameter/Knob	Effect
State Estimator	KL divergence ball	$c$	Higher $c$: more conservative estimate
Cost Optimization	Standard MPC QP	$Q_k$, $R_k$	Standard trade-offs
Robustness-Performance	Tuning $c$	$c$	Improved robustness, higher control input magnitude, less smoothness
Feasibility/Stability	Recursive feasibility, input-to-state stability	Terminal set/weights	Robust constraint satisfaction

The robust Kalman filter-MPC integration delivers an effective, low-complexity means of improving resilience to modeling errors and disturbances for high-performance, constraint-satisfying control across a range of real-world applications (Zenere et al., 2017, Zenere et al., 2018).