Dual Adaptive MPC in Uncertain Systems

• Dual adaptive MPC is an advanced control framework that integrates regulation with active system identification by selecting inputs that improve future uncertainty reduction.
• It employs techniques like set-membership identification and tube-based robust MPC to balance exploration and exploitation for optimal performance.
• Empirical studies show that dual adaptive MPC can reduce tracking error by up to 30% while ensuring recursive feasibility and robust constraint satisfaction.

Dual adaptive MPC is a class of adaptive model predictive control in which the control input is selected for a dual purpose: it regulates the plant while also improving the informativeness of future identification. In the formulations reported for uncertain linear systems, this “dual effect” is made explicit by combining online uncertainty reduction—often through set-membership identification—with an MPC objective that anticipates how present inputs change future uncertainty and therefore future worst-case performance (Parsi et al., 2020, Parsi et al., 2022). Subsequent work extends the same basic idea to quasi-Linear Parameter Varying systems, output-feedback tube MPC, Gaussian-process-based stochastic formulations, and approximation architectures that adapt controllers online as model knowledge evolves (Mulagaleti et al., 31 Mar 2026, Dey et al., 22 May 2026, Filabadi et al., 17 Dec 2025, Hose et al., 2024).

1. Conceptual basis and the dual effect

The defining feature of dual adaptive MPC is that control affects both system performance and the informativity of identification. In the linear time-invariant setting with bounded disturbances and parametric uncertainty in the state-space matrices, this is formulated as an MPC law that explicitly accounts for how future control actions reduce parameter uncertainty and thereby alter future closed-loop cost (Parsi et al., 2020).

A standard point of departure is the uncertain model

$x_{k+1} = A(\theta)x_k + B(\theta)u_k + w_k,$

where $\theta$ is an unknown constant parameter vector in a known polytope and $w_k \in \mathbb{W}$ is a bounded disturbance (Parsi et al., 2020). The dual effect is then not a qualitative intuition but an optimization object: exploratory inputs are desirable only insofar as they improve future robust performance.

This separates dual adaptive MPC from passive adaptive MPC. In the passive case, the controller optimizes regulation with respect to the current uncertainty description but does not optimize for future informativeness. In the active case, the controller predicts how an input-dependent measurement will shrink the feasible parameter set and uses that prediction inside the receding-horizon cost (Parsi et al., 2020).

A related distinction concerns terminology. In this literature, “dual” refers to dual control, not merely to optimization in the dual domain. It is therefore conceptually different from dual fast-gradient solvers for MPC and from dual-mode robust MPC constructions, even though those methods also use the word “dual” (Ferranti et al., 2015, Meng, 2022).

2. Uncertainty sets, identification, and predicted learning

A central mechanism in dual adaptive MPC is online set-membership identification. For linear uncertain systems, the feasible parameter set is updated by intersecting the previous set with the non-falsified set induced by recent measurements:

$$\Delta_k := \left\{\theta \in \mathbb{R}^p \mid x_{t+1} - A(\theta)x_t - B(\theta)u_t \in \mathbb{W},\; t=k-s..k-1 \right\},$$

$\Theta_k := \Theta_{k-1} \cap \Delta_k.$

This guarantees that the feasible parameter set contains the true parameter $\theta^*$ (Parsi et al., 2020).

The distinctive step in dual adaptive MPC is to propagate not only the state uncertainty but also the future uncertainty reduction that would result from planned inputs. In the active exploration formulation, a predicted parameter set $\hat{\Theta}_k$ is computed from predicted measurements using a parameter estimate $\hat{\theta}_k$ and the candidate control input. A predicted state tube is then built under this reduced uncertainty for a short horizon $\hat N \le N$ (Parsi et al., 2020). This makes future learning input-dependent.

An exact version of this idea is developed through strong duality. In that formulation, strong duality is used to reformulate the set-membership equations exactly within the MPC optimization, so the controller can account exactly for how future inputs affect future admissible parameter sets (Parsi et al., 2022). The paper emphasizes that this exact embedding avoids set approximations or ad hoc intersections, but introduces bilinear constraints and therefore a nonconvex optimization problem.

Beyond full-state linear settings, related identification mechanisms appear in broader adaptive MPC frameworks. For qLPV systems, a constrained Extended Kalman Filter jointly estimates states and parameters and constrains the estimate to lie in a time-varying polytope encoding recursive feasibility of the subsequent tube MPC (Mulagaleti et al., 31 Mar 2026). For output-feedback linear systems, an adaptive observer provides point estimates of the system state, model parameters, and initial condition while jointly updating sets containing the true parameters and initial state (Dey et al., 22 May 2026). These output-feedback constructions do not by themselves define dual control, but they supply the identification layer on which dual adaptive MPC can be built.

3. Tube-based robustification and formal guarantees

Most rigorous dual adaptive MPC formulations rely on robust tube MPC to maintain safety while exploration is performed. In the basic LTI construction, the optimization is posed over tube translations and scalings, with state tubes of the form

$$\mathbb{X}_{l|k} = \{ z_{l|k} \} \oplus \alpha_{l|k}\mathbb{X}_0,$$

so that all admissible trajectories under disturbance and parameter uncertainty remain inside the tube (Parsi et al., 2020). Robust state and input constraints are enforced for all admissible parameters and disturbances.

In the exact set-membership reformulation, the dual adaptive MPC method can be implemented using homothetic tube and flexible tube parameterizations of state tubes (Parsi et al., 2022). Homothetic tubes are parameterized as translated and scaled copies of a fixed polytope, whereas flexible tubes use variable polytopes with fixed hyperplane shapes. The reported trade-off is that homothetic tubes are less conservative and achieve better performance, while flexible tubes are computationally faster and more scalable but more conservative (Parsi et al., 2022).

For qLPV systems, the tube machinery is expressed in polytopic geometry. The dynamics are embedded in a convex polytopic inclusion, polytopic tubes are constructed so that mapped tubes under all vertex matrices and full disturbance remain in the successor tube, and the terminal set is a robustly invariant set (Mulagaleti et al., 31 Mar 2026). Stability is established via an ISS-Lyapunov argument, and recursive feasibility is preserved by constraining the estimator update to remain in the feasible set.

Adaptive-tube formulations further relax a classical restriction of robust tube MPC. In output-feedback MPC with adaptive tubes, constraint tightening, terminal ingredients, and tube geometry are updated as parameter and state estimates evolve, and the method does not require a common quadratically stabilizing linear feedback gain across the parametric uncertainty set (Dey et al., 22 May 2026). Recursive feasibility and robust exponential stability are established there. A plausible implication is that dual adaptive MPC built on adaptive tubes can reduce the conservatism associated with fixed uncertainty sets and common-gain assumptions.

4. Objective design: regulation, exploration, and information gain

The most characteristic element of dual adaptive MPC is the design of the cost function. In the active exploration formulation for uncertain LTI systems, the objective combines a predicted worst-case cost over the predicted state tube for stages $\theta$0 with the usual worst-case cost over the robust state tube for stages $\theta$1:

$\theta$2

Because the predicted tube depends on the anticipated reduction of the parameter set, minimizing this cost makes exploration performance-oriented rather than purely geometric (Parsi et al., 2020).

The exact set-membership formulation preserves the same philosophy. Its predicted worst-case cost is defined over a predicted state tube generated using the predicted uncertainty sets, and the paper states that this induces performance-oriented exploration: exploratory actions are attractive only when the resulting information gain yields tangible performance improvement through tighter tubes and lower predicted worst-case cost (Parsi et al., 2022).

An alternative realization appears in dual MPC for qLPV systems, where the control input is explicitly decomposed as

$\theta$3

Here, $\theta$4 is the output of a robust tube-based MPC for reference tracking and $\theta$5 is an exploration input restricted to $\theta$6, while $\theta$7 is restricted to $\theta$8 (Mulagaleti et al., 31 Mar 2026). The scalar $\theta$9 becomes a direct exploration–exploitation tuning parameter. The exploration sequence is designed using persistency-of-excitation criteria such as maximizing

$w_k \in \mathbb{W}$0

so that identification is accelerated without abandoning robust tracking (Mulagaleti et al., 31 Mar 2026).

In stochastic formulations, the exploration term is information-theoretic rather than worst-case geometric. In the active-inference GP controller, the expected free energy objective takes the form

$w_k \in \mathbb{W}$1

where the mutual-information term captures expected uncertainty reduction from the planned trajectory (Filabadi et al., 17 Dec 2025). This gives a probabilistic realization of dual control: the controller minimizes regulation cost while maximizing information gain.

5. Representative formulations and architectural variants

The literature now spans several technical realizations of dual or dual-adjacent adaptive MPC. The following summary organizes the main variants reported in the cited papers.

Approach	Main mechanism	Reported properties
LTI dual adaptive MPC (Parsi et al., 2020)	Predicted parameter set and predicted state tube in the cost	Active exploration, robust constraint satisfaction
Exact set-membership dual adaptive MPC (Parsi et al., 2022)	Strong-duality reformulation of set-membership inside MPC	Exact uncertainty update, recursive feasibility
qLPV dual MPC (Mulagaleti et al., 31 Mar 2026)	Tube MPC plus explicit excitation input $w_k \in \mathbb{W}$2	Recursive feasibility, stability, improved tracking
GP dual MPC with active inference (Filabadi et al., 17 Dec 2025)	Expected free energy with mutual-information term	Stochastic optimal control for dual controller design
Output-feedback adaptive tubes (Dey et al., 22 May 2026)	Adaptive observer plus evolving tube geometry	Recursive feasibility, robust exponential stability
Parameter-adaptive AMPC (Hose et al., 2024)	Neural policy plus sensitivity network for online correction	Stability guarantee, no retraining
Risk-aware adaptive robust MPC (Li, 15 Jul 2025)	Learned prediction-error set plus adaptive safety margin	Chance-constraint satisfaction up to a user-defined risk level

These variants differ primarily in how they represent uncertainty and in how they make exploration consequential for control. Set-membership formulations couple identification and control through predicted feasible parameter sets; qLPV formulations inject an explicit excitation channel; GP-based formulations quantify epistemic uncertainty probabilistically; adaptive-tube and risk-aware schemes update the geometry of robustness online. This suggests that dual adaptive MPC is less a single algorithm than a family of receding-horizon designs centered on controlled uncertainty reduction.

Approximate and neural implementations broaden this family further. A parameter-adaptive approximate MPC architecture uses two neural networks—$w_k \in \mathbb{W}$3 for the nominal control input and $w_k \in \mathbb{W}$4 for the sensitivity of the optimal input to parameters—and applies the online correction

$w_k \in \mathbb{W}$5

The paper states that this approach is analogous in spirit to dual adaptive MPC because the control policy is modified in real time as system understanding evolves (Hose et al., 2024).

A second neural direction appears in bilinear Mamba-Koopman Neural MPC, which introduces control-dependent coupling into latent dynamics and is described as important for dual adaptive MPC because it enables within-horizon adaptation, operator-level correction, and improved robustness under stale-plan execution (Pagi et al., 6 May 2026). Although this work is not a set-membership tube-MPC formulation, it targets the same underlying problem: control laws that adapt as the effective model changes.

6. Performance, computation, and embedded realization

Performance gains reported for dual adaptive MPC are typically measured against passive adaptive MPC or regulation-only baselines. In the LTI active exploration study, the closed-loop cost decreased from 6.04 for passive adaptive MPC to 4.49 for $w_k \in \mathbb{W}$6 and 4.21 for $w_k \in \mathbb{W}$7, corresponding to reductions of 25% and 30%, respectively (Parsi et al., 2020). The same study reports average solver times of 0.042 s for passive adaptive MPC, 0.89 s for $w_k \in \mathbb{W}$8, and 1.0 s for $w_k \in \mathbb{W}$9, making explicit the computational price of deeper exploration horizons.

The exact set-membership study reports that both exact dual controllers outperform passive or approximate counterparts, that homothetic exact set-membership achieves the lowest mean cost, and that all controllers robustly satisfy input and state constraints for all realizations and disturbances (Parsi et al., 2022). The significance is methodological as much as numerical: better performance is attributed to exact input-dependent uncertainty updates rather than to mere parameter-set shrinkage.

For qLPV systems, numerical results show that increasing the exploration budget $$\Delta_k := \left\{\theta \in \mathbb{R}^p \mid x_{t+1} - A(\theta)x_t - B(\theta)u_t \in \mathbb{W},\; t=k-s..k-1 \right\},$$0 initially accelerates parameter convergence and improves steady-state tracking and out-of-sample model error, but overly large $$\Delta_k := \left\{\theta \in \mathbb{R}^p \mid x_{t+1} - A(\theta)x_t - B(\theta)u_t \in \mathbb{W},\; t=k-s..k-1 \right\},$$1 degrades performance because over-excitation disrupts tracking (Mulagaleti et al., 31 Mar 2026). The reported best trade-off occurs at $$\Delta_k := \left\{\theta \in \mathbb{R}^p \mid x_{t+1} - A(\theta)x_t - B(\theta)u_t \in \mathbb{W},\; t=k-s..k-1 \right\},$$2, with an approximately 50% reduction in tracking error and model error compared to no adaptation, and an average QP solve time of 8 ms.

Real-time implementation has also become a theme. The parameter-adaptive approximate MPC controller was demonstrated on two physical cartpole systems using the same pre-trained network on a severely resource-constrained STM32G474 microcontroller with 170 MHz and 96 kB RAM, with full evaluation time below 2 ms (Hose et al., 2024). No retraining or dataset regeneration was required when transferring between the two systems; only the parameter vector was changed at deployment.

Risk-aware adaptive robust MPC shows a different computational architecture: a hierarchical dual-timescale scheme with a medium-frequency GP-based active learning engine for a learned prediction-error set and a low-timescale adaptive safety-margin update law (Li, 15 Jul 2025). On a benchmark DC-DC converter under non-stationary parametric uncertainties, the framework reportedly achieves the target empirical risk level precisely and yields significantly lower average cost than robust and stochastic MPC baselines.

7. Scope, adjacent concepts, and common misconceptions

A frequent misconception is to equate any adaptive MPC with dual adaptive MPC. The literature does not support that equivalence. Passive adaptive MPC updates uncertainty sets or model parameters but ignores how present inputs improve future identification; dual adaptive MPC incorporates that learning effect into the objective itself, whether via predicted worst-case tubes, explicit excitation channels, or information-theoretic terms (Parsi et al., 2020, Filabadi et al., 17 Dec 2025).

A second misconception is to conflate “dual adaptive MPC” with “dual-mode MPC.” Dual-mode robust MPC for non-holonomic mobile robots combines robust MPC with a local nonlinear robust control law applied within a specified terminal region in order to reduce computation burden and improve tracking accuracy (Meng, 2022). That is a mode-switching architecture for robustness and terminal control, not a dual-control treatment of exploration versus exploitation.

A third misconception is terminological: “dual” in dual adaptive MPC does not refer to dual optimization algorithms. The parallel dual fast gradient method solves linear MPC by time-splitting, dual decomposition, and adaptive constraint tightening to certify suboptimality, recursive feasibility, and closed-loop stability (Ferranti et al., 2015). Its use of the dual domain is algorithmic, whereas dual adaptive MPC uses “dual” to denote the coupled objectives of control and learning.

Current adjacent directions indicate an expansion rather than a replacement of the classical tube-based viewpoint. Output-feedback adaptive tubes provide a modular path toward dual adaptive MPC with evolving tube geometry (Dey et al., 22 May 2026); GP-active-inference controllers encode exploration through mutual information (Filabadi et al., 17 Dec 2025); learned uncertainty quantification and adaptive safety margins realize a dual-timescale adaptation of robustness and risk (Li, 15 Jul 2025); and neural approximate controllers use sensitivities or control-dependent latent dynamics to adapt online without full re-optimization (Hose et al., 2024, Pagi et al., 6 May 2026). This suggests that the core identity of dual adaptive MPC is preserved across implementations: receding-horizon control that treats learning as a first-class control objective while retaining formal guarantees or tractable approximations thereof.