Adaptive Lyapunov-Constrained MPC

• Adaptive Lyapunov-Constrained MPC is a control framework that embeds Lyapunov-based decrease constraints within MPC to guarantee closed-loop stability.
• It employs real-time adaptation mechanisms such as horizon tuning and constraint tightening to balance performance, robustness, and computational efficiency.
• The approach is effective for nonlinear, uncertain, and fault-tolerant systems, ensuring safety with practical implementations in robust and iterative control settings.

Adaptive Lyapunov-Constrained Model Predictive Control (LMPC) is a class of advanced predictive control algorithms in which stability and/or performance guarantees are achieved by embedding Lyapunov-like decrease or contraction constraints into the Model Predictive Control (MPC) optimization, with on-line adaptation mechanisms that dynamically tune the prediction horizon, model class, or other controller parameters in response to real-time closed-loop performance, uncertainty, and system status. This methodology has found wide application in robust control of nonlinear, uncertain, or iterative systems, as well as in settings where safety, fault tolerance, and computational efficiency are paramount.

1. Stability Guarantees and Lyapunov Integration

The integration of Lyapunov-based constraints within MPC is fundamental to establishing closed-loop stability in both standard and adaptive LMPC schemes. The core mechanism relies on imposing a decrease condition for an appropriate finite-horizon value function $V_N(x)$ (or other control Lyapunov function, CLF) along the closed-loop trajectory: $$V_N(x(i)) \ge V_N(x(i+1)) + \alpha \cdot l(x(i), \mu_N(x(i))),$$ where $l$ is the stage cost, $\mu_N$ is the (possibly suboptimal) control law, and $\alpha \in (0,1]$ quantifies the degree of suboptimality (Jahn et al., 2011). When adaptive horizons $N_i$ are used, Lyapunov decrease must hold with respect to the instantaneous prediction horizon.

Theoretical results establish that, under a relaxed Lyapunov inequality and enhanced stabilizing assumptions (to relate different prediction horizons), the closed-loop cost is bounded relative to the infinite-horizon optimal solution even as the controller adapts: $$\alpha_C V_\infty(x(n)) \leq \alpha_C V_\infty^{(\mu_{N_i})}(x(n)) \leq V_{N^*}(x(n)) \leq V_\infty(x(n)),$$ with $\alpha_C$ encapsulating local performance factors (Jahn et al., 2011).

In systems with explicit terminal costs and feedbacks (often derived from CLFs), adaptive MPC approaches such as Adaptive Horizon MPC (AHMPC) verify that the candidate trajectory enters regions where Lyapunov descent is ensured, relaxing the need for global off-line terminal invariant sets (Krener, 2016, Krener, 2019).

2. Adaptation Mechanisms

Adaptation in LMPC focuses on modifying controller parameters in real time to guarantee specified stability or suboptimality margins:

• Horizon Adaptation: Algorithms dynamically adjust the prediction horizon $N$ to satisfy a prescribed lower bound $\overline{\alpha}$ on Lyapunov decrease. If $\alpha(N_i)$ exceeds the threshold, the horizon is reduced; otherwise, it is extended until the decrease condition is reestablished (Jahn et al., 2011, Krener, 2016, Krener, 2019).
• Constraint Tightening and Model Updating: In uncertain or iterative settings, constraint tightening is reduced as the uncertainty set shrinks (e.g., via set-membership adaptation) (Bujarbaruah et al., 2018). In basis function adaptive MPC, the identification set is refined as informative data accrue, and models are persistently explored for excitation (Tanaskovic et al., 2013).
• Online Fault Adaptation: For fault-tolerant systems (e.g., AUVs), a Bayesian multi-model estimator identifies system faults, and the current system model—parameterized in terms of actuation effectiveness and alignment—is injected into the LMPC, ensuring the Lyapunov decrease constraint under the active mode (Liu et al., 21 Sep 2025).

These adaptation mechanisms permit online trading between performance, computational effort, and stability, without relying on conservatively chosen, globally fixed design margins.

3. Formulations and Contraction Constraints

There are several distinct yet related ways LMPC formulations encode Lyapunov contraction:

Approach	Lyapunov Constraint Type	Comments
Alpha-Decrease (relaxed Lyapunov)	$$V_{N}(x_k) \ge V_{N}(x_{k+1}) + \overline{\alpha} l(x_k,u_k)$$	Works for fixed or adaptive horizon (Jahn et al., 2011)
Terminal CLF-based	$V_f(x_{N}) \le V_f(x_{N+1}) - \alpha(|x_{N}|)$	Verified over planned (extended) trajectory (Krener, 2016, Krener, 2019)
Direct CLF decrease (tracking LMPC)	$$V(\xi_{k+1}) - V(\xi_{k}) \le -\alpha \|\tilde{x}_k\|^2$$	Used for robust AUV tracking/fault adaptation (Liu et al., 21 Sep 2025)
Safe Set / Terminal Cost Descent	$J_{t+1}^{LMPC} - J_t^{LMPC} \le -h(x_t, u_t) < 0$	Used for learning/iterative LMPC (Rosolia et al., 2016, Rosolia et al., 2019)

Contraction conditions are imposed as hard constraints or are verified online (as in adaptive horizon and sample-based LMPCs) to ensure monotonicity or bounded suboptimality.

4. Extension to Uncertainties, Constraints, and Faults

Adaptive LMPC is particularly well-suited to scenarios involving:

• Uncertain or Time-varying Models: Indirect-adaptive MPC techniques use parameter-dependent Lyapunov functions and robust control invariant (RCI) sets, maintaining both constraint satisfaction and input-to-state stability with respect to the estimation error (Cairano, 2015). Adaptive MPC with robust output feedback can use dual-stage optimization (nominal tracking and exploration for model set reduction) (Tanaskovic et al., 2013).
• Nonlinear and Constrained Problems: Data-driven approaches such as Koopman LMPC transform nonlinear dynamics into (bi)linear forms in lifted spaces, enabling explicit quadratic Lyapunov constraints in high dimensions (Narasingam et al., 2020, Dony, 13 May 2025). Convex parameterizations via Bézier curves ensure entire reference trajectories satisfy continuous constraints, while low-level CLF trackers maintain robust tubes in the presence of discretization and uncertainty (Csomay-Shanklin et al., 2022).
• Iterative and Learning Tasks: Learning MPCs employ safe sets and data-driven cost-to-go functions updated via trajectories from previous iterations. Monotonic improvement and recursive feasibility are guaranteed by Lyapunov-like decreases in the optimal cost function (Rosolia et al., 2016, Rosolia et al., 2019, Thananjeyan et al., 2020).

For fault-tolerant systems, e.g., autonomous underwater vehicles (AUVs), combining Bayesian failure diagnosis with adaptive LMPC enables graceful and stable mode transitions, coupling real-time model updates with Lyapunov-constrained prediction (Liu et al., 21 Sep 2025).

5. Practical Algorithms and Performance Trade-offs

The practical effectiveness of Adaptive Lyapunov-Constrained MPC is demonstrated through:

• Adaptive resource usage: Real-time adaptation of the prediction horizon or constraint tightening enables significant computational savings compared to worst-case fixed-horizon designs (Jahn et al., 2011, Krener, 2016).
• Robustness in constraint satisfaction: Enforcement of Lyapunov contraction as a hard constraint provides rigorous guarantees of recursive feasibility, even under model mismatch, learning, or actuator saturation (Liu et al., 21 Sep 2025, Csomay-Shanklin et al., 2022).
• Improved closed-loop performance: Adaptive learning LMPC, integrating data from iterative executions, achieves monotonic cost reduction and fast convergence to task-optimal or minimum-time solutions, even on nonlinear or time-varying systems (Rosolia et al., 2019, Mittal et al., 2020).
• Fault tolerance: Real-time model identification (via UKF-IMM, Bayesian selection, or similar) combined with a blended LMPC controller achieves rapid detection and seamless accommodation of component failures (Liu et al., 21 Sep 2025).

Notable practical considerations include the required online solution of QP/SOCPs or mixed-integer programs (for basis selection or safe set management), need for efficient implementation in multi-rate architectures, and the interplay between robustness margins and computational tractability.

6. Mathematical Foundations and Key Formulas

Essential mathematical elements include:

• Relaxed Lyapunov inequality (generic horizon):

$$V_N(x(i)) \ge V_N(x(i+1)) + \alpha \cdot l(x(i), \mu_N(x(i)))$$

• A posteriori suboptimality estimate:

$$\alpha V_\infty(x(i)) \leq \alpha V_\infty^{(\mu_N)}(x(i)) \leq V_N(x(i)) \leq V_\infty(x(i))$$

• Contraction in tracking/fault adaptation:

$$V(\xi_{k+1}) - V(\xi_k) \le -\alpha \| \tilde{x}_k \|^2$$

• Safe set and cost-to-go in learning LMPC:

$$J^{LMPC}_t(x_{t+1}) - J^{LMPC}_t(x_t) \le -h(x_t, u_t)$$

• Koopman-Lyapunov quadratic derivative condition:

$$\dot{V}(z) = z^\top (P\Lambda + \Lambda^\top P)z + u z^\top (PB + B^\top P)z < 0$$

These core formulations anchor the design and analysis of LMPC algorithms across a broad range of application domains.

7. Impact and Future Directions

Adaptive Lyapunov-Constrained MPC bridges theoretical guarantees of stability and performance with practical requirements of efficiency, robustness, and adaptability.

Future research directions highlighted in the literature include:

• Scalable high-dimensional learning: Integration of adaptive Koopman operator identification, neural Lyapunov function approximation, and data-driven safe set management for large-scale, uncertain, or partially observed systems (Mittal et al., 2020, Dony, 13 May 2025).
• Fault-tolerant and safety-critical control: Embedding formal diagnosis and resilience mechanisms within the LMPC architecture for cyber-physical systems and autonomous vehicles (Liu et al., 21 Sep 2025).
• Multi-rate and hierarchical control: Further development of strategies that integrate multi-timescale planning and execution, ensuring both rigorous constraint satisfaction and responsiveness in complex environments (Csomay-Shanklin et al., 2022).
• Hybrid learning–model-based approaches: Exploiting the strengths of model-free or reinforcement learning with formal Lyapunov-constrained MPC for improved adaptability and provable safety (Mittal et al., 2020).

Together, these directions underscore Adaptive Lyapunov-Constrained MPC as a rapidly maturing paradigm, combining structural system-theoretic guarantees with the flexibility required for next-generation autonomous and safety-critical systems.