Robustness guarantees and general solvers for coordinated deformation-flight control of morphing aircraft via learned representations and coupled state-dependent Riccati equations

Che and Wu (IEEE TAES 2025) proposed a coordinated controller for morphing aircraft in which a domain-adversarially invariant meta-learning (DAIML) stage learns a morphing-condition-independent representation of the unknown dynamics, the interaction between flight control and structural deformation is modeled as a two-player noncooperative differential game, and the feedback Nash equilibrium is obtained from a pair of coupled state-dependent Riccati equations (CSDREs) solved by Lyapunov iterations. The framework deliberately disregards measurement/model noise and representation (approximation) errors, its solver is proven convergent only under pointwise stabilizability/detectability for a low-dimensional weakly-coupled CSDRE pair, the model class excludes deformation-dependent input matrices, and validation is confined to simulation on a single aircraft model. This problem asks to close those gaps: establish quantitative robustness and closed-loop stability guarantees for the learned-representation CSDRE game controller under noise and approximation error, develop broadly applicable and computationally efficient solvers for coupled SDREs beyond the current coverage, and extend the model class to systems in which wing deformation affects the flight control inputs. All three of these gaps remain open.

Background

A morphing aircraft (MA) changes its wing geometry (for example wingspan) in flight to adapt aerodynamic characteristics to the flight condition and mission. Because structural deformation alters the aerodynamic parameters, the deformation dynamics and the flight dynamics are coupled, and the two subsystems can pursue different objectives: the flight-control agent commands elevator and throttle, while a morphing agent commands the wingspan rate, and each optimizes its own cost. Che and Wu (2025) formulate this interaction as a two-player noncooperative differential game (a game in which players have distinct cost functions and reach a feedback Nash equilibrium, meaning no player can lower its cost by unilaterally changing its strategy) over a learned model of the unknown dynamics. In their pipeline, an offline stage uses domain adversarially invariant meta-learning (DAIML — a technique that trains a neural-network feature map that is invariant across operating domains, here morphing conditions, so that a small number of samples suffices to adapt to a new condition) to decompose the unknown dynamics into a morphing-condition-independent representation function $\Phi(x)$ times a morphing-condition-dependent vector of linear coefficients $a(\xi)$; online, the current coefficient vector is retrieved by least-squares adaptation and a classifier. The state-dependent Riccati equation (SDRE) approach factorizes a nonlinear affine system pointwise in the state into a linear-like form $\dot{x}=A(x)x+B(x)u$ and solves an algebraic Riccati equation at each state, yielding a nonlinear feedback that suboptimally stabilizes the system; for two coupled players this becomes a pair of coupled SDREs (CSDREs), each containing the other player's Riccati solution. Building on Lyapunov iterations for coupled algebraic Riccati equations of Nash games (Li and Gajic 1995) and CSDRE control of mechatronic systems (Wang, Yaz, and Schneider 2018), Che and Wu prove convergence of their iteration under Assumption 2 — pointwise stabilizability and detectability of the factorized pairs — and note that analytical solutions of the coupled iterations are unavailable except in simplistic low-dimensional cases, so the CSDREs must be solved numerically at every sampled state for real-time operation; they state outright that no universally effective approach for solving CSDREs currently exists. The survey of Çimen (2012) documents that SDRE control is a standard nonlinear design route, but the coupled two-player case remains methodologically thin. Validation in Che and Wu (2025) is a simulation on a single longitudinal five-state morphing-aircraft model, and the framework assumes the input matrix is unaffected by wingspan deformation. The authors themselves list as unresolved the impact of measurement noise and representation errors on the game model and controller, and (in the journal version) the extension to systems where deformation affects the control inputs.

Problem Statement

The problem is to complete the coordinated deformation-flight control framework of Che and Wu (2025) so that it carries provable guarantees and practical applicability, while preserving its architecture (an offline-learned morphing-condition-independent representation of the unknown dynamics plus an online two-player noncooperative differential game whose feedback Nash strategies are computed from a pair of coupled state-dependent Riccati equations). Concretely: (1) Robustness — model the error sources present in any deployment (measurement noise, model mismatch, and the approximation error of the learned representation $\Phi(x)$ and adapted coefficients $a(\xi)$) and quantify how they propagate through the CSDRE solutions $(P_u(x), P_a(x))$ to the closed-loop dynamics, establishing guarantees such as error bounds or input-to-state-type stability of the closed-loop game equilibrium, together with mitigation or certification mechanisms. (2) Solvers — determine whether coupled SDRE pairs arising in such games admit broadly applicable, computationally efficient numerical solution methods beyond the regime covered by the source's Lyapunov-iteration algorithm (which is proven only under pointwise stabilizability/detectability and is computationally heavy because the equations are re-solved at every sampled state), for example under stronger coupling, higher state dimension, weaker assumptions, or stochastic disturbances, with proven convergence and characterized per-step cost. (3) Model class — extend the framework beyond affine systems whose input matrix is unaffected by wingspan deformation, to systems where deformation modifies the control-effectiveness matrix. These components jointly serve one objective — turning the simulation-grade coordinated deformation-flight game controller into a certifiable, solver-supported, deployable control methodology — and the three gaps (1)-(3) are jointly in scope, so a resolution of any one of them alone does not settle the other two.

References

1. Che, Hao-Chi; Wu, Huai-Ning (2025). 'Coordinated Control of Deformation and Flight for Morphing Aircraft via Meta-Learning and Coupled State-Dependent Riccati Equations.' IEEE Transactions on Aerospace and Electronic Systems. DOI 10.1109/TAES.2025.3598801, https://doi.org/10.1109/TAES.2025.3598801; arXiv:2501.05102, https://arxiv.org/abs/2501.05102
2. Wang, Jun-Wei; Wang, Feng; Wu, Huai-Ning; Liu, Zhou-Yang (2025). 'Safe RL-Based Adaptive Cooperative Game Control of Wing Deformation and Flight State Tracking for Morphing Hypersonic Vehicles.' IEEE Transactions on Aerospace and Electronic Systems. DOI 10.1109/TAES.2025.3560610, https://doi.org/10.1109/TAES.2025.3560610
3. Che, Hao-Chi; Wu, Huai-Ning (2026). 'Approximate Optimal Control for Morphing Aircraft via Attention Meta-Learning and Continual Learning.' IEEE Transactions on Neural Networks and Learning Systems. DOI 10.1109/TNNLS.2026.3687995, https://doi.org/10.1109/TNNLS.2026.3687995
4. Che, Hao-Chi; Wu, Huai-Ning (2026). 'Meta-learning-based approximate optimal control for morphing aircraft.' Engineering Applications of Artificial Intelligence. DOI 10.1016/j.engappai.2026.115978, https://doi.org/10.1016/j.engappai.2026.115978
5. Meng, Xiangrui; Zhang, Kun (2026). 'Optimal Parallel Control Scheme of Variable Wingspan Aircraft with Input Delay.' DOI 10.1109/CSIS-IAC70275.2026.11584988, https://doi.org/10.1109/CSIS-IAC70275.2026.11584988
6. Fu, Qien; Sun, Changyin (2025). 'Attention-Based Multi-Objective Control for Morphing Aircraft.' Biomimetics 10(5):280. DOI 10.3390/biomimetics10050280, https://doi.org/10.3390/biomimetics10050280
7. Wang, Xin; Yaz, Edwin E.; Schneider, Susan C. (2018). 'Coupled State-Dependent Riccati Equation Control for Continuous Time Nonlinear Mechatronics Systems.' Journal of Dynamic Systems, Measurement, and Control 140(11):111013. DOI 10.1115/1.4040295, https://doi.org/10.1115/1.4040295
8. Li, T-Y.; Gajic, Z. (1995). 'Lyapunov Iterations for Solving Coupled Algebraic Riccati Equations of Nash Differential Games and Algebraic Riccati Equations of Zero-Sum Games.' In New Trends in Dynamic Games and Applications, Birkhäuser Boston, pp. 333-351. DOI 10.1007/978-1-4612-4274-1_17, https://doi.org/10.1007/978-1-4612-4274-1_17
9. Çimen, Tayfun (2012). 'Survey of State-Dependent Riccati Equation in Nonlinear Optimal Feedback Control Synthesis.' Journal of Guidance, Control, and Dynamics 35(4):1025-1047. DOI 10.2514/1.55821, https://doi.org/10.2514/1.55821