The proposed robust invertibility framework is illustrated through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.
Abstract
This paper proposes a new notion of robust invertibility for nonlinear dynamical systems, and introduces constructive parameterizations of recurrent neural network which are robustly invertible by design. We define robust invertibility as the existence of a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input-output gains (the system is bi-Lipschitz), implying that both forward prediction and input reconstruction are robust to signal perturbations and initial-state mismatch. We construct robustly invertible recurrent models via series composition of static orthogonal layers and dynamic layers satisfying a strong input-output monotonicity property, and provide a differentiable neural network parameterizations in the form of the bi-Lipschitz recurrent equilibrium network (BiLipREN). Additionally, composition with dynamic orthogonal layers yields a nonlinear minimum-phase/all-pass (a.k.a. inner--outer) factorization. We illustrate the utility of the framework through a series of application examples in data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows, illustrating its utility for robust control, trajectory optimization, and generative modeling of complex trajectory distributions.
This paper derives sufficient conditions for input-dependent contraction and formally establish an input-to-state contraction property under bounded external excitations, and develops a novel deep learning framework that seamlessly incorporates time-varying control inputs while ensuring incremental exponential convergence via input-dependent contraction regularization.
This work first learns an implicit spectral predictor using Observation Spectral Filtering using Observation Spectral Filtering, a convex method that competes with the best linear observer for the system, and applies spectral-to-LDS distillation to convert this predictor into an explicit recurrent linear dynamical system.
Liane Galanti, Devan Shah, Shlomo Fortgang et al.· 0 citations
This paper considers robust nonadaptive regulation for general nonlinear systems in an output-feedback setting with arbitrarily high relative degree. We develop a nonadaptive design that combines an input-driven filter and a generic internal model with a recursive backstepping law, thereby recasting the regulation problem as the robust input-to-state stabilization of an augmented error system. Unlike adaptive schemes, the proposed method does not rely on linearly parameterized regressors and does not require the construction of Lyapunov functions having merely nonpositive derivatives. Under standard assumptions on the exosystem, including purely imaginary and simple eigenvalues, together with a minimum-phase input-to-state stability condition on the internal dynamics, we establish global asymptotic regulation and derive explicit, verifiable inequalities for selecting the design gains. The resulting nonadaptive framework guarantees convergence of the estimation and tracking errors even when the controlled-system dynamics are complex or only partially known. The effectiveness of the theoretical results is demonstrated using a benchmark controlled Duffing system.
Shimin Wang, M. Guay, Richard D. Braatz· 0 citations
In this work, we develop a projection neural network based on a second-order dynamical model (SO-PDM) for solving inverse variational inequality problems (IVIPs) in Hilbert spaces. The proposed framework incorporates inertial and damping components, resulting in improved convergence behavior while ensuring feasibility through a projection operator. Under the Lipschitz continuity assumption on the operator, the proposed SO-PDM admits a unique global trajectory. Under the additional strong monotonicity assumption and suitable parameter conditions, convergence to the unique solution of the IVIP is established. A discrete-time formulation is derived via a finite-difference scheme, leading to a projection-based inertial algorithm with relaxation. Under suitable parameter conditions, the algorithm is shown to converge linearly to the unique solution of the IVIP, and under an additional parameter condition, the global asymptotic stability of the continuous-time SO-PDM is established via Lyapunov analysis. Furthermore, a numerical comparison in a higher-dimensional setting shows that the proposed algorithm converges faster and attains higher accuracy than the existing first-order projection method. Numerical experiments further confirm the effectiveness and stability of the proposed SO-PDM, including its application to sparse signal recovery in compressed sensing.
Vajahat Karim Khan, M. Sarfaraz, Hafiz Farooq Ahmad et al.· Mathematics· 0 citations