[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85789-en":3,"doc-seo-85789-105":29,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":4,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85789,8796095462418,"Noah","https://ap-avatar.wpscdn.com/avatar/80000253c1241d02b47?x-image-process=image/resize,m_fixed,w_180,h_180&k=1778826106357471780",8,"Research & Report","Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion Based Control to Generative Trajectory Modelling","This paper develops robust invertibility for nonlinear dynamical systems and provides constructive recurrent neural network parameterizations that are robustly invertible by design. Robust invertibility is defined via a causal inverse system where both forward and inverse dynamics are contracting and exhibit bounded incremental input–output gains, yielding a bi-Lipschitz property. Forward prediction and input reconstruction remain stable under signal perturbations and initial-state mismatches. Robust models are built using orthogonal static layers and monotone dynamic layers, forming the bi-Lipschitz recurrent equilibrium network (BiLipREN), with applications in internal model control, surrogate loss learning, and generative trajectory modeling.","Robustly Invertible Nonlinear Dynamics and the BiLipREN: From Inversion-Based Control to Generative Trajectory Modelling  \nYurui Zhang, Ruigang Wang and Ian R. Manchester  \n.  \narXiv :2607 . 10026v1 [ ee ss . SY] 10 Jul 2026  \nAbstract—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.  \nIndex Terms—Stability of Nonlinear Systems, Contraction Theory, Neural Networks, Inverse Optimal Control, Generative Modelling.  \nI. INTRODUCTION  \nTHE notion of dynamic system invertibility plays a fun  \ndamental role in control theory and applications. In the absence of any uncertainty, an ideal inverse of a system provides a perfect feedforward controller, while in the feedback setting robust control was famously characterized by Zamesin terms of existence of an approximate inverse [1] .  \nInvertible nonlinear functions play a similarly fundamental role in machine learning, especially generative modelling through the concept of a normalizing flow [2]: an invertible mapping between a simple probability distribution (a multivariate normal) and a complex distribution (e.g. the distribution of images of a certain object) .  \nIn this paper, we introduce new notions of invertibility for nonlinear dynamical systems and new recurrent neural  \nThis work was supported in part by the Australian Research Council through projects DP230101014 and IH210100030 .  \nThe authors are with the Australian Centre for Robotics (ACFR), and the School of Aerospace, Mechanical and Mechatronic Engineering, The University of Sydney, Australia, {yurui . zhang, [ruigang.wang](ruigang.wang) , [ian.manchester](ian.manchester}@sydney.edu.au)[}](ian.manchester}@sydney.edu.au)[@sydney.edu.au](ian.manchester}@sydney.edu.au)  \nnetwork model classes with guaranteed robust invertibility, and illustrate their utility in examples spanning classical control techniques to modern generative modelling.  \nA. System Inversion and Control  \nMany practical techniques in control design and signal processing are based in some way on dynamic system inversion. System inverses are used for feedforward compensation in precision motion control (e.g. [3]–[5]) and for digital predistortion compensation in electronics (e.g. [6]–[8]) . System inversion is used a component in feedback schemes such as nonlinear dynamic inversion for flight control (e.g. [9]–[12]) and internal model control schemes which have been widely applied in the process industries (e.g. [13], [14]) . Other works such as [12],[15] developed robust output feedback regulators for certain classes of invertible nonlinear MIMO systems.  \nA linear system y = G (z)u with all poles and zeros strictly inside the unit circle (or left-half-plane in continuous-time) is called a minimum-phase system. F","cbCaipn2GOu94alM","https://ap.wps.com/l/cbCaipn2GOu94alM","pdf",3265380,1,15,"English","en",105,"# Introduction\n## System Inversion and Control\n## Surrogate Loss Learning","[{\"question\":\"What does “robust invertibility” mean in the proposed framework?\",\"answer\":\"It means there exists a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input–output gains, making the overall system bi-Lipschitz.\"},{\"question\":\"How does BiLipREN guarantee robust invertibility by construction?\",\"answer\":\"It is built by series composition of static orthogonal layers and dynamic layers that satisfy a strong input–output monotonicity property, yielding a differentiable bi-Lipschitz recurrent equilibrium network.\"},{\"question\":\"Where can the framework be applied beyond robust control?\",\"answer\":\"The paper illustrates use cases including data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows for generative modeling of complex trajectory 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does “robust invertibility” mean in the proposed framework?","Question",{"text":74,"@type":75},"It means there exists a causal inverse system such that both the forward and inverse systems are contracting and have bounded incremental input–output gains, making the overall system bi-Lipschitz.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does BiLipREN guarantee robust invertibility by construction?",{"text":79,"@type":75},"It is built by series composition of static orthogonal layers and dynamic layers that satisfy a strong input–output monotonicity property, yielding a differentiable bi-Lipschitz recurrent equilibrium network.",{"name":81,"@type":72,"acceptedAnswer":82},"Where can the framework be applied beyond robust control?",{"text":83,"@type":75},"The paper illustrates use cases including data-driven internal model control, dynamic surrogate loss learning, and signal-space normalizing flows for generative modeling of complex trajectory 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