Neural-network (NN) controllers are increasingly used in nonlinear control systems, but their highly nonlinear behavior makes them difficult to explain and verify, raising trustworthiness concerns in safety- and mission-critical applications. A key step toward certifiable trustworthiness is to find a Forward-Invariant Set (FIS): a state-space region such that any trajectory starting inside remains inside. If the FIS excludes unsafe states, safety can be guaranteed for initial states within it. Finding an analytically proven FIS for a given AI-controlled system with a fixed controller is difficult. We propose a framework that uses an Invertible Neural Network (INN) to transform the original state space into a latent space where a regular-shaped FIS is more likely to exist. We train the INN so that a preferred hyper-rectangular candidate becomes invariant in the latent space, then formally verify it. We prove that, whenever verification succeeds, both the latent-space candidate and its inverse-transformed counterpart in the original state space are analytically proven FISs. We evaluate the approach on 45 AI-controlled systems across three representative control testbeds. Our method finds certified FISs for all 45 systems, whereas an adapted state-of-the-art baseline finds none. It is also faster on 40 of the 45 systems, and the centers of the resulting FISs roughly match domain-expert preferences.
Figures & tables
Fig. 1 : Solution heuristics: a hyper-rectangular FIS constructed and proven in the latent state space maps to a nonlinear FIS in the original state space.
Fig. 2 : Training subsets: Zinr , Zb , Zadv , and Zrep .
We propose a framework for training neural network controllers with certified robust forward invariant polytopes. First, we parameterize a family of lifted control systems in a higher dimensional space, where the original neural controlled system evolves on an invariant subspace of each lifted system. We use interval analysis and neural network verifiers to further construct a family of lifted embedding systems, carefully capturing the knowledge of this invariant subspace. If the vector field of any lifted embedding system satisfies a sign constraint at a single point, then a certain convex polytope of the original system is robustly forward invariant. Treating the neural network controller and the lifted system parameters as variables, we propose an algorithm to train controllers with certified forward invariant polytopes in the closed-loop control system. Through two examples, we demonstrate how the simplicity of the sign constraint allows our approach to scale with system dimension to over 50 states, and outperform state-of-the-art Lyapunov-based sampling approaches in runtime.
Akash Harapanahalli, Samuel Coogan
School of Electrical and Computer Engineering, Georgia Institute of Technology
In this paper, we address the certification of datadriven feedforward control for periodic tracking of unknown nonlinear systems under partial state measurements. To this end, we adopt an invertible neural network (INN) as a surrogate for the unknown system. This choice allows us to bypass solving a nonconvex inversion problem, eliminating the associated inversion errors and reducing tracking error certification to a surrogate modeling problem. We then apply conformal prediction to provide finite-sample probabilistic guarantees on the surrogate modeling error which, through the derived tracking error bound, yield marginal certificates on feedforward tracking error. Finally, we demonstrate the approach on a DC-motor-driven mechanical load with nonlinear friction.
Berk Altiner, Rajasree Sarkar, Arunava Banerjee +2
Department of Mechanical Engineering, University of Minnesota–Twin Cities, Minneapolis, MN 55455, USA · DEVCOM Army Research Laboratory, Aberdeen Proving Ground, Aberdeen, MD 21005, USA
Implicit neural controllers (INCs) are static feedback laws that are evaluated through an algebraic fixed point {equation}; they include as special cases neural network controllers. We propose a so-called implicit representation of neural networks as a key enabling device that exposes the controller as a trainable linear interconnection closed through a known static activation map, thereby making well-posedness and Lyapunov/IQC analysis mathematically easy to handle. For finite-dimensional LTI plants, we first develop a rigorous analysis theory for a given INC, including Perron--Frobenius and norm conditions for well posedness, LMI/IQC certificates for exponential stability, and LMIs for discounted infinite-horizon quadratic performance. We then formulate synthesis as a certification-compatible heuristic search: training is carried out under explicit well-posedness constraints, implicit-differentiation formulas provide gradients, and the resulting controller is accepted only after independent post-training LMIs or regional admissibility checks are feasible. Finally, we establish constrained-control separation results: for a specific scalar unstable plant with hard actuator bounds, an INC achieves a strictly smaller discounted infinite-horizon cost than any admissible finite-order dynamic linear controller. Additional results cover quadratic state-input costs, comparison with linear static output feedback, and computable upper/lower-bound certificates. Numerical examples illustrate the mechanism and the resulting certified performance.