LSTM Neural Networks: Input to State Stability and Probabilistic Safety Verification

The goal of this paper is to analyze Long Short Term Memory (LSTM) neural networks from a dynamical system perspective. The classical recursive equations describing the evolution of LSTM can be recast in state space form, resulting in a time-invariant nonlinear dynamical system. A sufficient condition guaranteeing the Input-to-State (ISS) stability property of this class of systems is provided. The ISS property entails the boundedness of the output reachable set of the LSTM. In light of this result, a novel approach for the safety verification of the network, based on the Scenario Approach, is devised. The proposed method is eventually tested on a pH neutralization process.Comment: Accepted for Learning for dynamics & control (L4DC) 202

Reachability-based Trajectory Design

Autonomous mobile robots have the potential to increase the availability and accessibility of goods and services throughout society. However, to enable public trust in such systems, it is critical to certify that they are safe. This requires formally specifying safety, and designing motion planning methods that can guarantee safe operation (note, this work is only concerned with planning, not perception). The typical paradigm to attempt to ensure safety is receding-horizon planning, wherein a robot creates a short plan, then executes it while creating its next short plan in an iterative fashion, allowing a robot to incorporate new sensor information over time. However, this requires a robot to plan in real time. Therefore, the key challenge in making safety guarantees lies in balancing performance (how quickly a robot can plan) and conservatism (how cautiously a robot behaves). Existing methods suffer from a tradeoff between performance and conservatism, which is rooted in the choice of model used describe a robot; accuracy typically comes at the price of computation speed. To address this challenge, this dissertation proposes Reachability-based Trajectory Design (RTD), which performs real-time, receding-horizon planning with a simplified planning model, and ensures safety by describing the model error using a reachable set of the robot. RTD begins with the offline design of a continuum of parameterized trajectories for the plan- ning model; each trajectory ends with a fail-safe maneuver such as braking to a stop. RTD then computes the robot’s Forward Reachable Set (FRS), which contains all points in workspace reach- able by the robot for each parameterized trajectory. Importantly, the FRS also contains the error model, since a robot can typically never track planned trajectories perfectly. Online (at runtime), the robot intersects the FRS with sensed obstacles to provably determine which trajectory plans could cause collisions. Then, the robot performs trajectory optimization over the remaining safe trajectories. If no new safe plan can be found, the robot can execute its previously-found fail-safe maneuver, enabling perpetual safety. This dissertation begins by presenting RTD as a theoretical framework, then presents three representations of a robot’s FRS, using (1) sums-of-squares (SOS) polynomial programming, (2) zonotopes (a special type of convex polytope), and (3) rotatotopes (a generalization of zonotopes that enable representing a robot’s swept volume). To enable real-time planning, this work also de- velops an obstacle representation that enables provable safety while treating obstacles as discrete, finite sets of points. The practicality of RTD is demonstrated on four different wheeled robots (using the SOS FRS), two quadrotor aerial robots (using the zonotope FRS), and one manipulator robot (using the rotatotope FRS). Over thousands of simulations and dozens of hardware trials, RTD performs safe, real-time planning in arbitrary and challenging environments. In summary, this dissertation proposes RTD as a general purpose, practical framework for provably safe, real-time robot motion planning.PHDMechanical EngineeringUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttp://deepblue.lib.umich.edu/bitstream/2027.42/162884/1/skousik_1.pd

State Controllability Analysis for Irreversible Systems Using Set Theory

[ES] Los sistemas irreversibles han sido poco estudiados en el marco de la teoría de control, a pesar de que una de las aplicaciones relevantes de los mismos es el control de los procesos por lotes, los cuales son irreversibles. Por lo tanto, en este artículo se propone un método para analizar la controlabilidad de estado de estos sistemas mediante la teoría de conjuntos, extensible también a los procesos por lotes. Para ello, se proponen las definiciones de Conjunto Reversible y Conjunto de Trayectorias Controlables, ambas para sistemas no lineales; este último conjunto permite el análisis de controlabilidad de estado de los sistemas irreversibles. Adicionalmente, se propone un algoritmo que permite calcular dichos conjuntos desde el conocimiento de la dinámica del sistema. La propuesta es aplicada a un problema de referencia de un proceso por lotes, con lo cual se obtienen resultados de simulación que evidencian las ventajas de la misma para analizar cuantitativamente la controlabilidad de estado de los sistemas irreversibles.[EN] The irreversible systems have been little studied within the control theory framework, although one of their relevant cases is the batch process control problem. Therefore, in this work a method for analysing state controllability of irreversible systems is proposed. The method uses set theory and its extension to batch processes. Definitions for Reversible Set and Controllable Trajectories Set, both for nonlinear systems, are given in order to analyze state controllability for irreversible systems. Additionally, an algorithm for calculating mentioned sets from the dynamic process knowledge is proposed. The proposal is applied to a batch process benchmark. 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Limitaciones de la Controlabilidad de Estados en los Procesos por Lotes. Información tecnológica, 23(5), 97-108. doi:10.4067/s0718-07642012000500010Hermann, R., & Krener, A. (1977). Nonlinear controllability and observability. IEEE Transactions on Automatic Control, 22(5), 728-740. doi:10.1109/tac.1977.1101601Kerrigan, E. C., & Maciejowski, J. M. (s. f.). Invariant sets for constrained nonlinear discrete-time systems with application to feasibility in model predictive control. Proceedings of the 39th IEEE Conference on Decision and Control (Cat. No.00CH37187). doi:10.1109/cdc.2001.914717Lee, K. S., & Lee, J. H. (2003). Iterative learning control-based batch process control technique for integrated control of end product properties and transient profiles of process variables. Journal of Process Control, 13(7), 607-621. doi:10.1016/s0959-1524(02)00096-3Limon, D., Alamo, T., & Camacho, E. F. (2005). Enlarging the domain of attraction of MPC controllers. 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