Computational Techniques for Stochastic Reachability

As automated control systems grow in prevalence and complexity, there is an increasing demand for verification and controller synthesis methods to ensure these systems perform safely and to desired specifications. In addition, uncertain or stochastic behaviors are often exhibited (such as wind affecting the motion of an aircraft), making probabilistic verification desirable. Stochastic reachability analysis provides a formal means of generating the set of initial states that meets a given objective (such as safety or reachability) with a desired level of probability, known as the reachable (or safe) set, depending on the objective. However, the applicability of reachability analysis is limited in the scope and size of system it can address. First, generating stochastic reachable or viable sets is computationally intensive, and most existing methods rely on an optimal control formulation that requires solving a dynamic program, and which scales exponentially in the dimension of the state space. Second, almost no results exist for extending stochastic reachability analysis to systems with incomplete information, such that the controller does not have access to the full state of the system. This thesis addresses both of the above limitations, and introduces novel computational methods for generating stochastic reachable sets for both perfectly and partially observable systems. We initially consider a linear system with additive Gaussian noise, and introduce two methods for computing stochastic reachable sets that do not require dynamic programming. The first method uses a particle approximation to formulate a deterministic mixed integer linear program that produces an estimate to reachability probabilities. The second method uses a convex chance-constrained optimization problem to generate an under-approximation to the reachable set. Using these methods we are able to generate stochastic reachable sets for a four-dimensional spacecraft docking example in far less time than it would take had we used a dynamic program. We then focus on discrete time stochastic hybrid systems, which provide a flexible modeling framework for systems that exhibit mode-dependent behavior, and whose state space has both discrete and continuous components. We incorporate a stochastic observation process into the hybrid system model, and derive both theoretical and computational results for generating stochastic reachable sets subject to an observation process. The derivation of an information state allows us to recast the problem as one of perfect information, and we prove that solving a dynamic program over the information state is equivalent to solving the original problem. We then demonstrate that the dynamic program to solve the reachability problem for a partially observable stochastic hybrid system shares the same properties as for a partially observable Markov decision process (POMDP) with an additive cost function, and so we can exploit approximation strategies designed for POMDPs to solve the reachability problem. To do so, however, we first generate approximate representations of the information state and value function as either vectors or Gaussian mixtures, through a finite state approximation to the hybrid system or using a Gaussian mixture approximation to an indicator function defined over a convex region. For a system with linear dynamics and Gaussian measurement noise, we show that it exhibits special properties that do not require an approximation of the information state, which enables much more efficient computation of the reachable set. In all cases we provide convergence results and numerical examples

Human Motion Trajectory Prediction: A Survey

With growing numbers of intelligent autonomous systems in human environments, the ability of such systems to perceive, understand and anticipate human behavior becomes increasingly important. Specifically, predicting future positions of dynamic agents and planning considering such predictions are key tasks for self-driving vehicles, service robots and advanced surveillance systems. This paper provides a survey of human motion trajectory prediction. We review, analyze and structure a large selection of work from different communities and propose a taxonomy that categorizes existing methods based on the motion modeling approach and level of contextual information used. We provide an overview of the existing datasets and performance metrics. We discuss limitations of the state of the art and outline directions for further research.Comment: Submitted to the International Journal of Robotics Research (IJRR), 37 page

Decision-Making Under Uncertainty: Beyond Probabilities

This position paper reflects on the state-of-the-art in decision-making under uncertainty. A classical assumption is that probabilities can sufficiently capture all uncertainty in a system. In this paper, the focus is on the uncertainty that goes beyond this classical interpretation, particularly by employing a clear distinction between aleatoric and epistemic uncertainty. The paper features an overview of Markov decision processes (MDPs) and extensions to account for partial observability and adversarial behavior. These models sufficiently capture aleatoric uncertainty but fail to account for epistemic uncertainty robustly. Consequently, we present a thorough overview of so-called uncertainty models that exhibit uncertainty in a more robust interpretation. We show several solution techniques for both discrete and continuous models, ranging from formal verification, over control-based abstractions, to reinforcement learning. As an integral part of this paper, we list and discuss several key challenges that arise when dealing with rich types of uncertainty in a model-based fashion

A variational approach to path estimation and parameter inference of hidden diffusion processes

We consider a hidden Markov model, where the signal process, given by a diffusion, is only indirectly observed through some noisy measurements. The article develops a variational method for approximating the hidden states of the signal process given the full set of observations. This, in particular, leads to systematic approximations of the smoothing densities of the signal process. The paper then demonstrates how an efficient inference scheme, based on this variational approach to the approximation of the hidden states, can be designed to estimate the unknown parameters of stochastic differential equations. Two examples at the end illustrate the efficacy and the accuracy of the presented method.Comment: 37 pages, 2 figures, revise

Online Model Learning of Buildings Using Stochastic Hybrid Systems Based on Gaussian Processes

Dynamical models are essential for model-based control methodologies which allow smart buildings to operate autonomously in an energy and cost efficient manner. However, buildings have complex thermal dynamics which are affected externally by the environment and internally by thermal loads such as equipment and occupancy. Moreover, the physical parameters of buildings may change over time as the buildings age or due to changes in the buildings’ configuration or structure. In this paper, we introduce an online model learning methodology to identify a nonparametric dynamical model for buildings when the thermal load is latent (i.e., the thermal load cannot be measured). The proposed model is based on stochastic hybrid systems, where the discrete state describes the level of the thermal load and the continuous dynamics represented by Gaussian processes describe the thermal dynamics of the air temperature. We demonstrate the evaluation of the proposed model using two-zone and five-zone buildings. The data for both experiments are generated using the EnergyPlus software. Experimental results show that the proposed model estimates the thermal load level correctly and predicts the thermal behavior with good performance