Noise-Enabled Observability of Nonlinear Dynamic Systems Using the Empirical Observability Gramian Nathan D. Powel A dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy University of Washington 2016 Reading Committee: Kristi Morgansen, Chair Juris Vagners Mehran Mesbahi Program Authorized to Offer Degree: Aeronautics & Astronautics c Copyright 2016 Nathan D. Powel University of Washington Abstract Noise-Enabled Observability of Nonlinear Dynamic Systems Using the Empirical Observability Gramian Nathan D. Powel Chair of the Supervisory Committee: Professor Kristi Morgansen Aeronautics & Astronautics While control actuation is well understood to influence the observability of nonlinear dy- namical systems, actuation of nonlinear stochastic systems by process noise has received comparatively little attention in terms of the effects on observability. As noise is present in essentially all physically instantiated systems, complete analysis of observability must account for process noise. We approach the problem of process-noise-induced observability through the use of a tool called the empirical observability Gramian. We demonstrate that the empirical observability Gramian can provide a unified approach to observability anal- ysis, by providing sufficient conditions for weak observability of continuous-time nonlinear systems, local weak observability of discrete-time nonlinear systems, and stochastic observ- ability of continuous-time stochastic linear systems with multiplicative noise. The empirical observability Gramian can be used to extend notions of stochastic observability that depend explicitly on linear systems structure to nonlinear stochastic systems. We use Monte Carlo methods to analyze the observability of nonlinear stochastic systems with noise and control actuation. TABLE OF CONTENTS Page List of Figures . iii List of Tables . iv Chapter 1: Introduction . .1 1.1 Literature Review . .4 1.2 Contributions . 11 1.3 Organization . 11 Chapter 2: Observability Background . 13 2.1 Linear Observability . 13 2.2 Nonlinear Continuous-Time Observability . 15 2.3 Nonlinear Discrete-Time Observability . 18 Chapter 3: Continuous-Time Observability . 20 3.1 Control Explicit Empirical Observability Gramian . 20 3.2 Limit " ! 0.................................... 22 3.3 Finite " ...................................... 26 Chapter 4: Bounds on Filter Information . 32 4.1 Fisher Information Matrix . 32 4.2 Filter Information Bounds . 36 Chapter 5: Discrete-Time Observability . 43 5.1 Discrete-Time Empirical Observability Gramian . 43 5.2 Limit " ! 0.................................... 44 5.3 Finite " ...................................... 47 i Chapter 6: Stochastic Observability . 52 6.1 Stochastic Empirical Observability Gramian . 53 6.2 Expected Value of the Empirical Observability Gramian . 54 6.3 Stochastic Observability . 68 6.4 Noise as Modeling Error . 73 Chapter 7: Simulation . 75 7.1 Control and Noise Affine Dynamics . 76 7.2 Unicycle Dynamics . 79 7.3 Quadrotor Dynamics . 82 Chapter 8: Conclusion . 87 Bibliography . 93 ii LIST OF FIGURES Figure Number Page 1.1 Stochastically actuated unicycle . .3 3.1 Fr´echet derivative maximization domain . 27 4.1 Fisher information matrix condition number bounds . 35 7.1 Observability metrics versus noise amplitude for a noise affine system . 78 7.2 Relative noise and control impact on observability of a noise affine system . 79 7.3 Trajectory and heading ensemble analysis for a noise actuated unicycle . 81 7.4 Observability metrics versus noise amplitude for a unicycle system . 81 7.5 Relative noise and control impact on observability of a unicycle system . 82 7.6 Observability metrics versus noise amplitude for a quadrotor system . 85 iii LIST OF TABLES Table Number Page 7.1 Quadrotor parameters . 84 iv ACKNOWLEDGMENTS This dissertation would not have been possible without the assistance and support of my advisor, Professor Kristi Morgansen. Her guidance set me on the path to this accomplishment and helped me to become a more proficient and independent researcher, and without the robotic fish to catch my eye, I might not have ended up in control theory at all. My thanks also go out to Jake, Brian, Natalie, Atiye, and everyone else in the Nonlinear Dynamics and Controls Lab for assistance during the production of this dissertation. I appreciated your help with edits, advice, and acting as a sounding-board when working out particularly tricky problems. I also owe many thanks to the patience and support of my wife, Ashley. Her relentless support and unfailing understanding throughout this (long) process is deeply appreciated. Without her assistance I would not be where, or who, I am today. Last, but not least, I would like to thank the Air Force Office of Scientific Research (AFOSR grant FA-9550-07-1-0528), Office of Naval Research (ONR MURI N000141010952), and Allan and Inger Osberg (Allan and Inger Osberg Endowed Presidential Fellowship in Engineering) for providing the funding for this research. v DEDICATION To my friend, my partner, my love: my wife Ashley. vi 1 Chapter 1 INTRODUCTION Sensing and estimation are pervasive in everyday life, in both engineered systems, such as thermostats, cell phones, and automobiles, and in biological systems, such as fruit fly halteres, cat whiskers, and human perception. For stability and control, engineered systems, in particular, rely heavily on the ability to sense and estimate otherwise hidden variables, and may be constrained in their design and performance by what they can estimate from a fixed set of sensors. The aim of this dissertation is to provide new tools for the analysis of the limits of sensing in nonlinear systems and to analyze the effects of noise in system dynamics on the sensing of hidden variables. Given that noise is ubiquitous in natural and engineered systems, we believe that a better understanding of noise in nonlinear systems could lead to substantial improvements in sensing and estimation. The property of a system that denotes whether all states of the system can be estimated from the output of that system (measurements or sensor readings) is called observability. Observability is important because modern control relies on knowledge of system states for controllers to react on. In theory, if a system is observable, then an estimator can be designed to determine its states. However, if a system is not observable, then some states cannot be determined from the output of the system by any estimator. For deterministic linear systems with known dynamics, observability can be determined quickly and easily, and observability and control of the system are decoupled. Checking observability for deterministic nonlinear systems, however, can be significantly more difficult, and choice of control input is well known to influence observability. For example, during its seminal crossing of the Atlantic ocean, the Aerosonde autonomous unmanned vehicle was required to periodically perform S-turn maneuvers in order to obtain full observability of its 2 air-relative velocity because the vehicle was equipped with a Pitot wind-speed sensor and GPS but no compass [1]. To clarify this property of nonlinear systems, consider an example system which we will build upon as we progress through the results in this dissertation: a planar unicycle vehicle, i.e., a vehicle constrained to pivot about its center and move forward or backwards along its current heading angle. Let us assume that we have control over the vehicle's linear acceler- ation (along the heading direction) and rotational rate and are able to measure the position of the vehicle, but we cannot directly measure its heading. When the vehicle is stationary, the heading is not observable unless an acceleration input is applied, demonstrating that observability can be a state and/or control dependent property in nonlinear systems. Adding stochastic elements to our analysis can complicate the question of observabil- ity further. Noise is often treated as an undesirable property to be suppressed or worked around when possible. However, noise may have beneficial effects on many kinds of systems. White Gaussian noise is intentionally injected into systems (through the control inputs) to provide persistent excitation for system identification [2], and the phenomenon of stochastic resonance, in which the addition of process noise can amplify or enhance some determinis- tic behavior of a system, has been observed in systems ranging from climate models [3, 4] to neuron signal transmission models [5{9]. As we show below, noise driven actuation can improve observability as well. Furthermore, noise is ubiquitous in physically instantiated systems; no physical system is truly deterministic if we examine it closely enough. For example, aerodynamic turbulence, vibration, electrical noise, thermal fluctuations, quantum dynamical effects, and other un- modeled dynamics or incompletely understood physics can appear in our models as process noise [10]. We also note that noise can actuate dynamical systems in much the same way as control (which, as we showed above, can be important in nonlinear systems) but, in many systems, noise may be present in states that we cannot actuate directly by the control input. This ubiquity and potential for system actuation beyond the reach of control justify the additional complexity of including noise in observability analysis. 3 Figure 1.1: The unicycle actuated by noise in the acceleration term will make a random walk along the dashed line, which passes