Mathematics of Autonomy

The following sections are included:

• Preface
• Acknowledgments
• Glossary of Acronyms
• Glossary of Symbols
• Contents

In this Chapter we firstly introduce the concept of trusted autonomous systems. Then we define a rigorous model of trusted autonomy in the form of cyber-physical-cognitive (CPC) autonomy. Finally, we provide the necessary technical preliminaries for comprehensive reading of the monograph.

The following sections are included:

• Introduction to Port-Hamiltonian Modeling of Multi-Physical Networks
  • PHS Background
  • Informal PHS Description
  • Gradient Operator and Gradient Descent
  • PHS Definition
  • First PHS Example: An LCL-Circuit
  • Poisson Structure
  • Open Port-Hamiltonian Systems
  • Interconnection of Port-Hamiltonian Systems
  • Including Dissipation
  • Dirac Structure
  • Composition of Dirac Structures
  • Control by Interconnection
  • Passive Control Systems
  • Second PHS Example: Mass-Spring-Damper System
• Dirac Structures on Directed Graphs
  • Dirac Structures
  • Directed Graphs or Digraphs
  • Dirac Structures on Digraphs
  • PHS with Dirac Structures on Digraphs
• Category-Theoretic Abstraction: Deductive Reasoning on Graphs
  • Digraphs as Deductive Systems
  • Cartesian Closed Deductive Systems and Categories
  • Basics of Topos Theory and Intuitionistic Logic

Recently, we have proposed in [IY16] a rigorous model for prediction and control of a large-scale joint swarm of unmanned ground vehicles (UGVs) and unmanned aerial vehicles (UAVs), performing an autonomous land-air operation. In that Chapter, we have also introduced a need for a cognitive supervisor for the high-dimensional distributed multi-robotic system. Its primary task is to have a bird’s-eye view of the situation across the joint land-air operation, and based on the GPS locations of both the target and all included robots, to provide them with good 2D and 3D attractor fields so that they can reach the proximity of the target in the shortest possible time. The purpose of the present Chapter is to develop a rigorous model for this cognitive supervisor, based on recent discoveries in brain science that show us how humans navigate in 2D environments and how bats navigate in 3D environments, and to couple this with a meta-cognitive supervisor model that allows vehicles to reason about actions and construct simple plans…

In this Chapter we will propose a number of mathematical tools from quantum information and quantum computation adapted for modern theoretical autonomy. In particular, we will focus on quantum-mechanical tools suitable for analysis and design of the CPC-autonomy, based on the fundamental concept of quantum entanglement, expressed in the form of a certain composite wave function |Ψ(t)〉 equivalent to the composite transition (probability) amplitude 〈Out(t_fin) | In(t_init)〉.

The following sections are included:

• Modern Brain Models: Deep Learning Neural Networks
  • Introduction to Deep Learning
  • Deep Belief Networks (DBNs) using Restricted Boltzmann Machines (RBMs)
  • Recurrent Neural Nets (RNNs)
  • Convolutional Neural Networks (ConvNets)
• TensorFlow: The State-of-the-Art in Machine Learning
• Tensor Decompositions for Deep Representation Learning
  • Multi-Task Representation Learning: Shallow and Deep
  • Basics of Tensor Factorization
  • Knowledge Sharing Between the Tasks
  • Tensor Decompositions
  • Deep Multi-Task Representation Learning
• Generalized Tensor Decompositions in ConvNets
  • Introducing ConvNets
  • Tensors in ConvNets
  • Generalized Tensor Decompositions
  • A Typical ConvNet Architecture
  • ConvNet Classification
  • Grid Tensors for Shallow and Deep ConvNets
  • From Tensors to Matrices in ConvNets

The following sections are included:

• Brief Introduction to Perceptual Control Theory (PCT)
• Predecessors of PCT: Wiener’s Cybernetics, Beinstein’s Neural Control and Gardner’s Cognitive Control
  • Wiener’s Cybernetics and Linear Control Theory
  • Primary Control Example: Inverted Pendulum Balance
  • Bernstein’s Neural Control and Motion Pattern Architecture
  • Gardner’s Cognitive Control: Cognitive Behavior and Adaptation
• PCT Fundamentals
  • Controlled Variables in Psychology
  • Marken’s PCT Tracking Tests
• PCT Approach to Inverted Pendulum Balance
• PCT in Psychotherapy: Method of Levels
• PCT versus Brooks’ Subsumption Architecture
• PCT Alternative 1: Lewinian Psycho–Physical Group Dynamics
• PCT Alternative 2: Model Predictive Control
  • MPC Application: Control of a Rotational Spacecraft Model
  • MPC-Based Mean-Field Games for Multi-Agent CPC-Autonomy
• PCT Alternative 3: Synergetics Approach to CPC-Autonomy
  • Nonequilibrium Phase Transitions
• A Model-Free PCT Alternative: Adaptive Fuzzy Inference for Human-Like Decision and Control
  • Motivation: Why Adaptive Fuzzy Inference?
  • Standard Fuzzy Control Example: Balancing an Inverted Pendulum
  • History and Basics of Fuzzy Logic
  • Fuzzy Inference System
  • Fuzzy Control Basics
  • Two Detailed Fuzzy Control Examples
  • Conclusion: When to Use Adaptive Fuzzy Inference?
  • Mathematical Takagi–Sugeno Fuzzy Dynamics

In this Chapter¹ [note: The work presented in this Chapter has been performed in collaboration with Dr. Sharon Boswell, Ag Group Leader Behavior and Control, Decision Sciences, Joint and Operations Analysis Division, Defence Science and Technology Group, Australia.] we extend the modern theoretical concept of the 2D quantum turbulence (see [Tsu06, TKK10, IR12]), experimentally confirmed in Bose-Einstein condensates (BEC, see [PS03]), to derive the analytical (closed-form) model of the 3D turbulent wind flow, which we call the “wind canon.” This wind canon is used as a soft attrition weapon against a team of UAVs flying in an urban environment, in which each UAV has a sophisticated collision avoidance system based on a 3D fuzzy-logic collision-avoidance controller (described below). In response to the wind turbulence, the team exhibits the behavior called fuzzy entanglement.

The following sections are included:

• Bayesian Probability Basics
• Kalman’s State-Space LQR/LQG Control Systems.
  • State-Space Formulation for Linear MIMO Systems
  • Linear Stationary Systems and Operators
  • Kalman’s LQR/LQG Controller
• Kalman Filtering Basics
  • Classical (Linear) Kalman Filter
  • Extended (Nonlinear) Kalman Filter
  • Unscented Kalman Filter
  • Ensemble Kalman Filter and Nonlinear Estimation
• General Bayesian Filter and Cognitive Control
  • Bayesian Filter
  • Cognitive Dynamic and Control
  • Bayesian Programming Framework with Robotic Applications
• Particle Filters: Superior Estimation Models for CPC-Autonomy
  • Particle Filtering Basics
• Low-Dimensional FastSLAM Algorithms
• High-Dimensional FastSLAM Algorithms

In this Chapter,¹ [note: The work presented in this chapter has been performed in collaboration with Dr. Martin Oxenham, Decision Sciences Branch, Joint and Operations Analysis Division, Defence Science and Technology Group, Australia.] a general model is presented for a universal large-scale fleet/swarm of all unmanned vehicles, including Aerial Vehicles (UAVs), Ground Vehicles (UGVs), Sea Vehicles (USVs) and Underwater Vehicles (UUVs), as a Kähler super-dynamics extension of our recent work [IY16]. Based on Newton-Euler dynamics of each vehicle, a control system for the universal autonomous fleet is designed in both Lagrangian and Hamiltonian form, to predict and control all degrees-of-freedom of each individual vehicle in the fleet (regardless of the fleet size). As an application, a 3D simulation is developed, presenting a search-and-rescue operation in an urban environment. The Appendix defines the formalism for the associated continuous system, representing a very large fleet, given in the form of the Kähler–Ricci flow (or, complex Monge–Ampère equation).

Here we present a general computational tensor framework for socio-econo-political sciences. It is based on geometric computational methods validated in theoretical physics, robotics, control theory and various kinds of networks (electrical, neural, biokinetic, ‘small worlds’, etc.) The basic underlying structure is that of a high-dimensional configuration space, technically called the ‘smooth manifold’, which is a set of all system’s degrees-of-freedom, formalized as a non-linear deformation of a (linear) vector space. Once the configuration manifold of a system under consideration (e.g., any kind of a ‘social game’, together with its complex decision strategies, or ‘tactics’) is defined as a ‘nonlinear vector space’, then a variety of computational tools, the so-called ‘tensors’, can be defined on it (here we apply the basic computational concept of Wolfram’s Mathematica: ‘everything is a tensor’, including, spinors, twistors, exterior differential systems, chains and cochains). These tensors include a variety of ‘socio-economic forces’, which can be derived as gradients of ‘socio-political energies’, upon which the system can be optimized using standard (Euler-Lagrangian) variational techniques, etc. The psycho-social basis for this ‘tensors-on-a-manifold’ formalism is classical Lewinian ‘field theory in social science’, in which the collective configuration space is a ‘topological product’ of individual ‘life spaces’.

The following sections are included:

• Classical Artificial Neural Networks as Simplistic Brain Models
  • Biological Versus Artificial Neural Nets (ANNs)
  • Most Popular Classical Discrete ANNs
  • Most Popular Classical Continuous ANNs
  • Recurrent Neural Nets (RNNs)
  • Grossberg’s Adaptive Resonance Theory
  • Hopfield’s Associative RNNs
  • Kosko’s Bidirectional Competitive RNNs
  • Support Vector Machines (SVMs)
  • Spiking Neural Nets as Axonal Brain Models
• Current Research in AI and Supercomputing
  • Strong AI vs. Weak AI
  • IBM’s Watson and TrueNorth vs. Top Supercomputers

The following sections are included:

• Bibliography
• Index