A Categorical Theory of Hybrid Systems by Aaron David Ames B.A. (University of St. Thomas) 2001 B.S. (University of St. Thomas) 2001 A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Engineering—Electrical Engineering and Computer Sciences in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Shankar Sastry, Chair Professor Mariusz Wodzicki Professor Alberto Sangiovanni-Vincentelli Fall 2006 The dissertation of Aaron David Ames is approved: Chair Date Date Date University of California, Berkeley Fall 2006 A Categorical Theory of Hybrid Systems Copyright 2006 by Aaron David Ames Abstract A Categorical Theory of Hybrid Systems by Aaron David Ames Doctor of Philosophy in Engineering—Electrical Engineering and Computer Sciences University of California, Berkeley Professor Shankar Sastry, Chair This dissertation uses the formalism of category theory to study hybrid phenomena. One be- gins with a collection of “non-hybrid” mathematical objects that have been well-studied, together with a notion of how these objects are related to one another; that is, one begins with a category C of the non- hybrid objects of interest. The objects being considered can be “hybridized” by considering hybrid objects over C consisting of pairs (D,A) where D is a small category of a specific form, termed a D-category, which encodes the discrete structure of the hybrid object and A : D C → is a functor encoding its continuous structure. The end result is the category of hybrid objects over C, denoted by Hy(C). In Part I, the foundations for the theory of hybrid objects are established. After reviewing the basics of category theory, inasmuch as they will be needed in this dissertation, D-categories are formally introduced. Hybrid objects over a general category C are then defined along with the corresponding no- tion of a category of hybrid objects. Elementary properties of categories of this form are discussed. We then proceed to relate the formalism of hybrid objects to hybrid systems in their classical form, the end re- sult of which is a categorical formulation of hybrid systems together with a constructive correspondence between classical hybrid systems and their categorical counterpart. Finally, executions or trajectories of both classical and categorical hybrid systems are introduced, and they are related to one another—again in a constructive fashion. Part II applies the categorical theory of hybrid objects to obtain novel results related to the re- duction and stability of hybrid systems. The geometric reduction of simple hybrid systems is first con- sidered, e.g., conditions are given on when robotic systems undergoing impacts can be reduced. As an application of these results, it is shown that a three-dimensional bipedal robotic walker can be reduced to a two-dimensional bipedal walker; the result is walking gaits in three-dimensions based on correspond- ing walking gaits for a two dimensional biped—walking gaits that simultaneously stabilize the walker to 1 the upright position. Using hybrid objects, the reduction results for simple hybrid systems are general- ized to general hybrid systems; to do so, many familiar geometric objects—manifolds, differential forms, et cetera—are first “hybrizied.” The end result is a hybrid reduction theorem much in the spirt of the clas- sical geometric reduction theorem. This part of the dissertation concludes with a partial characterization of Zeno behavior in hybrid systems. A new type of equilibria, Zeno equilibria, is introduced and sufficient conditions for the stability of these equilibria are given. Since the stability of these equilibria correspond to the existence of Zeno behavior, the end result is sufficient conditions for the existence of Zeno behavior. The final portion of this dissertation, Part III, lays the groundwork for a categorical theory, not of hybrid systems, but of networked systems. It is shown that a network of tagged systems correspond to a network over the category of tagged systems and that taking the composition of such a network is equiva- lent to taking the limit; this allows us to derive necessary and sufficient conditions for the preservation of semantics, and thus illustrates the possible descriptive power of categories of hybrid and network objects. Professor Shankar Sastry Dissertation Committee Chair 2 To my mother, Catherine A. Bartlett, for always nurturing my talents, giving me the strength to pursue them and the wisdom to know how. i Contents Table of Contents ii List of Figures iv List of Tables vi Introduction vii I Foundations 1 1 Hybrid Objects 2 1.1 Categories . 4 1.2 D-categories . 13 1.3 Hybrid Objects . 24 1.4 Cohybrid Objects . 31 1.5 Network Objects . 35 2 Hybrid Systems 38 2.1 Hybrid Systems . 39 2.2 Hybrid Intervals . 52 2.3 Hybrid Trajectories . 57 II Hybrid Systems 65 3 Simple Hybrid Reduction & Bipedal Robotic Walking 66 3.1 Simple Hybrid Lagrangians & Simple Hybrid Mechanical Systems . 69 3.2 Simple Hybrid Systems . 74 3.3 Hybrid Routhian Reduction . 79 3.4 Simple Hybrid Reduction . 88 3.5 Bipedal Robotic Walking . 98 4 Hybrid Geometric Mechanics 117 4.1 Hybrid Differential Forms . 121 4.2 Hybrid Lie Groups and Algebras . 126 4.3 Hybrid Momentum Maps . 131 4.4 Hybrid Manifold Reduction . 137 4.5 Hybrid Hamiltonian Reduction . 141 ii 5 Zeno Behavior & Hybrid Stability Theory 147 5.1 Zeno Behavior . 149 5.2 First Quadrant Hybrid Systems . 152 5.3 Zeno Behavior in DFQ Hybrid Systems . 156 5.4 Stability of Zeno Equilibria . 166 5.5 Going Beyond Zeno . 178 III Networked Systems 188 6 Universally Composing Embedded Systems 189 6.1 Universal Heterogeneous Composition . 191 6.2 Equivalent Deployments of Tagged Systems . 199 6.3 Networks of Tagged Systems . 203 6.4 Universally Composing Networks of Tagged Systems . 206 6.5 Semantics Preserving Deployments of Networks . 210 Appendix 213 A Limits 213 A.1 Products ..................................................213 A.2 Equalizers and Pullbacks . 218 A.3 Limits....................................................220 A.4 Limits in Categories of Hybrid Objects . 223 Future Directions 227 Bibliography 229 Index 238 iii List of Figures 1.1 The edge and vertex sets for a D-category. 16 2.1 The bouncing ball. 40 2.2 The hybrid model of a bouncing ball. 41 2.3 The hybrid model of a two water tank system. 43 2.4 The hybrid manifold for the bouncing ball. 46 2.5 The hybrid manifold for the water tank system. 47 2.6 A graphical illustration of an execution. 59 2.7 Positions and velocities over time of an execution of the bouncing ball hybrid system. 61 3.1 Ball bouncing on a sinusoidal surface. 71 3.2 Pendulum on a cart. 72 3.3 Spherical pendulum mounted on the floor. 74 3.4 Positions y1 vs. y2 and velocities over time of HBµ . ......................... 86 3.5 Positions and velocities over time, as reconstructed from the reduced system HCµ . 88 3.6 Reconstruction of the reduced spherical pendulum: position of the mass and angular veloc- itiesovertime................................................ 97 3.7 Two-dimensional bipedal robot. 107 3.8 Three-dimensional bipedal robot. 110 3.9 θns, θs and ϕ over time for a stable walking gait and different initial values of ϕ. 113 3.10 Phase portraits for a stable walking gait. The black region shows the initial conditions of ϕ and ϕ˙ that converge to the origin. 114 3.11 The initial configuration of the bipedal robot. 115 3.12 A walking sequence for the bipedal robot. 116 4.1 A trajectory of the two-dimensional (completed) bouncing ball model. 120 5.1 Zeno behavior that effectively makes a program (Matlab) halt. 150 5.2 An example of Genuine Zeno behavior (left). An example of Chattering Zeno behavior (right).151 ball 5.3 The hybrid system HFQ ..........................................153 5.4 The phase space of the diagonal system given in Example 5.3 for c 1 (left) and c 1 = = − (right). 159 5.5 A simulated trajectory of the two tank system given in Example 5.4. 165 5.6 Zeno equilibria for a FQ hybrid system. 168 5.7 A graphical representation of the “local” nature of relatively globally asymptotically stable Zeno equilibria. 170 5.8 Convergence to a Zeno equilibria. 171 5.9 Graphical verification of the properties (I) (III) for the bouncing ball hybrid system for the − first domain (left) and the second domain (right). 175 iv 5.10 A graphical illustration of the construction of the functions P1 and P2 for the bouncing ball. 176 5.11 A Lagrangian hybrid system: HL.....................................179 5.12 A completed hybrid system: HL. ....................................181 5.13 Simulation gets stuck at the Zeno point. Velocities over the time (top), displacements over the time (middle) and displacement on the x3 direction vs. the displacement on the x2 di- rection (bottom). 183 5.14 Simulation goes beyond the Zeno point. Velocities over the time (top), displacements over the time (middle) and displacement on the x3 direction vs. the displacement on the x2 di- rection (bottom). 184 5.15 Simulation of the Cart that goes beyond the Zeno point. Velocities over the time (top) and displacements over the time (bottom). 186 5.16 Simulation of the Pendulum goes beyond the Zeno point. Angular velocities over the time (top) and angles over the time (bottom). 187 6.1 A graphical representation of a behavior of a tagged system. 193 v List of Tables 0.1 Chapter dependency chart. ix 1.1 Important.