UBC Theses and Dissertations

UBC Theses Logo

UBC Theses and Dissertations

Stabilization via smooth partitions, transversality and graphs Glaum, Michael 1993

Your browser doesn't seem to have a PDF viewer, please download the PDF to view this item.

Item Metadata

Download

Media
831-ubc_1993_fall_glaum_michael.pdf [ 3.35MB ]
Metadata
JSON: 831-1.0079675.json
JSON-LD: 831-1.0079675-ld.json
RDF/XML (Pretty): 831-1.0079675-rdf.xml
RDF/JSON: 831-1.0079675-rdf.json
Turtle: 831-1.0079675-turtle.txt
N-Triples: 831-1.0079675-rdf-ntriples.txt
Original Record: 831-1.0079675-source.json
Full Text
831-1.0079675-fulltext.txt
Citation
831-1.0079675.ris

Full Text

STABILIZATION VIA SMOOTH PARTITIONS, TRANSVERSALITY AND GRAPHS By Michael Glaum B. Math., University of Waterloo, 1991  A THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF MASTER OF SCIENCE  in THE FACULTY OF GRADUATE STUDIES DEPARTMENT OF MATHEMATICS INSTITUTE OF APPLIED MATHEMATICS  We accept this thesis as conforming to the required standard  THE UNIVERSITY OF BRITISH COLUMBIA  August 1993 © Michael Glaum, 1993  In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of th is thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission.  (Signature)  Department of  ^V\ Ck+t'N e  The University of British Columbia Vancouver, Canada  Date  DE-6 (2/88)  4.0995+- 110,, )99?)  CS  Abstract  With the aim of circumventing the difficulty in constructing Liapunov functions, a strategy for the design of static stabilizing feedback control laws of nonlinear systems is proposed. The basic method is to partition the state space and to find all controls so that the closed-loop dynamics are transverse or coincident to the partition edges. Stability is analyzed through the directed graph whose vertices are the subsets of the partition and whose arcs are consistent with the transversality. A strategy for the choice of partition is proposed using computable Pfaffian systems.  11  Table of Contents  ii  Abstract^ List of Figures^  vii  Acknowledgement^  x  1  Introduction  1  1.1  Systems, Feedback and Design ^  1  1.2  Liapunov's Method ^  4  1.3  Overview ^  5  2  Modelling  6  Control Theory Fundamentals  7  2.1  Engineering Motivation ^  7  2.2  States, Signals and Systems ^  7  2.2.1^Dynamical Systems ^  8  2.2.2^Input/Output Systems ^  9  2.3  Feedback ^  2.4  Example - The Simple Robot Arm  2.5  Tracking ^  14  2.6  Stabilization within Control Design ^  16  10  111  ^  11  3  II 4  Differential Geometry in Control Theory  18  3.1  Overview ^  18  3.2  Example - The Spherical Pendulum ^  20  3.3  Example - The Simple Robot Arm ^  23  3.4  Tentative Formalism ^  25  3.5  Example - Controlled Constrained Motion ^  27  3.6  Formalism ^  29  3.6.1^Nominal Stabilization of Nonlinear Control Systems ^  31  3.6.2^Example §3.5 continued  31  ^  Design  33  Smooth Partitions, Transversality and Graphs  34  4.1  Motivation ^  34  4.2  Nonlinear Control Systems Modelled as Finite-State Machines ^  35  4.3  Definitions ^  36  4.3.1^Smooth Partitions  37  4.4  4.5  ^  4.3.2^Transversality ^  37  4.3.3^Example ^  38  4.3.4^Graphs of Framed Dynamical Systems ^  40  4.3.5^Example §4.3.3 continued  40  ^  4.3.6^Liapunov Functions ^  41  Liapunov's Theorem ^  42  4.4.1^Example §4.3.3 continued  51  ^  Liapunov Functions for Control Systems ^  iv  51  4.6 Design of Coincident and Transverse Edges and Face-Monotone Dynamics of Control Systems ^  51  4.7 Graphs of Control Systems ^  52  4.8 Example: the 2-integrator ^  53  4.9 A Proposed Method for the Design of Framed Dynamics ^ 54 4.9.1 Example §4.6.1 continued 4.10 Example-A Liapunov Method. ^ 4.10.1 Example: the n-integrator. .  ^56 57 ^58  5 Construction of Smooth Partitions and Liapunov Functions^61  5.1 Tentative Construction ^  61  5.2 Pfaffian Systems ^  62  5.2.1 A Classical Linear Pfaffian Problem ^  62  5.2.2 The Nonlinear 'OR' Pfaffian Problem ^  63  5.3 Transversality Multifunctions ^ 5.3.1 Example: the 2-integrator ^  64 65  5.4 Liapunov Functions ^  66  5.5 Smooth Partitions ^  68  5.6 Example ^  68  6 Conclusions^  74  Appendices^  76  A Prerequisite Mathematical References^  76  A.1 Basics ^  76  A.1.1 Basic Topology ^  76  A.1.2 Algebra ^  76  A.2 Geometric Analysis ^  77  A.2.1 Advanced Calculus = Local Differential Geometry ^ 77 A.2.2 Manifolds ^  77  A.2.3 Bundles ^  77  A.2.4 Algebraic Structures over Manifolds ^  77  A.2.5 Topological Dynamics ^  78  A.2.6 Exterior Differential Systems  ^78  A.2.7 Geometric Mechanics ^  78  A.2.8 Functional Analysis ^  78  A.3 Control and System Theory ^  78  A.4 Miscellaneous ^  79  A.4.1 Graph Theory ^  79  A.4.2 Automata Theory ^  79  A.4.3 Multifunctions ^  79  B Cited Theorems^  81  B.1 Long Tubular Flows ^  81  B.2 Flow-Connectedness ^  82  B.3 Location of Zeros ^  84  B.4 Non-triviality of the Tangent bundle to the 2-Sphere ^ 84 B.5 Contractability of Domains of Attraction ^  85  B.6 Frobenius' Theorem ^  85  B.7 Input-Output Stability via Asymptotic Stability ^  85  B.8 LaSalle's Invariance Principle ^  86  Bibliography^  88  vi  List of Figures  1.1 Components within the plant  ^2  1.2 A closed-loop interconnection of plant and controller.  ^3  1.3 A compensation interconnection between plant and controller.  ^3  1.4 Orbits converging on level curves of a Liapunov function  ^4  1.5 A partition of the state space and a superimposed direction field together implying a graph  ^5  2.6 A typical industrial robot.  ^7  2.7 A simplified motor component.  ^8  2.8 A dynamical system or flow on a plane and its geometric meaning. . . . ^9 2.9 Graphic representation of an input/output system ^ 10 2.10 A feedback loop interconnecting the plant and controller ^ 11 2.11 A controlled-observed robot arm ^  12  2.12 Free body diagram of the mass m ^  12  2.13 Open-loop robot arm ^  13  2.14 A robot arm with vertically up-motionless stable. ^ 14 2.15 The open-loop tracking strategy of tandem connection with an inverse. ^ 15 2.16 A compensator plant controller interconnection ^ 2.17 A reference input r(.) and the compensated arm's response (2.11)  15  OH satisfying ^16  2.18 A)Compensated I/O system and b)associated dynamical system ^ 17 3.19 The spherical pendulum ^ vii  20  3.20 The spherical coordinate system (3.16)  ^21  3.21 The second spherical coordinate system (3.22) ^  22  3.22 A robot arm before a) and after c) winding b) ^  23  3.23 Closed-loop dynamics (3.26) under (3.28) with three orbits (ir, 0),  r+, r_  not attracted to (0, 0)  ^25  3.24 A vector field f E X(M) defined on a manifold M ^  26  3.25 A commutative diagram for the tentative definition ^ 27 3.26 Mass in constrained to move on a manifold N propelling itself by a force in the tangent space. ^ 3.27 Commutative diagram for f : P 4 TM ^ -  27 30  4.28 Level sets of a Liapunov function and the orbits of a closed-loop dynamical system converging on them ^ 4.29 An example of a finite-state machine ^  35 35  4.30 a) the direction field of a dynamical system across some triangles, and b) the associated finite state machine. ^  36  4.31 Flows a) transverse to, b)coincident with, and c)neither transverse to nor coincident with an edge. ^ 4.32 Phase portrait of (4.44) ^  38 39  4.33 a) A smooth partition framing the flow (4.44) and b) the superposition of the direction field.  ^39  4.34 The graph associated with the flow (4.42) and the smooth partition in Figure 4.33 ^  40  4.35 Conceivable situations that Liapunov's theorem excludes ^ 44 4.36 Geometry of a tubular flow around -y ^  48  4.37 Rectified flow in a subtube from F_ to F+ ^  48  viii  ^  4.38 Geometry to exclude closed trajectories.  ^  50  4.39 The graph associated with the control system (4.57) and the smooth par^  tition in Figure 4.33  54  4.40 An attracting invariant manifold which contains an attracting equilibrium ^  point. 5.41 A normal field and a surface normal to it. ^ 5.42 Foliation of a selection of^ 5.43 Foliations of selections of  57  63 66 ^71  N-02  5.44 A smooth partition for (5.102)  ^71  5.45 The graph associated with (5.102) and Figure 5.43 ^ 72 5.46 A stable subgraph of the graph in Figure 5.44 ^  73  B.47 A flow rectified by a diffeomorphism (I) ^  81  B.48 Rectified flow past a transverse submanifold ^  82  B.49 E attracts M and attracts E B.50 A flow with an attracting homoclinic orbit  ix  82 83  Acknowledgement  I am indebted to Drs Wayne Nagata and Frank Karray for all their help but especially for their encouragement in preparing this thesis. Many of my colleagues have contributed to my education and this thesis, but I would like to acknowledge David Austin, Alan Boulton, Andrew Hare, Charlie Horn, Alex Kachura, Djun Kim, Philip Loewen, Alan Lynch, Sheena McRae, Kirsten Morris, Rob Ritchie, Stephen Smith and John Wainwright for their academic suggestions. Finally, it is a pleasure to thank Gfil Civelekoklu, Lynn Van Coller, David Fisher, Martin Fry, Margie Grier, Chris Piggott, the Glaum family, the Wiley family, Canada, coffee, Gustav Mahler, NSERC, Dimitri Shostakovich, the Yukon, Acer Saccharum, Carya ovata, Juglans nigra, Pinus strobus, Populus tremuloides, Pseudotsuga menziesii and Quercus alba.  x  Chapter 1  Introduction  Advances in engineering, science and technology have had a profound effect on Canadian society. Popular examples are the tangible advances such as the invention of plastics or the use of computers. A more pervasive but inconspicuous advance, however, was the invention of the idea of active control, for example, feedback. Indeed, feedback is used in walkmans, in telecommunications, for cruise-control and anti-lock brakes in automobiles, in Tokomak reactors, and even in our own physiological processes. The necessary choice of mathematical model and control strategy, however, has a crucial effect on the performance of systems using active control. The classical linear models and control strategies suffice in many instances, but for advanced systems with inherent nonlinearity, or where performance is essential, the more realistic but difficult nonlinear models and strategies are necessary. In this thesis we propose a control strategy for engineering systems modelled by nonlinear differential equations. 1.1 Systems, Feedback and Design A typical example of a nonlinear system is a robot arm. Such a system must contain actuator and sensor components to control and observe itself, as in Figure 1.1. In the example of a robot arm, typical actuator and sensor hardware include motors and potentiometers respectively, placed at the joints of the arm. A motor accepts a voltage input 1  Chapter 1. Introduction^  2  Figure 1.1: Components within the plant. signal to influence the arm, and a potentiometer outputs a voltage signal representing some observation of the arm. As stated, the arm may be influenced, e.g. made to perform a task, by selecting an input signal to relay to the actuator. By augmenting the arm with a controller to automatically make this selection, the robot arm may then be made to "control itself". A controller's choice of instantaneous input is often based on the fed back output observed from the sensors. The controller may then be summarized in a feedback law which associates a control with an observed output. Using such a strategy, the control problem within system design may be isolated as the choice of the feedback law and the analysis of its influence on the "closed-loop" system. The choice of feedback law is based on a mathematical input/output model, called the plant P, representing the effect of a control signal u on the output signal y. Controllers are often implemented using digital computers so that the controller may be reprogrammed easily. It is important to note that it is the mathematically designed software executing on the computer which actually "controls" the system. An input/output model for systems such as robot arms may be derived via Newtonian, Lagrangian or Hamiltonian mechanics. In the simplest situation, the model is realized by nonlinear but smooth deterministic differential equations in a finite number of variables,  Chapter I. Introduction^  3  Figure 1.2: A closed-loop interconnection of plant and controller.  Figure 1.3: A compensation interconnection between plant and controller. that is, a class accommodated within the sequel. A basic requirement of the closed-loop system is global stability: intuitively, that a prescribed orbit, e.g. an equilibrium point, attracts all other orbits, or bounded reference inputs r produce bounded outputs y, respective of Figures 1.2 and 1.3. Other design specifications implicitly assume stability or are essentially implied by stability, for example tracking. In conclusion, we have isolated within engineering system design the mathematical problem of choosing a feedback law for a system given an Input/Output model realized by a system of nonlinear differential equations so that the closed-loop is stable, i.e. the  Nominal Stabilization Problem.  Chapter 1. Introduction^  4  Figure 1.4: Orbits converging on level curves of a Liapunov function. 1.2 Liapunov's Method While there is no general method for the construction of stabilizing feedback control laws of nonlinear systems, the most viable is Liapunov's method [AV]. This method is contingent upon the designer being able to find a function satisfying algebraic hypotheses relating the geometry and dynamics of the flow. The deficiency of this method is that there is no algorithm for constructing such a function. In this thesis, we re-interpret the algebraic "definiteness" hypotheses of Liapunov's method in terms of the geometric notion of transversality to a submanifold. The concepts of partition, transversality and graphs are then used to give a geometric version of Liapunov's method. Intuitively, the stability relationship between geometry and dynamics in Liapunov's classical method is implied as in Figure 1.4, while for the method of this thesis, by the graph implied by a direction field superimposed over a partition of the state space, as in Figure 1.5. Differential geometry gives a convenient, if not necessary, language to formulate the above concepts.  Chapter 1. Introduction^  5  Figure 1.5: A partition of the state space and a superimposed direction field together implying a graph. 1.3 Overview In chapters two and three we review the background for modelling mechanical systems, specifically, control theory fundamentals and geometric mechanics respectively. Some of the mechanical examples of chapter three appear to give a new perspective of exhibiting phenomena discussed only within the modern theory of differential geometry. Chapter four outlines the stated nonlinear design strategy. The author is not aware of the method appearing anywhere in the literature, but there is some evidence that similarities exist with the theory of hybrid systems [LS]. The aim of this thesis is not to present a definitive algorithm but to initiate the use of certain geometric methods. In chapter five, strategies are presented for constructing partitions and Liapunov functions. Some mathematical prerequisites are reviewed in the appendices.  Part I Modelling  6  Chapter 2 Control Theory Fundamentals  2.1 Engineering Motivation Consider a programmable robot arm as in Figure 2.6. Such a robot must have actuators, for example, simplified motor components as in Figure 2.7.  Question: 2.1.1 In figure 2.7, what voltage do we apply to the motor, say, to have the arm 2 track a predetermined trajectory? This engineering tracking problem may be taken as one motivation for the mathematics of control and system theory.  2.2 States, Signals and Systems In this section we will contrast the "dynamical system" and "input/output" models discussed in this thesis, using the arm in the previous section as an example.  Figure 2.6: A typical industrial robot. 7  Chapter 2. Control Theory Fundamentals^  8  Figure 2.7: A simplified motor component. 2.2.1 Dynamical Systems Once programmed, the complete arm is an example of a mechanical system, with measurable idealized physical properties such as instantaneous relative angular displacement, velocity and acceleration, mass, etc. The arm is, furthermore, dynamic, in that some of the measurements may evolve over time. Depending on the implementation, the arm may be deterministic, meaning that if all the physical properties of the arm were measured at one instant, there exist models which could predict all future measurements very well. Associated with such models is the concept of state, specifically, the least amount of instantaneous information x which uniquely determines all other measurable physical properties. The set M of all attainable states is called the state space. For the arm above, the set M := R2 of all paired attainable relative angular displacements and velocities at any instant may be taken as the state space, depending upon the implementation of the arm and controller. The evolution of the system is then a path xc : R -4 R2(= M) in the state space, identified by the initial state e E R2 at time 0. We summarize this information by defining the map ç: R x M M by  (t , e) := xe (t)^  (2.1)  for all t E R. The map 0 is called a dynamical system or flow. The geometric meaning  Chapter 2. Control Theory Fundamentals^  9  Figure 2.8: A dynamical system or flow on a plane and its geometric meaning. of 0 is, on one hand, that t^0(t, xo) models the evolution of the arm when initialized at a state xo E M, and on the other hand, that x 1-4 0(to,x) summarizes the evolution of the arm after time to over all initial states x E M. See Figure 2.8. The dual interpretation of a flow is useful for defining stability, asymptotic stability, invariant sets, etc. The reader is asked to refer to [HS] for a discussion of these and further concepts which occur frequently in this thesis. 2.2.2 Input/Output Systems  Before the arm is programmed, however, we must discuss how the external concepts of input, output and causality effect on the internal dynamics of the arm. The voltage applied to the motor, the input, and the measured angular displacement and velocity of the arm, the output, as vector-valued functions of time, are examples of signals. The transformation from input signal to output signal, as in the arm, is called an input/output system, or I/O system for short.  In figures, signals and input/output systems are represented by directed lines and black-boxes respectively, as in Figure 2.9, and are often given names like P or C for  Chapter 2. Control Theory Fundamentals ^  Co n iTE).\ ••btrottl  Motor, ^ robe:* cm-t.4 art 6, Tcyt-e.n-t-;e>tAk4er'  10  ^9 0 IDSeNc4Ibr, ■w■ kS  Figure 2.9: Graphic representation of an input/output system. plant (the arm in our case) or controller. This suggests we should model the algebraic structure of the (initialized) I/O motorarm-potentiometer system by an operator  P: C(R, R) —+ C (R, R2) : v(.)^(  et) 9(.)  (2 .2)  mapping a continuous voltage signal to a smooth angular displacement and velocity signal. These linear operators are frequently realized by systems of differential equations or, through Laplace transforms, by matrices of complex functions. We will consistently use the former representation in this thesis. The reader is asked to refer to [Vid] for a discussion of (UBIBO) I/O stability, the I/O version of dynamical systems stability: intuitively, that a bounded control signal produces a bounded output signal. 2.3 Feedback One strategy for implementing control is through the use of feedback, that is, basing our choice of instantaneous control on the past and present observed output. Under this  Chapter 2. Control Theory Fundamentals^  11  Figure 2.10: A feedback loop interconnecting the plant and controller. scheme the control is often effected on the plant through a I/O system called the controller C connected to the plant as in Figure 2.10 resulting in an autonomous dynamical system exhibiting closed loop dynamics. -  Feedback control theory is concerned with the design of the controller C and the analysis of its influence on the closed-loop dynamics 2.4 Example  -  The Simple Robot Arm  Feedback is particularly useful for stabilizing an unstable equilibrium of a mechanical system, using the controller to compensate for perturbations and the tendency to move away. Following [So], consider the vertical robot arm modelled as a point mass m at the end of a rigid massless rod of length 1 with a motor at the pivot supplying a variable torque u and a potentiometer observing the angular displacement and velocity as in Figure 2.11. Suppose we wish to stabilize the arm against gravity and perturbations into a vertically upward motionless state (say while other robot arm components perform a task). Assume that the potentiometer measures angular displacements in a continuous fashion. That is, if the arm rotates 27r radians clockwise, then 27r is added to the displacement.  Chapter 2. Control Theory Fundamentals ^  12  Figure 2.11: A controlled-observed robot arm.  Figure 2.12: Free body diagram of the mass m. As such, let 0 E R denote the clockwise angular displacement of the arm from vertically up. From Figure 2.12 and Newton's second law, the governing equation is u m/Ö = mg sin 0 + — / '^  (2.3)  Dr in terms of the state vector (xi, x2) := (OM E R2,  o  1 )u-  (2.4)  m12  The equation (2.4) is a model for the open-loop robot arm and potentiometer: given a control u(.) : [0, oo) —> R in LI, (2.4) has a unique solution x : [0, co) —> R2 as in  Chapter 2. Control Theory Fundamentals^  13  Figure 2.13: Open-loop robot arm. Figure 2.13 ( see [An]). Using a primitive version of the technique of feedback linearization [Is], we now change control variables to linearize (2.4). Specifically, let ft  := mlg(—xi + sin xi) + u^  (2.5)  so that on substitution, (2.4) becomes, in standard form, 0 1 )^)^0 +^1 }IL 0 X2 1^ M12  (2.6)  As an aside, note that it not always possible to linearize control systems. If we apply a linear feedback law, say  fi(xi, x2) := (—a — 0)  (2.7) \ X2 I  via Figure 2.10, then the associated closed-loop dynamics is obtained by substituting (2.7) into (2.6). We calculate  01) g n v^X2 1  0 1^(—a 13) X1 ( X2 ) n—Tri  (2.8)  Chapter 2. Control Theory Fundamentals^  14  Figure 2.14: A robot arm with vertically up-motionless stable.  Or (2.9)  —m12 We must choose a and so that (2.9), the closed-loop dynamics, will have have (0,0) as a stable equilibrium point, i.e. the coefficient matrix must has eigenvalues with negative real part. Necessary and sufficient conditions are easily seen to be > mgl, > 0.  (2.10)  In conclusion, if we choose any a and 13 satisfying (2.10), the robot arm of the closedloop dynamical system in Figure 2.14 will rise to the vertically upward motionless state and return there when perturbed.  2.5 Tracking Recall the tracking problem of § 2.1. If we model the motor+arm+potentiometer by the plant in Figure 2.13, then the simplest strategy would be to connect P in tandem with an "inverse" Q as in figure 2.15 (derived, say, by substituting x1 = r, x2 = i into (2.4) and solving for u, where r is the reference to be tracked ). There are fundamental math-  Chapter 2. Control Theory Fundamentals ^  15  Figure 2.15: The open-loop tracking strategy of tandem connection with an inverse.  Figure 2.16: A compensator plant controller interconnection. ematical and engineering difficulties with this strategy (causality, sensitivity, robustness) [Ka]. Alternatively, the stabilizer of the arm in the previous example may be used to solve the tracking problem motivated in the question of § 2.1. Specifically, if P and C represent the plant and controller of Figures 2.13 and 2.14, then associated with the stable dynamical system in Figure 2.10 is the  compensated I/O system in Figure 2.16 which takes  the desired trajectory r(.) to be tracked as input, properly scaled by m12, and outputs the angular displacement and velocity 9(.),O(.). It follows from (2.4) that these signals satisfy the differential equation  10 ) (^ xi )  0  g a —— ml2  m12  + r.  (2.11)  x2  For example, with a and # chosen so that  g a — —^= —1,^  / m/2^m /2  = —2^  (2.12)  Chapter 2. Control Theory Fundamentals^  16  Figure 2.17: A reference input r(.) and the compensated arm's response OH satisfying (2.11). and for the reference trajectory for t < 0  0  r(t) :=  1 for 0 < t < — 2 2 1 1 —t2 —2t + — for — < t < 1 2 2 1 for 1 < t  -1-t2  (2.13)  a simulation of the response of the arm is given in Figure 2.17. The stability of this I/O system is asserted in the next section. 2.6 Stabilization within Control Design In most popular design strategies, for example I1' control [DFT] [BB}, the assumption -  is often made that the plant is I/O stable. Thus, to apply these powerful methods to unstable plants we must first compensate them, that is, find a controller K0 so that  Chapter 2. Control Theory Fundamentals^  17  Figure 2.18: A)Compensated I/O system and b)associated dynamical system. the I/O system P(/ — K0P)-1 in Figure 2.18a) is I/O stable. But this I/O system  P(I — K0P)-1 would be stable if the closed-loop dynamical system in Figure 2.18b) is asymptotically stable, as argued by theorem B.7.2 of the appendix. Furthermore, any stabilizer would suffice as any undesired response could be undone through feedback so no regard need be paid towards the design specifications. Note that controllers used in Hc° are typically "observer-based", so we will assume an input to state model [So] for our plants in the sequel. For example, if the potentiometer -  -  in example 2.4 did not observe the complete state, then an observer would be used to asymptotically estimate the remaining state variables, and this estimated state would be used in the control law. In summary, within most control design problems we may isolate the nominal stabilization problem: given a plant P find a controller K0 so that the closed-loop system in figure 2.18b) is asymptotically stable. In this thesis, we will propose a framework for constructing such a controller for nonlinear but smooth plants P. We will now discuss the plants accommodated within this framework.  Chapter 3  Differential Geometry in Control Theory  3.1 Overview We now turn to an issue in the modelling of classical mechanical systems, spècifically, the structure of the state and control space. This chapter is intended to familiarize the reader with the basics of classical mechanics and differential geometry. The majority of this chapter is not referred to in subsequent chapters, and thus may be skipped on a first reading. The classical Lagrangian model [Go], for systems without control, is coordinate-based. One chooses n "generalized" coordinates xi E R, together with associated n "generalized" velocities i, E R, and define a Lagrangian function L : R2n ÷ R via energy arguments —  based on physical laws or models. The evolution of the mechanical system is then modeled by the Euler-Lagrange equations  d far, } 0L0  (3.14)  aii^axi— for 1 < i < n. The form of these differential equations is, most generally,  F(x,  =  0  (3.15)  for a function F: R3n- + R . Unfortunately, for systems such as the spherical pendulum —  discussed below, there are some singular states (x, th) for which (3.15) cannot be solved for i uniquely. The initial-value problem (3.15) with initial condition at one of these points thus has no unique solution. 18  Chapter 3. Differential Geometry in Control Theory^  19  More importantly, in a punctured neighborhood of these singular states, that is, where we can solve for i, say as  1  =  f(x,i),  (3.16)  the vector field f is typically discontinuous. In summary, flow computation is not a well-posed problem in the sense of Hadamard [Ga] for such an example. (Furthermore, numerical approximation of the differential equation in a neighborhood of these points is prone to error.) As this indeterminacy is not observed in physical reality, the classical model of Euler-Lagrange is inappropriate for some problems. The physical states at which this indeterminacy occurs depends, suspiciously, on the choice of coordinates. In fact, we may recover an adequate well-posed model by "combining" two overlapping coordinate charts, with determinacy being implied in at least one coordinate chart, at each physical state. Differential geometry is the conceptual framework for such a model. It should be noted, however, that there are many other motivations for the use of differential geometry in mechanics. A few are illustrated in the following examples. Important motivations not mentioned here include nonholonomic systems [vW] and the theory of families of linear systems [HK]. In subsequent chapters we will speak of arbitrary subsets defining a partition of the state space and thus we will continue to use the language of differential geometry in the sequel. Abortive discussions and hints of the following examples and formalism may be found throughout the existing literature. Example §3.3 was taken almost verbatim from [So], but the other examples appear to give an original perspective and motivation. Example §3.5 was especially contrived to exhibit a phenomena hinted at in [Ba]. The formalism §3.6 is intended to be tentative, following [Br].  Chapter 3. Differential Geometry in Control Theory^  20  vtassew +o move on  "Sc."ffor }Figure 3.19: The spherical pendulum 3.2 Example - The Spherical Pendulum  Consider the spherical pendulum as in Figure 3.19, where the joint J has "two-degrees of freedom". The natural candidates for the generalized coordinates are the spherical coordinates (0, 0) E R2 in equations (3.17) and in Figure 3.20. x= cos sin 0 y= sin 9 sin 0  ^  (3.17)  z=^cos 0 For a specific configuration (0, 0), the potential energy of the mass m is PO, := mg cos 0, up to an additive constant. Differentiating (3.17) with respect to time along a trajectory (OH, OH), we have that the velocity v of the mass m in rectangular coordinates at a specific state (0,0,0,  (.k) is  v= (—sinesin0O-1-cosOcos44,cosOsin0O+sinOcos0i/),—sin0ik), ^(3.18) 1 using the notation sin 0 e := (sin 0) é, so the kinetic energy function —mil v113 may be 2^  Chapter 3. Differential Geometry in Control Theory ^  21  Figure 3.20: The spherical coordinate system (3.17). shown to be K(0,0,0, ())) = ;-m(sin2 0 42 + 42 ) •  Therefore, the Lagrangian L = K  —  (3.19)  P is  L(0, 0,0, i) = fli, (sin2 0 42 + CO) - mg cos (/).  (3.20)  Thus, the Euler-Lagrange equations are d  Tit [sin2 0 é]  = 0  -d(ik) - sin 0 cos 0 0 — g sin 0  =0  dt  Or  sin2 0 e =^—2 sin 0 cos 0 è ik 0^= sin q5 cos 0 02 + g sin 0.  (3.21)  Notice that any initial-value problem (3.21), 0(0) = 00, e(0) = eo, 0(0) = nor, 0(0) = 4o, n E Z does not have a unique solution as §(0) is not well-defined. Indeed, when 0 = mr, n E Z, 0 has no physical meaning. Furthermore, deleting these hyperplanes  Chapter 3. Differential Geometry in Control Theory ^  22  Figure 3.21: The second spherical coordinate system (3.23). = fir, n E Z, we may write (3.21) as —2cot0a  1 — sin 20 02 g sin 0. 2  (3.22)  The right-hand side vector field, through cot 0, is discontinuous. Thus any numerical solution based on these equations will be very prone to error near these hyperplanes. Again, the physical points at which indeterminacy occurs depends on the choice of coordinates. In the coordinate system (3.17), these physical states are vertically up or down, of any velocity. By a choice of a second superimposed coordinate system (3.23) say in Figure 3.21, x =^cos 02  y=  COS 02 sin 02^  (3.23)  = sin 02 sin 02  we obtain a second model si n2 02 tj.2 = —2 sin 02 cos 02 U2d)2 — g COS 02 sin 02 (i.;42^sin 02 COS 02 03 - g sin 02 COS 02.  (3.24)  Chapter 3. Differentia] Geometry in Control Theory ^  23  Figure 3.22: A robot arm before a) and after c) winding b). Numerical simulation of the evolution of the mass m requires both (3.21) and (3.24). We use the former on {(0, 0, 0, 0) : 10  —  nirl > c for all n} and the later on {(02, 02, 02, 02)  102 — nir j > € for all n}, for some € > 0, converting between them when one chart ends, via (3.17), (3.23) and their derivatives. This example illustrates the need for a conceptual framework dealing with multiple coordinate systems. This theory exists as the differential geometry of manifolds and vector bundles. The non-well posed phenomena occurred in this example because the natural state space TS2, the tangent bundle to the sphere S2 ( see § B.4.1) is not diffeomorphic to the classical Euler-Lagrange state space IV, with the discontinuity being reflected in equation (3.22). Indeed, now is an appropriate time to review the motivated differential geometric approach to mechanics described in [Ar2] and [Mar]. 3.3 Example  -  The Simple Robot Arm  Following [So], recall the simple robot arm of example of § 2.4 for the scenario of Figure 3.22 where the arm is externally rotated by 27r radians clockwise, n E Z, from equilibrium and released.  24  Chapter 3. Differential Geometry in Control Theory^  In implementations where there is no physical memory in the system of the number of complete revolutions from some reference, we would consider the wound state as returned to equilibrium. The unwinding by 27r radians counterclockwise would be unnecessary, so we seek a strategy different from that in § 2.4. In our current implementation, the natural configuration manifold of the arm is the circle SI rather than R. It may be shown that the dynamics/control of the arm is again governed by the differential equations w  0=  :g- si^u n0+ 1^m12  4:4) =  (3.25)  but rather on the state space manifold TS' = Si- x R, with coordinates 0 E Sl, and w E R.  As a consequence of theorem B.5.1, however, we have, following [So], Proposition 3.3.1 There does not exist a continuous static feedback law k : S' x R --+ R stabilizing (3.25). That is, there is no continuous k for which the differential equation  0= CO =  w  7 sin 0 + k(0,w) m12 g  (3.26)  on the manifold S' x R has a point (00, 0) as a global attractor. Proof: If such a k did exist, by continuous dependence on initial conditions [Ail], it  would define a continuous flow on SI x R. Indeed, the flow is maximally defined by the hypothesis the there is a global attractor. Furthermore, by that hypothesis, S1 x R is a domain of attraction of the flow. Thus, by the theorem B.5.1, S1 x R would be contractable, which is a contradiction. 0 There is no agreement in the literature on how to construct a stabilizer K in view of the previous proposition. We must use discontinuous feedback laws and/or suffice to  25  Chapter 3. Differential Geometry in Control Theory^  Figure 3.23: Closed-loop dynamics (3.25) under (3.27) with three orbits (it, not attracted to (0, 0).  o),r+,r_  attract only a portion of S' x R. In regards to the second alternative, as suggested by the damped vertical planar pendulum, any  K (0 , w) := a sin 9 + 13w^  (3.27)  with a < — mlg and 0 < 0 will stabilize (3.25) to (0,0) except for three orbits, as in Figure 3.23. 3.4 Tentative Formalism In summary, the first example § 3.2 demonstrates that for some mechanical systems, the differential geometric framework of [Mar] is necessary. The second example § 3.3 demonstrates that the topological structure of the state space is a consequence of the design specifications. By analogy with the development of linear control theory from the theory of linear differential equations, it seems fitting at this time to speculate on the extension of the  26  Chapter 3. Differential Geometry in Control Theory ^  Figure 3.24: A vector field f E X(M) defined on a manifold M. framework of [Mar] necessary to model the previous control examples, following [Br]: (H1) we model the state and control spaces on differentiable manifolds, say M and U. Thus the state+control space (which consists of the attainable states and the admissible controls while at that state) is modeled on a topological product manifold M x U. For later purposes, we note its trivial bundle structure : M x U^M: (x, u)1.-+ x. (H2) the effect of the control u(.) : R -÷ U on the state x(.):R—>Mis modeled by a differential equation i(t) = f (x(t),u(t))^  (3.28)  for some function (section)/ : M x U -4 TM ( i.e. satisfying f (x, u) E TM). The geometric meaning of f (or rather, just a vector field with no controls) is given in Figure 3.24. As regularity hypotheses, we assume that f is continuous and the diagram in Figure 3.25 commutes, where the tangent bundle 7rT : TM -+ M x maps a tangent vector to the point at which it is based at. The next example, however, does not fit this framework. Although it may appear contrived, it is a simplified idealization of a phenomena observed in the study of satellites.  Chapter 3. Differential Geometry in Control Theory ^  Mx  27  ---TM  -rr N  ZrrT  Figure 3.25: A commutative diagram for the tentative definition.  Figure 3.26: Mass m constrained to move on a manifold N propelling itself by a force in the tangent space. 3.5 Example - Controlled Constrained Motion Consider a mass m constrained to move on a frictionless embedded 2-submanifold N of R3. While at n E N, the mass is assumed to be able to apply to itself, i.e. by its own propulsion, any force in Tn.1■1, the tangent plane to N at n. See Figure 3.26. If we model m by a point, the configuration space is N. We might suspect that the control space is U := R2. Indeed, at each configuration n E N, we must map a control (u1, u2) E R2 to a vector in TEN, the propulsion force, as in Figure 3.26.  Chapter 3. Differential Geometry in Control Theory ^  28  Thus we must define a bijection (I) : N x R2 -4 TN, that is 4)(n, u) E TN for all n E N,u E R2. This map (I), coordinating the effect of propulsion control, is used in deriving the governing equations ± = f(x,u),x E TN, u E R2, for the mass m. The point of this example is that via theorem B.4.1, for N diffeomorphic to say S2, (I) and thus f cannot be continuous, thus violating hypothesis (H2) above. Thus the state+control space of this example does not have the topological product structure (TN) x U between state and control spaces. Instead,  Proposition 3.5.1 The state+control space has the natural rank two non-trivial vector bundle structure if : p*T N -4 TN given by the pullback of the natural configuration+control tangent bundle structure it : TN -- N under the natural state space tangent bundle projection p: TN -4 N onto the configuration manifold.  Proof: We stated earlier the hypothesis that the set of admissible controls while at a configuration n E N is TEN. Therefore, the configuration+control manifold has a natural bundle structure it : TN -4 N where, for any configuration n E N in the base manifold, the fibre 7r-1(n) := TN consists of the admissible controls. Likewise, the state space has a natural tangent bundle projection p : TN -4 N [Mar] onto the configuration manifold so the fact that the set of admissible controls while at a state x depends only on the configuration p(x) implies that the state+control space has a natural pullback vector bundle structure Fr : p*TN —> TN given algebraically by Fr-1(x) := 7-1(1)(x)) (3.29) = Tp(z)N,^  (3.30)  which may be read as the set of admissible controls while at a state x is the set of admissible controls while at its configuration p(x).  ^P  Chapter 3. Differential Geometry in Control Theory ^  29  The non-triviality of the vector bundles follows from the non-triviality of TN -4 N for N diffeomorphic to S2 §B.4.1. 0 Thus, for mechanical systems where there does not exist a consistent admissible control space defined across the state space, due to the non-Euclidean topology of the later, one must resort to defining separate admissible control spaces for each coordinate chart of the state space, i.e. trivialize the state+control space.  3.6 Formalism We now state the formalism and problems motivated by the examples of the preceding sections and chapters. We will only discuss Input-to-State models, i.e. where there is no output space, in view of the "observer-based" strategy described in § 2.6. • The state space is modeled on a finite-dimensional Ce°- manifold M with corners. Typically M is a tangent or cotangent bundle. • The set of conceivable controls U(x) while at a state x is assumed to have the structure of a Lie group G, with the admissible controls a subset of U(x). The state+control space  :=  U  {x}  x  U(x)  ^  (3.31)  xEAI  is assumed to have the structure of a principal G-bundle 7r : P^M: {x} x U(x)^x^(3.32)  i.e. with typical fibres the (state in product with the) conceivable controls. The notation P should not be confused with the notation P for a plant. • The law governing the dynamics/control is encoded ^in a^map  f : P > TM^ —  (3.33)  Chapter 3. Differential Geometry in Control Theory ^  30  P'-,--i-fri 1r \ Z—rr P^M^Tti  Figure 3.27: Commutative diagram for f : P -4 TM for which the diagram in figure 3.27 commutes. Geometrically, given a state+control, f specifies a tangent vector based at the state. Thus f defines, locally, the usual differential equation ± = f (x, u).^  (3.34)  We will not discuss the mathematics of this approach, but rather assign meaning to f only when used in a feedback loop as below. • A (memoryless nonlinearity) feedback control law is a section 4) : M -4 P of the state+control bundle P, i.e. such that r o 4) = lAf. Geometrically, at each state x E Al, c13(x) E {x} x U(x) specifies a choice of control. • The closed-loop dynamics associated with exerting a feedback law 4, is encoded in f o 4). Indeed, f o 4. : M —> TM is a section of the tangent bundle to M and thus defines an autonomous differential equation ± = f o 4)(x)^ and thus the closed-loop dynamical system indirectly.  (3.35)  Chapter 3. Differential Geometry in Control Theory^  31  3.6.1 Nominal Stabilization of Nonlinear Control Systems  Within this context then, the nominal stabilization problem is to find a (memoryless nonlinearity) feedback law 4) : M —> P such that the vector field f o 4, E X(M) of the associated closed-loop dynamics has a globally asymptotically stable equilibrium. This is the basic problem of this thesis. It will be abbreviated by the subsection number §3.6.1. 3.6.2 Example § 3.5 continued.  As argued before, the configuration space is S2, the state space has a bundle structure  p : TS2 -4 S2, the configuration+control space has a bundle structure 7r : TS2 --+ S2, and thus the state+control space has a pullback bundle structure Fr : p*TS2 4 TS2. -  Consider an arbitrary local coordinate chart (U, (X1, X2)) for S2, (Xi , X2) : U^R2,^  (3.36)  say stereographic projection. The state space has an induced coordinate patch  aa X ax2"  (TU, (Xi, X2, _--71-, -----)) denoted rather by (Xi, X2, Xi, X2) :  TU --÷ R.^  (3.37)  Likewise the configuration+control space TS2 has an induced coordinate patch  aa ax,' ax)"  (TU, (Xi, X2, denoted rather by (X1, X2, Ui, U2) :  TU^R4.^  The pullback bundle 7r*TS2 has thus an induced coordinate patch  (71-*TU, (X1, X2, -.'''.1., -2, Ul, U2))  (3.38)  Chapter 3. Differential Geometry in Control Theory ^  32  :=^X2) -k1 .k21 U11 U2) : r*TU -+ R. ^(3.39)  The double tangent bundle TTS2 has an induced coordinate chart  ^v.(TTU, (Xi,• ^a^a^a X21 .‘..11 )2) =7)  a )) . ax2' aX1' a)2  In terms of these local coordinates the evolution of the system is encoded via (H4) above with  __.^a^a^IL, a •^u2 a ,„ .1, ,„ , .^ f 0 y,^-1,,.,„: -2, .2, xi— + x2— + ^+ ^°xi^ax2 m °X,^ax2 as a tangent vector in To-i(x, ^,i2)TS2.  Part II Design  33  Chapter 4  Smooth Partitions, Transversality and Graphs  4.1 Motivation In the classical Liapunov method §B.8 for the design of a memoryless stabilizing feedback control law of a nonlinear control system  th = f(x, u), in the local coordinate form of §3.6.1, we must find a differentiable function V : M R with nested compact sub-level sets and a feedback control law u : M -+ U such that the closed-loop dynamics converge on the level-sets of V as in Figure 4.28. Intuitively then, the closed-loop dynamics is asymptotically stable. Separately achieving each hypothesis is relatively easy, but together these hypotheses typically conflict. In the sequel, we will propose a method based on a different relation between geometry and closed-loop dynamics than that given in Figure 4.28. We will refer to geometric concepts, like submanifold and transversality, rather than the classical algebraic concepts such as positive-definiteness. Admittedly, cleverness is needed to apply this method in practice, so simpler linear examples are used initially to illustrate the method. A genuinely nonlinear system for which standard nonlinear methods appear inconclusive is studied in the next chapter. It appears that the design strategy proposed in this chapter is original, although the cited concepts, such as partitions, transversality, graphs, framing and Liapunov functions, are all based upon classical versions. A related analysis stressing combinatorial issues 34  Chapter 4. Smooth Partitions, Transversality and Graphs ^  35  Figure 4.28: Level sets of a Liapunov function and the orbits of a closed-loop dynamical system converging on them.  Figure 4.29: An example of a finite-state machine. over the choice of partition is given in [Hsu].  4.2 Nonlinear Control Systems Modelled as Finite State Machines -  In [So] the claim is (implicitly) made that the finite-state machines (FSMs) of automata theory are instances of control systems. Instead of a differentiable manifold we have a finite set of points S := {A, B, C, .. .} as state space and instead of a differential equation with controls we have a finite set of transitions, say, T := {A -- B,B 4 C, C -4 A, ...} -  brought about by inputs (controls) U := {u1, u2, u3, ...} to the machine, as in Figure 4.29. As a rough analogy of the subsequent strategy, we will attempt to model nonlinear  Chapter 4. Smooth Partitions, Transversality and Graphs^  36  Figure 4.30: a) the direction field of a dynamical system across some triangles, and b) the associated finite state machine. control systems by FSMs (S, T). Each state A E S will represent a subset ,T(A) of the state space on the nonlinear system, and each transition A B will be brought about by a control law which forces each state x E .1(A) into .F(B), as in Figure 4.30. In this way the sub-level sets of the classical Liapunov theorem are replaced by arbitrary geometric subsets of the state-space and the Liapunov functions are now used to assert the transition from one subset to another. The FSM merely codifies the relation between the geometry and the dynamics of the system. Such a relation is already present in the classical theorem, (as in Figure 4.28) so no conceptual change is proposed here. One important distinction, however, is that some conclusions can be made by working with one (vs. a foliation of) level set(s) of a Liapunov function. In the first section we will clarify the geometric concepts using definitions. For example, graphs rather than FSMs should be used to model the relation between geometry and dynamics. Furthermore, differential geometry will continue to be used in order to define transversality. 4.3 Definitions In this section we propose definitions used to clarifying the subsequent strategy.  Chapter 4. Smooth Partitions, Transversality and Graphs^  37  4.3.1 Smooth Partitions Definition 4.3.1 Given an n-manifold M with corners §A.2.2, such as the state space of a nonlinear control system, we mean by a smooth partition P := {Fi} a partition of M into n-manifolds Fi with corners, called faces, so that Fi n F is either empty or an m-manifold with corners with 1 < m < n, for each i j. The edge set  ap of P is defined to be {Fi n F; :F, n Fi is a (n — 1)-manifold with  corners), and the elements of this set are called edges. The smooth partitions are assumed to be locally finite in that only a finite number of faces intersect any bounded subset of M. Each face F E P is assumed to be a closed set in the topology of M. For example, the real line R may be partitioned up into • • • U [-2, —1] U [-1, 0] U [0, 1] U [1,2] U • • • . ^(4.40) Note that the neighboring faces [n, n + 1] and [n + 1, n + 2] intersect at the edge in + 11. More classical definitions of partition assume the topological or analytic structure of a simplex [Fl] or an analytic stratification [Su] respectively, for the component sets. 4.3.2 Transversality Definition 4.3.2 Following [AR. I, suppose that M is a smooth manifold, E is a codimension 1 embedded submanifold of Al without boundary and f E X(M) is a continuous vector field on Al. We say f is transverse to E if span( f (x)) TrE = T  ^  (4.41)  for all x E E. Likewise, we say f is coincident with E if f (x) E TrE  ^  (4.42)  Chapter 4. Smooth Partitions, Transversality and Graphs ^  38  Figure 4.31: Flows a) transverse to, b)coincident with, and c)neither transverse to nor coincident with an edge. for all x E E. See Figure 4.31.  Definition 4.3.3 Suppose that M is a smooth manifold, P is a smooth partition of M and f E X(M) is a continuous vector field on M. We say that the dynamical system defined by f is framed by P if, for every edge E E 02, f is either transverse to or coincident with the (n — 1)-interior of E, exclusively. Associated with a smooth framed flow is a disjoint partition of M into intP,aTP,acP and 00T2 each consisting of the union of the interior of the faces of 7', the open edges for which the flow is transverse, the closed edges for which the flow is coincident, and the boundary of transverse edges not intersecting coincident edges.  4.3.3 Example Consider the linear dynamical system Xi =^x2 X2 = —2x1 — 3x2  (4.43)  which has phase portrait in Figure 4.32. As such consider the smooth partition P = • • , Flo} in Figure 4.33.  Chapter 4. Smooth Partitions, Transversality and Graphs ^  39  Figure 4.32: Phase portrait of (4.43).  Figure 4.33: a) A smooth partition framing the flow (4.43) and b) the superposition of the direction field.  Chapter 4. Smooth Partitions, Transversality and Graphs^  40  Figure 4.34: The graph associated with the flow (4.43) and the smooth partition in Figure 4.33. Clearly the dynamical system is framed by the smooth partition. 4.3.4 Graphs of Framed Dynamical Systems Definition 4.3.4 A smooth partition P framing a dynamical system defines a graph  G := (V, T). The faces of the smooth partition comprise the set V of the graph, V := P, and two vertices F1, F2 E V = P have a directed arc F1 -4 F2 E T (respectively, an undirected arc (F1, F2) E T ) between them if  (Hi) the common edge E := F1 n F2 0 0 is a non-empty (n —1)-submanifold and (H2) the flow is transverse to the (n — 1)-interior of E from F1 to F2 (respectively is coincident with E).  4.3.5 Example § 4.3.3 cont'd. The graph associated with the linear system (4.43) and the smooth partition in Figure 4.33 is given in Figure 4.34. Note that it has both directed and undirected arcs.0  Chapter 4. Smooth Partitions, Transversality and Graphs ^  41  In contrast to § 4.1, the more general concept of a graph is used to analyze the relation between geometry and control. Indeed, in example § 4.3.3 above, the graph 4.34 could not be associated with the state and transitions of a FSM as described in § 4.1, since there exist distinct states in F5 which evolve into F6 and F7 respectively, in violation of the determinism hypothesis of FSMs.  4.3.6 Liapunov Functions Definition 4.3.5 Following Thal a function V : C-2 —> R of class C(0) fl C1(12), 0 0 C2 C M open, is said to be a Liapunov function of an autonomous dynamical system defined by a vector field f E X (M), if the Lie-derivative V : SI -4 R is nowhere zero, V (x) 0 0, for every x E C2. Recall that the Lie derivative of V with respect to a flow 0 is given by the derivative of V along the trajectories of 0, i.e.  1.7(x) := -a aiV(q5(t,x))1t=o  ^  (4.44)  or, using the chain rule, by V (x) := dV (x) f (x) (4.45) which does not require an explicit formula for the orbits. Thus V 0 0 may be interpreted as saying that V is consistently increasing or decreasing along the trajectories, or alternatively, as saying that the orbits intersect the level curves of V transversely. The following lemma is basic to the proofs of most Liapunov theorems.  Lemma 4.3.6 Consider a dynamical system 0(.,•) : R x M —> M with forward orbit  r+ trapped by a compact set F C M, f_f_ C F. Suppose further that there exists a class C(F)r1C1(intF) Liapunov function V : F -+ R on the interior of F, satisfying V (x) < 0 for all x E intF. Then 0 0 w(r) C OF.  ^ ^  Chapter 4. Smooth Partitions, Transversality and Graphs^  42  Proof taken from [HS] and [La]: As F is a compact trapping set for fk, we have that 0 0 w(r+) C F, and that V o r+ : R+ -÷ R is well-defined. By definition 4.3.5, V o r+ is non-increasing and bounded below (as V is continuous on the compact set F). Thus there exists acER such that  v(r+(t)) -+ cast^oo.^ (4.46) Now for any y E co(r+) there exists a sequence tn^oo such that r+(t) -÷ y and thus v(r+(t)) -+ V(y), all limits as n^00, so by the uniqueness of limits of subsequences and by (4.46), V(y) = c for every y E w(r+).^ (4.47) Suppose that there exists a y E w(r+) n intF. Then as w-limits are invariant,  F(y) c w(F) so by (4.47), V(0(t, y)) = c for all t. Differentiating with respect to time at t = 0 we may conclude from (4.44) that  a a-t-v(0(t,y))1t=0 =1./(y) =0^(4.48) which contradicts the fact that V(y) <0, as y E intF. In conclusion w(r+) n intF = 0. 0 4.4 Liapunov's Theorem In this section we propose a version of Liapunov's theorem for asserting the asymptotic stability of a flow via the analysis of its graph associated with a smooth partition. These results are used in the design of control systems but not for the analysis of specific dynamical systems due to the nature of the hypotheses. See § 4.6. A flow cb. framed by a smooth partition P is face-monotone on P if for every face  F E P there exists a Liapunov function on intF.  Chapter 4. Smooth Partitions, Transversality and Graphs^  43  Theorem 4.4.1 (A version of Liapunov's theorem) Suppose that {0t}tER is a dynamical system framed by a smooth partition P with associated graph  g  satisfying the  hypotheses (H1) 0 has an equilibrium point ,  (H2) g  is Q -stable as in definition A.4.1 where Q identifies the vertices associated with  faces containing the equilibrium point Q:={FEP:"XEF}, (H3) 0 is face-monotone on P , and (H4) for every state x E acP U 0,9TP , that is on a coincident edge or only on the boundary of transverse edges, either x is attracted to the equilibrium point along acP U aeTP or else there exists a face F E P and a T> 0 such that 0(T, x) E intF, (H5)  the only 0i-connected subset of acP U aaTP is the equilibrium point ±. itself, (referring to § B.2)  (H6) for unbounded faces F E P, the associated Liapunov function V : F -4 R is unbounded in that V (x) —> oo as Isi --4 oo in F, but V (x) < 0 for all x E intF, and finally  (H7) (I) is smooth and autonomous. Then ±- is globally asymptotically stable for 0. 0  Hypothesis (H6) may be replaced by analogues like (H6)' for unbounded faces F E P the associated Liapunov function V : F -4 R is bounded below but 1./(x) -4 —oo as lxj -+ oo in F.  Chapter 4. Smooth Partitions, Transversality and Graphs^  44  Figure 4.35: Conceivable situations that Liapunov's theorem excludes. Such algebraic hypotheses could be avoided by partitioning M into an infinite number of bounded faces. The reader should realize that the proof is not immediate as we must exclude situations as in Figure 4.35. Proof of theorem 4.4.1: Assume (H1)-(H7) all hold and let x E M be an arbitrary but fixed initial state. We seek to show that w(x) = M with the aid of some preliminary lemmas. Lemma 4.4.2 If the forward orbit F+ (z) of an arbitrary state z E M is trapped by  a face F E P then co(z) = {-±}. Proof of lemma 4.4.2: By hypothesis (H3), there exists a Liapunov function V: F —> R on intF. Without loss of generality, assume V < 0 on intF. By definition 4.3.5 then, it follows that t e-+ V (0(t, z)) is non-increasing and thus from hypothesis (H6) that for an arbitrary e > 0,  F, := fy E F :17(y) < V(z) + €} is a compact trapping set for r+(z). Thus from lemma 4.3.6 we may conclude that  o 0 w(z) c aF, c aF.  Chapter 4. Smooth Partitions, Transversality and Graphs ^  45  To prove otherwise, suppose that there exists a y E OT F n ce(z) . Then r(y) c w(z), as w-limit sets are invariant sets. By Theorem B.1.1, however, r(y) F so w(y) F, which contradicts r+(z) C F. Thus 0 0 w(z) C OcF U 0,9TF. By Theorem B.2.2 of § B.2 and hypothesis (H5) we may thus conclude that w(z)  = 4LO  Lemma 4.4.3 Given a state y E M and a face F E P to which y belongs, either Al) w(y) =  41 or  A2) there exists T, V E R, 0 < < v and a F' E P such that cb(r,y) F and 0(v, y) E intF' Proof of lemma 4.4.3: As t 1-4 4(t, x) is continuous, F is closed and 0(0, y) E F we have either that i) 4(t, y) E F for all t > 0 and thus (Al) w(y) = M by lemma 4.4.2, or ii) there exists a r> 0 such that z := 0(7-, y) F. To prove otherwise, suppose that r±(z) is trapped by aTp. Then as aTP is a locally finite union of open codimension 1 submanifolds, there exists a T> 0 and a unique open transverse edge E E aTp such that 0(t, z) E E for all 0 < t < T. Differentiating with respect to time at t = 0 we see that f (z) E TEE, where f is the vector field defining 0. This contradicts the fact that E is a transverse edge. In conclusion, T(z) fl [acP u aoTp U intP) 0 0.^(4.49) If I'+(z) n int'P 0 then there exists a T> 0 and a F' E P such that 4)(T, z) E intF' and thus (A2) q!)(r T, y) E intF'.  46  Chapter 4. Smooth Partitions, Transversality and Graphs^  If instead  r+(z) n intP then by (4.49), r+ (z) n[acPuaaT'P]  = 0 (4.50)  0 0 so there exists a T > 0 such that  4)(T, z) E acP U OaTP. Then by hypothesis (H4), alternative Al) holds as (4.50) excludes the second alternative of that hypothesis (H4). 0 Lemma 4.4.4 For the previously fixed but arbitrary state x E M either  Al) w(x) = {} or A2) there exist sequences (v1)>1, (i)> in R with vi <  < Vi+i  for all i > 1 and  a sequence (Fi)i>i of faces in P such that 0(v1,x) E intFi but O(ri,x) Fi for all i > 1. Proof of lemma 4.4.4: The proof is by induction on the index i of alternative  A2). To start, there exists a face F E P such that x E F as P is a partition. Lemma 4.4.3 applied at y = x, however, implies that either alternative Al) holds or that there exists a > 0 and a face F1 E P such that 0(v1, x) E Assume that we have concluded by induction that either Al) holds or that there exists finite, void or finite, and finite sequences (vi)1<i<„, (  1<i<n-1  and (F1)1<1<„ in R, R and P respectively satisfying the conclusion A2) for only 1 < i < n. Lemma 4.4.3 applied at y = 0(v„, x), however, implies that either Al) w(4(v„,x)) = w(x) = M or that there exists 0 <  T <  v and  Fn+1 E P such that 0(r, 0(vn, x))^Fn but 0(vAvn, x)) E^In the case of the second alternative then, there exists vn+i := v + v„, r„ := r + satisfying v„ <  r,, < lin+110(11n+11 X)  The proof follows by induction. 0  E intFn+i and  0(7-n X) Fri •  Chapter 4. Smooth Partitions, Transversality and Graphs ^  47  Lemma 4.4.5 Suppose that there exists an arc -y := {4(t, z) : 0 < t < T}, faces F_, F+ E P and arER such that -y is not closed, z E intF_,O(T, z) E intF+, 0 < T < T and q5(r,z) t% F_. Then F_ 0 F+ and there is a directed path in the  associated graph g from F_ to F+.  Proof of lemma 4.4.5: By theorem B.1.1 there exists a tubular flow (CZ, 4)) of the flow 0 with 7 C a Then there exists x_, x+ E I, yo E In-1 such that (1)(z) = (x--, Yo), 4)(0(T, z)) = (x+, Yo). But the family of cubes B(p) := {q E Rn : lig — Piico <r}, r > 0,p E Rn is a base for the Euclidean topology on B". Thus, as 4) is a diffeomorphism, and intF_ and intF+ are neighborhoods for z and (1)(T, z) respectively, there exists a r > 0 such that  .13;.' (4)(z)) c In (4.51) B77,1 (4.(cb(T, z)))  c,.. In  4)-1(B77(4)(z)))  c  (4.52)  intF_ (4.53)  4)-1(B,7(4)(0(T, z)))) C intF+, (4.54) all as in figure 4.36. Consequently, define the rectified tube E := [x_, x] x B,7-1(yo), the tube base {x_} x /37z-1(yo), the projection 7r_ : E -4 O_E : (x, y) 1 4 -  (x_, y),  and the tubular flow 4-2, (i) for 0 by -S.4 := 4)-1(E) and ii) := (1)1s=1, all as in figures 4.37. Now the orbits of the induced flow 4).0 on E are parallel to those of the constant flow (1, 0). Thus for every (n — 1)-dimensional coincident edge E E acP, we have that r_.(i)(E n fi) is a (n — 2)-dimensional submanifold with  Chapter 4. Smooth Partitions, Transversality and Graphs ^  Figure 4.36: Geometry of a tubular flow around 7.  Figure 4.37: Rectified flow in a subtube from F_ to F.  48  Chapter 4. Smooth Partitions, Transversality and Graphs^  49  corners in 8_ E. Likewise, aaTP is a a locally finite union of at most (n — 2)dimensional submanifolds with corners in M. Thus it may be shown then that  D := _(1)((5cP U NTT') n is a finite union of at most (n — 2)-dimensional submanifolds with corners in a__E. By dimensionality arguments, D is a proper subset of  a_E. Note that  open transverse edges E E aTp induce open transverse edges (i(E n S7/) of the induced flow  CO, as (I) is a diffeomorphism.  Let q E 0 \ D be arbitrary. Then the induced arc (t , q) : 0 < t < intersects only induced interior faces and induced open transverse edges in E. Specifically, by the locally finite hypothesis of smooth partitions, there exists a finite sequence (Pi)1(<N of faces in P and a finite sequence (pi)0<j<N in R satisfying 0 = po <  P1< • • • < /IN  = T and  for pi_I < t < pi, for all 1 < j < N. By the original hypotheses of this lemma, however, we have that N> 1, Pi = F_, and EN = F+. To be consistent with the induced flow, we must have that Pj associated graph  -4  frj+i is a directed path in the  g, for each 1 < j < N. In conclusion, there is a non-trivial  directed path from F_ to F. If F_ = F+, however, then there would be a cycle in the associated graph  g, which contradicts hypothesis (H2). 0  Lemma 4.4.6 If alternative A2) of lemma 4.4.4 holds, then Fi Fi+i and there is a directed path in the associated graph  g from Fi to Fi+1, for each i > 1.  Chapter 4. Smooth Partitions, Transversality and Graphs ^  50  Figure 4.38: Geometry to exclude closed trajectories. Proof of lemma 4.4.6: Let  := {cb(t,x)^<t < v+1}.  To prove otherwise, assume that -yi is a closed trajectory, say with period > 0. Set z := (vi, x) and F_ := F1, so z E intF_. Choose a c > 0 sufficiently small so that 0(—e, z) E intF_, and set T := o- — c, and F+ := F_ so we have that ck(T, z) E intF+ , and -y := {q5(t, z)  : 0 < t < T} is a compact but not  closed arc. Finally, 0(7-i — z) (Z' F_, sa in figure 4.38. By lemma 4.4.5 applied to -y then, we conclude that F_ F+, which contradicts the construction  F_ = F1 = F+. Therefore, the original arc -yi is not closed. Thus applying lemma 4.4.5 to the original -yi, with z Fi,F+ := F1+1,T := vi+i —  Vi,7 := -  := 0(vi,x), F_ :=  vi, we may conclude that Fi F1+1  and that there is a nontrivial directed path in  G from Fi to F1+1.  Completion of the proof of theorem 4.4.1: If alternative A2) of lemma 4.4.4 holds then by lemma 4.4.6, there is an infinite directed path in the associated graph Thus alternative Al), tv(x)  g,  which contradicts hypothesis (H2).  = M, must hold. 0  Chapter 4. Smooth Partitions, Transversality and Graphs^  51  4.4.1 Example § 4.3.3 cont'd Recall the system (4.43) of example § 4.3.3 framed by the smooth partition P := {F1, • • • , F10} of Figure 4.33a), and with associated graph Here Q := {F1,F4,F5,F6, &Flo} and indeed  g  g  in Figure 4.34.  is Q-stable (H2). The flow is  face-monotone (H3) as demonstrated by the choice of (linear!) Liapunov functions V(xl, x2) := —x1 on F1 U F2 U F3 U F4, V (Xi, X2) := 2x1 + 3x2 on F5, V (Xi, X2) := x1 on F6 U F7 U F8 U F9, V (Xi, X*2) := -2x1 -  3x2 on F10. Hypothesis (H6) is clearly satisfied.  The flow on the union of coincident edges (F1 n F4) U (F2 n F3) U (F6 and [(F3 U F4)  n F5] u [(F8 u F9) n F10]  n F9) u (F7 n Fs)  is topologically equivalent to the flow i = —x on  R, so hypotheses (H4) and (H5) are satisfied. 4.5 Liapunov Functions for Control Systems Definition 4.5.1 A function V :  fi —* R of class C() n C1(12), 0 //cm open, is  said to be a Liapunov function of a control system th = f (x ,u) if the x-Lie-derivative V : CI x U --4 R defined by V (x ,u) := dV (x) f (x ,u) has at every state x E f2 an admissible control u = u(x) E U for which it is nonzero: Vx E Q, 3u E U so that f/(x, u) #0. 4.6 Design of Coincident and Transverse Edges and Face-Monotone Dynamics of Control Systems In contrast to dynamical systems, one may design the invariant manifolds of a control system and algebraically find the feedback law realizing them. Specifically, consider a  Chapter 4. Smooth Partitions, Transversality and Graphs^  52  nonlinear control problem i = f (x, u)^  (4.55)  and a given submanifold E of the state space M defined by E = ix E M : V (x) = Vol for some known function V : M --+ R of class CI. If x(.) is an arbitrary closed-loop trajectory constrained to satisfy V(x(t)) = Vo for t E R then differentiating we see by the chain rule and (4.55) that  av av f,(x(t),u)+ • • . + ax, z(t)^ax„  fn(x(t),u) = 0.^(4.56)  Thus if we can solve the system  av^av  V(x,u)= a- -z.(x)f,(x,u)+ • • • + --a-zi(x)f„(x,u) = 0,^(4.57) say via §B.3, for a u = u(x) E U for each x E E, then by the implicit function theorem we can define a class C' control law u : E —> U realizing the response with E invariant. Likewise, we can try to find a control law so that the closed-loop is transverse to E by replacing the equality in (4.57) by an inequality. Finally, suppose we have a Liapunov function V: F --+ R for a control system on a face F as in definition 4.3.5. If we can find a class Cl function u: F -+ U such that  V (x,u(x)) 0 0 for each x E F, then the closed-loop response of the control system under that feedback law u will be face-monotone on F with V as a Liapunov function. 4.7 Graphs of Control Systems Definition 4.7.1 Consider a nonlinear control system § 3.6.1 and a smooth partition P of the state space M. Again we define an associated graph  g=  (V, T). Specifically,  53  Chapter 4. Smooth Partitions, Transversality and Graphs ^  the faces of P are the vertices V of the graph, and two vertices F1, F2 E directed arc F1 -4 F2 E T (respectively an undirected arc (F1, (H1) the common edge E :=F1 (H2)  V = P have a  F2) E T ) between them if  n F2 0 0 is a non-empty (n — 1)-submanifold and  there exists a continuous control law ulE E ^U defined on E for which the flow is transverse across the (n-1)-interior of E from F1 to F2. (respectively is coincident with E ).  When (H2) is met for every edge in  op  we say that P frames the control system.  Recall that hypothesis (H2) may be checked using the techniques of §4.6. 4.8 Example: the 2-integrator.  Consider the linear dynamical system il  = x2^  i2 = u  (4.58)  and the smooth partition P = {F1, ^, Flo} in Figure 4.33a) again. The associated graph is given in Figure 4.39. For example, on the edge F1 n F4, we have that W(xl , x2) := x1 + x2 is constant. We may thus use this function to study hypothesis (H2) of the definition of 4.7 via §4.6. Specifically, Tk(xi, x2, u) = x2 + u, so if we choose u = u(x) > —x2 (respectively u = u(x) < —x2, respectively u = u(x) = —x2) on F1 n F4, the resulting closed-loop dynamics will be transverse across that edge from F1 to F4 (respectively, transverse to that edge from F4 to F1, respectively coincident with that edge). Furthermore, on the 1-interior of the edge F1 n F2, 11(Xl, x2) := x1 is constant, and u) = x2 > 0 so any closed-loop dynamics is transverse to that edge from F2 to F1. Again V(xl, x2) := x1 is a Liapunov function of the control system for each face.  Chapter 4. Smooth Partitions, Transversality and Graphs^  54  Figure 4.39: The graph associated with the control system (4.58) and the smooth partition in Figure 4.33.  4.9 A Proposed Method for the Design of Framed Dynamics Given a nonlinear control system, consider the problem of selecting a feedback control law for which the closed-loop dynamics is stable. When modelled on the examples and graphs discussed in the previous sections, this selection is reflected in the transition from Figure 4.39 for the control system to the stable subgraph in Figure 4.34 for the closed-loop dynamical system. The first part of the design strategy proposed in this chapter is then, loosely, 1. Select a smooth partition of the state space framing the control system (using the ideas of chapter 5). 2. Construct the associated graph. 3. Select a subgraph 0 with exactly one arc (directed or undirected) between faces with a common edge, so that hypothesis (H2) of theorem 4.4.1 is satisfied. If it cannot be satisfied, retry the previous steps.  Chapter 4. Smooth Partitions, Transversality and Graphs^  55  4. Realize a class C' control /clap on OP as per §4.6 with the transversality/ coincidence of 0. For example, if F1 -4 F2 is a directed arc in Q then realize a law /Cb FI n F2 on F1 n F2 with F1 n F2 a transverse edge of the closed-loop dynamics in the direction F1 —* F2 5. Check hypotheses (H4), (H5). If they are not satisfied retry the previous steps. 6. Realize a control law klintp on intP with face-monotone closed-loop dynamics as per § 4.6. That is, for each face F E P, (a) choose a Liapunov function V of the controls system on intF in the sense of definition 4.5.1 (say using the methods of chapter 5) (b) check that 1.7(x, lejap (x)) < 0 for all x E OF, referring to the control law constructed in step 4, (It is the consistency of parity, rather that the actual sign that is important, so > 0 would be just as good) (c) find a control /char on intF such that the closed-loop dynamics is facemonotone on F with V as a Liapunov function § 4.3.6, say via §4.5.1, and so that k(x) x  aF  k(x) :=^ klintgx) x E intF  (4.59)  is of class Cl, and (d) check hypotheses (H3) and (H6), retrying the previous steps as necessary. 7. Apply Theorem 4.4.1. Admittedly, the choice of smooth partition in step 1 has a crucial effect on the success of later steps in the algorithm. In practice, then, the algorithm would have to be repeated for successive refinements of the partition. Note that the presence of redundant edges only complicates the combinatorial problem in step 3.  56  Chapter 4. Smooth Partitions, Transversality and Graphs^  4.9.1 Example § 4.8 cont'd  We return to examples §4.3.3, §4.3.5, §4.4.1 and §4.8 to illustrate this design strategy. From equation (4.58) and the smooth partition 4.33a), we construct the associated graph in Figure 4.39. Here Q := {F11 F49 F5, F6, F9, F10} . A subgraph with exactly one arc between neighboring faces satisfying hypothesis (H2) is given in figure 4.34. A control law which realizes that subgraph need only satisfy u <0 on the 1-interior of F3 fl F4 and F5 n (F6 u F7),  u>  0 on F8 n F9 and F10 n(F1u F2), u = —x2 on xi +x2 = 0 and u = —2x2  on 2x1 + x2 = 0. For example, W(x) := x2 is constant on F3 n F4, with W(x, u) = u, so if we want the transition F3 --+ F4 we should have the orbits go against gradW, i.e. W < 0, and thus u <0 is adequate. Likewise, W(x) := x1 + x2 is constant on Fi with W(x, u) = x2+ u, so if (F1, F4) is an undirected arc, i.e. F1  n F41  n F4 is a coincident edge,  then we should have W = 0 so u = —x2 is adequate. As argued in § 4.4.1, hypotheses (H4) and (H5) are satisfied. On the faces F1 U F2 U F3 U F4 U F6 U F7 U F8 U F9 define the Liapunov function V(xi, x2) := xi so V(xi , x2) = x2. Then regardless of how we interpolate u1731, the closed-loop dynamics will be face-monotone with these Liapunov functions. On the face F5 define the Liapunov function  V(xi, x2) := 2x1 +3x2,^  (4.60)  noting that V —+ oo as x oo in F5. Then the Lie derivative is V(xi, x2) = 2x2 + 3u,^  (4.61)  and it may be shown on substitution that consistently V < 0 on OF5 according to the 2 previous conditions, so we require that u < --x2 on intF5 to satisfy (H3)&(H6). Likewise 3 for F10. One feedback law satisfying these conditions is u := —2x1 — 3x2.  Chapter 4. Smooth Partitions, Transversality and Graphs ^  57  Figure 4.40: An attracting invariant manifold which contains an attracting equilibrium point. 4.10 Example-A Liapunov Method.  As an alternative to Figure 4.28, we consider the next simplest relation between geometry and dynamics as in Figure 4.40. This section, though a mere special case of the rest of the thesis, could be interpreted as an independent method. Consider a nonlinear control system i = f (x, u)^  (4.62)  for x E M, u E U as in §3.6.1. Suppose that there exists a function V : M -4 R with associated Lie derivative  V: Mx U -+ R: (x,u)I-4 dV(x)f(x,u)^(4.63) such that for every state x E M, there are (not necessarily unique) controls k+, ko, k_ E U such that '(x, k) >0  (4.64)  1./(x, ko) = 0  (4.65)  V (x , k_) < 0  (4.66)  Chapter 4. Smooth Partitions, Transversality and Graphs ^  58  as realized via §B.3. Then the level set V-1(0) may be made an invariant set by setting u = ko(x) satisfying (4.65) for x E V 1(0). Suppose further that for some choice of such -  ko, the induced flow on V-1(0),  ± = f(x,ko(x)), x E V 1(0)  ^  -  (4.67)  is globally asymptotically stable to some equilibrium point 2 E V 1(0). Then a candidate -  feedback control law is Ik_(x) for V(x) > 0 u(x) :=^ko(x) for V(x) = 0^  (4.68)  k+(x) for V(x) < 0, where k_, k+ satisfy (4.66) and (4.64) respectively, i.e. V < 0 and V > 0. Such a control law is easy to implement as it only involves function evaluations, and not function inversions [Is].  4.10.1 Example: the n-integrator. Consider the n-integrator ii = X2 (4.69) in-1 = Xn  in = U. Although a stabilizer may be derived almost immediately using classical linear strategies, we use this example to illustrate the method of §4.10. As a candidate Liapunov function we take V(xl, • • • , xn) = 0 - x = Igixi + • • • + Azxn.^ (4.70) The associated Lie-derivative 1.7(x, u) = 01x2 +... + ign-ixn + Onu  ^  (4.71)  ^  Chapter 4. Smooth Partitions, Transversality and Graphs ^  59  can be made positive, negative or zero at any x E Rn provided fin 0 0. Now 1.7(X, U) =  0n-1  /3nx2^Xn 0^u = —I2-1 Pn  (4.72)  and  /32^I3n (4.73) flu^Pu provided /5.1^0, and the closed-loop dynamics on the invariant plane (4.73) under the V(x) = 0 41* xi =  —  —X2 — • • • —^  control law (4.72) is given, in the coordinates x2, • ,x„, by  x2  = OC3  (4.74) in-1 = in 01 fin—i in =^7,—x2^ 0 Xn • Pn^In —  By stability analysis for linear dynamical systems, (4.74) will be asymptotically stable if Oi  n—1) z—1  (4.75)  for 1 < i < n. Note that (x2, • • • , xn) -4 0 as t^oo implies x1^0 as t -4 oo while on the plane (4.73). We could now construct any law as in (4.68), but rather here we will find a linear feedback law of that form. Specifically, a linear control law u = k • x = kixi + • + knxx satisfying (4.72) on the plane (4.73) may be shown to be, most generally, of the form ^u(x)  fin_i 02^01 fl =^+ (—k1 — —)x2+••• + (--41 —)xn.^(4.76) 131^On^131^On  Substituting into (4.71), we conclude that 1.7(x, u(x)) =^+•• • + On-ixn + Onkixi + (137 -2-ak1 — #i)x2 + • • •^(4.77) flu 02 (4.78) — 13n—i)xn 131 = OnOliciV(x).  (4.79)  Chapter 4. Smooth Partitions, Transversality and Graphs^  60  Consequently, if #1, O'n are as in (4.75) and k1 = —1, say, then V < 0 when V > 0 and vice-versa. In conclusion, we have constructed the stabilizing feedback control law -xl - (t31 + 02)x2 - • • - - (3.-1  =  + 0.)x.  ^  (4.80) (4.81)  Chapter 5  Construction of Smooth Partitions and Liapunov Functions  In this chapter we discuss an algebraic approach to the coupled problem of constructing smooth partitions and Liapunov functions. To construct and refine smooth partitions, we rather construct submanifolds of codimension 1, transverse to or coincident with the flow, and piece these together so that they become edges of smooth partitions. The strategy proposed here is to make the edges either isolated level sets of scalar functions or the subsets of state space where there is a change in the structure of the class of all local Liapunov functions. Differential conditions for a general submanifold to be transverse to or coincident with some closed-loop response are used to classify both strategies. Aside from Frobenius' Theorem, the author is not aware of the ideas presented in this chapter appearing anywhere in the established literature.  5.1 Tentative Construction As motivated above, consider the problem of determining the class of all local Liapunov functions of a given flow. The condition that the trajectories of a flow defined by a vector field f  a^a  E X(M)  (locally f (x) = fi(x) , + • • • + fn(x) ) are transverse to a hypothetical function may axi^ax„ be given in terms of its gradient 1-form w (locally w(x) = wi(x)dxi + - • • + wn(x)dx„), as, —  —  w(x)f(x) = wi(x)h(x) + • • • +w(x)f(x) 0 O.^(5.82)  61  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^62  Thus if we can solve (5.82) for wi, • • • , con, we have identified candidates for the Liapunov functions through their gradients. Furthermore, "changes" in the class of solutions of (5.82), as one varies x, will identify candidates for edges of smooth partitions. We are thus led to the problem of recovering scalar functions from their gradients. Alternatively, we seek to recover the level sets of the scalar function, i.e. the Pfaffian problem.  5.2 Pfaffian Systems 5.2.1 A Classical Linear Pfaffian Problem. In the classical problem [vW], one attempts to recover a (foliation of) codimension 1 submanifolds N, of a larger manifold M, from its normal field, as in figure 5.41. At every point x E M, M the larger manifold, suppose we have given a normal vector  w(x) E (TM)*, i.e. a cotangent vector, so co defines a 1-form. We seek a submanifold N such that  (w(x),TxN) = w(x)(TzN) = 0,  (5.83)  that is, TzAT C kerw(x), for all x E N, say passing through some given (n-2)-dimensional submanifold. A necessary and sufficient condition for existence is afforded by Frobenius' Theorem. For details, see theorem B.6.1. Roughly, if co A dw = 0 then there exist scalar functions A, V: M --+ R such that co = AdV and A > 0, so the submanifolds N are the level sets {x E M : V(x) = V0}. The Pfaffian systems we will be considering, however, are nonlinear.  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^63  Figure 5.41: A normal field and a surface normal to it. 5.2.2 The Nonlinear 'OR' Pfaffian Problem In this variant of the classical Pffafian problem, one seeks a codimension-1 submanifold N satisfying w(TN) = 0  ^  (5.84)  for some w E Ar(x) for each x E N, where M(x) is just a given set of co-vectors, Ar(x) C TM. The set-valued function N. is an example of a multifunction. For example, the flow a = 1, i2 = 0 is transverse to a 1-d submanifold N of R2 if TN span(1, 0), for all x E N, or equivalently, N satisfies the nonlinear Pfaffian problem above with Ar(x) = {widxi w2dx2 : w1 0}. ^ (5.85) A 1-form w satisfying w(x) E N.(x) for all x E M is said to be a selection of the multifunction N. Existence of solutions to the nonlinear Pfaffian problem may likewise be asserted by Frobenius' Theorem. Specifically, if N is a codimension 1 submanifold of M solving the nonlinear Pfaffian problem, let the associated co-vector in N.(x) satisfying (5.84) be w(x), for each x E N. Thus if we have a foliation {Nr}fER of these codimension-1 submanifolds, we have defined a 1-form w on all of M. The foliation N,. satisfies the classical linear  Chapter 5. Construction of Smooth Partitions and Liapunov Functions ^64  Pfaffian problem with that 1-form w(x). Examples of this reasoning will be given later. 5.3 Transversality Multifunctions The following definition is used to classify the transversality of a control system to a submanifold as one varies an admissible control u E U. Definition 5.3.1 Let U 0 0 be a set. A function g : U -4 R is said to be of sign class Sc, where 0 0 C C {-1,0, +1}, if {sgn g(u) : u E U} = C. We also define the transversality classes  '0' := Sol, '1' := S{_i} U S(441, '01' := 44,0} U So,1-11, '2' :---- S{ 1,1 1}, -  -  '02' := S{-1,o,1-1}, as compound sign classes.0 Let TM := (TM) M) \ {0} denote the nonzero covectors. Definition 5.3.2 Consider a nonlinear control system th = f(x,u)^  (5.86)  as in § 3.6.1, with U denoting the constraint set of admissible controls. Let C denote one of the transversality classes above. Define the C-transversality multifunction  .N(x) := 1w E TixM : u 1 4 w( f (x,u)) is of class C}.^(5.87) -  To calculate the transversality multifunctions in concrete examples, it may be useful to define the transversality function  Y: (x, u, w) 1 4 w( f (x,u))^ -  and apply a strategy similar to that in §B.3.  (5.88)  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^65  5.3.1 Example: the 2-integrator.  Consider the 2-integrator i = f(x, u) defined on M := ir,U := R, using canonical coordinates, by f (xi, x2,u) :=  X2  " U ).  (5.89)  For any w E Ti14., say w = w1dx1 + w2dx2, (w1, w2) 0 0, we have that T(X, IL, CJ) =  WiX2  +  (4.12U.  (5.90)  The function u i-+ T(w, x , u), for w2 0 0, is always of class '02', but for w2 = 0 is of class '0' or '1' according to whether x2 is zero or non-zero. Thus the transversality multifunctions are No (  xi ) =  {(.4.) E tM : w = widxi,wi 0 0}^if^x2 = 0  X2  0^if x2 0 0  Ari x1 (  0^if x2 = 0  )=1 /V02^= {W E TM : W =  ICJ E T;M : co = widx174.01 0 0}^if^x2 0 0  X2  a r^X1  (^)  WidX1  X2  + W2dX2, W2^0}-  (5.91)  (5.92)  (5.93)  It is easily shown that the only maximal solutions of the nonlinear 'OR' Pfaffian problem with  Ar = .N; are the half-lines {L±,A}AER given by L+,A := { (A, x2) : x2 > 01, LmA := {(A, x2) : x2 < 0}.^(5.94)  Again, L±,A were constructed to be the edges of smooth partitions. Across these edges the flow is consistently in one direction, as in Figure 5.42.  Chapter 5. Construction of Smooth Partitions and Liapunov Functions ^66  •  e  L  Figure 5.42: Foliation of a selection of NI. 5.4 Liapunov Functions Consider the use of a Liapunov function V : F^R to assert that some closed-loop response of a control system is face-monotone across a face of a smooth partition. Conditions that a scalar function is indeed a Liapunov function are afforded by the transversality multifunctions. Proposition 5.4.1 Consider a nonlinear control system th = f (x , u) with transversality multifunctions Arc. Let F be a nonempty open subset of the state space. (Cl) If there is a Liapunov function V : F -4 R on F then the gradient 1-form dV is a selection of Ar1U .Aroi U A.r2 U A'2 on F. (C2) Conversely, a selection w of the multifunction  Ni uNO1 u.V2U.A[02 on F satisfying  Frobenius' condition w A dw = 0 defines a Liapunov function on F. Specifically,  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^67  by Frobenius' Theorem, there exist functions A,V : F —* R such that A > 0 and w = AdV and in this case V is a Liapunov function. Proof: Suppose that V : F --* R is a Liapunov function. By definition 4.3.5, there  exists a feedback law k : F -4 U so that  1.1 (x , k (x)) := dV (x) f (x , k(x)) 0 0.^(5.95) Thus, for every x E F, u i— dV (x) f (x , u)  ^  (5.96)  is not of class '0', so by definition 5.3.2,  dV(x) E (.,Af u .Aloi U N2 Li N.02)(X)•^(5.97) Conversely, suppose that w is a selection of the multifunction Ni U./Vol UN; U.A/02 on  F satisfying Frobenius' condition w A dw = 0. Then by Frobenius' Theorem B.6.1, there exists functions A, V : F -4 R such that A > 0 and w = AdV. Now as w is a selection of the said multifunction, for every x E F there exists a u(x) E U such that  w( f (x , u(x)) 0 0. Thus by the positivity of A, V(x, u(x)) = dV (x) f (x , u(x))  0 0,  i.e. that V is a Liapunov function on F. 0 In critical cases, just the existence of sections of NI is sufficient. Corollary 5.4.2 Suppose that a nonlinear control system framed by a smooth partition  P has a selection of Ail on a face F  e P satisfying Probenius' condition. Then the control  system has face-monotone closed-loop dynamics regardless of the control law defined on the interior of the face F. Proof: Suppose that such a selection exists. By the proof of the second part of propo-  sition 5.4.1, there exists a function V : F -4 R such that dV(x) E A/1(x) for all  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^68  x E F, so by definition 5.3.2, V(x, u) = dV(x)f (x, u) 0 for all x E F and u E U. Thus by definition 4.3.5, V is a Liapunov function for any closed-loop dynamics on  F. O. For example see § 4.8, with face F1 and the smooth selection co := dxi of Ari. The hypothesis of § 4.10 may be reformulated as saying that there is a global selection of A'2.  5.5 Smooth Partitions Very few useful comments can be made for deciding on an effective smooth partition of a control system. At least, one strategy for edge construction is, as stated above, to use isolated level sets of Liapunov functions. Alternatively, in example 5.3.1, there is a change in the structure of the transversality multifunctions along the line x2 = 0. This makes it a candidate for an edge. Off this edge, construction of Liapunov functions or edges of smooth partitions is not affected by the dynamics of the system. It is instructive to calculate the transversality multifunctions of various control systems to observe the structure of local Liapunov functions.  5.6 Example Stabilize, about (0, 0),  ii = x2, (xi, x2) E R2^ i2 = xiu, u E R.  (5.98)  Note that this system does not have relative degree [Is] defined at (0, 0), so by Theorem 2.6 of [Is], the state space exact linearization problem is not solvable, i.e., feedback linearization is inconclusive, and this problem is genuinely nonlinear. The author is not  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^69  aware of a time-invariant static stabilizing feedback control law having been previously derived for this system. It seems virtually impossible, however, to show Liapunov's classical method must fail. Nevertheless, the partition utilized in this example could not be associated with discrete level sets of a classical Liapunov function. To determine the transversality multifunctions, we again define the T function  Y:^n2, xi,x2, u) 1-4 n1x2 + n2xiu, for (ni, n2) 0 (0,0), (xi, x2) E R2, u E R, as per (5.88). Then, u 1—> T(ni,n2,0,0,u) = ni(0) + n2(0)u = 0 is of class '0', for all (ni, n2) E R2; •  T(ni3O,x1,0,u) = ni(0) + 0(xi)u = 0  is of class '0', for all n1 E R, x1 E R;  u 1-4 T (0 , n2,0 x2, u) (02) n2(0)u = 0 is of class '0', for all n2 E  R, x2 E R;  u^T(721, n2, 0, x2, u) = n1x2 n2(0)u = n1x2 is of class '1', for all n1^0,x2 0 0; T(ni, 0, xl, x2, u) = n1x2 0(xi)u = n1x2 is of class '1', for all n1 0 0,x2 0 0; T(ni,n2,xi,x2,u) = nix2 n2x1u is of class '02', for all (ni, n2) 0 (0,0), x1^0,x2 E R.  (5.99)  Chapter 5. Construction of Smooth Partitions and Liapunov Functions ^70  Thus it may be shown that  7 ,x 1,x2,,R2^if Aro(xi, x2) =  Ali (xi, x2) =  1^  (xi,  x2) = 0  = nidxi : n1 0 0}  if^xi 0 0,x2 = 0  = n2dx2 : n2 0 0}  if^x1 = 0,x2 0 0  (5.100)  if^xi 0 0, x2 0 0,  0  if x2 = 0  0  (5.101)  {n = nidxi + n2dx2 : ni 0 0}  if^xi = 0, x2 0 0  in = nidxi : ni 0 0}  if^xi 0 0,x2 0 0,  2(x1, x2) ..  1  0^if x1 = 0  (5.102)  {nidxi + n2dx2 : n2 0 0} if xi 0 0.  The structure of the transversality multifunctions changes on the coordinate axis, so they are candidate edges. Clearly, the nonlinear Pfaffian problem with Al = No does not have any solutions, while for  Al = Ni, we have again the two families  by (5.94). Likewise, the nonlinear Pfaffian problem with the foliations Viol, OE R1 { cr,± li, 0  }  {/„,±}„ER given  Al = Á102 has, amongst others,  r>0 7 {po,± } 0>o , where  := {(x110) : x1 E R},  Cr,+  (5.103)  {(x1, Vr2 — xT) : x1 E (0, r)} ,  (5.104)  Cr,_  :=  {(xi, —Vr2 — xi) :x1 E (—r,0)},  (5.105)  Po,+  :=  {(xi, x? +  [3) :x1 < 01,  (5.106)  Po,- := {(x1,—x? — 13) : x1 > 0},  (5.107)  as in figure 5.43. Consider the smooth partition in Figure 5.44 constructed from the edges ho, and  P2n,-I- , P2n,— 1 C2n+1,-1- 5 C2n +1,— 1  ure 5.45.  where n > 0 in Z, and the associated graph in fig-  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^71  Figure 5.43: Foliations of selections of  A/02.  Figure 5.44: A smooth partition for (5.98).  72  Figure 5.45: The graph associated with (5.98) and ( 5.44). A subgraph satisfying hypothesis (H2) of Theorem 4.4.1 is given in Figure 5.46. We wish to find a control law so that the closed-loop dynamics is as framed in that graph. By section § 4.6, it need only satisfy the conditions,  u(xi , x2) = 2x2 (5.108) on the edges p2„,± for n > 0, u(xl , x2  ) = —1  (5.109)  on the edges c2n+1,± for n > 0, and u(xi, 0) < 0 for x1 0 0. The first two conditions were obtained by differentiating the defining equations for the p and c submanifolds, and solving for u. Consider any feedback control law satisfying these conditions. The closed-loop dynamics is face-monotone (H3) by proposition 5.4.2 for the Liapunov function V(xi, x2) := xl, which also satisfies hypothesis (H6). Hypothesis (H5) is clear as the flow on po,+ Upo,_ is topologically equivalent to the flow i =  —  x on R. Hypotheses (H4) is also clear. There-  fore, by Theorem 4.4.1, the closed-loop dynamics will be globally asymptotically stable.  Chapter 5. Construction of Smooth Partitions and Liapunov Functions^73  Figure 5.46: A stable subgraph of the graph in Figure 5.45.  Chapter 6  Conclusions  All methods for nonlinear systems compromise scope and viability. For example, Liapunov's method [AV] has a very general scope but is difficult to apply in practice, while feedback linearization [Is] limits itself to affine systems yet is almost entirely computable. The method proposed in §4.9 and chapter five for the construction of static stabilizing feedback laws for nonlinear but smooth control systems modelled by a system of deterministic differential equations is also no exception. The algorithm strictly contains Liapunov's method, and appears to also accommodate 1. I/O systems [So], that is, systems with outputs rather than complete detection of state, 2. systems with non-Euclidean geometry, such as those in chapter three, and 3. discrete-time systems. A cursory inspection of the proof of theorem 4.4.1, however, will reveal dependency in its hypotheses. Thus there is inefficiency in the intermediate hypotheses checking steps 2-6 of algorithm §4.9. Furthermore, weakened hypotheses and conclusion would permit the design of closed-loop dynamics (3.25)-F(3.27) in figure 3.23, where the equilibrium point (0, 0) attracts only a dense set. In conclusion, the author believes that a revision of theorem 4.4.1, reflecting a more efficient algorithm of greater scope is feasible. In the algorithm's current form §4.9, however, the viability of step 1, the choice of effective partition, is not clear. The author believes that this problem may be partially 74  Chapter 6. Conclusions^  75  circumvented by embedding the existing algorithm in a recursive loop, where the partition is refined at each iteration. Such an algorithm would, arguably, constitute a significant improvement over Liapunov's classical method. Theorem 4.4.1 asserts the global asymptotic stability of flows whose associated vector fields are of class globally C1. Strictly speaking then, theorem 4.4.1 cannot be used to prove that a piecewise Cl feedback law designed using algorithm 4.9 actually is a stabilizer. I strongly believe, although I have not checked all the details, that this smoothness hypothesis (H7) of theorem 4.4.1 may be weakened to accommodate piecewise CI vector fields, that is, algorithm 4.9 may be extended to the design of piecewise Cl feedback laws. Finally, the ideas of this thesis could be used to study other problems. For example, the graphs in §4.7 could be used to study tracking directly and not through stabilization.  Appendix A Prerequisite Mathematical References  It is pedagogically impossible to develop the mathematical prerequisites of this thesis in an appendix, as is standard in geometric control theory texts [Is], [So]. Rather we outline the logistics involved in learning the fundamental ideas of the prerequisites through the following lists, comments and citations. A rather complete encyclopedic reference is [CD].  A.1 Basics A.1.1 Basic Topology Refer to [Mu][Si] for definitions of the following concepts which appear repeatedly in various contexts throughout the thesis. Sets, elements; topological structure on a set: open and closed sets; closure, interior and boundary; basis and neighborhoods; relative topologies; metric spaces; continuity and convergence.  A.1.2 Algebra Refer to [HK][HS] for definitions of the following concepts. Vector spaces; operators; tensor products; modules; exterior products; inverses; eigenvalues and eigenvectors; dual and tensor spaces; ideals; groups; group actions.  76  Appendix A. Prerequisite Mathematical References^  77  A.2 Geometric Analysis A.2.1 Advanced Calculus = Local Differential Geometry Refer to [CS] for notation, definition and theorems concerning Vector-valued functions f : Rn^Rm; differentiability; Jacobians; implicit function theorem. A.2.2 Manifolds Refer to [Bo][vV1/4/ for definitions concerning Topological manifolds; coordinate charts; differentiable manifolds, atlases and compatibility; submanifolds, immersions, embeddings and codimension; Lie groups; mappings between manifolds. Note that the basic geometric structures of this thesis are differentiable manifolds with corners. The distinction is that each point has a neighborhood which is homeomorphic to a relatively open subset of [0, +oor. See [KS][Do]. A.2.3 Bundles Refer to [vIV][CD] for definitions, examples and theorems concerning Product manifolds; vector bundles; vector bundle morphisms; group actions; principal fibre bundles; fibre bundle morphisms; bundle of frames; connections; sections of bundles; pullback of bundles. A.2.4 Algebraic Structures over Manifolds Refer to [vW][F11. Tangent spaces, cotangent spaces tensor spaces; tangent cotangent and tensor bundles; vector fields; differential forms; differentiation as a bundle map.  Appendix A. Prerequisite Mathematical References ^  78  A.2.5 Topological Dynamics  Refer to [Arl][HS][Wi]. Vector fields defining differential equations on manifolds; Lie derivatives; flows; equilibrium points; linearization; stability, asymptotic stability; invariant manifolds. A.2.6 Exterior Differential Systems  Refer to [vW]. Forms; Exterior differential systems; codistributions; integral-manifolds; exactness and closedness; integrability; ideals. A.2.7 Geometric Mechanics  Refer to [Go][Ar2][Mar]. State; Newtonian, Lagrangian, and Hamiltonian mechanics; modelling state spaces by vector bundles and governing equations by differential equations on manifolds. A.2.8 Functional Analysis  Refer to [NS] Hilbert and Banach spaces; LP spaces; linear operators; nonlinear operators; inverses; realization by differential equations. A.3 Control and System Theory  Refer to [So][Ka][0g]. Linear systems; signals, systems and states; feedback; input and output; closedloop; stabilization and compensation; tracking; design specifications; controllers; Poleplacement.  Appendix A. Prerequisite Mathematical References^  79  A.4 Miscellaneous A.4.1 Graph Theory Refer to [Lu][Ha] for definitions, examples, notation and algorithms concerning Directed and undirected graphs; arcs, vertices; directed paths; tree-growing algorithms. In this thesis we propose a graph theory version of "attractivity":  Definition A.4.1 A graph g^(V, T) with directed and undirected arcs and a finite set of distinguished vertices Q C V is said to be Q-stable if (H1) there is no infinite directed path in V. That is, there does not exist a sequence Itili>i of directed arcs in T and vertices {c} >i in V, such that ai_1 4 ai, for all i > 1, and (H2) for every vertex a in V \Q, there is a vertex 3 in V such that a -4 is a directed arc in T. Intuitively, Q is the "global attractor" for such a graph as each directed path is finite and ( every continuation ) terminates at a vertex in Q. Note that hypothesis (111) excludes cycles and paths diverging away from Q.  A.4.2 Automata Theory Refer to [So]. Finite-state machines; state, input and transitions; determinism.  A.4.3 Multifunctions A function Al M^P such that the value Al (x) of the function at a point x e M is actually a set is called a multifunction. Thus the range^of Al is a set of sets. If  Appendix A. Prerequisite Mathematical References ^  80  AT (x) c W, for every x E M, that is P is a subset of the power set of W, then we write  N. : M W .  Appendix B Cited Theorems  B.1 Long Tubular Flows Following [PdM], a tubular flow for a vector field X E Xr(M) on a smooth manifold M of dimension n is a pair (I2, (I)) where C2 is an open set in M and (I) is a Cr diffeomorphism of SI onto a cube In := I X In-1 = {(x, y) E R x R1 : lx1 < 1, Ily11,,,, < 1} which take trajectories of X in 12 to the straight lines I x {y} C I x Theorem B.1.1 (Long Tubular Flow [PdM]) Let -y C M be an arc of a trajectory of X that is compact and not closed. Then there exists a tubular flow (f2, cD) of X such that -y C ft, as in figure B.47.  Since the transyersality of a vector field X to a submanifold E is preserved under diffeomorphism, it follows from the Long Tubular Flow theorem that qualitatively all such configurations look as in figure B.48.  Figure B.47: A flow 0 rectified by a diffeomorphism (D. 81  82  Appendix B. Cited Theorems^  Figure B.48: Rectified flow past a transverse submanifold.  Figure B.49: E attracts M and attracts E. B.2 Flow-Connectedness Recall from the proof of theorem 4.4.1 that we had a local invariant submanifold E of  M such that for every orbit I'm in M, w(FM) C E, while for every orbit rE in E, w(rE) = {1} [ figure B.49 }. Then, using additional hypotheses on the flow on E we concluded that for every orbit riti in M, w(FM) = Additional hypotheses are indeed necessary as demonstrated in the counter-example X1 = X2 = 21'1 -3x  — px2(x? —^ix3)  M := {(x1,x2) E R2 : x1 > 0, H(  2  3'  -  0) < H(xi,x2) < 0}  Appendix B. Cited Theorems^  83  Figure B.50: A flow with an attracting homoclinic orbit. E := {(x1,x2) E R2 : x1 > 0, H(xl, x2) = 0},^(B.112) where 1 2  (B.113)  H(Xl, X2) := -X9+ X31 2 -^ A  as illustrated in figure B.50 where coal = E for every orbit  r in M \ E, when jz > 0.  In the case that E is 2-dimensional, we could use the hypothesis that the only cyclegraph in E is itself and apply the fundamental theorem on w-limit sets. A higherdimensional analogue is afforded in "flow-connectedness". Specifically, Definition B.2.1 aDFD Suppose that X is a compact metric space and T: X -4 X is a homeomorphism. We say that X is T -connected if for every closed proper subset A of X we have that TA  n CA 0 0,^  (B.114)  where CA denotes the closure (in X) of the compliment of A in X. Theorem B.2.2 aDFD A necessary and sufficient condition that X be an w-limit set under a flow {0t}tER is that X be 44-connected.  84  Appendix B. Cited Theorems^  B.3 Location of Zeros  Given a set M C Rn, a subset E C M defined as the level set of a differentiable function V: M^R, U C Rm, consider the problem, as in §4.6 and §4.10, of finding, if possible, for each x E E, values u_,uo,u+ E U such that f(x, u_)  < 0,  (B.115)  f (x , uo)  = 0,  (B.116)  f (x , u+)  > 0.  (B.117)  Aside from ad-hoc reasoning, this problem may be effectively solved on a computer. Specifically, discretize E and U, say via some fx),J1AEA C E and fu,,I,EE C U respectively. Then, for each A E A, if Isgnf (xA,u,) :• E El^  (B.118)  is 1-11,101, 1+11, 1-1,01, {0, +1} we speculate that we can solve (B.115), (B.116), (B.117), (B.116)&(B.116), and (B.116)Sz(B.117) only, respectively, for x = x), with ad-  equate values u_, uo, u+ constructed automatically. If (B.118) is only {-1, +1} then by the implicit function theorem, (B.115) (B.116) and (B.117) can be solved, and an adequate value of uo may be realized by Newton's method. The interpolation of the values uo from the discretization follows from the implicit function theorem. B.4 Non-triviality of the Tangent bundle to the 2-Sphere Proposition B.4.1 The tangent bundle T-7-, : TS2 -4 S2 : Xn ^n (Xn E TnS2 ,n E S2) is not trivial. That is, there is no continuous bzjection (I) : S2 x R2 -+ TS2 for which = rT o (I), where ri : S2 x R2 -* S2:  (n, u) 1-4 n.  For a proof see any modern differential geometry text such as [v\V].  Appendix B. Cited Theorems^  85  B.5 Contractability of Domains of Attraction  We paraphrase Theorem 4.8.14 from [So]. Theorem B.5.1 A domain of attraction of a continuous flow 0 : R x M -4 M on a topological manifold must be a contractable set. By continuous we mean that 0 is a continuous function.  B.6 Frobenius' Theorem  We paraphrase [vW]. Theorem B.6.1 Suppose that M is a differentiable manifold on which we are given a one-form w : U .— T*U defined some non-empty open subset of M. Then there exists a scalar function f(.) : U -4 1 :I such that Nk := {X E  U : f (x) = k, df (x) 0 0}  are integral manifolds for w for each k E rangef , , (that is TzNk C kerw(x) for every x E Nk,) if, and only if, w A dw = 0 on U.  B.7 Input-Output Stability via Asymptotic Stability  Following [Vid], consider an Input-Output system i(t) = f(t,x(t),u(t)) ^  y(t) = g (t , x (t), u(t))  (B.119)  86  Appendix B. Cited Theorems^  with 0 as an equilibrium point, f(t, 0,0) = 0 and g(t, 0,0) = 0 for all t E R, flu.° is of class C' and f and g are locally Lipschitz at (x, u) = (0,0). Definition B.7.1 The system (B.119) is said to be small signal Li-stable with finite gain and zero bias if there exists constants r> 0 and -yp, < oo such that  x(0) = 0, Ilu(t)ilp < rp, u E LP^  (B.120)  y E LP, blip^7pllullp.^  (B.121)  Theorem B.7.2 Suppose that x = 0 is an exponentially stable equilibrium point of the unforced system i(t) = f (t , x (t) , 0) .^  (B.122)  Then (B.119) is small signal LP-stable with finite gain and zero bias for each p E [0, +co].  B.8 LaSalle's Invariance Principle Definition B.8.1 A Local Liapunov function for a nonlinear control system (with state space modelled on a differentiable manifold M) relative to an equilibrium point x° is a continuous function V: M --+ R for which there exists a neighborhood 0 of x° such that the following properties hold: 1. V is proper at x0, that is, the sub-level set Ix E  M : V (x) < el is a compact  subset of 0 for each € > 0 small enough. 2. V is positive definite on 0, that is, V (x°) = 0 and V (x) > 0 for each x E 0,x 0  x°. 3. For each initial state e E 0 there exists a response x(.) : R+ .— M, x(0) = e of the control system to some admissible control such that ti-4 V (x(t)) is non-increasing and non-constant on the interval where x(t) E 0.  Appendix B. Cited Theorems^  87  The function V is furthermore, a global Liapunov function, if it satisfies (2) and (3) above with 0 = M and if V is globally proper, that is, the sub-level set {x E M V (x) < L} is compact for each L > 0. Theorem B.8.2 If there exists a local (respectively global) Liapunov function of a nonlinear control system relative to an equilibrium point x0, then the control system is locally (respectively globally) asymptotically controllable. That is, for each neighborhood V of x° there exists a neighborhood W of x° such that each x E W can be asymptotically controlled to x° without leaving V (respectively and also every x E Al can be asymptotically controlled to x°.) We say z can be asymptotically controlled to y without leaving V if there exists an admissible response x(.) : R+ -- M such that x(0) = z , lirn x(t) = y t-).o. and x(t) E V for all t > 0.  Bibliography  [AR] R. Abraham, J. Robbins. (1967) Transversal Mappings and Flows. New York: Benjamin. [AV] J.K. Aggarwal, M. Vidyasagar. (1977) Nonlinear Systems: Stability Analysis. Stroudsburg: Hutchinson & Ross. [An] V.I. Arnold. (1973) Ordinary Differential Equations. Cambridge: MIT Press. [Ar2] V.I. Arnold. (1989) Mathematical Methods of Classical Mechanics. New York: Springer-Verlag. [Ba] S.P. Banks. (1988) Mathematical Theories of Nonlinear Systems. New York: Prentice-Hall. [Bo] W. Boothby. (1975) An Introduction to Differentiable Manifolds and Riemannian Geometry. New York: Academic Press. [BB] S.P. Boyd, C.H. Barratt. (1991) Linear Controller Design: Limits of Performance. Prentice-Hall: Englewood Cliffs. [Br] R.W. Brockett. (1977) "Control theory and analytical mechanics." in The 1976 Ames Research Center (NASA) Conference on Geometric Control Theory. ( eds: C. Martin, R. Hermann.) Brookline Mass: MathSci Press. [CD] Y. Choquet-Bruhat, C. DeWitt-Morette, M. Dillard-Bleick. (1977) Analysis, Manifolds and Physics Part I: Basics. Amsterdam: North-Holland. [CS] L.J. Corwin, R.H. Szczarba. (1982) Multivariable Calculus. New York: Marcel Dekker. [Do] A. Douady. (1964) "Varietes a bord angleux et voisinages tubulaires." in Seminaire Henri Cartan, 14e armee: 1961-1962, no 1. Paris: Secretariat Mathematiques. [DF] Y.N. Dowker, F.G. Friedlander. "On the limit sets in dynamical systems." Proc. London Math Soc. 53, vol 3. (1954) [DFT] J. Doyle, B. Francis, A. Tannenbaum. (1992) Feedback Control Theory. MacMillan.  88  Bibliography^  89  [Fl] H. Flanders. (1963) Differential Forms with Applications to the Physical Sciences. New York: Dover. [Ga] P.R. Garabedian. (1964) Partial Differential Equations. New York: Chelsea. [Go] H. Goldstein. (1950) Classical Mechanics. Reading: Addison-Wesley. [Ha] M. Hall. (1967) Combinatorial Theory. Waltham: Blaisdell. [HK] M. Hazewinkel, R.E. Kalman. (1976) "On invariants, canonical forms, and moduli for linear constant, finite-dimensional dynamical systems". Lecture Notes Econ.Math. System Theory 131, 48-60. [Hsu] C.S. Hsu. (1987) Cell-to-Cell Mapping: A Method of Global Analysis for Nonlinear Systems. New York: Springer-Verlag. [HS] M. Hirsch, S. Smale. (1974) Differential Equations, Dynamical Systems and Linear Algebra. New York: Academic Press. [HK] K. Hoffman, R. Kunze. (1961) Linear Algebra. Englewood Cliffs: Prentice-Hall. [Is] A. Isidori. (1989) Nonlinear Control Systems: An Introduction. Berlin: SpringerVerlag. [Ka] T. Kailath. (1980) Linear Systems. Englewood Cliffs: Prentice-Hall. [KS] R.C. Kirby, L.C. Siebenmann. (1977) Foundational Essays on Topological Manifolds, Smoothings, and Triangulations. Princeton: Princeton University Press. [La] J.P.LaSalle. (1976) The Stability of Dynamical Systems. Philadelphia: SIAM. [LS] G.A. Lafferiere, E.D. Sontag. (1993) "Remarks on control Lyapunov functions for discontinuous stabilizing feedback". to appear in IEEE Conference on Decision and Control 1993. [Lu] C.L.Liu. (1968) Introduction to Combinatorial Mathematics. New York: McGrawHill. [Mar] J E Marsden. (1992) Lectures on Mechanics. Cambridge: Cambridge University Press. [Mu] J.R. Munkres. (1975) Topology: A First Course. Englewood Cliffs: Prentice-Hall. [NS] A.W. Naylor, G.R. Sell. (1971) Linear Operator Theory in Engineering and Science. New York: Holt, Rinehart and Winston.  Bibliography^  90  [0g] K. Ogata. (1970) Modern Control Enginering. Englewood Cliffs: Prentice-Hall. [PdM] J. Palais, W. de Melo. (1982) Geometric Theory of Dynamical Systems: An Introduction. New York: Springer-Verlag. [Si] G. F. Simmons. (1963) Introduction to Topology and Modern Analysis. New York: McGraw-Hill. [So] E.D. Sontag. (1990) Mathematical Control Theory. New York: Springer-Verlag. [Su] H.J. Sussman. (1978) "Analytic stratifications and control theory." Proc. Int. Congress of Mathematicians, Helsinki, 78, 865-871. [Vid] M. Vidyasagar. (1993) Nonlinear Systems Analysis. Englewood Cliffs: PrenticeHall. [vW] C. von Westenholz. (1978) Differential Forms in Mathematical Physics. Amsterdam: North-Holland. [Wi] S. Wiggins. (1990) Introduction to Applied Nonlinear Dynamical Systems and Chaos. New York: Springer-Verlag.  

Cite

Citation Scheme:

        

Citations by CSL (citeproc-js)

Usage Statistics

Share

Embed

Customize your widget with the following options, then copy and paste the code below into the HTML of your page to embed this item in your website.
                        
                            <div id="ubcOpenCollectionsWidgetDisplay">
                            <script id="ubcOpenCollectionsWidget"
                            src="{[{embed.src}]}"
                            data-item="{[{embed.item}]}"
                            data-collection="{[{embed.collection}]}"
                            data-metadata="{[{embed.showMetadata}]}"
                            data-width="{[{embed.width}]}"
                            async >
                            </script>
                            </div>
                        
                    
IIIF logo Our image viewer uses the IIIF 2.0 standard. To load this item in other compatible viewers, use this url:
http://iiif.library.ubc.ca/presentation/dsp.831.1-0079675/manifest

Comment

Related Items