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

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STABILIZATION VIASMOOTH PARTITIONS, TRANSVERSALITY AND GRAPHSByMichael GlaumB. Math., University of Waterloo, 1991A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinTHE FACULTY OF GRADUATE STUDIESDEPARTMENT OF MATHEMATICSINSTITUTE OF APPLIED MATHEMATICSWe accept this thesis as conformingto the required standardTHE UNIVERSITY OF BRITISH COLUMBIAAugust 1993© Michael Glaum, 1993(Signature)C SIn presenting this thesis in partial fulfilment of the requirements for an advanceddegree at the University of British Columbia, I agree that the Library shall make itfreely available for reference and study. I further agree that permission for extensivecopying of th is thesis for scholarly purposes may be granted by the head of mydepartment or by his or her representatives. It is understood that copying orpublication of this thesis for financial gain shall not be allowed without my writtenpermission.Department of ^V\ Ck+t'N e The University of British ColumbiaVancouver, CanadaDate  4.0995+- 110,, )99?) DE-6 (2/88)AbstractWith the aim of circumventing the difficulty in constructing Liapunov functions, a strat-egy for the design of static stabilizing feedback control laws of nonlinear systems isproposed.The basic method is to partition the state space and to find all controls so that theclosed-loop dynamics are transverse or coincident to the partition edges. Stability isanalyzed through the directed graph whose vertices are the subsets of the partition andwhose arcs are consistent with the transversality. A strategy for the choice of partitionis proposed using computable Pfaffian systems.11Table of ContentsAbstract^ iiList of Figures viiAcknowledgement^ x1 Introduction 11.1 Systems, Feedback and Design ^ 11.2 Liapunov's Method ^ 41.3 Overview^ 5Modelling 62 Control Theory Fundamentals 72.1 Engineering Motivation ^ 72.2 States, Signals and Systems 72.2.1^Dynamical Systems ^ 82.2.2^Input/Output Systems 92.3 Feedback 102.4 Example - The Simple Robot Arm ^ 112.5 Tracking ^ 142.6 Stabilization within Control Design ^ 161113 Differential Geometry in Control Theory 183.1 Overview^ 183.2 Example - The Spherical Pendulum ^ 203.3 Example - The Simple Robot Arm 233.4 Tentative Formalism ^ 253.5 Example - Controlled Constrained Motion ^ 273.6 Formalism ^ 293.6.1^Nominal Stabilization of Nonlinear Control Systems ^ 313.6.2^Example §3.5 continued ^ 31II Design 334 Smooth Partitions, Transversality and Graphs 344.1 Motivation ^ 344.2 Nonlinear Control Systems Modelled as Finite-State Machines ^ 354.3 Definitions ^ 364.3.1^Smooth Partitions ^ 374.3.2^Transversality 374.3.3^Example ^ 384.3.4^Graphs of Framed Dynamical Systems ^ 404.3.5^Example §4.3.3 continued ^ 404.3.6^Liapunov Functions ^ 414.4 Liapunov's Theorem 424.4.1^Example §4.3.3 continued ^ 514.5 Liapunov Functions for Control Systems ^ 51iv4.6 Design of Coincident and Transverse Edges and Face-Monotone Dynamicsof Control Systems ^  514.7 Graphs of Control Systems ^  524.8 Example: the 2-integrator  534.9 A Proposed Method for the Design of Framed Dynamics ^ 544.9.1 Example §4.6.1 continued  ^564.10 Example-A Liapunov Method. ^  574.10.1 Example: the n-integrator. .  ^585 Construction of Smooth Partitions and Liapunov Functions^615.1 Tentative Construction ^  615.2 Pfaffian Systems  625.2.1 A Classical Linear Pfaffian Problem ^  625.2.2 The Nonlinear 'OR' Pfaffian Problem  635.3 Transversality Multifunctions ^  645.3.1 Example: the 2-integrator  655.4 Liapunov Functions ^  665.5 Smooth Partitions  685.6 Example ^  686 Conclusions^ 74Appendices^ 76A Prerequisite Mathematical References^ 76A.1 Basics ^  76A.1.1 Basic Topology ^  76A.1.2 Algebra ^  76A.2 Geometric Analysis ^  77A.2.1 Advanced Calculus = Local Differential Geometry ^ 77A.2.2 Manifolds ^  77A.2.3 Bundles  77A.2.4 Algebraic Structures over Manifolds ^  77A.2.5 Topological Dynamics ^  78A.2.6 Exterior Differential Systems ^78A.2.7 Geometric Mechanics ^  78A.2.8 Functional Analysis  78A.3 Control and System Theory ^  78A.4 Miscellaneous ^  79A.4.1 Graph Theory ^  79A.4.2 Automata Theory  79A.4.3 Multifunctions^  79B Cited Theorems^ 81B.1 Long Tubular Flows ^  81B.2 Flow-Connectedness  82B.3 Location of Zeros ^  84B.4 Non-triviality of the Tangent bundle to the 2-Sphere ^ 84B.5 Contractability of Domains of Attraction ^  85B.6 Frobenius' Theorem^  85B.7 Input-Output Stability via Asymptotic Stability ^  85B.8 LaSalle's Invariance Principle ^  86Bibliography^ 88viList of Figures1.1 Components within the plant ^21.2 A closed-loop interconnection of plant and controller.  ^31.3 A compensation interconnection between plant and controller.  ^31.4 Orbits converging on level curves of a Liapunov function ^41.5 A partition of the state space and a superimposed direction field togetherimplying a graph ^52.6 A typical industrial robot.  ^72.7 A simplified motor component.  ^82.8 A dynamical system or flow on a plane and its geometric meaning. . . .  ^92.9 Graphic representation of an input/output system ^ 102.10 A feedback loop interconnecting the plant and controller  112.11 A controlled-observed robot arm ^  122.12 Free body diagram of the mass m  122.13 Open-loop robot arm ^  132.14 A robot arm with vertically up-motionless stable. ^  142.15 The open-loop tracking strategy of tandem connection with an inverse. ^ 152.16 A compensator plant controller interconnection ^  152.17 A reference input r(.) and the compensated arm's response OH satisfying(2.11)  ^162.18 A)Compensated I/O system and b)associated dynamical system^ 173.19 The spherical pendulum ^  20vii3.20 The spherical coordinate system (3.16)  ^213.21 The second spherical coordinate system (3.22) ^  223.22 A robot arm before a) and after c) winding b)  233.23 Closed-loop dynamics (3.26) under (3.28) with three orbits (ir, 0), r+, r_not attracted to (0, 0)  ^253.24 A vector field f E X(M) defined on a manifold M ^ 263.25 A commutative diagram for the tentative definition  273.26 Mass in constrained to move on a manifold N propelling itself by a forcein the tangent space. ^  273.27 Commutative diagram for f : P -4 TM^  304.28 Level sets of a Liapunov function and the orbits of a closed-loop dynamicalsystem converging on them ^  354.29 An example of a finite-state machine^  354.30 a) the direction field of a dynamical system across some triangles, and b)the associated finite state machine. ^  364.31 Flows a) transverse to, b)coincident with, and c)neither transverse to norcoincident with an edge. ^  384.32 Phase portrait of (4.44)  394.33 a) A smooth partition framing the flow (4.44) and b) the superposition ofthe direction field.  ^394.34 The graph associated with the flow (4.42) and the smooth partition inFigure 4.33 ^  404.35 Conceivable situations that Liapunov's theorem excludes^ 444.36 Geometry of a tubular flow around -y^  484.37 Rectified flow in a subtube from F_ to F+  48viii4.38 Geometry to exclude closed trajectories.  ^504.39 The graph associated with the control system (4.57) and the smooth par-tition in Figure 4.33  ^544.40 An attracting invariant manifold which contains an attracting equilibriumpoint.  ^575.41 A normal field and a surface normal to it. ^  635.42 Foliation of a selection of^ 665.43 Foliations of selections of N-02 ^715.44 A smooth partition for (5.102)  ^715.45 The graph associated with (5.102) and Figure 5.43 ^  725.46 A stable subgraph of the graph in Figure 5.44  73B.47 A flow rectified by a diffeomorphism (I) ^  81B.48 Rectified flow past a transverse submanifold  82^B.49 E attracts M and attracts E    82B.50 A flow with an attracting homoclinic orbit    83ixAcknowledgementI am indebted to Drs Wayne Nagata and Frank Karray for all their help but especiallyfor their encouragement in preparing this thesis.Many of my colleagues have contributed to my education and this thesis, but I wouldlike to acknowledge David Austin, Alan Boulton, Andrew Hare, Charlie Horn, AlexKachura, Djun Kim, Philip Loewen, Alan Lynch, Sheena McRae, Kirsten Morris, RobRitchie, Stephen Smith and John Wainwright for their academic suggestions.Finally, it is a pleasure to thank Gfil Civelekoklu, Lynn Van Coller, David Fisher, Mar-tin 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 Quercusalba.xChapter 1IntroductionAdvances in engineering, science and technology have had a profound effect on Canadiansociety. Popular examples are the tangible advances such as the invention of plastics orthe use of computers.A more pervasive but inconspicuous advance, however, was the invention of the ideaof active control, for example, feedback. Indeed, feedback is used in walkmans, intelecommunications, for cruise-control and anti-lock brakes in automobiles, in Tokomakreactors, and even in our own physiological processes.The necessary choice of mathematical model and control strategy, however, has acrucial effect on the performance of systems using active control. The classical linearmodels and control strategies suffice in many instances, but for advanced systems withinherent nonlinearity, or where performance is essential, the more realistic but difficultnonlinear models and strategies are necessary.In this thesis we propose a control strategy for engineering systems modelled bynonlinear differential equations.1.1 Systems, Feedback and DesignA typical example of a nonlinear system is a robot arm. Such a system must containactuator and sensor components to control and observe itself, as in Figure 1.1. In theexample of a robot arm, typical actuator and sensor hardware include motors and poten-tiometers respectively, placed at the joints of the arm. A motor accepts a voltage input1Chapter 1. Introduction^ 2Figure 1.1: Components within the plant.signal to influence the arm, and a potentiometer outputs a voltage signal representingsome observation of the arm.As stated, the arm may be influenced, e.g. made to perform a task, by selectingan input signal to relay to the actuator. By augmenting the arm with a controller toautomatically make this selection, the robot arm may then be made to "control itself". Acontroller's choice of instantaneous input is often based on the fed back output observedfrom the sensors. The controller may then be summarized in a feedback law whichassociates a control with an observed output.Using such a strategy, the control problem within system design may be isolatedas 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 maybe reprogrammed easily. It is important to note that it is the mathematically designedsoftware 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 bynonlinear but smooth deterministic differential equations in a finite number of variables,Chapter I. Introduction^ 3Figure 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, thata prescribed orbit, e.g. an equilibrium point, attracts all other orbits, or bounded ref-erence inputs r produce bounded outputs y, respective of Figures 1.2 and 1.3. Otherdesign specifications implicitly assume stability or are essentially implied by stability, forexample tracking.In conclusion, we have isolated within engineering system design the mathematicalproblem of choosing a feedback law for a system given an Input/Output model realizedby a system of nonlinear differential equations so that the closed-loop is stable, i.e. theNominal Stabilization Problem.Chapter 1. Introduction^ 4Figure 1.4: Orbits converging on level curves of a Liapunov function.1.2 Liapunov's MethodWhile there is no general method for the construction of stabilizing feedback controllaws of nonlinear systems, the most viable is Liapunov's method [AV]. This method iscontingent upon the designer being able to find a function satisfying algebraic hypothesesrelating the geometry and dynamics of the flow. The deficiency of this method is thatthere is no algorithm for constructing such a function.In this thesis, we re-interpret the algebraic "definiteness" hypotheses of Liapunov'smethod in terms of the geometric notion of transversality to a submanifold. The con-cepts of partition, transversality and graphs are then used to give a geometric version ofLiapunov's method.Intuitively, the stability relationship between geometry and dynamics in Liapunov'sclassical method is implied as in Figure 1.4, while for the method of this thesis, by thegraph implied by a direction field superimposed over a partition of the state space, as inFigure 1.5.Differential geometry gives a convenient, if not necessary, language to formulate theabove concepts.Chapter 1. Introduction^ 5Figure 1.5: A partition of the state space and a superimposed direction field togetherimplying a graph.1.3 OverviewIn chapters two and three we review the background for modelling mechanical systems,specifically, control theory fundamentals and geometric mechanics respectively. Some ofthe mechanical examples of chapter three appear to give a new perspective of exhibitingphenomena discussed only within the modern theory of differential geometry.Chapter four outlines the stated nonlinear design strategy. The author is not awareof the method appearing anywhere in the literature, but there is some evidence thatsimilarities exist with the theory of hybrid systems [LS]. The aim of this thesis is not topresent a definitive algorithm but to initiate the use of certain geometric methods.In chapter five, strategies are presented for constructing partitions and Liapunovfunctions. Some mathematical prerequisites are reviewed in the appendices.Part IModelling6Chapter 2Control Theory Fundamentals2.1 Engineering MotivationConsider 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 thearm 2 track a predetermined trajectory?This engineering tracking problem may be taken as one motivation for the mathematicsof control and system theory.2.2 States, Signals and SystemsIn this section we will contrast the "dynamical system" and "input/output" modelsdiscussed in this thesis, using the arm in the previous section as an example.Figure 2.6: A typical industrial robot.7Chapter 2. Control Theory Fundamentals^ 8Figure 2.7: A simplified motor component.2.2.1 Dynamical SystemsOnce programmed, the complete arm is an example of a mechanical system, withmeasurable idealized physical properties such as instantaneous relative angular displace-ment, velocity and acceleration, mass, etc. The arm is, furthermore, dynamic, in thatsome of the measurements may evolve over time. Depending on the implementation, thearm may be deterministic, meaning that if all the physical properties of the arm weremeasured at one instant, there exist models which could predict all future measurementsvery well. Associated with such models is the concept of state, specifically, the leastamount of instantaneous information x which uniquely determines all other measurablephysical 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 displace-ments and velocities at any instant may be taken as the state space, depending upon theimplementation of the arm and controller. The evolution of the system is then a pathxc : 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 meaningChapter 2. Control Theory Fundamentals^ 9Figure 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 initializedat a state xo E M, and on the other hand, that x 1-4 0(to,x) summarizes the evolutionof 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 theseand further concepts which occur frequently in this thesis.2.2.2 Input/Output SystemsBefore the arm is programmed, however, we must discuss how the external concepts ofinput, output and causality effect on the internal dynamics of the arm. The voltage ap-plied to the motor, the input, and the measured angular displacement and velocity of thearm, the output, as vector-valued functions of time, are examples of signals. The trans-formation from input signal to output signal, as in the arm, is called an input/outputsystem, or I/O system for short.In figures, signals and input/output systems are represented by directed lines andblack-boxes respectively, as in Figure 2.9, and are often given names like P or C forChapter 2. Control Theory Fundamentals^ 10Co n iTE).\••btrottlMotor,^ robe:* cm-t.4art 6,Tcyt-e.n-t-;e>tAk4er'^90 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 motor-arm-potentiometer system by an operatorP: C(R, R) —+ C (R, R2) : v(.)^(et)9(.)(2 .2)mapping a continuous voltage signal to a smooth angular displacement and velocitysignal. These linear operators are frequently realized by systems of differential equationsor, through Laplace transforms, by matrices of complex functions. We will consistentlyuse the former representation in this thesis.The reader is asked to refer to [Vid] for a discussion of (UBIBO) I/O stability, theI/O version of dynamical systems stability: intuitively, that a bounded control signalproduces a bounded output signal.2.3 FeedbackOne strategy for implementing control is through the use of feedback, that is, basingour choice of instantaneous control on the past and present observed output. Under thisChapter 2. Control Theory Fundamentals^ 11Figure 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 con-troller C connected to the plant as in Figure 2.10 resulting in an autonomous dynamicalsystem exhibiting closed-loop dynamics.Feedback control theory is concerned with the design of the controller C and theanalysis of its influence on the closed-loop dynamics2.4 Example - The Simple Robot ArmFeedback is particularly useful for stabilizing an unstable equilibrium of a mechanicalsystem, using the controller to compensate for perturbations and the tendency to moveaway.Following [So], consider the vertical robot arm modelled as a point mass m at the endof a rigid massless rod of length 1 with a motor at the pivot supplying a variable torque uand 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 verticallyupward motionless state (say while other robot arm components perform a task).Assume that the potentiometer measures angular displacements in a continuous fash-ion. That is, if the arm rotates 27r radians clockwise, then 27r is added to the displacement.Chapter 2. Control Theory Fundamentals^ 12Figure 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 verticallyup.From Figure 2.12 and Newton's second law, the governing equation isum/Ö = mg sin 0 + —/ '^ (2.3)Dr in terms of the state vector (xi, x2) := (OM E R2,1 )u-om12(2.4)The equation (2.4) is a model for the open-loop robot arm and potentiometer: givena control u(.) : [0, oo) —> R in LI, (2.4) has a unique solution x : [0, co) —> R2 as in0 1 )^)^0+^1 }IL01 M12X2(2.6)(2.7)0 1 )g nv^X2101^(—a 13) X1( X2 )n—Tri(2.8)Chapter 2. Control Theory Fundamentals^ 13Figure 2.13: Open-loop robot arm.Figure 2.13 ( see [An]).Using a primitive version of the technique of feedback linearization [Is], we now changecontrol variables to linearize (2.4). Specifically, letft := mlg(—xi + sin xi) + u^ (2.5)so that on substitution, (2.4) becomes, in standard form,As an aside, note that it not always possible to linearize control systems. If we apply alinear feedback law, sayfi(xi, x2) := (—a — 0)\ X2 Ivia Figure 2.10, then the associated closed-loop dynamics is obtained by substituting (2.7)into (2.6). We calculateOr(2.9)—m12Chapter 2. Control Theory Fundamentals^ 14Figure 2.14: A robot arm with vertically up-motionless stable.We must choose a and so that (2.9), the closed-loop dynamics, will have have (0,0) asa stable equilibrium point, i.e. the coefficient matrix must has eigenvalues with negativereal 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 closed-loop dynamical system in Figure 2.14 will rise to the vertically upward motionless stateand return there when perturbed.2.5 TrackingRecall the tracking problem of § 2.1. If we model the motor+arm+potentiometer by theplant in Figure 2.13, then the simplest strategy would be to connect P in tandem withan "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^ 15Figure 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 solvethe tracking problem motivated in the question of § 2.1. Specifically, if P and C representthe plant and controller of Figures 2.13 and 2.14, then associated with the stable dynam-ical system in Figure 2.10 is the compensated I/O system in Figure 2.16 which takesthe desired trajectory r(.) to be tracked as input, properly scaled by m12, and outputsthe angular displacement and velocity 9(.),O(.). It follows from (2.4) that these signalssatisfy the differential equation0g a— —ml2xi10 ) (^)+ r.m12 x2(2.11)For example, with a and # chosen so thatg a— —^= —1,^/2= —2^ (2.12)/ m/2^mfor t < 01for 0 < t < —21for —2 < t < 1for 1 < t(2.13)Chapter 2. Control Theory Fundamentals^ 16Figure 2.17: A reference input r(.) and the compensated arm's response OH satisfy-ing (2.11).and for the reference trajectory0-1-t2r(t) := 2 1—t2 —2t + —21a simulation of the response of the arm is given in Figure 2.17. The stability of this I/Osystem is asserted in the next section.2.6 Stabilization within Control DesignIn most popular design strategies, for example I-1' control [DFT] [BB}, the assumptionis often made that the plant is I/O stable. Thus, to apply these powerful methods tounstable plants we must first compensate them, that is, find a controller K0 so thatChapter 2. Control Theory Fundamentals^ 17Figure 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 systemP(I — K0P)-1 would be stable if the closed-loop dynamical system in Figure 2.18b) isasymptotically stable, as argued by theorem B.7.2 of the appendix. Furthermore, anystabilizer would suffice as any undesired response could be undone through feedback sono regard need be paid towards the design specifications.Note that controllers used in Hc° are typically "observer-based", so we will assume aninput-to-state model [So] for our plants in the sequel. For example, if the potentiometerin example 2.4 did not observe the complete state, then an observer would be used toasymptotically estimate the remaining state variables, and this estimated state would beused in the control law.In summary, within most control design problems we may isolate the nominal sta-bilization problem: given a plant P find a controller K0 so that the closed-loop systemin figure 2.18b) is asymptotically stable.In this thesis, we will propose a framework for constructing such a controller fornonlinear but smooth plants P. We will now discuss the plants accommodated withinthis framework.Chapter 3Differential Geometry in Control Theory3.1 OverviewWe 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 thereader with the basics of classical mechanics and differential geometry. The majority ofthis chapter is not referred to in subsequent chapters, and thus may be skipped on a firstreading.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 argumentsbased on physical laws or models. The evolution of the mechanical system is then modeledby the Euler-Lagrange equationsd far, } 0L0aii^axi—for 1 < i < n. The form of these differential equations is, most generally,(3.14)F(x, = 0 (3.15)for a function F: R3n- —+ R . Unfortunately, for systems such as the spherical pendulumdiscussed below, there are some singular states (x, th) for which (3.15) cannot be solvedfor i uniquely. The initial-value problem (3.15) with initial condition at one of thesepoints thus has no unique solution.18Chapter 3. Differential Geometry in Control Theory^ 19More importantly, in a punctured neighborhood of these singular states, that is, wherewe can solve for i, say as1 = f(x,i), (3.16)the vector field f is typically discontinuous. In summary, flow computation is not awell-posed problem in the sense of Hadamard [Ga] for such an example. (Furthermore,numerical approximation of the differential equation in a neighborhood of these pointsis prone to error.) As this indeterminacy is not observed in physical reality, the classicalmodel of Euler-Lagrange is inappropriate for some problems. The physical states at whichthis indeterminacy occurs depends, suspiciously, on the choice of coordinates. In fact, wemay recover an adequate well-posed model by "combining" two overlapping coordinatecharts, with determinacy being implied in at least one coordinate chart, at each physicalstate. Differential geometry is the conceptual framework for such a model.It should be noted, however, that there are many other motivations for the use ofdifferential geometry in mechanics. A few are illustrated in the following examples.Important motivations not mentioned here include nonholonomic systems [vW] and thetheory of families of linear systems [HK].In subsequent chapters we will speak of arbitrary subsets defining a partition of thestate space and thus we will continue to use the language of differential geometry in thesequel.Abortive discussions and hints of the following examples and formalism may be foundthroughout 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].vtassew+o moveonChapter 3. Differential Geometry in Control Theory^ 20"Sc."ffor }-Figure 3.19: The spherical pendulum3.2 Example - The Spherical PendulumConsider the spherical pendulum as in Figure 3.19, where the joint J has "two-degreesof 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 0y= sin 9 sin 0^(3.17)z=^cos 0For 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 atrajectory (OH, OH), we have that the velocity v of the mass m in rectangular coordinatesat a specific state (0,0,0, (.k) isv= (—sinesin0O-1-cosOcos44,cosOsin0O+sinOcos0i/),—sin0ik),^(3.18)1using the notation sin 0 e := (sin 0) é, so the kinetic energy function —mil v1132^may bedTit [sin2 0 é] = 0Chapter 3. Differential Geometry in Control Theory^ 21Figure 3.20: The spherical coordinate system (3.17).shown to beK(0,0,0, ())) = ;-m(sin2 0 42 + 42 ) •Therefore, the Lagrangian L = K — P isL(0, 0,0, i) = fli, (sin2 0 42 + CO) - mg cos (/).Thus, the Euler-Lagrange equations are(3.19)(3.20)d- (ik) - sin 0 cos 0 0 — g sin 0 = 0dtOrsin2 0 e =^—2 sin 0 cos 0 è ik(3.21)0^= sin q5 cos 0 02 + g sin 0.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, when0 = mr, n E Z, 0 has no physical meaning. Furthermore, deleting these hyperplanesChapter 3. Differential Geometry in Control Theory^ 22Figure 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 0.2The right-hand side vector field, through cot 0, is discontinuous. Thus any numericalsolution 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 ofcoordinates. In the coordinate system (3.17), these physical states are vertically up ordown, of any velocity. By a choice of a second superimposed coordinate system (3.23)say in Figure 3.21,x =^cos 02y= COS 02 sin 02^ (3.23)= sin 02 sin 02(3.22)we obtain a second modelsi 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^ 23Figure 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 conceptualframework dealing with multiple coordinate systems. This theory exists as the differentialgeometry of manifolds and vector bundles.The non-well posed phenomena occurred in this example because the natural statespace TS2, the tangent bundle to the sphere S2 ( see § B.4.1) is not diffeomorphic tothe classical Euler-Lagrange state space IV, with the discontinuity being reflected inequation (3.22). Indeed, now is an appropriate time to review the motivated differentialgeometric approach to mechanics described in [Ar2] and [Mar].3.3 Example - The Simple Robot ArmFollowing [So], recall the simple robot arm of example of § 2.4 for the scenario of Fig-ure 3.22 where the arm is externally rotated by 27r radians clockwise, n E Z, fromequilibrium and released.Chapter 3. Differential Geometry in Control Theory^ 24In implementations where there is no physical memory in the system of the number ofcomplete revolutions from some reference, we would consider the wound state as returnedto 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, thenatural 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 thedifferential equations0=4:4) =w:g- si^un 0 + 1 m12(3.25)but rather on the state space manifold TS' = Si- x R, with coordinates 0 E Sl, andw 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 --+ Rstabilizing (3.25). That is, there is no continuous k for which the differential equation0=CO =wg k(0,w) 7 sin 0 + m12(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], itwould define a continuous flow on SI x R. Indeed, the flow is maximally definedby 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 Rwould be contractable, which is a contradiction. 0There is no agreement in the literature on how to construct a stabilizer K in viewof the previous proposition. We must use discontinuous feedback laws and/or suffice toChapter 3. Differential Geometry in Control Theory^ 25Figure 3.23: Closed-loop dynamics (3.25) under (3.27) with three orbits (it, o),r+,r_not attracted to (0, 0).attract only a portion of S' x R. In regards to the second alternative, as suggested bythe damped vertical planar pendulum, anyK (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 inFigure 3.23.3.4 Tentative FormalismIn summary, the first example § 3.2 demonstrates that for some mechanical systems,the differential geometric framework of [Mar] is necessary. The second example § 3.3demonstrates that the topological structure of the state space is a consequence of thedesign specifications.By analogy with the development of linear control theory from the theory of lineardifferential equations, it seems fitting at this time to speculate on the extension of theChapter 3. Differential Geometry in Control Theory^ 26Figure 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 andU. Thus the state+control space (which consists of the attainable states andthe admissible controls while at that state) is modeled on a topological prod-uct 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 adifferential equationi(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). Thegeometric meaning of f (or rather, just a vector field with no controls) is givenin Figure 3.24. As regularity hypotheses, we assume that f is continuous and thediagram in Figure 3.25 commutes, where the tangent bundle 7rT : TM -+ Mx 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 appearcontrived, it is a simplified idealization of a phenomena observed in the study of satellites.Chapter 3. Differential Geometry in Control Theory^ 27Mx ---TM-rrZrrTNFigure 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 inthe tangent space.3.5 Example - Controlled Constrained MotionConsider a mass m constrained to move on a frictionless embedded 2-submanifold N ofR3. While at n E N, the mass is assumed to be able to apply to itself, i.e. by its ownpropulsion, 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 thecontrol 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^ 28Thus we must define a bijection (I) : N x R2 -4 TN, that is 4)(n, u) E TN for alln E N,u E R2. This map (I), coordinating the effect of propulsion control, is used inderiving 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 sayS2, (I) and thus f cannot be continuous, thus violating hypothesis (H2) above. Thusthe 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 vec-tor bundle structure if : p*T N -4 TN given by the pullback of the natural configura-tion+control tangent bundle structure it : TN -- N under the natural state space tangentbundle projection p: TN -4 N onto the configuration manifold.Proof: We stated earlier the hypothesis that the set of admissible controls while at aconfiguration n E N is TEN. Therefore, the configuration+control manifold has anatural bundle structure it : TN -4 N where, for any configuration n E N in thebase 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 whileat a state x depends only on the configuration p(x) implies that the state+controlspace has a natural pullback vector bundle structure Fr : p*TN —> TN given alge-braically byFr-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 ofadmissible controls while at its configuration p(x).Chapter 3. Differential Geometry in Control Theory^ 29The non-triviality of the vector bundles follows from the non-triviality of TN -4 Nfor N diffeomorphic to S2 §B.4.1. 0Thus, for mechanical systems where there does not exist a consistent admissible con-trol 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 chartof the state space, i.e. trivialize the state+control space.3.6 FormalismWe now state the formalism and problems motivated by the examples of the precedingsections and chapters. We will only discuss Input-to-State models, i.e. where there isno 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 thestructure of a Lie group G, with the admissible controls a subset of U(x). Thestate+control space^P := U {x} x U(x)^(3.31)xEAIis assumed to have the structure of a principal G-bundle7r : P^M: {x} x U(x)^x (3.32)i.e. with typical fibres the (state in product with the) conceivable controls. Thenotation P should not be confused with the notation P for a plant.• The law governing the dynamics/control is encoded ^in a^mapf : P —> TM (3.33)Chapter 3. Differential Geometry in Control Theory^ 30P'-,--i-fri1r \ Z—rrP^M^TtiFigure 3.27: Commutative diagram for f : P -4 TMfor 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 meaningto f only when used in a feedback loop as below.• A (memoryless nonlinearity) feedback control law is a section 4) : M -4 P of thestate+control bundle P, i.e. such that r o 4) = lAf. Geometrically, at each statex 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 inf o 4). Indeed, f o 4. : M —> TM is a section of the tangent bundle to M and thusdefines an autonomous differential equation± = f o 4)(x)^ (3.35)and thus the closed-loop dynamical system indirectly.(TU, (Xi, X2, -a aax,' ax)"Chapter 3. Differential Geometry in Control Theory^ 313.6.1 Nominal Stabilization of Nonlinear Control SystemsWithin this context then, the nominal stabilization problem is to find a (memorylessnonlinearity) feedback law 4) : M —> P such that the vector field f o 4, E X(M) of theassociated closed-loop dynamics has a globally asymptotically stable equilibrium.This is the basic problem of this thesis. It will be abbreviated by the subsectionnumber §3.6.1.3.6.2 Example § 3.5 continued.As argued before, the configuration space is S2, the state space has a bundle structurep : 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 patcha a(TU, (Xi, X2, _--71-, -----))X ax2"denoted rather by(Xi, X2, Xi, X2) : TU --÷ R.^ (3.37)Likewise the configuration+control space TS2 has an induced coordinate patchdenoted rather by(X1, X2, Ui, U2) : TU^R4.^ (3.38)The pullback bundle 7r*TS2 has thus an induced coordinate patch(71-*TU, (X1, X2, -.'''.1., -2, Ul, U2))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. •^a^a^a .a ))(TTU, (Xi, X21 .‘..11 )2) =7) ax2' aX1' a)2In terms of these local coordinates the evolution of the system is encoded via (H4) abovewith,.,„: ,„ ,„ , .^__.^a^a^IL, a •^u2 af 0 y,^-1, -2, .1, .2, xi— + x2— +^+^°xi^ax2 m °X,^ax2as a tangent vector in To-i(x,^,i2)TS2.Part IIDesign33Chapter 4Smooth Partitions, Transversality and Graphs4.1 MotivationIn the classical Liapunov method §B.8 for the design of a memoryless stabilizing feedbackcontrol law of a nonlinear control systemth = f(x, u),in the local coordinate form of §3.6.1, we must find a differentiable function V : M Rwith nested compact sub-level sets and a feedback control law u : M -+ U such that theclosed-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 hypothesestypically conflict. In the sequel, we will propose a method based on a different relationbetween geometry and closed-loop dynamics than that given in Figure 4.28. We willrefer to geometric concepts, like submanifold and transversality, rather than the classicalalgebraic concepts such as positive-definiteness.Admittedly, cleverness is needed to apply this method in practice, so simpler linearexamples are used initially to illustrate the method. A genuinely nonlinear system forwhich standard nonlinear methods appear inconclusive is studied in the next chapter.It appears that the design strategy proposed in this chapter is original, although thecited concepts, such as partitions, transversality, graphs, framing and Liapunov functions,are all based upon classical versions. A related analysis stressing combinatorial issues34Chapter 4. Smooth Partitions, Transversality and Graphs^ 35Figure 4.28: Level sets of a Liapunov function and the orbits of a closed-loop dynamicalsystem 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 MachinesIn [So] the claim is (implicitly) made that the finite-state machines (FSMs) of automatatheory are instances of control systems. Instead of a differentiable manifold we have afinite set of points S := {A, B, C, .. .} as state space and instead of a differential equationwith 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 nonlinearChapter 4. Smooth Partitions, Transversality and Graphs^ 36Figure 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 thestate space on the nonlinear system, and each transition A B will be brought aboutby 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 arbi-trary geometric subsets of the state-space and the Liapunov functions are now used toassert the transition from one subset to another. The FSM merely codifies the relationbetween the geometry and the dynamics of the system. Such a relation is already presentin 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 workingwith 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 ex-ample, graphs rather than FSMs should be used to model the relation between geometryand dynamics. Furthermore, differential geometry will continue to be used in order todefine transversality.4.3 DefinitionsIn this section we propose definitions used to clarifying the subsequent strategy.Chapter 4. Smooth Partitions, Transversality and Graphs^ 374.3.1 Smooth PartitionsDefinition 4.3.1 Given an n-manifold M with corners §A.2.2, such as the state spaceof a nonlinear control system, we mean by a smooth partition P := {Fi} a partitionof M into n-manifolds Fi with corners, called faces, so that Fi n F is either empty oran 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 withcorners), and the elements of this set are called edges.The smooth partitions are assumed to be locally finite in that only a finite number offaces intersect any bounded subset of M. Each face F E P is assumed to be a closed setin 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 asimplex [Fl] or an analytic stratification [Su] respectively, for the component sets.4.3.2 TransversalityDefinition 4.3.2 Following [AR. I, suppose that M is a smooth manifold, E is a codi-mension 1 embedded submanifold of Al without boundary and f E X(M) is a continuousvector field on Al.  We say f is transverse to E ifspan( f (x)) TrE = T^(4.41)for all x E E. Likewise, we say f is coincident with E iff (x) E TrE^(4.42)Chapter 4. Smooth Partitions, Transversality and Graphs^ 38Figure 4.31: Flows a) transverse to, b)coincident with, and c)neither transverse to norcoincident 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 Mand f E X(M) is a continuous vector field on M. We say that the dynamical systemdefined by f is framed by P if, for every edge E E 02, f is either transverse to orcoincident with the (n — 1)-interior of E, exclusively.Associated with a smooth framed flow is a disjoint partition of M into intP,aTP,acPand 00T2 each consisting of the union of the interior of the faces of 7', the open edgesfor which the flow is transverse, the closed edges for which the flow is coincident, and theboundary of transverse edges not intersecting coincident edges.4.3.3 ExampleConsider the linear dynamical systemXi =^x2(4.43)X2 = —2x1 — 3x2which 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^ 39Figure 4.32: Phase portrait of (4.43).Figure 4.33: a) A smooth partition framing the flow (4.43) and b) the superposition ofthe direction field.Chapter 4. Smooth Partitions, Transversality and Graphs^ 40Figure 4.34: The graph associated with the flow (4.43) and the smooth partition inFigure 4.33.Clearly the dynamical system is framed by the smooth partition.4.3.4 Graphs of Framed Dynamical SystemsDefinition 4.3.4 A smooth partition P framing a dynamical system defines a graphG := (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, anundirected 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 iscoincident 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 Fig-ure 4.33 is given in Figure 4.34. Note that it has both directed and undirected arcs.0Chapter 4. Smooth Partitions, Transversality and Graphs^ 41In contrast to § 4.1, the more general concept of a graph is used to analyze the relationbetween geometry and control. Indeed, in example § 4.3.3 above, the graph 4.34 couldnot be associated with the state and transitions of a FSM as described in § 4.1, sincethere exist distinct states in F5 which evolve into F6 and F7 respectively, in violation ofthe determinism hypothesis of FSMs.4.3.6 Liapunov FunctionsDefinition 4.3.5 Following Thal a function V : C-2 —> R of class C(0) fl C1(12), 0 0C2 C M open, is said to be a Liapunov function of an autonomous dynamical systemdefined 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 derivativeof V along the trajectories of 0, i.e.1.7(x) := -aiV(q5(t,x))1t=o^ (4.44)or, using the chain rule, byV (x) := dV (x) f (x) (4.45)which does not require an explicit formula for the orbits. Thus V 0 0 may be inter-preted as saying that V is consistently increasing or decreasing along the trajectories, oralternatively, 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 orbitr+ trapped by a compact set F C M, f_f_ C F. Suppose further that there exists a classC(F)r1C1(intF) Liapunov function V : F -+ R on the interior of F, satisfying V (x) < 0for all x E intF. Then 0 0 w(r) C OF.Chapter 4. Smooth Partitions, Transversality and Graphs^ 42Proof taken from [HS] and [La]: As F is a compact trapping set for fk, we have that0 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 continuouson the compact set F). Thus there exists acER such thatv(r+(t)) -+ cast^oo. (4.46)Now for any y E co(r+) there exists a sequence tn^oo such that r+(t) -÷ y andthus v(r+(t)) -+ V(y), all limits as n^00, so by the uniqueness of limits ofsubsequences 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 totime at t = 0 we may conclude from (4.44) thataa-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. 04.4 Liapunov's TheoremIn this section we propose a version of Liapunov's theorem for asserting the asymptoticstability 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 ofspecific 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 faceF E P there exists a Liapunov function on intF.Chapter 4. Smooth Partitions, Transversality and Graphs^ 43Theorem 4.4.1 (A version of Liapunov's theorem) Suppose that {0t}tER is a dy-namical system framed by a smooth partition P with associated graph g satisfying thehypotheses(H1) 0 has an equilibrium point ,(H2) g is Q -stable as in definition A.4.1 where Q identifies the vertices associated withfaces containing the equilibrium pointQ:={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 theboundary of transverse edges, either x is attracted to the equilibrium point alongacP 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 isunbounded 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. 0Hypothesis (H6) may be replaced by analogues like(H6)' for unbounded faces F E P the associated Liapunov function V : F -4 R isbounded below but 1./(x) -4 —oo as lxj -+ oo in F.Chapter 4. Smooth Partitions, Transversality and Graphs^ 44Figure 4.35: Conceivable situations that Liapunov's theorem excludes.Such algebraic hypotheses could be avoided by partitioning M into an infinite numberof bounded faces. The reader should realize that the proof is not immediate as we mustexclude situations as in Figure 4.35.Proof of theorem 4.4.1: Assume (H1)-(H7) all hold and let x E M be an arbitrarybut fixed initial state. We seek to show that w(x) = M with the aid of somepreliminary lemmas.Lemma 4.4.2 If the forward orbit F+ (z) of an arbitrary state z E M is trapped bya face F E P then co(z) = {-±}.Proof of lemma 4.4.2: By hypothesis (H3), there exists a Liapunov functionV: F —> R on intF. Without loss of generality, assume V < 0 on intF. Bydefinition 4.3.5 then, it follows that t e-+ V (0(t, z)) is non-increasing and thusfrom 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 concludethato 0 w(z) c aF, c aF.Chapter 4. Smooth Partitions, Transversality and Graphs^ 45To prove otherwise, suppose that there exists a y E OT F n ce(z) . Then r(y) cw(z), as w-limit sets are invariant sets. By Theorem B.1.1, however, r(y) Fso 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 thatw(z) = 4LOLemma 4.4.3 Given a state y E M and a face F E P to which y belongs, eitherAl) w(y) = 41 orA2) there exists T, V E R, 0 < < v and a F' E P such that cb(r,y) F and0(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 Fwe have either thati) 4(t, y) E F for all t > 0 and thus (Al) w(y) = M by lemma 4.4.2, orii) 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 aTPis a locally finite union of open codimension 1 submanifolds, there exists aT> 0 and a unique open transverse edge E E aTp such that 0(t, z) E Efor all 0 < t < T. Differentiating with respect to time at t = 0 we seethat f (z) E TEE, where f is the vector field defining 0. This contradictsthe 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 that4)(T, z) E intF' and thus (A2) q!)(r T, y) E intF'.Chapter 4. Smooth Partitions, Transversality and Graphs^ 46If insteadr+(z) n intP = 0 (4.50)then by (4.49), r+ (z) n[acPuaaT'P] 0 0 so there exists a T > 0 such that4)(T, z) E acP U OaTP. Then by hypothesis (H4), alternative Al) holdsas (4.50) excludes the second alternative of that hypothesis (H4). 0Lemma 4.4.4 For the previously fixed but arbitrary state x E M eitherAl) w(x) = {} orA2) there exist sequences (v1)>1, (i)> in R with vi < < Vi+i for all i > 1 anda sequence (Fi)i>i of faces in P such that 0(v1,x) E intFi but O(ri,x) Fi forall i > 1.Proof of lemma 4.4.4: The proof is by induction on the index i of alternativeA2). 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) EAssume that we have concluded by induction that either Al) holds or thatthere exists finite, void or finite, and finite sequences (vi)1<i<„, ( 1<i<n-1and (F1)1<1<„ in R, R and P respectively satisfying the conclusion A2) foronly 1 < i < n. Lemma 4.4.3 applied at y = 0(v„, x), however, implies thateither Al) w(4(v„,x)) = w(x) = M or that there exists 0 < T < v andFn+1 E P such that 0(r, 0(vn, x))^Fn but 0(vAvn, x)) E^In thecase of the second alternative then, there exists vn+i := v + v„, r„ := r +satisfying v„ < r,, < lin+110(11n+11 X) E intFn+i and 0(7-n X) Fri •The proof follows by induction. 0Chapter 4. Smooth Partitions, Transversality and Graphs^ 47Lemma 4.4.5 Suppose that there exists an arc -y := {4(t, z) : 0 < t < T}, facesF_, 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 theassociated graph g from F_ to F+.Proof of lemma 4.4.5: By theorem B.1.1 there exists a tubular flow (CZ, 4)) ofthe 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 abase for the Euclidean topology on B". Thus, as 4) is a diffeomorphism, andintF_ and intF+ are neighborhoods for z and (1)(T, z) respectively, there existsa r > 0 such that.13;.' (4)(z)) c In (4.51)B77,1 (4.(cb(T, z))) c,.. In (4.52)4)-1(B77(4)(z))) c 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 infigures 4.37.Now the orbits of the induced flow 4).0 on E are parallel to those of theconstant flow (1, 0). Thus for every (n — 1)-dimensional coincident edge E EacP, we have that r_.(i)(E n fi) is a (n — 2)-dimensional submanifold withChapter 4. Smooth Partitions, Transversality and Graphs^ 48Figure 4.36: Geometry of a tubular flow around 7.Figure 4.37: Rectified flow in a subtube from F_ to F.Chapter 4. Smooth Partitions, Transversality and Graphs^ 49corners 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 thenthatD := _(1)((5cP U NTT') nis a finite union of at most (n — 2)-dimensional submanifolds with corners ina__E. By dimensionality arguments, D is a proper subset of a_E. Note thatopen transverse edges E E aTp induce open transverse edges (i(E n S7/) of theinduced 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 existsa finite sequence (Pi)1(<N of faces in P and a finite sequence (pi)0<j<N in Rsatisfying 0 = po < P1< • • • < /IN = T andfor 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 withthe induced flow, we must have that Pj -4 frj+i is a directed path in theassociated graph g, for each 1 < j < N. In conclusion, there is a non-trivialdirected path from F_ to F. If F_ = F+, however, then there would be acycle in the associated graph g, which contradicts hypothesis (H2). 0Lemma 4.4.6 If alternative A2) of lemma 4.4.4 holds, then Fi Fi+i and thereis a directed path in the associated graph g from Fi to Fi+1, for each i > 1.Chapter 4. Smooth Partitions, Transversality and Graphs^ 50Figure 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 sufficientlysmall so that 0(—e, z) E intF_, and set T := o- — c, and F+ := F_ so we havethat ck(T, z) E intF+ , and -y := {q5(t, z) : 0 < t < T} is a compact but notclosed arc. Finally, 0(7-i — z) (Z' F_, sa in figure 4.38. By lemma 4.4.5 appliedto -y then, we conclude that F_ F+, which contradicts the constructionF_ = F1 = F+. Therefore, the original arc -yi is not closed.Thus applying lemma 4.4.5 to the original -yi, with z := 0(vi,x), F_ :=Fi,F+ := F1+1,T := vi+i — Vi,7 := - vi, we may conclude that Fi F1+1and 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 infinitedirected path in the associated graph g, which contradicts hypothesis (H2).Thus alternative Al), tv(x) = M, must hold. 0Chapter 4. Smooth Partitions, Transversality and Graphs^ 514.4.1 Example § 4.3.3 cont'dRecall 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 g in Figure 4.34.Here Q := {F1,F4,F5,F6, &Flo} and indeed g is Q-stable (H2). The flow isface-monotone (H3) as demonstrated by the choice of (linear!) Liapunov functionsV(xl, x2) := —x1 on F1 U F2 U F3 U F4, V (Xi, X2) := 2x1 + 3x2 on F5, V (Xi, X2) := x1 onF6 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 n F9) u (F7 n Fs)and [(F3 U F4) n F5] u [(F8 u F9) n F10] is topologically equivalent to the flow i = —x onR, so hypotheses (H4) and (H5) are satisfied.4.5 Liapunov Functions for Control SystemsDefinition 4.5.1 A function V : fi —* R of class C() n C1(12), 0 //cm open, issaid to be a Liapunov function of a control systemth = f (x ,u)if the x-Lie-derivative V : CI x U --4 R defined byV (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 Dynam-ics of Control SystemsIn contrast to dynamical systems, one may design the invariant manifolds of a controlsystem and algebraically find the feedback law realizing them. Specifically, consider aChapter 4. Smooth Partitions, Transversality and Graphs^ 52nonlinear control problemi = f (x, u)^ (4.55)and a given submanifold E of the state space M defined by E = ix E M : V (x) = Volfor some known function V : M --+ R of class CI. If x(.) is an arbitrary closed-looptrajectory constrained to satisfy V(x(t)) = Vo for t E R then differentiating we see bythe chain rule and (4.55) thatf,(x(t),u)+ • • . +ax, z(t)^ax„Thus if we can solve the systemav avfn(x(t),u) = 0.^(4.56)av^avV(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 theoremwe 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 Eby replacing the equality in (4.57) by an inequality.Finally, suppose we have a Liapunov function V: F --+ R for a control system on aface F as in definition 4.3.5. If we can find a class Cl function u: F -+ U such thatV (x,u(x)) 0 0for each x E F, then the closed-loop response of the control system under that feedbacklaw u will be face-monotone on F with V as a Liapunov function.4.7 Graphs of Control SystemsDefinition 4.7.1 Consider a nonlinear control system § 3.6.1 and a smooth partitionP of the state space M. Again we define an associated graph g = (V, T). Specifically,Chapter 4. Smooth Partitions, Transversality and Graphs^ 53the faces of P are the vertices V of the graph, and two vertices F1, F2 E V = P have adirected arc F1 -4 F2 E T (respectively an undirected arc (F1, F2) E T ) between them if(H1) the common edge E :=F1 n F2 0 0 is a non-empty (n — 1)-submanifold and(H2) there exists a continuous control law ulE E^U defined on E for which theflow is transverse across the (n-1)-interior of E from F1 to F2. (respectively iscoincident 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 systemil = x2^ (4.58)i2 = uand the smooth partition P = {F1,^, Flo} in Figure 4.33a) again. The associated graphis 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 (respectivelyu = u(x) < —x2, respectively u = u(x) = —x2) on F1 n F4, the resulting closed-loopdynamics will be transverse across that edge from F1 to F4 (respectively, transverse tothat 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, andu) = 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^ 54Figure 4.39: The graph associated with the control system (4.58) and the smooth parti-tion in Figure 4.33.4.9 A Proposed Method for the Design of Framed DynamicsGiven a nonlinear control system, consider the problem of selecting a feedback controllaw for which the closed-loop dynamics is stable. When modelled on the examples andgraphs discussed in the previous sections, this selection is reflected in the transition fromFigure 4.39 for the control system to the stable subgraph in Figure 4.34 for the closed-loopdynamical 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 theideas of chapter 5).2. Construct the associated graph.3. Select a subgraph 0 with exactly one arc (directed or undirected) between faceswith a common edge, so that hypothesis (H2) of theorem 4.4.1 is satisfied. If itcannot be satisfied, retry the previous steps.Chapter 4. Smooth Partitions, Transversality and Graphs^ 554. Realize a class C' control /clap on OP as per §4.6 with the transversality/ coin-cidence of 0. For example, if F1 -4 F2 is a directed arc in Q then realize a law/Cb FI nF2 on F1 n F2 with F1 n F2 a transverse edge of the closed-loop dynamics inthe direction F1 —* F25. 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 asper § 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 ofdefinition 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 lawconstructed in step 4, (It is the consistency of parity, rather that the actualsign that is important, so > 0 would be just as good)(c) find a control /char on intF such that the closed-loop dynamics is face-monotone on F with V as a Liapunov function § 4.3.6, say via §4.5.1, andso thatk(x) x aFk(x) :=^ (4.59)klintgx) x E intFis 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 successof later steps in the algorithm. In practice, then, the algorithm would have to be repeatedfor successive refinements of the partition. Note that the presence of redundant edgesonly complicates the combinatorial problem in step 3.Chapter 4. Smooth Partitions, Transversality and Graphs^ 564.9.1 Example § 4.8 cont'dWe 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 graphin Figure 4.39. Here Q := {F11 F49 F5, F6, F9, F10} . A subgraph with exactly one arcbetween neighboring faces satisfying hypothesis (H2) is given in figure 4.34. A controllaw which realizes that subgraph need only satisfy u <0 on the 1-interior of F3 fl F4 andF5 n (F6 u F7), u > 0 on F8 n F9 and F10 n(F1u F2), u = —x2 on xi +x2 = 0 and u = —2x2on 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 n F41with W(x, u) = x2+ u, so if (F1, F4) is an undirected arc, i.e. F1 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 functionV(xi, x2) := xi so V(xi , x2) = x2. Then regardless of how we interpolate u1731, theclosed-loop dynamics will be face-monotone with these Liapunov functions. On the faceF5 define the Liapunov functionV(xi, x2) := 2x1 +3x2,^ (4.60)noting that V —+ oo as x oo in F5. Then the Lie derivative isV(xi, x2) = 2x2 + 3u,^ (4.61)and it may be shown on substitution that consistently V < 0 on OF5 according to the2previous conditions, so we require that u < --x2 on intF5 to satisfy (H3)&(H6). Likewise3for F10. One feedback law satisfying these conditions is u := —2x1 — 3x2.Chapter 4. Smooth Partitions, Transversality and Graphs^ 57Figure 4.40: An attracting invariant manifold which contains an attracting equilibriumpoint.4.10 Example-A Liapunov Method.As an alternative to Figure 4.28, we consider the next simplest relation between geometryand dynamics as in Figure 4.40. This section, though a mere special case of the rest ofthe thesis, could be interpreted as an independent method.Consider a nonlinear control systemi = 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 withassociated Lie derivativeV: 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 Usuch 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^ 58as realized via §B.3. Then the level set V-1(0) may be made an invariant set by settingu = ko(x) satisfying (4.65) for x E V-1(0). Suppose further that for some choice of suchko, 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 candidatefeedback control law isIk_(x) for V(x) > 0u(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 acontrol law is easy to implement as it only involves function evaluations, and not functioninversions [Is].4.10.1 Example: the n-integrator.Consider the n-integratorii = X2(4.69)in-1 = Xnin = 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 functionwe takeV(xl, • • • , xn) = 0 - x = Igixi + • • • + Azxn.^(4.70)The associated Lie-derivative1.7(x, u) = 01x2 +...  + ign-ixn + Onu^(4.71)Chapter 4. Smooth Partitions, Transversality and Graphs^ 59can be made positive, negative or zero at any x E Rn provided fin 0 0. Now0n-11.7(X, U) = 0^u = —I2-1/3nx2^XnPn(4.72)and/32^I3nV(x) = 0 41* xi = — —X2 — • • • —^ (4.73)flu Puprovided /5.1^0, and the closed-loop dynamics on the invariant plane (4.73) under thecontrol law (4.72) is given, in the coordinates x2, • ,x„, byx2 = OC3in-1 = inin =^017,—x2^fin—i 0 Xn •Pn In(4.74)By stability analysis for linear dynamical systems, (4.74) will be asymptotically stable ifn — 1 )Oiz — 1(4.75)for 1 < i < n. Note that (x2, • • • , xn) -4 0 as t^oo implies x1^0 as t -4 oowhile on the plane (4.73). We could now construct any law as in (4.68), but ratherhere we will find a linear feedback law of that form. Specifically, a linear control lawu = k • x = kixi + • + knxx satisfying (4.72) on the plane (4.73) may be shown to be,most generally, of the form02^01 fl fin_i^u(x) =^+ (—k1 — —)x2+••• + (--41 —)xn.^(4.76)131^On^131^OnSubstituting into (4.71), we conclude that^1.7(x, u(x)) =^+•• • + On-ixn + Onkixi + (137-2-ak1 — #i)x2 + • • •^(4.77)flu02— 13n—i)xn131= OnOliciV(x).(4.78)(4.79)Chapter 4. Smooth Partitions, Transversality and Graphs^ 60Consequently, if #1, O'n are as in (4.75) and k1 = —1, say, then V < 0 when V > 0 andvice-versa.In conclusion, we have constructed the stabilizing feedback control law-xl - (t31 + 02)x2 - • • - - (3.-1 + 0.)x.^(4.80)= (4.81) Chapter 5Construction of Smooth Partitions and Liapunov FunctionsIn this chapter we discuss an algebraic approach to the coupled problem of constructingsmooth partitions and Liapunov functions. To construct and refine smooth partitions,we rather construct submanifolds of codimension 1, transverse to or coincident with theflow, 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 scalarfunctions or the subsets of state space where there is a change in the structure of theclass of all local Liapunov functions. Differential conditions for a general submanifold tobe transverse to or coincident with some closed-loop response are used to classify bothstrategies.Aside from Frobenius' Theorem, the author is not aware of the ideas presented in thischapter appearing anywhere in the established literature.5.1 Tentative ConstructionAs motivated above, consider the problem of determining the class of all local Liapunovfunctions of a given flow.The condition that the trajectories of a flow defined by a vector field f E X(M)a^a(locally f (x) = fi(x) —, + • • • + fn(x) — ) are transverse to a hypothetical function mayaxi 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)61Chapter 5. Construction of Smooth Partitions and Liapunov Functions^62Thus if we can solve (5.82) for wi, • • • , con, we have identified candidates for the Lia-punov functions through their gradients. Furthermore, "changes" in the class of solutionsof (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 Pfaffianproblem.5.2 Pfaffian Systems5.2.1 A Classical Linear Pfaffian Problem.In the classical problem [vW], one attempts to recover a (foliation of) codimension 1submanifolds 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 vectorw(x) E (TM)*, i.e. a cotangent vector, so co defines a 1-form. We seek a submanifold Nsuch 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)-dimensionalsubmanifold.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 functionsA, 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^63Figure 5.41: A normal field and a surface normal to it.5.2.2 The Nonlinear 'OR' Pfaffian ProblemIn this variant of the classical Pffafian problem, one seeks a codimension-1 submanifoldN satisfyingw(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 R2if TN span(1, 0), for all x E N, or equivalently, N satisfies the nonlinear Pfaffianproblem above withAr(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 themultifunction N.Existence of solutions to the nonlinear Pfaffian problem may likewise be asserted byFrobenius' Theorem. Specifically, if N is a codimension 1 submanifold of M solving thenonlinear 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 linearChapter 5. Construction of Smooth Partitions and Liapunov Functions^64Pfaffian problem with that 1-form w(x). Examples of this reasoning will be given later.5.3 Transversality MultifunctionsThe following definition is used to classify the transversality of a control system to asubmanifold 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 signclass Sc, where 0 0 C C {-1,0, +1}, if {sgn g(u) : u E U} = C. We also define thetransversality 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.0Let TM := (TM) \ {0} denote the nonzero covectors.Definition 5.3.2 Consider a nonlinear control systemth = f(x,u)^ (5.86)as in § 3.6.1, with U denoting the constraint set of admissible controls. Let C denote oneof 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 usefulto define the transversality functionY: (x, u, w) 1-4 w( f (x,u))^ (5.88)and apply a strategy similar to that in §B.3.Chapter 5. Construction of Smooth Partitions and Liapunov Functions^655.3.1 Example: the 2-integrator.Consider the 2-integrator i = f(x, u) defined on M := ir,U := R, using canonicalcoordinates, by" U ).For any w E Ti14., say w = w1dx1 + w2dx2, (w1, w2) 0 0, we have thatT(X, IL, CJ) = WiX2 + (4.12U.f (xi, x2,u)X2 := (5.89)(5.90)The function u i-+ T(w, x , u), for w2 0 0, is always of class '02', but for w2 = 0 isof class '0' or '1' according to whether x2 is zero or non-zero. Thus the transversalitymultifunctions arexi {(.4.) E tM : w = widxi,wi 0 0}^if^x2 = 0No) =( (5.91)X2 0^if x2 0 0Ari (x1) = 10^if x2 = 0(5.92) X2a r^X1ICJ E T;M : co = widx174.01 0 0}^if^x2 0 0/V02 = {W E TM : W = WidX1 + W2dX2, W2^0}-(^) (5.93)X2It is easily shown that the only maximal solutions of the nonlinear 'OR' Pfaffian problemwith Ar = .N; are the half-lines {L±,A}AER given byL+,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 edgesthe flow is consistently in one direction, as in Figure 5.42.Chapter 5. Construction of Smooth Partitions and Liapunov Functions^66•eLFigure 5.42: Foliation of a selection of NI.5.4 Liapunov FunctionsConsider the use of a Liapunov function V : F^R to assert that some closed-loopresponse of a control system is face-monotone across a face of a smooth partition. Con-ditions that a scalar function is indeed a Liapunov function are afforded by the transver-sality multifunctions.Proposition 5.4.1 Consider a nonlinear control systemth = f (x , u)with transversality multifunctions Arc. Let F be a nonempty open subset of the statespace.(Cl) If there is a Liapunov function V : F -4 R on F then the gradient 1-form dV isa 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 satisfyingFrobenius' condition w A dw = 0 defines a Liapunov function on F. Specifically,Chapter 5. Construction of Smooth Partitions and Liapunov Functions^67by Frobenius' Theorem, there exist functions A,V : F —* R such that A > 0 andw = 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, thereexists a feedback law k : F -4 U so that1.1 (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 onF 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 aselection of the said multifunction, for every x E F there exists a u(x) E U such thatw( 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. 0In 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 partitionP has a selection of Ail on a face F e P satisfying Probenius' condition. Then the controlsystem has face-monotone closed-loop dynamics regardless of the control law defined onthe 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 allChapter 5. Construction of Smooth Partitions and Liapunov Functions^68x 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 onF. 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 selectionof A'2.5.5 Smooth PartitionsVery few useful comments can be made for deciding on an effective smooth partition ofa control system. At least, one strategy for edge construction is, as stated above, to useisolated level sets of Liapunov functions.Alternatively, in example 5.3.1, there is a change in the structure of the transversalitymultifunctions along the line x2 = 0. This makes it a candidate for an edge. Off thisedge, construction of Liapunov functions or edges of smooth partitions is not affected bythe dynamics of the system.It is instructive to calculate the transversality multifunctions of various control sys-tems to observe the structure of local Liapunov functions.5.6 ExampleStabilize, about (0, 0),ii = x2, (xi, x2) E R2^(5.98)i2 = xiu, u E R.Note that this system does not have relative degree [Is] defined at (0, 0), so by Theorem2.6 of [Is], the state space exact linearization problem is not solvable, i.e., feedbacklinearization is inconclusive, and this problem is genuinely nonlinear. The author is notChapter 5. Construction of Smooth Partitions and Liapunov Functions^69aware of a time-invariant static stabilizing feedback control law having been previouslyderived for this system. It seems virtually impossible, however, to show Liapunov'sclassical method must fail. Nevertheless, the partition utilized in this example could notbe associated with discrete level sets of a classical Liapunov function.To determine the transversality multifunctions, we again define the T functionY:^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 = 0is of class '0', for all (ni, n2) E R2;• T(ni3O,x1,0,u) = ni(0) + 0(xi)u = 0is of class '0', for all n1 E R, x1 E R;u 1-4 T (0 , n2,0 x2, u) (02) n2(0)u = 0is of class '0', for all n2 E R, x2 E R;u^T(721, n2, 0, x2, u) = n1x2 n2(0)u = n1x2is of class '1', for all n1^0,x2 0 0;T(ni, 0, xl, x2, u) = n1x2 0(xi)u = n1x2is of class '1', for all n1 0 0,x2 0 0;T(ni,n2,xi,x2,u) = nix2 n2x1uis of class '02', for all (ni, n2) 0 (0,0), x1^0,x2 E R.(5.99)if^xi 0 0,x2 = 0if^x1 = 0,x2 0 0if^xi 0 0, x2 0 0,Aro(xi, x2) == nidxi : n1 0 0}= n2dx2 : n2 0 0}07 1,x2, ,R2^if (xi, x2) = 0,x(5.100)Chapter 5. Construction of Smooth Partitions and Liapunov Functions^70Thus it may be shown that0 if x2 = 0Ali (xi, x2) = {n = nidxi + n2dx2 : ni 0 0}1^in = nidxi : ni 0 0} if^xi = 0, x2 0if^xi 0 0,x2 0 00, (5.101)(5.102)12(x1, x2) .. {nidxi + n2dx2 : n2 0 0} if xi 0 0.The structure of the transversality multifunctions changes on the coordinate axis, sothey are candidate edges. Clearly, the nonlinear Pfaffian problem with Al = No does nothave any solutions, while for Al = Ni, we have again the two families {/„,±}„ER givenby (5.94). Likewise, the nonlinear Pfaffian problem with Al = Á102 has, amongst others,the foliations Viol, OE R1 { cr,± } r>0 7 {po,± } 0>o , whereli, 0 := {(x110) : x1 E R}, (5.103)Cr,+ {(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 where n > 0 in Z, and the associated graph in fig-ure 5.45.0^if x1 = 0Chapter 5. Construction of Smooth Partitions and Liapunov Functions^71Figure 5.43: Foliations of selections of A/02.Figure 5.44: A smooth partition for (5.98).72Figure 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. Wewish 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 conditionswere obtained by differentiating the defining equations for the p and c submanifolds, andsolving for u.Consider any feedback control law satisfying these conditions. The closed-loop dy-namics 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^73Figure 5.46: A stable subgraph of the graph in Figure 5.45.Chapter 6ConclusionsAll methods for nonlinear systems compromise scope and viability. For example, Lia-punov's method [AV] has a very general scope but is difficult to apply in practice, whilefeedback 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 stabiliz-ing feedback laws for nonlinear but smooth control systems modelled by a system ofdeterministic differential equations is also no exception.The algorithm strictly contains Liapunov's method, and appears to also accommodate1. I/O systems [So], that is, systems with outputs rather than complete detection ofstate,2. systems with non-Euclidean geometry, such as those in chapter three, and3. discrete-time systems.A cursory inspection of the proof of theorem 4.4.1, however, will reveal dependency inits hypotheses. Thus there is inefficiency in the intermediate hypotheses checking steps2-6 of algorithm §4.9. Furthermore, weakened hypotheses and conclusion would permitthe design of closed-loop dynamics (3.25)-F(3.27) in figure 3.23, where the equilibriumpoint (0, 0) attracts only a dense set. In conclusion, the author believes that a revisionof 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 ofeffective partition, is not clear. The author believes that this problem may be partially74Chapter 6. Conclusions^ 75circumvented by embedding the existing algorithm in a recursive loop, where the partitionis refined at each iteration. Such an algorithm would, arguably, constitute a significantimprovement over Liapunov's classical method.Theorem 4.4.1 asserts the global asymptotic stability of flows whose associated vectorfields are of class globally C1. Strictly speaking then, theorem 4.4.1 cannot be usedto prove that a piecewise Cl feedback law designed using algorithm 4.9 actually isa stabilizer. I strongly believe, although I have not checked all the details, that thissmoothness hypothesis (H7) of theorem 4.4.1 may be weakened to accommodate piecewiseCI vector fields, that is, algorithm 4.9 may be extended to the design of piecewise Clfeedback 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 APrerequisite Mathematical ReferencesIt is pedagogically impossible to develop the mathematical prerequisites of this thesis inan appendix, as is standard in geometric control theory texts [Is], [So]. Rather we outlinethe logistics involved in learning the fundamental ideas of the prerequisites through thefollowing lists, comments and citations. A rather complete encyclopedic reference is [CD].A.1 BasicsA.1.1 Basic TopologyRefer to [Mu][Si] for definitions of the following concepts which appear repeatedly invarious contexts throughout the thesis.Sets, elements; topological structure on a set: open and closed sets; closure, interiorand boundary; basis and neighborhoods; relative topologies; metric spaces; continuityand convergence.A.1.2 AlgebraRefer to [HK][HS] for definitions of the following concepts.Vector spaces; operators; tensor products; modules; exterior products; inverses; eigen-values and eigenvectors; dual and tensor spaces; ideals; groups; group actions.76Appendix A. Prerequisite Mathematical References^ 77A.2 Geometric AnalysisA.2.1 Advanced Calculus = Local Differential GeometryRefer to [CS] for notation, definition and theorems concerningVector-valued functions f : Rn^Rm; differentiability; Jacobians; implicit functiontheorem.A.2.2 ManifoldsRefer to [Bo][vV1/4/ for definitions concerningTopological manifolds; coordinate charts; differentiable manifolds, atlases and com-patibility; submanifolds, immersions, embeddings and codimension; Lie groups; mappingsbetween manifolds.Note that the basic geometric structures of this thesis are differentiable manifolds withcorners. The distinction is that each point has a neighborhood which is homeomorphicto a relatively open subset of [0, +oor. See [KS][Do].A.2.3 BundlesRefer to [vIV][CD] for definitions, examples and theorems concerningProduct manifolds; vector bundles; vector bundle morphisms; group actions; principalfibre bundles; fibre bundle morphisms; bundle of frames; connections; sections of bundles;pullback of bundles.A.2.4 Algebraic Structures over ManifoldsRefer to [vW][F11.Tangent spaces, cotangent spaces tensor spaces; tangent cotangent and tensor bun-dles; vector fields; differential forms; differentiation as a bundle map.Appendix A. Prerequisite Mathematical References^ 78A.2.5 Topological DynamicsRefer to [Arl][HS][Wi].Vector fields defining differential equations on manifolds; Lie derivatives; flows; equi-librium points; linearization; stability, asymptotic stability; invariant manifolds.A.2.6 Exterior Differential SystemsRefer to [vW].Forms; Exterior differential systems; codistributions; integral-manifolds; exactnessand closedness; integrability; ideals.A.2.7 Geometric MechanicsRefer to [Go][Ar2][Mar].State; Newtonian, Lagrangian, and Hamiltonian mechanics; modelling state spacesby vector bundles and governing equations by differential equations on manifolds.A.2.8 Functional AnalysisRefer to [NS]Hilbert and Banach spaces; LP spaces; linear operators; nonlinear operators; inverses;realization by differential equations.A.3 Control and System TheoryRefer to [So][Ka][0g].Linear systems; signals, systems and states; feedback; input and output; closed-loop; stabilization and compensation; tracking; design specifications; controllers; Pole-placement.Appendix A. Prerequisite Mathematical References^ 79A.4 MiscellaneousA.4.1 Graph TheoryRefer to [Lu][Ha] for definitions, examples, notation and algorithms concerning Directedand 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 setof 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 sequenceItili>i of directed arcs in T and vertices {c} >i in V, such that ai_1 4 ai, for alli > 1, and(H2) for every vertex a in V \Q, there is a vertex 3 in V such that a -4 is a directedarc in T.Intuitively, Q is the "global attractor" for such a graph as each directed path isfinite 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 TheoryRefer to [So].Finite-state machines; state, input and transitions; determinism.A.4.3 MultifunctionsA function Al M^P such that the value Al (x) of the function at a point x e M isactually a set is called a multifunction. Thus the range^of Al is a set of sets. IfAppendix A. Prerequisite Mathematical References^ 80AT (x) c W, for every x E M, that is P is a subset of the power set of W, then we writeN. : M W .Appendix BCited TheoremsB.1 Long Tubular FlowsFollowing [PdM], a tubular flow for a vector field X E Xr(M) on a smooth manifold Mof dimension n is a pair (I2, (I)) where C2 is an open set in M and (I) is a Cr diffeomorphismof SI onto a cube In := I X In-1 = {(x, y) E R x R1 : lx1 < 1, Ily11,,,, < 1} which taketrajectories of X in 12 to the straight lines I x {y} C I xTheorem B.1.1 (Long Tubular Flow [PdM]) Let -y C M be an arc of a trajectoryof X that is compact and not closed. Then there exists a tubular flow (f2, cD) of X suchthat -y C ft, as in figure B.47.Since the transyersality of a vector field X to a submanifold E is preserved underdiffeomorphism, it follows from the Long Tubular Flow theorem that qualitatively allsuch configurations look as in figure B.48.Figure B.47: A flow 0 rectified by a diffeomorphism (D.81Appendix B. Cited Theorems^ 82Figure B.48: Rectified flow past a transverse submanifold.Figure B.49: E attracts M and attracts E.B.2 Flow-ConnectednessRecall from the proof of theorem 4.4.1 that we had a local invariant submanifold E ofM 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 weconcluded that for every orbit riti in M, w(FM) =Additional hypotheses are indeed necessary as demonstrated in the counter-exampleX1 = X2= 21'1 -3x — px2(x? —^ix3)2M := {(x1,x2) E R2 : x1 > 0, H( -3' 0) < H(xi,x2) < 0}Appendix B. Cited Theorems^ 83Figure B.50: A flow with an attracting homoclinic orbit.E := {(x1,x2) E R2 : x1 > 0, H(xl, x2) = 0},^(B.112)where1 2H(Xl, X2) := -X9- + X31 -2 -^A (B.113)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 cycle-graph in E is itself and apply the fundamental theorem on w-limit sets. A higher-dimensional 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 isa homeomorphism. We say that X is T -connected if for every closed proper subset A ofX we have thatTA 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 setunder a flow {0t}tER is that X be 44-connected.Appendix B. Cited Theorems^ 84B.3 Location of ZerosGiven a set M C Rn, a subset E C M defined as the level set of a differentiable functionV: 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 thatf(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, ifIsgnf (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 bythe implicit function theorem, (B.115) (B.116) and (B.117) can be solved, and an ade-quate value of uo may be realized by Newton's method. The interpolation of the valuesuo from the discretization follows from the implicit function theorem.B.4 Non-triviality of the Tangent bundle to the 2-SphereProposition 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^ 85B.5 Contractability of Domains of AttractionWe 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 ona topological manifold must be a contractable set. By continuous we mean that 0 is acontinuous function.B.6 Frobenius' TheoremWe paraphrase [vW].Theorem B.6.1 Suppose that M is a differentiable manifold on which we are given aone-form w : U .— T*U defined some non-empty open subset of M. Then there exists ascalar function f(.) : U -4 1 :I  such thatNk := {X E U : f (x) = k, df (x) 0 0}are integral manifolds for w for each k E rangef , , (that isTzNk C kerw(x)for every x E Nk,) if, and only if,w A dw = 0on U.B.7 Input-Output Stability via Asymptotic StabilityFollowing [Vid], consider an Input-Output systemi(t) = f(t,x(t),u(t))^(B.119)y(t) = g (t , x (t), u(t))Appendix B. Cited Theorems^ 86with 0 as an equilibrium point, f(t, 0,0) = 0 and g(t, 0,0) = 0 for all t E R, flu.° is ofclass 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 finitegain and zero bias if there exists constants r> 0 and -yp, < oo such thatx(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 theunforced systemi(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 PrincipleDefinition B.8.1 A Local Liapunov function for a nonlinear control system (withstate 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° suchthat the following properties hold:1. V is proper at x0, that is, the sub-level set Ix E M : V (x) < el is a compactsubset 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 0x°.3. For each initial state e E 0 there exists a response x(.) : R+ .— M, x(0) = e of thecontrol system to some admissible control such that ti-4 V (x(t)) is non-increasingand non-constant on the interval where x(t) E 0.Appendix B. Cited Theorems^ 87The 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 MV (x) < L} is compact for each L > 0.Theorem B.8.2 If there exists a local (respectively global) Liapunov function of a non-linear control system relative to an equilibrium point x0, then the control system is locally(respectively globally) asymptotically controllable. 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