Citation
Lyapunov stability analysis applied to systems of Lagrange equations using the energy metric algorithm

Material Information

Title:
Lyapunov stability analysis applied to systems of Lagrange equations using the energy metric algorithm
Creator:
Belmonte, Richard David
Place of Publication:
Denver, CO
Publisher:
University of Colorado Denver
Publication Date:
Language:
English
Physical Description:
viii, 114 leaves : illustrations ; 29 cm

Subjects

Subjects / Keywords:
Lyapunov functions ( lcsh )
Lagrange equations ( lcsh )
Algorithms ( lcsh )
Algorithms ( fast )
Lagrange equations ( fast )
Lyapunov functions ( fast )
Genre:
bibliography ( marcgt )
theses ( marcgt )
non-fiction ( marcgt )

Notes

Bibliography:
Includes bibliographical references (leaves 101-108).
Thesis:
Submitted in partial fulfillment of the requirements for the degree, Master of Science, Department of Electrical Engineering, Department of Computer Science and Engineering.
Statement of Responsibility:
by Richard David Belmonte.

Record Information

Source Institution:
|University of Colorado Denver
Holding Location:
|Auraria Library
Rights Management:
All applicable rights reserved by the source institution and holding location.
Resource Identifier:
22927339 ( OCLC )
ocm22927339

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PAGE 4

was

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V V

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stable stability analy sis stability determination, Stability determination stability

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stability, stare, -abilis stability stable equilibrium; stable. unstable.

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stability in-the-sense-of-

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Lyapunov Lyapunov stability. Lyapunov's

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Jecond method direct method. Jecond method region 0/ Jtability. jirJt method Jecond method.

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large dynamical systems, large-scale systems intercon nected systems, complex dynamical systems, composite syste1n3, decen tralized systems.

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interconnected 8Y8-tem8, large-8cale sY8tem partitioned, large 8pace structure

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t t .... .. i .. .. {\I\

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:i:

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5.8 Z.

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state function of Lagrange;

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n 2n

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Q

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differential form; one-form, l-form

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one-form vector. vector column-vector) one-form row vector bra-ket bra, row-vector ket, column vector contravariant covariant duals

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w w n One-forms (also Pfaffian-forms exterior differential forms. exterior calculus. also global analysis manifold the ory

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k k n R3. line integral J A dx B dy C dz w A dx B dy C dz surface integral J J Pdxdy Qdydz Rdzdx a P dx dy Q dy dz R dz dx volume integral J J J Hdxdydz f3 H dxdydz n

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8M w (k k f w=fdw. IBM w=O. IBM (k M exact closed M(x,y) N(x, y)dy exact form w exact fJ w) w dfJ. w closed dw w dXi

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-y x w dx dy x +y x +y -y x ( x2 y2' x2 y2) path-independent.

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exterior product, wedge prod uct. ,x ,x x

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B0B0BOl B0B0BOl (Bx Bx O(Bx ax l ) 0 (axl ax o curiO a02 aOl axl ax2 second method direct method)

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ro bust control. robustness;

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also Q

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Step n xnl Step dt direction fields: n(n Step Step

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Step I, elbow-path, taxicab-metric r:C1 i(:C2=:C3=:Cn=O) V= Jo V Step V taxicab-geometry,

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x x x Wn X wn dt X2 dX2 -d (-wnxl-2(wn X 2)/X2 v W LiXl (W;Xl 2(WnX2) dXl LiX2 X2 dX2 nxl X 2 V X2 8X2

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V also (a x) X a
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w b X2 dXI X2 dX2 V(x) c(l 2xi V(x) -(a b V(x) V(x) V V X2 b

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X2 V a b -Vex) Vstable 2X2 a b V

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x

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V

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V

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x

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Operational Transform Method, Step Step n n Step dx i [wi(Xl,X2,".Xn)] Jo Wi(Xt,X2, ... ,Xi-l,ei,0, ... ,0)dei.

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ATp+PA=-Q

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v _eT qjj V V 2V(x) xTRx 2V(x) -xTQx.

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simple pendulum plane pendulum. w2 x wx

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x

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6 D Pm Eex E' q E' q Pm

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Mh as Pm (0-0 Vex) b 00 00 ) 00 ] a a

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(f3/ 2 a det det 2(aM f32/2) a f32/2M a f3 e88ian matrix 8 2 8 2 /8X 28X 8 2 /8X18X2 8 2 V det (p a det det 2aM f3

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Vq) Vq aXI {3XIX2 2M X 2 Vq v,. 8V 8V q X2, X2 XIX2 a V a 2M l'total Vquadratic 'Vperiodic V (3.73) V -(a Do) M b = Do a

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a_a (3=a, (3=a V=Yc

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V /J

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Variable Gradient Method V {3Xlx2 Wm Vm Wm {3XIX2. m m -a 2axl (3X2, -a x2 r r10 Wm (2a6 (3X2) d6 de2. dt Wm

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-2a -13 -a). j3XIX2 -a) -j3, (3X2 dXl V VV (allxl a12x2) a21xl a22x 2

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(a12 {3a a= 2M' O<{3
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phase-locked loop PLL) lock-on lock-in V V cycle slipping V

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generalized coordinates, qi, generalized velocities U U(ql' q2, ... qn; t)

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Lagrangian, Lagrange function, C, Euler-Lagrange equation Lagrange equations of motion, (ac) a.c d a a (i=1,2, ... t qi 2n Legendre transformation Hamilton's equations of motion. generalized momentum a.c aT Pi

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Hamiltonian qiPi i Hamilton's canonical equations n 81-l 8Pi 81-l canonical adjoints. 2n phase space,

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Hamilton's Principle. it' it' -U)dt (C)dt to to (to, tt) C(x, x', t) dt extremum. to E +U

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Itl Y .1')

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2n n 2n

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Lit N LPi( T, state function

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Q T Jo

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n Mi, li. qi. n n F(t)

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M2

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(q), q2, ... qn)T n n diag(M1 M2 Mn) n n (F},F2 Fn n k2(q2 ql)] q2 )[F2 k2 ( q2 ql) k3q2] klXI k2(X3 Xl)]

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k 2(q2 k2(X3 2 S2 Ml k2(X3 dXl X2 dX2 J (k2 k3) 2 W2 2M2 X3 X4 M2 2 X 3 -Xl+-X2 aXl a X 2 V X3 X2

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k2 M2 XlX4 sgn(x4)=-sgn(xl) (4.51) 2 V; l 2 ) 2Ml k2 Xl! X2 X3, X4

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V

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k a kIm, 1m

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(-a

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2+a

PAGE 105

The Math Works, Inc. matrix laboratory

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Lyapunov exponent vector Lyapunov function

PAGE 109

Mechanics, Manifolds, Tensors, and Applications, Mathematical Methods Physicists nonlinear systems, Int. Contr., The World Scale Systems, Specialist Techniques in Engineering Mathematics, IEEE Automat. Contr.,

PAGE 110

Applied differential geometry, Structural Modelling Optimization: A Methodology of Engineering Control, Anal ysis, Manifolds, Physics, Optimization Nonsmooth Analysis, Control System Analysis Design: Conventional IEEE Automat. Contr., Robust Control,(a Differential Catastrophe Theory for Scientists Engineers, Classical Mechanics, Principles of Dynamics,

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Differential Topology, Lectures Partial Dif ferential Equations, IEEE Automat. Contr., IEEE Int. Large Scale Sys. Symp., Work Session in Lyapunov's method, Large Scale Systems: Modeling and Control Nonlinear Ordinary Differential Equations, Basic ASME, Appl. Math., Taxicab Geometry: An Adventure in Non-Euclidean Geometry, The Variational Principles Mechanics, Adaptive Control: The Model Reference Approach,

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Probleme General Stabilit6 Mouvement, Annales Faculte Sciences Toulose, An nals Mathematics Studies, Study IEEE Automat. Contr., Real-Time Control Electric Power IEEE Automat. Contr., Classical Dynamics Particles The World Scale Dynamics Control Flexible craft, Methods Analytical Dynamics, IEEE Automat. Contr., Qualitative Analysis Large-Scale Dynamical Proc. IEEE Scale Sys.

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IEEE Trans. Contr., Engineering and Configurations of Sta tions and Platforms, Civilian Stations and the U.S. in Modern Control Engineering, IEEE Trans. Automat. Contr., IEEE Contr., Catastrophe Theory and it Applications, From Being Becoming, IEEE Trans. IFAC Symp., Stability and Robustness of MuItivanable Feedback Systems,

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IEEE Power App. Sys., An introduction to Catastrophe Theory, Differential A Heuristic Introduction, State Functions Linear Control Systems, Geometrical Methods of Mathematical Physics, Large-Scale Dynamical Systems: Stability Struc ture, Automatica, The Space Station, Matrix Logic, Nonlinear Differential Equations, Mechanics, Control Dynamic Systems, Elements of Hamiltonian Mechanics, Instabilities Catastrophes in Science Engineering,

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Procs. IFAC 5th World Systems Analysis, Principles of Coherent Communication, Int. Systems Sci., IEEE Automat. Contr., Acta Tecbnica Csav, IEEE Automat. Contr., IEEE Automat. Contr., IEEE Automat. Contr., Schaum's Outline Tbeory Problems Lagrangian Dynamics, Stations Platforms,

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Advanced Mathematics, IEEE Automat. IEEE Automat. Contr., Principles of

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Stability in-the-sense of Lyapunov Lyapunov stability stable in the sense of Lyapunov b to) to) b t) t to state-function state-functions of: Lagrange, Lyapunov, and Pontryagin. state-function definite, semi-definite, indefinite

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V positive (negative) definite positive (negative) semi-definite K V indefinite quadratic form TIn) TIn X2 Tnn Xn

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Stability in-the-sense of Lyapunov f(x), f(O) asymptotically stable in-the-large V, Lyapunov function. n

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det[Tn1 det det deteR) T22 T23 T33

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(see Lagrange reduction method

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X2 X2, Xn), (X2' X3,"" Xn), (Xn-17 Xn),