Tutorial on Lyapunov
104 Pages
English
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Tutorial on Lyapunov's Stability

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104 Pages
English

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Tools for Analysis of Nonlinear Systems: Lyapunov’s MethodsStanis ław H. ŻakSchool of Electrical andComputer EngineeringECE 675Spring 20081‡‡‡‡‡‡OutlineNotation using simple examples of dynamical system modelsObjective of analysis of a nonlinear systemEquilibrium pointsStabilityLyapunov functionsBarbalat’s lemma2A Spring-Mass Mechanical Systemx---displacement of the mass from the rest position3‡‡‡‡Modeling the Mass-Spring SystemAssume a linear mass, where k is the linear spring constantApply Newton’s law to obtainDefine state variables: x =x and x =dx/dt1 2The model in state-space format:4‡‡Analysis of the Spring-Mass System ModelThe spring-mass system model is linear time-invariant (LTI)Representing the LTI system in standard state-space format5Modeling of the Simple PendulumThe simple pendulum6‡‡The Simple Pendulum ModelApply Newton’s second lawJ θ = −mgl sin θwhere J is the moment of inertia,2J = mlCombining givesgθ = − sin θl7‡‡‡State-Space Model of the Simple PendulumRepresent the second-order differential equation as an equivalent system of two first-order differential equationsFirst define state variables,x =θ and x =d θ/dt1 2Use the above to obtain state–space model (nonlinear, time invariant)8"‡‡Objectives of Analysis of Nonlinear SystemsSimilar to the objectives pursued when investigating complex linear systemsNot interested in detailed solutions, rather one seeks to ...

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Tools for Analysis of Nonlinear
Systems: Lyapunov’s Methods
Stanis ław H. Żak
School of Electrical and
Computer Engineering
ECE 675
Spring 2008
1‡





Outline
Notation using simple examples of
dynamical system models
Objective of analysis of a nonlinear
system
Equilibrium points
Stability
Lyapunov functions
Barbalat’s lemma
2A Spring-Mass Mechanical System
x---displacement of the mass from
the rest position
3‡



Modeling the Mass-Spring System
Assume a linear mass, where k is the
linear spring constant
Apply Newton’s law to obtain
Define state variables: x =x and x =dx/dt
1 2
The model in state-space format:
4‡

Analysis of the Spring-Mass System
Model
The spring-mass system model is linear
time-invariant (LTI)
Representing the LTI system in standard
state-space format
5Modeling of the Simple Pendulum
The simple pendulum
6





The Simple Pendulum Model
Apply Newton’s second law
J θ = −mgl sin θ
where J is the moment of inertia,
2
J = ml
Combining gives
g
θ = − sin θ
l
7‡


State-Space Model of the Simple
Pendulum
Represent the second-order differential
equation as an equivalent system of two
first-order differential equations
First define state variables,
x =θ and x =d θ/dt
1 2
Use the above to obtain state–space
model (nonlinear, time invariant)
8"


Objectives of Analysis of Nonlinear Systems
Similar to the objectives pursued when
investigating complex linear systems
Not interested in detailed solutions, rather
one seeks to characterize the system
behavior---equilibrium points and their
stability properties
A device needed for nonlinear system
analysis summarizing the system
behavior, suppressing detail
9‡

Summarizing Function (D.G.
Luenberger, 1979)
A function of the system state
vector
As the system evolves in time,
the summarizing function takes
on various values conveying
some information about the
system
10