Nonlin Syst Lecture

7/24/2019 Nonlin Syst Lecture

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NNOONNLLIINNEEAAR R DDYYNNAAMMIICCSS

C C o o u u r r s s e e o o b b j j e e c c t t i i v v e e s s : :

Almost all dynamical systems are, to some extend, non-linear.Whereas in the curriculum for undergraduate studies such non-linearities have been ignored, the

course Nonlinear Dynamics focuses on the effects that non-linearities have in such systems.

This course has two main objectives:

1. to provide the mathematical fundamentals for reasoning about stability and control ofnonlinear systems in a formal way, and

2. to provide numeri cal tools for analyzing the behavior of nonlinear systems and for designingnonlinear controllers.

The theory will be illustrated by numerous examples of systems whose simplified linear behavior isknown from past courses.

The content will be mathematical with illustrative examples taken from general engineering systems

(from mechanical, electrical, chemical and aeronautical engineering, as well as from bioengineering and finance).

Course web-site: http://www.ac.tuiasi.ro/pntool/nonlin_sys/

http://www.ac.tuiasi.ro/pntool/nonlin_sys/





Nonl inear Dynami cs – Lecture 1

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R R e e l l a a t t i i o o n n b b e e t t w w e e e e n n c c o o u u r r s s e e o o b b j j e e c c t t i i v v e e s s a a n n d d c c u u r r r r i i c c u u l l u u m m o o b b j j e e c c t t i i v v e e s s : :

Prerequisites for this course include an understanding of undergraduate calculus, linear algebra,

differential equations and linear control methods.The student should also be able to use MATLAB as basic simulation and analysis software.

The class e-mail account ([email protected]) will consist of the students’ e-mail addresses

and will be used twofold:

it can be used by the students to freely ask colleagues questions regarding course topics

it gives the teacher the possibility to send announcements to everyone enrolled in this class

To be included in this group please send an e-mail to [email protected]

G G r r a a d d i i n n g g b b a a s s i i s s : :

Continuous evaluation: (quality of laboratory activity) Percentage in final grade: 20% Interim tests: (2 open book written test) Percentage in final grade: 20%

Other assignments: (1 small project in theoretical and numerical investigation of a NS)

Percentage in final grade: 20% Final evaluation: Percentage in final grade: 40%

Assignment(s):

2 theoretical subjects; no access to documentation; percentage: 40%;

2 practical problems; documentation allowed; percentage: 60%;

Students are allowed to sustain the exam only after passing the laboratory classes.

mailto:[email protected]










3

C C o o u u r r s s e e c c o o n n t t e e n n t t : :

I. Introduction1. Linear vs. Nonlinear Phenomena

a. Phenomena of Nonlinear Systems (Multiple Equilibria, Limit Cycles, Bifurcations,Chaos).

b. Examples of Simple Nonlinear Systemsc. Qualitative and Quantitative Aspects

2. Planar (Second-Order) Dynamical Systemsa. Phase Plane Techniques

b. Limit Cycles - Bendixson Criterion

c. Multiple Equilibria - Index Theoryd. Bifurcation - Fold, Pitch, Fork, Hopf, Saddle Connection

3. Mathematical Preliminariesa. Linear Algebra - Vector Spaces, Norms, Contraction Mapping Theorem

b. Differential Equations and Vector Fields

c. Existence and uniqueness of solutions for differential equations




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II. Stability Theory1. Lyapunov Stability and Instability

a. Definitions of State-Space Stability

b. Basic Lyapunov Stability Theorems, Basic Instability Theorems, Converse LyapunovStability Theorems

c. Exponential Stability Theorems- LaSalle Principle

d. Stability of Linear Systems vs Nonlinear Systemse. Frequency Domain Analysis: Absolute Stability, Circle Criterion, Popov Criterion

2. Input-Output Stability

a. Definitions of Input-Output Stability

b. Small Gain Theoremc. Passivity and Passivity Theoremsd. Harmonic Balance and Describing Functions

e. Connections between Input - Output and State Space Stability




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III. Feedback Control1. SISO Systems

a. Input-Output Linearization

b. Full-State Linearizationc. Zero Dynamics, Stabilization, Tracking

2. MIMO Systems

a. Linearization by Static State Feedback b. Full State Linearization

c. Dynamic Extension

3. Nonlinear Design Tools

a. Back-stepping techniques b. Nonlinear Observersc. Sliding Mode Control and introduction to differential geometry




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P P a a r r t t i i a a l l b b i i b b l l i i o o g g r r a a p p h h y y / / r r e e a a d d i i n n g g l l i i s s t t

required reading:

1.

H.K. Khalil, Nonlinear Systems, 3rd Edition, Prentice Hall, Upper Saddle River, NJ, 2002.

2.

M. Voicu, Systems Theory (in Romanian), Editura Academiei Române, Bucureşti, 2008. 3.

* Documentation for laboratory applications in electronic format.

optional reading:1. A. Astolfi, D. Karagiannis, R. Ortega, Nonlinear and Adaptive Control with Applications,

Springer-Verlag London Limited, 2008.2. A. Bacciotti and L. Rosier, Liapunov Functions and Stability in Control Theory, 2nd Edition,

Springer-Verlag, Berlin Heidelberg, 2005.

3.

X. Liao, L.Q. Wang, and P. Yu, Stability of Dynamical Systems, Elsevier, Amsterdam, 2007.4.

A.N. Michel, L. Hou, D. Liu, Stability of Dynamical Systems: Continuous, Discontinuous,

and Discrete Systems, Birkhäuser, Boston, 2007.5. A.N. Michel, K. Wang, and B. Hu, Qualitative Theory of Dynamical Systems. The Role of

Stability Preserving Mappings, 2nd Edition, Marcel Dekker, Inc., New York, 2001.

6. J. Reyn, Phase Portraits of Planar Quadratic Systems, Springer Science+Business Media,LLC, New York, 2007.

7.

F. Blanchini, and S. Miami. Set-Theoretic Methods in Control, Birkhäuser, Boston, Basel,Berlin, 2008.

8.

L. Gruyitch, J.-P. Richard, P. Borne, and J.-C. Gentina. Stability Domains. Chapman & Hall /

CRC: Boca Raton, 2004.

http://www.acad.ro/

http://www.acad.ro/




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D D y ynnaammiicc s s y y s st t eemm iinn ccoonnt t iinnuuoouu s s t t iimmee

input:

1

p

p

u

u

u state variables:

1

n

n

x

x

x , output:

1

q

q

y

y

y

state equati on:

1 1 1 1

2 2 1 1

1 1

, ,..., , ,...,, ,..., , ,...,

, ,

, ,..., , ,...,

n p

n p

n n n p

x f t x x u u x f t x x u u

t

x f t x x u u

x f x u , : n p n f

initial condition: 0 0 0 0( ) , ,t t t t x x

output equation: , ,t y h x u , : n p q

h

0 0 0 0

( ) , ( ), ( )( ) , ,

( ) , ( ), ( )

t t t t t t t t

t t t t

x f x u x x

y h x u

SystemInput Output

u y

state, x




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L Liinneeaar r s s y y s st t eemm

Definition 1. The function : n m

f is linear with respect to independent variable n x if and

only if it satisfies two conditions:

1 . Additivity : 1 2 1 2( ) ( ) ( ) f x x f x f x , 1 2, n x x

2. Homogeneity : ( ) ( ) f x f x , n x and all scalars .

The system is l inear under three assumptions:

(i) Additivity of zero-input and zero-state response

(ii)

Linearity in relation to initial conditions (linearity of zero-input response)(iii)

Linearity in relation to inputs (linearity of zero-state response').

The systems which don't satisfy the conditions (i), (ii) and (iii) are nonlinear systems.

System

Input Output

u y

state, x

Sys. Sys. 1u 2u1 y 2 y

= Sys.1 2u u 1 2 y y




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N N oonnl l iinneeaar r s s y y s st t eemm:: ssttaattee vvaar r iiaa b bllee:: nx ,, iinn p puutt

pu ,, oouutt p puutt qy

general form: unforced state:

, ,

, ,

:

:

n p n

n p q

t

t

x f x u

y h x u

f

h

,

: n n

t

x f x

f

autonomous (time invariant)

,

,

:

:

n p n

n p q

x f x u

y h x u

f

h

0 0 0 0( ) , ,

: n n

t t t t

x f x

x x

f

L Liinneeaar r s s y y s st t eemm

time variant time invariant unforced

( ) ( )

( ) ( )

t t

t t

x A x B u

y C x D u

x Ax Bu

y Cx Du

x Ax

n nA , n pB ,q nC , q pD

initial condition: 0 0 0 0( ) , ,nt t t t x x




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E E x xii s st t eennccee aannd d U U nniiqquueennee s s s s oo f f S S ool l uut t iioonn s s

,t x f x

: n n f is piecewise continuous in t and locally Lipschitz in x over the domain of interest

,t f x is piecewise conti nuous in t on an interval J if for every bounded subinterval 0 J J ,

f is continuous in t for all 0t J , except, possibly, at a finite number of points where f may have

finite-jump discontinuities

,t f x is locally L ipschitz in x at a point 0x if there is a 0r so that in the neighborhood

0 0( , ) || ||n N r r x x x x , ,t f x satisfies the Lipschitz condition with some 0 L

|| ( , ) ( , ) || || ||t t L f x f y x y , 0t .

,t f x is locally L ipschitz in x on a domain (open and connected set) n if it is locally

Lipschitz at every point 0 x .




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When 1n and ( ) f f x the Lipschitz condition is:

| ( ) ( ) |,

| |

f x f y L x y

x y

On the plot of ( ) f x , a straight line joining any two points cannot have a slope whose absolute value

is greater than L.

Any function ( ) f x that has inf ini te slope at some point is not locally L ipschitz at that point.

A discontinuous function is not locall y Li pschitz at the points of discontinuity.

L Leemmmmaa::

Consider the state equation

(E)

0 0

,

( )

t

t

x f x

x x

Let ,t f x be piecewise continuous in t and locally Lipschitz in x at 0x , for all 0 1[ , ]t t t .

Then, there is 0 such that the state equation (E) has a unique soluti on over 0 0[ , ]t t :

0 0 0 0( ) ( ; , ), [ , ].t t t t t t x x x ♦

Without the local Lipschitz condition, we cannot ensure uniqueness of the solution.

The lemma is a local result because it guarantees existence and uniqueness of the solution over

an interval 0 0[ , ]t t , but this interval might not include a given interval 0 1[ , ]t t .

The solution may cease to exist after some time.




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GGl l oobbaal l E E x xii s st t eennccee aannd d U U nniiqquueennee s s s s oo f f S S ool l uut t iioonn s s

,t x f x

,t f x is globally L ipschitz in x on n if there is a 0 L so that

|| ( , ) ( , ) || || ||t t L f x f y x y , , n

x y .

If ,t f f x and its partial derivatives i j f x are continuous for all nx , then:

,t f x is globally L ipschitz in x if and only if i j f x are globally bounded, uniformly in t .

( , )0 : , , , , 1,...,ni

j

f t M M t i j n

x

x x .

L Leemmmmaa::


(E) 0 0, , ( )t t x f x x x .

Let ,t f x be piecewise continuous in t and globally Lipschitz in x , for all 0 1[ , ]t t t .

Then, the state equation (E) has a unique soluti on over 0 1[ , ]t t :

0 0 0 1( ) ( ; , ), [ , ].t t t t t t x x x ♦

The global Lipschitz condition is satisfied for linear systems of the form ( ) ( )t t x A x g .

The global Lipschitz condition is rather restrictive for general nonlinear systems.




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L Leemmmmaa::


(E) 0 0, , ( )t t x f x x x .

Let ,t f x be piecewise continuous in t and Lipschitz in x , for all0

t t and all x in a domain

(open and connected set) n .

Let W be a compact subset of and suppose that every solution of (E) with 0 W x lies

entirely in W .

Then, the state equation (E) has a unique soluti on that is defined for all 0t t :

0 0 0( ) ( ; , ),t t t t t x x x . ♦

other problems:

Dependence on Initial Conditions and Parameters

The Maximal Interval of Existence

E E qquuiil l iibbr r iiuumm p pooiinnt t s s

A pointn x is said to be an equilibrium point of ,t x f x if

0 0( ) ( ) ,t t t t

x x x x . ♦

For the autonomous system x f x , the equilibrium points are the real solutions of the equation

0f x .




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An equilibrium point could be isolated , that is, there are no other equilibrium points in its vicinity, or

there could be a continuum of equil ibri um points .

A linear system x Ax can have:

an isolated equilibrium point at 0x (if A is nonsingular) or

a continuum of equilibrium points in the null space of A (if A is singular).

It cannot have multiple isolated equilibrium points , for if ax and bx are two equilibrium points,

then by linearity any point on the line (1 )a b x x connecting ax and bx will be an

equilibrium point.

A nonlinear state equation can have multiple isolated equilibrium points.For example, the state equation

1 2

2 1 2sin ( , 0)

x x

x a x bx a b

has equilibrium points at 1 2, 0 x n x , for 0, 1, 2,...n




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L Liinneeaar r ii z z aat t iioonn

A common engineering practice in analyzing a nonlinear system is to linearize it aboutsome nominal operating point and analyze the resulting linear model.

What are the limitations of linearization?

Since linearization is an approximation in the neighborhood of an operating point, it canonly predict.

The “local” behavior of the nonlinear system in the vicinity of that point. It cannot predictthe “nonlocal” or “global” behavior.

There are “essentially nonlinear phenomena” that can take place only in the presence of

nonlinearity.

N N oonnl l iinneeaar r P P hheennoommeennaa

Finite escape time

Multiple isolated equilibrium points

Limit cycles

Subharmonic, harmonic, or almost-periodic oscillations

Chaos

Multiple modes of behavior




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Finite escape time

Example: The differential equation

20, (0) 0 x x x x

has the solution

0

20

0 1

0

1 1,

( ) , 0,1 x

dxdt t c c

x x x

x

x t t x t

finite escape time:0

1 f t

x




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Uniqueness problems




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N N o o n n l l i i n n e e a a r r s s y y s s t t e e m m s s e e x x a a m m p p l l e e s s

11.. P P eennd d uul l uumm eeqquuaat t iioonn

l - length of the rod, m - mass of the bob

ideal model : no friction force

sinml mg

state space model: 1 x , 2 x

1 2 1 2

2 22 1 2 1sin sin ,

x x x x

g g x x x xl l

equilibrium points:

21 1

1 22 21

0sin 0 , 0, 1, 2,...

00 0sin 0

x x x n n

x x g x x x

l

two points on the circle: (0,0) , ( , 0)

conservative system : given an initial push it will oscillate forever




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the fri ction force resisting the motion is proporti onal to the speed of the bob with a coefficient of

friction k ( 0k , small)

sinml mg kl

state space model: 1 x , 2 x :1 2

2 1 2sin

x x

g k x x x

l m

equilibrium points:1

1 2 2

, 0, 1, 2,...0

0

x n n x x

x




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C C oommmmoonn nnoonnl l iinneeaar r iit t iiee s s

Nonlin Syst Lecture

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