introduction to nonlinear dynamical systems and nonlinear ...1 introduction to nonlinear dynamical...

1

Introduction to Nonlinear Dynamical Systems and Nonlinear Control

Strategies

Presenter: Dr. Bingo Wing-Kuen LingLecturer, Centre for Digital Signal Processing,

Department of Electronic Engineering, King’s College London.

Email address: [email protected]

2

Nonlinear SystemsBehaviors of nonlinear systemsDefinition of chaosDefinitions of symbolic dynamicsExamples of systems governed by symbolic dynamics

Challenge Problems and Some SolutionsSigma delta modulatorsDigital filters with two’s complement arithmeticPerceptrons

Impulsive ControlFuzzy ControlConclusionsQuestions and Answers

Outline

3

Nonlinear SystemsBehaviors of nonlinear systems

Behaviors depend on the initial conditions.( ) ( ) ( )( )txtx

dttdx

−= 1

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time t

y(t)

4

Nonlinear SystemsBehaviors of nonlinear systems

Behaviors could be very complicated.( ) ( )( ) ( ) ( )tztbxtxxatx −−= 0&( ) ( )( ) ( ) ( )tztdytyycty +−= 0&( ) ( ) ( ) ( )( )tfytextztz −=&

5

Nonlinear SystemsDefinition of chaos (Devaney)

Sensitive to initial conditionsAn interval map has sensitive dependence to initial conditions if there exists a , such that, for any , and any neighborhood of , there exists , and an iteration such that

.That is, no matter how small the neighborhood, the map has the ability that a small error can grow terribly large. This is often referred to as the Butterfly effect, due to the thought-experiment described by Edward Lorenz.

II →:F0>δ

Ix∈ Ω xΩ∈y 0≥n

( ) ( ) δ>− yx nn FF

6

Nonlinear Systems

0 100 200 300 400 500 600 700 800 900 1000-1.5

-1

-0.5

0

0.5

1

1.5

2

Time index k

x 1(k

)

7

Nonlinear Systems

Topological transitiveAn interval map is topologically transitive if for any two open sets , there exists such that .This is equivalent to the statement that there exists a dense orbit. In other words, the state vector can end up almost anywhere in phase space.

II →:FIVU ⊂, 0>k

( ) ØF ≠∩VUk

8

Nonlinear Systems

-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

9

Nonlinear SystemsThe periodic orbits are dense.

Any 2 of the above forces the third to follow.

10

Nonlinear Systems

Definition of symbolic dynamicsSymbolic dynamics is a kind of system dynamics that some signals in the system are multi-levelled.Denote as the time index, as the system state vector, as the system input, as the symbolic vectors, as a set of symbols. Then the system can be represented by the following state space equation:

where

( )kx

( ) ( ) ( ) ( )( )kkkkfk ,,,1 usxx =+( ) ( )( ) Skgk ∈= xs

( )ks( )kuk

S

11

Examples of systems governed by symbolic dynamicsSigma delta modulatorsDigital filters with two complement arithmeticPerceptronsPhase lock loopsTurbo decodersetc…

Nonlinear Systems

12

Examples of systems governed by symbolic dynamicsSigma delta modulators

Let

Then

Nonlinear Systems

F(z) Qu(k)y(k) s(k)

( )∑

∑

=

−

=

−

= N

i

ii

N

j

jj

za

zbzF

0

1

( ) ( ) ( )( )

( ) ( ) ( )( ) ( )∑

∑∑

=

==

−−−−−=

−−−=−

N

jjj

N

jj

N

ii

jkyajksjkuba

ky

jksjkubikya

10

10

1

13

( )

( )( )

( )

( )( )

( ) ( )

( ) ( )( ) ( ) ⎥

⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−−−−−

−−−

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

+

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−−

−

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

=

⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢

⎣

⎡

−

+−

112200

00

12100

0

0010

1

1

0

1

0

0

1

0

kskuksku

NksNku

ab

ab

kyky

Nky

aa

aaky

ky

Nky

N

N

M

LLL

LLL

MM

MM

LLL

M

LLL

LL

OOM

MOOOM

L

M

Nonlinear Systems

14

Let

then ( ) ( ) ( )[ ] ( ) ( )[ ]TTN kyNkykxkxk 1,,,,1 −−≡≡ LLx( ) ( ) ( )[ ]TkuNkuk 1,, −−≡ Lu

( ) ( ) ( )[ ] ( )( ) ( )( )[ ]TTN kyQNkyQksksk 1,,,,1 −−≡≡ LLs

( )⎩⎨⎧−

≥≡

otherwisey

yQ1

01

Nonlinear Systems

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

−−

≡

0

1

0

1000

0010

aa

aaN LLL

LL

OOM

MOOOM

L

A

15

thenwhere

( ) ( ) ( ) ( )( )kkkk suBAxx −+=+1

⎥⎥⎥⎥⎥⎥⎥

⎦

⎤

⎢⎢⎢⎢⎢⎢⎢

⎣

⎡

≡

0

1

0

00

00

ab

abN LLL

LLL

MM

MM

LLL

B

Nonlinear Systems

( ) ( )( ) { } { } { }1,11,11,1 −××−×−∈= LkQk xs

16

Nonlinear Systems

17

Examples of systems governed by symbolic dynamicsDigital filters with two complement arithmetic

Nonlinear Systems

z-1

z-1a

b

Accumulator f(•)( )ku ( )ky

18

Nonlinear SystemsThe digital filter can be described by the following nonlinear state-space difference equation:

( )( )

( )( ) ( ) ( )( ) ⎟

⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡++

=⎥⎦

⎤⎢⎣

⎡++

kukaxkbxfkx

kxkx

21

2

2

1

11

19

Nonlinear Systemswhere f(•) is the nonlinear function associated with the two’s complement arithmetic

1 3-1-3

1

-1

f(v) linesStraight

v

overflow No OverflowOverflow

20

Nonlinear Systems

Let

and m is the minimum integer satisfying

Then

( ) ( )( ) ( ){ }

( ) { }mmks

xxxxIkxkx

k

ab

,,1,0,1,,

11,11:,

10

21212

2

1

LL −−∈

21

Nonlinear Systemsis called symbolic sequences.( )ks

( ) ( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )M

311110

131

21

21

21

22

Nonlinear SystemsFor example:

( ) [ ] ( ) 0 and 5.0 ,1,6135.06135.00 ==−=−= kuabTx( ) ( ) ( )( ) ( ) ( )

( ) ( ) ( ) ( )( ) ( ) ( )( ) ( ) ( )( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( )LLM

M

11001000

9161.019 and 1180839.118189311.018 and 1170689.11717

0166184.016160157597.01515

9982.015 and 1140018.11414

011534.011009203.000

221

221

21

21

221

21

21

−−=

==⇒−=+−=−=⇒=+

=⇒=+=⇒−=+

−=−=⇒=+

=⇒−=+=⇒=+

s

xsaxbxxsaxbx

saxbxsaxbx

xsaxbx

saxbxsaxbx

valuecomplement sTwo'

23

Nonlinear Systems

-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x1

x2

24

Nonlinear Systems

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

x1

x2

25

Nonlinear Systems

-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

26

Nonlinear Systems

-0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

x1

x2

27

Nonlinear Systems

28

Examples of systems governed by symbolic dynamicsPerceptrons

Nonlinear Systems

( )kx1

( )kxd

( )kw1

( )kwdM

( )kw0

Q ( )ks

29

Assume that there are N training feature vectors and the dimension of the feature vectors is d. Denote xi(k) as the training features for i=1,2,…,d and for k=0,1,…,N-1. Denote x(k)=[1, x1(k), …, xd(k)]T as the training featurevectors for k=0,1,…,N-1. Denote w(k)=[w0(k), w1(k), …, wd(k)]T as the weights of the perceptrons, t(k) as the desired output of the perceptron, s(k) is the real output of the perceptron. Suppose that the weights of the perceptron is updated using the perceptron learning algorithm, then:

( ) ( ) ( ) ( ) ( )kksktkk xww2

1 −+=+

Nonlinear Systems

30

Nonlinear Systems

-2-1.5

-1-0.5

0

-1

0

1

2

3

4-1.5

-1

-0.5

0

0.5

1

1.5

w0w1

w2

31

Sigma delta modulatorsStability depends on the initial condition.

Challenge Problems and Some Solutions

( ) 75.0=ku

( ) [ ]T0,0,0,0,00 =x( ) 5432154321

0025.10075.50075.100025.10519584.140015.640497.1037420

−−−−−

−−−−−

−+−+−+−+−

=zzzzz

zzzzzzF

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20

-10

0

10

20

30

40

50

60

Time index k

y 1(k

)

32


Sigma delta modulators

( ) 75.0=ku

0 2000 4000 6000 8000 10000 12000-2

-1.5

-1

-0.5

0

0.5

1

1.5

2 x 1011

Time index k

x 5(k

)

( ) [ ]T0,0,0,0,001.00 =x( ) 5432154321

0025.10075.50075.100025.10519584.140015.640497.1037420

−−−−−

−−−−−

−+−+−+−+−

=zzzzz

zzzzzzF

33



34



-250 -200 -150 -100 -50 0 50 100 150 200 250-250

-200

-150

-100

-50

0

50

100

150

200

250

x1

x 2

35

Sigma delta modulatorsStability depends on the input signals.


( ) 75.0=ku

( ) [ ]T0,0,0,0,00 =x( ) 5432154321

0025.10075.50075.100025.10519584.140015.640497.1037420

−−−−−

−−−−−

−+−+−+−+−

=zzzzz

zzzzzzF

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-20

-10

0

10

20

30

40

50

60

Time index k

y 1(k

)

36



( ) 76.0=ku

( ) [ ]T0,0,0,0,00 =x( ) 5432154321

0025.10075.50075.100025.10519584.140015.640497.1037420

−−−−−

−−−−−

−+−+−+−+−

=zzzzz

zzzzzzF

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000-6

-4

-2

0

2

4

6

8x 107

Time index k

y 1(k

)

37


Sigma delta modulatorsSuppose that and . Define

Assume that . Suppose that either is strictly stable,

or marginally stable and the frequency spectrum of the input of the loop filter does not contain an impulsive located at the natural frequency of the loop filter. If is strictly stable for , then

the sigma delta modulator is globally stable.

( ) 1−=NNbaQ 0aaN >

( )( )zKF

zFK ++→ 1lim

0

( )( )zKF

zF+1

[ ]⎭⎬⎫

⎩⎨⎧

38

Digital filters with two’s complement arithmeticDigital filters with two’s complement arithmetic could exhibit various behaviors.


-0.4 -0.2 0 0.2 0.4 0.6 0.8 1-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

0 10 20 30 40 50-1

-0.5

0

0.5

1

Time index k

s(k)

39

Digital filters with two’s complement arithmetic


-0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

x1

x2

0 10 20 30 40 50-1

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0

Time index k

s(k)

40

Digital filters with two’s complement arithmetic


-1 -0.5 0 0.5 1-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

x1

x2

0 10 20 30 40 50-2

-1.5

-1

-0.5

0

0.5

1

Time index k

s(k)

41


Digital filters with two’s complement arithmeticWhat are the conditions correspond to different behaviors?Define

⎥⎦

⎤⎢⎣

⎡−

≡θθθθ

cossinsincos

Â

( ) ( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛⎥⎦

⎤⎢⎣

⎡−+

−≡ −11

22 0asckk xTx 1)

2cos a≡θ

⎥⎦

⎤⎢⎣

⎡≡

θθ sincos01

T

( )00 ss ≡

⎥⎦

⎤⎢⎣

⎡−

≡∗11

2 acx

( ) ℜ∈≥= ckcku ,0for

42


For the type I trajectory, the following three statements are equivalent each other:

( ) ( ) ( )⎭⎬⎫

⎩⎨⎧

−+

−≤⎟⎠⎞

⎜⎝⎛ +−∈ ∗−

asc

csc

22

120:00 00 xxTxx 1

( ) ( ) 0for 1 ≥=+ kkk xAx )))

( ) 0for 0 ≥= ksks

43


For the type II trajectory, the following three statements are equivalent each other:

for andsuch that for and

for

0≥k

( ) ( ) ( )( ){ }∞

∗

∞

∗− −≤−∈ iii xxxTxx1 1:00

( ) ( )kk iMi xAx))) =+1

( ) ( )isiMks =+M∃ 0≥k 1,,1,0 −= Mi L1,,1,0 −= Mi L

1,,1,0 −= Mi L

44


For the type III trajectory, the following three statements are equivalent each other:

There is an elliptical fractal pattern exhibited on the phase plane.The symbolic sequences are aperiodic.The set of initial conditions iswhere

which is also an elliptical fractal set.

( ) ( )( ) ( ) ( ){ }MkisisiD iiM +=−≤−= ∞∗∗− and 1:0 1 xxxTxI

MMDI

∀

\2

45

PerceptronsSuppose that the set of training feature vectors are linearly separable. Then, by using the perceptron training algorithm, the weights of the perceptrons will converge to a fixed point.When the set of training feature vectors are not linearly separable, will the weights be bounded?If it is bounded, under what conditions will the weights exhibit limit cycle behaviors and under what conditions will the weights exhibit chaotic behaviors?


46

PerceptronsIf and such that ,then such that andSuppose that and are co-prime and areintegers. That is . Then is periodic with period if and only if


( ) 10 +∗ ℜ∈∃ dw 0~ ≥∃B ( ) Bk ~≤∗w 0≥∀k0≥′′∃B ( ) Bk ′′≤w 0≥∀k ( ) 10 +ℜ∈∀ dw

1q 2q M NNqMq 21 = ( )n∗w

( ) ( )( ) ( )( ) ( )∑−=

∗

=+++−+1

0 2

M

j

T

jkMjkMjkMQjkMt 0xxw

M

47

Conventional controller generates control signals fed into the input of the plant.If the plant is not controllable, then the plant will lose control.The control force usually lasts forever.

Impulsive Control

48

Since different initial conditions may correspond to different stability conditions, if the initial conditions are moved to another positions, then the plant will be automatically stablized and the control force can be removed from the plant.

Impulsive Control

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-3

-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

Time t

y(t)

49

Control signals are not fed into the input of the plant, so controllability is not an issue.The impulsive controller is usually implemented via a reset circuit.

Impulsive Control

50

Impulsive Control

51

Impulsive Control

52

In crisp set, either x∈A or x∉A.A fuzzy set is a set that x∈A associated with a fuzzy membership function μA(x)∈[0,1]. If μA(x)=0, then x∉A. If μA(x)=1, then x∈A.In the traditional binary logics, all combinational logics can be represented by combinations of complement, intersection and union of binary variables because all Boolean equations can be represented by sum of products or product of sums.Similarly, there are three fundamentals operations in fuzzy logics and these operations represent the traditional complement, intersection and union operations.

Fuzzy Control

53

For the fuzzy complement operations, the fuzzy set A maps to a fuzzy set Ā with the fuzzy membership function satisfying the properties c(0)=1, c(1)=0, and c(x1)>c(x2) if x1

54

For the fuzzy union operations, the fuzzy sets A and B maps to a fuzzy set A∪B with the fuzzy membership functionsatisfying s(1,1)=1, s(0,a)=s(a,0)=a, s(a,b)=s(b,a), s(s(a,b),c)=s(a,s(b,c)) and s(a,b)≤s(a’,b’) if a≤a’ or b≤b’. The common fuzzy union operation is s(a,b)=max(a,b).A fuzzy relationship Q in U1×U2×…Un is defined as Q≡{(u1×u2×…un),μQ(u1×u2×…un): (u1×u2×…un)∈U1×U2×…Un}, where μQ:U1×U2×…Un→[0,1].

Fuzzy Control

( ) ( ) ( )( )xxsx BABA μμμ ,≡U

55

If a variable can take words in natural languages as its values, it is called a linguistic variable, where the words are characterized by fuzzy sets. A linguistic variable is characterized by (X,T,U,M), where X is the name of the linguistic variable, T is the set of linguistic values that X can take, U is the actual physical domain in which the linguistic variable X takes its quantitative values, and M is a set of semantic rules which relates each linguistic values in T with a fuzzy set.

Fuzzy Control

56

For an example, X is the speed of a car, T={slow,medium,fast}, U=[0,Vmax], and M consists of three fuzzy membership functions that describes slow, medium and fast.A fuzzy if then rule is a conditional statement expressed in theform of “IF fuzzy proposition 1 (FP1), then fuzzy proposition 2 (FP2).”.There are two types of fuzzy propositions, an atomic fuzzy proposition and a compound fuzzy proposition.An atomic fuzzy proposition is a single statement “x is A”, in which x is a linguistic variable and A is a linguistic value of x.A compound fuzzy proposition is a composition of atomic fuzzy propositions using the connectives “and”, “or” and “not” which represent fuzzy intersection, fuzzy union and fuzzy complement, respectively. For an example, “x is A and x is not B.” The fuzzy if then rule is a fuzzy implication. The common fuzzy implications are Mamdani implication

Fuzzy Control

( ) ( ) ( )( )yxyx FPFPQMM 21 ,min, μμμ ≡

57

A fuzzy system usually consists of three parts: fuzzifier, fuzzy engine and defuzzifier.A fuzzifier is a system that maps the inputs of a system to fuzzy inputs associated with fuzzy member functions.A fuzzy engine is a system that maps the input fuzzy membership functions to output member functions.A defuzzifier is a system that maps the fuzzy outputs associated with fuzzy member functions to outputs of the system.

Fuzzy Control

58

There are three common types of fuzzy systems, pure fuzzy systems, Takagi-Sugeno-Kang (TSK) fuzzy systems and fuzzy systems with fuzzifier and defuzzifier.The pure fuzzy systems map the input fuzzy sets to output fuzzy sets via a fuzzy inference engine which consists of a set of fuzzy if then rules.The TSK fuzzy systems formulate the fuzzy if then rules via a weighted average fuzzy inference engine.Fuzzy systems with fuzzifier and defuzzifier employ the fuzzifier to map the real valued variables into fuzzy sets and the defuzzifier transforms fuzzy sets into real valued variables.

Fuzzy Control

59

Dynamics of nonlinear systems depend on initial conditions and input signals.Chaotic systems is sensitive to initial conditions, topological transitive and consisting of rich frequency spectra.Symbolic dynamics is a kind of dynamics that signals in the systems are multi-levelled.Many real systems, such as digital filters with two’s complement arithmetic, sigma delta modulators and perceptrons, are governed by symbolic dynamics.Global stability conditions of sigma delta modulators and boundedness conditions of perceptrons are discussed. Conditions for digital filters with two’s complement arithmetic exhibiting various behaviors are presented.Impulsive control is to generate an impulsive or reset the system states directly so that no further control action is required and the systems will be automatically stablized.Fuzzy control to control the systems via fuzzy rules.

Conclusions

60

Questions and Answers

Thank you!

Let me think…

Bingo

Introduction to Nonlinear Dynamical Systems and Nonlinear Control StrategiesOutlineNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsNonlinear SystemsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsChallenge Problems and Some SolutionsImpulsive ControlImpulsive ControlImpulsive ControlImpulsive ControlImpulsive ControlFuzzy ControlFuzzy ControlFuzzy ControlFuzzy ControlFuzzy ControlFuzzy ControlFuzzy ControlConclusionsQuestions and Answers

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