-
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
Challenge Problems and Some Solutions
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
Sigma delta modulators
Challenge Problems and Some Solutions
-
34
Sigma delta modulators
Challenge Problems and Some Solutions
-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.
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
)
-
36
Sigma delta modulators
Challenge Problems and Some Solutions
( ) 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
Challenge Problems and Some Solutions
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.
Challenge Problems and Some Solutions
-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
Challenge Problems and Some Solutions
-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
Challenge Problems and Some Solutions
-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
Challenge Problems and Some Solutions
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
Challenge Problems and Some Solutions
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
Challenge Problems and Some Solutions
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
Challenge Problems and Some Solutions
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?
Challenge Problems and Some Solutions
-
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
Challenge Problems and Some Solutions
( ) 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