Nonlinear control

(1)

١

Nonlinear control

ﺱﺭﺪﻣ : ﺩﻮﻨﺷﻮﺧ ﺪﻴﺠﻤﻟﺍﺪﺒﻋ ﻥﺍﻮﻨﻋ ﻝﺮﺘﻨﮐ

ﯽﻄﺧﺮﯿﻏ

ﻝﻭﺍ ﻞﺼﻓ :

ﻪﻣﺪﻘﻣ ) ﻝﺮﺘﻨﮐ ﻪﭽﺨﻳﺭﺎﺗ (

1868 First article of control on governor by Maxwell 1877 Routh stability criterion

1892 Liapunov stability condition 1895 Hurwitz stability condition 1922 The first form of PID controller 1932 Nyquist

1943 Neural Networks 1945 (1938) Bode 1947 Nichols 1948 Root locus

1949 Wiener optimal control research

1950 Adaptive Control, Sliding mode control

(2)

٢

ﻪﻣﺪﻘﻣ 1955 Kalman filter and controllability observability analysis

1956 Artificial Intelligence

1957 Bellman optimal and adaptive control 1960-1963 Kalman filter

1962 Pontryagin optimal control 1962 Stability of nonlinear systems 1965 Fuzzy set

1972 Vidyasagar multi-variable optimal control and Robust control

1981 Doyle Robust control theory 1990 Neuro-Fuzzy

ﻪﻣﺪﻘﻣ

Control Liapunov Functions Feedback linearization Sliding mode control

Back stepping (1987-1999)

؟ﻥﻮﻨﮐﺎﺗ

ﯽﺒﻴﮐﺮﺗ ﯼﺎﻫ ﺵﻭﺭ Robust Adaptive

Intelligent adaptive/ Artificial /NN Model Switching

ﻝﺮﺘﻨﮐ ﯼﺎﻫ ﻢﺘﺴﻴﺳ Multi Agent

(3)

٣

ﯽﮑﻴﻣﺎﻨﻳﺩ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﯼﺭﺍﺪﻳﺎﭘ ﻡﻮﻬﻔﻣ ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ

: ﻥﺁ ﺭﺩ ﻥﺪﻧﺎﻣ ﯽﻗﺎﺑ ﻭ ﻪﻄﻘﻧ ﻦﻳﺍ ﺯﺍ ﺖﮐﺮﺣ ﻉﻭﺮﺷ

؟ﯽﻫﺎﮔﺪﻳﺩ ﻪﭼ ﺯﺍ ﯼﺭﺍﺪﻳﺎﭘ

ﯽﺗﺍﺫ ﺮﻴﻏ ﺎﻳ ﯽﺗﺍﺫ ﯽﻣﻮﻬﻔﻣ )

ﯼﺎﻫ ﻢﺘﺴﻴﺳ Affine

(

؟ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﯼﺭﺍﺪﻳﺎﭘ ﺎﻳ ﻢﺘﺴﻴﺳ ﯼﺭﺍﺪﻳﺎﭘ•

ﻪﻣﺪﻘﻣ ) ﻪﻳﺎﭘ ﺚﺣﺎﺒﻣ ﯼﺭﻭﺁﺩﺎﻳ (

ﻑﻮﻧﺎﭘﺎﻴﻟ ﯼﺭﺍﺪﻳﺎﭘ ﻩﺎﮔﺪﻳﺩ• ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﺮﻫ ﯼﺍﺮﺑ ﺮﮔﺍ ﺖﺳﺍ ﺭﺍﺪﻳﺎﭘ x=0

> 0 ﺪﺷﺎﺑ ﻪﺘﺷﺍﺩ ﺩﻮﺟﻭ

> 0 ﻪﮐ

:

(0) < ≫ < ε, ∀ ≥ 0

ﯼﺭﺍﺪﻳﺎﭘ•

(4)

۴

ﯼﺭﺍﺪﻳﺎﭘ ﻩﺎﮔﺪﻳﺩ• ﻑﻮﻧﺎﭘﺎﻴﻟ

ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﺖﺒﺜﻣ ﺭﺍﺪﻘﻣ ﮏﻳ ﻭ ﺪﺷﺎﺑ ﻑﻮﻧﺎﭘﺎﻴﻟ ﺭﺍﺪﻳﺎﭘ ﺮﮔﺍ ﺖﺳﺍ ﯽﺒﻧﺎﺠﻣ ﺭﺍﺪﻳﺎﭘ x=0

ﺪﺷﺎﺑ ﻪﺘﺷﺍﺩ ﺩﻮﺟﻭ ﺮﻳﺯ ﺕﺭﻮﺻ ﻪﺑ ﺖﺑﺎﺛ :

ﯽﺒﻧﺎﺠﻣ ﯼﺭﺍﺪﻳﺎﭘ • Asymptotic Stability

ﯼﺮﺳﺍﺮﺳ ﯼﺭﺍﺪﻳﺎﭘ • Global Stability

• ﺮﻫ ﯼﺍﺯﺍ ﻪﺑ ﺮﮔﺍ ﺖﺳﺍ ﯼﺮﺳﺍﺮﺳ ﯼﺭﺍﺪﻳﺎﭘ ﺭﺩ ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﺪﺷﺎﺑ ﻖﻘﺤﻣ ﻉﻮﺿﻮﻣx(0)

.

ﯼﺭﺍﺪﻳﺎﭘ ﻩﺎﮔﺪﻳﺩ• ﻑﻮﻧﺎﭘﺎﻴﻟ

ﻝﺩﺎﻌﺗ ﻪﻄﻘﻧ ﺖﺑﺎﺛ ﻭ ﺖﺒﺜﻣ ﺮﻳﺩﺎﻘﻣ ﺮﮔﺍ ﺖﺳﺍ ﻲﻳﺎﻤﻧ ﺭﺍﺪﻳﺎﭘ x=0

c, k, ﻪﺘﺷﺍﺩ ﺩﻮﺟﻭ

ﻪﮐ ﺪﻨﺷﺎﺑ :

Exponentially stability

ﻲﻳﺎﻤﻧ ﯼﺭﺍﺪﻳﺎﭘ ﯽﻄﺧ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﺎﺑ ﻥﺁ ﻁﺎﺒﺗﺭﺍ

(5)

۵

ﻑﻮﻧﺎﭘﺎﻴﻟ ﯼﺭﺍﺪﻳﺎﭘ ﻩﺎﮔﺪﻳﺩ• ﻢﻴﻘﺘﺴﻣ ﺮﻴﻏ ﺵﻭﺭ ،ﯽﻣﻮﻫﻮﻣ ﺭﻮﺤﻣ ﭗﭼ ﺖﻤﺳ ﺭﺩ ﻩﮋﻳﻭ ﺮﻳﺩﺎﻘﻣ ﺭﺍﺮﻘﺘﺳﺍ ،ﯽﻄﺧ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﯼﺯﺎﺳ ﯽﻄﺧ ﯽﻣﻮﻫﻮﻣ ﺭﻮﺤﻣ ﯼﻭﺭ ﻭ ﺖﺳﺍﺭ ﺖﻤﺳ ﻢﻴﻘﺘﺴﻣ ﺵﻭﺭ ﻊﺑﺎﺗ ﺮﮔﺍ ﺕﺭﻮﺻ ﻪﺑ V

: →

ﻭ ﻪﺘﺳﻮﻴﭘ ﻭ ﻩﺪﺷ ﻒﻳﺮﻌﺗ

ﻭ ﺪﺷﺎﺑ ﻦﻴﻌﻣ ﺖﺒﺜﻣ ﺰﻴﻧ ﻭ ﺪﺷﺎﺑ ﺮﻳﺬﭘ ﻖﺘﺸﻣ

̇ ≤ 0

ﻪﻄﻘﻧ ﺖﺳﺍ ﺭﺍﺪﻳﺎﭘx=0

.

ﺮﮔﺍ .ﺩﻮﺑ ﺪﻫﺍﻮﺧ ﯽﺒﻧﺎﺠﻣ ﺭﺍﺪﻳﺎﭘ x=0 ﻪﻄﻘﻧ ̇ < 0

ﻪﻣﺪﻘﻣ

ﻪﻴﻀﻗ• ﯼﺭﺍﺪﻳﺎﭘ دﻮﺷ ﯽﻣ ﻦﯿﻌﻣ ﻪﻤﯿﻧ ﯽﻔﻨﻣ فﻮﻧﺎﭘﺎﯿﻟ ﻊﺑﺎﺗ ﻖﺘﺸﻣ ﻪﮐ يدراﻮﻣ رد دﺮﺑرﺎﮐﻝﺎﺳﻻ . Invariant set ﺎﯾ ادروﺎﻧ يﺎﻫ ﻪﻋﻮﻤﺠﻣ ﻒﯾﺮﻌﺗ ﻪﻋﻮﻤﺠﻣ ﮏﯾ: ﯽﮑﯿﻣﺎﻨﯾد ﻢﺘﺴﯿﺳ ﻪﺑ ﺖﺒﺴﻧ ار M

Xd=f(x) ﺪﻨﻣﺎﻧ ﺮﯿﻐﺘﻣﺎﻧ ﻪﻋﻮﻤﺠﻣ

ﺮﮔا : 0 ∈ ⇒ ∈

ﺮﮕﯾد ترﺎﺒﻋ ﻪﺑ رد ﻪﻟدﺎﻌﻣ ﻞﺣ ﺮﮔا ﻪﮐ ﺖﺳا ﯽﻃﺎﻘﻧ ﻪﻋﻮﻤﺠﻣM

راﺮﻗ نآ ردt=0

رد هراﻮﻤﻫ هﺎﮕﻧآ دﺮﯿﮔ ﺪﻧﺎﻣ ﯽﻣ ﯽﻗﺎﺑ M

.

(6)

۶

ﻪﻣﺪﻘﻣ

بﻮﺴﺤﻣ ادروﺎﻧ ﻪﻋﻮﻤﺠﻣ ﮏﯾ دﻮﺧ لدﺎﻌﺗ ﻪﻄﻘﻧ اﺬﻟ لدﺎﻌﺗ ﻪﻄﻘﻧ ﻪﺑ ﻒﯾﺮﻌﺗ ﺖﻫﺎﺒﺷ• دﻮﺷ ﯽﻣ . يﺪﺣ يﺎﻫ ﻞﮑﯿﺳ ياﺮﺑ ﻒﯾﺮﻌﺗ ﻪﻌﺳﻮﺗ•

ﻝﺎﺳﻻ ﻪﻴﻀﻗ• ﺪﯿﻨﮐ ضﺮﻓ

: →

و ﺪﺷﺎﺑ ﺮﯾﺬﭘ ﻖﺘﺸﻣ و ﻪﺘﺳﻮﯿﭘ ﻊﺑﺎﺗ ﮏﯾ

⊂ ﻪﻟدﺎﻌﻣ ﻞﺣ ﻪﺑ ﺖﺒﺴﻧ ﺮﯿﻐﺘﻣﺎﻧ ﻪﻋﻮﻤﺠﻣ ﮏﯾ Xd=f(x)

ﺰﯿﻧ و ﺪﺷﺎﺑ

̇ ≤ 0 رد

وM

= : , ̇ = 0 و

رد ﺮﯿﻐﺘﻣﺎﻧ ﻪﻋﻮﻤﺠﻣ ﻦﯾﺮﺘﮔرﺰﺑN لﺎﺣ ،ﺖﺳاE

رد ﻪﮐ ﯽﻠﺣ ﺮﻫ ﻪﺑ ﯽﻧﺎﻣز ﺖﯾﺎﻬﻧ ﯽﺑ رد دﻮﺷ عوﺮﺷ M

دﻮﺷ ﯽﻣ ﻢﺘﺧ N

. لﺎﺜﻣ و ﺎﯾاﺰﻣ ،دﺮﺑرﺎﮐ هﻮﺤﻧ

ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ Hyper stability

ﻊﺟﺮﻣ ﻝﺪﻣ ﯽﻘﻴﺒﻄﺗ ﻝﺮﺘﻨﮐ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﺭﺩ ﺩﺮﺑﺭﺎﮐ ﺎﻳ ﯽﻟﺎﻌﻓﺮﻴﻏ Passivity

ﺯﺍ • ۱۹۶۱ ﺖﺳﺍ ﻩﺪﺷ ﻪﺋﺍﺭﺍ .

ﺖﺳﺍ ﻩﺪﺷ ﺎﻨﺑ ﻢﺘﺴﻴﺳ ﺭﺩ ﯼﮊﺮﻧﺍ ﺶﻫﺎﮐ ﺱﺎﺳﺍ ﺮﺑ • .

ﺩﺭﺍﺩ ﻁﺎﺒﺗﺭﺍ ﯽﻄﺧﺮﻴﻏ ﻢﺘﺴﻴﺳ ﯽﺟﻭﺮﺧ ﻭ ﯼﺩﻭﺭﻭ ﺎﺑ ﺎﻬﻨﺗ • .

Linear

Nonlinear

(7)

٧

ﯽﻟﺎﻌﻓﺮﻴﻏ ﻭ ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﻒﻳﺮﻌﺗ ﺮﻳﺩﺎﻘﻣ ﺎﺑ ﺮﮔﺍ ﺪﻨﻣﺎﻧ ﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﺍﺭ ﻭﺮﺑﻭﺭ ﻢﺘﺴﻴﺳ ﯼﺎﻫ ﻞﺣ ﻪﻤﻫ ﯼﺍﺮﺑ ﺐﻳﺍﺮﺿ ﺖﺒﺜﻣ ﯽﻘﻴﻘﺣ X(x(t0),t ) ﻢﻴﺷﺎﺑ ﻪﺘﺷﺍﺩ

:

ﻻﺎﺑ ﻪﻄﺑﺍﺭ ﺮﮕﻳﺩ ﻥﺎﻴﺑ :

≤

ﻪﻄﺑﺍﺭ ﺪﺷﺎﺑ ﯽﻣ ﯽﻟﺎﻌﻓﺮﻴﻏ ﻁﺮﺷ ﻥﺎﻤﻫ X

) . ﻢﺘﺴﻴﺳ ﯼﮊﺮﻧﺍ (

W=f(v,t,T) ) t ( u ) t ( D ) t ( x ) t ( c y

) t ( u ) t ( B ) t ( x ) t ( A x

+

=

+

=

) ) t ( x ( ) t (

x <δ ₀ +γ

u v

w

ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﺮﻴﺴﻔﺗ

=

ﻪﻠﺻﺎﻓ ﺭﺩ ﻩﺪﺷ ﮏﻠﻬﺘﺴﻣ ﯼﮊﺮﻧﺍ t0,t1

+ ﻥﺎﻣﺯ ﺭﺩ ﺩﻮﺟﻮﻣ ﯼﮊﺮﻧﺍ t1

ﺭﺩ ﻪﻴﻟﻭﺍ ﻩﺪﺷ ﻩﺮﻴﺧﺫ ﯼﮊﺮﻧﺍ +t0

ﻪﻠﺻﺎﻓ ﺭﺩ ﺝﺭﺎﺧ ﺯﺍ ﻢﺘﺴﻴﺳ ﻪﺑ ﻩﺪﺷ ﻞﻘﺘﻨﻣ ﯼﮊﺮﻧﺍ t0,t1

∫

+

=

+ ₀ ²

2

1 ) D x(t ) uvdt

t ( x

Linear

Nonlinear

w v

u

(8)

٨

ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﺮﻴﺴﻔﺗ ﻥﻮﭼ D>0 :

ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﻁﺮﺷ ﻖﺒﻃ ﻭ

ﻪﺠﻴﺘﻧ ﺭﺩ : ﺪﺣﺍﻭ ﺮﺑﺍﺮﺑ ﺎﺘﻟﺩ

∫

≤ ₀ ² +

2

1) x(t ) uvdt

t ( x

γ2

∫

^uvdt<

2 2 0 2

1 ) ≤ x(t ) +γ t

( x

γ t x t x

γ t x γ t x t

x

+

+ +

)

≤ ( )

⇒ (

) ) (

≤( )

≤ ( ) (

0 1

2 0 2 2

0 2 1

ﯼﺭﺍﺪﻳﺎﭘﺍﺮﻓ ﻂﻳﺍﺮﺷ :

ﺪﺷﺎﺑ ﯽﻘﻴﻘﺣ ﺖﺒﺜﻣ ﯽﻄﺧ ﻢﺘﺴﻴﺳ - .

PR

ﺪﺷﺎﺑ ﺭﺍﺮﻗﺮﺑ ﺮﻳﺯ ﯼﮊﺮﻧﺍ ﯼﻭﺎﺴﻣﺎﻧ - :

ﯽﻄﺧﺮﻴﻏ ﻥﺎﻤﻟﺍ ﯼﺩﻭﺭﻭ ﺮﮔﺍ ﻥﺁ ﯽﺟﻭﺮﺧ ﻭv

ﻢﻳﺭﺍﺩ ﺪﺷﺎﺑw ) :

ﺱﻼﮐ ﻊﺑﺍﻮﺗ (p

≥ − ﺎﻳ

≤

Linear

Nonlinear

w v

u

(9)

٩

ﺖﺒﺜﻣ ﯽﻄﺧ ﻊﺑﺍﻮﺗ ﻥﺩﻮﺑ ﯽﻘﻴﻘﺣ

ﻊﺑﺎﺗ ﻢﻴﺷﺎﺑ ﻪﺘﺷﺍﺩ ﺮﮔﺍ ﻢﻴﻣﺎﻧ ﯽﻣ ﯽﻘﻴﻘﺣ ﺖﺒﺜﻣ ﺍﺭG(s)

:

Re(G(s))>=0 for all Re(s)>=0

ﻝﺎﺜﻣ : G(s)=1/(s+1) S=a+jb, G(a+jb)=1/(a+jb+1)=(a+1-jb)/((a+1)^2+b^2) Re(G(s))=(a+1)/((a+1)^2+b^2)

For all a>=0 Re(G(s))>=0  G(s) is Positive real

ﻪﻣﺪﻘﻣ ) ﯽﻘﻴﺒﻄﺗ ﻝﺮﺘﻨﮐ ﺮﺑ ﯼﺍ ﻪﻣﺪﻘﻣ (

ﻒﻠﺘﺨﻣ ﯽﮑﻴﻣﺎﻨﻳﺩ ﯼﺎﻫ ﻢﺘﺴﻴﺳ• LTI ﯽﺳﺎﻨﺷﺭﺎﮐ ﻩﺭﻭﺩ ﻝﺮﺘﻨﮐ/ LTV/ NLTI/ NLTV

ﻝﺮﺘﻨﮐ ﻑﺍﺪﻫﺍ :

ﯼﺭﺍﺪﻳﺎﭘ ﺏﻮﻠﻄﻣ ﺩﺮﮑﻠﻤﻋ  ﺵﺎﺸﺘﻏﺍ ﻭ ﺰﻳﻮﻧ ﻑﺬﺣ ﻡﺍﻮﻗ ﺎﻳ ﯼﺭﺍﺪﻳﺎﭘ ﺐﺳﺎﻨﻣ ﻪﻴﺷﺎﺣ 

ﺎﻫﺮﮕﻠﻤﻋ ﺩﻭﺪﺤﻣ ﺢﻄﺳ ﺎﻫ ﯽﮔﺪﺷ ﺖﻔﺟ ﻑﺬﺣ

(10)

١٠

ﯽﻨﻴﻌﻣﺎﻧ ﻡﻮﻬﻔﻣ• (uncertainty)

ﯽﮑﻴﻣﺎﻨﻳﺩ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﺭﺩ

ﯽﮑﻴﻣﺎﻨﻳﺩ ﻢﺘﺴﻴﺳ ﮏﻳ ﺭﺩ ﺮﻴﻴﻐﺗ ﻭ ﯽﻨﻴﻌﻣﺎﻧ ﺯﻭﺮﺑ ﻪﺑ ﻒﻠﺘﺨﻣ ﯼﺎﻫ ﻩﺎﮕﻧ ﻥﺎﻣﺯ ﺐﺴﺣ ﺮﺑ ﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﺮﻴﻴﻐﺗ )

ﻦﻴﻌﻣﺎﻧ ﻭ ﻦﻴﻌﻣ (

ﯼﺭﺎﺘﺧﺎﺳ ﻩﺎﮕﻧ ﻭ ﻖﻴﻗﺩ ﺮﻴﻏ ﯼﺯﺎﺴﻟﺪﻣ )

ﯼﺮﺘﻣﺍﺭﺎﭘ ( ﯼﺭﺎﺘﺧﺎﺳ ﺮﻴﻏ ﻭ

) ﻩﺪﺸﻧ ﻝﺪﻣ ﮏﻴﻣﺎﻨﻳﺩ (

Parametric or Structural and Non-structural ﻩﺪﺷ ﯽﻄﺧ ﯼﺎﻫ ﯽﻄﺧ ﺮﻴﻏ ﻭ ﻖﻴﻗﺩ ﺮﻴﻏ ﯼﺯﺎﺴﻟﺪﻣ

ﺵﺎﺸﺘﻏﺍ ﻭ ﺰﻳﻮﻧ ﺩﻭﺭﻭ )

ﯽﺟﺭﺎﺧ ﺎﻳ ﯽﻠﺧﺍﺩ (

ﻥﺁ ﻥﺍﺰﻴﻣ ﻭ ﺵﺎﺸﺘﻏﺍ ﻉﻮﻗﻭ ﻥﺎﻣﺯ

ﻡﻭﺎﻘﻣ ﻪﺑ ﯼﺭﺎﺘﺧﺎﺳ ﺮﻴﻏ ﻭ ﯽﻘﻴﺒﻄﺗ ﻝﺮﺘﻨﮐ ﻪﺑ ﯼﺭﺎﺘﺧﺎﺳ ﯽﻨﻴﻌﻣﺎﻧ ﻊﺟﺍﺮﻣ ﯽﺧﺮﺑ ﺭﺩ ﺖﺳﺍ ﻩﺪﺷ ﻩﺩﺍﺩ ﻉﺎﺟﺭﺍ .

ﻢﻳﺯﺍﺩﺮﭘ ﯽﻣ ﻪﻟﺎﺴﻣ ﻦﻳﺍ ﻪﺑ ﺮﺗ ﯼﺩﺎﻴﻨﺑ ﺮﮕﻳﺩ ﯽﻫﺎﮕﻧ ﺭﺩ .

ﻲﻨﻴﻌﻣﺎﻧ  matched ﻭ

unmatched

Xd=(A+d1)X+(B+d2)U ﺎﻳ

Xd=AX+BU+d1+d2

(11)

١١

ﺮﻴﻈﻧ ﯽﻟﻮﻤﻌﻣ ﺭﻮﺨﺴﭘ ﻝﺮﺘﻨﮐ•

؟ﺪﻫﺩ ﯽﻣ ﺏﺍﻮﺟ ﺎﺠﮐ ﺎﺗ PID

ﻝﺎﺜﻣ ﺎﺑ ﺵﻭﺎﮐ ﮏﺷﻮﻣ ﮏﻳ ﻭﺎﻳ ﻝﺎﻧﺎﮐ ﯽﮑﻴﻣﺎﻨﻳﺩ ﻢﺘﺴﻴﺳ :

ﻩﺪﻨﻨﮐ ﻝﺮﺘﻨﮐ

4 PID

3 2

2 1

a s a s

a s a r

r  

 



ﻪﺘﺴﺑ ﻭ ﺯﺎﺑ ﻪﻘﻠﺣ ﺦﺳﺎﭘ•

0 5 10 15 20 25 30

4 4.2 4.4 4.6 4.8 5 5.2 5.4 5.6 5.8 6

0 5 10 15 20 25 30

-1 0 1 2 3 4 5

(12)

١٢

ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﺦﺳﺎﭘ• ۱

ﺮﻴﺧﺎﺗ ﻪﻴﻧﺎﺛ

) ﺏﻮﻠﻄﻣﺎﻧ ﺩﺮﮑﻠﻤﻋ ﺎﻣﺍ ﯼﺭﺍﺪﻳﺎﭘ ﻆﻔﺣ (

ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﺦﺳﺎﭘ•

ﺖﻣﻼﻋ ﻥﺎﻤﻫ ﺎﺑ ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﺐﻳﺮﺿ ﺮﻴﻴﻐﺗ ﺎﺑ ) ﺏﻮﻠﻄﻣﺎﻧ ﺩﺮﮑﻠﻤﻋ ﺎﻣﺍ ﯼﺭﺍﺪﻳﺎﭘ ﻆﻔﺣ (

0 5 10 15 20 25 30

-5 -4 -3 -2 -1 0 1 2 3 4 5

0 5 10 15 20 25 30

-1 0 1 2 3 4 5

ﺮﻴﻴﻐﺗ ﺎﺑ ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﺦﺳﺎﭘ• ﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﺯﺍ ﯽﮑﻳ ﺖﻣﻼﻋ )

ﯼﺭﺍﺪﻳﺎﭘﺎﻧ (

ﻪﺘﺴﺑ ﻪﻘﻠﺣ ﺦﺳﺎﭘ•

ﯼﺮﻴﮔ ﻩﺯﺍﺪﻧﺍ ﺰﻳﻮﻧ ﻝﺎﻤﻋﺍ ﺎﺑ

) ﺏﻮﻠﻄﻣﺎﻧ ﺩﺮﮑﻠﻤﻋ ﺎﻣﺍ ﯼﺭﺍﺪﻳﺎﭘ ﻆﻔﺣ (

0 5 10 15 20 25 30

0 0.5 1 1.5 2 2.5

3x 10⁵

0 5 10 15 20 25 30

-1 0 1 2 3 4 5

(13)

١٣

؟ﺪﻨﺷﺎﺑ ﻢﻫ ﺎﺑ ﻪﻤﻫ ﺮﮔﺍ ﺪﺷ ﻡﺎﺠﻧﺍ ﻞﻘﺘﺴﻣ ﺭﻮﻃ ﻪﺑ ﺮﻴﻴﻐﺗ ﺩﺭﺍﻮﻣ ﺯﺍ ﮏﻳﺮﻫ ﻝﺎﻤﻋﺍ• ﻢﺘﺴﻴﺳ ﯼﺎﻫﺮﺘﻣﺍﺭﺎﭘ ﺭﺩ ﻲﻳﺰﺟ ﺮﻴﻴﻐﺗ ﻭ ﺮﻴﺧﺎﺗ ﻥﺎﻣﺰﻤﻫ ﻝﺎﻤﻋﺍ

0 5 10 15 20 25 30

-100 -50 0 50 100 150 200

ﯼﺭﺎﺘﺧﺎﺳ ﻉﻮﻧ ﺯﺍ ﯽﻨﻴﻌﻣﺎﻧ• )

ﻞﻳﺪﺒﺗ ﻊﺑﺎﺗ ﺯﺍ ﯽﻌﻗﺍﻭ

ﻪﺳ ﻪﺒﺗﺮﻣ ﻭﺩ ﻪﺒﺗﺮﻣ ﯼﺍﺮﺑ ﺎﻣ ﻭ ﻩﺩﻮﺑ

ﻢﻳﺍ ﻩﺩﺮﮐ ﯽﺣﺍﺮﻃ (

5 10 15 20 25 30

-6 -4 -2 0 2 4 6

8 3rd order

2nd order

(14)

١۴

ﻢﺘﺴﻴﺳ ﯼﺮﮔ ﯽﻄﺧ ﺮﻴﻏ• ﯽﻄﺧ ﻢﺘﺴﻴﺳ ﺭﺩ ﻥﺁ ﻑﺬﺣ ﻭ ﯽﻄﺧﺮﻴﻏ ﻢﺘﺴﻴﺳ ﺭﺩ ﯽﮔﺪﺷ ﺖﻔﺟ - Coupling

ﺎﻬﻧﺁ ﺶﻘﻧ ﻭ ﻻﺎﺑ ﻪﺒﺗﺮﻣ ﯼﺎﻫ ﻡﺮﺗ - High order terms

ﻞﻣﺎﺷ ﯽﻄﺧ ﺮﻴﻏ ﯼﺎﻫ ﻢﺘﺴﻴﺳ ﯼﺎﻫ ﯽﮔﮋﻳﻭ - :

ﺵﺮﭘ ﻩﺪﻳﺪﭘ  Jumping

ﺏﻮﺷﺁ ﻪﻳﺪﭘ  Chaos

ﯼﺪﺣ ﻞﮑﻴﺳ  Limit cycle

ﮏﻴﻧﻮﻣﺭﺎﻫﺮﻳﺯ ﻩﺪﻳﺪﭘ  Sub-harmonic

ﮏﻳﺮﺘﻣﺍﺭﺎﭘ ﮏﻳﺮﺤﺗ  Parametric excitation

ﻥﺪﺷ ﻪﺘﺴﺑ ﻭ ﺯﺎﺑ ﻩﺪﻳﺪﭘ 

ﻩﺪﺷ ﻪﺋﺍﺭﺍ ﯼﺎﻫ ﻞﺣ ﻩﺍﺭ• :

ﻡﻭﺎﻘﻣ ﻝﺮﺘﻨﮐ  /LTI

ﺩﺮﮑﻠﻤﻋ ﺖﻳﺩﻭﺪﺤﻣ /

ﯼﺭﺍﺪﻳﺎﭘ ﻭ ﯽﺿﺎﻳﺭ ﯽﻧﺎﺒﻣ

ﯼﻮﻗ ﺭﺎﻴﺴﺑ ﯽﻘﻴﺒﻄﺗ ﻝﺮﺘﻨﮐ  /NLTV

ﯽﺒﻳﺮﻘﺗ ﺩﻭﺪﺤﻣﺎﻧ ﺩﺮﮑﻠﻤﻋ /

ﯽﻧﺎﺒﻣ

ﺩﻮﺟﻮﻣ ﯽﺿﺎﻳﺭ ﯽﺒﻳﺮﻘﺗ ﺩﻭﺪﺤﻣﺎﻧ ﺩﺮﮑﻠﻤﻋ ﺪﻨﻤﺷﻮﻫ ﻝﺮﺘﻨﮐ /

Soft

computing ﻒﻴﻌﺿ ﯼﺭﺍﺪﻳﺎﭘ ﯽﻧﺎﺒﻣ ﻭ ﺕﺎﻴﺿﺎﻳﺭ ﺎﻫ ﺵﻭﺭ ﻪﻤﻫ ﺮﺑ ﯽﺸﺷﻮﭘ ﯽﻄﺧﺮﻴﻏ ﯼﺎﻫ ﺵﻭﺭ ﻩﺎﮕﻳﺎﺟ /

ﯽﻫﺎﮔﺪﻳﺩ

ﯼﺩﺮﺑﺭﺎﮐ

(15)

١۵

ﻊﺟﺍﺮﻣ Applied nonlinear control, Slotin, 1991.

Nonlinear systems, H. Khalil, 3rd edition, 2002.

Nonlinear oscillation, A. H. Nyfeh,

Adaptive Control, K. J. Astrom, Wesley, 1995.

Adaptive Control Tutorial, P. Ioannou, B. Fidan, SIAM, Advanced in Design and Control, 2006

Adaptive Control, L. D. Landau, R. Lozano, Springer-Verlag London Limited 2011 -

ﺱﺭﺩ ﯽﻠﺻﺍ ﻦﺘﻣ )

۵۰ ﻡﺮﺗ ﻥﺎﻳﺎﭘ ﻥﺎﺤﺘﻣﺍ ﺭﺩ ﺪﺻﺭﺩ (

ﯼﺭﺍﺬﮔ ﺵﺯﺭﺍ -

ﯼﺯﺎﺳ ﻪﻴﺒﺷ )

۴۰ ﺎﻫ ﻩﮊﻭﺮﭘ ﺭﺩ ﺪﺻﺭﺩ

( (ﯽﻟﺎﺳرا ﻪﻟﺎﻘﻣ)

-

ﻖﻴﻘﺤﺗ ) ۱۰ ﺭﺎﻨﻴﻤﺳ ﺎﻳ ﻩﮊﻭﺮﭘ ﺐﻟﺎﻗ ﺭﺩ ﺪﺻﺭﺩ (

ﯼﺯﺎﺴﻟﺪﻣ ﻭ ﻪﻟﺎﻘﻣ ﯼﻮﺠﺘﺴﺟ

۹ • ﻞﻣﺎﺷ ﻩﺮﻤﻧ ۲ ﻩﺮﻤﻧ ) ﺭﺎﻨﻴﻤﺳ ﺎﻳ ﻦﻳﺮﻤﺗ (

۷ ﻡﺮﺗ ﻥﺎﻳﺎﭘ ﻩﺮﻤﻧ

ﺱﺭﺩ ﻊﺟﺍﺮﻣ ﻭ ﺕﺎﺤﻴﺿﻮﺗ ﺮﻳﺎﺳ ﺎﻫ ﻞﺼﻓﺮﺳ

Nonlinear control

Nonlinear control

=

∫

∫

∫

ﯽﺷﺰﻐﻟ ﺩﻮﻣ ﻝﺮﺘﻨﮐ •

ﺐﻘﻋ ﻪﺑ ﻡﺎﮔ ﻭ ﺭﻮﺨﺴﭘ ﯼﺯﺎﺳ ﯽﻄﺧ •

ﯽﻘﻴﺒﻄﺗ ﻝﺮﺘﻨﮐ •

http://wp.kntu.ac.ir/khoshnood