2. Mathematical Description of Systemsocw.snu.ac.kr/sites/default/files/NOTE/9203.pdfPerception and...

Perception and Intelligence LaboratorySchool of Electrical Engineering at SNU

2. Mathematical Description of Systems

Linear System Representation

Causality and Lumpedness

Linear Systems

Linear Time-Invariant (LTI) Systems

Linearization

Examples of Linear Systems

Discrete-Time Systems

1Linear Systems



Differential Equation

Impulse Response

Transfer Function

State Space Equation

2Linear Systems

0

( ) ( , ) ( )

y( ) G( ) u( )

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

t

ty t G t u d

s s s

x t A t x t B t u ty t C t x t D t u t

τ τ τ=

=

= += +

∫

&

( ) ( ) ( 1) ( ) ( 1)( ) ( ( ), ( ),..., ( ), ( ), ( ),..., ( ))n n n n ny t f y t y t y t u t u t u t− −=



Definition 2.1: The state of a system at time is the information at that, together with the input , for , determines uniquely the output for all .

3Linear Systems

0( )x t 0t0t ( )u t 0t t≥

0t t≥( )y t

Voltage Source

( )u t ( )y tLi CvOutput Voltage

0 0

0 0

1 0 2 0

( ), ( )

( ), ( )

( ), ( )

L Ci t v t

y t y t

x t x t

&

Example

( )y t ( )u t0t

can be uniquely determined for any input if initial values of induction current and capacitor voltage at

State:


Causality and Lumpedness

Causal System (Nonanticipatory System)The current output depends on only the past and current inputs but not on the future inputs

4Linear Systems

Lumped System (Finite Dimensional System)The number of state variables is finiteIf infinite, Distributed System

Voltage Source

( )u t ( )y tLi CvOutput Voltage


Linear Systems

Linear System Satisfying superposition principle: additivity + homogeneity

5Linear Systems

00

0

1 1 0 2 2 01 1 0 2 2 0 0

1 1 0 2 2 0 0

( )For ( ), , 1, 2

( ),

( ) ( )then ( ) ( ),

( ) ( ),

ii

i

x ty t t t i

u t t t

x t x ty t y t t t

u t u t t tα α

α αα α

⎫⇒ ≥ =⎬≥ ⎭

+ ⎫⇒ + ≥⎬+ ≥ ⎭


Linear Systems

Zero-input Response

6Linear Systems

00

0

( )( ),

( ) 0, zi

x ty t t t

u t t t⎫⇒ ≥⎬= ≥ ⎭

Zero-state Response

00

0

( ) 0( ),

( ), zs

x ty t t t

u t t t= ⎫

⇒ ≥⎬≥ ⎭

00

0

( )( ) ( ),

( ), zi zs

x ty t y t t t

u t t t⎫⇒ + ≥⎬≥ ⎭

By additivity

Response = Zero-input Response+Zero-state Response


Linear Systems

Input-Output DescriptionAssume initial state is zero.Define piecewise continuous function of input:

7Linear Systems

( ) ( ) ( )

where0,

( ) 1/ ,

0,

i ii

i

i i i

i

u t u t t t

t t

t t t t t

t t

δ

δ

Δ

Δ

≈ − Δ

⎧ <⎪

− = Δ ≤ < + Δ⎨⎪ ≥ + Δ⎩

∑ ( )u t

t

( )iu t

Δ

1( ) ( ) ( ) ( )i i i iu t t t u t u tδΔ − Δ = Δ =Δ


Linear Systems

Input-Output Description

8Linear Systems

Let be the output for the input , i.e. ( , )ig t tΔ ( )it tδΔ −

( ) ( , )i it t g t tδΔ Δ− →

By homogeneity

( ) ( ) ( , ) ( )i i i it t u t g t t u tδΔ Δ− Δ → Δ

By additivity

( ) ( ) ( , ) ( )

( ) ( )

i i i ii i

t t u t g t t u t

u t y t

δΔ Δ− Δ → Δ

≈ ≈

∑ ∑


Linear Systems


9Linear Systems

The output for the input

0

( ) ( , ) ( )

( ) lim ( , ) ( )

( ) ( , ) ( )

i ii

i ii

y t g t t u t

y t g t t u t

y t g t u dτ τ τ

Δ

ΔΔ→

∞

−∞

≈ Δ

= Δ

=

∑

∑

∫

( )y t ( )u t

where

( ) ( , ) : Impulse Responset g tδ τ τ− →


Linear Systems


10Linear Systems

Causal

0

0

0

( ) ( , ) ( )

( , ) ( )

( , ) ( ) ( , ) ( )

( , ) ( )

t

t t

t

t

t

y t g t u d

g t u d causal

g t u d g t u d

g t u d relaxed

τ τ τ

τ τ τ

τ τ τ τ τ τ

τ τ τ

∞

−∞

−∞

−∞

=

= ⇐

= +

= ⇐

∫∫∫ ∫

∫

Relaxed at :initial state at is 0

( , ) 0, for t<g t τ τ=

0t 0t


Linear Systems


11Linear Systems

MIMO System

0

( ) ( , ) ( )t

tt t dτ τ τ= ∫y G u

where

11 1

21 22

1

( , ) ... ( , )( , ) ( , ) ... ...

( , )... ...( , ) ... ( , )

p

q qp

g t g tg t g t

t

g t g t

τ ττ τ

τ

τ τ

⎡ ⎤⎢ ⎥⎢ ⎥=⎢ ⎥⎢ ⎥⎢ ⎥⎣ ⎦

G


Linear Time Invariant Systems

12Linear Systems

Linear Time Invariant (LTI)A system is said to be time invariant if

and any , we have

00

0

( )( ),

( ),x t

y t t tu t t t

⎫⇒ ≥⎬≥ ⎭

T

00

0

( )( ),

( ),x t T

y t T t t Tu t T t t T

+ ⎫⇒ − ≥ +⎬− ≥ + ⎭



13Linear Systems

If the system is LTI,

( , ) ( , )( , 0) (let =- )( )

g t g t T Tg t Tg t

τ ττ τ

τ

= + +

= −

≡ −

Output of LTI system

0 0( ) ( ) ( ) ( ) ( )

t ty t g t u d g u t dτ τ τ τ τ τ= − = −∫ ∫



14Linear Systems

Example) unity-feedback system2 3

3

1

( ) ( 1) ( 2) ( 3) ...

( 3)i

g t a t a t a t

a t

δ δ δ

δ∞

=

= − + − + − +

= −∑Output

0 01

1

( ) ( ) ( ) ( ) ( )

( )

t ti

i

i

i

y t g t u d a t i u d

a u t i

τ τ τ δ τ τ τ∞

=

∞

=

= − = − −

= −

∑∫ ∫

∑Unit time delay

a( )u t ( )y t


Linear Systems

Transfer Function

15Linear Systems

Laplace transform of

s

0 0

s

0 0

s s

0 0

y(s) ( ( ) ( ) )

( ( ) ( ) )

( ) ( ) ,

y(s) g(s) u(s)

t t

t

v

g t u d e dt relaxed

g t u d e dt causality

g v e dv u e d v t t vτ

τ τ τ

τ τ τ

τ τ τ τ

∞ −

∞ ∞ −

∞ ∞− −

= − ⇐

= − ⇐

= ⇐ = − = +

=

∫ ∫∫ ∫∫ ∫

s

0

g(s) : ( )

g(s) : ( ) ( ) t

g t

g g t e dt∞ −= = ∫L

(s) (s) (s)=y G uTransfer Function Matrix: MIMO case


Linear Systems

Properness of Transfer Function

16Linear Systems

g(s) ( ) / ( )N s D s=

g(s) proper deg ( ) deg ( )g( ) zero or constant

D s N s− ⇔ ≥⇔ ∞ =

g(s) strictly proper deg ( ) deg ( )g( ) zero

D s N s− ⇔ >⇔ ∞ =

g(s) biproper deg ( ) deg ( )g( ) non-zero constant

D s N s− ⇔ =⇔ ∞ =

g(s) improper deg ( ) deg ( )| g( ) |

D s N s− ⇔ <⇔ ∞ = ∞


Linear Systems

Properness of Transfer Function Matrix

17Linear Systems

(s) is (strictly) proper if all entries are (strictly) proper−G

1

(s) is biproper if (s) is square and

both (s) and (s) are proper−

−GG

G G


Linear System

State Space Equation

18Linear Systems

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

x t Ax t Bu ty t Cx t Du t

= += +

&

Laplace Transform

sx(s) - x(0) x(s) u(s)y(s) x(s) u(s)

A BC D

= += +

Which implies1 1

1 1

x(s) (sI ) x(0) (sI ) u(s)y(s) (sI ) x(0) (sI ) u(s) u(s)

A A BC A C A B D

− −

− −

= − + −

= − + − +

Transfer Function Matrix1(s) (sI )C A B D−= − +G


Linearization

Nonlinear System

19Linear Systems

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

x t f x t u t ty t h x t u t t

==

&

Linearization at an operating point 0 0( ), ( )x t u t

0 0( ) ( ) ( ), ( ) ( ) ( )x t x t x t u t u t u t= + = +

0 0 0

0 00 0

( ) ( ) ( ) ( , , )

( , , ) (.)

x t x t x t f x x u u tf ff x u t x u Ox u

= + = + +

∂ ∂= + + +

∂ ∂

&& &

0 0

( )

where = , (.)

x t Ax Buf fA x B u Ox u

= +∂ ∂

= +∂ ∂

&

⇒


Implementation and Examples

Op-Amp Circuit Implementation: P. 17Figure 2.7,

20Linear Systems

Examples: (p. 18 -29)Cart with inverted pendulum, Satellite in orbit, Hydraulic tanks,RLC circuits



Sampling Period: T

21Linear Systems

[ ] ( ), [ ] ( )[ ] ( )

u k u kT y k y kTx k x kT

= =

=

Linear System: homogeneity, additivityResponse= Zero-state response + Zero-input response

Impulse Sequence 1 if 0

[ ]0 if 0

kk

kδ

=⎧= ⎨ ≠⎩



22Linear Systems

Input Sequence [ ] [ ] [ ]

mu k u m k mδ

∞

=−∞

= −∑

[ ] [ , ]k m g k mδ − →

By homogeneity[ ] [ ] [ , ] [ ]k m u m g k m u mδ − →

By additivity

Impulse Response Sequence

[ ] [ ] [ , ] [ ]m m

k m u m g k m u mδ − →∑ ∑Input-Output Description

[ ] [ , ] [ ]m

y k g k m u m= ∑0

[ ] [ ] [ ]k

my k g k m u m

=

⇒ = −∑

CausalRelaxedLTI



23Linear Systems

Z-Transform

0y(z) ( [ ]) [ ] k

ky k y k z

∞−

=

= = ∑Z

Discrete Transfer Function

( )

0 0

( )

0 0

0 0

y(z) ( [ ] [ ])

[ ] [ ]

[ ] [ ]

g(z) u(z)

k m m

k m

k m m

k m

l m

l m

g k m u m z z

g k m z u m z

g l z u m z

∞ ∞− − −

= =

∞ ∞− − −

= =

∞ ∞− −

= =

= −

= −

=

=

∑ ∑

∑ ∑

∑ ∑



24Linear Systems

Z-Transform of

[ 1] [ ] [ ] [ ] [ ][ ] [ ] [ ] [ ] [ ]

x k A k x k B k u ky k C k x k D k u k+ = +

= +

State-Space Equations

[ 1]x k +

( 1)

0 0

1

( [ 1]) [ 1] [ 1]

( [ ] [0] [0])

(x(z) [0])

k k

k k

l

l

x k x k z z x k z

z x l z x x

z x

∞ ∞− − +

= =

∞−

=

+ = + = +

= + −

= −

∑ ∑

∑

Z



25Linear Systems

Z-Transform of State-Space Equations

1 1

1 1

x( ) [0] x( ) u( )y( ) x( ) u( )x( ) ( I ) [0] ( I ) u( )

( ) ( I ) [0] ( ( I ) ) u( )

z z zx A z B zz C z D zz z A zx z A B z

y z C z A zx C z A B D z

− −

− −

− = += +

= − + −

= − + − +

If zero initial state1( ) ( ( I ) ) u( )y z C z A B D z−= − +

Transfer Function1( ) ( I )z C z A B D−= − +G



26Linear Systems

Example: compound interest calculation

Interest: 0.015%[0] 1, [ ] 0, 1, 2,[ ] (1.00015) k

u u i ig k

= = =

=

Impulse Response

Output

0[ ] (1.00015) [ ]

kk m

my k u m−

=

= ∑Transfer Function

1

0 0

1

g( ) (1.00015) = (1.00015 )

1= 1 1.00015 1.00015

k k k

k kz z z

zz z

∞ ∞− −

= =

−

=

=− −

∑ ∑


System Type Internal Description

External Description

Distributed, Linear(Causal, Relaxed)

Lumped, Linear(Causal, Relaxed)

Distributed, Linear,Time-invariant

Rumped, Linear,Time-invariant

Concluding Remarks

27Linear Systems

Example: compound interest calculation

( ) ( )( ) ( )

x A t x B t uy C t x D t u= += +

&

0

( ) ( , ) ( )t

ty t g t u dτ τ τ

−= ∫

0

( ) ( , ) ( )t

ty t g t u dτ τ τ

−= ∫

0( ) ( ) ( )

y(s) = G(s) u(s)

ty t g t u dτ τ τ= −∫

x Ax Buy Cx Du= += +

&0

( ) ( ) ( )y(s) = G(s) u(s)

ty t g t u dτ τ τ= −∫

2. Mathematical Description of Systemsocw.snu.ac.kr/sites/default/files/NOTE/9203.pdfPerception and...

Documents

Transcript of 2. Mathematical Description of Systemsocw.snu.ac.kr/sites/default/files/NOTE/9203.pdfPerception and...