Meiling chensignals & systems1 Lecture #2 Introduction to Systems.

meiling chen signals & systems 1

Lecture #2

Introduction to Systems


systemA system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.


Example of system


System interconnection


System properties

• Causality

• Linearity

• Time invariance

• Invertibility


CausalityA system is said to be causal if the present value of the output signal depends only on the present or past values of the input signal.


Causal and noncausal system

Example: distinguish between causal and noncausal systems in the following:

t

)(tu

1 2

(1) Case I )()( tuty

t

)(ty

12Noncausal system

0)(

0)(1

tybut

tutwhen


(2) Case II

causal system

)()( tuty

t

)(ty

1 2

Delay system

(3) Case III )2()()( tututy

causal systemAt present past


(4) Case IV )2()()( tututy

noncausal systemAt present future

(5) Case V

noncausal system

stepunitistuiftuty )()()( 2

t

)(ty

0)(

0)(0

tybut

tutwhen


Linearity

)(1 tx

A system is said to be linear in terms of the system input x(t) and the system output y(t) if it satisfies the following two properties of superposition and homogeneity.

Superposition ：

Homogeneity ：

)(1 ty )(2 tx )(2 ty

)()( 21 txtx )()( 21 tyty

)(1 tx )(1 ty )(1 tax )(1 tay


Example 1.19

][][ nnxny

][][

][][

]}[][{][

][][][

][][][][

][][][][

][][

21

21

21

21

222

111

nbynay

nbnxnanx

nbxnaxnny

nbxnaxnxlet

nnxnynxnxlet

nnxnynxnxlet

nnxny

linear system

][nx ][ny


Example 1.20

)1()()( txtxty

)()1()()1()()(

)()(

)1()()(

)()(

12

112

11

1

111

1

tyatxtxataxtaxty

taxtxlet

txtxty

txtxlet

)()( 1 tayty Non linear system

)(tx )(ty


Properties of linear system :

(1)

(2)


Time invarianceA system is said to be time invariant if a time delay or time advance of the input signal leads to an identical time shift in the output signal.

)(txTime invariant

system

)(ty

)( 0ttx

0t

)( 0tty

0t


Example 1.18

)(

)()(

tR

txty

0),()(

)(

)()(

)(

)(

)(

)()(

)()(

)(

)()(

0201

0

011

0122

012

11

tfortytty

ttR

ttxtybut

tR

ttx

tR

txty

ttxtx

tR

txty

Time varying system

)(tx )(ty


Invertibility

)(tx )(tx)(ty

A system is said to be Invertible if the input of the system can be recovered from the output.

H Hinv

)}({)( txHty )}({)( tyHtx inv

)}}({{)}({ txHHtyH invinv


Example 1.15

)()( 0ttxty )(tx )(ty

IHH

ttxH

ttxH

inv

inv

)(

)(

0

0

Inverse system

Example 1.16)(tx )(ty

)()( 2 txty


LINEAR TIME-INVARIANT (LTI) SYSTEMS:

A basic fact: If we know the response of an LTI system to some inputs, we actually know the response to many inputs

System identification


example

)(3)()()(2)( txtxtytyty

The system is governed by a linear ordinary differential equation (ODE)

Linear time invariant system

)(tx )(ty

)]()([])()([2])()([

)]()(2)([)]()(2)([

)](3)([)](3)([

)(3)(3)()()]()([3])()([

)(3)()()(2)(

)(3)()()(2)(

212121

222111

2211

21212121

22222

11111

tbytaytbytaytbytay

tytytybtytytya

txtxbtxtxa

txbtxatxbtxatbxtaxtbxtax

txtxtytyty

txtxtytyty

linearity


LTI System representations

1. Order-N Ordinary Differential equation2. Transfer function (Laplace transform)3. State equation (Finite order-1 differential equations) )

1. Ordinary Difference equation2. Transfer function (Z transform)3. State equation (Finite order-1 difference equations)

Continuous-time LTI system

Discrete-time LTI system


)()()()(

2

2

tutydt

tdyRC

dt

tydLC

constantsOrder-2 ordinary differential equation

Continuous-time LTI system

1

1

)(

)(

)()()()(

2

2

RCsLCssU

sY

sUsYsRCsYsYLCs

Transfer function

Linear system initial rest

1

12 RCsLCs

)(sU )(sY


)(1

0

)(

)(10

)(

)(

2

1

12

1 tutx

tx

tx

tx

LR

LC

dt

tdytx

tytxlet

)()(

)()(

2

1

)()(1

)()(

)()(

122

21

tutxLC

txL

Rtx

txtx

)(tx )(tx)(tu

A


System response: Output signals due to inputs and ICs.

1. The point of view of Mathematic:

2. The point of view of Engineer:

3. The point of view of control engineer:

Homogenous solution )(tyh Particular solution )(ty p＋

＋

＋ Zero-state response )(ty zsZero-input response )(ty zi

Natural response )(tyn Forced response )(ty f

Transient response Steady state response


1)0(

,1)0(,0,)(3)(

4)( 2

2

2

dt

dyytety

dt

tdy

dt

tyd t

Example: solve the following O.D.E

(1) Particular solution: )()]([ tutyp

tp

pp etydt

tdy

dt

tyd 22

2

)(3)(

4)(

tp etylet 2)(

tp

tp etyetythen 22' 4)(2)(

13)2(44 2222 tttt eeee

tp etyhavewe 2)(


(2) Homogenous solution: 0)]([ tyh0)(3)(4)( tytyty hhh

tth BeAety 3)(

)()()( tytyty hp have to satisfy I.C. 1)0(

,1)0( dt

dyy

1)0()0(1)0(

1)0()0(1)0(

ph

ph

yydt

dy

yyy

tth eety 3

2

1

2

5)(


(3) zero-input response: consider the original differential equation with no input.

1)0(,1)0(0,0)(3)(4)( zizizizizi yyttytyty

0,)( 321 teKeKty tt

zi

21

21

3)0(

)0(

KKy

KKy

zi

zi

1

2

2

1

K

K

0,2)( 3 teety ttzi

zero-input response


(4) zero-state response: consider the original differential equation but set all I.C.=0.

0)0(,0)0(0,)(3)(4)( 2 zizi

tzszszs yytetytyty

tttzs eeCeCty 23

21)(

023)0(

01)0(

21

21

CCy

CCy

zs

zs

2

12

1

2

1

C

C

tttzs eeety 23

2

1

2

1)(

zero-state response


(5) Laplace Method:

1)0(

,1)0(,0,)(3)(

4)( 2

2

2

dt

dyytety

dt

tdy

dt

tyd t

2

1)(3)0(4)(4)0()0()(2

ssYyssYysysYs

12

5

2

1

32

1

342

15

)(2

ssssss

ssY

ttt eeesYty

2

5

2

1)]([)( 231


Complex response

Zero state response Zero input response

Forced response(Particular solution)

Natural response(Homogeneous solution)

Steady state response Transient response

ttt eeety

2

5

2

1)( 23

tttzs eeety 23

2

1

2

1)( 0,2)( 3 teety tt

zi

ttt eeety

2

5

2

1)( 23

tp ety 2)( tt

h eety 3

2

1

2

5)(

Meiling chensignals & systems1 Lecture #2 Introduction to Systems.

Documents

Transcript of Meiling chensignals & systems1 Lecture #2 Introduction to Systems.