lecture #2
DESCRIPTION
Lecture #2. Introduction to Systems. system. A 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. Causality. - PowerPoint PPT PresentationTRANSCRIPT
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Lecture #2
Introduction to Systems
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systemA system is an entity that manipulates one or more signals to accomplish a function, thereby yielding new signals.
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Example of system
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System interconnection
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System properties
• Causality • Linearity• Time invariance• Invertibility
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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.
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Causal and noncausal systemExample: distinguish between causal and noncausal systems in the following:
t
)(tu
1 2
(1) Case I )()( tuty
t
)(ty
12 Noncausal system
0)(0)(1
tybuttutwhen
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(2) Case II
causal system
)()( tuty
t
)(ty
1 2
Delay system
(3) Case III )2()()( tututy
causal systemAt present past
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(4) Case IV )2()()( tututy
noncausal systemAt present future
(5) Case V
noncausal system
stepunitistuiftuty )()()( 2
t
)(ty
0)(0)(0
tybuttutwhen
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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
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Example 1.19
][][ nnxny
][][][][]}[][{][][][][
][][][][][][][][
][][
21
21
21
21
222
111
nbynaynbnxnanxnbxnaxnnynbxnaxnxlet
nnxnynxnxletnnxnynxnxlet
nnxny
linear system
][nx ][ny
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Example 1.20
)1()()( txtxty
)()1()()1()()(
)()()1()()(
)()(
12
112
11
1
111
1
tyatxtxataxtaxty
taxtxlettxtxtytxtxlet
)()( 1 tayty Non linear system
)(tx )(ty
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Properties of linear system :
(1)
(2)
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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
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Example 1.18
)()()(
tRtxty
0),()()()()(
)()(
)()()(
)()()()()(
0201
0
011
0122
012
11
tfortyttyttRttxtybut
tRttx
tRtxty
ttxtxtRtxty
Time varying system
)(tx )(ty
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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
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Example 1.15
)()( 0ttxty )(tx )(ty
IHH
ttxH
ttxH
inv
inv
)(
)(
0
0
Inverse system
Example 1.16)(tx )(ty
)()( 2 txty
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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
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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
tbytaytbytaytbytaytytytybtytytya
txtxbtxtxatxbtxatxbtxatbxtaxtbxtax
txtxtytytytxtxtytyty
linearity
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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
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)()()()(2
2
tutydt
tdyRCdt
tydLC
constantsOrder-2 ordinary differential equation
Continuous-time LTI system
11
)()(
)()()()(
2
2
RCsLCssUsY
sUsYsRCsYsYLCs
Transfer function
Linear system initial rest
11
2 RCsLCs)(sU )(sY
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)(10
)()(10
)()(
2
11
2
1 tutxtx
txtx
LR
LC
dttdytx
tytxlet)()(
)()(
2
1
)()(1)()(
)()(
122
21
tutxLC
txLRtx
txtx
)(tx )(tx)(tu
A
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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
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1)0(
,1)0(,0,)(3)(
4)( 2
2
2
dtdy
ytetydt
tdydt
tyd t
Example: solve the following O.D.E
(1) Particular solution: )()]([ tutyp
tp
pp etydt
tdydt
tyd 22
2
)(3)(
4)(
tp etylet 2)(
tp
tp etyetythen 22' 4)(2)(
13)2(44 2222 tttt eeee
tp etyhavewe 2)(
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(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
21
25
)(
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(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(
KKyKKy
zi
zi
12
2
1
KK
0,2)( 3 teety ttzi
zero-input response
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(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
CCyCCy
zs
zs
2121
2
1
C
C
tttzs eeety 23
21
21)(
zero-state response
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(5) Laplace Method:
1)0(
,1)0(,0,)(3)(
4)( 2
2
2
dtdy
ytetydt
tdydt
tyd t
21)(3)0(4)(4)0()0()(2
ssYyssYysysYs
125
21
321
342
15)( 2
ssssss
ssY
ttt eeesYty
25
21)]([)( 231
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Complex response
Zero state response Zero input response
Forced response(Particular solution)
Natural response(Homogeneous solution)
Steady state response Transient response
ttt eeety
25
21)( 23
tttzs eeety 23
21
21)( 0,2)( 3 teety tt
zi
ttt eeety
25
21)( 23
tp ety 2)( tt
h eety 3
21
25
)(