Signal and Systems Chapter 9: Laplace Transform Motivation and - - PowerPoint PPT Presentation

Signal and Systems

Chapter 9: Laplace Transform

• Motivation and Definition of the (Bilateral) Laplace

Transform

• Examples of Laplace Transforms and Their Regions of

Convergence (ROCs)

• Properties of ROCs
• Inverse Laplace Transforms
• Laplace Transform Properties
• The System Function of an LTI System
• Geometric Evaluation of Laplace Transforms and Frequency

Responses

Motivation for the Laplace Transform

 CT Fourier transform enables us to do a lot of things, e.g.

• Analyze frequency response of LTI systems
• Sampling
• Modulation

 Why do we need yet another transform?  One view of Laplace Transform is as an extension of the

Fourier transform to allow analysis of broader class of signals and systems

 In particular, Fourier transform cannot handle large (and

important) classes of signals and unstable systems, i.e. when

−∞ ∞ |𝑦(𝑢)| 𝑒𝑢 = ∞

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 2

Motivation for the Laplace Transform (continued)

 In many applications, we do need to deal with unstable

systems, e.g.

• Stabilizing an inverted pendulum
• Stabilizing an airplane or space shuttle
• Instability is desired in some applications, e.g. oscillators and

lasers

 How do we analyze such signals/systems? Recall from Lecture

#5, eigenfunction property of LTI systems:

 𝑓𝑡𝑢is an eigenfunction of any LTI system  𝑡 = 𝜏 + 𝑘𝜕 can be complex in general

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 3

The (Bilateral) Laplace Transform



𝑦(𝑢) ↔ 𝑌(𝑡) =

−∞ ∞ 𝑦(𝑢)𝑓−𝑡𝑢𝑒𝑢 = 𝑀{𝑦(𝑢)

 s = σ+ jω is a complex variable – Now we explore the full range

• f 𝑡

 Basic ideas: 1.

𝑌(𝑡) = 𝑌(𝜏 + 𝑘𝜕) =

−∞ ∞ 𝑦(𝑢)𝑓−𝜏𝑢]𝑓−𝑘𝜕𝑢𝑒𝑢 = 𝐺{𝑦(𝑢)𝑓−𝜏𝑢

2.

A critical issue in dealing with Laplace transform is convergence:—X(s) generally exists only for some values of s, located in what is called the region of convergence(ROC): 𝑆𝑃𝐷 = {𝑡 = 𝜏 + 𝑘𝜕 so that

−∞ ∞ |𝑦 𝑢 𝑓−𝜏𝑢|𝑒𝑢 < ∞

3.

If 𝑡 = 𝑘𝜕 is in the ROC (i.e. σ= 0), then 𝑌(𝑡)|𝑡=𝑘ω = 𝐺{𝑦(𝑢)

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 4

absolute integrability needed absolute integrability condition

Example #1:



) 𝑦1(𝑢) = 𝑓−𝑏𝑢𝑣(𝑢 (a – an arbitrary real or complex number)



𝑌1 𝑡 =

−∞ ∞ 𝑓−𝑏𝑢𝑣 𝑢 𝑓−𝑡𝑢𝑒𝑢 = ∞ 𝑓− 𝑡+𝑏 𝑢 = − 1 𝑡+𝑏 𝑓− 𝑡+𝑏 ∞ − 1

 This converges only if Re(s+a) > 0, i.e. Re(s) > -Re(a) 

𝑌1 𝑡 =

1 𝑡+𝑏 , ℜ𝑓 𝑡 > −ℜ𝑓{𝑏

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 5

Example #2:



) 𝑦2(𝑢) = −𝑓−𝑏𝑢𝑣(−𝑢 𝑌2(𝑡) = −

−∞ ∞ 𝑓−𝑏𝑢𝑣(−𝑢)𝑓−𝑡𝑢 𝑒𝑢

= −

−∞ 0 𝑓−(𝑡+𝑏)𝑢𝑒𝑢

=

1 𝑡+𝑏 𝑓−(𝑡+𝑏)𝑢|−∞

=

1 𝑡+𝑏 [1 − 𝑓(𝑡+𝑏)∞

 This converges only if Re(s+a) < 0, i.e. Re(s) < -Re(a) 

𝑌2(𝑡) =

1 𝑡+𝑏 , ℜ𝑓{𝑡} < −ℜ𝑓{𝑏 Same as 𝑌1(s), but different ROC

 Key Point (and key difference from FT): Need both X(s) and

ROC to uniquely determine x(t). No such an issue for FT.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 6

Graphical Visualization of the ROC

 Example1:

𝑌1(𝑡) =

1 𝑡+𝑏 , ℜ𝑓{𝑡} > −ℜ𝑓{𝑏

𝑦1(𝑢) = 𝑓−𝑏𝑢𝑣(𝑢) → 𝑠𝑗𝑕ℎ𝑢 − 𝑡𝑗𝑒𝑓𝑒

 Example2:

𝑌2(𝑡) =

1 𝑡+𝑏 , ℜ𝑓{𝑡} < −ℜ𝑓{𝑏

𝑦2(𝑢) = −𝑓−𝑏𝑢𝑣(−𝑢) → 𝑚𝑓𝑔𝑢 − 𝑡𝑗𝑒𝑓𝑒

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 7

Rational Transforms

 Many (but by no means all) Laplace transforms of interest to us

are rational functions of s (e.g., Examples #1 and #2; in general, impulse responses of LTI systems described by LCCDEs), where X(s) = N(s)/D(s), N(s),D(s) – polynomials in s

 Roots of N(s)= zeros of X(s)  Roots of D(s)= poles of X(s)  Any x(t) consisting of a linear combination of complex

exponentials for t > 0 and for t < 0 (e.g., as in Example #1 and #2) has a rational Laplace transform.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 8

Example #3



) 𝑦(𝑢) = 3𝑓2𝑢𝑣(𝑢) − 2𝑓−𝑢𝑣(𝑢 𝑌(𝑡) =

∞ 3𝑓2𝑢 − 2𝑓−𝑢]𝑓−𝑡𝑢𝑒𝑢

𝑌(𝑡) = 3

∞ 𝑓−(𝑡−2)𝑢𝑒𝑢 − 2 ∞ 𝑓−(𝑡+1)𝑢𝑒𝑢

𝑌(𝑡) =

3 𝑡−2 − 2 𝑡+1 = 𝑡+7 𝑡2−𝑡−2 , ℜ𝑓{𝑡} > 2

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 9

Laplace Transforms and ROCs

 Some signals do not have Laplace Transforms (have no ROC)  𝑏)𝑦(𝑢) = 𝐷𝑓−𝑢 for all t since

−∞ ∞ |𝑦(𝑢)𝑓−𝜏𝑢|𝑒𝑢 = ∞ for all 𝜏

 𝑐)𝑦(𝑢) = 𝑓𝑘𝜕0𝑢 for all t

) 𝐺𝑈: 𝑌(𝑘ω) = 2𝜌𝜀(𝜕 − 𝜕0

−∞ ∞ |𝑦(𝑢)𝑓−𝜏𝑢|𝑒𝑢 = −∞ ∞ 𝑓−𝜏𝑢 𝑒𝑢 = ∞ for all 𝜏

X(s) is defined only in ROC; we don’t allow impulses in LTs

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 10

Properties of the ROC

 The ROC can take on only a small number of different

forms

1.

1) The ROC consists of a collection of lines parallel to the jω-axis in the s-plane (i.e. the ROC only depends on σ).Why?

−∞ ∞ |𝑦(𝑢)𝑓−𝑡𝑢|𝑒𝑢 = −∞ ∞ |𝑦(𝑢)𝑓−𝜏𝑢| 𝑒𝑢 < ∞ depends

• nly on

} 𝜏 = ℜ𝑓{𝑡

2.

If X(s) is rational, then the ROC does not contain any

• poles. Why?

Poles are places where D(s) = 0 ⇒ X(s) = N(s)/D(s) = ∞ Not convergent.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 11

More Properties

 If x(t) is of finite duration and is absolutely integrable, then the ROC is

the entire s-plane.

 𝑌(𝑡) =

−∞ ∞

𝑦(𝑢)𝑓−𝑡𝑢𝑒𝑢 =

𝑈

1

𝑈

2 𝑦(𝑢)𝑓−𝑡𝑢𝑒𝑢 < ∞ 𝑗𝑔

𝑈

1

𝑈

2 |𝑦(𝑢)| 𝑒𝑢 < ∞ Book Chapter#: Section# Computer Engineering Department, Signal and Systems 12

ROC Properties that Depend on Which Side You Are On - I

 If x(t) is right-sided (i.e. if it is zero before some time), and

if Re(s) = 𝜏0is in the ROC, then all values of s for which Re(s) > 𝜏0 are also in the ROC.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 13

ROC is a right half plane (RHP)

ROC Properties that Depend on Which Side You Are On -II

 If x(t) is left-sided (i.e. if it is zero after some time), and if

Re(s) = 𝜏0 is in the ROC, then all values of s for which Re(s) < 𝜏0 are also in the ROC.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 14

ROC is a left half plane (LHP)

Still More ROC Properties

 If x(t) is two-sided and if the line Re(s) = 𝜏0 is in the ROC,

then the ROC consists of a strip in the s-plane

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 15

Example:

 𝑦(𝑢) = 𝑓−𝑐|𝑢|

Book Chapter#: Section# Computer Engineering Department, Signals and Systems 16

Intuition?

 Okay: multiply by

constant (𝑓𝜏𝑢) and will be integrable

 Looks bad: no 𝑓𝜏𝑢

will dampen both sides

Example (continued):

 𝑦(𝑢) = 𝑓𝑐𝑢𝑣(−𝑢) + 𝑓−𝑐𝑢𝑣(𝑢) ⇒ −

1 𝑡−𝑐 , ℜ𝑓{𝑡} < 𝑐 + 1 𝑡+𝑐 , ℜ𝑓{𝑡} > −𝑐

 Overlap if 𝑐 > 0 ⇒ 𝑌 𝑡 =

−2𝑐 𝑡2−𝑐2 , with ROC:

 What if b < 0? ⇒No overlap ⇒ No Laplace Transform

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 17

Properties, Properties

 If X(s) is rational, then its ROC is bounded by poles or extends to

• infinity. In addition, no poles of X(s) are contained in the ROC.

 Suppose X(s) is rational, then a)

If x(t) is right-sided, the ROC is to the right of the rightmost pole.

b)

If x(t) is left-sided, the ROC is to the left of the leftmost pole.

 If ROC of X(s) includes the jω-axis, then FT of x(t) exists.

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 18

Example:

 Three possible ROCs

x(t) is right-sided ROC: III No x(t) is left-sided ROC: I No x(t) extends for all time ROC: II Yes

Book Chapter#: Section# Computer Engineering Department, Signal and Systems 19

Fourier Transform exists?

Computer Engineering Department, Signal and Systems 20 Computer Engineering Department, Signal and Systems 20

Inverse Laplace Transform

} ) ( { , ) ( ) (

t st

e t x F ROC j s dt e t x s X



 

    

     

Fix σ ∈ ROC and apply the inverse Fourier



   

     

 

d e j X e t x

t j t

) ( 2 1 ) (



   

     

 

d e j X t x

t j ) (

) ( 2 1 ) (

But s = σ + jω (σ fixed)⇒ ds =jdω



   



    



j j stds

e s X j t x ) ( lim 2 1 ) (

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 21

Inverse Laplace Transforms Via Partial Fraction Expansion and Properties

Example: 3 5 , 3 2 2 1 ) 2 )( 1 ( 3 ) (            B A s B s A s s s s x Three possible ROC’s — corresponding to three different signals Recall sided

• left

) ( } { , 1 t u e a s e a s

at

      



1 , { } ( ) right-sided

at

e s a e u t s a



    

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 22

ROC I: — Left-sided signal.

               

 

t t u e e t u Be t u Ae t x

t t t t

as Diverges ) ( 3 5 3 2 ) ( ) ( ) (

2 2

ROC II: — Two-sided signal, has Fourier Transform. ROC III: — Right-sided signal.

            

 

t u(t) e e t u Be t u Ae t x

t t t t

as Diverges 3 5 3 2 ) ( ) ( ) (

2 2

Book Chapter 9 : Section 2

diverge not Does ) ( 3 5 ) ( 3 2 ) ( ) ( ) (

2 2

            

 

t u e t u e t u Be t u Ae t x

t t t t

 Many parallel properties of the CTFT, but for Laplace transforms we need

to determine implications for the ROC

 For example:

Book Chapter 9 : Section 2 Computer Engineering Department, Signal and Systems 23

Properties of Laplace Transforms

Linearity

) ( ) ( ) ( ) (

2 1 2 1

s bX s aX t bx t ax   

ROC at least the intersection of ROCs of X1(s) and X2(s) ROC can be bigger (due to pole-zero cancellation)

) ( ) ( ) ( Then and ) ( ) ( E.g.

2 1 2 1

       s X t bx t ax b a t x t x

⇒ ROC entire s-

Computer Engineering Department, Signal and Systems 24

Time Shift

) ( as ROC same ), ( ) ( s X s X e T t x

sT 

 

? 2 } { , 2 : Example

3

     s e s e s

T t t t u e s e s e

t sT

      

 

| ) ( 2 } { , 2

2

3   T

) 3 ( 2 } { , 2

) 3 ( 2 3

     

 

t u e s e s e

t s

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 25

Time-Domain Differentiation

 

   

 

       

 

j j j j st st

ds e s sX j dt t dx ds e s X j t x ) ( 2 1 ) ( , ) ( 2 1 ) (



ROC could be bigger than the ROC of X(s), if there is pole-zero

• cancellation. E.g.,

) (

• f

ROC the containing ROC with ), ( ) ( s X s sX dt t dx 

plane

• s

entire ROC 1 1 ) ( ) ( } { , 1 ) ( ) (          s s t dt t dx s e s t u t x 

s-Domain Differentiation

X(s) ds s dX t tx as ROC same with , ) ( ) (  

(Derivation is similar to ) d s dt 

a s e a s a s ds d t u te at              



} { , ) ( 1 1 ) ( E.g.

2

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 26

Convolution Property

x(t)

y(t)=h(t)＊x(t) h(t)

For Then

) ( ) ( ) ( ) ( ) ( ), ( ) ( ), ( ) ( s X s H s Y s H t h s Y t y s X t x     

• ROC of Y(s) = H(s)X(s): at least the overlap of the ROCs of H(s) & X(s)
• ROC could be empty if there is no overlap between the two ROCs

E.g.

x(t)=etu(t),and h(t)=-e-tu(-t)

• ROC could be larger than the overlap of the two.

) ( ) ( * ) ( t t h t x  

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 27

The System Function of an LTI System

x(t)

y(t) h(t)

function system the ) ( ) (   s H t h

The system function characterizes the system ⇓ System properties correspond to properties of H(s) and its ROC A first example:



  

   dt t h ) ( stable is System



axis the includes

• f

ROC jω H(s)

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 28

Geometric Evaluation of Rational Laplace Transforms

Example #1: A first-order zero

a s s X   ) (

1

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 29

Example #2: A first-order pole

) ( 1 1 ) (

1 2

s X a s s X   

) ( ) ( ) | ) ( | log log (or ) ( 1 ) (

1 2 1 2 1 2

s X s X s X (s)| |X s X s X         Still reason with vector, but remember to "invert" for poles Example #3: A higher-order rational Laplace transform ) ( ) ( ) (

1 1 j P j i R i

s s M s X       

 

j P j i R i

s s M s X       

  1 1

) (

 

 

        

R i p j j i

s s M s X

1 1

) ( ) ( ) (  

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 30

First-Order System

Graphical evaluation of H(jω)

    1 } { , / 1 / 1 1 1 ) (        s e s s s H ) ( 1 ) (

/

t u e t h

t 







 

) ( 1 ) (

/

t u e t s

t  

 

       / 1 1 1 / 1 / 1 ) (      j j j H

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 31

Bode Plot of the First-Order System

                            



                        1/ 2 / 1/ 4 / ) ( tan ) ( 1/ / 1 1/ 2 / 1 1 ) / 1 ( / 1 ) ( 1/ j 1/ ) (

1 2 2

j H j H j H

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 32

Second-Order System

e(poles) e{s} ROC 2 ) (

2 2 2

     

n n n

s s s H    1   



complex poles — Underdamped

1  

1  

 

double pole at s = −ωn — Critically damped 2 poles on negative real axis — Overdamped

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 33

Demo Pole-zero diagrams, frequency response, and step response of first-order and second-order CT causal systems

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 34

Bode Plot of a Second-Order System

Top is flat when ζ= 1/√2 = 0.707 ⇒a LPF for ω < ωn

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 35

Unit-Impulse and Unit-Step Response of a Second- Order System

No oscillations when ζ ≥ 1 ⇒ Critically (=) and

• ver (>) damped.

Book Chapter 9 : Section 2

Computer Engineering Department, Signal and Systems 36

First-Order All-Pass System

• 1. Two vectors have

the same lengths 2.

) ( } { , ) (        a a s e a s a s s H

                 a a j H              ~ 2 / 2 ) ( ) (

2 2 2 2 1

Book Chapter 9 : Section 2

CT System Function Properties

x(t)

y(t) h(t)

) ( ) ( ) ( s X s H s Y 

H(s) = “system function” 1) System is stable   ROC of H(s) includes jω axis 2) Causality => h(t) right-sided signal => ROC of H(s) is a right-half plane



  

  dt t h ) (

Question: If the ROC of H(s) is a right-half plane, is the system causal? E.x.

sided

• right

) ( 1 } { , 1 ) ( t h s e s e s H

sT

     

T t t t T t t sT

t u e s L s e L t h

      

                 | ) ( 1 1 1 ) (

1 1

at ) (

) (

   

 

t T t u e

T t

Non-causal

37

Properties of CT Rational System Functions

a) However, if H(s) is rational, then The system is causal ⇔ The ROC of H(s) is to the right of the rightmost pole

b) If H(s) is rational and is the system function of a causal system, then

The system is stable ⇔ jω-axis is in ROC ⇔ all poles are in the LHP

38

Checking if All Poles Are In the Left-Half Plane

) ( ) ( ) ( s D s N s H 

Poles are the roots of D(s)=sn+an-1sn-1+…+a1s+a0 Method #1: Calculate all the roots and see! Method #2: Routh-Hurwitz – Without having to solve for roots.

 

2 1 1 2 1 2 2 3 1 1 2

and , ,

• rder
• Third

,

• rder
• Second
• rder
• First

LHP the in are roots all that so Condition Polynomial a a a a a a a s a s a s a a a s a s a a s             

39

Initial- and Final-Value Theorems

If x(t) = 0 for t < 0 and there are no impulses or higher order discontinuities at the origin, then

) ( lim ) ( s sX x

s   

 ) ( lim ) ( s sX x

s

 

Initial value Final value

If x(t) = 0 for t < 0 and x(t) has a finite limit as t → ∞, then

40

Applications of the Initial- and Final-Value Theorem

For n-order of polynomial N(s), d – order of polynomial D(s)

• Initial value：
• Final value：

              

  

1 1 finite 1 ) ( lim ) ( n d n d n d s sX x

s

? ) ( 1 1 ) ( E.g.   



x s s X

) ( ) ( ) ( s D s N s X 

at poles No ) ( lim ) ( lim ) ( If        

 

s s X s sX x

s s

41

LTI Systems Described by LCCDEs

 

 



N k M k k k k k k k

dt t x d b dt t y d a ) ( ) (

 

 

  

N k M k k k k k k k k

s X s b s Y s a s dt d s dt d ) ( ) ( , property ation differenti

• f

use Repeated ：

) ( where ) ( ) ( ) (

Rational

    

 

 

  

N k k k M k k k

s a s b s H s X s H s Y

roots of numerator ⇒ zeros roots of denominator ⇒ poles

ROC =? Depends on: 1) Locations of all poles. 2) Boundary conditions, i.e. right-, left-, two-sided signals.

42

System Function Algebra

Example: A basic feedback system consisting of causal blocks

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

2

s Y s H s X s Z s X s E    

 

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

2 1 1

s Y s H s X s H s E s H s Y   

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

2 1 1

s H s H s H s X s Y s H    

More on this later in feedback

ROC: Determined by the roots of 1+H1(s)H2(s), instead of H1(s)

43

Block Diagram for Causal LTI Systems with Rational System Functions Example：

) ( ) ( ) ( s X s H s Y 

2 3 6 4 2 ) (

2 2

     s s s s s H

) 6 4 2 ( 2 3 1

2 2

           s s s s

— Can be viewed as cascade of two systems.

) ( 2 3 1 ) ( Define

2

s X s s s W    ：

) ( 2 ) ( 3 ) ( ) (

• r

rest at initially ), ( ) ( 2 ) ( 3 ) (

2 2 2 2

t w dt t dw t x dt t w d t x t w dt t dw dt t w d      

) ( 6 ) ( 4 ) ( 2 ) ( ) ( ) 6 4 2 ( ) ( Similarly

2 2 2

t w dt t dw dt t w d t y s W s s s Y      

44

Example (continued) Instead of

) ( 6 4 2 2 3 1 ) ( ) (

2 2

t y s s s s t x s H

  

   

We can construct H(s) using:

) ( 6 ) ( 4 ) ( 2 ) ( ) ( 2 ) ( 3 ) ( ) (

2 2 2 2

t w dt t dw dt t w d t y t w dt t dw t x dt t w d      

Notation: 1/s—an integrator

45

Note also that Cascade 1 ) 1 ( 2 2 3 1 3 2 1 2 ) (                                    s s s s s s s ) (s s H

connection parallel 1 8 2 6 2       s s

PFE

Lesson to be learned：There are many different ways to construct a system that performs a certain function.

46

The Unilateral Laplace Transform (The preferred tool to analyze causal CT systems described by LCCDEs with initial conditions)

Note: 1) If x(t) = 0 for t < 0, 2) Unilateral LT of x(t) = Bilateral LT of x(t)u(t-) 3) For example, if h(t) is the impulse response of a causal LTI system then, 4) Convolution property: If x1(t) = x2(t) = 0 for t < 0, Same as Bilateral Laplace transform

 



  

  ) ( ) ( ) ( t x dt e t x s

st

UL X

) ( ) ( s s X X 

) ( ) ( s s H H 

 

) ( ) ( ) ( ) (

2 1 2 1

s x t x t x X X UL  

47

Differentiation Property for Unilateral Laplace Transform ) ( ) ( ) ( ) ( ) (



  



x s s dt t dx s t x X X

Initial condition! Derivation： Note：



 



       ) ( ) ( dt e dt t dx dt t dx

st

UL

   

  

 

) (

| ) ( ) (

st s st

e t x dt e t x s       

X

) ( ) (



  x s sX

integration by parts ∫f．dg=fg－∫g．df

) ( ' ) ( ) ( ) ( ' ) ( )) ( ) ( ( ) ( ) (

2 2 2    

          

 

x sx s s x dt t dx x s s s dt t dx dt d dt t x d X UL X       

48

Use of ULTs to Solve Differentiation Equations with Initial Conditions

Example： Take ULT：

) ( ) ( , ) ( ' , ) ( ) ( ) ( 2 ) ( 3 ) (

2 2

t u t x y y t x t y dt t dy dt t y d         

 

s s s s s s s

dt dy dt y d

         

               

) ( 2 ) ) ( ( 3 ) (

2 2

2

Y Y Y

UL UL

           

                    ZSR s s s ZIR s s s s s s ) 2 )( 1 ( ) 2 )( 1 ( ) 2 )( 1 ( ) 3 ( ) (              Y

ZIR — Response for zero input x(t)=0 ZSR — Response for zero state, β=γ=0, initially at rest

49

Example (continued)

• Response for LTI system initially at rest (β = γ = 0 )
• Response to initial conditions alone (α = 0). For example:

) ( ) 2 )( 1 ( 1 ) ( ) ( ) ( s H s s s s s        Y H

) , 1 ( ) ( ' , 1 ) ( ), input no ( ) (     

 

  y y t x

, 2 ) ( 2 1 1 2 ) 2 )( 1 ( 3 ) (

2

           

 

t e e t y s s s s s s

t t

Y

50