ELEN90055 Control Systems Worksheet 2
Semester 2
Laplace transforms – key concepts
1. Calculating Laplace transform and inverse Laplace transform: Method 1. Using the following equations:
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∞ −st L{y(t)}=Y(s)= e y(t)dt
−1 1 σ+j∞ st
Laplace transforms table.1
We might need to first perform a partial fraction expansion and then we find expres- sions that can be found in the Laplace transform table. There are two different ways to perform partial fraction expansion: (i) cross multiplication, (ii) the following direct formulas:
i. Partial fraction expansion for systems with distinct poles:
A transfer function G(s) of the following general form where m < n (which means G(s) is strictly proper)
{Y(s)}=y(t)=2πj
Method 2. (Preferred method) Using the Laplace transform table and the properties of
b sm+···+bs+b ∑bisi G(s)=m 1 0= i=0
ansn +···+a1s+a0 n
Bk = lim (s−αk)G(s). s→αk
1These tables can be found in almost all linear control textbooks. For example, see the attached tables from “Modern Control Engineering” by K. Ogata, 5th edition.
can be expanded as
k=1 k where the residues Bk are computed as
n Bk G(s) = ∑ s − α ,
ii. Partial fraction expansion (general formula):
If the transfer function has repeated poles, G(s) can be written as (we continue to assume m < n)
G(s)= i=0 . n
∏ (s−αk)rk k=1
Then the partial fraction expansion representation is
nrk Bkl G(s)= ∑∑(s−αk)l,
k=1 l=1 where the residues Bkl are computed as
r r −l (s−αk) kG(s) .
drk−l ds k
(rk − l)! s→αk
Tutorial problems
1. Find the Laplace transform of the following functions:
(a∗) (b∗) (c∗)
f(t)=1+2t+t2+δ(t)
f(t)=(t+3)2
f(t)=tsint
Hint: use the multiplication by time property of Laplace transform.
f(t)=sin2t f(t)=e−tsin2t
Youcancheckyouranswerstotheaboveproblemsusingthe‘laplace’MATLAB com- mand. For example, for part (d) you can use the following MATLAB code:
>> laplace(sin(t)2)
You may also need to use the ‘simplify’ command in MATLAB.
2. Find the time signal f(t) corresponding to the following Laplace transforms:
(a∗)F(s)= (b∗) F(s)=
(d∗) F(s)=s2+2s+3
3s+2 = 3s+2
s2 +4s+20 (s+2)2 +16
(s+1)3 (e)F(s)= 1 = 1
s4 −1 (s−1)(s+1)(s2 +1)
Hint: Perform partial fraction expansion.
(f) F(s) = s2−1 (s2 +1)2
Hint: First find the inverse Laplace transform of G(s) = s s2 +1
and then use the multi-
plication by time property of Laplace transform: d G(s) = −L {tg(t)}. Verify your ds
answer using MATLAB: >> syms s
>> ilaplace((s2 − 1)/(s2 + 1)2) 3
3. Solve the following differential equations using Laplace transforms: (a∗) y ̈(t)−2y ̇(t)+4y(t)=0; y(0)=1, y ̇(0)=2
(b) y ̈(t)+y ̇(t) = sint; y(0) = 1, y ̇(0) = 2
Hint: make sure the transfer function is strictly proper before doing partial fraction expansion.
4. Find limt→∞ f(t) for functions with the following Laplace transforms: (a) F(s)= 10
s(s+1)(s+2) (b) F(s)= 2
s(s−2)(s+2)
Hint: You may solve this problem using two different methods: (i) first find f(t) and
then find limt→∞ f(t). (ii) use the final value theorem, but make sure the conditions of the theorem are satisfied.
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