• The transfer function describes how the input (in the Laplace domain) is transferred to the output
EG1110 SIGNALS AND SYSTEMS
(in the Laplace domain).
• The transfer function does not give explicit information about initial conditions
Transfer functions
It gives an easy way to compute the forced response
• Laplace Transforms useful for describing systems
• G(s) = L−1 {g(t)}
– Differential equations → algebraic equations
where g(t) is the impulse response of G.
– Convolution → multiplication
• For LTI (linear-time-invariant) systems input-output relationship:
Y (s) = G(s)U (s)
G(s) =
n(s)
d(s)
− transfer function
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• Location of poles (in particular) are important:
Poles and zeros
• Transfer function G(s) contains information through its poles and zeros
– Stability
• Typical transfer function:
– Speed of response
– Transient characteristics
G(s) =
n(s)
d(s)
(1)
• Location of zeros:
– Affects nature of transient response, but not settling time
Values of s such that n(s) = 0 - zeros
– Does not affect stability!
Values of s such that d(s) = 0 - poles
– Affects how the system behaves with feedback
i.e. G(s) = 0 when n(s) = 0
and G(s) = ∞ when d(s) = 0
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General form of transfer function
Locations of poles and zeros
• Transfer functions are generally of the form
• For real systems, our transfer function always is...
....a ratio of two polynomials with real coefficients
(s + z1)(s + z2) . . .
G(s) =
=
(s + p1)(s + p2) . . .
Qm
i=1 (s + zi )
Qn
j=1 (s + pj )
(2)
i.e
where n ≥ m (the order of the numerator is less than or equal to that of the denominator).
• “Order” of transfer function given by the maximum power of s in the denominator.
• Using partial fractions we can always expand G(s) as
G(s) =
G(s) =
s+a
s2 + bs + c
Coefficients a, b, c are all real.
• This implies that our poles and zeros will always be
A2
A1
+
+ ... + C
s + p1 s + p2
– real
– complex conjugate
This is a consequence of our system being linear
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Effect of pole locations
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Example
jω
Motion of mass in a plane described by
stable,
oscillatory
f (t) = M ẍ(t) + B ẋ(t) + Kx(t)
unstable
stable,
damped
Taking Laplace transforms (and assuming zero initial conditions)
σ
X(s)
1
=
F (s) M s2 + Bs + K
stable,
oscillatory
unstable
Hence can write transfer function as
G(s) =
n(s)
1
=
d(s) M s2 + Bs + K
Hence system has
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• No zeros!
Poles and Stability
• Poles at M s2 + Bs + K = 0 i.e
s=
−B ±
√
B 2 − 4M K
2M
• Consider a first order system with transfer function
(3)
Behaviour of system varies, depending on values of system parameters:
• Poles are negative and real if
B 2 > 4M K,
B > 0,
B>
√
G(s) =
1
,
s+a
a>0
• Inverse Laplace transform (impulse response) therefore is
g(t) = e−at
B 2 − 4M K
Thus response to impulse gradually decays.
• One pole is postive and real, one pole is negative and real if
√
B 2 > 4M K, B > 0, B < B 2 − 4M K
• Consider a first order system with transfer function
• Poles are complex if
• Inverse Laplace transform (impulse response) therefore is
2
G(s) =
B < 4M K
1
,
s−a
a>0
g(t) = eat
• etc. etc.
Thus response to impulse gradually grows exponentially.
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• Thus we have the following characterisation of stability:
• More typically as G(s) is normally described as
IF G(s) HAS ALL POLES IN THE LEFT HALF COMPLEX PLANE, IT IS STABLE
G(s) =
n(s)
=
d(s)
Qm
i=1 (s
Qn
j=1 (s
+ zi )
+ pj )
• ....and instability
IF G(s) HAS AT LEAST ONE POLE IN THE RIGHT HALF COMPLEX PLANE, IT IS UN-
and we can write this as
G(s) =
A1
A2
+
+ ...
s + p1 s + p2
• Taking the inverse Laplace transform we have
g(t) = A1e−p1t + A2e−p2t + . . .
STABLE
• An interesting case occurs when G(s) has poles on the imaginary axis - these systems are called
marginally or critically stable.
Given an input of finite duration, the output will not diverge, nor will the former equilibrium point
be returned to.
• Therefore if a single pi is negative, the impulse response will be exponentially growing i.e system
will be unstable!
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– ζ > 1 - system is overdamped. Typically a slow response
Special case: second order systems
• Second order systems (no zeros) have convenient way to be written (called a standard form)
G(s) =
wn2
s2 + 2ζωns + ωn2
(4)
• Thus a second order system can be characterised by two parameters:
– ζ - damping ratio. Dictates oscillatory quality of response and stability
– ωn - mainly responsible for speed of response
• Parameter of particular interest is damping ratio, ζ:
– ζ < 0 -system unstable, pole in RHP.
– ζ = 0 -system is oscillator
– 0 < ζ < 1 - system exhibits decaying oscillations
– ζ = 1 - system is “critically” damped (fastest rise time without overshoot)
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