EG1110 SIGNALS AND SYSTEMS
Useful aspects of feedback
• Can provide accurate tracking (i.e. in face of disturbances etc)
- minimises error
Feedback and stability margins
• Can stabilise unstable systems
Revision
Problematic aspects of feedback
• Feedback affects stability of a system
• Can de-stabilise stable systems (!)
– Can improve or degrade open-loop stability
• Can be sensitive to parameters of open-loop systems
• Closed-loop stability not trivially related to open-loop stability
– Systems with same open loop poles may have different closed loop poles.
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Illustrative example
Case 1: Perfectly known system
u(s)
Let α = 1. Time constant of open-loop system (no feedback) is
+
_
G(s)
y(s)
T = 1/0.1 = 10seconds
i.e. system is slow to respond (due to step, impulse....)
y(s) = G(s)[u(s) − y(s)]
G(s)
=
u(s)
1 + G(s)
Using feedback though, closed loop transfer function is
α
1
G(s)
=
=
1 + G(s) s + α + 0.1 s + 1.1
Thus time constant of closed loop system is
Say G(s) is given by
G(s) =
α
s + 0.1
T = 1/1.1 ≈ 0.9seconds
We consider two cases
i.e. much faster than open-loop - response improved
(feedback good)
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Case 2: Imperfectly known system
Notes
Let α = 1 ± δ, where δ is an uncertain parameter rangining between 0 and 1.2.
• In open-loop any value of δ would always result in a stable system
Then in closed loop
G(s)
α
1±δ
=
=
1 + G(s) s + α + 0.1 s + 1.1 ± δ
Imagine δ = 1.2, then a possible closed-loop transfer function would be
G(s) =
– δ would have no affect on open-loop poles.
– ⇒ uncertainty has no affect on stability
• In closed-loop system, uncertainty affects stability
−0.2
s − 0.1
i.e we have made the closed-loop system unstable by feedback
(feedback bad)
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Stability Margins
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Consider closed-loop transfer function
• Suppose we have a closed-loop system which is stable
G(s)
1 + G(s)
– How do we ensure that it is insensitive to ‘modelling errors’?
– Are there ways to measure how sensitive the closed-loop is to modelling errors?
• Motivates the use of robust margins.
• Boundary of stability is when G(s) = −1. At this point system becomes unstable.
• In frequency domain
G(jω) = −1
⇒
|G(jω)|ejφ = −1
where


φ = arg[G(jω)] = tan−1 
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
=(G(jω)) 

<(G(jω))
For |G(jω)|ejφ = −1, this requires
Gain margin
– |G(jω)| = 1
– φ(jω) = −180
The amount of extra open-loop gain needed to make the closed-loop
system unstable when phase is -180 degrees
o
• Thus we can define a stability margin in terms of “how far” the system’s frequency response is
• Let ω180 be the frequency at which φ is -180 degrees
i.e. φ(ω180 ) = −180
from the point -1.
• Closed-loop stability margins determined by open-loop frequency response.
• Then, typically, |G(jω180 )| = α < 1
• This implies we would need an extra open-loop gain of 1/α to make the closed loop system
unstable.
• Thus the gain margin is
GM =
1
1
=
α |G(jω180 )|
(or −20 log10 |G(jω180 )| in dB)
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Phase margin
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Notes
The amount of extra open-loop phase needed to make the closed-loop
system unstable when the gain unity
• Let ωc be the frequency at which |G(jω)| = 1
• The foregoing descriptions of gain and phase margins are simplified.
• When systems have multiple cross over points, their definition becomes a little more complex
• Unstable open-loop systems also have gain reduction margins
i.e. |G(jωc)| = 1
• Gain/phase margins will be described in more detail in Second Year Control
• Then at this frequency (typically), the phase would be such that
φ(ωc) < −180o
• Thus it follows that
−180 = φ(ωc) − P M
⇒ −P M = −180 − φ(ωc)
⇒ P M = 180 + φ(ωc)
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