5. Robustness
RobustnessRobustness
• Stability robustness• Performance robustness• Performance robustness
Internal Stability of Feedback Systems
• The poles of the above four transfer functions must satisfy one of the following conditions:
• The poles of the four transfer functions must satisfy one of the following three conditions:
SISO SystemsSISO Systems
( ) poles of ( ) cancelled1 ( ) ( )
K s K sK s G s
⇒+1 ( ) ( )
( ) poles of ( ) cancelled1 ( ) ( )
K s G sG s G s
G K
+
⇒+1 ( ) ( ) consider only 1 ( ) ( ) 0G s K s
G s K s+
∴ + =
MIMO SystemsMIMO Systems
if ( ), ( ) stable,G s K s
{ }the system is internally stable if and only ifdet ( ) ( ) 0I G s K s+ ={ }det ( ) ( ) 0have negative real parts.
I G s K s+ =
SISO Nyquist Stability CiterionSISO Nyquist Stability Citerion
• A SISO is internally stable if all the zeros of 1+G(s)K(s) have negative real parts.
• Argument Principle: If a closed contour g pencircling(in the clockwise direction) the right half plane is mapped through g p pp g1+G(s)K(s), the resulting contour will encircle the origin times in c p zN N N= −gthe counter clockwise direction.
c p z
Example 5 1Example 5.1
1, 1 0c p zN N N= = ⇒ =
stablec p z
∴
Example 5 2Example 5.2
Gain & Phase Margins for SISOGain & Phase Margins for SISO
Unstructured UncertaintyUnstructured Uncertainty
• Additive unstructured uncertainty• Multiplicative unstructured • Multiplicative unstructured
uncertaintyF db k t i t• Feedback uncertainty
Additive UncertaintyAdditive Uncertainty
Multiplicative UncertaintyMultiplicative Uncertainty
Feedback UncertaintyFeedback Uncertainty
PerturbationsPerturbations
Example 5 5Example 5.5
• Input multiplicative uncertainty• 0 5% below 10rad/sec-accurate• 0.5% below 10rad/sec accurate• 50% above 1000rad/sec-inaccurate
Example 5 5Example 5.5
0.5( 10)s +( 1000)s +
0.5 100=1000.00520log100 40dB=40 / 2 20 /
0 100 : 0 005 0 5
dB dec dB dec
g gω
=+
= = → =0 : 0.005 0.50 1000
g gω = = → =+
Stability Robustness AnalysisStability Robustness Analysis
Stability Robustness AnalysisStability Robustness Analysis
• BIBO stable if the inverse is finiteBIBO stable if the inverse is finite
For all possible ( ) 1
if 1 the inverse is finite
s
N∞
Δ ≤
<if 1, the inverse is finited dy wN
∞<
( ) 0.1 10
( ) 0.2 5d d
d d
y w
y w
s N
s N
∞ ∞
∞ ∞
Δ = → ≠
Δ = → ≠
( ) 0.5 2d d
d dy ws N∞
∞ ∞Δ = → ≠
( ) 0.9 1.11111
( ) 1 1d dy ws N
N
∞ ∞Δ = → ≠
Δ
…
( ) 1 1d dy ws N
∞ ∞Δ = → ≠
Small Gain TheoremSmall Gain Theorem
Small Gain Theorem for SISO System
( ) 1sΔ ≤( ) 11 ( ) ( ) 0
ss G s
Δ ≤
−Δ =
Example 5 6Example 5.6
Example 5 6Example 5.6