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