cauchys teorém

12 – Frequency‐domain CTRL design
Martin Hromcik
Automatic control 2012
9‐III‐12
Nyquist’s stability criterion
Automatické řízení - Kybernetika a robotika
Cauchy’s theorem of argument, for general function of complex variable ... .
Corollary: Nyquist test (for closed‐loop stability assessment, based on open‐loop frequency response characteristics). OL = L = b/a, CL = 1/(1+L) = a/(a+b)
• critical point ‐1 encircled clockwise by the OL Nyquist graph as many times as is # CL unstable poles ‐ # CL unstable zeros (=OL unstable poles)
• alternatiuvely: # CL unstable poles = ...
• counterclockwise ‐> counted with negative sign
Nyquist’s stability criterion:
CL stable # of encirclements of ‐1 counterclockwise = # unstable OL poles
Special case – OL stable:
CL stable Ù OL Nyquist does not encircle ‐1
Michael Šebek
Pr‐ARI‐12‐2012
2
Gain Margin
Automatické řízení - Kybernetika a robotika
ω180
L( jω)
K =3
K =2
K =1
o
ω180
o
•
o
phase crossover frequency ω180o :∠L( jω180o ) = −180
•
Gain Margin: GM = 1 L( jω180o )
•
•
that simple only for OL stable systems ... for unstable OL: same idea, more careful treatment (more more crossovers, GM’s for each of them, ...) Michael Šebek
3
Phase Margin
Automatické řízení - Kybernetika a robotika
L( jω)
ωC
ωC : L( jωC ) = 1 = 0 dB
•
Gain crossover frequency
•
Phase Margin:
•
Time‐delay equivalent formulation:
Michael Šebek
PM = 180o + ∠L( jωC )
θmax = PMrad ωc = (π 180) PMdeg ωc
ARI‐12‐2012
4
GM and PM
Automatické řízení - Kybernetika a robotika
• GM: robustness w.r.t. gain variations; typical requirement GM > 2 (=6dB)
• PM: robustness w.r.t. delays in the loop; typical requirement PM >30°
1−
ωc
ω180
GM [dB]
1
GM
1
GM
ω180
PM
PM
Michael Šebek
ωc ω180
ARI‐12‐2012
ωc
5
Reversed case ...
Automatické řízení - Kybernetika a robotika
<= ... stabilizing stable OL ...
1 GM
PM
L( jω)
... stabilizing unstable OL ... =>
PM
L( jω )
1 GM
Michael Šebek
ARI‐12‐2012
6
GM and more crossovers ...
Automatické řízení - Kybernetika a robotika
•
GM interval ( Kmin , Kmax )
• if CL stable for L(s) = L0 (s)
• then CL remains stable
for all
L(s) = kL0 (s)
kmin < k < kmax
(
•
0 ≤ kmin ≤ 1
)
1 ≤ kmax ≤ ∞
CL unstable (already) for
L(s) = kmin L0 (s)
L(s) = kmax L0 (s)
Michael Šebek
ARI‐12‐2012
7
PM and more crossovers
Automatické řízení - Kybernetika a robotika
(φmin , φmax )
•
PM interval
•
•
If CL stable for L(s) = L0 (s)
then CL remains stable for all L( s ) = e − jφ L0 ( s )
•
φmin < φ < φmax
and becomes unstable for and
•
L( s ) = e − jφmin L0 ( s )
L( s ) = e − jφmax L0 ( s )
Note that −π ≤ φmin ≤ 0
0 ≤ φmax ≤ π
Michael Šebek
ARI‐12‐2012
8
Co je špatného na klasických pojmech ?
Automatické řízení - Kybernetika a robotika
• GMs and PMs are most common indicators of controllers robustness in engineering practice. • Note however that certain GMand PM “safe” systems can be though destabilized easily by combined small gain and phase variation: L ( s ) = kL0 ( s )e − jφ
k ∈ [ k min , k max ], φ ∈ [φmin , φmax ]
• Solution (generalization of GM/PM concepts): distance (norm) of L from the critical point ‐1. Hinf robust control ... Michael Šebek
ARI‐12‐2012
9
CL frequency response
Automatické řízení - Kybernetika a robotika
OL vs. CL frequency response:
• CL:
L( s )
T ( s) =
1 + L( s )
• approximately
1 for L( s ) large
T (s) ≈
L( s ) for L( s ) small
• typically L( s )
• therefore
T (s) ≈
Michael Šebek
L( s )
T ( s)
ω << ωc
ω >> ωc
large for high freqs
small for low freqs
1
L( s )
for low freqs
for high freqs
ARI‐12‐2012
ωc
ω << ωc
ω >> ωc
10
Crossover frequency behavior
Automatické řízení - Kybernetika a robotika
•
• PM 90°
• PM 45°
•
L( s ) ≈ 1 ,
around ωc ,
depends on PM
T ( s)
∠L( jωc ) = −90o
T ( jωc ) = 1
∠L( jωc ) = −135o
T ( jωc ) ≈ 1.31
2 ≈ 0.707
for 2nd order system:
PM = arctan
Mp =
Mp
2ς
−2ς 2 + 1+ 4ς 4
1
2ς 1− ς 2
Michael Šebek
ARI‐12‐2012
11
Bandwidth and crossover frequency
Automatické řízení - Kybernetika a robotika
P
P
•
20
dB
M ,ω
bandwidth = those frequencies (of harmonic 10 M(ω) = T( jω)
signals) wich the CL is capable to track crossover
ωc
reasonably ... 1
0.7
typically, CL = low pass filter:
good tracking for ω small
ω
( T ≈ 1), but not for large ( T ≈ 1)
0.1
bandwidth
formal definition of classical control: bandwidth = frequency whee the output has half energy of input, y 2 = 0.5u 2
ω ω
ω
Hence
Y ( jω ) = ½ U ( jω ) = 0.707 U ( jω ) ⇔ Y ( jω ) dB = U ( jω ) dB − 3dB
bandwidth defines “responsivenes” of CL, speed ...
higher bandwidth – more aggressive control law – implies faster transients, but often higher sensitivity to parameter changes and noises. lower bandwidth typically opposite. ωc : T ( jωc ) = 1 (0 dB)
Crossover frequency:
Resonance peak:
M P = max{ T ( jω ) }, ω P : M P = T ( jω P )
abs
•
0
−3
BW
•
•
•
•
•
•
P
Michael Šebek
ARI‐12‐2012
c
12
− 20