Frequency Frequencyresponse responseplots plots Bode Bodeplots plots Nyquist Nyquistplots plots MM-and andN-circles N-circles Nichols Nicholsplots plots Dutch Institute of Systems and Control
Quantitative QuantitativeFeedback FeedbackTheory Theory Root Rootlocus locus
Steady-state errors-1 r
F
+
C
−
Tracking behavior: Assume
Response
P
tn r (t ) = 1(t ) n!
y
rˆ( s ) =
L( s ) y$ ( s ) = F ( s ) r$( s ) 11+4 L(2 s )44 4 3 H ( s)
Tracking error
εˆ ( s ) = rˆ( s ) − yˆ ( s ) = [1 − H ( s )] rˆ( s )
Dutch Institute of Systems and Control
1 s n +1
Steady-state errors-2 Steady-state tracking error ε ∞( n ) = lim ε (t ) = lim sεˆ ( s ) = lim t →∞
s →0
If F(s)=1 (no prefilter) then 1 − H ( s) = ε ∞( n )
1 1 + L( s)
= lim
1
s →0 s n [1 + L ( s )]
Dutch Institute of Systems and Control
s →0
1 − H (s) sn
Type k system A feedback system is of type k if L( s ) =
Lo ( s ) s
k
,
Then ε ∞( n ) = lim s →0
1 s n [1 + L( s )]
for 0 ≤ n < k 0 s for n = k = lim k = 1/ Lo (0) s →0 s + L ( s ) o ∞ for n > k k −n
Dutch Institute of Systems and Control
Lo (0) ≠ 0
Steady-state errors-3
Dutch Institute of Systems and Control
Integral control-1 Integral control: Lo ( s ) Design the closed-loop system such that L( s ) = s L (s) Type k control: L( s ) = o sk Results in good steady-state behavior Also: sk 1 = = O( s k ) for S (s) = 1 + L ( s ) s k + Lo ( s ) Dutch Institute of Systems and Control
s→0
Integral control-2 Type k control: S ( s ) = O( s k ) for Hence if
tn v (t ) = 1 (t ), n!
s→0
vˆ( s ) =
1 s n +1
then the steady-state error is zero if n < k (rejection) k = 1: Integral control: Rejection of constant disturbances k = 2: Type-2 control: Rejection of ramp disturbances Etc. Dutch Institute of Systems and Control
Integral control-3 Integral control: L( s) =
Lo ( s ) s
k
= P ( s )C ( s )
The loop has integrating action of order k “Natural” integrating action is present if the plant transfer function has one or several poles at 0 If no natural integrating action exists then the compensator needs to provide it
Dutch Institute of Systems and Control
Integral control-4 “Pure” integral control:
PI control:
PID control:
1 C ( s) = sTi
1 C ( s) = g 1 + sTi 1 C ( s ) = g sTd + 1 + sTi
Ziegler-Nichols tuning rules Dutch Institute of Systems and Control
Internal model principle Asymptotic tracking if model of disturbance is included in the compensator
Francis, D.A. and Wonham, W.M., (1975) The internal model principle for linear multivariable regulators, Applied Mathematics and Optimization, vol 2, pp. 170-194
Dutch Institute of Systems and Control
Frequency response plots s t o l p ls o h c i N
Bode plots
t s i u q Ny lots p Dutch Institute of Systems and Control
Bode plots-1 Bode plot: doubly logarithmic plot of |L(jω)| versus ω semi logarithmic plot of arg L(jω) versus ω
L( jω ) =
Dutch Institute of Systems and Control
ωo2 ( jω )2 + 2ζ oωo ( jω ) + ωo2
Bode plots-2 Helpful technique: By construction of the asymptotic Bode plots of elementary first- and second-order factors of the form jω + α
and
( jω ) 2 + 2ζ 0ω o2 ( jω ) + ω o2
The shape of the Bode plot of ( jω − z1 )( jω − z2 ) L ( jω − zm ) L ( jω ) = k ( jω − p1 )( jω − p2 ) L ( jω − pm )
may be sketched Dutch Institute of Systems and Control
Nyquist plots Nyquist plot: Locus of L(jω) in the complex plane with ω as parameter Contains less information than the Bode plot if ω is not marked along the locus L( jω ) =
Dutch Institute of Systems and Control
ωo2 ( jω )2 + 2ζ oωo ( jω ) + ωo2
M- and N-circles-1 r
+ −
L
y
Closed-loop transfer function: L H= =T 1+ L
M-circle: Locus of points z in the complex plane where z = M 1+ z
N-circle: Locus of points z in the complex plane where z =N arg 1+ z Dutch Institute of Systems and Control
M- and N-circles-2
Dutch Institute of Systems and Control
Nichols plots Nichols plot: Locus of L(jω) with ω as parameter in the log magnitude versus argument plane
L( jω ) =
ωo2 ( jω ) 2 + 2ζ oωo ( jω ) + ω o2
Dutch Institute of Systems and Control
Nichols chart: Nichols plot with M- and N-loci included
Fr eq do uen m cy ain
Ti
do me m ain
Classical design specifications Rise time, delay time, overshoot, settling time, steady-state error of the response to step reference and disturbance inputs; error constants
Dutch Institute of Systems and Control
Bandwidth, resonance peak, roll-on and roll-off of the closed-loop frequency response and sensitivity functions; stability margins
Classical design techniques
Lead, lag, and lag-lead compensation (loopshaping) (Root locus approach) (Guillemin-Truxal design procedure) Quantitative feedback theory QFT (robust loopshaping)
Dutch Institute of Systems and Control
Classical design techniques Rules for loopshaping Change open-loop L(s) to achieve certain closed-loop specs first modify phase then correct gain
Dutch Institute of Systems and Control
Lead compensation Lead compensation: Add extra phase in the cross-over region to improve the stability margins Typical compensator: “Phase-advance network” C ( jω ) = α