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Dutch Institute of Systems and Control

Overview Ch. Ch. 2. 2. Classical Classical control control system system design design Introduction Introduction

Classical Classicaldesign designtechniques techniques

Steady-state Steady-stateerrors errors

Classical Classicaldesign designspecifications specifications

Type Typekksystems systems

Lead, Lead,lag, lag,lead-lag lead-lagcompensation compensation

Integral Integralcontrol control

Guillemin-Truxal Guillemin-Truxalmethod method

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

1 + jω T , 0

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