Design Methods for Control Systems

Chapter 2 Classical Control System Design

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

Classical Classicaldesign designtechniques techniques

Classical Classicaldesign designspecifications specifications

Type Typekksystems systems

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

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 )

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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 )]

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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

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Lo (0) ≠ 0

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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

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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

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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 ) =

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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 ) =

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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

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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

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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

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 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)

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Classical design techniques Rules for loopshaping  Change open-loop L(s) to achieve certain closed-loop specs  first modify phase  then correct gain

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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