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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

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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

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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 ) =

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

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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

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

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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 <α <1 1 + jωα T

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Lead/lag compensator

C ( jω ) = α Dutch Institute of Systems and Control

1 + jω T 1 + jωα T

Lag compensation Lag compensation: Increase the low frequency gain without affecting the phase in the cross-over region Example: PI-control: 1 + jω T C ( jω ) = k jω T

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Lead-lag compensation Lead-lag compensation: Joint use of lag compensation at low frequencies phase lead compensation at crossover

Lead, lag, and lead-lag compensation are always used in combination with gain adjustment

Dutch Institute of Systems and Control

Notch compensation (inverse) Notch filters: suppression of parasitic dynamics additional gain at specific frequencies Special form of general second order filter

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Notch compensation s2 s + 2 β1 + 1 2 u ω ω1 H = = 21 s s ε + 2β 2 +1 2 ω2 ω2

“Notch”-filter :ω1= ω2

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Notch compensation

ampl.

fase

β1 β2

0°

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Root locus method-1

Important stage of many designs: Fine tuning of gain compensator pole and zero locations Helpful approach: the root locus method (use rltool!)

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Root locus method-2 ( s − z1 )( s − z2 ) L ( s − zm ) N (s) L( s) = =k D(s) ( s − p1 )( s − p2 ) L ( s − pn )

−

L

Closed-loop characteristic polynomial χ (s) = D(s) + N (s) = ( s − p1 )( s − p2 )L ( s − pn ) + k ( s − z1 )( s − z2 )L ( s − zm )

Root locus method: Determine the loci of the roots of χ as the gain k varies

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Root locus method-3 χ ( s ) = ( s − p1 )( s − p2 ) L ( s − pn ) + k ( s − z1 )( s − z2 ) L ( s − zm )

Rules: For k = 0 the roots are the open-loop poles pi For k → ∝ a number m of the roots approach the open-loop zeros zi. The remaining roots approach ∝ The directions of the asymptotes of those roots that approach ∝ are given by the angles 2i + 1 π , i = 0, 1, L , n − m − 1 n−m Dutch Institute of Systems and Control

Root locus method-4 The asymptotes intersect on the real axis in the point (sum of open-loop poles) − (sum of open-loop zeros) n−m

Those sections of the real axis located to the left of an odd total number of open-loop poles and zeros on this axis belong to a locus The loci are symmetric with respect to the real axis ....

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Root locus method-5 L( s) =

k s ( s + 2)

L( s) =

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k ( s + 2) s ( s + 1)

L( s) =

k s ( s + 1)( s + 2)

Guillemin-Truxal method-1 r +

−

C

y P

Closed-loop transfer function: PC H= 1 + PC

Procedure: Specify H 1 H Solve the compensator from C = ⋅ P 1− H Dutch Institute of Systems and Control

Guillemin-Truxal method-2 Example: Choose H (s) =

am s m + am −1s m −1 + L + a0

s n + an −1s n −1 + L + am s m + am −1s m −1 + L + a0

This guarantees the system to be of type m + 1 How to choose the denominator polynomial? Well-known options: Butterworth polynomials Optimal ITAE polynomials Dutch Institute of Systems and Control

Butterworth and ITAE polynomials Butterworth polynomials Choose the n left-half plane poles on the unit circle so that together with their right-half plane mirror images they are uniformly distributed along the unit circle ITAE polynomials Place the poles so that ∞

∫ t e(t ) dt 0

is minimal, where e is the tracking error for a step input Dutch Institute of Systems and Control

Butterworth and ITAE

m=0 Dutch Institute of Systems and Control

Guillemin-Truxal method-3 Disadvantages of the method: Difficult to translate the specs into an unambiguous choice of H. Often experimentation with other design methods is needed to establish what may be achieved. In any case preparatory analysis is required to determine the order of the compensator and to make sure that it is proper The method often results in undesired pole-zero cancellation between the plant and the compensator

Dutch Institute of Systems and Control

Quantitative feedback theory QFT-1

Ingredients of QFT: For a number of selected frequencies, represent the uncertainty regions of the plant frequency response in the Nichols chart Specify tolerance bounds on the magnitude of T Shape the loop gain so that the tolerance bounds are never violated

Dutch Institute of Systems and Control

QFT-2 Example: Plant

P( s) =

g s 2 (1 + sθ )

Nominal parameter values: Parameter uncertainties:

g = 1, θ = 0 0.5 ≤ g ≤ 2, 0 ≤ θ ≤ 0.2

Tentative compensator: k + sTd C (s) = , k = 1, Td = 1.414, To = 0.1 1 + sTo Dutch Institute of Systems and Control

QFT-3 Responses of the nominal design Specs on |T | Frequency [rad/s]

Tolerance band [dB]

0.2 1 2 5 10

0.5 2 5 10 20

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Uncertainty regions Uncertainty regions for the nominal design The specs are not satisfied Additional requirement: The critical area may not be entered Dutch Institute of Systems and Control

QFT-4 Design method: Manipulate the compensator frequency reponse so that the loop gain satisfies the tolerance bounds avoids the critical region Preparatory step 1: For each selected frequency, determine the performance boundary Preparatory step 2: For each selectedfrequency, determine the robustness boundary

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Performance and robustness boundaries Nominal plant frequency response Robustness boundaries Performance boundaries

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QFT-5 Design step: Modify the loop gain such that for each selected frequency the corresponding point on the loop gain plot lies above and to the right of the corresponding boundary For the case at hand this may be accomplished by a lead compensator of the form 1 + sT1 C (s) = 1 + sT2

Step 1: Set T2 = 0, vary T1 Step 2: Keep T1 fixed, vary T2 Dutch Institute of Systems and Control

QFT-6

Eventual design: T1 = 3 T2 = 0.02

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QFT-7 Responses of the redesigned system

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Prefilter design-1 2½-degree-of-freedom configuration Closed-loop transfer function NF H= Fo Dcl

Co r

F X

Fo e −

Y X

+ +

For the present case: Dcl ( s ) = 0.02( s + 0.3815)( s + 2.7995)( s + 46.8190) N (s) = 1 Dutch Institute of Systems and Control

u

P

z

Prefilter design-2 Use the polynomial F to cancel the (slow) pole at –0.3815, and let ω o2 1 2 Fo ( s ) = 2 1 , , = = ω ζ o o 2 2 s + 2ζ oω o s + ω o Perturbed responses

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