Specifications and closed-loop control

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Specificationsand closed-loop control


Specifications:Basic terminology

Resolution: value of smallest resolvable displacement (step) of the component; it differs from the resolution of the measurementAccuracy (error): difference between the (mean) actual motion and the ideal displacement of the deviceRepeatability (precision): range of deviations of the actual displacement for the same nominal (error free) displacement


Specifications:Basic terminology

Resolution: largest of the smallest steps the device can make (smallest programmable step) during point-to-point motionAccuracy (error): maximum error between any 2 points in the work volume, i.e. difference between the mean value of the reached positions and the nominal positionRepeatability (precision): error between a number of successive attempts to move to the same position


How to reach specifications

Precise actuation only?

Move until you think you have reached the target position (called „open-loop control“)

This works only sometimes.For precision system even more seldom.

SystemActuation Position, force, etc.


Problems with open-loop control:example 1

The actuation is either not accurate (but you could calibrate) or not precise.

A

D

12 bits actuation, position with higher resolution needed

Example: The resolution of your actuation is not sufficient



The system may be affected by external disturbances

Affected byweight of mass(causes unwanteddisplacement)

Not affected

Example: The mass spring system at the right is affected by the weight



The system is slightly different from what imagined

Example: The spring constant k of the mass spring system is different from what assumed. The same force causes a different displacement.

Example: The spring constant k varies with the displacement. Increasing the force does not increase the displacement proportionally.



You may want to reach the end position faster (dynamics is not satisfactory)

Example: When system shows slightly damped oscillations, like in flexure systems.



The system is unstable: the desired position can not be achieved

Example: The magnetically levitated ball is attracted by the magnet if too close to magnet, falls down if too far from the magnet.


Closed-loop control

Use information from measurements!

•Build error e between reference r (desired value) and measured value y•Use this error somehow to influence the actuation u

controller processue yr

Many problems can be solved withclosed-loop control


Problems solved withclosed-loop control: example 1

The actuation is either not accurate or not precise.

Example: It is possible to obtain a high resolution reference voltage using high resolution AD converters, even with low resolution DA converters.

Low passfilter

DA

ADcontroller

+-

Highresolutionvoltage!



The system may be affected by external disturbances

Affected byweight of massNo unwanteddisplacement

Not affected

Example: The extra displacement caused by the weight can be perfectly compensated in any position



The system is slightly different from what imagined

Example: The closed-loop control can take care of varying of uncertain spring constant values k



You may want to reach the end position faster (dynamics is not satisfactory)

Example: Closed-loop control can effectively damp oscillations, even change bandwith.



The open-loop system is unstable

Example: The magnetically levitated ball can be kept floating at a given position


Limitations inclosed-loop control systems

However, not everything is possible

Example: closed-loop control cannot cure deficiencies in the system repeatability or in the measurement accuracy

Example: With given measurement noise, the achievable bandwidth is limited (trade-off)

Example: Stability and performance are competing (trade-off)


Closed-loopControl Systems


Plant with controller

r: reference signal

u: command signal

w: measured signal

y: output signal

d: disturbance

Controller Plantyur

w

d


Control problem

Given Plant G(s), find controller C(s) such thatclosed-loop system with transfer function

satisfies behavior specifications


Feedforward scheme

Closed-loop system with transfer function


Advantages ofFeedforward scheme

• Poles still determined by G(s) and C(s) Closed-loop bandwidth may be reduced, which improves noise rejection (see later)

• F2(s) already generates the necessary actuation, C(s) is responsible only for deviations better response, less problems, better accuracy

• F1(s) „conditions“ the reference by filtering it (same must be done by F2(s)) Sharp changes in R(s) don‘t cause abrupt changes in actuation


Measurable disturbancecompensation scheme


Advantages ofCompensation scheme

• Closed-loop behavior still determined by Gyu(s) and C(s) Closed-loop bandwidth may be reduced, which improves noise rejection (see later)

• H(s) already generates the necessary actuation, C(s) is responsible only for deviations better response to disturbance changes, less problems, better accuracy


Trade-offs


General issues


Setup

Objective: output y should follow desired or commanded signal r while not using too much control effort u!

disturbances

Measurement noise

Plant uncertainty

Robust performance=command following+disturbance rejection,despite of plant variations


Major transfer functions

• Loop gainL(s)=P(s).C(s)

• Sensitivity functionS(s)=1/(1+L(s))=1/(1+P(s).C(s))

• Complementary sensitivity functionT (s)=L(s)/(1+L(s))=P(s).C(s)/(1+P(s).C(s))


Nominal signal response (1)

Output

y(s) = S(s)[d0(s) + P(s)di(s)]+ T(s)[r(s) à ñ(s)]

S(s) reduces the influence of disturbances

T(s) non–zero: measurement noise affects the output (unlike open-loop)



Performance error

eP(s)

eP(s) = S(s)[r(s) à do(s) à P(s)di(s)]à T(s)ñ(s)

+ -

S(s) small with respect to reference and disturbances

T(s) small with respect to noise


V(s) = à T(s)di(s) + P(s)T(s)[r(s) à ñ(s) à do(s)]


Command input(actuation)

1. V(s) increases as P(s) decreases in frequency (usual case), unless T(s) also decreases or r(s), do(s) and (s) also decrease

2. As (s) increases in frequency, T(s) must decrease bandwidth limitation!


Limitation 1

• high-frequency measurement noise (t) or disturbances di(t) or do(t) and

• Limitations on the desired control magnitude v(t)

Limitation of bandwidth on theclosed-loop system


Influence of plant variations


Sensitivity to plant variations

Plant variations

causes variations

|S(j)|<1 feedback decreases effect

|S(j)|>1 feedback increases effect


Robust stability

T(s)

Suppose we know (uncertainty is limited)

Then stability requires

(justification e.g. with Nyquist diagram, not presented here)

jT(j ! )j < WT(! )1

jÉ (j ! )j < WT(! )


Summary of specifications


Specifications

So, because of plant variations...

• S(j) should be small ( ) maxizimize benefits (disturbance rejection, command following, performance)

• T(j) should be small ( ) minimize costs (noise amplification, instability, control effort, ...)

jS(j ! )j < WS(! )1

jT(j ! )j < WT(! )1


Limitation 2

But

So, specifications are tradeoffs!

S(j) and T(j) can’t be both small at same

S(s) +T(s) = 1+C(s)P(s)1 + 1+C(s)P(s)

C(s)P(s) = 1


Limitation 2

However

Disturbance rejection and command following arimportant at low frequencies S(j) small at low frequencies

Uncertainty and measurement noise important ahigh frequencies T(j) small at high frequencies


Open-loop specifications


Specifications on Loop gain (1)

For low frequency

jL (j ! ) ý 1 =) S(j ! ) = 1+L(j ! )1 ù L (j ! )

1 = L(j ! )à 1

jS(j ! )j < Ws(j ! )1 =) jL (j ! )j > Ws(j ! )


jT(j ! )j < WT(j ! )1 =) jL (j ! )j < Wà 1

T (j ! )


For high frequency

jL (j ! )j ü 1 =) T(j ! ) = 1+L(j ! )L (j ! ) ù L(j ! )



At crossover

jL (j ! )j ù 1 ^ \ L(j ! ) ù æù

jS(j ! )j ý 1 ^ jT(j ! )j ý 1

Undesirable

Condition on phase


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0

Pha

se (

deg)

Summary of specifications

0dB=1


Bode Gain-Phase Relation

\ L (j ! 0) à \ L (0) = ù1R

à 11

d÷dlogjL j logj! à ! 0

! +! 0jd÷

If no pole and right half-plane zeros

where

÷= log ! 0!

ð ñ

Phase completely determined by (weighted)Integral of derivative of log amplitude

In fact20n db/decade decrease \ L (j ! 0) ù à n2

ù


Bode Gain-Phase Relation (2):crossover slope

As angle at crossover must be between – and

Bode rate decrease must be 20 db/decade at crossover

20dB/decade


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Bode Gain-Phase Relation (3)crossover slope

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! High decrease rate


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same high decrease ratespecs 100dB higher

Bode Gain-Phase Relation (4)crossover slope, narrower specs


Bode sensitivity integral


Bode Sensitivity integral

If relative degree at least 2 (2 more poles than zeroes)and no right-half plane poles (stable plant)

Tradeoff between sensitivity properties in different frequency ranges

R01logjS(j ! )jd! = 0


Bode Sensitivity integral

S = 1+L(s)1 for L(s) = s(s+1)

1

Green surfacec exactly compensates red one


Bode sensitivity integral:different designs

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-60

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Magnitu

de (

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)

Bode Diagram

Frequency (rad/sec)


0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-40

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de (

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)

Bode Diagram

Frequency (rad/sec)

Bode sensitivity integral:bandwidth limitations


Bode Sensitivity integralunstable plant

If relative degree at least 2 (2 more poles than zeroes)

Tradeoff between sensitivity properties in different frequency ranges

Sensitivity higher because of effort (cost) in stabilizing systems

R01logjS(j ! )jd! = ù

Pi=1Np Re(pi)


Quantization and Noise


• Arithmetic operations

• Coefficient quantization

• Converter resolution

• Other noise sources

Sources of quantization errors


Quantizationin arithmetic operations


Fixed-point arithmetic

N =P

j=àmnà 1 bj á2j = (bnà 1. . .b0 ï bà 1bà 2. . .bàm)2 bj 2 f0;1g

C+1 bit normalized representation with fictitious binary point

with

Example


Quantization infixed-point arithmetic

Quantization error introduced in multiplication,not in addition.

0.011 X 0.010 =0.001 (C=3, truncation)

Product quantized through rounding or truncation

Attention to overflow with addition (not with multiplication, because of normalization)

0.100 + 0.100 =?? overflow


Floating-point arithmetic

N =M â 2E Mantissa M and exponent E in fixed-point, normalized representation (0.5 M 1)


Quantization error introduced in multiplication AND in addition.Product and addition quantized through rounding or truncationof the Mantissa M (not of exponent E).

0.11X2000 + 0.10X2111 = 0.10X2111

0.10X2000 x 0.11X2111 = 0.01X2111

Attention to overflow and underflow

Quantization infloating-point arithmetic

(M 1â 2E 1)(M 2â 2E 2) = (M 1â M 2) â 2(E 1+E 2)

(M 1â 2E 1) + (M 2â 2E 2) = (M 1â 2E 1) + (M 02â 2E 1) = (M 1+M 0

2) â 2E 1


Coefficient quantization


Roundoff of parameters

H(z) = z4+2:689z3+3:3774z2+2:3823z+0:6942z3+1:584z2+1:2769z+0:5642

H(z) = z4+2:625z3+3:375z2+2:375z+0:625z3+1:5z2+1:25z+0:5

Transfer function with complex poles at 0.4965±j0.8663

Coefficient truncation with 3 bits after the comma:

Transfer function with two poles at 1.08!


Converter quantization


A/D converter quantization

E(e2) = 12q2

E(e2) = 3q2

Noise modeled assumed as uniformly distributed over the values

Rounding with step

Truncation with step q= 2à nbits

q= 2à nbits


Other noise sources

•Electrical noise in signals (from supply, thermal noise, …)

•Interferences

•Non-ideal behaviour of components (e.g. settling time of converters)

•Many, many sources


Propagation ofquantization noise


Propagation of quantization error

+

-x

xq yu


Stochastic quantization error

Example


General noise propagation

• Consider RMS value of noise

• Transfer function from noise to point of interest determines the influence of the noise

• In presence of multiple noise sources, total variance at output is sum of variance from single sources

= i2


Sample rate selection


Fast/slow sampling?

High frequency:• Better performance• cost of design-to-cost avoided

Low frequency:• lower cost components• more time for complex algorithms

Lowest frequency that meets specifications


Frequency (rad/sec)

Ph

ase

(d

eg

); M

ag

nitu

de

(d

B)

Bode Diagrams

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0

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30From: U(1)

10-1 100 101-150

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0

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To:

Y(1

)

Frequency glossary

! s! b! r

Sampling frequency [rad/s]

Closed-loop bandwidth [rad/s]

Open-loop resonant frequency [rad/s]

! r! b ! s


Sampling theorem‘s limits

Reconstruction of time-domain signals from rã ! yã

! b! s > 2

Condition on closed-loop bandwidth!


0 2 4 6 8

0

0.5

1

OU

TP

UT

S

Wb*t (rad)

11.3 (a) Ws/Wb = 4

--o-- X1--x-- X2----- U/4

0 2 4 6 8

0

0.5

1

OU

TP

UT

SWb*t (rad)

11.3 (b) Ws/Wb = 8

--o-- X1--x-- X2----- U/4

0 2 4 6 8

0

0.5

1

OU

TP

UT

S

Wb*t (rad)

11.3 (c) Ws/Wb = 20

--o-- X1--x-- X2----- U/4

0 2 4 6 8

0

0.5

1

OU

TP

UT

S

Wb*t (rad)

11.3 (d) Ws/Wb = 40

--o-- X1--x-- X2----- U/4

Smoothness

One of many selection criteria

6ô ! b! s ô 40


100

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102

1

1.2

1.4

1.6

1.8

2

2.2

2.4

2.6

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3Fig. 11.5

SAMPLING MULTIPLE, Ws/Wb

DIS

CR

ET

E/C

ON

TIN

UO

US

R

MS

Sensitivity wrt random plant disturbances

Sometimes „the“ criterion for sample rate selection

Degrading with respect tocontinuous system

Example:

! b! s > 20

x

+

Gc(s) 1/s2yur e

-+

d

+


100

101

102

1

1.2

1.4

1.6

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2

2.2

2.4

2.6

2.8

3Fig. 11.6

Sampling multiple, Ws/Wb

Dis

cre

te/c

on

tinu

ou

s R

MS

--------- no quantization

-x--x--x- 9 bit word size

-*--*--*- 8 bit word size

-o--o--o- 7 bit word size

Disturbance sensitivitywith observer and quantization

! b! s > 20

x+

1/s2yu

d

+

Same example:

Quantization worsens disturbance sensitivity


Sensitivity to parameter variations

Example: Aircraft with fuselage bendingPlant: s=-2j13.5 and bending mode with =0.01 and r=25rad/sObjective: rigid body motions: s=-16j10 (notch for bending mode)

Robust methods

Improvements with higher f


Measurement noise andantialiasing filters

+Gc(z) Plant

u yr e

-+

d

+ZOH

Example:

1Hz + 60Hz sinewavessampled at 28Hz

0 1 2-1.5

-1

-0.5

0

0.5

1

1.5Fig. 11.12, (a) signal + noise

time (sec)0 1 2

-1.5

-1

-0.5

0

0.5

1

1.5(b) 28 Hz sampled (a)

time (sec)

0 1 2-1.5

-1

-0.5

0

0.5

1

1.5(c) prefiltered (a)

time (sec)0 1 2

-1.5

-1

-0.5

0

0.5

1

1.5(d) 28 Hz sampled (c)

time (sec)

Filter


100

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1

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2

2.2

2.4

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3Fig. 11.13

SAMPLING MULTIPLE, Ws/Wb

DIS

CR

ET

E/C

ON

TIN

UO

US

R

MS

-o----o- Wp/Wb = 20

-*----*- Wp/Wb = 10

-+----+- Wp/Wb = 5

-x----x- Wp/Wb = 2

Measurement noise andantialiasing filters

+Gc(z) Plant

u yr e

-+

d

+ZOH

Example:

1st order filter

Filter

! p

x

! b! s = 25 for 20% degrading


Noise and prefilters

0 20 40 60 80 100

0

0.5

1

no

no

ise

0 20 40 60 80 100

0

0.5

1

no

ise

0 20 40 60 80 100

0

0.5

1

w.

filte

r

0 20 40 60 80 100

0

0.5

1

syst

em

+fil

ter


Measurement noise andanti-aliasing filters:Conclusions

+Gc(z) Plant

u yr e

-+

d

+ZOH

Filter

1. Prefilters useful

2. Bandwidth approaching closed-loop bandwidth

3. Include prefilter in plant model before controller design! b! p ù 2


Elements of the closed-loop


Choice of elements

What can we change?• System (higher resolution converters, better sensors,

better layout, lower temperature, ...)• Control algorithm (controller, sampling frequency,

tradeoffs: intuitive examplereduce noise by averaging in time)


Elements of the closed-loop

• Process (mechanical system)• Sensors• Actuators• Computing electronics


Process

The process is characterized by the following properties• Linearity• Time-invariance• Stability (right or left half-plane poles)• Minimum “phaseness” (right or left half-plane zeroes)• Bandwidth• ...• ...


ProcessLinearity and time-invariance (LTI)

If properties satisfied, many design methods available.

If not, consider variation as plant uncertainties

The larger the uncertainties, the smaller T(s) should be, thus• less stability or• more actuation, or• more amplification of measurement noise

Suggestion: take/construct LTI system or linearize it


ProcessStability and Minimum “phaseness”

Unstable system (right half-plane poles) is more difficult to control (higher sensitivity)

Suggestion: Live with fact that performance must be worse

Non minimum phase system (right half-plane zeroes) more difficult to control (higher sensitivity)

Suggestion: Change sensor placement (may render the plant minimum phase) or live with the worse performance


ProcessBandwidth

Raising bandwidth requires higher actuation

Consider

Roll-of of T(s) at higher frequency than roll-off of P(s)

means higher value of V(s)

Suggestion: Design open-loop bandwidth as high as closed-loop bandwidth or use high performance actuation (large values at high frequencies)



Sensors

Sensors are characterized by the following properties• Range (functional requirement)• Resolution• Accuracy• Precision• Sensitivity• Linearity• Response time• Bandwidth• Signal-to-noise ratio• ...


SensorsResolution, Sensitivity and Signal-to-noise ratio

Low resolution and low sensor sensitivity inject noise into the closed-loop

Suggestion: Analyze the noise as indicated before and consider sensitivity function constraints on noise attuenuation


SensorsAccuracy, Precision and Linearity

Low accuracy, precision and linearity can be considered as plant uncertainties

Suggestion: Analyze the uncertainties and consider complementary sensitivity function constraints on plant perturbations


SensorsResponse time

Noticeable response time causes delay in loop.

Suggestion: Choose bandwidth short enough that delay causes no problems, or consider delay in controller design (methods like Smith predictor or model predictive control)


Actuators

Actuators are characterized by the same properties of sensors (for which the suggestions would be similar), with the addition of • Continuous power output• Peak output


ActuatorsPeak output, continuous power

Peak output (force, torque) and continuous power are determined by the nominal trajectory and, because of

by the noise and disturbances affecting the loop (margin to be considered)

Suggestion: Analyze the needs in terms of nominal trajectories and calculate additional need due to noise and disturbances



Computing electronics is characterized by the following properties •Power ~ speed x word length x instruction power•Real-time properties•Other properties which do not affect the system performance (like type of interfaces, ...)

Computing electronics


Computing electronicssampling time

High computing power allows for fast sampling frequencies

Suggestion: Asses the needs in term of sampling time with criteria seen before

Note: Resolution and sampling time influence each other (higher sampling times may reduce the effect of low resolution actuation)


Computing electronicsjitter

Non regular sampling time affect the behavior of the closed loop

Suggestion: Choose sampling time 10 times larger than jitter

Documents

Specifications and closed-loop control