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Specifications and closed-loop control. Specifications: Basic terminology. Resolution: value of smallest resolvable displacement (step) of the component; it differs from the resolution of the measurement - PowerPoint PPT Presentation
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SUPSI DTI ProgettazioneControllori
Specificationsand closed-loop control
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
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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.
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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
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Problems with open-loop control:example 2
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
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Problems with open-loop control:example 3
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.
SUPSI DTI ProgettazioneControllori
Problems with open-loop control:example 4
You may want to reach the end position faster (dynamics is not satisfactory)
Example: When system shows slightly damped oscillations, like in flexure systems.
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Problems with open-loop control:example 5
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.
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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
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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!
SUPSI DTI ProgettazioneControllori
Problems solved withclosed-loop control: example 2
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
SUPSI DTI ProgettazioneControllori
Problems solved withclosed-loop control: example 3
The system is slightly different from what imagined
Example: The closed-loop control can take care of varying of uncertain spring constant values k
SUPSI DTI ProgettazioneControllori
Problems solved withclosed-loop control: example 4
You may want to reach the end position faster (dynamics is not satisfactory)
Example: Closed-loop control can effectively damp oscillations, even change bandwith.
SUPSI DTI ProgettazioneControllori
Problems solved withclosed-loop control: example 5
The open-loop system is unstable
Example: The magnetically levitated ball can be kept floating at a given position
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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)
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Closed-loopControl Systems
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Plant with controller
r: reference signal
u: command signal
w: measured signal
y: output signal
d: disturbance
Controller Plantyur
w
d
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Control problem
Given Plant G(s), find controller C(s) such thatclosed-loop system with transfer function
satisfies behavior specifications
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Feedforward scheme
Closed-loop system with transfer function
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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
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Measurable disturbancecompensation scheme
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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
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Trade-offs
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General issues
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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
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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))
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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)
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Nominal signal response (2)
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
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V(s) = à T(s)di(s) + P(s)T(s)[r(s) à ñ(s) à do(s)]
Nominal signal response (3)
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!
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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
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Influence of plant variations
SUPSI DTI ProgettazioneControllori
Sensitivity to plant variations
Plant variations
causes variations
|S(j)|<1 feedback decreases effect
|S(j)|>1 feedback increases effect
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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(! )
SUPSI DTI ProgettazioneControllori
Summary of specifications
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
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Open-loop specifications
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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 ! )
SUPSI DTI ProgettazioneControllori
jT(j ! )j < WT(j ! )1 =) jL (j ! )j < Wà 1
T (j ! )
Specifications on Loop gain (2)
For high frequency
jL (j ! )j ü 1 =) T(j ! ) = 1+L(j ! )L (j ! ) ù L(j ! )
SUPSI DTI ProgettazioneControllori
Specifications on Loop gain (3)
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
SUPSI DTI ProgettazioneControllori
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
-270
-225
-180
-135
-90
-45
0
Pha
se (
deg)
Summary of specifications
0dB=1
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
Bode Gain-Phase Relation (3)crossover slope
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
! High decrease rate
SUPSI DTI ProgettazioneControllori
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
-250
-200
-150
-100
-50
0
50
Mag
nitu
de (
dB)
100
102
104
106
same high decrease ratespecs 100dB higher
Bode Gain-Phase Relation (4)crossover slope, narrower specs
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Bode sensitivity integral
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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
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Bode Sensitivity integral
S = 1+L(s)1 for L(s) = s(s+1)
1
Green surfacec exactly compensates red one
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Bode sensitivity integral:different designs
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-60
-50
-40
-30
-20
-10
0
10
20
30
Magnitu
de (
dB
)
Bode Diagram
Frequency (rad/sec)
SUPSI DTI ProgettazioneControllori
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-40
-35
-30
-25
-20
-15
-10
-5
0
5
10
Magnitu
de (
dB
)
Bode Diagram
Frequency (rad/sec)
Bode sensitivity integral:bandwidth limitations
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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)
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Quantization and Noise
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• Arithmetic operations
• Coefficient quantization
• Converter resolution
• Other noise sources
Sources of quantization errors
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Quantizationin arithmetic operations
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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
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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
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Floating-point arithmetic
N =M â 2E Mantissa M and exponent E in fixed-point, normalized representation (0.5 M 1)
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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
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Coefficient quantization
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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!
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Converter quantization
SUPSI DTI ProgettazioneControllori
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
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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
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Propagation ofquantization noise
SUPSI DTI ProgettazioneControllori
Propagation of quantization error
+
-x
xq yu
SUPSI DTI ProgettazioneControllori
Stochastic quantization error
Example
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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
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Sample rate selection
SUPSI DTI ProgettazioneControllori
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
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Frequency (rad/sec)
Ph
ase
(d
eg
); M
ag
nitu
de
(d
B)
Bode Diagrams
-20
-10
0
10
20
30From: U(1)
10-1 100 101-150
-100
-50
0
50
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
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Sampling theorem‘s limits
Reconstruction of time-domain signals from rã ! yã
! b! s > 2
Condition on closed-loop bandwidth!
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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
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100
101
102
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
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
+
SUPSI DTI ProgettazioneControllori
100
101
102
1
1.2
1.4
1.6
1.8
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
100
101
102
103
1
1.2
1.4
1.6
1.8
2
2.2
2.4
2.6
2.8
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
Elements of the closed-loop
SUPSI DTI ProgettazioneControllori
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)
SUPSI DTI ProgettazioneControllori
Elements of the closed-loop
• Process (mechanical system)• Sensors• Actuators• Computing electronics
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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• ...• ...
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
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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)
V(s) = à T(s)di(s) + P(s)T(s)[r(s) à ñ(s) à do(s)]
SUPSI DTI ProgettazioneControllori
Sensors
Sensors are characterized by the following properties• Range (functional requirement)• Resolution• Accuracy• Precision• Sensitivity• Linearity• Response time• Bandwidth• Signal-to-noise ratio• ...
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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
SUPSI DTI ProgettazioneControllori
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)
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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
SUPSI DTI ProgettazioneControllori
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
V(s) = à T(s)di(s) + P(s)T(s)[r(s) à ñ(s) à do(s)]
SUPSI DTI ProgettazioneControllori
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
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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)
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Computing electronicsjitter
Non regular sampling time affect the behavior of the closed loop
Suggestion: Choose sampling time 10 times larger than jitter