1. Digital Control. System Models. M. Sami Fadali. Professor of Electrical
Engineering. University of Nevada. 2. Outline. • Model of ADC. • Model of DAC.
Outline • Model of ADC. • Model of DAC. • Model of ADC, analog subsystem and DAC. • Systems with transport lag. • Examples
Digital Control System Models M. Sami Fadali Professor of Electrical Engineering University of Nevada 1
Common Digital Control System Configuration Computer or Microprocessor
DAC
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ADC Model Assume that (i) ADC outputs are exactly equal in magnitude to their inputs. (i.e. neglect quantization errors). (ii) ADC yields digital output instantaneously. (iii) Sampling is perfectly uniform. Model ADC as ideal sampler with sampling period T.
Analog Subsystem
ADC
T
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DAC Model
Transfer Function of ZOH
Assume that DAC (i)outputs are exactly equal in magnitude to their inputs.
Use impulse sampling and Laplace transform of a unit pulse
(ii) yields an analog output instantaneously. (iii) outputs are constant over each sampling period. zero-order-hold: piecewise constant analog output.
_
first-order hold constructs signals in terms of straight lines
_
second-order hold constructs them in terms of parabolas.
Zero-order Hold
kT
kT
(k+1)T
kT
(k+1)T
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Time Response
Effect of Sampler on Transfer Function of a Cascade In the s-domain
Change order and variables of integration U(s)
H1(s)
X(s)
H2(s)
Y(s)
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Convolving
Punch Line
& CT Signal
Equivalent impulse response for cascade = the convolution of cascaded impulse responses. • Impossible to separate convolved time function Repetitions of the CT function each in the location of an impulse in the train
• Sample the output
U(s)
U*(s) H(s)
Y(s)
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Sampled Output
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Transfer Function for Cascade
• Change the order of summation and integration
• Cascade not separated by samplers: components cannot be separated after sampling. • Cascade separated by samplers: each block has a sampled output and input as well as a z-domain transfer function. blocks not separated:
• Sample and transform
blocks separated: ∗
∗
∗ 11
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Example
(a) Systems Directly Connected
Find the equivalent sampled impulse response sequence and the equivalent z-transfer function for the cascade of the two analog systems with sampled input
• Impulse response • Sampled impulse response
(a) If the systems are directly connected. (b) If the systems are separated by a sampler.
• z-transfer function
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(b) Systems Separated By Samplers
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TF for DAC, Analog Subsystem, ADC Combination
• Partial fractions
L
• Impulse response sequence
T kT
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Zero-order Hold
kT
(k+1)T
Analog Subsystem 16
Sample & z-Transform
Impulse Response of DAC and Analog Subsystem 1.5
• Sampled impulse response
1.0
1.0
• z-transfer function of DAC (zero-order hold), analog subsystem, ADC (ideal sampler) cascade
0.8
Positive Step 0.5
0.6
0.0
0.4
0.5
Z
0.2
Delayed Negative Step 1.0 1.5 2
4
6
8
1 time s
(a) Response of analog system to step inputs.
∗
Z L
0.0
0
∗
0.2 0
2
4
6
8
•
10
time s
Simplify Notation Z
(b) Response of analog system to a unit pulse input. 17
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Solution
Example
/
• Voltage divider
Find the z-domain transfer function for ADC, DAC, and the shown series R-L circuit with the inductor voltage as output.
/
Z
• Use R
Z
L Vin
Vo
/ /
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/ 20
Solution
Example Find the z-domain transfer function of an amplifier, an armature controlled DC motor, ADC and DAC if the s-domain transfer function of the motor and amplifier is
• Partial fraction expansion
⁄ 21
Systems with Transport Lag
Solution (Cont.)
• Transfer function for systems with a transport delay
Z
• Use
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Z
= positive integer. Example: time delay = 3.1 s,
⁄
⁄
⁄
1 1
⁄
1
⁄
and
1
∗
⁄
Z L 23
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Use Modified z-Transform
Delay Theorem
Z
∗
Z _
• Causal time function • Sampling & shifting
• Define the transfer function • Advance theorem of Laplace transforms
Z _ Z
∗
• Shifting a step function has no effect on samples, shifting a decaying exponential does.
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Modified z-Transform
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Example
Z
If the sampling period is 0.1 s, determine the z-transfer function for the ADC, DAC and analog transfer function
• Transfer function
.
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Solution
CT Function
• Delay in terms of the sampling period • Integer
1 0.9 0.8 0.7
• Partial fraction expansion
0.6 0.5
• z-transfer function
.
0.4
.
0.3 0.2
.
0.1 0 0
0.5
1
1.5
2
t s 29
Block Diagram of Single-loop Digital Control System R(z) +
U(z)
E(z) C(z)
GZAS(z)
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Closed-loop Transfer Function Y(z)
• CL characteristic equation
• Closed-loop System Poles • Roots of the characteristic equation (selected for desired time response specifications as in s-domain design) 31
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MATLAB Commands
MATLAB Example
z-transfer function for ADC, DAC & analog subsystem.
G(s) = (2s2+4s+3)/(s3+4s2+6s+8)
>> g= tf(num, den) % Enter analog transfer function >> gd= c2d(g,T, 'method') % Sampling period T
>> g = tf( [2,4,3], [1,4,6,8]) >> gd= c2d(g, 0.1, 'zoh')
'method' = method to obtain the digital transfer function
% zero-order hold
>> gd = c2d(g, 0.1, 'foh') % first-order hold >> gd = c2d(g, 0.1, 'impulse') % z-transform T 33
Transport Delay >> g=tf(3,[1,3],'InputDelay', 0.31) Transfer function: 3 exp(-0.31*s) * ------s+3 >> c2d(g,.1) Transfer function: 0.2366 z + 0.02256 z^(-4) * ----------------------z - 0.7408
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