SUPSI DTI. Automazione. Digital Control of. Dynamic Systems. Silvano Balemi.
University of Applied Sciences of Southern Switzerland. Manno, 2004 ...
SUPSI
DTI
Automazione
Digital Control of Dynamic Systems Silvano Balemi University of Applied Sciences of Southern Switzerland Manno, 2004
SUPSI
DTI
Automazione
Discrete-time signals
DTI
Automazione
Step response of a sampled system
Step Response From: U(1)
0.35
0.3
To: Y(1)
0.25
Amplitude
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0.2
0.15
0.1
0.05
0
0
0.1
0.2
0.3
0.4
0.5
Time (sec.)
0.6
0.7
0.8
0.9
1
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Sample and hold
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Sampling
Multiplication with a train of unit impulses (operation is linear but time-variant)
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Train of impulses and its Fourier expansion
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Sampled signal
with
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Spectrum of Sampled signal
SPECTRUM OF A SIGNAL 2 |R|
1.5
1
0.5
0 -10
1.5
-8
-6
-4
-2
0
2
4
6
8
10
SPECTRUM OF THE SAMPLED SIGNAL |R*| AND ITS COMPONENTS
1
w1
0.5
0 -15
-10
-5
0
5
10
15
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Automazione
Hold
Linear operation 1(t)
1(t − T)
Impulse response of a ZOH
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Z transform
Laplace transformation with
where The z transform corresponds to the sequence with the function
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Relation between different transforms
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Z transform: Examples and properties
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Examples of z transforms
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Some transformations
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Linearity Delay Anticipation Damping Product Initial value End value
Automazione
Properties of the z transform
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z transform
1. From the Laplace transformation Factorization
Using „primitives“ k G(s) = s+a
g(t) = k á e àaát
G(z) = k á zàezàaáT
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Inverse z transform
1. Inverse trasform via factorization
2. Inverse transform via recursion
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Sampled Systems
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Discrete-time Transfer function from time domain
No transfer function between u and y but between u* and y*
and with variable substitution l=k-m
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Discrete-time Transfer function from frequency domain
with variable substitution m=k+n
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Transfer function with ZOH
Gzoh(z)
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Example Transfer function with ZOH
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State space representation
u constant from 0 to T from
Transfer function
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Description of Linear Time-invariant Discrete-time Systems
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Stability of sampled systems
x
x x x
x x
x
x
x
x x
x
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Step responses
x
x
x
x
x
x
x
x
x
x
x x x
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Digital controller
Digital part
Closed-loop sampled systems
Cont.-time process
Analog part
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model of program
Closed-loop Discrete-time system (2)
model of D/A conv
model of process
Gzoh(z)
model of A/D conv
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Example: system stability
≈ 0.09
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Example of a program for a controller
U(z)
G c(z) = E(z) = z 2z+3 +z+1 (z 2 + z + 1) á U(z) = (z + 3) á E(z) (1 + z à1 + z à2) á U(z) = (z à1 + 3 á z à2) á E(z) U(z) + z à1 á U(z) + z à2 á U(z) = z à1 á E(z) + 3 á z à2 á E(z) {u k} + {u kà1} + {u kà2} = {e kà1} + 3 á {e kà2} u k + u kà1 + u kà2 = e kà1 + 3 á e kà2 u k = à u kà1 à u kà2 + e kà1 + 3 á e kà2
