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Discrete-time models and control
Silvano Balemi University of Applied Sciences of Southern Switzerland
Zürich, 2009-2010
Discrete-time signals
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Step response of a sampled system
Sample and hold
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Multiplication with a train of unit impulses (operation is linear but time-variant)
Sampling
Train of impulses and its Fourier expansion
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Sampled signal
with
Spectrum of Sampled signal
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Hold
Linear operation
Impulse response of a ZOH
1(t)
1(t - T)
Z transform
where
Laplace transformation with
The z transform corresponds to the sequence
with the function
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Relation between different transforms
Z transform: Examples and properties
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Examples of z transforms
Some transformations
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Properties of the z transform
Linearity
Delay
Anticipation
Damping
Product
Initial value
End value
z transform
1. From the Laplace transformation
Factorization
Using „primitives“
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Inverse z transform
1. Inverse trasform via factorization
2. Inverse transform via recursion
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
Discrete-time Transfer function from frequency domain
with variable substitution m=k+n
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Transfer function with ZOH
Gzoh(z)
Example Transfer function with ZOH
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State space representation
u constant from 0 to T
Transfer function
from
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
Step responses
x
x
x
x
x x
x x
x x
x
x
x
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Closed-loop Control
Closed-loop sampled systems
Digital part Analog part
Digital controller
Cont.-time process
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Closed-loop Discrete-time system (2)
model of A/D conv
model of program
model of D/A conv
model of process
Gzoh(z)
Example: system stability
≈ 0.09
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Example of a program for a controller
Example of a program for a controller: C-code
ek_1=0; ek_2=0; uk_1=0; uk_2=0; while TRUE { yk=read_yk(); ek=yrefk-yk; uk=-uk_1-uk_2+ek_1-3*ek_2; write(uk); uk_2=uk_1; uk_1=uk; ek_2=ek_1; ek_1=ek; }
C-code
Minimize control delay!
ek_1=0; ek_2=0; uk_1=0; uk_2=0; while TRUE { uk=-uk_1-uk_2+ek_1-3*ek_2; yk=read_yk(); write(uk); ek=yrefk-yk; uk_2=uk_1; uk_1=uk; ek_2=ek_1; ek_1=ek; }
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Control design
• Controller designed in discrete-time domain
• Controller designed in continuous-time domain and then transformed into discrete-time domain
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Discrete-time controllers: design of GTOT(z)
• Same order of GTOT(z) as of GZOH(z)
• Numerator of GTOT(z) with order n-1
• all zeroes at –1
• Possible amplification for static error reduction
• Controller obtained from process GZOH(z) and from GTOT(z)
Choice of GTOT(z) and calculation of Gc (z)
Discrete-time controllers: design of GTOT(z) Example
Desired closed-loop discrete-time poles
Plant
Closed-loop tr. function
Controller
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Discrete-time controllers: deadbeat control
• Choice of
starting from
• Controller obtained with
All poles at the origin
The fastest controller of the west
Discrete-time controllers: deadbeat control Example
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Discrete-time controllers: Transformation of poles
Transformation
with
Example:
Discrete-time controllers: Discrete PID equivalent
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Discrete-time controllers: bilinear transformation
Transformation
Stretching of the band –π/π onto the s-plane
Π
-Π
1 2 3 4 5
Discrete-time controllers: bilinear transformation
, pole at -b
1.
2.
3.
4.
5.
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Pole assignement: Polynomial approach
Characteristic polynomial
compared with desired characteristic polynomial gives 2nC+1 variables for nC+nG unknowns
Example: first (second) order controller is sufficient for control of second (third) order system
Controllability
Property of a system to reach any given state from the origin in a finite time through an appropriate input signal
Controllability matrix indicates controllability if full rank
Controllable subspace
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Pole assignment: State-feedback controller
If system satisfies a property called controllability
state feedback yields
Any chosen set of closed-loop poles can be obtained through an appropriate matrix K
Observability
Property of a system to estimate the value of the states looking at the inputs and at the outputs
Observability matrix indicates observability if full rank
unobservable subspace
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Observer/estimator
If system satisfies a property called observability
feedback with L yields state error system satisfying
Any chosen set of poles for the error system can be obtained through an appropriate matrix L
State estimate
State-feedback controller with static error compensation
Controller for plant with extended matrices
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discrete-time controllers: continuous or discrete-time design?
Gc(s)
G(s)
Gc(z)
GZOH(z)
continuous-time design
discrete-time design
discrete-time approximation
discrete-time modeling
Gc(w)
G(w)
Saturations and Wind-up
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Control Wind-up
Actuation signal Output signal
PID controller with Anti-Wind-up
Or limitation of output
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Anti-Wind-up through saturated feedback and FIR filter implementation
All signals bounded!
Anti-Wind-up through saturated feedback and IIR filter implementation
F(z) is polynomial in z-1 with well stable poles (inside unit circle) Case F(z)=1 corresponds to previous case.
If Anti-windup measure is too fast (actuation signal may jump from bound to bound) slow-down with low-pass filter F(z)
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Anti-Wind-up in state-feedback controllers
Anti-Wind-up through saturated feedback for state-feedback controllers
All signals bounded!
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Anti-Wind-up through saturated feedback for state-feedback controllers
F(z) is polynomial in z-1 with stable poles (inside unit circle)