dcs-33 discretized controller design by...
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Spring 2021
數位控制系統Digital Control Systems
DCS-33
Discretized Controller –
Design by Emulation
Feng-Li Lian
NTU-EE
Feb – Jun, 2021
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DCS33-EmulationDesign-2
Feng-Li Lian © 2021Introduction: CT and DT Plant-Controller
G(s)
u(t) y(t)
C(s)
e(t)r(t)
G(z)
u[k] y[k]
C(z)
e[k]r[k]
G(s) C(s)
G(z) C(z)
G(s) C(s)
G(z) C(z)
Transform CT plant into DT plant
By DT plant, design DT controller
By CT plant, design CT controller
Transform CT controller into DT controller
Emulation
Discrete DesignDirect Design
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DCS33-EmulationDesign-3
Feng-Li Lian © 2021Basic Design Concept
Basic principles of low-order controller design
Anderson 1993B.D.O. Anderson, “Controller Design: Moving from Theory to Practice,” IEEE Control Systems Magazine, 13(4), pp. 16-25, Aug. 1993
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DCS33-EmulationDesign-4
Feng-Li Lian © 2021Introduction: CT and DT Plant-Controller
Study in Digital Control Systems
• Controller Design of Digital Control Systems
– Design Process
> Discrete Design:
» CT plant -> DT plant -> DT controller
> Emulation:
» CT plant -> CT controller -> DT controlle
> Direct Design: (B.D.O. Anderson, 1992 Bode Prize Lecture)
» CT plant -> DT controller
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DCS33-EmulationDesign-5
Feng-Li Lian © 2021Outline
Discrete Design
By Transfer Function
By State Space
Design by Emulation
Tustin’s Method or bilinear approximation
Matched Pole-Zero method (MPZ)
Modified Matched Pole-Zero method (MMPZ)
Digital PID-Controllers
Techniques for Enhancing the Performance
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DCS33-EmulationDesign-6
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Tustin’s method:
Franklin et al. 2002
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DCS33-EmulationDesign-7
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Tustin’s method:
Franklin et al. 2002
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DCS33-EmulationDesign-8
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Tustin’s method:
sysDs = tf( 10*[0.5 1], [0.1 1] );
sysDz = c2d( sysDs, 0.025, ‘tustin’);
Franklin et al. 2002
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DCS33-EmulationDesign-9
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Tustin’s method:
Franklin et al. 2002
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DCS33-EmulationDesign-10
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
MATLAB’s command: c2d
C2D Conversion of continuous-time systems to discrete time.
SYSD = C2D( SYSC, TS, METHOD )
converts the continuous system SYSC to a discrete-time system SYSD
with sample time TS.
The string METHOD selects the discretization method among the
following:
'zoh' Zero-order hold on the inputs.
'foh' Linear interpolation of inputs (triangle appx.)
'tustin' Bilinear (Tustin) approximation.
'prewarp' Tustin approximation with frequency prewarping.
The critical frequency Wc is specified last as in
C2D( SysC, Ts, 'prewarp', Wc)
'matched' Matched pole-zero method (for SISO systems only).
MathWorks: c2d: https://www.mathworks.com/help/control/ref/c2d.html
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DCS33-EmulationDesign-11
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Differentiation & Tustin Approximation:
Forward difference: (Euler’s method)
Backward difference:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-12
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Trapezoidal method: (Tustin, bilinear)
Differentiation & Tustin Approximation:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-13
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Properties of Approximations:
Forward
Backward
Trapezoidal
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-14
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Stability of Approximations:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-15
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Approximations introduce frequency distortion
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-16
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Approximations introduce frequency distortion
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-17
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Approximations introduce frequency distortion
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-18
Feng-Li Lian © 2021Design by Emulation – Tustin’s Method
Tustin with Prewarping:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-19
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
1. Map poles and zeros according to the relation
2. If the numerator is of lower order than the denominator,
add powers of (z+1) to the numerator
until numerator and denominator are of equal order
3. Set the DC or low-frequency gain of D(z) = that of D(s)
Franklin et al. 2002
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DCS33-EmulationDesign-20
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• Case 1:
Franklin et al. 2002
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DCS33-EmulationDesign-21
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• Case 2:
Franklin et al. 2002
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DCS33-EmulationDesign-22
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• The same power of z in the num & den of D(z):
Franklin et al. 2002
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DCS33-EmulationDesign-23
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• Space station attitude digital controller
Franklin et al. 2002
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DCS33-EmulationDesign-24
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• Space station attitude digital controller
Franklin et al. 2002
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DCS33-EmulationDesign-25
Feng-Li Lian © 2021Design by Emulation – MPZ
Matched Pole-Zero (MPZ) method:
• Space station attitude digital controller
Franklin et al. 2002
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DCS33-EmulationDesign-26
Feng-Li Lian © 2021Design by Emulation – MMPZ
Modified Matched Pole-Zero (MMPZ) method:
• u[k+1] depends only on e[k], but not e[k+1]
Franklin et al. 2002
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DCS33-EmulationDesign-27
Feng-Li Lian © 2021Design by Emulation – MMPZ
Comparison of Digital Approximation Methods:
Franklin et al. 2002
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DCS33-EmulationDesign-28
Feng-Li Lian © 2021Design by Emulation – Digital PID-Controllers
The “textbook” version of the PID-controller:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-29
Feng-Li Lian © 2021Design by Emulation – Digital PID-Controllers
Modification of Linear Response:
Astrom & Wittenmark 1997
small ---> s Td
Two-input-&-one-output controller:
A pure derivative cannot be,
and should not be, implemented:
Because amplification of measurement noise
Derivative gain should be limited
large ---> N
N = 3 ~ 20
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DCS33-EmulationDesign-30
Feng-Li Lian © 2021Design by Emulation – Digital PID-Controllers
Discrete-time PID:
Astrom & Wittenmark 1997
One popular approximation:
(No Approximation)
(Forward)
(Backward)
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DCS33-EmulationDesign-31
Feng-Li Lian © 2021Design by Emulation – Digital PID-Controllers
Discrete-time PID:
Astrom & Wittenmark 1997
Can be written as:
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DCS33-EmulationDesign-32
Feng-Li Lian © 2021Design by Emulation – Digital PID-Controllers
Coefficients in different approximations:
Astrom & Wittenmark 1997
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DCS33-EmulationDesign-33
Feng-Li Lian © 2021Outline
Discrete Design
By Transfer Function
By State Space
Design by Emulation
Tustin’s Method or bilinear approximation
Matched Pole-Zero method (MPZ)
Modified Matched Pole-Zero method (MMPZ)
Digital PID-Controllers
Techniques for Enhancing the Performance