Digital Control Emulation Techniques Lecture Overview
Delve into the world of digital control with a focus on emulation techniques discussed in a CSE416 lecture. Learn about controller emulation, forward Euler approximation, system models, Z-transforms, pole locations, and stability considerations.
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CSE416: DIGITAL CONTROL Lecture 04: Emulation Techniques Dr. Ahmed Mahmoud, 21/10/2019
Controller Emulation Controller emulation is to use a discrete equivalent of a continuous controller in the design of a digital controller. For a PI controller in continuous-time domain: ? ? ? = ??? ? + ?? ? ? ?? ? The frequency domain representation of the controller is: ? ? = ??+?? ? ? ? For digital realization of such a controller, the proportional part does not pose any problems, however the integrator requires approximation. The integrator can be approximated by Euler s forward rule: ? 1 ? ?? ? ? ?? ?? ? ? ?? ? ?=0 If the sampling period T is sufficiently small, this approximation yields a controller similar to the continuous one. 2 21/10/2019 CSE416: DIGITAL CONTROL
The Drain Tank Example The differential equation model of the system is: ? ? ? = ? ? ? ? The system transfer function is: The unit step response is: ? ? = 1 ? ? ? ? ? 3 21/10/2019 CSE416: DIGITAL CONTROL
Forward Euler Approximation Forward Euler rule approximate derivate by: Approximate the drain tank DE: The approximated difference equation model is: 4 21/10/2019 CSE416: DIGITAL CONTROL
Forward Euler Approximation Behavior ? ?= ?.? ? ?= ?.? ? ?= ? ? ?= ?.? ? ?= ? 5 21/10/2019 CSE416: DIGITAL CONTROL
Z-Transform of Forward Euler The difference equation of the system is: By applying Z transform: ?? ? 1 ? ? ? =? ??(?) ? The transfer function is: ? ? ? =? ?= ? 1 ? ? The discrete equivalent has a pole at ? = 1 ? ? The continuous pole location is fixed at ? = 1 ? 6 21/10/2019 CSE416: DIGITAL CONTROL
Forward Euler Pole Location ? ?= ? ? ?= ?.? ? ?= ? ? ?= ?.? ? ?= ?.? 7 21/10/2019 CSE416: DIGITAL CONTROL
Forward Euler Poles Mapping Forward Euler maps ? ? 1 ? ? 0 ? = ? + ?? 1 1 0 ? 2 -1 ? For achieving stability in discrete-time approximation: 2 ?< 1 ?< 0 ? ?< 2 Accuracy and stability of forward Euler approximation depends on sampling time. 8 21/10/2019 CSE416: DIGITAL CONTROL
Backward Euler Approximation Backward Euler rule approximates derivate by: Approximate the drain tank DE: The approximated difference equation model is: 9 21/10/2019 CSE416: DIGITAL CONTROL
Backward Euler Approximation Behavior ? ?= ?.? ? ?= ?.? ? ?= ? ? ?= ?.? ? ?= ? 10 21/10/2019 CSE416: DIGITAL CONTROL
Z-Transform of Backward Euler The difference equation of the system is: By applying Z transform: 1 +? ? ? ? 1? ? =? ??(?) ? The transfer function is: ? ? ? =? ?= 1 +? ? 1 ? 1 The discrete equivalent has a pole at ? = 1+? ? The continuous pole location is fixed at ? = 1 ? 11 21/10/2019 CSE416: DIGITAL CONTROL
Backward Euler Pole Location ? ?= ?.? ? ?= ? ? ?= ? ? ?= ?.? ? ?= ?.? 12 21/10/2019 CSE416: DIGITAL CONTROL
Backward Euler Poles Mapping Forward Euler maps ? ? 1 ?? ? ? ? = ? ?? 1 0 1 1 2 ? 1 3 2 ? For backward Euler stability, if the continuous pole is stable, then the discrete equivalent pole is also stable. 13 21/10/2019 CSE416: DIGITAL CONTROL
Trapezoidal Rule (Tustin Transformation) The trapezoidal rule uses centered differences to approximate derivatives. Consider the following differential equation: ? ? = ? ? . It has the following transfer function: ? ? ? ?=1 ? . It can be approximated with trapezoidal rule as: ? ? ? ? 1 =? ? +?(? 1) ? 2 The Z transfer function is: ? ? ? ?=? ?+1 2(? 1) . The trapezoidal rule maps ? 2(? 1) ? ?+1 . 14 21/10/2019 CSE416: DIGITAL CONTROL
Which to Use for Implementation? Forward (explicit) Euler approach is numerically not efficient . Complex algorithms designed for efficient numerical integration are not applicable to real-time control systems. Tustin transformation is often used in practice to produce a satisfactory closed-loop system behavior. 15 21/10/2019 CSE416: DIGITAL CONTROL
Example: Emulation of a Lead Controller 1 ?2 is controlled by ? ? = 0.23?+1 Consider a double integrator plant ? ? = which is realized on a digital controller with ? = 1?. Apply Euler s forward transform. ? ? 1 ? ? 0.23 ? 1 + 1 ?+1 ? ? 1 + 1= 0.23? 2 ? ? ? = ? ? ? ? ?? ? = 0.6? 0.4 ? ? . ? ? = 0.6? ? 0.4? ? 1 . 16 21/10/2019 CSE416: DIGITAL CONTROL
Effect of Sampling Time on Control Performance 1 Consider a system: ? 10?+1 . The controller is lead compensator defined as: 10?+1 ?+1 . The controller is sampled fast with ? = 0.05s, and slow with ? = 0.5? 17 21/10/2019 CSE416: DIGITAL CONTROL
