1. z-Plane Analysis of Discrete-Time Control Systems
Chapter 3
z-Plane Analysis of
Discrete-Time
Control Systems

2. Chap. 3 z-Plane Analysis of Discrete-Time Control Systems
Introduction
Backgrounds necessary for the analysis and design of discrete-time control systems in the z plane are presented.
The main advantage of the z transform method: it enables us to apply conventional continuous-time design methods to discrete-time systems.
The chapter covers:
Mathematical representation of the sampling operation
The convolution integral method for obtaining the z transform
The sampling theorem based on the fact that the Laplace transform of the sampled signal is periodic
Mathematical modeling of digital controllers in terms of pulse transfer function
Realization of digital controllers and digital filters
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3. Impulse Sampling and Data Hold
A fictitious sampler
The output of the sampler is a train of impulses.
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4. Impulse Sampling and Data Hold
Impulse Sampling (cont.)
Let’s define a train of unit impulses:
The sampler may be considered a modulator with
The modulating signal: the input x(t)
The carrier : the train of unit impulses T(t)
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5. Impulse Sampling and Data Hold
Impulse Sampling (cont.)
The Laplace transform of x*(t)
If we define or
Hence we may write
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6. Impulse Sampling and Data Hold
Impulse Sampling (cont.)
Summary
If the continuous-time signal x(t) is impulse sampled in a periodic manner, the sampled signal may be represented by
The Laplace transform of the impulse-sampled signal x*(t) has been shown to be the same as the z transform of signal x(t) if
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7. Impulse Sampling and Data Hold
Data-Hold Circuits
Data-hold: a process of generating a continuous-time signal h(t) from a discrete-time sequence x(kT).
A hold circuit approximately reproduces the signal applied to the sampler.
The simplest data-hold: zero-order hold (clamper)
Note that signal h(kT) must equal x(kT):
n-th order hold
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8. Impulse Sampling and Data Hold
Zero-Order Hold
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9. Impulse Sampling and Data Hold
Zero-Order Hold (cont.)
A real sampler and zero-order hold
Since
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10. Impulse Sampling and Data Hold
Zero-Order Hold (cont.)
Mathematical model: an impulse sampler and transfer function
From the figure
Since
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11. Impulse Sampling and Data Hold
Transfer function of First-Order Hold
n-th order hold
1-st order hold
By applying the condition that
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12. Impulse Sampling and Data Hold
Transfer function of First-Order Hold (cont.)
Derivation of the transfer function
Taking the Laplace transform
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13. Impulse Sampling and Data Hold
Transfer function of First-Order Hold (cont.)
Derivation of the transfer function (cont.)
The transfer function
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14. Reconstructing Original Signals from Sampled Signals
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15. Reconstructing Original Signals from Sampled Signals
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16. Reconstructing Original Signals from Sampled Signals
Sampling Theorem
If the sampling frequency is sufficiently high compared with the highest-frequency component involved in the continuous-time signal, the amplitude characteristics of the continuous-time signal may be preserved in the envelope of the sampled signal.
To reconstruct the original signal from a sampled signal, there is a certain minimum frequency that the sampling operation must satisfy.
We assume that x(t) does not contain any frequency components above 1 rad/sec.
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17. Reconstructing Original Signals from Sampled Signals
Sampling Theorem
The frequency spectrum:
If s, defined as 2/T is greater than 21, where 1 is the highest-frequency component present in the continuous-time signal x(t), then the signal x(t) can be reconstructed completely from the sampled signal x*(t).
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18. Reconstructing Original Signals from Sampled Signals
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19. Reconstructing Original Signals from Sampled Signals
Ideal Low-pass filter
The ideal filter attenuates all complementary components to zero and will pass only the primary component.
If the sampling frequency is less than twice the highest-frequency component of the original continuous-time signal, even the ideal filter cannot reconstruct the original continuous-time signal.
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20. Reconstructing Original Signals from Sampled Signals
Ideal Low-pass filter Is NOT Physically Realizable
For the ideal filter an output is required prior to the application of the input to the filter – physically not realizable.
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21. Reconstructing Original Signals from Sampled Signals
Frequency-Response Characteristics of the ZOH
Transfer function of ZOH
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22. Reconstructing Original Signals from Sampled Signals
Frequency-Response Characteristics of the ZOH(cont.)
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23. Reconstructing Original Signals from Sampled Signals
Frequency-Response Characteristics of the ZOH(cont.)
The comparison of the ideal filter and the ZOH.
ZOH is a low-pass filter, although its function is not quite good.
The accuracy of the ZOH as an extrapolator depends on the sampling frequency.
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24. Reconstructing Original Signals from Sampled Signals
Folding
The phenomenon of the overlap in the frequency spectra.
The folding frequency (Nyquist frequency): N
In practice, signals in control systems have high-frequency components, and some folding effect will almost always exist.
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25. Reconstructing Original Signals from Sampled Signals
Aliasing
The phenomenon that the frequency component ns 2 shows up at frequency 2 when the signal x(t) is sampled.
To avoid aliasing, we must either choose the sampling frequency high enough or use a prefilter ahead of the sampler to reshape the frequency spectrum of the signal before the signal is sampled.
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26. Reconstructing Original Signals from Sampled Signals
Hidden Oscillation
An oscillation existing in x(t) between the sampling periods.
For example, if the signal
is sampled at t=0, 2/3, 4 /3,…, then the sampled signal will not show the frequency component with =3 rad/sec.
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27. The Pulse Transfer Function
Convolution Summation
For the continuous time-system
For the discrete-time system
For a physical system a response cannot precede the input
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28. The Pulse Transfer Function
Convolution Summation (cont.)
The value of the output y(t) at the sampling instants t=kT are given by
Since we assume that x(t)=0 for t <0
It is noted that if G(s) is a ratio of polynimials in s and if the degree of the denominator polynomial exceeds that of the numerator polynomial only by 1 the output y(t) is discontinuous.
Convolution summation
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29. The Pulse Transfer Function
Convolution Summation (cont.)
In analyzing discrete-time control systems it is important to remember that the system response to the impulse-sampled signal may not portray the correct time-response behavior of the actual system unless the transfer function G(s) of the continuous-time part of the system has at least two more poles than zeros, so that
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30. The Pulse Transfer Function
The z transform of y(kT)
Pulse transfer function
to the Kronecker delta input
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31. The Pulse Transfer Function
Starred Laplace Transform of the Signal involving both Ordinary and Starred Laplace Transform
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32. The Pulse Transfer Function
General Procedures for Obtaining Pulse Transfer Functions
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33. The Pulse Transfer Function
Pulse Transfer Function of Cascaded Elements
Note that
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34. The Pulse Transfer Function
Pulse Transfer Function of Closed-Loop Systems
Refer to Table 3-1
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35. The Pulse Transfer Function
Table 3-1: Five typical configurations for closed-loop discrete-time control systems
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36. The Pulse Transfer Function
Pulse Transfer Function of a Digital Controller
The input to the digital controller is e(k) and the output is m(k)
The z transform of the equation
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37. The Pulse Transfer Function
Closed-loop Pulse Transfer Function of a Digital Control System
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38. The Pulse Transfer Function
Pulse Transfer Function of a Digital PID Controller
The PID control action in analog controllers
Discretization of the equation to obtain the pulse transfer function
Define
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39. The Pulse Transfer Function
Pulse Transfer Function of a Digital PID Controller(cont.)
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40. The Pulse Transfer Function
Obtaining response between consecutive sampling instants
Laplace transform method
Modified z transform method
State-space method
Laplace Transform Method
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41. Realization of Digital Controllers and Digital Filters
A computational algorithm that converts an input sequence of numbers into an output sequence in such a way that the characteristics of the signal are changed in some prescribed fashion.
The block diagram realization of digital filters using delay elements, adders, and multipliers will be discussed.
The general form of the pulse transfer function
PID controller
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42. Realization of Digital Controllers and Digital Filters
Direct Programming
From the diagram
Total number of delay element: m+n
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43. Realization of Digital Controllers and Digital Filters
Standard Programming
The number of delay elements is minimized.
from n+m to n
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44. Realization of Digital Controllers and Digital Filters
Standard Programming (cont.)
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45. Realization of Digital Controllers and Digital Filters
Standard Programming (cont.)
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46. Realization of Digital Controllers and Digital Filters
Standard Programming (cont.)
It is important to have a good level of accuracy. There are three sources of errors affect the accuracy.
The error due to the quantization of the input signal into a finite number of discrete levels.
The error due to the accumulation of round-off errors in the arithmetic operations in the digital system.
The error due to quantization of the coefficients of the pulse transfer function.
For decomposing higher-order pulse transfer functions in order to avoid coefficient sensitivity problem, the following three approaches are commonly used:
Series programming
Parallel programming
Ladder programming
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47. Realization of Digital Controllers and Digital Filters
Series Programming
A series connection of first-order and/or second-order pulse transfer functions.
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48. Realization of Digital Controllers and Digital Filters
Parallel Programming
Expanding the pulse transfer function into partial fractions.
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49. Realization of Digital Controllers and Digital Filters
Parallel Programming (cont.)
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50. Realization of Digital Controllers and Digital Filters
Ladder Programming
Expanding the pulse transfer function into the continued-fraction form
Define
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51. Realization of Digital Controllers and Digital Filters
Ladder Programming (cont.)
Note that
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52. Realization of Digital Controllers and Digital Filters
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53. Realization of Digital Controllers and Digital Filters
Infinite-Impulse Response Filter and Finite-Impulse Response Filter
Digital filters may be classified according to the duration of the impulse response.
Infinite-Impulse Response Filter
Not all ai’s are zero.
An infinite number of nonzero samples.
Also called recursive filter
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54. Realization of Digital Controllers and Digital Filters
Infinite-Impulse Response Filter and Finite-Impulse Response Filter (cont.)
Finite-Impulse Response Filter
The impulse response is limited to a finite number of samples defined over a finite range of time intervals.
The impulse response sequence is finite.
Also called nonrecursive filter
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55. Realization of Digital Controllers and Digital Filters
Realization of a Finite-Impulse Response Filter
Suppose we decide to employ the N immediate past values of the input and the current input.
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56. Realization of Digital Controllers and Digital Filters
Realization of a Finite-Impulse Response Filter (cont.)
The characteristics of the finite-impulse response filter:
The finite-impulse response filter is nonrecursive. Because of the lack of feedback, the accumulation of errors in past outputs can be avoided in the processing of the signal.
Implementation of the finite-impulse response filter does not require feedback, so the direct programming and standard programming are identical. Also, implementation may be achieved by high-speed convolution using the FFT.
The poles of the pulse transfer function of the FIR filter are at the origin, and therefore, it is always stable.
If the input signal involves high-frequency components, then the number of delay elements needed in the finite-impulse response filter increases and the amount of time delay becomes large.
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