1. President UniversityErwin SitompulSMI 7/1 Dr.-Ing. Erwin Sitompul President University Lecture 7 System Modeling and Identification http://zitompul.wordpress.com

2. President UniversityErwin SitompulSMI 7/2 Chapter 4Dynamical Behavior of Processes Construct an s-Function model of the interacting tank-in- series system and compare its simulation result with the simulation result of the component model from Homework 2. For the tanks, use the same parameters as in Homework 2. The required initial conditions are: h 1,0 = 20 cm, h 2,0 = 40 cm. v1v1 qiqi h1h1 h2h2 v2v2 q1 q1 a1 a1 a2 a2 qo qo Homework 6

3. President UniversityErwin SitompulSMI 7/3 The interacting tank-in-series system can be described by these differential equation: s-Function of Interacting Tank-in-Series Chapter 4Dynamical Behavior of Processes v1v1 qiqi h1h1 h2h2 v2v2 q1 q1 a1 a1 a2 a2 qo qo

4. President UniversityErwin SitompulSMI 7/4 Chapter 4Dynamical Behavior of Processes s-Function of Interacting Tank-in-Series

5. President UniversityErwin SitompulSMI 7/5 Chapter 4Dynamical Behavior of Processes s-Function of Interacting Tank-in-Series Direct comparison between component model and s-function model

6. President UniversityErwin SitompulSMI 7/6 Computer-Controlled Systems Chapter 5Discrete-Time Process Models Computer-controlled system indicates that the control law is calculated by computer. The feedback scheme of such system is shown below: D/A: Digital-to-analog A/D: Analog-to-digital S/H: Sample-and-hold T s : Sampling time, sampling period k: Integer, ≥ 0 A/D

7. President UniversityErwin SitompulSMI 7/7 Chapter 5Discrete-Time Process Models The control error e(kT s ) is given as the difference between the set point signal w(kT s ) and the controlled process output y(kT s ), in digital form, in times specified by the sampling period T s. The computer interprets the signal e(kT s ) as a sequence of numbers and given the control law, it generates a new sequence of control signals u(kT s ) The discretized process represents a system with the input being the sequence of u(kT s ) and the output being the sequence of y(kT s ). Sampled Data System

8. President UniversityErwin SitompulSMI 7/8 Classification of Signals Chapter 5Discrete-Time Process Models Continuous-time signals or analog signals: defined for every value of time they take on in a continuous interval (t 0,t 1 ). In other words, at any given instant an analog signal can take any value. For example, the signal x(t) = sin(t), − ∞ < t < ∞. Discrete-time signals: defined only at specific values of time. These time instants need not be equidistant, but in practice they are usually taken at equally spaced intervals. In other words, the time variable of the signal can take only certain values. The amplitude of the signal can be continuous i.e., can take any value. For example, x(t) = sin(nt), n = 0,1,2,... n. The process of converting an analog signal to discrete-time signal is called sampling. A discrete-time signal is sometimes called a sampled signal. Discrete-valued signals or digital signal: arise when the discrete signals are quantized. A quantized signal assumes only discrete amplitude values. In other words, in these signals both the amplitude and time variable can take only certain values.

9. President UniversityErwin SitompulSMI 7/9 Classification of Signals Chapter 5Discrete-Time Process Models : continuous-time signal (analog signal) : discrete-time signal (sampled signal) : discrete-valued signal (digital signal)

10. President UniversityErwin SitompulSMI 7/10 A/D Converter Chapter 5Discrete-Time Process Models The transformation of a continuous-time signal to a discrete-time signal is done by the A/D converter. A/D

11. President UniversityErwin SitompulSMI 7/11 Chapter 5Discrete-Time Process Models D/A Converter D/A converter with a sample-and-hold implements the transformation of a discrete-time signal to a continuous-time signal that is constant within one sampling period.

12. President UniversityErwin SitompulSMI 7/12 Chapter 5Discrete-Time Process Models S/H Element A possible realization of sample-and-hold is the zero-order hold with the transfer function of the form: The sampling time T s should be chosen in a way so that the process dynamics can be captured correctly. High frequency continuous-time signals require high sampling frequency (f s ), or equivalently, low sampling period T s.

13. President UniversityErwin SitompulSMI 7/13 Chapter 5Discrete-Time Process Models Sampling Period With small sampling period we may captures the dynamics of a system better, but the computational load will be heavier. On the other hand, system with large sampling period may require low computational demand, but useful information might be lost. In order to avoid loss of information but still capture the process dynamics correctly, the following inequality must hold: where T sin is the lowest oscillation period of sinusoidal component of the sampled signal. Nyquist-Shannon Sampling Theorem If a function x(t) contains no frequencies higher than β cycle-per- second, then it is completely determined by giving its ordinates at a series of points spaced 1/2β seconds apart.

14. President UniversityErwin SitompulSMI 7/14 Chapter 5Discrete-Time Process Models Loss of Information Due To Sampling

15. President UniversityErwin SitompulSMI 7/15 Chapter 5Discrete-Time Process Models Ideal Sampler Let us now investigate properties of an ideal sampler. Its output variable y * can be represented as a periodic sequence of δ functions as follows: Let us define ω s = 2 π/T s, and therefore Representation in Fourier Series

16. President UniversityErwin SitompulSMI 7/16 Chapter 5Discrete-Time Process Models Ideal Sampler The output variable of the ideal sampler can then be written as: The Fourier transform of this function if y(0) = 0 is given as:

17. President UniversityErwin SitompulSMI 7/17 Chapter 5Discrete-Time Process Models Ideal Sampler The spectral density function of the variable y(t) is |Y(jω)|, while the spectral density of the sampled signal y * (t) is given as: Substituting s for jω, Sampling result = Sum of series of original signal, shifted nω s away from the original frequency

18. President UniversityErwin SitompulSMI 7/18 Chapter 5Discrete-Time Process Models Ideal Sampler Spectral density of original signal y(t) Spectral density of sampled signal y * (t) ω c : critical frequency ω s : sampling frequency

19. President UniversityErwin SitompulSMI 7/19 Chapter 5Discrete-Time Process Models If ω c is smaller than or equal to half of the sampling frequency, the spectral density of |Y * (jω)| is composed of spectra of |Y(jω)| shifted to the right and left, nω s away. There are no overlapping. If ω c is larger than half of the sampling frequency, then the spectral density of |Y * (jω)| consists of spectra |Y(jω)| shifted to the right and left, nω s away also. But now, there is overlapping. Hence the spectral density of the signal |Y * (jω)| is distorted. ω ωsωs –ω s 0 If ω s < 2ω c, then overlapping occurs. Original signal cannot be reconstructed from the sampled signal. Ideal Sampler

20. President UniversityErwin SitompulSMI 7/20 Chapter 5Discrete-Time Process Models Choosing The Sampling Period The sampling period choice is rather a problem of experience than some exact procedure. Basically, sampling period has a strong influence on dynamic properties of the controlled system, as well as the whole closed-loop system. The following rule of thumbs can be used to determine the sampling period of first- and second-order system: 1 st order τ/4 < T s < τ/2 2 nd orderT n /20 < T s < T n /4, T n = 2π/ω n

21. President UniversityErwin SitompulSMI 7/21 Let us again consider an ideal sampler, as shown below. This sampler implements the transformation of a continuous-time signal f(t) to an impulse modulated signal f * (t). Chapter 5Discrete-Time Process Models Z -Transform Individual impulses appear on the sampler output in the sampling times kT s, k = 0, 1, 2,... and are equal to functions f(kT s ), k = 0, 1, 2,... This impulse modulated signal containing a sequence of impulses is denoted by f * (t), which can be expressed as: The Laplace transform of this function is:

22. President UniversityErwin SitompulSMI 7/22 Chapter 5Discrete-Time Process Models Z -Transform Let us introduce a new variable Then we can write The Z -transform can now be defined as: Z -transform is mathematically equivalent to Laplace transform and differs only in the argument. Z -transform exists only if some z exists such that the series converges for k → ∞.

23. President UniversityErwin SitompulSMI 7/23 Chapter 5Discrete-Time Process Models Properties of Z -Transform Shifting Theorem Initial Value Theorem Final Value Theorem Given the Z -transform of a function, we can find the value of the function in time domain using the inverse Z -transform, but only for each value of sampled time, t = kT s.

24. President UniversityErwin SitompulSMI 7/24 Chapter 5Discrete-Time Process Models Table of Z -Transform

25. President UniversityErwin SitompulSMI 7/25 Chapter 5Discrete-Time Process Models Example Prove the table for the Z -transform of Recalling the formula to calculate the sum of infinite geometric series Then

26. President UniversityErwin SitompulSMI 7/26 Homework 7 Chapter 4Dynamical Behavior of Processes 1.Find the values y(kT) for k = 0 to 4, when 2.We have a function Using a partial fraction expansion of Y(s) and the table given on previous slide, find Y(z) when T s = 0.1 s. NEW