ME280 “Fractional Order Mechanics” Fractional Order Modeling: Part-1

1. ME280 “Fractional Order Mechanics” Fractional Order Modeling: Part-1 YangQuan Chen, Ph.D., Director, MESA (Mechatronics, Embedded Systems and Automation)Lab MEAM/EECS, School of Engineering, University of California, Merced E: yqchen@ieee.org; or, yangquan.chen@ucmerced.edu T: (209)228-4672; O: SE1-254; Lab: Castle #22 (T: 228-4398) 10/22/2013. Thursday 09:00-10:15, KL217

2. Integer-Order Modeling • Introduction • Preliminaries • Non-parametric methods • Transient Analysis • Impulse Response • Step Response • Frequency Domain • Frequency Response Analysis • Fourier Analysis • Correlation Analysis • Pseudo Random Binary Sequence ME280 "Fractional Order Mechanics" @ UC Merced

3. Parameter estimation of dynamic models • Tailor-made & Ready-made models • Linear ready-made models • Linear regression • Least square estimation • Identification Experiment • Choice of input, sampling frequency etc. • Identification under closed loop • Post treatment of data • Model Quality • Test of linearity • Residual analysis • Choice of model structure ME280 "Fractional Order Mechanics" @ UC Merced

4. References • Ljung, L, System Identification – Theory for the users, Prentice Hall, 1987 • Ljung, L, The MATLAB System Identification Toolbox, The Mathworks Inc. http://www.mathworks.com • Ljung, L and Glad, T, Modeling of Dynamic Systems, Prentice Hall, 1994 • Soderstrom, T, System Identification, Prentice Hall 1989 • Astrom, KJ and Wittenmark, B, Adaptive Control, Chapter 3, Addison Wesley, 1989 • Matlab System Identification Toolbox • Read the tutorial chapter of the User’s Guide. • Run the Demos. ME280 "Fractional Order Mechanics" @ UC Merced

5. What is System Identification? Means to get relevant system description from observed data Resultant system description Model Definition of a System Just an object whose properties we want to study Most things in our environment can be classified as a system Mechanical Devices Electrical Circuits Chemical Process Ecology Stock Market Human Being …… Introduction ME280 "Fractional Order Mechanics" @ UC Merced

6. Introduction • Static versus Dynamic Systems • Static System: • The output of the system depends only on the current input • Dynamic System: • The output depends on the input and past values of the input and output • Our interest lies in the dynamic systems only • Simple Example of a dynamic system: • An RC circuit • ‘y(t)’ depends on input ‘u(t)’ from = 0 to = T

7. How to study the properties of a System? Direct means: Perform an experiment and reach a conclusion based on the results obtained We may have to take “trial and error” approach Can an experiment be always performed? It may be too expensive to perform an experiment It may be dangerous The system is yet to be constructed If, for some reason, the experiment can not be performed Make a model of the system The model should represent certain desirable aspects of the system We study the model to understand the characteristics of the system Introduction ME280 "Fractional Order Mechanics" @ UC Merced

8. Application of Models: Prediction Halley’s comet: was seen in 1531, 1607 and 1682 In 1704, Halley predicted that the comet would re-appear in 1758 That prediction is still in use Control Design of a feedback controller begins with the model of the dynamics system to be controlled Fault diagnosis Models used to indicate faults Simulation based training Training using the actual system may be expensive or dangerous Aircraft simulator Space vehicle simulator Introduction ME280 "Fractional Order Mechanics" @ UC Merced

9. Introduction • Different forms of the model • Mental or cognitive model: • Knowing the characteristics of a good friend • Verbal model: • Demand-Supply-Price relationship • If demand goes up, supply remaining constant, price goes up • Physical model: • 3-D models of an apartment shown to the customer • Aircraft/ship model used in flow tunnels • Mathematical model: • Relationship between input and output shown using mathematics • We can use well-defined equations (parametric) or graphs (non-parametric) ME280 "Fractional Order Mechanics" @ UC Merced

10. Given a mathematical model, how do we understand the properties of the system Analytical solution Possible for simple models If a first order ODE is used to represent the dynamics, we can solve the ODE for the response of the system Difficulty increases with increase in model order and with presence of non-linearity Simulation Use the power of digital computers to simulate the behavior of the system Powerful computers are affordable these days and have found wide use for simulation Introduction ME280 "Fractional Order Mechanics" @ UC Merced

11. In order to perform simulation, we need a mathematical model in hand How can we get the model? Two approaches for model building Physical modeling Use laws of nature (Newton’s law, Kirchoff’s law) We still need to know the values of the parameters (such as mass, resistance etc.) Identification Use observations from the system in order to fit a model to the system Introduction ME280 "Fractional Order Mechanics" @ UC Merced

12. Preliminaries • Linear Systems: • The response to a scaled sum of admissible inputs is the scaled sum of the responses to the individual inputs • Time Invariant Systems: • Systems whose structures remain fixed with time • The response of the system does not depend on absolute time • Causal Systems: • “No cause, then no effect” • Output at any point of time depends on the inputs up to that point of time only • Non-anticipative • Most important class of dynamic systems considered is the class of linear, time-invariant, causal systems

13. Continuous-time Systems Input and output signals of the system are time-continuous Most of the systems around us are continuous time system Underlying time-structure is defined on the continuum of real numbers Discrete-time Systems Signals are defined at discrete point of time Inventory of a car dealer, census data etc. are natural examples of discrete time systems Sampling of a continuous time signal generates a discrete sequence of data Studying a continuous time system with the help of digital techniques gives a discrete time system Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

14. How to describe a continuous time system? Differential equations Transfer functions State space equations Frequency response Impulse response Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

15. Differential Equations The input-output relationship is described by a differential equation of order n: A spring-mass system can be described by the following 2nd order system Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

16. State Space Equations Rewriting the differential equation of order n into a system of n first-order differential equations Let us take the second order differential equation describing the mass-spring system We define 2 state variables: the displacement and rate of change of displacement Then we can write Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

17. Preliminaries • In a more general form for state space description: • ‘x’ is the state vector (nx1), ‘y’ is the output vector (px1), and ‘u’ is the input vector (mx1) • A (nxn), B (nxm), C (pxn), and D (pxm) matrices describes the dynamics of the system. Often, D=0 • This general description is for a system with ‘m’ inputs and ‘p’ outputs • For a Single Input Single Output (SISO) system, m=1, p=1 • B is a column vector and C is a row vector ME280 "Fractional Order Mechanics" @ UC Merced

18. Preliminaries • Transfer Function • If U(s) and Y(s) are the Laplace Transform of the input and output signals u(t) and y(t), the system can be described by an algebraic equation • Differentiation in time-domain is equivalent to multiplying by ‘s’ in s-domain • The mass-spring system dynamics can be re-written as ME280 "Fractional Order Mechanics" @ UC Merced

19. Preliminaries • For a SISO system, the transfer function is in general a rational function of ‘s’: • Zeros are the values of ‘s’ where G(s)=0. These are the roots of the numerator polynomial of the transfer function • Poles are the values of ‘s’ where G(s)=. These are the roots of the denominator polynomial of the transfer function • State space description is related to the transfer function description through ME280 "Fractional Order Mechanics" @ UC Merced

20. Frequency Response Steady state response of an LTI system to sinusoidal input If a stable LTI system (all its poles are strictly in the negative half of the complex plane) is excited by a sinusoidal input of frequency  rad/sec, the output will be a sinusoid of the same frequency, with possible change is magnitude and phase The function ‘G(j)’ is the freq. response of the system The frequency response is related to the transfer function through evaluation of the later on the imaginary axis Bode plots are used for graphical representation of frequency response Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

21. Impulse Response The impulse response of an LTI system is its output (function of time) when the system is subject to an impulse input Consider a system having transfer function G(s) Taking Inverse Laplace Transform, * represents the convolution operation When u(t) is an impulse Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

22. Preliminaries • Transfer function ‘G(s)’ of a system is related to the impulse response ‘g(t)’ through Laplace Transform • Transfer function and the impulse response in graphical representation are non-parametric representation of the dynamic system • All the forms of system description can be converted from one form to another

23. Preliminaries • Discrete-Time Systems • Signals are defined at discrete point of time • Inherent discrete system or a sampled continuous system • Discrete-time model defines the relationship between the input and output of a system, where these signals are represented by two sequences • Discrete-time system can be described in the ways similar to those used in continuous time: • Difference equation • State space • Transfer function • Frequency response • Impulse response ME280 "Fractional Order Mechanics" @ UC Merced

24. Difference Equation Specific case: K=3, M=3 State space description: Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

25. Transfer Function If the input and output sequences of a system have the following z-Transform Then, they are related by G(z) is the transfer function of the discrete time system For single-input single-output system, G(z) is a rational function of ‘z’ The roots of the numerator are the zeros of the TF The roots of the denominator are the poles of the TF Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

26. Preliminaries • For multi-input multi-output system, G(z) is a matrix of TFs • In Laplace transform, ‘s’ relates to the differential operator and in z-transform, ‘z’ relates to the difference or shift operator • TF and difference equation are inter-changeable • Discrete transfer function is related to the discrete state space description by – ME280 "Fractional Order Mechanics" @ UC Merced

27. Frequency response The frequency response of a discrete time system is obtained from its transfer function, by evaluating the later for It is clear that the frequency response of a discrete system depends on the sampling interval (sampling period) ‘T’ The sampling process effectively maps the frequency line into a circle where the frequency must be represented by a repeating angular variable Angular frequency of 2/T is indistinguishable from zero frequency Impulse response Response of a discrete time system when input is an unit impulse Its related to the transfer function through z-transform Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

28. Preliminaries • Effect of sampling: • What happens when we try to get a discrete model of a continuous time system from the sampled data of its input and output? • Time domain representation of discrete-time signal • Discrete-time sequence {x[n]} is generated by periodically sampling a continuous-time signal at uniform time intervals • Aliasing: • A continuous time sinusoidal signal of higher frequency acquires the identity of a sinusoidal sequence of lower frequency after sampling ME280 "Fractional Order Mechanics" @ UC Merced

29. Preliminaries • A family of continuous time sinusoids leads to identical sampled signals • The example shown here for 3 Hz, 7 Hz, and 13 Hz sinusoids sampled at 10 Hz • Effect of sampling can also be analyzed using signal spectrum

30. Preliminaries • Spectrum of a signal describes the frequency content of it • Frequency domain representation of a signal • Transform domain representation • Fourier Series • If the signal is periodic with a period of ‘T’ and extends in time from - to + , then it can be written as the sum of sines and cosines

31. Preliminaries • Fourier Transform: • Fourier series expansion is too restrictive • Practical signals are not always periodic, and are not available over infinite range of time • Let us take the example of an isolated pulse • We can’t get Fourier series • However, it can be approximated to be part of a periodic signal, as shown in the lower figure • Approximation is improved by increasing the period

32. Preliminaries • Fourier Transform: • As the period increases, • Corresponding fundamental frequency in Fourier Series becomes smaller • Frequency difference between consecutive terms in the series becomes smaller • In the limit of infinite ‘T’ • The approximation becomes perfect • The sum of the series can be converted to an integral over d • Discrete coefficient converts to a continuous function • Fourier transform and its inverse are expressed by the following integrals, ME280 "Fractional Order Mechanics" @ UC Merced

33. Preliminaries • Fourier Transform: • Fourier Transform of unit impulse, (t) • Taking the inverse Fourier transform, ME280 "Fractional Order Mechanics" @ UC Merced

34. Preliminaries • Power Spectrum of a signal • Square of the magnitude of the Fourier transform of the signal • The spectrum is an even function for the frequency axis and, therefore, plotted only on the positive frequency axis • Phase information is lost in the squaring process, different time domain function may have the same power spectrum. • It is generally not possible to regenerate the time signal from the power spectrum alone ME280 "Fractional Order Mechanics" @ UC Merced

35. Preliminaries • Discrete-Time Fourier Transform • The discrete-time Fourier transform of a sequence x[n] is defined by, • For all practical purposes, the data will be of finite length • For length-N sequence, only N values of X(ej), called the frequency samples, at N distinct frequency pints are sufficient to determine x[n] • Discrete Fourier Transform (DFT) ME280 "Fractional Order Mechanics" @ UC Merced

36. Preliminaries • Effect of sampling on signal spectrum • Consider a continuous time signal ‘xc(t)’ • Its Fourier transform is, • A sequence x[n] is generated by sampling this signal at regular interval of ‘T’ • Discrete time Fourier Transform of the sequence is • We want to know the relations between these two spectra ME280 "Fractional Order Mechanics" @ UC Merced

37. Preliminaries • Sampling operation can be treated as multiplication of the continuous time signal by a periodic impulse train • There are two different ways to find the frequency spectrum of the sampled signal • We shall omit the details, and put only the results in the next page

38. Preliminaries • The continuous time Fourier transform of the sampled sequence is, • This is a periodic function with frequency same as the sampling frequency

39. Sampling Theorem: If the spectrum of the continuous time signal is zero outside the Nyquist limit, given below, the spectrum of the sampled signal is the same as the original one If the continuous time signal has frequency component lying outside this limit, they are folded within this band and mis-interpreted as lower frequency signal. This is aliasing. Pre-filtering before sampling ensures insignificant energy in the frequency band outside Nyquist interval Preliminaries ME280 "Fractional Order Mechanics" @ UC Merced

40. Preliminaries • Disturbances • We can not perform analysis of a system’s characteristics by concentrating only on the input and output signals • There are signals which are beyond our control  Disturbances • Sources of disturbance: • Measurement Noise • Uncontrollable input • The disturbance in a system may come from various sources, but their effect can be combined into one disturbance signal • Undisturbed signal z(t) corrupted by disturbance v(t) • When studying the model, for example through simulation, we must give realistic values to the disturbances • Can we provide a mathematical description for the disturbance? ME280 "Fractional Order Mechanics" @ UC Merced

41. Preliminaries • The disturbance signal ‘v(t)’ is often described as the output of a linear system, whose input is white noise • Disturbance Model • White Noise is a special class of stochastic signal, characterized by a sequence of N uncorrelated, identically distributed stochastic variable with mean and covariance function • The covariance functions of white noise obeys ME280 "Fractional Order Mechanics" @ UC Merced

42. Transient Analysis (Time domain method) Impulse Response Analysis Step Response Analysis Impulse Response Weighting function Fundamental in the description of a linear system Complete characterization of the input-output map of a linear time invariant system Nonparametric Method: Transient Analysis ME280 "Fractional Order Mechanics" @ UC Merced

43. Nonparametric Method: Transient Analysis • Impulse Response Analysis • Transient response analysis with an impulse applied at the input of the system • The observation at the output of a system when the input is an impulse, is the impulse response of the system ME280 "Fractional Order Mechanics" @ UC Merced

44. Nonparametric Method: Transient Analysis • Impulse Response Analysis (practical problems) • Ideal Impulse can not be used in practice • Approximated by a narrow pulse ME280 "Fractional Order Mechanics" @ UC Merced

45. Nonparametric Method: Transient Analysis • Impulse Response Analysis (Discrete Time System) • Sampled signal from a continuous time system • For • the input sequence • the output sequence • If the sampling period ‘T’ is normalized to 1, the sequences are • Output of a discrete time system is given by the following • g(k) is the discrete impulse response. Sometime it is written as gk ME280 "Fractional Order Mechanics" @ UC Merced

46. Nonparametric Method: Transient Analysis • Discrete unit impulse signal is defined as • If the input to a system is discrete unit impulse, then the output is • But • The output of the system, whose input is unit impulse, is impulse response of the system ME280 "Fractional Order Mechanics" @ UC Merced

47. Nonparametric Method: Transient Analysis • In presence of disturbance / measurement noise • The output of the system to an impulse input is • If the noise level is high, the measured output deviates from the true impulse response of the system ME280 "Fractional Order Mechanics" @ UC Merced

48. Nonparametric Method: Transient Analysis • Problem to find static gain • Consider 2 weighting functions • The impulse responses of these two with sampling interval 1 are shown • The two output sequences are virtually indistinguishable

49. Nonparametric Method: Transient Analysis • We continue with the same two weighting functions, and take their step responses, as shown here. • There is a considerable difference in the step response • Step response is integral of impulse response. Even small differences in impulse response accumulate to give significant difference in step response • This example illustrates the problem of finding accurate estimates of the static properties (low frequency properties) from an impulse response

50. Nonparametric Method: Transient Analysis • Choice of sampling frequency • Let’s consider two other weighting functions • Their impulse responses are shown • Significant difference in initial response • An adequate assessment of the initial response would require much more frequent sampling • Synchronization between the generation of input and recording of response is also important to assess the initial response ME280 "Fractional Order Mechanics" @ UC Merced