A TWO-STAGE METHOD FOR SYSTEM IDENTIFICATION FROM TIME SERIES A Thesis Presented to The Faculty of the Fritz J. and Dolores H. Russ College of Engineering and Technology Ohio University In Partial Fulfillment of the Requirements for the Degree Master of Science by Kenneth Allan Nadsady March, 1998 Acknowledgements I would like to thank Dr. Dennis Irwin, my thesis advisor, for his inspiration, guidance, and occasional but caring sternness without which I would have never completed this thesis. I would also like to thank the rest of my Professors at Ohio University, especially Dr. Jerrell Mitchell, Dr. Doug Lawrence, Dr. Larry Snyder, and Dr. Brian Manhire for their help and support throughout my academic experience in Athens. Special thanks go to Denise Ragan and Janelle Baney for the fantastic job they do helping graduate students with the mundane things. Their cheerful attitude and ability to make me laugh during tense times helped tremendously. I would also like to thank my graduate student pals, especially Kramer, Eddie T., Dave Burge, Mark Duncan, and John Shramko. Their success in obtaining the Master of Science Degree helped me to see the light at the end of the tunnel. I am deeply indebted to Paul and Beth Stocker, for providing the funding that supported me in my graduate studies. Lastly, I wish to thank my family for all their love and support throughout my educational career. I can only hope that I will be able to use the knowledge and experience I have gained to help each of them in some way. I dedicate my graduate work in loving memory of my Grandfather, Joe Case. ii Table of Contents Acknowledgements . Table of Contents . ii List of Figures . iii Chapter 1: Introduction ............................. 1 1.1. Motivation for This Thesis . 1 1.2. Thesis Conventions and Organization . 2 Chapter 2: Two-Stage Identification Algorithm . .. 5 2.1. Finding a Valid State Vector Sequence. ................ .. 5 2. 1.1. Improving Computational Efficiency 17 2.2. Finding the System Matrices . .. 20 Chapter 3: Application of Spectral Estimation to the Time Series . 22 3.1. Computing Frequency Response via Spectral Estimation 22 3.2. Incorporating Averaging for Noise Rejection . 24 Chapter 4: Application to a Constructed Example . 26 4. 1. Description of the Example System . 26 4.2. Simulation . 29 4.3. Results from Example System Data. ....... 31 Chapter 5: Application to a Real Sampled-Data System 36 5.1. ACES System Description 36 5.2. ACES Data Description. ................ 37 5.3. Results from ACES Sampled Data 39 Chapter 6: Conclusions and Recommendations . 47 Bibliography 50 Appendices 52 iii List of Figures Figure (4.1): Hankel Matrix Singular Values. .. 32 WT Figure (4.2): Model vs. Experimental Frequency Response, H11(d ). •••••• 33 Figure (4.3): Model vs. Experimental Frequency Response, H12(~wT). .•...• 33 WT Figure (4.4): Model vs. Experimental Frequency Response, H21(i ). •••••• 34 WT Figure (4.5): Model vs. Experimental Frequency Response, H22(d ). •••••• 34 Figure (4.6): Model Frequency Response Relative Error. .. 35 Figure (5.1): Marshall Space Flight Center Large Space Structure Ground Test Facility (ACES Configuration). 37 Figure (5.2): ACES Hankel Matrix Singular Values. 41 Figure (5.3): BGx/AGSx pair 56th Order Model vs. Experimental Frequency Response. .. 43 Figure (5.4): BGy/AGSx pair 56th Order Model vs. Experimental Frequency Response. 44 Figure (5.5): BGx/AGSy pair 56th Order Model vs. Experimental Frequency Response. .. 45 Figure (5.6): BGy/AGSy pair 56th Order Model vs. Experimental Frequency Response. .. 46 1 Chapter 1: Introduction As technology advances, physical systems with which scientists and engineers must deal become increasingly sophisticated. At the same time present economic conditions result in the demand for higher productivity and performance. For these reasons and others, the need for accurate models based on measured observations from the systems is increasing. Models may be used to shorten product development time, decrease development cost, or otherwise provide a means of testing or analysis in a convenient, nondestructive way. In fact, a major part of the engineering field deals with how to·produce good designs based on mathematical models (Ljung, 1987). System identification deals with the problem of building mathematical models of dynamical systems based on observed data from the systems. Flexible mechanical structures can be particularly difficult to model, especially in cases where the structure is itself constructed from many smaller flexible members. This work focuses on the investigation of a two-stage system identification technique and how well this technique performs when applied to systems that exhibit the lightly­ damped modal behavior that is characteristic of flexible structures. 1.1. Motivation for This Thesis The purpose of this thesis and the research upon which it is based is to aid in the development and testing of tools that may be used to compute mathematical models of dynamical systems. In particular, the need for accurate mathematical 2 modeling tools for linear, time-invariant systems with lightly-damped modes is addressed. The fundamental theorems and relationships contained in this thesis come from the work of Moonen, et. ale (1989). The primary advantage of this technique is that it computes explicitly a state­ space realization directly from time-domain data. Other algorithms often require an intermediate calculation step and cannot use time-domain data directly. For example, the Eigenvalue Realization Algorithm (ERA) requires the pulse response of the system as input. If the pulse response is not available, as is often the case, it must be calculated from the time .or frequency domain data. The required pulse response is usually obtained by first applying spectral estimation techniques to the time-domain data to get the frequency response, and then applying inverse Fourier transforms to the frequency domain data (Medina, p42). 1.2. Thesis Conventions and Organization Most of the computations were performed and all of the graphs were generated using the numeric computation and graphics software package, MATLAB®, from The Math Works, Incorporated. MATLAB combines a large library of built-in functions with external libraries of script-files and script-functions (called m-files and m­ functions, respectively) that are compiled at run-time, resulting in a flexible software package that well serves the type of computations required in this research. Matrix, vector, and scalar quantities are not distinguishable by notation but by the context in which they appear. However, submatrices are denoted with the colon 3 notation used by Golub and Van Loan (1989) and the MATLAB User's guide (1992). All MATLAB built-in commands and external m-files and m-functions are capitalized when referred to by name in the text, but when a command is given as entered on the command line it is lower case bold script. Discrete-time signals can be represented as sequences of numbers (or possibly sequences of vectors). For example a sequence of numbers x, in which the nth number in the sequence is denoted x[n], can be formally defmed as x = {x[n]}, n = 1,2, ... ,00. However, for convenience either x or x[n] will be used when referring to the entire sequence. The context should guide the reader to the proper interpretation. In general, the first element of a sequence need not be associated with the index n = 1, since by a change of variables another .index m can be found such that this is so. For example if x = {x[n]}, n = a,a+l, ... ,oo, then X = {x[m]}, m = 1,2, ... 00, ifm = n-a+ 1. Often, such sequences arise from periodic sampling of an analog signal, say xa(t), - 00 < t < 00. In this case, the numeric value of the nth number in the sequence is equal to the value of the analog signal at time nT where T is the sampling interval. For example, x[n] = xa(nT), - 00 < n < 00. Chapter 1 of this thesis is this introductory chapter. Chapter 2 introduces an algorithm that approaches the system identification problem in two stages. In the first stage, a sequence is found that represents consecutive state-space vectors corresponding to the consecutive input and output vectors observed from the system. The state vector sequence found is consistent with the input and output vector 4 sequences. The second stage involves computing a state-space realization of the system from the state and I/O vector sequences. Chapter 3 discusses spectral estimation as a means to experimentally generate a model for comparison and/or validation purposes. The algorithm is applied to an ideal sampled-data system in Chapter 4. Application of the method to an actual sampled-data system appears in Chapter 5. Finally, Chapter 6 draws some conclusions concerning the method and gives some ideas for future work. 5 Chapter 2: Two-Stage Identification Algorithm It is desired to find a state-space realization (A,B,C,D) for the discrete, linear, time-invariant, multi-input, multi-output (MIMO) system described by the following difference equations, shown in matrix form: x[k + 1] = Ax[k] + Bu[k] (2.1) y[k] = Cx[k] + Du[k] where x E lRnx1, y E WX1, U E Rmxl, A E Rnxn, B E Rnxm, C E W xn, and D E wxm. Only a finite record of the input and output data sequences are known. For example, u[k] and y[k] are known for k=O, 1,... ,N. The identification algorithm contains two basic steps. First a valid state-vector sequence is found as a basis for the intersection of the row spaces of two block Hankel matrices, constructed with the input/output I/O time series data. Next the system matrices are found as the least-squares solution to a set of linear equations, formed from the manipulation of the state-vector sequence found in the first step and partitions of the I/O data-containing block Hankel matrices.