APPROXIMATE ANALYSIS OF NONLINEAR STOCHASTIC SYSTEMS By SHARIF ABDULLAH ASSAF II Bachelor of Engineering American University of Beirut Beirut, Lebanon 1964 Master of Science Oklahoma State University Stillwater, Oklahoma 1972 Submitted to the Faculty of the Graduate College of the Oklahoma State University in partial fulfillment of the requirements for the Degree of DOCTOR OF PHILOSOPHY May, 1975 T~ I 'I 'JS-.D A JSJ'/a... ~~~ . ·l ..... '" OKLAHOMA STATE UNIVERSITY LIBRARY MA.V .1 2 1976 APPROXIMATE ANALYSIS OF NONLINEAR STOCHASTIC SYSTEMS Thesis Approved: Dean of the Graduate College ii PREFACE This study is concerned with the development of a method of calcu lating the statistics of processes in nonlinea:r stochastic systems. The main objective is to evaluate the statistical characteristics of the pro cesses with sufficient accuracy. The accurate values are used to check the accuracy of the method of statistical linearization (MSL). The MSL is a very general and powerful method of nonlinear systems analysis and synthesis. The only drawback of the method has bee.n the lack of a means of checking its accuracy and, therefore, its applicability. The method developed in this study fulfills these needs, to a large extent. I wish to express my appreciation to my major adviser, Dr. Larry Zirkle, for his guidance and assistance throughout this study. Appreci ation is also expressed to the chairman of the thesis corrnnittee, Dr. Richard Lowery, for his encouragement and counsel; to Dr. Henry Sebesta and iJr. Craig Sims, members of the committee, who provided invaluable assistance and criticism throughout my program. I would like to acknowledge the financial support recei ve.d from the School of Mechanical Engineering a.s a graduate re,search assistant and from the Center for Systems Science, iii TABLE OF CONTENTS Chapter Page I. INTRODUCTION 1 Background • • • • • • • . 1 Purpose and Method of Investigation 3 Summary of Main Results . • • • • • • • • • • • 5 II. THE METHOD OF STATISTICAL LINEARIZATION 7 Introduction • • • • • . • • • • • 7 A General Formulation of the MSL 9 The BMSL • • • • • • • • • • • • • 15 Multidimensional Nonlinearities • • • • 17 Applications • • • • • • • • • 23 Stationary Systems . • • 23 o • • • • • • Non-stationary Systems 26 SuD1IIlary • • • • • • • • • • • 28 e • • • • • III. A METHOD OF NONLINEAR ANALYSIS 29 In troduc tj.on • • • • • • 29 A Basic Result • • • • • • 32 Equations for the System Statistics 34 The Moments Equations • • • • 34 Equations for the Characteristic Functions 37 The Semi-invariants Equations 39 The Correlation Functions Equations 41 Computation of the System Statistics 44 A Computational Aspect • • • • • • 46 IV. APPLICATIONS 48 Introduction 48 Application to Zero-Mean Systems • • • • • • • 48 Theoretical Investigation • • 50 Experimental Investigation 64 Application to Non-Zero-Mean Systems 83 v. APPLICABILITY AND DESIGN APPLICATIONS OF THE BMSL 100 Introduction • • • • • . •• 100 Applicability of the BMSL 100 A Modified MSL- • • • • • • 103 iv Chapter Page Design Applications • • • • • • • • 108 Optimum Linear Compensation • • • 108 Optimum Nonlinear Compensation • 111 Other Applications 112 VI. CONCLUSION 113 Conclusions • • • • 113 Problems for Further Investigation 116 A SELECTED BIBLIOGRAPHY • • • • • • • • • • • • • . 119 APPENDIX - APPROXIMATE ANALYTIC REPRESENTATION OF PROBABILITY DENSITY FUNCTIONS • • • • . • • • 122 Introduction • 122 Univariate Distributions 123 Representation by a Series 123 Multivariate Distributions • • • • 144 Series Representation of Multivariate Density Functions • • • • • • • • 145 Probability Density Function of a Random Process • • • • • • • • • . • • • • • • • . • • • 158 Probability Density Function of a Vector of Random. Processes • • • • • • • • • • • • • • • • • 168 Summary and Conclusions • • • • • • • • • . 170 v LIST OF FIGURES Figure Page 1. Correlation Functions of the Output of the Cubic Nonlinearity and tis Equivalent Linear Models • . • . . • . 18 2. Correlation Functions of the Output of the Relay Nonlinearity and its Equivalent Linear Models . . . . • . . 19 3. Spectral Density Functions of the Output of the Relay Nonlinearity and its Equivalent Linear • . . . . . • 20 Models 4. A Closed-Loop System of General Form- Stationary . • . . 24 5. (a) A Simple Nonlinear Feedback System 49 (b) The Equivalent Unit Feedback System • 49 6. A Simple First Order Nonlinear System . 50 7. Nonlinear Characteristics Used with the First Order System with White Noise Input •. 51 8. Input-Error Curves, First Order System with White Noise Input- Cubic 55 9. Amplitude Probabi.lity Density Functions of Error Signal in First Order System- Cubic . • . • • , , . . , 56 Input Error Curves. First Order System with White Noise Input- Ideal Relay • , , • • • , • • , . 57 11. Amplitude Probab.i.lity Dens.ity Functions of Error Signal in First Order System - Ideal Relay 58 12. Input-Error Curves. First Order System with White Noise Input-Relay with Dead Zone , . , . . . 59 13. Amplitude Probability Density Functions of Error Signal in First Order System- Relay with Dead Zone. (a) b=S (b) b•lO ....••... 60 14. Spectral Density Functions of Error Signal in First Order System-Ideal Relay •..... 63 vi Figure Page 15. Nonlinear Feedback Systems Used in Experimental Investigation. (a) First Order (b) Second Order • • . . • 65 16. Nonlinear Characteristics Used in Experimental Investigation , . 65 17. Input-Error Curveso First Order System with Filtered White Noise Input- Cubic Nonlinearity • . . • • • 70 18. Amplitude Probability Density Functions and Histograms of Error Signal in First Order System- Cubic Nonlinearity • . . . • . . . • • • • . . . . 71 19. Input-Error Curfes, First Order System with Filtered White Noise Input- Relay with Dead Zone • • • . . 74 20. Amplitude Probability Density Functions and Histograms of Error Signal in First Order System- Relay with Dead Zone • • • • 75 21. Input-Error Curves. First Order System with Filtered White Noise Input- Limiter Nonlinearity 78 22. Amplitude Probability Density Functions and Histograms of Error Signal in First Order System-Limiter Nonlinearity . , . • • • • . . • • • • • • 79 23. Input-Error Curves. Second Order System with Filtered White Noise Input- Cubic Nonlinearity • • • • • . 84 24. Amplitude Probability Density Functions and Histogram of Error Signal in Second Order System- Cubic Nonlinearity • . . • • • • . . • . • • • 85 25. Input-Error Curves. Second Order System with Filtered White Noise Input - Relay with Dead zone 86 (l • e. " e e • !! • (I (!, • Cl !! ., • 26. Amplitude Probability Density Functions and Histogram of Error Signal in Second Order System- Relay with Dead Zone • . , . . , • . • . • . • . . 87 27. Input-Error Curves. Second Order System with Filtered White Noise Input- Limiter Nonlinearity . . . . • 88 28. Amplitude Probability Density Functions and Histogram of Error Signal in Second Order System- Limiter • • • • • . • • • , • . • . . • . . . 89 29. A simple First Order Non-Zero-Mean System with White Noise Input • . • • • • • , . . • . . • • • • . . • 91 vii Figure Page 30. Input-Output Curves. First Order Non-Zero-Mean System 94 e • • • • • • • • • • • • • • • • 31. Amplitude Probability Density Functions of Output Signal in Non-Zero-Mean System • . • • . 95 32. Spectral Density Functions of Output in Non-Zero- Mean System . . • . . " • . . . . • • . • • • • 96 33. (a) Block Diagram of Original System for the Linear Compensation Problem • • • • • • 110 (b) Equivalent Open-Loop Configuration • • • • • • • 110 34. System Confi.g u.r a.t io.n . fo. r . th.e . N .o n.l i.n e.a r. C.o m.p .e n.sa t.io n. . . . . . Problem 112 35. Exact and Approximate Density .F u.n c.t io.n .s of a U.n i.fo r.m . . . . . Deistribution Over (-1,1) 138 36. Exact and Approximate Density Functions of a Triangular Distribution Over (-1,1) 138 37. Exact and Approximate Density Function of a Chi-Square Distribution • • • • • • • . • • • • • • • • • 139 38. Exact and Approximate Density Functions of a Rayleigh Distribution • • • • • 140 39. Regions of Unimodal Curves and Regions of Non Negative Ordinates • • • • • • • • • • 141 vii! LIST OF SYMBOLS a. Initial moment of order n n Quasi-moment of order n Mathematical expectation operator f(z) Characteristic function of a process Coefficient of skewness Y1 Yz Coefficient of excess G(j w) Transfer function of a linear system g(z) Log characteristic function of a process Chebychev-Hermite polynomials of order k k Static gain corresponding to the mean value 0 Static gain corresponding to centered component Matrix of statics gains for centered components >. Semi-invariant of order n n m Mean value of x x Central moment of order n p Covariance matrix p* (x) Normal density function of independent variables 0 p (x) Normal density function, general 0 p(x) Arbitrary density function ¢(.) Nonlinear characteristic of arbitrary form Statistical characteristic of a nonlinearity cp 0 R (.) Correlation function of x x ix