Name approximate methods for studying nonlinear systems. Analysis of nonlinear automatic control systems
- The harmonic linearization method in design is not linear systems automatic control.[Djv-10.7M] Edited by Yu.I. Topcheeva. Team of authors.
(Moscow: Mashinostroenie Publishing House, 1970. - Series “Nonlinear Automatic Control Systems”)
Scan: AAW, processing, Djv format: Ilya Sytnikov, 2014 - BRIEF CONTENTS:
Preface (5).
Chapter I. Theoretical foundations of the harmonic linearization method (E.P. Popov) (13).
Chapter II. A new form of harmonic linearization for control systems with nonlinear hysteresis characteristics (E.I. Khlypalo) (58).
Chapter III. Harmonic linearization method based on assessing the sensitivity of a periodic solution to higher harmonics and small parameters (A.A. Vavilov) (88).
Chapter IV. Determination of amplitude and phase frequency characteristics of nonlinear systems (Yu.I. Topcheev) (117).
Chapter V. Approximate frequency methods for analyzing the quality of nonlinear control systems (Yu.I. Topcheev) (171).
Chapter VI. Improving the accuracy of the harmonic linearization method (V.V. Pavlov) (186).
Chapter VII. Application of the harmonic linearization method to discrete nonlinear control systems (S.M. Fedorov) (219).
Chapter VIII. Application of the asymptotic method of N.M. Krylov and N.N. Bogolyubov in the analysis of nonlinear control systems (A.D. Maksimov) (236).
Chapter IX. Application of harmonic linearization to nonlinear self-tuning control systems (Yu.M. Kozlov, S.I. Markov) (276).
Chapter X. Application of the harmonic linearization method to nonlinear automatic systems with finite state machines (M.V. Starikova) (306).
Chapter XI. An approximate method for studying oscillatory processes and sliding modes in automatic systems with variable structure (M.V. Starikova) (390).
Chapter XII. An approximate study of a pulse-relay control system (M.V. Starikova) (419).
Chapter XIII. Determination of oscillatory processes in complex nonlinear systems with various initial deviations (M.V. Starikova) (419).
Chapter XIV. Application of the harmonic linearization method to systems with periodic nonlinearities (L.I. Semenko) (444).
Chapter XV. Application of the harmonic linearization method to systems with two nonlinearities (V.M. Khlyamov) (467).
Chapter XVI. Amplitude-phase characteristics of relay mechanisms with DC and DC motors alternating current, obtained using the harmonic linearization method (V.V. Tsvetkov) (485).
Applications (518).
Literature (550).
Alphabetical index (565).
Publisher's abstract: This book is part of a series of monographs devoted to nonlinear automatic control systems.
It systematically, quite comprehensively, sets out the theory of nonlinear automatic control systems, based on the method of harmonic linearization. The main attention is paid theoretical foundations harmonic linearization method and its practical applications to continuous, discrete, self-adjusting systems, as well as systems with finite state machines and a tunable structure. Ways to improve the accuracy of the harmonic linearization method by taking into account the influence of higher harmonics are considered. The proposed methods are illustrated with numerous examples.
The book is intended for scientists, engineers, teachers and graduate students of higher educational institutions dealing with automatic control issues.
Let's consider a chemical-technological object, the input of which receives a random signal And(/), and a random process is observed at the output at(/). When using correlation methods to identify linear objects with constant parameters, it is usually assumed (or the test signal is specially selected in this way) that the random functions and (t) And at (t) are stationary and stationary-pair related in a broad sense, i.e. their mathematical expectations are constant, and auto- and cross-correlation functions are functions of not two, but one argument equal to their difference.
When identifying nonlinear dynamic systems, the conditions for the normality of probability densities of functions and (t) And y(t) and their joint probability densities, as a rule, are not satisfied, i.e., the characteristics of an object are determined in conditions where the joint probability densities of the functions and (t) And at(/) are not Gaussian.
Therefore, the conditional probability density function y(t) relatively and (t) will also be non-Gaussian. Regression output random variable with respect to the input random function for given values of the arguments is generally nonlinear, and the correlation of functions And(0 and at (t) heteroscedastic.
Thus, to identify nonlinear objects, correlation methods operating with mathematical expectations and correlation functions of random processes are no longer enough. The error in solving the problem of identifying a nonlinear object using correlation methods used for linear systems is greater, the stronger the regression of functions y(t) relatively and (t) differs from linear and the greater the unevenness mathematical expectation conditional variances.
The problem of identifying nonlinear objects operating under conditions of random disturbances is a very complex mathematical problem, which is currently under development and is still far from being completed. Nevertheless, it is already possible to name a number of methods that, although cannot be considered exhaustive, provide a fairly good approximate solution to the problem of identifying nonlinear objects statistical methods. These methods include: 1) methods based on the use of dispersion and interdispersive functions of random processes; 2) linearization method nonlinear regression in areas of homoscedasticity of the mathematical expectation of the conditional variance of the function y(t) relatively and (t) 3) Wiener approach to identifying nonlinear systems; 4) a method for identifying nonlinear systems based on the use of the apparatus of conditional Markov processes.
Let's briefly look at each of the listed methods.
1. If the dependence between the values of random functions And(0 and at (t) nonlinear, then the correlation coefficient between the values of the random function can no longer serve sufficiently good criterion to measure the strength of the connection between them. Therefore, to characterize the connection between And And at are used
dispersion relations, which are determined through dispersion functions (2, 3].
Mutual dispersion function 0 yU (*, t) for real random functions y(t) And and (t) And auto-dispersion (dispersion) function G„ K (*, m) for a random process And(t) are determined by the relations
Where M( ) - symbol of mathematical expectation; M.
Based on the values defined above p ui, t| uk and R you can build a special TV criterion to test the hypothesis about the linearity of the relationship between signals y and and:
Where P- number of experiments; To- number of intervals in the correlation table. Let's check the hypothesis about the linearity of the relationship between y t And etc for the object discussed in §6.4. Function
N(t), constructed from the input and output implementations of the system, is shown in Fig. 8.2. In this case, the identification problem is reduced to searching for unknown parameters of the object, which are the coefficients of the operator in the Hilbert space. The signal at the system input is expanded into a series of Laguerre subfunctions:
with odds 
Rice. 8.3.
Rice. 8.4.
Here P-th Laguerre function g n(t) is constructed as a product of the Laguerre polynomial ln(t) to exponent:
Note that the Laplace image of the Laguerre polynomials based on (8.19) has the form
This shows that the necessary Laguerre coefficients can be obtained by passing the signal and (t) through a chain of linear dynamic links (see Fig. 8.3).
The operator of a nonlinear system is represented as an expansion in Ermnt polynomials:
which are orthogonal on the real axis - oo t. The Hermite functions are constructed from Hermite polynomials:
with the help of which the transition operator from the Laguerre coefficients of the input signal to the output signal is written in the form
Relationship (8.20) is valid for any nonlinear object and can be used as the basis for its identification. The identification method is greatly simplified if a special signal in the form of Gaussian white noise is applied to the input. In this case, the Laguerre functions are uncorrelated Gaussian random processes with equal variances. In this case, the determination of coefficients... To reduces to finding the cross-correlation function of the system output and the Hermite polynomials:
Determination of odds b(j... To completes the solution to the identification problem. General scheme calculations is shown in Fig. 8.4.
When solving problems of identifying chemical technological objects, the considered method has limited application for a number of reasons. The latter include, for example, difficulties arising when moving from coefficients b tj k to the technological parameters of the object. The method is not suitable for non-stationary systems. Difficulties in implementing this procedure during normal operation of the facility also reduce the effectiveness of the method. Finally, the need to truncate all operations associated with passages to the limit and the replacement of series with finite sums are sources of additional computational errors.
4. Another possible approach to constructing optimal filters for nonlinear systems is based on the use of the apparatus of conditional Markov processes. Let us consider the essence of this approach using a specific example.
EXAMPLE Let the useful signal be a rectangular pulse
the moment of appearance of which t on the segment 0 x T needs to be determined. Pulse height A 0 and its duration h are assumed to be known. The signal arriving at the object is and (t)=s(*)+m> (*) is the sum of the useful component s(0 and white noise w(*), which is described by the probability integral)
