title: Spectral Analysis of Time-Series Associated with Control Systems
Creators: name: Smith, Karl Leland
Publisher: name: Utah State University
Year: 1965

description: The progress of science is based to a large degree on experimentation. The scientist, engineer, or researcher is usually interested in the results of a single experiment only to the extent that he hopes to generalize the results to a class of similar experiments associated with an underlying phenomenon. The process by which this is done is called inductive inference and is always subject to uncertainty. The science of statistical inference can be used to make inductive inferences for which the degree of uncertainty can be measure in terms of probability. A second type of inference called deductive inference is conclusive. If the premises are true, deductive inference leads to true conclusions. Proving the theorems of mathematics is an example of deductive inference; while in the empirical sciences, inductive inference is used to find new knowledge.
In engineering and physical science, analytical , i.e., deterministic techniques have been developed to provide deductive descriptions of the real world. Sometimes the assumptions required to make deterministic techniques appropriate are too restrictive since no provision is made for stochastic or uncertainty involved in concluding real world situations. In these situations, the science of statistics provides a basis for generalizing the results of experiments associated with the phenomena of interest.
In order to make statistical inference sound, the experimenter must decide in advance which factors must be controlled in the experiment. The factors which are unknown or which cannot be controlled must be controlled by the device of randomization. Uncontrolled factors express themselves as experimental error in the experiment. Randomization is used in order to insure that the experimental error satisfies the probability requirements specified in the statistical model for the experiment, thereby making it possible for the experimenter to generalize the results of his experiment using significance and confidence probability statements.
Much of statistics is devoted to situations for which experiments are conducted according to schemes of restricted randomization. Therefore, the experimental errors are independent and are assumed to have a common, yet unknown, probability distribution that can be characterized by estimating the mean and the variance.
However, there are certain other types of experimental situations for which it is desirable to observe a physical phenomena with the observations ordered in time or space. The resulting observations can be called a time series.The experimental errors of a time series are likely to be correlated. Consequently, if an unknown probability distribution is to be characterized, covariances as well as the respective means and the variances must be estimated.
A time series resulting from observation of a given physical phenomena may exhibit dominant deterministic properties if the experiment can be well controlled. Or, the time series may exhibit dominant statistical properties if it is impossible or impractical to isolate and control various influencing factors. Generally an experiment will consist of both deterministic and statistical elements in some degree in a real world situation.
The procedures of analysis presented in Chapter III consider the statistical analysis of periodic and aperiodic digital (discrete) time series, in both the time and frequency domains, using Fourier analysis, covariance and correlation analysis, and the estimation of power and cross power spectral density functions.
Time ordered observations are important in the analysis of engineering systems. Certain characteristics of engineering systems are discussed in Chapter IV, and the input-output concept of control system engineering introduced. The input-output technique is not limited to control system engineering problems, but may be applicable in other areas of science also.
A deterministic method of ascertaining the output performance of an engineering system consists of subjecting the system to a sinusoidal input function of time, and then measuring the output function of time. If the engineering system is linear, the well-developed techniques are available for analysis; but if the system is nonlinear, then more specialized analysis procedures must be developed for specific problems.
In a broad sense, the frequency-response approach consists of investigating the output of a linear system to sinusoidal oscillations of the input. If the system of nonlinear, then the frequency-response approach must be modified; one such modification is the describing function technique. These techniques are also discussed in Chapter IV.
Under actual experimental conditions, the deterministic approach of subjecting a system to a sinusoidal input function for purposes of analysis is likely to be complicated by nonlinearities of the system and statistical characteristics of the data. The physical characteristics of the data will undoubtedly be obscured by random measuring errors introduced by transducers and recording devices, and uncontrollable environmental and manufacturing influences. Consequently, generalized procedures for analyzing nonlinear systems in the presence of statistical variation are likely to be required to estimate the input-output characteristics if the system is to work with inferential models applied to recorded data. Such procedures are presented in Chapter III and Chapter V.
In Chapter V the empirical determination from input-output rocket test data of a deterministic and statistical model for predicting rocket nozzle control system requirements is complicated by the fact that the control system is nonlinear and the nozzle data is non-stationary consisting of both systematic and random variation. The analysis techniques developed are general enough for analysis of other types of nonlinear systems.
If the nonlinear effect of coulomb friction can be estimated and the responses are adjusted accordingly, the nozzle system bears a close relationship to a linear second order differential equation consisting of an acceleration times moment of enertia component, a gas dynamic spring component and a viscous friction component. In addition, vibration loading is present in the data. Consequently, estimation of auto correlation and power spectral density functions is used to isolate these vibrations.
Analysis of the control system data is also considered in terms of auto correlations, and in terms of a power spectral density functions. Random input functions rather than sinusoidal input functions may be required under more general experimental conditions.
Chapter VI numerically illustrates the analysis procedures. The actual rocket test data used in developing the analysis was classified; consequently, only fictitious data are used in this paper to illustrate the procedures.
Chapter VIII is concerned with illustrating the procedures of Chapter III utilizing various time series data. The last part of Chapter VII is concerned with estimation of the power spectral function using techniques of multiple regression; i.e., the model of the General Linear Hypothesis. A definite limitation is the model assumption concerning the residual error of the model. The assumption concerning the error of the model can probably be made more tenable by suitable transformation of either the original time series data or the autocovariances. In any even the spectral function developed by assuming the model for the General Linear Hypothesis gives the same spectral function as defined in Chapter III. However, such quantities as the variance, tests of hypotheses and variance of the spectral function can now be estimated, if the assumptions concerning residual error are valid.
Chapter VIII summarizes the results of previous chapters.