Identification of Transfer Functions in Feedback System and Inverse Problem.

Identification of transfer function matrices in a feedback system is discussed by using the singular value decomposition of Hankel matrix from the viewpoint of inverse problems. A method of model reduction is considered, and selection criteria are proposed for identification of them. Transformation formula between open loop and closed loop transfer function matrices are determined from the feedback loop structure, and they are needed for identification of open loop transfer function matrices under such a condition where the feedback system is in a minimum phase. Though the identifiability of open loop transfer function matrices can be examined in the framework of innovation model equivalent to the feedback system, there are pole-zero cancellations in the identification of them. The method to reduce a model order of an open loop transfer function is discussed by using the singular value decomposition of a gramian given by the open loop transfer function with higher degree. To check reliability of the present algorithm, a simulation study is performed for an example.

Although the dynamic characteristics of the baroreflex system have been described by baroreflex transfer functions obtained from open-loop analysis, the predictability of time-series output dynamics from input signals, which should confirm the accuracy of system identification, remains to be elucidated. Moreover, despite theoretical concerns over closed-loop system identification, the accuracy and the predictability of the closed-loop spontaneous baroreflex transfer function have not been evaluated compared with the open-loop transfer function. Using urethane and α-chloralose anaesthetized, vagotomized and aortic-denervated rabbits (n = 10), we identified open-loop baroreflex transfer functions by recording renal sympathetic nerve activity (SNA) while varying the vascularly isolated intracarotid sinus pressure (CSP) according to a binary random (white-noise) sequence (operating pressure ± 20 mmHg), and using a simplified equation to calculate closed-loop-spontaneous baroreflex transfer function while matching CSP with systemic arterial pressure (AP). Our results showed that the open-loop baroreflex transfer functions for the neural and peripheral arcs predicted the time-series SNA and AP outputs from measured CSP and SNA inputs, with r2 of 0.8 ± 0.1 and 0.8 ± 0.1, respectively. In contrast, the closed-loop-spontaneous baroreflex transfer function for the neural arc was markedly different from the open-loop transfer function (enhanced gain increase and a phase lead), and did not predict the time-series SNA dynamics (r2; 0.1 ± 0.1). However, the closed-loop-spontaneous baroreflex transfer function of the peripheral arc partially matched the open-loop transfer function in gain and phase functions, and had limited but reasonable predictability of the time-series AP dynamics (r2, 0.7 ± 0.1). A numerical simulation suggested that a noise predominantly in the neural arc under resting conditions might be a possible mechanism responsible for our findings. Furthermore, the predictabilities of the neural arc transfer functions obtained in open-loop and closed-loop conditions were validated by closed-loop pharmacological (phenylephrine and nitroprusside infusions) pressure interventions. Time-series SNA responses to drug-induced AP changes predicted by the open-loop transfer function matched closely the measured responses (r2, 0.9 ± 0.1), whereas SNA responses predicted by closed-loop-spontaneous transfer function deviated greatly and were the inverse of measured responses (r, −0.8 ± 0.2). These results indicate that although the spontaneous baroreflex transfer function obtained by closed-loop analysis has been believed to represent the neural arc function, it is inappropriate for system identification of the neural arc but is essentially appropriate for the peripheral arc under resting conditions, when compared with open-loop analysis.

Identification of Transfer Function in a Feedback System and Autoregressive Analysis

Identification of transfer functions and statistical inverse problems

A method is presented for synthesizing a feedback system under the constraints that the open-loop transfer function must have specified K <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">v</inf> and contain real poles at prescribed locations. The method is based on examination of the inverse root-locus plot for the closed-loop poles and zeros. Algebraic equations are obtained for the open-loop pole and zero locations. Examples are given for systems through fourth order in which the resulting linear algebraic equations are readily solved for the required compensation poles and zeros.

Spontaneous stabilization of HTGRs without reactor scram and core cooling—Safety demonstration tests using the HTTR: Loss of reactivity control and core cooling

On closed-loop system identification using polyspectral analysis given noisy input–output time-domain data

From the viewpoint of stochastic inverse problem, identification of transfer functions in feedback systems has been examined by the previous papers[1][2]. In this paper it is applied for a new approach to electroencephalography (EEG). Some transfer functions between measurement regions of EEG are identified by our method of inverse problem in a stochastic feedback system. In our method for stochastic inverse problem an innovation model must be self-consistent: Since an innovation model equivalent to correlation functions has minimum phase and suitable properties from the theoretical considerations[3], innovation models with their properties are selected for dynamic analysis of EEG. Hence, a stochastic feedback model in the stationary process can be determined from EEG time series data of measurement regions, and we can obtain transfer functions between measurement regions.

Reactor noise in accelerator-driven systems (ADSs) is different from that in critical or radioactive source-driven subcritical systems because of the periodically pulsed source and its non-Poisson character. In two earlier papers, we developed a theory of ADS reactor noise, incorporating these features. The non-Poisson character of the source does not permit the use of the forward Kolmogorov equation or the Bartlette formula, two commonly used techniques in traditional noise theory. The method used in these papers was a probability-generating function combined with the linear character of the reactor noise in zero-power systems. In this paper we develop the Langevin approach to reactor noise in ADSs. Apart from being simpler, the Langevin approach allows treatment of feedback effects arising in ADSs with significant power as well as other noise sources, if any. We demonstrate that it is possible to obtain the correct expressions for various noise descriptors using this approach. The method is then applied to treat correlated non-Poisson pulsed sources with a finite pulse width including delayed neutrons. The present paper complements and expands our earlier discussions of ADS reactor noise.

Canadian experience in the area of reactor noise

A new approach to ranking the coolants of a liquid metal fast reactor (LMFR) by degree of preference is proposed and implemented. The approach involves comparing the optimal LMFR layouts with different coolants and compatible structural materials. The layouts were obtained in solving mathematical programming problems with restrictions in the same formulation. The problem is solved in three stages. At the first stage, on the basis of an elementary analysis of the physicochemical properties and cost, unacceptable metals with a high cost (rare metals), a small wide range of operating temperatures, a low boiling point, and high induced activity were excluded. The second stage involves obtaining the optimal layout of LMFR with different coolants. Then, using lexicographic methods for solving multicriteria problems, a coolant ranking procedure is carried out based on a comparison of the optimal LMFR layouts. The third stage involves the analysis of preferred options. Depending on the power and purpose of the reactor, the most preferred coolants are different. For medium and high power reactors, lead extracted from thorium ores is most preferred. For low power reactors, lead from polymetallic and lead ores is preferred. In such reactors, the use of lead of uranium ores is possible. For high- and very-high-temperature low power reactors, gallium and its alloys, including gallium-lead alloys, are preferred. For space reactors, a coolant based on the eutectic Na-K-Cs alloy is more preferred.

The aim of this work is to generalize to the case of weighted L 2 spaces some results about L 2 approximation by analytic and rational functions which are useful to perform the identification of unknown transfer functions of stable (linear causal time-invariant) systems from incomplete frequency data.KeywordsHardy spacesweighted rational approximationfrequency domain identification.

Of all the closed-loop poles of feedback control systems, the three or sometimes two closed-loop poles in the neighbourhood of the cross-over frequency ωc at which the open-loop transfer function, G(s) reduces to unity in magnitude plays the decisive role in shaping the major form of the transient response of the system. However, the evaluation of the closed-loop real poles of higher-order feedback systems in the vicinity of the cross-over frequency ωc (without use of a digital computer) is highly time-consuming and tedious. This paper presents a procedure entitled “Iterative deviation averaging method” for the rapid evaluation of the real-closed loop poles in the neighbourhood of ωc, The procedure is illustrated with a simple example.

We consider n-input n-output convolution feedback systems characterized by y = G * e and e = u - y, where the open-loop transfer function ĝ contains a finite number of unstable poles and the expansion ĝa(s) + ∑i=0 ĝ Gi exp(-sti). The latter is such that, i) ĝa is the Laplace transform of L1 functions, ii) t0 = 0, iii) the delay-times ti, for i = 1, 2, ..., are positive and iv) the coefficient matrices Gi form an absolutely convergent series, i.e. ∑i=0 ĝ |Gi| < ∞. Thus the subsystem represented by G is an open-loop unstable distributed parameter system. One theorem is obtained. It gives necessary and sufficient conditions for stability. A basic device is the following: the sum of the principal parts of the Laurent expansions of ĝ at the unstable poles is factored as a ratio of two right coprime polynomial matrices. There are two necessary and sufficient conditions, the first is the usual infimum one, i.e. infRe s ĝ 0 |det[I+ ĝ(s)]| > 0, and the second is required to prevent the closed-loop transfer function from being unbounded in some small neighborhood of each open-loop unstable pole. The latter condition has a nice interpretation in terms of McMillan degree theory. The modification of the theorem for the discrete-time case is immediate. The same is true if, instead of unity feedback, constant nonunity feedback represented by a nonsingular matrix is used. This allows us, through the use of the small gain theorem or the contraction mapping theorem, to consider non-linear systems, derived from the original one by introducing a non-linear time-varying gain in the forward loop.

The use of neural networks in system identification is an emerging field. Neural networks have become popular in recent years as a means to identify linear and non-linear systems whose characteristics are unknown. The success of sigmoidal networks in parameter identification has been limited. However, harmonic activation-based neural networks, recent arrivals in the field of neural networks, have shown excellent promise in linear and non-linear system parameter identification. They have been shown to have excellent generalization capability, computational parallelism, absence of local minima, and good convergence properties. They can be used in the time and frequency domain. This paper presents the application of a special class of such networks, namely Fourier Series neural networks (FSNN) to vehicle system identification. In this paper, the applications of the FSNNs are limited to the frequency domain. Two examples are presented. The results of the identification are based on simulation data. The first example demonstrates the transfer function identification of a two-degree-of freedom lateral dynamics model of an automobile. The second example involves transfer function identification for a quarter car model. The network set-up for such identification is described. The results of the network identification are compared with theory. The results indicate excellent prediction properties of such networks.