Weighted H 2 approximation of transfer functions

Optimal Approximation, Simulation and Analog Realization of the Fundamental Fractional Order Transfer FunctionThis paper provides an optimal approximation of the fundamental linear fractional order transfer function using a distribution of the relaxation time function. Simple methods, useful in systems and control theories, which can be used to approximate the irrational transfer function of a class of fractional systems for a given frequency band by a rational function are presented. The optimal parameters of the approximated model are obtained by minimizing simultaneously the gain and the phase error between the irrational transfer function and its rational approximation. A simple analog circuit which can serve as a fundamental analog fractional system is obtained. Illustrative examples are presented to show the quality and usefulness of the approximation method.

This paper describes a simple procedure for synthesizing an active distributed RC network which, by using dominant poles and zeros, realizes a very accurate approximation of an arbitrary stable rational transfer function. The network uses a single uniformly distributed RC line with taps spaced along its length. A linear combination of tap voltages is added to the input signal to form the driving voltage for the RC line; the output signal is also a linear combination of the tap voltages. The network offers a number of significant advantages. Since it realizes a nearly rational transfer function, the approximation problem can be conveniently solved using readily available results on rational function approximation. Also, the network uses only one uniform RC line, the transfer function can be changed simply by changing resistor values, and the frequency can be scaled by minor connection changes. Thus one standard network with minor modification is useful for a wide variety of applications. This paper develops the design procedure and derives the various sensitivity functions of importance. Two example designs are carried out: an approximation to a second-order low-pass transfer function and an approximation to a second-order band-pass transfer function with a Q of 100. The sensitivities for the examples are very reasonable and the measurements made on laboratory models indicate excellent agreement with theoretical predictions.

The approximation of a fractional order PIλDμ-controller transfer function using a chain fraction theory is considered. Analytical expressions for the approximation of s±α components of the transfer functions of PIλDμ-controllers were obtained through the application of the chain fraction theory. Graphs of transition functions and frequency characteristics of Dμ (α = μ = 0.5) and Iλ (α = λ = −0.5) parts for five different decomposition orders were obtained and analyzed. The results showed the possibility of applying the approximation of the PIλDμ-controller transfer function by the method of chain fractions with different valuesof λ and μ. For comparison, the transfer functions with the same order polynomials, obtained by the methods of Oustaloup transformation and chain fractions, were approximated for α = ±0.5. The analysis proved the advantages of using the chain fraction method to approximate the transfer function of the PIλDμ-controller. The performed approximation opens up the possibility of developing engineering methods for the technical implementation of PIλDμ-controllers. The accuracy of the same order transfer function approximation is higher when the method of chain fractions is used. It has been established that the adequacy of the frequency characteristics of the transfer functions obtained by the chain fraction method also depends on the approximation order.

A unified theory of identification in a feedback system is presented from the viewpoint of inverse problem on reactor noise. A power reactor described by a 4 block feedback model is expressed in terms of equivalent correlation functions as a 5 block feedback model with an additional loop, if one of subsystems of the reactor becomes unstable, however, the total system is stable by negative feedback effects, and if noise sources are not independent. The addition of a loop in the feedback system leads errors in estimation of open loop transfer functions. In the case, modified open loop transfer functions are identified instead of original ones. From the viewpoint of reactor diagnosis, notes on system identification are discussed on an open loop transfer function in a low power reactor and on effects of diagonalization in covariance space of noise sources.

Automated On-Line Frequency Domain Identification and Genetic Estimation of Plant Transfer Functions

A new tuning method for Fractional Order Controllers (FOC) based on Bode’s ideal transfer function as a reference model is proposed in this paper. The proposed fractional order controller parameters are obtained by a simple tuning technique in two steps. In the first step, the open loop reference fractional integrator is designed according to the desired performances, and then approximated by a rational transfer function using Charef’s or Oustaloup’s approximation method. In the second step, a standard pole placement technique is used to align each pole of the plant transfer function with the nearest one of the Rational Function Approximation (RFA). The FOC transfer function is then obtained by gathering the remained poles and zeros of the RFA. The most innovative character of the proposed method is its simplicity and its remarkable performances in terms of robustness towards the variation of the static gain. Simulation of some illustrative examples confirms and validates the proposed method.

Identification of Transfer Function in a Feedback System and Autoregressive Analysis

Identification of transfer functions and statistical inverse problems

This paper deals with the question of the existence of weak and strong common
\nquadratic Lyapunov functions (CQLFs) for stable discrete-time linear timeinvariant
\n(LTI) systems. The main result of the paper provides a simple characterisation
\nof pairs of such systems for which a weak CQLF of a given form exists
\nbut for which no strong CQLF exists. An application of this result to second
\norder discrete-time LTI systems is presented.

The approximation of linear time-invariant (LTI) systems by sampling series is a central task of signal processing. For the Paley-Wiener space PW1 π of bandlimited signals with absolutely integrable Fourier transform, it is known that there exist signals and stable LTI systems such that the canonical approximation process diverges. In this paper we analyze the structure of the sets of signals and systems creating divergence and show that both sets are jointly spaceable, i.e., contain subsets such that every linear combination of signals and systems from these subsets, which is not the zero element, leads to divergence. In signal processing applications the linear structure of the involved signal spaces is essential. Here, we show that the same linear structure also holds for the sets of signals and systems creating divergence.

An optimal algorithm for estimation of the parameters of rational transfer functions from prescribed impulse response data is presented. The proposed method is based on the minimization of the l/sub 2/-norm of the true fitting error and is uniformly applicable for rational models with arbitrary numbers of poles and zeros. Existing methods either modify the true nonlinear error criterion in the theoretical derivation or require the transfer function model to have exactly one less number of zeros than poles. The multidimensional nonlinear error criterion is decoupled into a purely linear and a nonlinear subproblem of reduced dimension. The inherent mathematical structure in the non-linear subproblem is exploited in formulating an efficient iterative computational algorithm for its minimization. The effectiveness of the algorithm is demonstrated with several simulation examples.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

The problem of estimating the transfer function of a linear system, together with the spectral density of an additive disturbance, is considered. The set of models used consists of linear rational transfer functions and the spectral densities are estimated from a finite-order autoregressive disturbance description. The true system and disturbance spectrum are, however, not necessarily of finite order. We investigate the properties of the estimates obtained as the number of observations tends to ∞ at the same time as the model order employed tends to ∞. It is shown that the estimates are strongly consistent and asymptotically normal, and an expression for the asymptotic variances is also given. The variance of the transfer function estimate at a certain frequency is related to the signal/noise ratio at that frequency and the model orders used, as well as the number of observations. The variance of the noise spectral estimate relates in a similar way to the squared value of the true spectrum.

The paper presents a systematic approach for finding the optimal set of predictors for consistent identification of a single transfer function in an acyclic dynamic network. It is assumed that the topology of the network is known, the forcing inputs are not measured, and the observations have positive additive costs. For a class of networks where the target node is not involved in a feedback loop, sufficient and necessary conditions are derived to consistently identify a certain transfer function via a multi-input single-output prediction error method. This enables designing a systematic graphical approach based on the notion of d-separation to look for an optimal set of predictors that minimizes an appropriate additive cost function. It is shown that the required conditions for consistency and optimality are equivalent to the notion of separation in an undirected graph resulted from systematically manipulating the graphical representation of the network. Then, some well-known algorithms from computer science can be used to find the optimal set of predictors.

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.

A new way of formulating a multipoint Pade approximant of a linear system transfer function is given. It is seen to be very flexible and computationally efficient. The expansion points can be a mixture of real, complex, purely imaginary and multiple points with no complex arithmetic required. The underlying idea of the method is to convert the rational approximation problem into one of polynomial approximation. This results in simple matrix multiplications and the solution of a set of linear equations to derive the reduced model. Examples are given to demonstrate the application of the technique.