Strictly Positive Real Condition Research Articles

Enhancing strict positive realness condition on families of polynomials by filter design

Some results on strict positive realness of families of polynomials are given. The main motivation for these results is the need for design criteria of filters ensuring the convergence of algorithms in the presence of uncertainty in the plant model in the area of identification and adaptive control. Two main results are given. The first provides analytical conditions under which a family of polynomials with zeros in a prescribed region of the complex plane is strict positive real or can be made strict positive real over an assigned region of the complex plane through the use of a suitable filter. The second is a design result providing a parameterization of a family of filters maximizing the region of the complex plane on which strict positive realness is achievable.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Noise cancellation with improved residuals

An effective adaptive scheme for noise cancellation when the signal to be recovered has known autocorrelation is presented. Two algorithms that exploit a special form of prior information are investigated. In this approach the desired signal is removed from the output feedback by linear prediction: the prior information used is the desired signal's autocorrelation. Knowing this, one can find a filter that whitens the desired signal. Screening the error feedback through this filter removes most of the desired signal energy, reducing its interference with the coefficient update. This is the basis for the first algorithm discussed, namely, the least-mean-square algorithm with augmented predictor (LMS-AP) proposed by Orgren et al. (1986). In many applications the whitening filter may not be strictly positive real (SPR). In such cases a different algorithm is needed; one which is assuredly convergent regardless of the satisfaction of the SPR condition. A modified LMS algorithm with augmented predictor (MLMS-AP) which provides such an alternative is proposed. >

A method for adaptive estimation of ARMA processes

When the noise process in adaptive identification of linear stochastic systems is correlated, and can be represented by a moving average model, extended least squares algorithms are commonly used, and converge under a strictly positive real (SPR) condition on the noise model. In this paper, we present an adaptive algorithm for the estimation of autoregressive moving average (ARMA) processes, and show that it is convergent without any SPR condition, and has a convergence rate of O({ loglog t)/t} 1 2 ) .

On a relaxation of SPR condition in parallel MRAS-continuous-time case

It is shown that the strict positive realness (SPR) condition for asymptotic stability of the continuous-time parallel model reference adaptive system (MRAS) can be removed provided that the parameter adaption law includes a proportional term and the information matrix is made sufficiently large. The present result a continuous-time counterpart of an earlier result (M. Tomizuka, ibid., vol.27, no.4, p.505-6, 1982) on the discrete-time parallel MRAS.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Adaptive stochastic filters with no strict positive real condition

A new adaptive stochastic filter structure is introduced which avoids the strict passivity test used as a sufficient condition for convergence required by existing adaptive schemes. The proposed algorithm consists of three stages. In the first stage, an autoregressive model is fitted and the residue obtained is used as an estimate of the noise. In the second stage, an autoregressive recursive moving average model is fitted using the residual of the first stage. A modified residual is then filtered using a parameter δ and the model obtained from the second stage to generate an improved estimate of the noise. In the third stage, this improved estimate of the noise is used to obtain a better autoregressive moving average model. It is shown that the proposed algorithm will also reduce the bias in the estimated parameters. The simulation results given show that the proposed filter compares favorably to the algorithm introduced by Mayne and Clark and also Landau. This filter is then applied to the adaptive line enhancement, sinusoidal detection, and adaptive spectral estimation problems to illustrate its usefulness.

Elimination of real positivity condition and error filtering in parallel MRAS

For a particular degree of persistent excitation, the commonly accepted strict positive realness (SPR) condition and associated error filtering in the design of model reference adaptive system (MRAS) are removed. It is shown that the parameter estimation vector remains bounded and the output error converges asymptotically to zero for a stable reference model. The proof proceeds with simple algebra, and does not rely on the commonly used hyperstability theorem and positivity concept.

A Simplified Convergence Analysis for the Parallel Identifier with an EEM

The design of parallel model reference adaptive systems (MRAS) lacks the mathematical advantages that make least squares performance criterion useful for estimation due to the dependence of the observation vector on the parameters of the adaptive system. In this case the design and convergence analysis are based on stability theories-Lyapunov's direct method and hyperstability theory. In this paper we present a proof for the asymptotic stability of the null solutions of the output error and parameter error vector for the parallel identifier with an “extended estimation model” (EEM). The proof does not rely on the mentioned stability theories and proceeds with simple algebra eliminatinq the association of an equivalent feedback system (EFS) and the strict positive realness (SPR) condition.

Convergence and rate of convergence of a recursive identification and adaptive control method which uses truncated estimators

A stochastic approximation-like method is used for the recursive identification of the coefficients in <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">y_{n}=\sum\min{1}\max{l_{1}}a_{i}y_{n-i}+\sum\min{0}\max{l_{2}}b_{i}u_{n-i}+ \sum\min{1}\max{l_{3}}c_{i}w_{n-i}+w_{n}</tex> , where <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{w_{n}}</tex> is a sequence of mutually independent and bounded random variables, and is independent of the bounded <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{u_{n}}</tex> . Such methods normally require the recursive estimation of the "residuals" or the <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">{w_{n}}</tex> , and the algorithms for doing this can be unstable if the parameter estimates are not close enough to their true values. The problem is solved here by use of a simple truncated estimator, which is probably what would be used in implementation in any, case. Then, under a stability, and strict positive real type condition, with probability 1 (w.p.1) convergence is proved and the rate of convergence is obtained. An associated adaptive control problem is also treated.

On satisfying strict positive real condition for convergence by overparameterization

Strict positive real (SPR) conditions for global convergence of stochastic adaptive control and recursive identification schemes have appeared in a number of analyses using different approaches. It is shown that this strict positive real condition on the unknown operator associated with the plant can be removed in every case by a sufficient overparameterization of the parameter vector.