On the accuracy in errors-in-variables identification compared to prediction-error identification

Errors-in-variables estimation problems for single-input–single-output systems with Gaussian signals are considered in this contribution. It is shown that the Fisher information matrix is monotonically increasing as a function of the input noise variance when the noise spectrum at the input is known and the corresponding noise variance is estimated. Furthermore, it is shown that Whittle’s formula for the Fisher information matrix can be represented as a Gramian and this is used to provide a geometric representation of the asymptotic covariance matrix for asymptotically efficient estimators. Finally, the asymptotic covariance of the parameter estimates for the system dynamics is compared for the two cases: (i) when the model includes white measurement noise on the input and the variance of the noise is estimated, and (ii) when the model includes only measurement noise on the output. In both cases, asymptotically efficient estimators are assumed. An explicit expression for the difference is derived when the underlying system is subject only to measurement noise on the output.     

Autotune identification for systems with right-half-plane poles and zeros

Autotuning based on relay feedback tests have received a great deal of attention recently. The relay feedback tests fail to generate sustained oscillation in some cases. For systems with RHP poles and/or zeros, incorrect autotune identification procedure may give erroneous information and unstable limit cycle. Frequency domain analyses provide a criteria for prediction of the existence of stable limit cycle. Furthermore, a procedure is given to ensure the success of a relay feedback test. The analyses and procedure are illustrated through open loop stable and unstable systems. Results show that, except for some rare occasion, successful relay feedback experiments can be obtained following the proposed procedure.     

Variance analysis of identified linear MISO models having spatially correlated inputs,with application to parallel Hammerstein models

This contribution concerns variance analysis of linear multi-input single-output models when the inputs are temporally white but where different inputs may be correlated. An expression is provided for the variance of a linearly parametrized estimate of the frequency response function from one block, i.e. from one input to the output. In particular, this expression reveals that the variance increases in one block when the number of estimated parameters in another block is increased, but levels off when the number of parameters in the other block reaches the number of parameters in the block in question. It also quantifies exactly how correlation between inputs affects the resulting accuracy and a graphical representation is provided for this purpose. The results are applicable to parallel MISO Hammerstein models when the nonlinearities are known and generalize an existing variance expression for this type of model.     

Achievable performance of multivariable systems with unstable zeros and poles

This paper examines the fundamental limitations imposed by unstable (right half plane; RHP) zeros and poles in multivariable feedback systems. We generalize previously known controller-independent lower bounds on the H     

Model reduction of systems with zeros interlacing the poles

The problem of reducing a system with zeros interlacing the poles (ZIP) on the real axis is considered. It is proved that many model reduction methods, such as the balanced truncation, balanced residualization, suboptimal and optimal Hankel approximations, inherit the ZIP property. Properties of the Hankel singular values of ZIP systems are also listed.     

Right half plane poles and zeros and design tradeoffs in feedback systems

This paper expresses limitations imposed by right half plane poles and zeros of the open-loop system directly in terms of the sensitivity and complementary sensitivity functions of the closed-loop system. The limitations are determined by integral relationships which must be satisfied by these functions. The integral relationships are interpreted in the context of feedback design.     

Stability tests and stabilization for piecewise linear systems based on poles and zeros of subsystems

This paper provides several stability tests for piecewise linear systems and proposes a method of stabilization for bimodal systems. In particular, we derive an explicit and exact stability test for planar systems, which is given in terms of coefficients of transfer functions of subsystems. Restricting attention to the bimodal and planar case, we show simple stability tests. In addition, we drive a necessary stability condition and a sufficient stability condition for higher-order and bimodal systems. They are given in terms of the eigenvalue loci and the observability of subsystems. All the stability tests provided in this paper are computationally tractable, and our results are applied to the stabilizability problem. We confirm the exactness and effectiveness of our approach by illustrative examples.     

The relationship between real poles and real zeros in SISO sampleddata systems

A relationship between real poles and real zeros of SISO (single-input, single-output) sampled data systems is presented, which is independent of the sampling period. The relation is stated in terms of a parity property involving the number of real zeros between any two real poles. This property allows the investigation of conditions for the preservation of stable or unstable zeros under sampling. A sufficient condition for stability of a sampled system with a continuous-time plant having all real zeros, and a sufficient condition for instability of a two-sampled system with an unstable plant is developed     

Relationship between poles and zeros of input–output and chain-scattering systems

A system is frequently represented by transfer functions in an input–output characterization. However, such a system (under mild assumptions) can also be represented by transfer functions in a port characterization, frequently referred to as a chain-scattering representation. Due to its cascade properties, the chain-scattering representation is used throughout many fields of engineering. This paper studies the relationship between poles and zeros of input–output and chain-scattering representations of the same system.     

Numerical conditioning and asymptotic variance of subspace estimates

New formulas for the asymptotic variance of the parameter estimates in subspace identification, show that the accuracy of the parameter estimates depends on certain indices of ‘near collinearity’ of the state and future input subspaces of the system to be identified. This complements the numerical conditioning analysis of subspace methods presented in the companion paper (On the ill-conditioning of subspace identification with inputs, Automatica, doi:10.1016/j.automatica.2003.11.009).     

Correlation-based tuning of decoupling multivariable controllers

The iterative method labelled correlation-based tuning (CbT) is considered in this paper for tuning linear time-invariant multivariable controllers. The approach allows one to tune some elements of the controller transfer function matrix to satisfy the desired closed-loop performance, while the other elements are tuned to mutually decouple the closed-loop outputs. Decoupling is achieved by decorrelating a given reference with the non-corresponding outputs. The controller parameters are calculated either by solving a correlation equation (decorrelation procedure) or by minimizing a cross-correlation function (correlation reduction). In addition, the preferred way of exciting a 2×2 system for CbT is investigated via the accuracy of the estimated controller parameters. It is shown that simultaneous excitation of both reference signals does not improve the accuracy of the estimated controller parameters compared to the case of sequential excitation. In fact, one must choose between low experimental cost (simultaneous excitation) and better accuracy of the estimated parameters (sequential excitation). The theoretical results are illustrated via three simulation studies.     

Assessing the quality of identified models through the asymptotic theory—when is the result reliable?

In this paper, the problem of estimating uncertainty regions for identified models is considered. A typical approach in this context is to resort to the asymptotic theory of Prediction Error Methods for system identification, by means of which ellipsoidal uncertainty regions can be constructed for the uncertain parameters.We show that the uncertainty regions worked out through the asymptotic theory can be unreliable in certain situations, precisely characterized in the paper.Then, we critically analyze the theoretical conditions for the validity of the asymptotic theory, and prove that the asymptotic theory also applies under new assumptions which are less restrictive than the usually required ones. Thanks to this result, we single out the classes of models among standard ones (ARX, ARMAX, Box-Jenkins, etc.) where the asymptotic theory can be safely used in practical applications to assess the quality of the identified model.These results are of interest in many applications, including iterative controller design schemes.     

MODIS time-series imagery for forest disturbance detection and quantification of patch size effects

The Moderate Resolution Imaging Spectroradiometer (MODIS) 250 m single day surface reflectance (MOD09GQK) and 16-day composite gridded vegetation index data (MOD13Q1) were used to detect forest harvest disturbance between 2000 and 2004 in northern Maine. A MODIS multi-date Normalized Difference Vegetation Index (NDVI) forest change detection map was developed from each MODIS data set. A Landsat TM/ETM+ change detection map was developed as a reference to assess the effect of disturbed forest patch size on classification accuracy (agreement) and disturbed area estimates of MODIS. The MODIS single day and 16-day composite data showed no significant difference in overall classification accuracies. However, the 16-day NDVI change detection map had marginally higher overall classification accuracy (at 85%), but had significantly lower detection accuracy related to disturbed patch size than the single day NDVI change detection map. The 16-day composite NDVI data achieved 69% detection accuracy and the single day NDVI achieved 76% when the disturbed patch size was greater than 20 ha. The detection accuracy increased to approximately 90% for both data sets when the patch size exceeded 50 ha. The R² (range 0.6 to 0.9) and slope (range 0.5 to 0.9) of regression lines between Landsat and MODIS data (based on forest disturbance percent of township) increased with the mean disturbed patch size of each township. The 95% confidence intervals of forest disturbance percent estimate for each township were narrow with less than 1% of each township at the mean MODIS forest disturbance level.     

The cost of complexity in system identification: The Output Error case

In this paper we investigate the cost of complexity, which is defined as the minimum amount of input power required to estimate the frequency response of a given linear time invariant system of order n with a prescribed degree of accuracy. In particular we require that the asymptotic (in the data length) variance is less or equal to γ over a prespecified frequency range [0,ω_B]. The models considered here are Output Error models, with an emphasis on fixed denominator and Laguerre models. Several properties of the cost are derived. For instance, we present an expression which shows how the pole of the Laguerre model affects the cost. These results quantify how the cost of the system identification experiment depends on n and on the model structure. Also, they show the relation between the cost and the amount of information we would like to extract from the system (in terms of ω_B and γ). For simplicity we assume that there is no undermodelling.     

A numerical investigation of direct and indirect closed-loop architectures for estimating nonminimum-phase zeros

ABSTRACT

This paper presents a numerical investigation of three direct architectures and three indirect architectures for identifying a plant operating in closed loop. Motivated by adaptive control of systems with nonminimum-phase (NMP) zeros and taking advantage of the fact that zeros are not moved by feedback, the performance metric is the accuracy of the estimates of the NMP zeros of the plant. Assuming known plant order, single-input, single-output, infinite-impulse-response models are constructed in the presence of process and sensor noise. Least squares provides the baseline estimation technique, and prediction error methods are used to account for correlation between the model input and noise. The goal is to compare the accuracy of the NMP-zero estimates obtained from each method and for each architecture.     

Regressor and structure selection in NARX models using a structured ANOVA approach

Regressor selection can be viewed as the first step in the system identification process. The benefits of finding good regressors before estimating complex models are especially clear for nonlinear systems, where the class of possible models is huge. In this article, a structured way of using the tool analysis of variance (ANOVA) is presented and used for NARX model (nonlinear autoregressive model with exogenous input) identification with many candidate regressors.     

An algebraic method to determine the common divisor,poles and transmission zeros of matrix transfer functions

A purely algebraic method which uses the matrix Routh algorithm and its reverse process of the algorithm is presented to decompose a matrix transfer function into a pair of right co-prime polynomial matrices or left co-primo polynomial matrices. The poles and transmission zeros of the matrix transfer function are determined from a pair of relatively prime polynomial matrices. Also, the common divisor of two matrix polynomials can be obtained by using the matrix Routh algorithm and the matrix Routh array.     

Effective low-Mach number improvement for upwind schemes

In this paper, we present an effective low-Mach number improvement for upwind schemes. The artificial viscosity of upwind schemes scales with $1∕Ma$ incurring a loss of accuracy for the Mach number approaching zero. The remedy is based on three steps: (i) the jump of the left and right states is split into the density diffusion part and velocity diffusion part; (ii) the velocity diffusion part is rescaled by multiplying a scaling function; (iii) the scaling function is only related to the local Mach number without the cut-off reference Mach number and meantime restricted by a shock senor. The resulting modification is very easily implemented and applied within Roe, HLL and Rusanov, etc. Then, asymptotic analysis and numerical experiments for a wide Mach number demonstrate that this novel approach is equipped with these attractive properties: (1) free from the cut-off global problem and damping checkerboard modes; (2) satisfying the correct $M{a}^2$ scaling of pressure fluctuations, the divergence constraint and a Poisson equation; (3) independent of Mach number in terms of accuracy; (4) better accurate and higher resolution in low Mach number regimes when compared with existing upwind schemes, and applicable for moderate or high Mach numbers. Thus, the proposed modification is expected to provide as an excellent and reliable candidate to simulate turbulent flows at all Mach numbers.