Polynomial Transformation Technique Research Articles

Identification of dual‐rate sampled errors‐in‐variables systems with time delays

AbstractFor identification of dual‐rate sampled errors‐in‐variables (EIV) systems with time delays, this paper utilizes the polynomial transformation technique and the redundant rule to derive the augmented dual‐rate model, and proposes a bias compensation‐based least squares (BC‐LS) algorithm. The basic idea is to obtain a biased parameter estimate by using the LS algorithm, then compensate the noise‐induced bias by the estimated noise variances, and finally estimate the time delay according to the unbiased parameter estimate and the given threshold. Considering that proper threshold selection has great influence on the performance of the BC‐LS identification algorithm for high noise level, the generalized orthogonal matching pursuit (GOMP) algorithm is introduced to simultaneously estimate the biased parameters and the time delay, and the bias compensation‐based GOMP (BC‐GOMP) algorithm is further proposed to identify the dual‐rate sampled EIV systems with time delays. The simulation examples demonstrate the effectiveness of the proposed two algorithms, and the BC‐GOMP algorithm performs better especially when the modeling dataset is small.

Outlier robust identification of dual‐rate Hammerstein models in the presence of unmodeled dynamics

AbstractThe article considers the outlier‐robust recursive least squares algorithm for identification of SISO (single‐input single‐output) dual‐rate Hammerstein model. The output sampling period is an integer multiple of the input sampling period. By using polynomial transformation technique we get a model which is suitable for parameter estimation for dual‐rate measurement data. After that deterministic dual‐rate Hammerstein model is extended with stochastic disturbance (which includes outliers) and unmodeled dynamics. Synthesis of identification algorithm is based on Newton–Raphson method which requires that loss function should be twice differentiable. Huber loss function, relevant for outliers treatment, has just first derivative. In order to overcome problem pseudo‐Huber loss function is introduced. This function behaves similarly to Huber loss function and has derivatives of arbitrary orders. Given recursive algorithm is based on synergy of Huber and pseudo‐Huber loss functions. Then we consider the robustness of algorithm. The main contributions of the article are: (i) determination of dual‐rate Hammerstein model which includes stochastic disturbance with outliers and unmodeled dynamics; (ii) derivation of robust recursive algorithm; (iii) determination of robustness for parameter estimate.

Maximum likelihood hierarchical least squares‐based iterative identification for dual‐rate stochastic systems

SummaryFor a dual‐rate sampled‐data stochastic system with additive colored noise, a dual‐rate identification model is obtained by using the polynomial transformation technique, which is suitable for the available dual‐rate measurement data. Based on the obtained model, a maximum likelihood least squares‐based iterative (ML‐LSI) algorithm is presented for identifying the parameters of the dual‐rate sampled‐data stochastic system. In order to improve the computation efficiency of the algorithm, the identification model of a dual‐rate sampled‐data stochastic system is divided into two subidentification models with smaller dimensions and fewer parameters, and a maximum likelihood hierarchical least squares‐based iterative (H‐ML‐LSI) algorithm is proposed for these subidentification models by using the hierarchical identification principle. The simulation results indicate that the proposed algorithms are effective for identifying dual‐rate sampled‐data stochastic systems and the H‐ML‐LSI algorithm has a higher computation efficiency than the ML‐LSI algorithm.

Maximum likelihood identification of dual‐rate Hammerstein output‐error moving average system

This study discusses the parameter estimation of the Hammerstein output-error moving average system using the dual-rate sampled data. The polynomial transformation technique is used to obtain the identification model of the discussed dual-rate sampled systems. The stochastic gradient optimisation method is an effective optimisation method. Compared with the Newton optimisation, it only needs to calculate the first derivative during the optimisation and the amount of calculation is relatively small. It is a good choice to use the stochastic gradient algorithm for the identification of Hammerstein dual-rate model after using the polynomial transformation technique. In order to improve the convergence speed, a maximum likelihood forgetting factor stochastic gradient identification algorithm is proposed by combining the maximum likelihood principle and the gradient search method. The convergence of the algorithm is analysed by using the stochastic process theory. Furthermore, in order to improve the estimation accuracy of the identification algorithm, a maximum likelihood multi-innovation forgetting factor stochastic gradient algorithm is proposed by using the multi-innovation identification theory. The effectiveness of the proposed algorithms is illustrated by a numerical simulation example and a water tank system.

Bias Correction-Based Recursive Estimation for Dual-Rate Output-Error Systems with Sampling Noise

This paper develops a bias correction-based recursive estimation algorithm for dual-rate output-error systems. The system output is subjected to both output noise and sampling noise. Using the polynomial transformation technique, the dual-rate output-error system is converted into an identification model where the sampled data can be directly applied. The noise variances of output noise and sampling noise are estimated by solving a nonlinear equation, which can minimize the estimation errors of noise variances. The simulation examples demonstrate that the proposed algorithm has higher parameter estimation accuracy in contrast to the auxiliary model-based recursive least-squares algorithm.

Dual-Rate Adaptive Decoupling Controller and Its Application to a Dual-Tank Water System

For a class of nonlinear multivariable dual-rate systems with strong couplings and uncertainties, a dual-rate adaptive decoupling controller with nonlinear feedforward compensation is proposed. First, a dual-rate model consisting of a linear model and a nonlinear term is derived by using the polynomial transformation technique. The unknown parameter matrices in the linear model are identified by using a proposed recursive least square algorithm and available dual-rate data, while the unknown nonlinear term is compensated by using the identification error. Based on the dual-rate model, the adaptive decoupling controller is then designed. The stability of the closed-loop system is presented. Finally, experiments are conducted on a dual-tank flow-level cascade control system. The results illustrate the effectiveness of the proposed method.

Recursive Least Squares Algorithm for Nonlinear Dual-rate Systems Using Missing-Output Estimation Model

In this paper, a recursive least squares algorithm is proposed for a class of nonlinear dual-rate systems. By using the missing-output estimation model, the unavailable outputs can be estimated. Then, the unknown parameters can be estimated from all the inputs and outputs. Compared with the polynomial transformation technique and the lifting technique, the unknown parameters can be estimated directly by using the missing-output estimation model, without increasing the number of parameters. The convergence analysis and the simulation results indicate that the proposed method is effective.

Multirate adaptive control of uncertain resonances beyond the Nyquist frequency in high-performance mechatronic systems

In this paper, an add-on multirate adaptive control scheme is proposed to compensate for uncertain mechanical resonant modes beyond the Nyquist frequency in mechatronic systems. The multirate adaptive controller is incorporated into the existing sampled-data control system sampled at a slow rate. To integrate well with the pre-designed feedback controller, an auxiliary error instead of the regulation error is utilized to design the multirate adaptive controller. By using a polynomial transformation technique and a recursive least-squares algorithm with a dead-zone, the proposed multirate adaptive control ensures boundedness of all the closed-loop signals and convergence of the auxiliary error. Our simulation and experimental results on the head-positioning servo control system of a hard disk drive demonstrate the effectiveness of our proposed compensation scheme in attenuating the vibration for uncertain resonances beyond the Nyquist frequency.

Highly Efficient Identification Methods for Dual-Rate Hammerstein Systems

This brief concerns parameter identification for a dual-rate Hammerstein CARMA system. By combining the polynomial transformation technique and the hierarchical identification principle, this brief transforms a dual-rate nonlinear Hammerstein CARMA system into a bilinear dual-rate identification model, and presents a hierarchical least squares algorithm to estimate the parameter vectors of the bilinear dual-rate identification model. Moreover, by using the key term separation principle, this brief transforms the dual-rate nonlinear Hammerstein CARMA system into a linear dual-rate identification model, and presents a key term separation based least squares algorithm to estimate the parameter vector of the linear dual-rate identification model. The two proposed methods possess higher computational efficiency compared with the previous over-parameterization least squares method in which many redundant parameters need estimating. The simulation results show the effectiveness of the two proposed algorithms.

Identification of Dual-Rate Sampled Hammerstein Systems with a Piecewise-Linear Nonlinearity Using the Key Variable Separation Technique

The identification difficulties for a dual-rate Hammerstein system lie in two aspects. First, the identification model of the system contains the products of the parameters of the nonlinear block and the linear block, and a standard least squares method cannot be directly applied to the model; second, the traditional single-rate discrete-time Hammerstein model cannot be used as the identification model for the dual-rate sampled system. In order to solve these problems, by combining the polynomial transformation technique with the key variable separation technique, this paper converts the Hammerstein system into a dual-rate linear regression model about all parameters (linear-in-parameter model) and proposes a recursive least squares algorithm to estimate the parameters of the dual-rate system. The simulation results verify the effectiveness of the proposed algorithm.

Multirate Adaptive Compensation of Uncertain Resonances Beyond the Nyquist Frequency

In this paper, an add-on multirate adaptive control scheme is proposed for compensation of uncertain mechanical resonant modes beyond the Nyquist frequency in mechatronic systems. A polynomial transformation technique is applied to obtain the mixed-rate model of the control system that is suitable for multirate adaptive control design. To preserve the performance of the pre-designed nominal controller, an auxiliary error is utilized in the parameter estimation process instead of tracking error. The proposed multirate adaptive control combined with a Recursive Least-Squares (RLS) algorithm and a dead-zone ensures boundedness of all the closed-loop signals and convergence of the auxiliary error. Our simulation results on the model of the Voice Coil Motor (VCM) actuator in a commercial Hard Disk Drive (HDD) demonstrate the effectiveness of our proposed compensation scheme in attenuating the mechanical resonances beyond the Nyquist frequency.

Several gradient parameter estimation algorithms for dual-rate sampled systems

This paper presents three identification methods for dual-rate sampled systems. The first method combines the stochastic gradient algorithm with the polynomial transformation technique, which can estimate the parameters of the identification model. The second method is the finite impulse response model based stochastic gradient algorithm, which can indirectly estimate the parameters of the dual-rate systems by using all the inputs and the available outputs. The third method is the missing output estimation model based stochastic gradient algorithm with a forgetting factor, which can directly estimate the parameters of the dual-rate systems by using all the inputs and all the outputs (include the estimated outputs). An example is provided to verify the effectiveness of the proposed methods.

Multi-innovation stochastic gradient algorithms for dual-rate sampled systems with preload nonlinearity

Since the stochastic gradient algorithm has a slower convergence rate, this letter presents a multi-innovation stochastic gradient algorithm for a class of dual-rate sampled systems with preload nonlinearity. The basic idea is to transform the dual-rate system model into an identification model which can use dual-rate data by using the polynomial transformation technique. A simulation example is provided to verify the effectiveness of the proposed method.

Time series AR modeling with missing observations based on the polynomial transformation

This paper focuses on parameter estimation problems of auto-regression (AR) time series models with missing observations. The standard estimation algorithms cannot be applied to such AR models with missing observations. The polynomial transformation technique is employed to transform the AR models into models which can be identified from available scarce observations, then the extended stochastic gradient algorithm is proposed to fit the time series with missing observations. The convergence properties of the proposed algorithm are analyzed and an example is given to test and illustrate the conclusions in the paper.

A modified stochastic gradient based parameter estimation algorithm for dual-rate sampled-data systems

In this paper, we propose a novel identification algorithm for a class of dual-rate sampled-data systems whose input–output data are measured by two different sampling rates. A polynomial transformation technique is employed to derive a mathematical model for such dual-rate systems. The proposed modified stochastic gradient algorithm has faster convergence rate than stochastic gradient algorithms for parameter identification using the dual-rate input–output data. Convergence properties of the algorithm are analyzed. Finally, illustrative and comparison examples are provided to verify the effectiveness and performance improvement of the proposed method.

Simulating Multivariate Nonnormal Data Using an Iterative Algorithm

Simulating multivariate nonnormal data with specified correlation matrices is difficult. One especially popular method is Vale and Maurelli's (1983) extension of Fleishman's (1978) polynomial transformation technique to multivariate applications. This requires the specification of distributional moments and the calculation of an intermediate correlation matrix such that when data are transformed, the target correlation matrix is reproduced. We present an alternative technique that involves sampling data from specified population distributions and identifying the intermediate correlation matrix through an iterative, trial-and-error process. We provide R program code to implement this technique and show that it can generate data under a wide range of conditions (e.g., with empirical samples, with discrete rather than continuous data, when distributional moments are undefined or outside the boundary conditions of other techniques). This approach could be useful in many contexts, especially Monte Carlo studies of multivariate statistics.

The residual based extended least squares identification method for dual-rate systems

In this paper, we focus on a class of dual-rate sampled-data systems in which all the inputs u ( t ) are available at each instant while only scarce outputs y ( q t ) can be measured ( q being an integer more than unity). To estimate the parameters of such dual-rate systems, we derive a mathematical model by using the polynomial transformation technique, and apply the extended least squares algorithm to identify the dual-rate systems directly from the available input–output data { u ( t ) , y ( q t ) } . Then, we study the convergence properties of the algorithm in details. Finally, we give an example to test and illustrate the algorithm involved.

The gradient algorithm for parameter and output estimation for dual-rate CARARMA systems

A recursive generalized extended stochastic forgetting gradient algorithm is used to identify the dual-rate stochastic systems based on the polynomial transformation technique. A time-varying forgetting factor is included to improve the rate of convergence. The intersample output estimation algorithm is also studied in the paper. Finally, a simulation example shows that the algorithm is excellent effective in parameter identification and output estimation.

Adaptive Digital Control of Hammerstein Nonlinear Systems with Limited Output Sampling

This paper is motivated by the practical control considerations that nonlinearity is abundant in industrial processes and output sampling rates are often limited due to hardware constraints. In particular, for a Hammerstein nonlinear sampled-data system in which the output sampling period is an integer multiple of the input updating period, we derive, by using a polynomial transformation technique, a mathematical model which is suitable for parameter estimation with dual-rate measurement data. Further, we present an adaptive control scheme for such a dual-rate nonlinear system; the parameter estimation-based adaptive algorithm can achieve virtually asymptotically optimal control and ensure that the closed-loop system is stable and globally convergent. The simulation results are included.

A GRADIENT BASED ADAPTIVE CONTROL ALGORITHM FOR DUAL‐RATE SYSTEMS

ABSTRACTIn this paper, using a polynomial transformation technique, we derive a mathematical model for dual‐rate systems. Based on this model, we use a stochastic gradient algorithm to estimate unknown parameters directly from the dual‐rate input‐output data, and then establish an adaptive control algorithm for dual‐rate systems. We prove that the parameter estimation error converges to zero under persistent excitation, and the parameter estimation based control algorithm can achieve virtually asymptotically optimal control and ensure the closed‐loop systems to be stable and globally convergent. The simulation results are included.