Recursive Least-squares Estimation Research Articles

Sliding-mode control combined with improved adaptive feedforward for wafer scanner

Abstract In this paper, a sliding-mode control method combined with improved adaptive feedforward is proposed for wafer scanner to improve the tracking performance of the closed-loop system. Particularly, In addition to the inverse model, the nonlinear force ripple effect which may degrade the tracking accuracy of permanent magnet linear motor (PMLM) is considered in the proposed method. The dominant position periodicity of force ripple is determined by using the Fast Fourier Transform (FFT) analysis for experimental data and the improved feedforward control is achieved by the online recursive least-squares (RLS) estimation of the inverse model and the force ripple. The improved adaptive feedforward is given in a general form of nth-order model with force ripple effect. This proposed method is motivated by the motion controller design of the long-stroke PMLM and short-stroke voice coil motor for wafer scanner. The stability of the closed-loop control system and the convergence of the motion tracking are guaranteed by the proposed sliding-mode feedback and adaptive feedforward methods theoretically. Comparative experiments on a precision linear motion platform can verify the correctness and effectiveness of the proposed method. The experimental results show that comparing to traditional method the proposed one has better performance of rapidity and robustness, especially for high speed motion trajectory. And, the improvements on both tracking accuracy and settling time can be achieved.

Distributed incremental bias-compensated RLS estimation over multi-agent networks

In this paper, we study the problem of distributed bias-compensated recursive least-squares (BC-RLS) estimation over multi-agent networks, where the agents collaborate to estimate a common parameter of interest. We consider the situation where both input and output of each agent are corrupted by unknown additive noise. Under this condition, traditional recursive least-squares (RLS) estimator is biased, and the bias is induced by the input noise variance. When the input noise variance is available, the effect of the noise-induced bias can be removed at the expense of an increase in estimation variance. Fortunately, it has been illustrated that distributed collaboration between agents can effectively reduce the variance and can improve the stability of the estimator. Therefore, a distributed incremental BC-RLS algorithm and its simplified version are proposed in this paper. The proposed algorithms can collaboratively obtain the estimates of the unknown input noise variance and remove the effect of the noise-induced bias. Then consistent estimation of the unknown parameter can be achieved in an incremental fashion. Simulation results show that the incremental BC-RLS solutions outperform existing solutions in some enlightening ways.

Improved Sample Value Adjustment for Synchrophasor Estimation at Off-Nominal Power System Conditions

The phasor measurement unit (PMU) commonly implements enhanced discrete Fourier transformation (EDFT) to compensate leakage and signal discontinuity errors at offnominal frequencies. However, EDFT results in a high total vector error (TVE) at large offnominal frequency and worsens with harmonics in power signals. Although the sample value adjustment (SVA) technique addressed such demerits, it generates an asymmetrical TVE profile. In addition, the frequency estimator was not taken into account in the previous publication on SVA. This paper proposes an improved sample value adjustment (ISVA) as a pre-DFT technique to further enhance the phasor estimation. The ISVA generates a symmetrical TVE profile and performed better than EDFT, SVA, and Taylor weighted least-square method at steady-state conditions. Subsequently, the ISVA is integrated with a recursive least-square frequency estimator, which enabled it to track dynamic signals as well. The algorithm is validated in MATLAB simulation and in a prototyped PMU. Static and dynamic test signals are based on the IEEE C37.118.1a-2014 standard.

Recursive Estimation in Large Panel Data Models: Theory and Practice

Bai (2009) proposes a recursive least-squares estimation method for large panel data models with unobservable interactive fixed effects, but the impact of recursion on the asymptotic properties of the least-squares estimators is not taken into account. In this paper, we extend Bai (2009) by investigating the recursive estimator asymptotically. In general, the asymptotic properties we establish for the recursive estimators largely complement the theory and practice of the recursive least-squares procedure suggested by Bai (2009). In particular, we show that consistency of the recursive estimator depends on three key points, consistency of the initial OLS estimator, the number of recursive steps and the endogeneity arising due to the dependence between regressors and interactive effects. Compared to the theoretical estimator in Bai (2009), such endogeneity affects the convergence rate of recursive least-squares estimators. Finite sample results are provided to illustrate our findings in detail.

Probabilistic damage detection of structures with uncertainties under unknown excitations based on Parametric Kalman filter with unknown Input

System identification and damage detection for structural health monitoring have received considerable attention. Various time domain analysis methodologies based on measured vibration data of structures have been proposed. Among them, recursive least-squares estimation of structural parameters which is also known as parametric Kalman filter (PKF) approach has been studied. However, the conventional PKF requires that all the external excitations (inputs) be available. On the other hand, structural uncertainties are inevitable for civil infrastructures, it is necessary to develop approaches for probabilistic damage detection of structures. In this paper, a parametric Kalman filter with unknown inputs (PKF-UI) is proposed for the simultaneous identification of structural parameters and the unmeasured external inputs. Analytical recursive formulations of the proposed PKF-UI are derived based on the conventional PKF. Two scenarios of linear observation equations and nonlinear observation equations are discussed, respectively. Such a straightforward derivation of PKF-UI is not available in the literature. Then, the proposed PKF-UI is utilized for probabilistic damage detection of structures by considering the uncertainties of structural parameters. Structural damage index and the damage probability are derived from the statistical values of the identified structural parameters of intact and damaged structure. Some numerical examples are used to validate the proposed method.

회전관성의 순환최소자승 추정을 이용한 모델 예견 기반 굴삭기의 충돌회피 알고리즘 개발

This paper proposes a model-prediction-based collision-avoidance algorithm for excavators for which the recursive-least-squares (RLS) estimation of the excavator's rotational inertia is used. To estimate the rotational inertia of the excavator, the RLS estimation with multiple forgetting and two updating rules for the nominal parameter and the forgetting factors was conducted based on the excavator-swing dynamics. The average value of the estimated rotational inertia that is for the minimizing effects of the estimation error was computed using the recursive-average method with forgetting. Based on the swing dynamics, the computed average of the rotational inertia, the damping coefficient for braking, and the excavator's braking angle were predicted, and the predicted braking angle was compared with the detected-object angle for a safety evaluation. The safety level defined in this study consists of the three levels safe, warning, and emergency braking. The analytical rotational-inertia-based performance evaluation of the designed estimation algorithm was conducted using a typical working scenario. The results of the safety evaluation show that the predictive safety-evaluation algorithm of the proposed model can evaluate the safety level of the excavator during its operation.

Cross-coupling integral adaptive robust posture control of a pneumatic parallel platform

A pneumatic parallel platform driven by an air cylinder and three circumambient pneumatic muscles was considered. Firstly, a mathematical model of the pneumatic servo system was developed for the MIMO nonlinear model-based controller designed. The pneumatic muscles were controlled by three proportional position valves, and the air cylinder was controlled by a proportional pressure valve. As the forward kinematics of this structure had no analytical solution, the control strategy should be designed in joint space. A cross-coupling integral adaptive robust controller (CCIARC) which combined cross-coupling control strategy and traditional adaptive robust control (ARC) theory was developed by back-stepping method to accomplish trajectory tracking control of the parallel platform. The cross-coupling part of the controller stabilized the length error in joint space as well as the synchronization error, and the adaptive robust control part attenuated the adverse effects of modelling error and disturbance. The force character of the pneumatic muscles was difficult to model precisely, so the on-line recursive least square estimation (RLSE) method was employed to modify the model compensation. The projector mapping method was used to condition the RLSE algorithm to bound the parameters estimated. An integral feedback part was added to the traditional robust function to reduce the negative influence of the slow time-varying characteristic of pneumatic muscles and enhance the ability of trajectory tracking. The stability of the controller designed was proved through Laypunov’s theory. Various contrast controllers were designed to testify the newly designed components of the CCIARC. Extensive experiments were conducted to illustrate the performance of the controller.

Diffusion Bias-Compensated RLS Estimation with Quantized Measurements over Adaptive Networks

We study the problem of distributed bias-compensated recursive least-squares (BCRLS) estimation with quantized measurements, which means that the nodes in adaptive networks collaborate to estimate a common deterministic parameter using quantized data. Traditional recursive least-squares (RLS) algorithms are biased when both the regressor and the output response are corrupted by stationary additive noise. And considering the limited bandwidth resources and the growing complexity in communication, the measurements transmitted in adaptive networks are always quantized in practical cases, which also contributes to the bias in estimation. Therefore, a distributed bias-compensated RLS algorithm is proposed in this paper to estimate the bias caused by both the background noise and the quantization noise and compensate for it cooperatively. Meanwhile, considering the advantages of diffusion strategies, the algorithm is developed in a diffusion fashion. Simulation results illustrate the good performance of the proposed algorithm.

Distributed Constrained Recursive Nonlinear Least-Squares Estimation: Algorithms and Asymptotics

This paper focuses on the problem of recursive nonlinear least squares parameter estimation in multi-agent networks, in which the individual agents observe sequentially over time an independent and identically distributed (i.i.d.) time-series consisting of a nonlinear function of the true but unknown parameter corrupted by noise. A distributed recursive estimator of the \emph{consensus} + \emph{innovations} type, namely $\mathcal{CIWNLS}$, is proposed, in which the agents update their parameter estimates at each observation sampling epoch in a collaborative way by simultaneously processing the latest locally sensed information~(\emph{innovations}) and the parameter estimates from other agents~(\emph{consensus}) in the local neighborhood conforming to a pre-specified inter-agent communication topology. Under rather weak conditions on the connectivity of the inter-agent communication and a \emph{global observability} criterion, it is shown that at every network agent, the proposed algorithm leads to consistent parameter estimates. Furthermore, under standard smoothness assumptions on the local observation functions, the distributed estimator is shown to yield order-optimal convergence rates, i.e., as far as the order of pathwise convergence is concerned, the local parameter estimates at each agent are as good as the optimal centralized nonlinear least squares estimator which would require access to all the observations across all the agents at all times. In order to benchmark the performance of the proposed distributed $\mathcal{CIWNLS}$ estimator with that of the centralized nonlinear least squares estimator, the asymptotic normality of the estimate sequence is established and the asymptotic covariance of the distributed estimator is evaluated. Finally, simulation results are presented which illustrate and verify the analytical findings.

Resource optimised reconfigurable modular parallel pipelined stochastic approximation‐based self‐tuning regulator architecture with reduced latency

Present self-tuning regulator architectures based on recursive least-square estimation are computationally expensive and require large amount of resources and time in generating the first control signal due to computational bottlenecks imposed by the calculations involved in estimation stage, different stages of matrix multiplications and the number of intermediate variables at each iteration and precludes its use in applications that have fast required response times and those which run on embedded computing platforms with low-power or low-cost requirements with constraints on resource usage. A salient feature of this study is that a new modular parallel pipelined stochastic approximation-based self-tuning regulator architecture which reduces the time required to generate the first control signal, reduces resource usage and reduces the number of intermediate variables is proposed. Fast matrix multiplication, pipelining and high-speed arithmetic function implementations were used for improving the performance. Results of implementation demonstrate that the proposed architecture has an improvement in control signal generation time by 38% and reduction in resource usage by 41% in terms of multipliers and 44.4% in terms of adders compared with the best existing related work, opening up new possibilities for the application of online embedded self-tuning regulators.

Single-frequency precise point positioning: an analytical approach

An analytical approach to single-frequency precise point positioning (PPP) is discussed in this paper. To obtain highest precision results, all biases must be eliminated or modelled to centimetre level. The use of the GRAPHIC ionosphere-free linear combination that is based on single-frequency phase and code observations eliminates the ionosphere bias; however, the rank deficient Gauss–Markov model is obtained. We explicitly determine rank deficiency of a Gauss–Markov model as a number of all ambiguity clusters, each of them defined as a set of all ambiguities overlapping in time. On the basis of S-transformation we prove that the single-frequency PPP represents an unbiased estimator for station coordinates and troposphere parameters, while it presents a biased estimator for ambiguities and receiver-clock error parameters. Additionally we describe the estimable parameters in each ambiguity cluster as the differences between ambiguity parameters and the sum of receiver-clock parameters with one of the ambiguities. We also show that any other particular solution on the basis of S-transformation is obtained only when the common least-squares estimation in single step is applied. The recursive least-squares estimation with parameter pre-elimination only determines the vector of unknowns as possible to transform through S-transformation, whereas the same does not hold for the cofactor matrix of unknowns. For a case study, we present our method on GPS data from 19 permanent stations (14 IGS and 5 EPN) in Europe, for 89 consecutive days in the beginning of 2013. The static case study revealed the precision of daily coordinates as 7.6, 11.7 and 19.6 mm for \(N\), \(E\) and \(U\), respectively. The accuracies of the \(N\), \(E\) and \(U\) components were determined as 6.9, 13.5 and 31.4 mm, respectively, and were calculated using the Helmert transformation of weighted-mean daily single-frequency PPP and IGb08 coordinates. The estimated convergence times were relatively diverse, expanding from 1.75 h (CAGL) to 5.25 h (GRAZ) for the horizontal position with the 10-cm precision threshold, and from 1.00 h (GRAS) to 3.25 h (BZRG) for the height component with a 20-cm precision threshold. The convergence times were shown to be strongly correlated to the remaining unmodelled biases in the GRAPHIC linear combination, primarily with multipath, where the correlation coefficient for the horizontal position was determined as \(\rho _P\) \(=\) 0.68 and for height as \(\rho _U\) \(=\) 0.85. The comparison to the model where raw observations are used (\(C\), \(L\)) and where the ionosphere bias is mitigated with global ionosphere models (GIM) revealed the supremacy of the proposed single-frequency PPP method based on the GRAPHIC linear combination in both the static and the semi-kinematic case study. In the static case study, the proposed single-frequency PPP model was superior both in terms of precision and accuracy. In the semi-kinematic case study, the usage of raw observations with GIM would improve results only when multipath and noise of code observations would prevail over the remaining ionosphere bias, i.e. after applying GIM.

An efficient algorithm for simultaneous identification of time-varying structural parameters and unknown excitations of a building structure

When a structure is being damaged under unknown excitations, structural parameters of the damaged elements are actually varying with time. Hence, the simultaneous identification of time-varying structural parameters and unknown excitations is an important task in structural health monitoring. Although some analytical methods for such identifications are available in the literature, they are complicated, time-consuming, or restrained by special requirements. This paper presents an efficient algorithm for identifying time-varying structural parameters of a building structure under unknown excitations. By projecting on to the column space of influence matrix of unknown excitations, the observation equation of the structural system with unknown excitations is transformed from a multiple linear regression equation to a simple linear regression equation. By further introducing a time-varying correction factor matrix, an analytical recursive least-squares estimation algorithm is developed for identifying unknown excitations and time-varying structural parameters such as stiffness and damping. The feasibility and accuracy of the proposed algorithm are finally demonstrated through numerical examples and comparison with the existing methods. The results clearly exhibit that the proposed algorithm can simultaneously identify unknown excitations and time-varying structural parameters efficiently and accurately.

Experimental verification of a GPC-LPV method with RLS and P1-TS fuzzy-based estimation for limiting the transient and residual vibration of a crane system

The paper deals with the problem of reducing the residual vibration and limiting the transient oscillations of a flexible and underactuated system with respect to the variation of operating conditions. The comparative study of generalized predictive control (GPC) and fuzzy scheduling scheme developed based on the P1-TS fuzzy theory, local pole placement method and interval analysis of closed-loop system polynomial coefficients is addressed to the problem of flexible crane control. The two alternatives of a GPC-based method are proposed that enable to realize this technique either with or without a sensor of payload deflection. The first control technique is based on the recursive least squares (RLS) method applied to on-line estimate the parameters of a linear parameter varying (LPV) model of a crane dynamic system. The second GPC-based approach is based on a payload deflection feedback estimated using a pendulum model with the parameters interpolated using the P1-TS fuzzy system. Feasibility and applicability of the developed methods were confirmed through experimental verification performed on a laboratory scaled overhead crane.

Research on the Complex Dynamic Characteristics and RLS Estimation’s Influence Based on Price and Service Game

Considering that the real competitions in service market contain two important factors, price and service, we build a dynamical price and service game model and study the complex dynamics of this bivariate game. Some special properties about the adjustment of service are noted by comparing our innovative bivariate game model with previous univariate game model. Besides, we discuss the stabilities of fixed points and compare the price and service game with price game. What is more, the recursive least-squares (RLS) estimation is introduced to substitute naive estimation; then the impacts of RLS estimation are studied by comparing it with naive estimation.

Adaptive frequency estimation of three-phase power systems

The frequency of a three-phase power system can be estimated by identifying the parameter of a second-order autoregressive (AR2) linear predictive model for the complex-valued αβ signal of the system. Since, in practice, both input and output of the AR2 model are observed with noise, the recursive least-squares (RLS) estimate of the system frequency using this model is biased. We show that the estimation bias can be evaluated and subtracted from the RLS estimate to yield a bias-compensated RLS (BCRLS) estimate if the variance of the noise is known a priori. Moreover, in order to simultaneously compensate for the noise on both input and output of the AR2 model, we utilize the concept of total least-square (TLS) estimation and calculate a recursive TLS (RTLS) estimate of the system frequency by employing the inverse power method. Unlike the BCRLS algorithm, the RTLS algorithm does not require the prior knowledge of the noise variance. We prove mean convergence and asymptotic unbiasedness of the BCRLS and RTLS algorithms. Simulation results show that the RTLS algorithm outperforms the RLS and BCRLS algorithms as well as a recently-proposed widely-linear TLS-based algorithm in estimating the frequency of both balanced and unbalanced three-phase power systems.

A Method for Averting Saturation From Series Transformers of Dynamic Voltage Restorers

This paper proposes a technique for preventing saturation in series transformers from dynamic voltage restorer (DVR) systems. The method consists in correcting the voltages which are injected through the transformers into the power system to compensate voltage sags. It restricts the compensating voltages during the sag whenever it predicts that a maximum limit for the flux linkage is about to be exceeded. The prediction is carried out at the beginning of a stabilized voltage sag. Moreover, the technique allows a certain level of sag compensation even when the estimated flux is expected to exceed the saturation limit. The voltage sag level and phase are computed through an adaptive recursive least squares (RLS). The RLS estimation incorporates a transient period before it achieves a stable state whenever there is a sag event. The DVR is not supposed to operate in this period. Therefore, this paper also outlines a simple procedure for detecting the RLS estimation stable level. Simulation of different scenarios of voltage sags and the results from the experimental implementation of a DVR show the effectiveness of the method.

Bias-compensation-based least-squares estimation with a forgetting factor for output error models with white noise

In this paper, the bias-compensation-based recursive least-squares (LS) estimation algorithm with a forgetting factor is proposed for output error models. First, for the unknown white noise, the so-called weighted average variance is introduced. With this weighted average variance, a bias-compensation term is first formulated to achieve the bias-eliminated estimates of the system parameters. Then, the weighted average variance is estimated. Finally, the final estimation algorithm is obtained by combining the estimation of the weighted average variance and the recursive LS estimation algorithm with a forgetting factor. The effectiveness of the proposed identification algorithm is verified by a numerical example.

INVERSE DYNAMIC INPUT ESTIMATION OF A SEISMIC SOIL–STRUCTURE INTERACTION SYSTEM

The innovative fuzzy weighting input estimation method is used to estimate the ground motion acceleration of a soil–structure interaction system in this study. The input estimation method is comprised of the Kalman filter without the input term and the fuzzy weighting recursive least square estimator. The recursive least squares estimator (RLSE) is weighted using the fuzzy weighting factor. The superior capabilities of this inverse method are demonstrated by solving the soil–structure interaction estimation problem. The precision of the proposed method is verified using the actual earthquake acceleration. The results show that this method has the advantages of stability and accuracy.

Adaptive robust motion trajectory tracking control of pneumatic cylinders with LuGre model-based friction compensation

Friction compensation is particularly important for motion trajectory tracking control of pneumatic cylinders at low speed movement. However, most of the existing model-based friction compensation schemes use simple classical models, which are not enough to address applications with high-accuracy position requirements. Furthermore, the friction force in the cylinder is time-varying, and there exist rather severe unmodelled dynamics and unknown disturbances in the pneumatic system. To deal with these problems effectively, an adaptive robust controller with LuGre model-based dynamic friction compensation is constructed. The proposed controller employs on-line recursive least squares estimation (RLSE) to reduce the extent of parametric uncertainties, and utilizes the sliding mode control method to attenuate the effects of parameter estimation errors, unmodelled dynamics and disturbances. In addition, in order to realize LuGre model-based friction compensation, the modified dual-observer structure for estimating immeasurable friction internal state is developed. Therefore, a prescribed motion tracking transient performance and final tracking accuracy can be guaranteed. Since the system model uncertainties are unmatched, the recursive backstepping design technology is applied. In order to solve the conflicts between the sliding mode control design and the adaptive control design, the projection mapping is used to condition the RLSE algorithm so that the parameter estimates are kept within a known bounded convex set. Finally, the proposed controller is tested for tracking sinusoidal trajectories and smooth square trajectory under different loads and sudden disturbance. The testing results demonstrate that the achievable performance of the proposed controller is excellent and is much better than most other studies in literature. Especially when a 0.5 Hz sinusoidal trajectory is tracked, the maximum tracking error is 0.96 mm and the average tracking error is 0.45 mm. This paper constructs an adaptive robust controller which can compensate the friction force in the cylinder.

The model order reduction using LS, RLS and MV estimation methods

Reducing the order of high-order systems eliminates the complexity of them. So, the system analysis and the controller design are done easily. Many model reduction methods have been presented up to now. In this paper, LS (Least squares), RLS (Recursive Least Squares) and MV (Minimum Variance) will be presented as model reduction methods. MV is an adaptive theory that we will use the concept of it to reduce the order of systems. This idea has not been presented so far. One advantage of this method over the other methods, is that it has several parameters to achieve the desired reduced system. The other advantages of this method are that the speed is more and the computations are less compared to the other methods. In this paper, after explaining the three methods, two illustrative examples are provided to demonstrate the applicability of them.