Extended Least-squares Algorithm Research Articles

Support Recovery and Parameter Identification of Multivariate ARMA Systems with Exogenous Inputs

Focusing on identification, support recovery, and parameter estimation of multivariate ARMA systems with exogenous inputs (ARMAX) is considered. It is assumed that sparsity appears not only in the unknown parameter matrix of the system inputs and outputs but also in the coefficients of system noises, and with exogenous inputs the system admits feedback control and thus the observation of the ARMAX system may be nonstationary. A two-stage sparse identification algorithm is introduced. First, the extended least squares (ELS) algorithm is applied to obtain an initial estimate for the unknown parameter matrix of the ARMAX system. Second, a convex optimization criterion is introduced based on norm with regularization, where adaptive weights generated from ELS algorithms are adopted in the optimization variables in the term. Third, it is proved that by optimizing the criterion, the sets of the zero and the nonzero entries of the parameter matrix, i.e., the support set of the ARMAX system, can be correctly identified with a finite number of observations, and estimates of the nonzero parameters converge to the true values with probability one. Fourth, the strictly positive realness (SPR) condition required by the ELS algorithm is relaxed by an overparametrization technique. The performance of the algorithm is testified through simulation examples.

A Robust Control Strategy With Perturbation Estimation for the Parrot Mambo Platform

This article addresses theoretical and practical challenges associated with a commercially available and ready-to-fly small-scale unmanned aircraft system (UAS) developed by Parrot SA: the Mambo quad rotorcraft. The dynamic model and the structure of the controller running onboard the UAS autopilot are not disclosed by its manufacturers. For this reason, a novel robust controller for discrete-time systems under time delays and input saturation is first developed for this platform. Then, three fundamental estimation and control challenges are addressed. The first challenge is the system identification of the X and Y translational dynamics of the UAS. To accomplish this goal, input-output data pairs are collected from different UAS platforms during real-time experimental flights. A group of dynamic models are identified from the data pairs through an extended least-squares algorithm. The obtained models are similar in nature but exhibit parametrical variations due to uncertainties in the fabrication process and different levels of wear and tear. Using a time-varying modeling approach, the second challenge addresses the development of a robust controller, which guarantees the stability of all the identified dynamic models. The third challenge addresses the development of a nonlinear controller enhanced with a perturbation estimation, which can reject, from the nominal model, the effects of model uncertainties and perturbations. These theoretical developments are presented in the form of two original theorems. The proposed strategies are ultimately validated in a set of real-time experiments, demonstrating their effectiveness and applicability.

Recursive Least-squares Estimation for Multivariable Systems Based on the Maximum Likelihood Principle

This paper studies the identification problem of multivariable controlled autoregressive moving average systems. For the case with a parameter matrix and an unmeasurable vector in the system identification model, we transform the model into several submodels based on the number of the outputs. A maximum likelihood-based recursive least-squares algorithm is derived to identify the parameters of each submodel. A multivariable recursive extended least-squares algorithm is provided as a comparison. The effectiveness of the proposed identification algorithm is verified by simulation examples.

Recursive identification for Hammerstein systems with diminishing excitation signals

In this paper, we consider the identification of Hammerstein systems where the nonlinearity is described by a combination of basis functions with unknown coefficients. The extended least squares (ELS) algorithm is applied to estimate the unknown parameters in the system. Contrary to the classical excitation signals for identification of Hammerstein systems, i.e., the periodic inputs or stationary random signals, here we choose a sequence of diminishing excitation signals as the system inputs. We prove that the strong consistency of the ELS algorithm still holds true and the convergence rate is obtained as well. A numerical example is given to verify the performance of the identification method.

Maximum Likelihood-Based Recursive Least-Squares Algorithm for Multivariable Systems with Colored Noises Using the Decomposition Technique

This paper considers the parameter estimation problems for a class of multivariable equation-error systems with colored noises. By using the decomposition technique, a multivariable system is transformed into several subsystems to reduce the computational burden, and a maximum likelihood-based recursive least-squares identification algorithm is developed for estimating the parameters of each subsystem. As a comparison, a multivariable recursive extended least-squares algorithm is presented. The analysis indicates that the proposed algorithm has lower computational complexity than the multivariable recursive extended least-squares algorithm, and the numerical simulation results demonstrate that the proposed method is effective.

Acceleration tracking control combining adaptive control and off-line compensators for six-degree-of-freedom electro-hydraulic shaking tables

An electro-hydraulic shaking table (EHST) is an essential experimental facility to simulate in real-time actual vibration situations. An adaptive controller combined with off-line compensators is proposed to improve the acceleration frequency bandwidth and tracking accuracy of a six-degree-of-freedom (6-DOF) EHST. A servo controller has been employed to implement acceleration closed-loop and coordinate control of the 6-DOF EHST. A recursive extended least-squares algorithm is employed to identify acceleration closed-loop transfer functions and a zero-phase-error tracking controller is used to design off-line inverse model compensators using the identified transfer functions. However, the off-line compensators cannot compensate in real-time varying dynamics of the 6-DOF EHST; so an online adaptive controller with a least-mean-squares (LMS) algorithm based on a delay compensator is employed. The proposed controller combines advantages of the off-line compensators and the online adaptive controller, which guarantees both a fast rate of convergence of the LMS algorithm and high-fidelity acceleration tracking accuracy of the 6-DOF EHST. Some experimental studies have been conducted on a 6-DOF EHST and experimental results show that acceleration tracking control performances, including the rate of convergence of the LMS algorithm and acceleration tracking accuracy, have been improved compared to a conventional three-variable controller and adaptive controllers.

Experimental evaluation of the parameter-based closed-loop transfer function identification for electro-hydraulic servo systems

Closed-loop systems of an electro-hydraulic servo system including position, acceleration, and force closed-loop systems and their closed-loop transfer functions based on parameter model are adaptive identified using a recursive extended least-squares algorithm. The position and force closed-loop tracking controllers are designed by a proportional–integral–derivative controller and are tuned by the position and force step signals. The acceleration closed-loop tracking controller is designed by a three-variable controller and the three states include position, velocity, and acceleration. Experimental results of the estimated position, acceleration, and force closed-loop transfer functions are performed on an actual electro-hydraulic servo system using xPC rapid prototyping technology, which clearly demonstrate the benefit of the adaptive identification method.

Acceleration waveform replication on six-degree-of-freedom redundant electro-hydraulic shaking tables using an inverse model controller with a modelling error

A redundant electro-hydraulic shaking table (REST) of six degrees-of-freedom (6 DOFs) with eight hydraulic actuators is an essential experimental tool in many industrial applications for real-time simulation of actual vibrations, such as structural vibration, earthquake simulation and fatigue testing. In order to obtain a high-fidelity acceleration waveform on the REST, a feed-forward inverse model (FFIM) controller with a modelling error compensator is proposed in this study. A recursive extended least-squares algorithm is employed to identify an acceleration closed-loop transfer function of the REST. A zero phase error compensation technology is employed to guarantee stability of the designed FFIM because the identified acceleration closed-loop transfer function is a typical non-minimum phase system and its direct inverse transfer function is unstable. The modelling error compensator is designed to compensate for the modelling error between the identified transfer function and the actual experimental REST, which deteriorates the acceleration waveform replication accuracy of the REST. A 6 DOF REST experimental system was used to verify the proposed controller. Experimental results demonstrated that the proposed controller gave satisfactory acceleration tracking performances on the REST.

A general multirate approach for direct closed-loop identification to the Nyquist frequency and beyond

Without injecting any external persistently exciting signals, a general multirate approach is proposed for direct closed-loop identification. First, an extended least-squares algorithm is designed to identify the mixed-rate model after polynomial transformation. Next, the identified mixed-rate model is used to compute the fast-rate model, from which system dynamics to the Nyquist frequency and beyond can be obtained. Identifiability is mathematically proven if the plant output sampling rate is non-integer multiples of the controller sampling rate. The effectiveness of the proposed approach is evaluated by both the simulation and experimental results in a hard disk drive.

Adaptive Tracking Games for Coupled Stochastic Linear Multi-Agent Systems: Stability, Optimality and Robustness

Distributed adaptive tracking-type games are investigated for a class of coupled stochastic linear multi-agent systems with uncertainties of unknown structure parameters, external stochastic disturbances, unmodeled dynamics, and unknown agents' interactions. The control goal is to make the states of all the agents converge to a desired function of the population state average (PSA). Due to the fact that only local information is available for each agent, the control is distributed. For the time-invariant parameter case, the extended least-squares algorithm, Nash certainty equivalence (NCE) principle, and certainty equivalence (CE) principle are used to estimate the unknown parameters and the PSA term, and to design adaptive control, respectively. Under some mild conditions, it is shown that the closed-loop system is almost surely uniformly stable with respect to the population number N; the estimate for the PSA term is strongly consistent; the adaptive control is almost surely an asymptotic Nash equilibrium. When the dynamics of each agent contains time-varying parameters and unmodeled dynamics, the projected least mean square (LMS) algorithm, NCE principle, and CE principle are adopted to estimate the unknown time-varying parameters, and the unknown PSA term, and to design robust adaptive control, respectively. In addition to stability of the closed-loop system and consistency of the PSA estimate, the control law is shown to be robust Nash equilibrium with respect to the unmodeled dynamics, the variation of the unknown parameters, and the external disturbances. Two numerical examples are given to illustrate the methods and results of this paper.

Stochastic modelling of plasma reflection during keyhole arcwelding

Keyhole arc welding (KAW), including the keyhole double-sided arcwelding process being developed at the University of Kentucky and keyholeplasma arc welding, can achieve much deeper narrower penetration than allother arc welding processes. If it could be controlled such that the heatinput and weld pool are minimized while at the same time the desired fullpenetration is guaranteed, it could become an effective yet affordabletechnology to improve productivity in welding thick materials. However, thekey in developing such a controlled KAW technology is the sensor which candetect the evolution of the keyhole. Preliminary study shows that the plasmareflection could lead to a practical yet accurate sensor. In this study, thedynamic behaviour of the plasma reflection is described using the reflectionarc angle (RAA). It is found that the RAA series can be considered anautoregressive moving-average (ARMA) process. The orders of the ARMA model aredetermined using auto-correlation and partial auto-correlation functions. Theparameters of the ARMA are recursively estimated using the extended leastsquares algorithm. It is found that the recursive estimates of the modelparameters change as the state of the keyhole changes. A discriminator hasbeen proposed to determine the state of the keyhole based on the recursiveestimates of the model parameters.

Adaptive estimation of HMM transition probabilities

This paper presents new schemes for recursive estimation of the state transition probabilities for hidden Markov models (HMMs) via extended least squares (ELS) and recursive state prediction error (RSPE) methods. Local convergence analysis for the proposed RSPE algorithm is shown using the ordinary differential equation (ODE) approach developed for the more familiar recursive output prediction error (RPE) methods. The presented scheme converges and is relatively well conditioned compared with the previously proposed RPE scheme for estimating the transition probabilities that perform poorly in low noise. The ELS algorithm presented is computationally of order N/sup 2/, which is less than the computational effort of order N/sup 4/ required to implement the RSPE (and previous RPE) scheme, where N is the number of Markov states. Building on earlier work, an algorithm for simultaneous estimation of the state output mappings and the state transition probabilities that requires less computational effort than earlier schemes is also presented and discussed. Implementation aspects of the proposed algorithms are discussed, and simulation studies are presented to illustrate the convergence and convergence rates.

A weighted adaptive one‐step‐ahead minimum variance controller based on the ELS algorithm

This paper deals with the design of a weighted adaptive one-step-ahead minimum variance controller based on the extended least squares (ELS) algorithm for a discrete time stochastic system. The stability of the closed-loop system and the convergence rates of the general adaptive tracking and estimation errors are respectively established under strictly positive real and minimum phase conditions. The best possible convergence rate of the average error between the predicted and desired outputs is obtained given some identifying condition and the above-stated conditions. No modification of the adaptive controller is made. © 1997 by John Wiley & Sons, Ltd.

Robustness and convergence of least-squares identification and adaptive tracking for stochastic systems

The adaptive conlrol for tracking a stochastic reference signal based upon the extended least-squares algorithm estimating unknown parameters of the modelled part in a stochastic syslem is recursively defined. It is shown that the closed-loop system is stable: the estimation error decreases as the unmodelled dynamics decays and the tracking error differs from the minimum plus a small value when the unmodelled dynamics is bounded in the average sense; the strong consistency of the estimates and asymptotical optimality of adaptive tracking are obtained when the unmodelled dynamics approaches zero in the average sense.

Rational model identification using an extended least-squares algorithm

A new least-squares-based parameter-estimation algorithm is derived for nonlinear systems which can be represented by a rational model defined as the ratio of two polynomial expansions of past system inputs, outputs and noise. Simulation results are included to illustrate the performance of the new algorithm.

Recent developments in optimal bounding ellipsoidal parameter estimation

The Optimal Bounding Ellipsoid (OBE) algorithms are viable alternatives to conventional adaptive filtering algorithms in situations where the noise does not satisfy the usual stationarity and whiteness assumptions. An example is shown in which the performance of an OBE algorithm is seen to be markedly superior to that of the recursive least-squares algorithm. Subsequently, an overview of some recent work in the area of OBE parameter estimation is presented. A lattice filter implementation of one particular OBE algorithm is first described. The extension of the OBE algorithm to the estimation of parameters of ARMA models is performed and the results of a convergence analysis are presented. It is demonstrated through a simulation example that the transient performance of the proposed algorithm is superior to that of the well-known extended least-squares algorithm.

ARMA parameter estimation using a novel recursive estimation algorithm with selective updating

The authors investigate an extension of a recursive estimation algorithm (the so-called optimal bounding ellipsoid or OBE algorithm) which features a discerning update strategy. In particular, an extension of the algorithm to ARMA (autoregressive moving-average) parameter estimation is presented along with a convergence analysis. The extension is similar to the extended least-squares (ELS) algorithm. However, the convergence analysis is complicated due to the discerning update strategy, which incorporates an information-dependent updating factor. The virtues of such an update strategy are more efficient use of the input data in terms of information processing and a modular adaptive filter structure which would facilitate the development of a parallel-pipelined signal processing architecture. It is shown that if the input noise is bounded and the moving-average parameters satisfy a certain magnitude bound, then the a posteriori prediction errors are uniformly bounded. With an additional persistence-of-excitation condition, the parameter estimates are shown to converge to a neighborhood of the true parameters, and the a priori prediction errors are shown to be asymptotically bounded. Simulation results show that the parameter estimation error for the EOBE algorithm is comparable to that for the ELS algorithm. >

Estimating lung mechanics of dogs with unilateral lung injury.

Extended least-squares algorithms using transpulmonary pressure and airway flow data from ventilatory waveforms were studied for their ability to track parameters of one- and two-compartment models of lung mechanics. A recursive extended least-squares algorithm with discounted measures estimated parameters of discrete-time models during synchronized intermittent mandatory ventilation. In tests on seven dogs developing oleic acid-induced unilateral hemorrhagic pulmonary edema, the one-compartment estimator responded rapidly and appropriately to changes in mechanics: compliance fell to 0.55 +/- 0.15 of its initial value and resistance rose by a factor of 1.8 +/- 0.5 in 3 h following injection of oleic acid. One-compartment parameter estimates revealed a difference between the airway resistance of inspiration and expiration. Two-compartment estimates were seldom physiologically plausible. The difference between inspiratory and expiratory resistance may have caused the two-compartment estimator to fail when applied to data from the entire respiratory cycle; when only expiratory data were used for estimation, the two-compartment estimates were meaningful. These estimates demonstrated increasing lung inhomogeneity after oleic acid was injected; at the end of 3 h, the ratio of the time constants of the two compartments ranged from 5 to 20 in six of the seven dogs. We conclude that the one- and two-compartment estimates may be combined to provide a meaningful assessment of lung mechanics.