Direct and indirect least squares methods in continuous-time parameter estimation

The discrete-time least squares approach is extended to the estimation of parameters in continuous nonlinear models. The resulting direct integral least squares (DILS) method is both simple and numerically efficient and it usually improves the mean-squared error of the estimates compared with the conventional indirect least squares (ILS) method. The biasedness of the DILS estimates may become serious if the sample points are widely spaced in time and/or the signal-to-noise ratio is low and so a continuous-time symmetric bootstrap (SB) estimator which removes this problem is described. The DILS, SB and ILS methods form a three-stage procedure combining the robustness and numerical efficiency of direct methods with the asymptotic unbiasedness of ILS procedures.     

A new method of parameter estimation and adaptive control of nonlinear systems with filterless least-squares

This paper is concerned with the parameter estimation and adaptive control problem for a sort of strict-feedback nonlinear systems with unknown parameters. In order to deal with the nonlinear functions, we develop a kind of least-squares estimator in the parameter estimation part. With such an estimator, we design the state-feedback control law and analyze the properties of the closed-loop system. To be concrete, the parameter estimator can guarantee the boundedness of parameter estimate error. At the same time, the adaptive control law is proposed to make the system states converge to zero and the equilibrium stable globally under some common assumptions. Finally, simulations are given to demonstrate the theoretical results.     

Constrained predictive pole-placement control with linear models

Predictive pole-placement (PPP) control is a continuous-time MPC using a particular set of basis functions leading to pole-placement behaviour in the unconstrained case. This paper presents two modified versions of the PPP controller which are each shown to have desirable stability properties when controlling systems with input, output and state constraints.     

Combined parameter and output estimation of dual-rate systems using an auxiliary model

For a dual-rate sampled-data system, an auxiliary model based identification algorithm for combined parameter and output estimation is proposed. The basic idea is to use an auxiliary model to estimate the unknown noise-free output (true output) of the system, and directly to identify the parameters of the underlying fast single-rate model from the dual-rate input-output data. It is shown that the parameter estimation error consistently converges to zero under generalized or weak persistent excitation conditions and unbounded noise variance, and that the output estimates uniformly converge to the true outputs. An example is included.     

Maximum likelihood parameter estimation algorithm for controlled autoregressive autoregressive models

A maximum likelihood parameter estimation algorithm is derived for controlled autoregressive autoregressive (CARAR) models based on the maximum likelihood principle. In this derivation, we use an estimated noise transfer function to filter the input–output data. The simulation results show that the proposed estimation algorithm can effectively estimate the parameters of such class of CARAR systems and give more accurate parameter estimates than the recursive generalized least-squares algorithm.     

Constrained data-driven optimal iterative learning control

A constrained optimal ILC for a class of nonlinear and non-affine systems, without requiring any explicit model information except for the input and output data, is proposed in this work. In order to address the nonlinearities, an iterative dynamic linearization method without omitting any information of the original plant is introduced in the iteration direction. The derived linearized data model is equivalent to the original nonlinear system and reflects the real-time dynamics of the controlled plant, rather than a static approximate model. By transferring all the constraints on the system output, control input, and the change rate of input signals into a linear matrix inequality, a novel constrained data-driven optimal ILC is developed by minimizing a predesigned objective function. The optimal learning gain is unfixed and updated iteratively according to the input and output measurements, which enhances the flexibility regarding modifications and expansions of the controlled plant. The results are further extended to the point-to-point control tasks where the exact tracking performance is required only at certain points and a constrained data-driven optimal point-to-point ILC is proposed by only utilizing the error measurements at the specified points only.     

Rapid parameter estimation algorithms using the ELS principle

Two computationally efficient algorithms for estimating the parameters of linear discrete-time systems are proposed. The algorithms are based on the extended least squares (ELS) principle. They are essentially a correlation version of the off-line ELS method that eliminate all the redundant computations, do not require construction and operations of large matrices and bypass the explicit evaluation of residuals. Examples are given to illustrate their feasibility and performance.     

Constrained RHC for LPV systems with bounded rates of parameter variations

This paper studies a stabilization problem of polytopically uncertain linear parameter varying systems with input constraints and bounded rates of parameter variations. In the framework of finite receding horizon control (RHC), a system containing “parameter” uncertainties is modified into a system with “parameter-incremental” uncertainties within each horizon. For the system modified in this manner, a robust RHC is designed by solving an optimization problem at each time instant. Based on the feasibility of the problem and the optimality of its solution, the closed-loop system stability is guaranteed. A numerical example is included to illustrate the validity of the results.     

A continuous-time framework for least squares parameter estimation

This paper proposes a continuous-time framework for the least-squares parameter estimation method through evolution equations. Nonlinear systems in the standard state space representation that are linear in the unknown, constant parameters are investigated. Two estimators are studied. The first one consists of a linear evolution equation while the second one consists of an impulsive linear evolution equation. The paper discusses some theoretical aspects related to the proposed estimators: uniqueness of a solution and an attractive equilibrium point which solves for the unknown parameters. A deterministic framework for the estimation under noisy measurements is proposed using a Sobolev space with negative index to model the noise. The noise can be of large magnitude. Concrete signals issued from an electronic device are used to discuss numerical aspects.     

Constrained state estimation for stochastic jump systems: moving horizon approach

We discuss the state estimation advantages for a class of linear discrete-time stochastic jump systems, in which a Markov process governs the operation mode, and the state variables and disturbances are subject to inequality constraints. The horizon estimation approach addressed the constrained state estimation problem, and the Bayesian network technique solved the stochastic jump problem. The moving horizon state estimator designed in this paper can produce the constrained state estimates with a lower error covariance than under the unconstrained counterpart. This new estimation method is used in the design of the restricted state estimator for two practical applications.     

Fast GLS algorithm for parameter estimation

A computationally efficient off-line algorithm for estimating the parameters of a linear discrete-time SISO system is presented. The algorithm is based on the generalized least-squares (GLS) principle. It is essentially a correlation version of the GLS method that (1) eliminates all the redundant computations, (2) does not require explicit evaluation of the residuals, (3) does not require explicit data filtering, and (4) eliminates the large storage requirement of the conventional off-line GLS algorithm.     

Least squares parameter estimation

This article demonstrates the application of least squares for the estimation of system parameters. Analytic as well as numerical approaches are described. The model of the system dynamics is assumed in the form of regression model. Solutions are discussed for the case of white noise and correlated noise corrupting the useful output signal of the system.     

用带约束的最小二乘法拟合平面圆曲线

研究各种拟合圆的方法,提出了一个圆的代数距离表示法的系数约束条件,在此约束条件下讨论圆的几何特征参数的估计问题,并给出了特征参数在约束条件下的最小二乘估计．实例验证表明,文中算法比一般最小二乘法具有更高的拟合精度．     

Nonlinear adaptive control system design with asymptotically stable parameter estimation error

The paper presents a new general method for nonlinear adaptive system design with asymptotic stability of the parameter estimation error. The advantages of the approach include asymptotic unknown parameter estimation without persistent excitation and capability to directly control the estimates transient response time. The method proposed modifies the basic parameter estimation dynamics designed via a known nonlinear adaptive control approach. The modification is based on the generalised prediction error, a priori constraints with a hierarchical parameter projection algorithm, and the stable data accumulation concepts. The data accumulation principle is the main tool for achieving asymptotic unknown parameter estimation. It relies on the parametric identifiability system property introduced. Necessary and sufficient conditions for exponential stability of the data accumulation dynamics are derived. The approach is applied in a nonlinear adaptive speed tracking vector control of a three-phase induction motor.     

State and parameter estimation of microbial growth processes

Based on a second order nonlinear model of microbial growth, two distinct problems—state estimation and parameter estimation—are considered. The first case assumes only one state is available from measurement and an asymptotic estimate of the other state is required. The second problem assumes both states are available, and considers the design of an estimator to provide asymptotic estimates of the growth system parameters.     

同参数估计对偶的自适应控制算法

本文把线性和非线性系统统一处理。从自适应控制算法与参数估计算法的对偶性出发,提出了自适应控制算法的一种统一格式。这种格式算法简单,并在一定的条件下,能使控制误差一致的足够小。     

On the use of constraints in least squares estimation and control

In this article, we examine the effect of constraints on estimation and control methods based on quadratic penalty functions. We begin with estimation theory and analyze how constraints alter the statistical properties of the least squares estimates. It is shown that constraints can be used to formulate maximum likelihood (MLE) and maximum a posteriori (MAP) estimators for a variety of unimodal distributions. This provides greater flexibility over the assumption of normality inherent in the MLE and MAP interpretation of traditional least squares. We discuss how these ideas apply to state space models of dynamic systems. Possible applications for controllers that handle constraints are also discussed. A parameter estimation example is given to demonstrate the potential for improved performance over unconstrained least squares.     

Adaptive parameter estimation for a general dynamical system with unknown states

This paper is concerned with the design of a state filter for a time‐delay state‐space system with unknown parameters from noisy observation information. The key is to investigate new identification algorithms for interactive state and parameter estimation of the considered system. Firstly, an observability canonical state‐space model is derived from the original model by linear transformation for the purpose of simplifying the model structure. Secondly, a direct state filter is formulated by minimizing the state estimation error covariance matrix on the basis of the Kalman filtering principle. Thirdly, once the unknown states are estimated, a state filter–based recursive least squares algorithm is proposed for parameter estimation using the least squares principle. Then, a state filter–based hierarchical least squares algorithm is derived by decomposing the original system into several subsystems for improving the computational efficiency. Finally, the numerical examples illustrate the effectiveness and robustness of the proposed algorithms.     

Nonlinear system identification with limited time data

With disturbances modeled by arbitrary solutions to a linear homogeneous differential equation, a least squares-equation error method is developed for parameter identification using data over a limited time interval which has application to certain classes of nonlinear and time varying systems. Examples include the Duffing, Hammerstein, Mathieu and Van der Pol equations together with a class of bilinear systems. The technique seeks to determine the parameters characterizing the disturbance modes in addition to the system parameters, based on the input-output data collected over the finite time interval. The approach circumvents the need to estimate unknown initial conditions through the use of a certain projection operator. Computational considerations are discussed and simulation results are summarized for the Van der Pol equation.