An approximation method for estimation in linear systems with parameter uncertainty

Estimation of the state variables of a linear system with parameter uncertainties is performed using an asymptotically unbiased linear minimum-variance recursive estimator in continuous time. Estimates of the parameters can be obtained simultaneously, but are found to be biased. By augmenting additional linear dynamic equations which represent an asymptotic expansion in the unknown parameters, a linear structure is formed which approximates the original nonlinear system. However, the initial conditions and additive process noise are not Gaussian. The convergence properties of the state variance for this expansion are illustrated analytically by a scalar dynamic system. The numerical aspects of this example illustrate the behavior of the actual variance of the error in the state estimate and the predicted error variance as the order of the approximation increases. For the vector state problem, only the multidimensional dynamic system in canonical form with a single output is developed. For ann-dimensional system withnunknown constant parameters, a first-order approximation requiresnadditional linear equations. This approach can be extended to correlated parameter processes.     

The discrete linear time invariant time-optimal control problem—An overview

A review of two decades of investigation into one of the fundamental problems of control theory, the discrete linear time invariant time optimal control problem, is presented. Two classes of multi-input time optimal controller, one based on system controllability and the other on eigenvalue assignment, are considered in some detail. Also discussed are the dual state reconstruction problem, the deadbeat control of linear systems with inaccessible state, and other extensions. The exposition is a unifying one and involves only that matrix algebra commonly encountered in control theory.     

Modal control of linear time invariant systems

A procedure is presented for the determination of a linear control law (for a system with one input) such that the resultant closed–loop System has specified eigenvalues. The design procedure is simpler than existing techniques.     

A note on sampling and parameter estimation in linear stochasticsystems

Numerical differentiation formulas that yield consistent least squares parameter estimates from sampled observations of linear, time invariant higher order systems have been introduced previously by Duncan et al. (1994). The formulas given by Duncan et al. have the same limiting system of equations as in the continuous time case. The formula presented in this note can be characterized as preserving asymptotically a partial integration rule. It leads to limiting equations for the parameter estimates that are different from the continuous case, but they again imply consistency. The numerical differentiation formulas given here can be used for an arbitrary linear system, which is not the case in the previous paper by Duncan et al     

一种新的线性系统模型参数估计方法

分析了使用区间分析进行参数估计的原因,由此在未知但有界（UBB）误差假设下,提出了线性系统模型参数估计的区间分析方法,同时,对Matlab区间算法工具箱INTLAB进行了简介,并将INTLAB应用于使用所提算法的算例中。通过数值算例,将此算法与Fogel椭球算法和最小二乘算法进行了比较,算例结果表明了所提算法的有效性和可行性。     

Modular approach to parameter estimation in geared and linear resonant systems

This paper proposes a unified approach to the estimation of the physical parameters defining both geared and linear resonant systems, namely the dead-zone, inertia, stiffness and damping parameters. Although the technique is based on discrete-time models, it allows extraction of the continuous-time parameters from the discrete coefficients. This is facilitated by a modular procedure, involving separate locked- and unlocked-load measurements. The difference equation coefficients associated with the discrete model become trivial functions of the physical system parameters, which are computed using predetermined polynomial approximations. The reduced-order estimation modules improve noise immunity and simplify the estimation routines. Very accurate experimental results verify the utility of the approach.     

On-line order estimation and parameter identification for linear stochastic feedback control systems (CARMA model)

This paper presents a new recursive estimate method for orders and coefficients of linear stochastic feedback control systems (CARMA model) under the assumption that the upper bounds of system orders are known. The strong consistency of the estimates for orders and coefficients is proved and the convergence rate of coefficient estimates to their true values is also obtained. The estimate algorithm is applied to adaptive tracking of the systems with unknown orders and unknown coefficients. The resulting closed-loop systems are then globally stable and the tracking sample mean square error is minimized as well. Simultaneously, the estimates of the adaptive tracking for orders and coefficients are also strongly consistent. The simulation results given here show that the new developed algorithms of both system identification and adaptive tracking are effective.     

线性控制系统的无界多面体不变集

本文利用非光滑分析方法,讨论了线性控制系统的无界多面体不变集问题.当无界多面体的极方向满足一定条件时,得到了该无界多面体为一类线性控制系统弱不变集的判别方法.然后在更一般的线性控制系统下给出了无界多面体为强不变集的充分条件.最后给出两个应用实例.     

A computational method for solving optimal control and parameter estimation of linear systems using Haar wavelets

In this article, a computational method based on Haar wavelet in time-domain for solving the problem of optimal control of the linear time invariant systems for any finite time interval is proposed. Haar wavelet integral operational matrix and the properties of Kronecker product are utilized to find the approximated optimal trajectory and optimal control law of the linear systems with respect to a quadratic cost function by solving only the linear algebraic equations. It is shown that parameter estimation of linear system can be done easily using the idea proposed. On the basis of Haar function properties, the results of the article, which include the time information, are illustrated in two examples.     

仿射不变参数化

仿射不变技术的目的是指图形在经历各种仿射变换后,它的某些性质保持不变,即这些性质本身不受变换的影响。将仿射不变技术用于参数化,由此得到的参数在各种仿射变换前后都保持不变。用这种参数做曲线插值时,可为实际应用带来很大的方便。以三次样条函数实现参数曲线,并通过计算实例对新方法与目前常用的参数化方法的插值结果进行了比较,在效率和显示结果上,新的方法是令人满意的。     

An LFT approach to parameter estimation

In this paper we consider a unified framework for parameter estimation problems. Under this framework, the unknown parameters appear in a linear fractional transformation (LFT). A key advantage of the LFT problem formulation is that it allows us to efficiently compute gradients, Hessians, and Gauss–Newton directions for general parameter estimation problems without resorting to inefficient finite-difference approximations. The generality of this approach also allows us to consider issues such as identifiability, persistence of excitation, and convergence for a large class of model structures under a single unified framework.     

Discrete pulse orthogonal functions for the analysis,parameter estimation and optimal control of linear digital systems

Discrete pulse orthogonal functions (DPOFs) are introduced, and their operational matrix is proposed to solve various control problems. The applications of DPOFs in digital control systems are analogous to those of block pulse functions (BPFs) in continuous systems. Applying the DPOFs, the approximate solutions of digital time-invariant systems can be easily obtained by a convenient algorithm. Three examples are presented to demonstrate the applications of DPOFs.     

An integrated system optimization and parameter estimation technique for hierarchical control of steady-state systems

This paper presents computer simulation results of an optimal adaptive algorithm (Brdy? and Roberts 1984) and develops a double iterative alternative. Both algorithms are optimal in the sense that Kuhn-Tucker necessary optimality conditions are satisfied. The aim of the latter is to reduce the number of times that information is required from the real system. Simulation results also show that it does not increase the total number of information exchanges during the iteration procedure and may even reduce it.     

Two-stage procedure for parameter estimation in large-scale interconnected linear zero-memory systems

The problem of parameter estimation in interconnected complex systems composed of linear zero-memory elements is considered. A two-stage scheme for estimating system parameters is proposed, and the convergence in probability and with probability one, of the parameter estimates to the true values of system parameters, is shown. Some computational aspects of the algorithm are discussed and its recursive version is provided. The rate of convergence is also studied.     

线性时不变控制系统中的互信息率和解析约束

针对受随机干扰的离散线性时不变系统的跟踪控制问题，采用信息论的方法讨论了系统获得有效跟踪性能的基本信息条件。从此条件出发，得出系统闭环传递函数设计的解析约束，表明了系统敏摩函数与补敏感函数的Bode积分间的关系。对于开环传递函数严格正则的系统，这一约束反映了开环不稳定零极点应具备的特性。     

Nonlinear parameter estimation by weighted linear associativememory with nonzero interception

The method of linear associative memory (LAM) has recently been applied in nonlinear parameter estimation. In the method of LAM, a model response, nonlinear with respect to the parameters, is approximated linearly by a matrix, which maps inversely from a response vector to a parameter vector. This matrix is determined from a set of initial training parameter vectors and their response vectors according to a given cost function, and can be updated recursively and adaptively with a pair of newly generated parameter-response vector. The advantage of LAM is that it can yield good estimation of the true parameter from a given observed response even if the initial training parameter vectors are far from the true values. In a previous paper, we have significantly improved the LAM method by introducing a weighted linear associative memory (WLAM) approach for nonlinear parameter estimation. In the WLAM approach, the contribution of each pair of parameter-response vector to the cost function is weighted in a way such that if a response vector is closer to the observed one then its pair plays more important role in the cost function. However, in both LAM and WLAM, the linear association is introduced with zero interceptions, which would not give an exact association even if the model function is linear and so will affect the efficiency of the estimations. In this paper, we construct a theory which introduces a linear association memory with a nonzero interception (WLAMB). The results of our estimation tests on two quite different models, Van der Pol equation and somatic shunt cable model, suggest that WLAMB can still significantly improve on WLAM.     

New finite-dimensional filters for parameter estimation ofdiscrete-time linear Gaussian models

The authors derive a new class of finite-dimensional recursive filters for linear dynamical systems. The Kalman filter is a special case of their general filter. Apart from being of mathematical interest, these new finite-dimensional filters can be used with the expectation maximization (EM) algorithm to yield maximum likelihood estimates of the parameters of a linear dynamical system. Important advantages of their filter-based EM algorithm compared with the standard smoother-based EM algorithm include: 1) substantially reduced memory requirements, and 2) ease of parallel implementation on a multiprocessor system. The algorithm has applications in multisensor signal enhancement of speech signals and also econometric modeling     

On guaranteed parameter estimation of a multiparameter linear regression process

This paper presents a sequential estimation procedure for the unknown parameters of a continuous-time stochastic linear regression process. As an example, the sequential estimation problem of two dynamic parameters in stochastic linear systems with memory and in autoregressive processes is solved. The estimation procedure is based on the least squares method with weights and yields estimators with guaranteed accuracy in the sense of the L_q-norm for fixed q≥2.The proposed procedure works in the mentioned examples for all possible values of unknown dynamic parameters on the plane R² for the autoregressive processes and on the plane R² with the exception of some lines for the linear stochastic delay equations. The asymptotic behaviour of the duration of observations is determined.The general estimation procedure is designed for two or more parametric models. It is shown that the proposed procedure can be applied to the sequential parameter estimation problem of affine stochastic delay differential equations and autoregressive processes of an arbitrary order.