Optimal Policy Response with Control Parameter and Intercept Covariance

Parameter uncertainty and the interaction between the uncertain parameters are important aspects of economic policy. In this work, I develop an analytical one-state variable, one-control variable model with two uncertain parameters (the control parameter and the intercept) and a nonzero covariance. I characterize the effect of changes in each of the covariance components on the optimal expected control variable. I found that the nature of the optimal policy maker’s response depends on the specific changing component of the covariance, the sign of the correlation coefficient and the sign of the optimal expected control variable when the covariance is zero. I obtain the conditions under which the effect of the covariance is considerable. This work complements previous studies by providing a complete set of cases and conditions for an aggressive or cautionary optimal policy maker’s response to changes in each covariance component. Finally, the importance of the analytical results is shown for the regulation of a stock pollutant leading to global warming.      

Least squares estimation of nonstationary covariance parameters in linear systems

Least squares estimation techniques are employed to overcome previous difficulties encountered in accurately estimating the state and measurement noise covariance parameters in linear stochastic systems. In the past accurate and rapidly converging covariance parameter estimates have been achieved with complex estimation algorithms only after specifying the statistical nature of the noise in the system and constraining the time variation of the covariance parameters. Weighted least squares estimation allows these restrictions to be removed while achieving near optimal accuracy using a filter on the same order of complexity as a Kalman filter. Allowing the covariance parameters to vary in as general a manner in time as the state in a linear discrete time stochastic system, and assuming that a Kalman filter is applied to this system using incorrect knowledge of the a priori statistics, it is shown how a covariance system is developed similar to the original system. Unbiased least squares estimates of the covariance parameters and of the original state are obtained without the necessity of specifying the distribution on the noise in either system. The accuracy of these estimates approaches optimal accuracy with increasing measurements when adaptive Kalman filters are applied to each system.     

Optimal instrumental-variable methods for identification of multivariable linear systems

The identification of multivariable linear systems using instrumental-variable (IV) methods is discussed. The emphasis is on accuracy properties. The IV estimates of multi-input-multi-output system parameters are proved to be asymptotically Gaussian distributed under weak conditions. An explicit expression for the covariance matrix of the parameter estimates is given. It is then shown how this matrix can be optimized by appropriately choosing the IV variant. The optimal accuracy so obtained is for some model structures equal to that corresponding to a prediction error method.     

Robust stochastic moment control via genetic-pole placement in communication network parameter setting

In this paper, the problems of stochastic robust approximate covariance assignment and robust covariance feedback stabilization, which are applied to variable parameters of additive increase/multiplicative decrease (AIMD) networks, are considered. The main idea of the developed algorithm is to use the parameter settings of an AIMD network congestion control scheme, where parameters may assign the desired network’s window covariance, with respect to the current network conditions. The aim is to search for the optimal AIMD parameters of a feedback gain matrix such that the objective functions defined via appropriate robustness measures and covariance assignment constraints can be optimized using an adaptive genetic algorithm (AGA). It is shown that the results can be used to develop tools for analyzing the behavior of AIMD communication networks. Quality of service (QoS) and other performance measures of the network have been improved by using the proposed congestion control. The accuracy of the controller is demonstrated by using MATLAB and NS software programs.     

Discrete-Time Risk-Sensitive Filters with Non-Gaussian Initial Conditions and Their Ergodic Properties

In this paper, we study asymptotic stability properties of risk-sensitive filters with respect to their initial conditions. In particular, we consider a linear time-invariant systems with initial conditions that are not necessarily Gaussian. We show that in the case of Gaussian initial conditions, the optimal risk-sensitive filter asymptotically converges to a suboptimal filter initialized with an incorrect covariance matrix for the initial state vector in the mean square sense provided the incorrect initializing value for the covariance matrix results in a risk-sensitive filter that is asymptotically stable, that is, results in a solution for a Riccati equation that is asymptotically stabilizing. For non-Gaussian initial conditions, we derive the expression for the risk-sensitive filter in terms of a finite number of parameters. Under a boundedness assumption satisfied by the fourth order absolute moment of the initial state variable and a slow growth condition satisfied by a certain Radon-Nikodym derivative, we show that a suboptimal risk-sensitive filter initialized with Gaussian initial conditions asymptotically approaches the optimal risk-sensitive filter for non-Gaussian initial conditions in the mean square sense. Some examples are also given to substantiate our claims.     

Specific optimal estimation

The goal of the specific optimal estimation problem is to achieve near-optimal (minimum variance) estimation using a structure that is easier to implement than the optimal solution. One chooses a reasonable configuration for the filter in which certain parameters are unspecified and then selects the parameters so that its performance is optimized. The problem is formulated as a two-point boundary-value problem resulting from consideration of the covariance of error of the estimate and application of the matrix-minimum principle. The examples presented indicate that near-optimal results can be obtained using a filter designed in this way.     

广义离散随机线性系统的最优递推滤波方法(Ⅱ)

本文对文献[1]给出的广义离散随机线性系统最优估计误差协方差阵进行了分析,在一定 条件下得到了误差协方差阵的上界和下界,继而讨论了由文献[1]给出的滤波器的稳定性.     

Prefiltering in iterative feedback tuning: optimization of the prefilter for accuracy

Iterative feedback tuning (IFT) is a data-based method for the tuning of restricted complexity controllers. At each iteration, an update for the controller parameters is estimated from data obtained partly from the normal operation of the closed loop system and partly from a special experiment, in which the output signal obtained under normal operation is fed back at the reference input. The choice of a prefilter for the input data to the special experiment is a degree of freedom of the method. In this note, the prefilter is designed in order to enhance the accuracy of the IFT update. The optimal prefilter produces a covariance of the new controller parameter vector that is strictly smaller than the covariance obtained with the standard constant prefilter.     

Comparative analysis of three approaches of antecedent part generation for an IT2 TSK FLS

Since extreme learning machine is a non-iterative estimation procedure, it is faster than gradient-based algorithms which are iterative. Moreover, the extreme learning machine does not have any design parameters such as learning rate, covariance matrix, etc. The rigorous proof of universal approximation of extreme learning machine with much milder conditions makes it a preferable choice in many different approaches. Although this algorithm is optimal for the parameters which appear linearly in the consequent part of interval type-2 fuzzy logic systems, it is not optimal for the parameters of the antecedent part as it uses random parameters. In this paper, heuristic optimization approaches such as genetic algorithm and artificial bee colony are used to optimize the parameters of the antecedent part of interval type-2 fuzzy logic systems. As these methods are global optimizers, there is less possibility that they will fall in a local minima and are suitable for the selection of the parameters of the antecedent part. A comparative analysis of the optimal parameters with the randomly and manually generated parameters is presented here using noise-free and noisy Mackey-Glass time series data sets and a real world data set. Simulation results support this idea over randomly and manually generated parameters.     

Improving kriging surrogates of high-dimensional design models by Partial Least Squares dimension reduction

Engineering computer codes are often computationally expensive. To lighten this load, we exploit new covariance kernels to replace computationally expensive codes with surrogate models. For input spaces with large dimensions, using the kriging model in the standard way is computationally expensive because a large covariance matrix must be inverted several times to estimate the parameters of the model. We address this issue herein by constructing a covariance kernel that depends on only a few parameters. The new kernel is constructed based on information obtained from the Partial Least Squares method. Promising results are obtained for numerical examples with up to 100 dimensions, and significant computational gain is obtained while maintaining sufficient accuracy.     

Comments on ‘ Dynamic feedback methodology for interconnected control systems ’ and ‘ Dynamic multilevel optimization for a class of non-linear systems ’

The paper considers the Kalman-Bucy filter for a linear system when the measurement noise covariance matrix is singular. It is shown that the problem of infimizing the square of a linear functional of the state estimation error is the dual of the optimal singular linear regulator problem. Furthermore there is an optimal reduced-order Kalman-Bucy filter for minimization of the trace of the state error covariance matrix, when all extremal controls for a dual regulator have finite order of singularity, and no Luenberger observer is needed. The proof is constructive. Necessary and sufficient conditions for the existence of a reduced-order optimal estimator are derived.     

An optimal controller reduction by covariance equivalent realizations

Based upon the theory of covariance equivalent realizations [1], this note presents a technique to obtain reduced-order controllers via an optimal projection of full-order LQG-controllers. This projection is optimal in the sense that it is a weighted least square approximation to a covariance equivalent projection of the full-order optimal closed-loop system. The technique, which is capable of extracting optimal, minimal LQG-controllers whenever they exist, is also shown to yield controllers that are equivalent to those obtained by component cost analysis [12].     

On Kalman filtering over fading wireless channels with controlled transmission powers

We study stochastic stability of centralized Kalman filtering for linear time-varying systems equipped with wireless sensors. Transmission is over fading channels where variable channel gains are counteracted by power control to alleviate the effects of packet drops. We establish sufficient conditions for the expected value of the Kalman filter covariance matrix to be exponentially bounded in norm. The conditions obtained are then used to formulate stabilizing power control policies which minimize the total sensor power budget. In deriving the optimal power control laws, both statistical channel information and full channel information are considered. The effect of system instability on the power budget is also investigated for both these cases.     

Optimal estimation and filtration under unknown covariances of random factors

The general schemes of linear estimation and filtration were considered on assumption of the unknown covariance matrix of random factors such as unknown parameters, measurement errors, and initial and external perturbations. A new criterion was introduced for the quality of estimate or filter. It is the level of damping random perturbations which is defined by the maximal value over all covariance matrices of the root-mean-square error normalized by the sum of variances of all random factors. The level of damping random perturbations was shown to be equal to the square of the spectral norm of the matrix relating the error of estimation and the random factors, and the optimal estimate minimizing this criterion was established. In the problem of filtration, it was shown how the filter parameters that are optimal in the level of damping random perturbations are expressed in terms of the linear matrix inequalities.     

Identification for control: Optimal input intended to identify a minimum variance controller

It is well known that the quality of the parameters identified during an identification experiment depends on the applied excitation signal. Prediction error identification using full order parametric models delivers an ellipsoidal region in which the true parameters lie with some prescribed probability level. This ellipsoidal region is determined by the covariance matrix of the parameters. Input design strategies aim at the minimization of some measure of this covariance matrix. We show that it is possible to optimize the input in an identification experiment with respect to a performance cost function of a closed-loop system involving explicitly the dependence of the designed controller on the identified model. In the present contribution we focus on finding the optimal input for the estimation of the parameters of a minimum variance controller, without the intermediate step of first minimizing some measure of the model parameter accuracy. We do this in conjunction with using covariance formulas which are not asymptotic in the model order, which is rather new in the domain of optimal input design. The identification procedure is performed in closed-loop. Besides optimizing the input power spectrum for the identification experiment, we also address the question of optimality of the controller. It is a wide belief that the minimum variance controller should be the optimal choice, since we perform an experiment for designing a minimum variance controller. However, we show that this may not always be the case, but rather depends on the model structure.     

Optimal linear stochastic systems with independent controllers

The problem of synthesis of optimal controllers for linear stochastic systems with independent control channels is considered. In the considered systems two independent controllers with independent measurement devices are used. The equations for optimal cost function, covariance matrix of the state estimation error and for the controlled system output are derived. It is shown that the application of a well-known separation principle or certainty equivalence principle results in controllers which are not optimal and which can be far from the optimality. It is also shown that improvement of the estimation accuracy by means of manipulation of the controller parameters can significantly improve the control performance as a whole. The numerical examples of calculation of the optimal independent controllers are given to demonstrate the obtained theoretical results and superiorities over controllers based on the separation principle. It is also demonstrated that the optimal controllers have smaller control gains for the control loops with larger observation noises, which can be characterized as cautious properties of the controllers.     

一种基于滤波参数在线辨识的鲁棒自适应滤波器

针对一类未建模动态和扰动下的非线性随机系统的状态估计问题,提出了一种基于 滤波参数在线辨识的鲁棒自适应滤波器.该算法通过极小化状态估计误差的方差同时正交化 相邻时刻的滤波残差,在线辨识状态预报误差和滤波残差的协方差,实现了对未建模动态和扰 动的自适应动态补偿,因此对未建模扰动具有很强的鲁棒性.仿真中研究了一个非线性随机时 滞系统,其参数存在缓变和突变,时滞会多次跳变,量测噪声发生了均值漂移和方差突变.算法 对时滞和参数的联合估计效果令人满意.     

Closed loop experiment design for linear time invariant dynamical systems via LMIs

All stationary experimental conditions corresponding to a discrete-time linear time-invariant causal internally stable closed loop with real rational system and feedback controller are characterized using the Youla-Kucera parametrization. Finite dimensional parametrizations of the input spectrum and the Youla-Kucera parameter allow a wide range of closed loop experiment design problems, based on the asymptotic (in the sample size) covariance matrix for the estimated parameters, to be recast as computationally tractable convex optimization problems such as semi-definite programs. In particular, for Box-Jenkins models, a finite dimensional parametrization is provided which is able to generate all possible asymptotic covariance matrices. As a special case, the very common situation of a fixed controller during the identification experiment can be handled and optimal reference signal spectra can be computed subject to closed loop signal constraints. Finally, a brief numerical comparison with closed loop experiment design based on a high model order variance expression is presented.     

Determination of the optimal number of features for quadratic discriminant analysis via the normal approximation to the discriminant distribution

Given the joint feature-label distribution, increasing the number of features always results in decreased classification error; however, this is not the case when a classifier is designed via a classification rule from sample data. Typically, for fixed sample size, the error of a designed classifier decreases and then increases as the number of features grows. The problem is especially acute when sample sizes are very small and the potential number of features is very large. To obtain a general understanding of the kinds of feature-set sizes that provide good performance for a particular classification rule, performance must be evaluated based on accurate error estimation, and hence a model-based setting for optimizing the number of features is needed. This paper treats quadratic discriminant analysis (QDA) in the case of unequal covariance matrices. For two normal class-conditional distributions, the QDA classifier is determined according to a discriminant. The standard plug-in rule estimates the discriminant from a feature-label sample to obtain an estimate of the discriminant by replacing the means and covariance matrices by their respective sample means and sample covariance matrices. The unbiasedness of these estimators assures good estimation for large samples, but not for small samples.Our goal is to find an essentially analytic method to produce an error curve as a function of the number of features so that the curve can be minimized to determine an optimal number of features. We use a normal approximation to the distribution of the estimated discriminant. Since the mean and variance of the estimated discriminant will be exact, these provide insight into how the covariance matrices affect the optimal number of features. We derive the mean and variance of the estimated discriminant and compare feature-size optimization using the normal approximation to the estimated discriminant with optimization obtained by simulating the true distribution of the estimated discriminant. Optimization via the normal approximation to the estimated discriminant provides huge computational savings in comparison to optimization via simulation of the true distribution. Feature-size optimization via the normal approximation is very accurate when the covariance matrices differ modestly. The optimal number of features based on the normal approximation will exceed the actual optimal number when there is large disagreement between the covariance matrices; however, this difference is not important because the true misclassification error using the number of features obtained from the normal approximation and the number obtained from the true distribution differ only slightly, even for significantly different covariance matrices.     

Selection of optimal AR spectral estimation method for internal carotid arterial Doppler signals using Cramer-Rao bound

In this study, Doppler signals recorded from internal carotid artery of 80 subjects were processed by PC-computer using autoregressive method and Doppler power spectra were obtained. The parameters of autoregressive method were estimated by different estimation methods such as Yule-Walker, covariance, modified covariance, Burg, least squares, and maximum likelihood estimation. Doppler spectra were then used to compare the applied estimation methods in terms of their frequency resolution and the effects in determination of stenosis in internal carotid arteries. The Cramer-Rao bounds were derived for the estimated autoregressive parameters of the internal carotid arterial Doppler signals and the performance evaluation of the estimation methods was performed using the Cramer-Rao bound values. Finally, the optimal autoregressive spectral estimation method for the internal carotid arterial Doppler signals was selected according to the computed Cramer-Rao bound values.