Direct Weight Optimization Research Articles

State-Space Kriging: A Data-Driven Method to Forecast Nonlinear Dynamical Systems

This letter presents a new method for modeling dynamical systems. The method uses historical data of the outputs to predict the evolution of the system. The proposed method is based on Direct Weight Optimization and the Kriging method. These data-based methods provide predictions as linear combinations of past outputs after solving a quadratic optimization problem. We introduce a novel methodology that we named <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">state-space Kriging</i> , which models the time evolution of the weighting parameters using a state-space formalism. In this way, the potential of Kriging, along with classical estimation methods, as the Kalman filter, can be leveraged to forecast the output of a nonlinear dynamical system. The optimization problems involved are easy to solve, and analytical solutions are provided. Some numerical examples and comparisons are provided to demonstrate the effectiveness of our proposal.

Introducing System Identification Strategy into Model Predictive Control

As system identification theory and model predictive control are belonged to two different research fields separately, so one gap exists between these two subjects. To alleviate this gap between them, one new idea proposed in this paper is to introduce system identification theory into model predictive control. As the most important element in model predictive control is the prediction of the output value for a nonlinear system, then the problem of deriving the prediction of the output value can be achieved by system identification theory. More specifically, a Bayesian approach is applied for the nonparametric estimation by modeling the prediction as realizations of zero mean random fields. Through comparing this kind of prediction corresponding to this Bayesian approach and the former direct weight optimization identification for nonlinear system, the authors see that if the unknown weights are chosen appropriately, these two approaches are equivalent to each other. Based on the obtained prediction of the output value, the authors substitute this prediction of the output value into one cost function of model predictive control, and then a quadratic programming problem with inequality constraints is formulated. When to solve this quadratic programming problem, a detailed process about how to derive its dual form is given. As the dual problem has a simple constraint set, it is amenable to the use of the common Gauss-Seidel algorithm, whose convergence can be shown easily. Finally, one simulation example confirms the proposed theoretical results.

Gradient-Based Particle Filter Algorithm for an ARX Model With Nonlinear Communication Output

A stochastic gradient (SG)-based particle filter (SG-PF) algorithm is developed for an ARX model with nonlinear communication output in this paper. This ARX model consists of two submodels, one is a linear ARX model and the other is a nonlinear output model. The process outputs (outputs of the linear submodel) transmitted over a communication channel are unmeasurable, while the communication outputs (outputs of the nonlinear submodel) are available, and both of the two-type outputs are contaminated by white noises. Based on the rich input data and the available communication output data, a SG-PF algorithm is proposed to estimate the unknown process outputs and parameters of the ARX model. Furthermore, a direct weight optimization method and the Epanechnikov kernel method are extended to modify the particle filter when the measurement noise is a Gaussian noise with unknown variance and the measurement noise distribution is unknown. The simulation results demonstrate that the SG-PF algorithm is effective.

Data-based predictive control via direct weight optimization

Abstract In this paper we propose a novel data-based predictive control scheme in which the prediction model is obtained from a linear combination of past system trajectories. The proposed controller optimizes the weights of this linear combination taking into account simultaneously performance and the variance of the estimation error. For unconstrained systems, dynamic programming is used to obtain an explicit linear solution of a finite or infinite horizon optimal control problem. When constraints are taken into account, the controller needs to solve online a quadratic optimization problem to obtain the optimal weights, possibly considering also local information to improve the performance and estimation. A simulation example of the application of the proposed controller to a quadruple-tank system is provided.

Improvement on Direct Weight Optimization Identification

Improvement on Direct Weight Optimization Identification

A New Structure Exploiting Derivation of Recursive Direct Weight Optimization

The recursive direct weight optimization method is used to solve challenging nonlinear system identification problems. This note provides a new derivation and a new interpretation of the method. The key underlying the note is to acknowledge and exploit a certain structure inherent in the problem.

Iterative Selection of Unknown Weights in Direct Weight Optimization Identification

To the direct weight optimization identification of the nonlinear system, we add some linear terms about input sequences in the former linear affine function so as to approximate the nonlinear property. To choose the two classes of unknown weights in the more linear terms, this paper derives the detailed process on how to choose these unknown weights from theoretical analysis and engineering practice, respectively, and makes sure of their key roles between the unknown weights. From the theoretical analysis, the added unknown weights’ auxiliary role can be known in the whole process of approximating the nonlinear system. From the practical analysis, we learn how to transform one complex optimization problem to its corresponding common quadratic program problem. Then, the common quadratic program problem can be solved by the basic interior point method. Finally, the efficiency and possibility of the proposed strategies can be confirmed by the simulation results.

A New Kernel-Based Approach for NonlinearSystem Identification

We present a novel nonparametric approach for identification of nonlinear systems. Exploiting the framework of Gaussian regression, the unknown nonlinear system is seen as a realization from a Gaussian random field. Its covariance encodes the idea of “fading” memory in the predictor and consists of a mixture of Gaussian kernels parametrized by few hyperparameters describing the interactions among past inputs and outputs. The kernel structure and the unknown hyperparameters are estimated maximizing their marginal likelihood so that the user is not required to define any part of the algorithmic architecture, e.g., the regressors and the model order. Once the kernel is estimated, the nonlinear model is obtained solving a Tikhonov-type variational problem. The Hilbert space the estimator belongs to is characterized. Benchmarks problems taken from the literature show the effectiveness of the new approach, also comparing its performance with a recently proposed algorithm based on direct weight optimization and with parametric approaches with model order estimated by AIC or BIC.

Piecewise linear solution paths with application to direct weight optimization

Recently, pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression. This paper presents a generalization of these algorithms to a wider class of problems. It is shown that the approach can be applied to the nonparametric system identification method, Direct Weight Optimization (DWO), and be used to enhance the computational efficiency of this method. The most important design parameter in the DWO method is a parameter ( λ ) controlling the bias-variance trade-off, and the use of parametric optimization with piecewise linear solution paths means that the DWO estimates can be efficiently computed for all values of λ simultaneously. This allows for designing computationally attractive adaptive bandwidth selection algorithms. One such algorithm for DWO is proposed and demonstrated in two examples.

Piecewise Linear Solution Paths for Parametric Piecewise Quadratic Programs

Recently, pathfollowing algorithms for parametric optimization problems with piecewise linear solution paths have been developed within the field of regularized regression. This paper presents a generalization of these algorithms to a wider class of problems, namely a class of parametric piecewise quadratic programs and related problems. By using pathfollowing algorithms that exploit the piecewise linearity, the entire solution paths can be very efficiently computed. Possible applications include design parameter selection for identification methods such as Direct Weight Optimization.

Recursive Direct Weight Optimization in Nonlinear System Identification: A Minimal Probability Approach

In this paper, a direct weight optimization method is proposed for nonlinear system identification based on a minimal probability idea. The approach has several quite attractive features and is very different from existing ones. It is optimal for any given number of finite data points and at the same time possesses asymptotic convergence. The estimator admits a closed form and no numerical optimization is needed. Theoretical analysis and numerical simulations show that the approach is a very competitive alternative to existing nonlinear identification methods.

DIRECT WEIGHT OPTIMIZATION FOR APPROXIMATELY LINEAR FUNCTIONS: OPTIMALITY AND DESIGN

The Direct Weight Optimization (DWO) approach to estimating a regression function is studied here for the class of approximately linear functions, i.e., functions whose deviation from an affine function is bounded by a known constant. Upper and lower bounds for the asymptotic maximum MSE are given, some of which also hold in the non-asymptotic case and for an arbitrary fixed design. Their coincidence is then studied. Particularly, under mild conditions, it can be shown that there is always an interval in which the DWO-optimal estimator is optimal among all estimators. Experiment design issues are also studied.

Nonlinear system identification via direct weight optimization

A general framework for estimating nonlinear functions and systems is described and analyzed in this paper. Identification of a system is seen as estimation of a predictor function. The considered predictor function estimate at a particular point is defined to be affine in the observed outputs and the estimate is defined by the weights in this expression. For each given point, the maximal mean-square error (or an upper bound) of the function estimate over a class of possible true functions is minimized with respect to the weights, which is a convex optimization problem. This gives different types of algorithms depending on the chosen function class. It is shown how the classical linear least squares is obtained as a special case and how unknown-but-bounded disturbances can be handled. Most of the paper deals with the method applied to locally smooth predictor functions. It is shown how this leads to local estimators with a finite bandwidth, meaning that only observations in a neighborhood of the target point will be used in the estimate. The size of this neighborhood (the bandwidth) is automatically computed and reflects the noise level in the data and the smoothness priors. The approach is applied to a number of dynamical systems to illustrate its potential.

A GENERAL DIRECT WEIGHT OPTIMIZATION FRAMEWORK FOR NONLINEAR SYSTEM IDENTIFICATION

The direct weight optimization (DWO) approach is a method for finding optimal function estimates via convex optimization, applicable to nonlinear system identification. In this paper, an extended version of the DWO approach is introduced. A general function class description — which includes several important special cases — is presented, and different examples are given. A general theorem about the principal shape of the weights is also proven.

Adaptive Dwo Estimator of a Regression Function

We address a problem of non-parametric estimation of an unknown regression function f : [-1/2,1/2] → R at a fixed point x0 ∊ (-1/2,1/2) on the basis of observations (xi, yi), i =l,.., n such that yi = f(xi) + ei, where ei ~ N(0, σ2) is unobservable, Gaussian i.i.d. random noise and xi ∊ [-1/2, 1/2] are given design points. Recently, the Direct Weight Optimization (DWO) method has been proposed to solve a problem of such kind. The properties of the method have been studied for the case when the unknown function f is continuously differentiable with Lipschitz continuous derivative having a priori known Lipschitz constant L. The minimax optimality and adaptivity with respect to the design have been established for the resulting estimator. However, in order to implement the approach, both L and σ are to be known. The subject of the submission is the study of an adaptive version of the DWO estimator which uses a data-driven choice of the method parameter L.

Local Modelling of Non linear Dynamic Systems Using Direct Weight Optimization

Local models and methods construct function estimates or predictions from observations in a local neighborhood of the point of interest. The bandwidth, i.e., how large the local neighborhood should be, is often determined based on asymptotic analysis. In this paper, an alternative, non-asymptotic approach that minimizes a uniform upper hound on the mean square error for a linear estimate is used. It is shown that the estimator is obtained from a quadratic program, that an automatic bandwidth selection is obtained, and that the approach can be seen as a local version of fitting affine models to data. Finally, the approach is applied to two benchmark systems.