186 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 47, NO. 1, FEBRUARY 2000 Deterministic Predictive Models for DC Voltage Reference Source Control Irena Nan˘ covska, Ljup˘ co Todorovski, Anton Jegli˘ c, and Du˘ san Fefer, Member, IEEE Abstract—In this paper, we use predictive models for voltage cor- rection in a high-precision dc voltage reference source (DCVRS). Voltage reference elements, which compose the DCVRS, are im- proved by implementing a control loop with built-in predictors. Thus, the sensitivity of the system is reduced and, thereby, the sta- bility of the DCVRS is improved. The predictive abilities of two different paradigms, neural-network-based predictors and differ- ence equation predictors obtained by equation discovery system LAGRAMGE, are compared. Index Terms—Equation discovery, machine learning, neural net- works, prediction methods, time series. I. INTRODUCTION T HE voltage correction of the dc voltage reference source (DCVRS) is used to decrease the sensitivity of the source to environmental conditions, as well as the technological im- perfection of the source implementation. The influence of envi- ronmental conditions has been efficiently reduced through the implementation of well-known prevention measures [1]. How- ever, this prevention measure can only partially eliminate sensi- tivity factors. There still remains some deterministic part of the voltage instability which can be modeled by predictive models. A control loop, based on predictive models, is added to the voltage reference elements, which compose the source to reduce its sensitivity. The control loop contributes to the enhancement of the robustness of the system and, thereby, the stability of the reference voltage (RV) Different predictive methods are used for forecasting the fu- ture value of a system variable on the basis of the present and past observations. Generally, predictive methods can be divided into stochastic and deterministic [2], [3]. The stochastic predic- tive methods assume that the observed system can be modeled as a stochastic process and the predictions are based on the sta- tistical properties of the observed system. On the other hand, the deterministic methods assume that the system behaves de- terministically. The predictions are based on the time-domain characteristics of the observed system [2]. We present two different deterministic paradigms for fore- casting: neural networks (NN’s) and equation discovery. NN’s used for prediction are characterized as black-box models, Manuscript received September 16, 1998; revised April 16, 1999. Abstract published on the Internet November 11, 1999. I. Nan˘ covska, A. Jegli˘ c, and D. Fefer are with the Laboratory for Process Con- trol and Measurements, Faculty of Electrical Engineering, University of Ljubl- jana, 1111 Ljubljana, Slovenia (e-mail: Irena.Nancovska@fe.uni-lj.si). L. Todorovski is with the Institute for Biomedical Informatics, Faculty of Medicine, University of Ljubljana, 1105 Ljubljana, Slovenia (e-mail: Ljupco.Todorovski@mf.uni-lj.si). Publisher Item Identifier S 0278-0046(00)01344-7. whereas difference equations obtained with equation discovery systems are transparent (white box). NN’s represent an emerging technology with some important characteristics, such as universal approximation (input–output mapping) and ability to learn from and adapt to their environ- ment [4]. In this paper, we use four types of NN’s with known computational power [5], [6]: 1) a supervised multilayer feed- forward network [multilayer perceptron (MP)]; 2) finite-dura- tion impulse response (FIR) MP; 3) recurrent network in real time; and 4) group method for data-handling network. Equation discovery systems explore the hypothesis space of all equations that can be constructed given a set of arithmetical operators, functions and variables, searching for an equation that fits the input data best. In this paper, we used equation dis- covery program LAGRAMGE [7] for discovering difference equa- tions. LAGRAMGE allows the use of domain-specific knowledge in the process of equation discovery. The structure of the discov- ered equations can be specified using context-free grammars. We used three different context-free grammars for discovering difference equations as predictive models. They define three different spaces of difference equations: linear, quadratic, and piecewise linear. In order to evaluate the predictive abilities of two described paradigms, we applied them to two other time-series prediction domains. The first is the well-known Lorenz attractor described by three differential equations and is a deterministic nonlinear dynamical system. The other time series is known as pink noise and is stochastic. It was generated using random function frac- tional Brownian motion (FBM) [8]. This paper is organized as follows. In Section II, we de- scribe the intelligent voltage reference element as a part of the DCVRS. The predictive models along with the methods for preprocessing of the time series are described in Section III. The results of applying the described techniques on three time-series data sets are presented in Section IV. Finally, Sec- tion V concludes with a summary of the results and directions for further work. II. DCVRS The DCVRS consists of solid-state voltage reference ele- ments, based on 7-V zener diodes LTZ1000. The LTZ1000 is a thermally compensated and stabilized monolithic component with an integrated temperature-sensing transistor and a heater. Although solid-state standards are mechanically robust, have low temperature coefficient of output, and have higher output voltages that minimize thermal dependence, they are still subject to various environmental influences and technological 0278–0046/00$10.00 © 2000 IEEE