On Parameter Estimation and Control of Time-Varying Stochastic Systems

Abstract

This thesis is about parameter estimation and control of time-varying stochastic systems. It can be divided into two parts.



The first part deals with an estimation algorithm commonly used when estimating parameters in time-varying stochastic systems, the Recursive Least Squares (RLS) algorithm with forgetting factor. The exact statistical properties for the RLS-estimator with forgetting factor are in most cases difficult to find, due to the complex dependency of the time-varying characteristics and on the forgetting factor. In the first part of this thesis, the RLS-estimator with forgetting factor is applied to different time-varying as well as stationary FIR-, AR- and ARX-structures and some distribution properties for the parameter estimates are derived.



A method to compute the exact distribution and moments of the RLS-estimator in time-varying Gaussian AR(1)-processes is presented. For stationary vector autoregressions and stationary ARX-models the asymptotic bias and covariance function of the RLS estimates are derived. The estimated covariance matrix of the parameter estimates is important when analyzing RLS with forgetting factor. The first moment of this estimate is calculated showing that the asymptotic bias is nonzero. Furthermore, the MSE for the parameter estimate is derived for time-varying FIR-models, giving a possibility to find an optimal forgetting factor in the RLS algorithm.



The second part concerns the problem on controlling time-varying stochastic systems. Optimal control of such systems is generally a very difficult task, which simultaneously must take the character of the unknown time-varying parameters and the fulfilment of the control action into account. The optimal controller action thus must have dual features. However, the optimal dual controller is in most cases impossible to derive, so suboptimal dual controllers must be used.



In the thesis a new optimal adaptive predictive controller (APC) for time-varying stochastic systems is presented that can be explicitely computed for arbitrary prediction horizons. Also a large simulation study of different suboptimal dual controllers is made. The study shows that the APC can successfully be used as a suboptimal dual controller.

Abstract (Swedish)

Popular Abstract in Swedish

Denna avhandling kan delas in i två delar. Den första delen behandlar en analys av en vanlig parameter estimations algoritm - Rekursiv Minska Kvadrat (RMK) algoritmen. Denna algoritm används ofta inom tekniska tillämpningar när man vill estimera modeller i realtid. T.ex. kan RMK användas för att finna modeller på hur framtida elförbrukning i Malmö (eller någon annan stad) kommer att se ut, så att elkraftbolagen vet hur mycket el som de måste producera om t.ex. en timme eller en dag. Vidare kan RMK även användas vid modellering av talsignaler i mobiltelefoni, t.ex. GSM. Problem uppstår när det visar sig att det man modellerar varierar med tiden (t.ex. elförbrukningen varierar med årstiderna och även med dygnets timmar ,samt tal varierar naturligtvis också). I avhandlingen beräknas hur bra man kan estimera sina tidsberoende parametrar under olika förutsättningar.



I andra delen av avhandlingen presenteras en styralgoritm som man kan använda för att styra system som störs av mycket brus och som varierar med tiden. Tillämpningarna kan man hitta inom många tekniska (och icke-tekniska) områden, allt från att styra raketer till att finna optimala köp- och sälj-strategier i aktiehandeln.

• author: Lindoff, Bengt
• supervisor: [unknown] [unknown]
• opponent: Prof Brockwell, Peter, Dept. of Statistics, Colorado State University, USA
• organization: Mathematical Statistics
• publishing date: 1997
• type: Thesis
• publication status: published
• subject: Probability Theory and Statistics
• keywords: Statistics, Adaptive Predictive Control, Adaptive Stochastic Control, Dual Control, Convergence Analysis, Quadratic Forms, Forgetting Factor, Recursive Least Squares, Recursive Estimation, Linear Systems, Time-Varying Stochastic Systems, operations research, programming, actuarial mathematics, Statistik, operationsanalys, programmering, aktuariematematik
• pages: 172 pages
• defense location: Sal A, Matematik-huset, Lund
• defense date: 1997-09-26 10:15:00
• external identifiers: other:ISRN: LUTFD2/TFMS-1010-SE
• ISBN: 91-628-2619-0
• language: English
• LU publication?: yes
• id: 53681129-a111-4fb6-80ff-24e8c94f9abc (old id 29543)
• date added to LUP: 2016-04-04 10:07:10
• date last changed: 2018-11-21 20:56:51