# Lund University Publications

## LUND UNIVERSITY LIBRARIES

### On Self-Tuning Regulators

and Wittenmark, Björn LU (2001) In Control Theory: Twenty-Five Seminal Papers (1932-1981)
Abstract
The problem of controlling a system with constant but unknown parameters is considered. The analysis is restricted to discrete time single-input single-output systems. An algorithm obtained by combining a least squares estimator with a minimum variance regulator computed from the estimated model is analyzed. The main results are two theorems which characterize the closed loop system obtained under the assumption that the parameter estimates converge. The first theorem states that certain covariances of the output and cross-covariances of the control variable and the output will vanish under weak assumptions on the system to be controlled. In the second theorem it is assumed that the system to be controlled is a general linear nth order... (More)
The problem of controlling a system with constant but unknown parameters is considered. The analysis is restricted to discrete time single-input single-output systems. An algorithm obtained by combining a least squares estimator with a minimum variance regulator computed from the estimated model is analyzed. The main results are two theorems which characterize the closed loop system obtained under the assumption that the parameter estimates converge. The first theorem states that certain covariances of the output and cross-covariances of the control variable and the output will vanish under weak assumptions on the system to be controlled. In the second theorem it is assumed that the system to be controlled is a general linear nth order system. It is shown that if the parameter estimates converge the control law obtained is in fact the minimum variance control law that could be computed if the parameters of the system were known. This is somewhat surprising since the least squares estimate is biased. Some practical implications of the results are discussed. In particular it is shown that the algorithm can be feasibly implemented on a small process computer.

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Reprint of paper from Automatica vol 9, pages 185-199 available at http:dx.doi.org/10.1016/0005-1098(73)90073-3 (restricted access) (Less)
author
organization
publishing date
type
Chapter in Book/Report/Conference proceeding
publication status
published
subject
in
Control Theory: Twenty-Five Seminal Papers (1932-1981)
editor
and
publisher
John Wiley & Sons
external identifiers
• scopus:0034851774
ISBN
0-7803-6021-4
language
English
LU publication?
yes
id
55481ed1-6f85-4125-aa9c-576e52febcff (old id 537831)
2007-09-12 20:13:50
date last changed
2018-05-29 10:50:37
```@inbook{55481ed1-6f85-4125-aa9c-576e52febcff,
abstract     = {The problem of controlling a system with constant but unknown parameters is considered. The analysis is restricted to discrete time single-input single-output systems. An algorithm obtained by combining a least squares estimator with a minimum variance regulator computed from the estimated model is analyzed. The main results are two theorems which characterize the closed loop system obtained under the assumption that the parameter estimates converge. The first theorem states that certain covariances of the output and cross-covariances of the control variable and the output will vanish under weak assumptions on the system to be controlled. In the second theorem it is assumed that the system to be controlled is a general linear nth order system. It is shown that if the parameter estimates converge the control law obtained is in fact the minimum variance control law that could be computed if the parameters of the system were known. This is somewhat surprising since the least squares estimate is biased. Some practical implications of the results are discussed. In particular it is shown that the algorithm can be feasibly implemented on a small process computer.<br/><br>
.<br/><br>
<br/><br>
Reprint of paper from Automatica vol 9, pages 185-199 available at http:dx.doi.org/10.1016/0005-1098(73)90073-3 (restricted access)},
author       = {Åström, Karl Johan and Wittenmark, Björn},
editor       = {Basar, Tamer},
isbn         = {0-7803-6021-4},
language     = {eng},
publisher    = {John Wiley & Sons},
series       = {Control Theory: Twenty-Five Seminal Papers (1932-1981)},
title        = {On Self-Tuning Regulators},
year         = {2001},
}

```