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Chapter 6
Fractional Order Error Models
With Parameter Constraints
Norelys Aguila-Camacho, Manuel A. Duarte-Mermoud and
´
Matı´as G. Mayol-Suarez
University of Chile, Santiago, Chile
6.1 INTRODUCTION
Error models (EM) were first introduced in adaptive control literature by
Narendra and Kudva (1974) and then in a unified form in Narendra (1976),
for the analysis of integer order (IO) adaptive systems. They have become an
important approach to face numerous adaptive control and identification pro-
blems. Four EM have been recognized in the IO case and completely stud-
ied, denoted as IOEM1 to IOEM4, where the adaptive laws (AL) are of IO
type. For details on these IOEM, the reader is referred to Chapter 6 of the
book by Narendra and Annaswamy (2005) where a complete study of them
is performed. This viewpoint has been successfully used for continuous-time
systems (Narendra and Kudva, 1974; Narendra, 1976; Narendra and
Annaswamy, 2005) (which will be the focus of this study), but it has also
been applied for discrete-time systems (Narendra and Lin, 1980) and even
for hybrid systems (systems with two different time-scales) (Narendra et al.,
1983, 1985). Its importance lies in the fact that the solution of any adaptive
control or identification problem can be readily obtained if the problem is
stated within this framework. The conditions under which stability and con-
vergence of IOEM are attained, were established in Narendra and
Annaswamy (2005) by defining suitable AL for each case.
A further step in the study of IOEM was done in 1988 when the analysis
of two IOEM, whose constant but unknown parameters are related through a
linear algebraic relationship, was completely solved in Duarte and Narendra
(1988a, 1996a), where conditions for stability and convergence of the con-
trol/identification error and parameter error were derived. It is well known
that in adaptive systems the ideal controller parameters and the true plant
parameters are related by linear algebraic equations through the certainty
equivalence principle or the Bezout identity. The importance of this subject
Mathematical Techniques of Fractional Order Systems. DOI: https://doi.org/10.1016/B978-0-12-813592-1.00006-4
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