J Control Theory Appl 2010 8 (4) 509–514
DOI 10.1007/s11768-010-8078-2
Asymptotic attenuation of a class of nonlinear
systems with unknown sinusoidal disturbances
Xiaoli QU
1
, Xiangbin LIU
2
, Mingxuan SUN
3
(1.Electronic Information Engineering College, Henan University of Science & Technology, Luoyang Henan 471003, China;
2.Advanced Control Systems Laboratory, School of Electronics and Information Engineering, Beijing Jiaotong University, Beijing 100044, China;
3.College of Information Engineering, Zhejiang University of Technology, Hangzhou Zhejiang 310032, China)
Abstract: In this paper, nonlinear observers are incorporated into the adaptive control to synthesize controllers for a
class of uncertain nonlinear systems with unknown sinusoidal disturbances which are presented in matched and unmatched
forms. In addition to magnitudes and phases, frequencies of the sinusoidal disturbances need not be known as well, so long
as the overall order is known. Nonlinear observers are constructed to eliminate the effect of unknown sinusoidal distur-
bances to improve the steady-state output tracking performance-asymptotic output tracking is achieved. The adaptation law
is used to obtain the estimate of all unknown parameters. The presented disturbance decoupling algorithms can deal with
matched and unmatched unknown sinusoidal disturbances.
Keywords: Adaptive control; Nonlinear systems; Backstepping; Disturbance attenuation
1 Introduction
Physical systems are subjected to various uncertain-
ties/disturbances, which lead to poor performances or even
instability. To achieve high performance in control systems,
one of the key steps in controller design is how to deal with
uncertainties and disturbances. In general, adaptive control
uses certainty equivalence principle and adaptation law to
compensate the impact of unknown constant parameters in
the systems guaranteeing asymptotic tracking of the control
system [1∼4].
The more information we investigate uncertainties/ dis-
turbances, the better control performance we can achieve.
To be specific, for the unknown unbiased sinusoidal sig-
nals generated by exosystems with known order, the internal
model principle is used to estimate the signal [5∼9]. How-
ever, the method requests the system to have matched dis-
turbances. In practice, systems are always affected by both
matched and unmatched disturbances. It is needed to deal
with such kinds of systems.
In this paper, the adaptive control (AC) incorporated with
nonlinear observers is proposed to deal with the uncertain
nonlinear systems with unknown sinusoidal disturbance.
Nonlinear observers are constructed to eliminate the im-
pact of unknown sinusoidal disturbance with known order
to improve the steady-state output tracking performance–
asymptotic output tracking is achieved when the system
is subjected to unknown sinusoidal disturbances only. The
adaptation law is used to estimate the unknown parameter
coming from the nonlinear observer dynamics.
The paper is organized as follows: the proposed adap-
tive controller for the first-order nonlinear system is pre-
sented in Section 2; the proposed adaptive controller for the
parametric-strict-feedback nonlinear system is presented in
Section 3. In Section 4, a design example simulation results
is presented. Conclusions are drawn in Section 5.
2 Adaptive control with a nonlinear observer
In this section, tracking control of a simple first-order sys-
tem will be used to illustrate the proposed adaptive control
with nonlinear observers. The system is assumed to be in
the following form:
˙x = u + φ(x)μ,
y = x,
(1)
where x ∈ R is the state of the system, y ∈ R is the output
of the system, u ∈ R is the control input, and φ(x) ∈ R
represents a known nonlinear function. μ is an unknown si-
nusoidal disturbance, generated by an unknown exosystem.
˙ω = Sω,
μ = L
T
ω,
(2)
in which we know nothing about exosystem matrices S and
L except the order or dimension of (2), i.e., ω ∈ R
s
where
s is a known number.
Assumption 1 The eigenvalues of S are with zero real
parts and distinct, and {S, L
T
} of the unknown exosystem
(2) observable. The signal μ is unaccessible to measure-
ment.
Let y
d
be the desired output trajectory, which is assumed
to be known, bounded with first-order derivatives. The con-
trol objective is to construct a control input u to make the
output y track y
d
with a prescribed accuracy in spite of var-
ious disturbances.
Received 5 May 2008; revised 2 March 2009.
This work was supported by the National Natural Science Foundation of China (No.60874041, 60834001), and the Youth Foundation of Henan
University of Science and Technology (No.13440018).
c
South China University of Technology and Academy of Mathematics and Systems Science, CAS and Springer-Verlag Berlin Heidelberg 2010