State estimates for non-linear SISO systems
B. Schwaller
*
, D. Ensminger
*
, J. Ragot
†
, B. Dresp
‡
*
Institut de M´ ecanique des Fluides et des Solides de Strasbourg, schwaller@convergence.u-strasbg.fr
†
Centre de Recherche en Automatique de Nancy, Jose.Ragot@ensem.inpl-nancy.fr
‡
Laboratoire de M´ ecanique et G´ enie Civil, dresp@lmgc.univ-montp2.fr
Abstract— In this study a new type of state observer for
dynamic systems containing non-linear polynomials is proposed.
The stability of the structure and also the uniform convergence
of the state estimates are analyzed. The simple and efficient
algebraic criteria of Naslin normal damping polynomials permit
synthesis of the parameters. A series of simulations illustrates
the proposed developments based on a Van der Pol equation.
I. INTRODUCTION
Since the studies of Luenberger [1], state observers have
been employed both in modeling and in the control or
identification of linear or non-linear systems. Amongst others
the studies of Bastin [2] relating to certain non-linear systems
transformable into a canonical form of observation may
be cited. In order to reduce the order and complexity of
dynamic systems, Stoev [3] employed non-linear observers
using splines. To permit estimation of a state vector for
systems with a lot of noise, Raoufi [4] employed a stochastic
approach. The chief difficulty encountered in employing
observers with non-linear systems is in setting up a method
to synthesize the parameters. It was to overcome this problem
that the technique of observers with high gains was proposed
by Tornamb` e [5], [6], [7], [8]. This approach consists of
reducing the observer error in a pre-determined range of am-
plitudes giving the possibility of extracting the parameters.
The aim of the present study is to propose a new type of
state observer for physical SISO systems, without having to
arbitrarily fix such a fluctuation band. The systems concerned
by this approach can be modeled by differential equations
containing non-linearities of the polynomial type. The aim
is to reconstruct an unbiased estimate of the state space of
the physical system from a set of parameters for the observer
and a single variable of measured output. The measurement
in question could be noisy and the noise may be correlated
or not. The physical systems concerned by our approach
are uniformly observable as understood by Gauthier [9] and
Hermann [10]. This means that the state of the modeled
system can be determined from its input u
0
(t) and its output
y(t). In the most general way this may be formalized by :
dx
i
(t)
dt
= x
i+1
(t) i =0 ...n - 1 (1a)
x
n
(t)=
ν-1
i=0
b
i
Γ
i
[u (t)] -
n-1
i=0
a
i
x
i
(t) -
ξ-1
i=n
a
i
Ψ
i
[x (t)]
(1b)
y(t)= x
θ
(t) (1c)
with :
• n : the order of the differential equation
• ν : the number of input functions used by the system
• Γ
i
[ u (t)] : the input functions are continuously deriv-
able, use the vector of successive derivatives u
T
(t)=
[u
0
(t),...u
n-1
(t)] as input variables and are linearly
independent
• x
i
(t) : the i
th
time derivative of x
0
(t) ; the state vector
is noted as x
T
(t)=[x
0
(t),...x
n-1
(t)]
• ξ - n : the number of the state vector functions x (t)
• Ψ
i
[ x (t)] : the state functions are continuously deriv-
able, use the vector x (t) as input variables, are linearly
independent and of a polynomial form
• a
i
,b
i
: the parameters of the physical system
• y(t) : the single measurable output
• θ : the index of the measured state variable between 0
and n - 1
As an example of such an SISO system, we may consider a
Van der Pol [11] system :
x
1
(t)=
dx
0
(t)
dt
(2a)
x
2
(t)= b
0
u
0
(t) - a
0
x
0
(t) - a
1
x
1
(t)
- a
2
Ψ
2
[ x (t)]+ a
3
Ψ
3
[ x (t)] (2b)
Ψ
2
[ x (t)]= x
1
(t) x
0
(t)
2
(2c)
Ψ
3
[ x (t)]= x
1
(t) (2d)
y(t)= x
0
(t) (2e)
with n =2,ν =1,ξ =4 ■
In § II-A the new observer will be described. A method
of calculation is proposed in § II-B proving that the state
vector of the observer uniformly converges towards that of
the physical system. In § II-C the same method is used to
demonstrate the stability of the observer. In § II-D deals
with the problem of synthesizing the parameters. Some
simulations illustrate the observation of the state of the
system (2) in § III.
II. OBSERVER
A. Description
The function Γ
i
[u (t)] (1b) may use the successive deriva-
tives u
1
(t),...u
n-1
(t) of u
0
(t). If this is the case, these
derivatives must be reconstructed to be accessible to the
state observer. To avoid time derivatives, the input signal
u is filtered by a low pass filter with multiple poles, with
16th Mediterranean Conference on Control and Automation
Congress Centre, Ajaccio, France
June 25-27, 2008
978-1-4244-2505-1/08/$20.00 ©2008 IEEE 1464