Frequency domain design of observers for linear systems using a ...

state feedback control and the observer in the frequency domain.
In this paper we intend to give a frequency domain description of full order
observers for linear systems where all measurements not corrupted by disturbances,
i.e noise - free output measurements, using the factorization approach
[1]. The procedure of designing such observers is obtained by using the famous
full order observer of type Luenberger designed in time domain [2] , then by
applying the factorization approach, where we propose some useful coprime factorization,
the desired representation in the frequency domain is derived. The
main reason of formulating the results of the time domain in the frequency one,
is the advantages that it presents for the observer-based control [13]. In fact,
in this case, the compensator is driven by the input and the output of the system.
So only the input-output behaviour of the compensator (characterized by
its transfer function) influences the properties of the closed-loop system. The
additional degrees of freedom given by the frequency approach can then be used
for robustness purpose for example [13].
The present work is organized as follows. The next section present the
problem that we propose to solve . Section 3 gives the time domain solution
for this full order problem that will be useful for our frequency domain design,
deduced from [2]. Section 4 deals with twofold . First we review some important
results of the factorization approach , and in a second time, we give the main
result of the paper by proposing a frequency domain description of full order
observer for linear systems . In Section 5 we present an illustrative example and
Section 6 concludes the paper.
2. Problem Statement
Consider the linear time-invariant multivariable system described by
ẋ(t) = Ax(t)+Bu(t) (1a)
y(t) = Cx(t) (1b)
Where x(t) ∈ n and y(t) ∈ m m ≤ n, are the state vector, the output
vector of the system, and u(t) ∈ q represents the input vector. The system
matrices A, B and C assumed to be known.
Problem :
Our aim is to design a full order observer, for system (1) in the frequency
domain using the factorization approach [1]. This frequency domain observer
will generates the estimate ˆx(s) for the Laplace Transform of the state vector
x(t) ,x(s) using the measurements y(s) and the inputs u(s).
3. Full Order observer in the time domain
Before giving the time domain design of the estimator , it is of interest to
recall the following definition:
Définition 1. [2]
2