This paper addresses the problem of simultaneous identification of linear discrete time delay multivariable systems. This problem involves both the estimation of the time delays and the dynamic parameters matrices. In fact, we suggest a new formulation of this problem allowing defining the time delay and the dynamic parameters in the same estimated vector and building the corresponding observation vector. Then, we use this formulation to propose a new method to identify the time delays and the parameters of these systems using the least square approach. Convergence conditions and statistics properties of the proposed method are also developed. Simulation results are presented to illustrate the performance of the proposed method. An application of the developed approach to compact disc player arm is also suggested in order to validate simulation results.
Time delay system identification has received great attention in the last years since time delay is a physical phenomenon which arises in most control loops industrial systems [
The identification of time delay systems is known to be a challenging identification problem because it involves both the estimation of dynamic parameters and time delay. Numerous methods have been proposed in the literature for the identification of time delay systems [
Among these methods, the graphical approach has been the most popular since it represents the first method proposed in the literature for the identification of continuous time delay systems [
Most of these approaches deal with the problem of the identification of singleinput singleoutput (SISO) time delay systems. However, the problem of multiinput multioutput (MIMO) time delay systems is one of the most difficult problems that represents an area of research where few efforts have been devoted in the past. The use of time delay approximation is extended to the MIMO case [
In this paper, we propose an alternative approach for the problem of simultaneous identification of linear discrete time delay multivariable systems. Indeed, we develop a new formulation of the problem allowing to define the time delay and the dynamic parameters in the same estimated vector and to build the corresponding observation vector. Then, we use this formulation to propose a new method to identify the time delays and the parameters of these systems using the least square approach. Convergence conditions and statistics properties of the proposed method are also developed. Simulation and experimental examples are presented to illustrate the effectiveness of the proposed methods and to compare their performance in terms of convergence and speed. Our approach presents several interesting properties which can be summarized as follows.
The simultaneous identification of the time delays and parameters matrices is achieved by a new formulation of the parameters matrices.
No a priori knowledge of the time delay is required. In fact, most of the publications assume the knowledge of the time delay variation range or the initial condition.
The consistency of recursive least square methods has received much attention in the identification literature. In this paper, the proof of the consistency of the estimates is established.
It can be used to deal with control adaptive purposes.
This paper is organized as follows. Section
In this paper, we address the problem of identification of square linear multivariable delay system with
The two polynomial matrices
The orders
The input sequences
The disturbance
The inputs, the outputs, and the noises are causal; that is,
In the following, we present three necessary definitions.
Operator
Operator
Operator
In this section, an extended least square algorithm for simultaneous online identification of time delays and parameter matrices is developed.
Equation (
On the other hand, the estimated output is described by the following relation:
Let us consider the prediction error
Since parameter matrix
To overcome this problem, we suggest considering the time delay matrix in parameter matrices
Moreover, we propose the use of the negative gradient of the error to obtain an appropriate observation vector which is given by
Replacing
An estimation
Using (
Using the matrix inversion lemma given by [
set
method:
number of input/output data.
For the system in (
Define the parameter estimation error vector:
Using (
Let us define now a nonnegative definite function:
Replacing
Substituting
Then,
Since
Since
Using this property
Now, let us define the matrix trace
We obtain, finally,
If we replace (
Consider the firstorder Taylor series expansion around the real matrix of
Since
The second partial derivative of the criterion with respect to the generalized matrix is
We now present a simulation example and an experimental validation to illustrate the performance of the proposed approach for the simultaneous identification of time delays and parameter matrices of square multivariable systems.
The objective of the simulation is to compare the efficiency of the proposed method (
The output is noisefree and the RLS method uses the true time delays.
The output is noise free and the RLS method uses the misestimated time delays.
The output is contaminated by additive noise and the RLS method uses the true time delays.
We consider a square linear multivariable discrete time delay system with two inputs and two outputs described by the following equation [
The two polynomials matrices
The time delay matrix
The proposed approach (DLSR) and the RLS algorithm are applied to estimate time delays and parameter matrices. The estimation starts with zero initial conditions. The obtained results are illustrated in Table
Simulation results: Case
True  Developed RLS  RLS  




















Known 


Evolution of the true () and the estimated ( ) delays: Case
Evolution of the true and the estimated parameters:
Evolution of the true and the estimated parameters:
Figures
A validation of the obtained model is presented in Figures
The evolution of the true and the estimated outputs
The evolution of the true and the estimated outputs
We apply the proposed approach (DLSR) and the RLS algorithm to estimate time delays and parameter matrices. The RLS algorithm uses misestimated time delays
Simulation results: Case
True  Developed RLS  RLS  
























Evolution of the true and the estimated parameters:
Evolution of the true and the estimated parameters:
Figures
The system's output is corrupted by additive zero mean white noises
The result of the simulation is given in Table
Simulation results: Case
True  Developed RLS  RLS  




















Known 


Figures
Evolution of the true () and the estimated ( ) delays: Case
Evolution of the true and the estimated parameters:
Evolution of the true and the estimated parameters:
A validation of the obtained model is presented in Figures
The evolution of the true and the estimated outputs
The evolution of the true and the estimated outputs
Based on Tables
the RLS method gives the better performance when the true time delays are used. However, it poorly performs for misestimated delays;
the proposed approach converges to the true delays with acceptable speed for the considered cases.
The experimental data from a mechanical construction of a CD player arm is considered. The system has two inputs that are forces of the mechanical actuators (
The data set contains
Evolution of input identification signals of CD arm.
Evolution of output identification signals of CD arm.
The system is described by the following equation:
The estimation starts with zero initial values for the parameter and the time delay matrices. Applying the proposed algorithm, we obtain
The estimated time delay matrix is
The evolution of the true () and the estimated ( ) outputs.
The evolution of the true () and the estimated ( ) outputs.
We can see clearly that the estimated output tracks fast and accurately the true output.
In this paper, we have addressed the problem of identification of linear discrete time delay multivariable systems. In fact, we have proposed a novel approach for the simultaneous identification of the unknown time delays and the parameter matrices of these systems. The proposed approach consists in constructing a linearparameter formulation that is used to estimate the time delays and the polynomial matrices using recursive least square algorithm. The obtained estimates were shown to be unbiased, and an expression for their covariance matrix was given. Numerical simulation and experimental test are presented to demonstrate the performance of the proposed approach.
Let us consider the shift operator and the backward difference given, respectively, by
The partial derivative of
We have
This work was supported by the Ministry of the Higher Education and Scientific Research in Tunisia.