Gradient Based Iterative Identification for Discrete-Time Delay Systems Sa¨ıda BEDOUI, Majda LTAIEF, Kamel ABDERRAHIM
National Engineering school of Gabes University of Gabes, Gabes 6029, Tunisia, Tel: +216 75 39 21 00 E-mail:
[email protected],
[email protected],
[email protected] Abstract—In this paper, the problem of identification of time delay systems is addressed. This problem involves both the estimation of the dynamic parameters and the identification of the time delay. In fact, we propose a gradient based iterative algorithm for time delay discrete systems using the hierarchical identification principle. This method consists in decomposing a nonlinear cost function into two simple cost functions in order to overcome the difficulties presented in nonlinear approaches. Simulation results are presented to illustrate the performance of our method and to compare it with an existing approach.
I.
I NTRODUCTION
Time delay systems still attract great attention despite the large number of analysis tools available in the literature [2]. This can be justified by the fact that the delay is an applied problem. It is also an old problem yet it is still open. A time delay system is a dynamic system whose temporal evolution depends not only on its present state, but also on its past state [2]. This phenomenon is present in most physical systems. It may be an inherent feature of the system as in the processes of transport of matter or energy or information. It may also be generated by the devices of control loops, such as response times of sensors and actuators, computation time of control laws and information transmission time in networks. This delay can be neglected if its value is too small for the system time constants. Otherwise, it cannot be neglected, and the dynamic representation of the system must be defined by a delay model. This model can be built using the physical laws that govern the functioning of the system to be modeled. This approach can lead to very complicated models that cause problems of exploitation and implementation [2], [3]. However, for engineering, a mathematical model must provide a compromise between accuracy and simplicity of implementation. A solution to this problem consists in using the identification approach which allows to build a mathematical model from input-output data [1]. The identification of time delay systems is known to be a challenging identification problem because it involves both the estimation of the dynamic parameters and the time delay. Several approaches have been proposed in the literature for the identification of time-delay systems [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15]. These methods can be classified in different ways depending on : the type of model time (continuous or discrete), the form of solution (linear or nonlinear), the type of data processing (batch or iterative or recursive), etc. In this paper, we address the problem of iterative identification of monovariable discrete time delay systems. Iterative identification methods have found applications in many fields,
such as signal processing and control, since they allow to estimate the parameters of linear and nonlinear systems in which the information vector contains unknown variables (unmeasured variables or unknown noise terms). The early iterative approaches proposed in the literature are based on parametrization techniques [10]. They include three main steps. The first consists in inserting a known time delay in the numerator of the model. The second allows to estimate the parameters of the system using a recursive algorithm. The last allows to deduce the time delay from zero coefficients of the numerator. In practice, it is difficult or rather impossible to have zero coefficients from experimental data. Indeed, we must set a threshold, which is a delicate task, mainly in the case of an output that is contaminated by additive noise. Another method is proposed in [16]. It consists, firstly, in using the recursive least square approach to identify the parameters assuming that the time delay is known, and secondly, in estimating the time delay, taking into account the results of the first step. The time delay may be identified either by maximizing the correlation function, or by minimizing the quadratic error. This method assumes that the domain range of the time delay is a priori known. We can also mention the method of Yang [24] which is based on non-linear integer programming. This method is used only to identify the first order time delay system. In our previous work, we have proposed two methods for the simultaneous identification of the time delay and dynamic parameters of monovariable time delay discrete systems. The first method is based on the least square approach [23]. This approach is characterized by a redundant unknown vector (i.e. whose elements are related to each other). The second method consists in minimizing a quadratic criterion using gradient approach [22]. The used criterion is a nonlinear function. It leads to a problem which is more difficult than that of linear systems. To overcome this problem, we propose a gradient based iterative algorithm for time delay discrete systems using the hierarchical identification principle [19], [20]. This paper is organized as follows. Section II presents the model and its assumptions. Section III recalls the gradientiterative method for the identification of time delay systems. In section IV, we propose a gradient based iterative algorithm for time delay discrete systems using the hierarchical identification principle. Performances of the proposed approach are evaluated through simulation results in section V. Section VI concludes the paper.
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II.
P ROBLEM
III.
STATEMENT
In the following, we address the problem of estimating the time delay and the parameters of the following system: A(q−1 )y(k) = q−d B(q−1 )u(k) + v(k)
(1)
where u(k) and y(k) are the system input and output, respectively, v(k) is a white noise, d is the time delay, A(q−1 ) and B(q−1 ) are two polynomials with the unit backward shift operator q−1 , [i.e. q−1 y(k) = y(k − 1)], defined by: A(q ) = 1 + a1q −1
−1
+ ... + ana q
−na
na
= 1 + ∑ ai q
We have proposed a method which consists in minimizing the cost function (3) using the negative gradient search. It leads to the following iterative algorithm of computing θˆG (k) as follows:
µ (k) ∂ J(k) θˆG (k) = θˆG (k − 1) − 2 ∂ θˆG (k − 1)
B(q−1 ) = b1 q−1 + ... + bnb q−nb = ∑ bi q−i
where
The expression of
i=1
Using the expression of A(q−1 ) and B(q−1 ), equation (1) can be rewritten as: nb
i=1
i=1
(2)
The polynomials A(q−1 ) and B(q−1 ) are coprime. The orders na and nb of the model are known. The input sequence {u(k)} is a stationary ergodic process, independent of v(k) and is persistently exciting. The disturbance v(k) is a sequence of independent, identically distributed random variable with zero mean and finite variance σv . The input, the output and the noise are causal, i.e. u(k) = 0, y(k) = 0 and v(k) = 0 for k ≤ 0.
A4. A5.
Problem statement: The goal is to develop an algorithm to estimate, simultaneously, the time delay d and the parameters {a1 , ..., ana , b1 , ..., bnb } by using the input/output measurement data {u(k), y(k)}.
(8)
∂ J(k) = −φG (k)e(k) ∂ θˆG (k − 1)
(9)
So, we obtain :
(3)
where d is the time delay and θ is the parameter vector given by: θ T := aˆ1 (k), aˆ2 (k), ..., aˆna (k), bˆ 1 (k), bˆ 2 (k), ..., bˆ nb (k) (4) e(k) is the prediction error given by: e(k) = y(k) − y(k) ˆ
(5)
and y(k) ˆ is the estimate of y(k) at iteration k defined by: na
nb
i=1
i=1
Using the approximation of Ln(q) ≈ 1 − q−1 (see appendix), we deduce the expression of φG (k): −y(k − 1) .. . −y(k − na) ˆ q−d(k−1) u(k − 1) .. .
φG (k) =
ˆ
nb
q−d(k−1) u(k − nb)
ˆ − ∑ bˆ i (k − 1)q−d(k−1) ∆u(k − i) i=1
where ∆u(k) = u(k) − u(k − 1).
In fact, we define the following cost function: J(θ , d) = e2 (k)
∂ J(k) is given by: ˆ ∂ θG (k − 1)
∂ J(k) ∂ e(k) = e(k) ˆ ˆ ∂ θG (k − 1) ∂ θG (k − 1) ∂ y(k) ˆ =− e(k) ˆ ∂ θG (k − 1)
The following assumptions are made: A1. A2. A3.
(7)
and µ (k) is the step size.
nb
na
METHOD
T ˆ θˆG (k) = aˆ1 (k), ..., aˆna (k), bˆ 1 (k), ..., bˆ nb (k), d(k)
−i
i=1
y(k) = − ∑ ai y(k − i) + ∑ bi u(k − i − d)
E XISTING
ˆ − 1)) y(k) ˆ = − ∑ aˆi (k − 1)y(k − i) + ∑ bˆ i (k − 1)u(k − i − d(k (6)
(10)
Thus, we have
θˆG (k) = θˆG (k − 1) + µ (k)φG (k)e(k)
(11)
ˆ The convergence of if all eigenvalues of θG is guaranteed symmetric matrix φG (k)φG (k)T are inside the unit circle. Therefore, the choice of µ (k) must satisfy the following condition: 2 0 < µ (k) < (12) kφG (k)φG (k)T k2 Proof The proof is in [22].
The implementation of the proposed identification algorithm is summarized as follows:
Nextly, minimizing the optimization problem in (14) leads ˆ as follows: to the iterative algorithm of computing d(k)
Algorithm 1 Step 1 : Initialization: set θˆG = θG0 and k = 0, Step 2 : Increment k, construct the observation vector φG (k) using (10) and choose µ (k) according to (12), Step 3 : Compute: θˆG (k) = θˆG (k − 1) + µ (k)φG(k)e(k) Step 4 : Stop the iteration if the identification data set is not available. Otherwise, let k = k + 1. Go back to step 2. IV.
T HE
and µ1 (k) is the step size or convergence factor to be given later.
ˆ = d(k ˆ − 1) − µ2 (k) ∂ J2 (k) d(k) ˆ − 1) 2 ∂ d(k
(18)
where µ2 (k) is the step size or convergence factor. Equation (18) can be rewritten as: ˆ = d(k ˆ − 1)− d(k) " # nb (19) ∂ ˆ −d(k−1) µ2 (k) q bi (k − 1)u(k − i) e(k)|θ ∑ ˆ − 1) ∂ d(k i=1
PROPOSED METHOD
The cost function J is a nonlinear function which leads to a more difficult problem than that of linear systems. To overcome this problem, we propose a solution based on the hierarchical identification approach [19]. In fact, we decompose the cost ˆ −1)) for fixed function J into two simple cost function J(θ , d(k ˆ ˆ ˆ d = d(k − 1) and J(θ (k), d) for fixed θ = θ (k). Consequently, ˆ − 1)) in θ and we minimize the quadratic functions J(θ , d(k ˆ J(θ (k), d) in d which are relatively easy. This is equivalent to minimizing the following two optimization problems:
Then, using the approximation of Ln(q) ≈ 1 − q−1 (see appendix), we have:
Problem 1. The optimization of θ : ˆ − 1) θˆ (k) = argmin( e2 (k) d(k−1) ) for fixed d(k ˆ
In order to guarantee the convergence of θˆ and d,ˆ µ1 (k) and µ2 (k) must satisfy:
Problem 2. The optimization of d: ˆ = argmin( e2 (k) ˆ ) for fixed θˆ (k) d(k) θ (k)
Using this idea, we define the following two cost functions: ˆ − 1)) = e2 (k) ˆ J1 (θ ) = J1 (θ , d(k d(k−1) 2 J2 (d) = J2 (θˆ (k), d) = e (k) θˆ (k)
(13) (14)
The negative gradient search method is then used to estimate the parameter vector θˆ and the time delay d.ˆ Minimizing the optimization problem in (13) leads to the iterative algorithm of computing θ (k) as follows :
So,
µ1 (k) ∂ J1 (k) θˆ (k) = θˆ (k − 1) − 2 ∂ θˆ (k − 1)
(15)
θˆ (k) = θˆ (k − 1) + µ1(k)φ (k) e(k)|d
(16)
where the observation vector is given by: −y(k − 1) .. . −y(k − na) ˆ φ (k) = q−d(k−1) u(k − 1) .. . ˆ
q−d(k−1) u(k − nb)
ˆ
nb
−µ2 (k)q−d(k−1) ∑ bi (k − 1)∆u(k − i)e(k)|θ where ∆u(k) = u(k) − u(k − 1).
P1: 0 < µ1 (k)
