An estimation of accuracy of Oustaloup approximation
Krzysztof Oprzędkiewicz1, Wojciech Mitkowski1 , Edyta Gawin2 1
AGH University of Science and Technology, Faculty of Electrotechnics, Automatics, Informatics and Biomedical Engineering Dept. of Automatics and Biomedical Engineering (kop,wojciech.mitkowski)@agh.edu.pl 2 High Vocational School in Tarnów, Polytechnic Institute
[email protected]
Abstract. In the paper a new accuracy estimation method for Oustaloup approximation is presented. Oustaloup approximation is a fundamental tool to describe fractional-order systems with the use of integer-order, proper transfer function. The accuracy of approximation can be estimated via comparison of impulse responses for plant and Oustaloup approximation. The impulse response of the plant was calculated with the use of an accurate analytical formula and it can be interpreted as a standard. Approach presented in the paper can be applied to effective tuning of Oustaloup approximant for given application (for example in FO PID controller). The use of proposed method does not require us to know time response of a modeled controller. The proposed methodology can be easily generalized to another known approximations. Results of simulations show that the good performance of approximation is reached for low order and narrow angular frequency range. Keywords: Fractional order transfer function, Oustaloup approximation,
1
An introduction
Fractional order models are able to describe a number of physical phenomena from area of electrotechnics (heat transfer, diffusion etc.) properly and accurate. Fractional–order approach can be interpreted as generalization of known integer-order models. Fractional order systems has been presented by many Authors ([1], [3], [9], [10]). An example of identification fractional order system can be found in [6], [7].
adfa, p. 1, 2011. © Springer-Verlag Berlin Heidelberg 2011
The proposal of generalization the Strejc transfer function model into fractional area was given in [11]. A modeling of fractional–order transfer function in MATLAB/SIMULINK requires us to apply integer order, finite dimensional, proper approximations. An important problem is to assign parameters of approximation correctly and to estimate its accuracy. The best known approximation presented by Oustaloup (see for example [4], [12]) bases on frequency approach. This is caused by a fact, that for fractional order systems the Bode magnitude plot can be drawn exactly and its parameters can be applied to approximants calculation. Additionally, for elementary fractional-order elements an analytical form of step and impulse responses is known (see [1], [14]). These responses can be applied as reference to estimate a correctness of built approximant. However, models obtained with the use of Oustaloup approximation are not always fully satisfying. This is caused by the fact that their accuracy is determined by proper selecting a frequency range and order of approximation. The goal of this paper is to discuss an application of a method proposed by authors in [8] to accuracy estimation of Oustaloup approximation. The presented approach uses analytical formulas of impulse response of elementary non integer order integrator, cost functions and numerical calculations proposed by the authors, done with the use of MATLAB. The approach shown in the paper can be also applied to effective selecting parameters of the Oustaloup approximation during modeling FO PID controller and another systems containing FO integrators. Additionally, it does not require the use a step response of modeled element. Particularly, in the paper the following problems will be presented:
Non-integer order integrator, The Oustaloup approximation, Cost functions describing the accuracy of approximation, Simulation results.
2
Non-integer order integrator
Let us consider an elementary non-integer order integrator described with the use of transfer function (1). This transfer function can be applied for example to model integral part of FO PID controller. G ( s)
1
(1)
s
In (1) R is a fractional-order of the plant. The analytical form of the impulse response ya(t) for plant described with the use of (1) is as follows (see[1, pp. 8,9]):
t 1 1 y a (t ) L1 1 s
(2)
where Γ(..) denotes complete Gamma function:
e x x 1 dx
(3)
0
Let us assume, that the impulse response described by (2) and (3) is the accurate response. This implies, that it can be applied as a standard to estimate the accuracy of approximation. 3
The Oustaloup approximation
The method proposed by Oustaloup (see for example [4], [12]) allows us to approximate an elementary non-integer order transfer function s with the use of a finite and integer-order approximation expressed as underneath:
s
kf
N
1
n1
1
s
n s
(4)
GORA ( s )
n
In (4) N denotes the order of approximation, μn and νn denote coefficients calculated as underneath: 1 l n n , n 1,...,N n 1 n , n 1,...,N 1
(5)
where: h l
N
h l
(6)
1 N
In (6) ω l and ω h describe the range of angular frequency, for which parameters are calculated. A steady-state gain kf is calculated to assure the convergence the step response of approximation to step response of the real plant in a steady state. Denote the impulse response of approximation (4) by yORA (t ) . It can be written as follows: yORA (t ) L1GORA (s)
(7)
The general form of the impulse response (7) is determined by poles and zeros of transfer function GORA (s) described by (4). They are real and different. This implies, that the general form of (7) can be easily expressed as follows: N
yORA (t ) k f
c e i
i 1
pi t
(8)
In (8) kf denotes the steady-state gain of the approximation, ci denote coefficients of transfer function (4) factorization. The impulse response (7) or (8) can be evaluated numerically with the use of MATLAB/SIMULINK. 4
Cost functions describing the accuracy of approximation
Let us assume that the impulse response ya(t) described by (2) is the accurate response. Let yORA (t ) denotes the impulse response of approximation, described by (7) or (8). Then the approximation error ea(t) can be defined as follows: ea (t ) ya (t ) yORA (t )
(9)
Furthermore, let us introduce the following cost functions, describing the accuracy of approximation: I max (, N ) max ea (t )) t
(10)
I 2 (, N ) ea2 (t )dt
(11)
0
In (10) and (11) ea(t) denotes the approximation error described by (9). Both cost functions (10) and (11) for given plant (described by ) are functions of approximation parameters: order N and angular frequency range from ωl to ωh . It can be expected, that increasing N for constant frequency range should increase an approximant quality, described by cost functions (10) and (11). However, results of simulations point that too high value of N can cause bad conditioning of a model and consequently, make it useless. The fastest method to check proper setting of the approximation parameters N and range of angular frequency described by values ωl and ωh is to calculate both proposed cost functions (10) and (11) as functions of approximation parameters: order and frequency range. An example
of such a tuning of the Oustaloup approximant with the use of simulations will be shown in the next section. 5
Simulation results
As an example let us consider the application of Oustaloup approximation to model the elementary fractional-order integrator described by (1). The calculations were run for time range from 0.05[s] to 0.5[s] with step 0.02[s]. Values of both cost functions (10) and (11) for different ranges of angular frequency from ωl to ωh, approximation order N and fractional order are given in Table 1. There were considered the following values of approximation parameters: N = 5, 10, 15, 25, = 0.2, 0.5, 0.9. Table 1. Values of cost functions (10) and (11) for different , ω l , ω h and N
Exp. No
Range of angular frequency
Fractional order
N 0.2
0.5
0.9
ωl
ωh
Imax
I2
Imax
I2
Imax
I2
1
0.1
10
5
0.3141
0.0031
0.1023
0.0047
2
0.01
100
5
0.4581
0.0062
0.2800
4.23e004 0.0027
3
.001
1000
5
0.4543
0.0066
0.2815
0.0030
0.0271
4
0.1
10
10
0.3027
0.0030
0.0965
0.0047
5
0.01
100
10
0.4561
0.0062
0.2802
3.8646e004 0.0027
6
.001
1000
10
0.4668
0.0065
0.2962
0.0030
0.0282
7
0.1
10
20
0.2997
0.0029
0.0951
0.0047
8
0.01
100
20
0.4560
0.0062
0.2800
3.7700e004 0.0027
9
.001
1000
20
0.4665
0.0064
0.2961
0.0030
0.0282
3.7088e006 2.9980e005 3.6842e005 3.7807e006 2.9964e005 3.7121e005 3.8009e006 2.9945e005 3.7116e005
0.0255
0.0255
0.0255
Let ea(t) be an approximation error described by (9). Exemplary diagrams ea(t) as a function of a time t for selected values of parameters from Table 1: , ωl, ωh and N are shown in Fig. 1 and Fig. 2.
Approximation error 0
-0.05
-0.1
-0.15
-0.2
-0.25
-0.3
-0.35 0.05
0.1
0.15
0.2
0.25 0.3 time [s]
0.35
0.4
0.45
0.5
Fig. 1. Approximation error ea(t) described by (9) for: =0.5, N=20 and: experiments no: 7 (+) ,8 (.), 9 (^), (variable range of angular frequency) Approximation error 0
-0.02
-0.04
-0.06
-0.08
-0.1
-0.12 0.05
0.1
0.15
0.2
0.25 0.3 time [s]
0.35
0.4
0.45
0.5
Fig. 2. Approximation error ea(t) described by (9) for: =0.5, N=5 (+) ,10 (.), 20 (^), experiments no: 1,4,7 respectively.
From Table 1. and Fig. 1 and Fig. 2 we can conclude at once, that the good approximant can be obtained with the use of approximation order N smaller than 20 and the narrower range of angular frequency between ωl and ωh. However, the more precise estimation of order N requires us to make next experiments. To estimate the approximation order N assuring the reasonable value of cost function (11) can be applied an approach presented in [2, pp 573,574] to estimate a model order. A criterion to determine an approximation order is the rate of change the cost function (11) as a function of order N: if the increase of N causes firstly big and next small improvement of calculated cost function, then this “threshold” value of N is a sensible value of approximation order. To estimate the order N the cost function (11) as a function of N was calculated. Calculations were done for ωl = 0.1, ωh =10, = 0.2, = 0.5 and N = 2..20. Results are shown in figure 3. -3
6
x 10
5
Cost function (11)
4
3
2
1
0
2
4
6
8
10 12 Approximation order N
14
16
18
20
Fig. 3. Cost function (11) as a function of approximation order N for = 0.2 (+) and = 0.5 (+),
From figure 3 we can conclude that the approximation order N greater than 8 does not significantly improve the cost function (11) for both
tested fractional orders of Oustaloup approximation. This allows us to formulate conclusion that the order of approximation assuring the good accuracy and simultaneously lowest possible computational complexity of Oustaloup approximation is equal 8. Additionally, it can be concluded that the accuracy of Oustaloup approximation improves with increasing fractional order . 6
Final conclusions
The final conclusions from the paper can be formulated as follows: In the paper the analysis of accuracy the Oustaloup approximation as a function of its parameters (order N and range of angular frequency from ωl to ωh) was presented. Different fractional-orders were also tested. The accuracy of Oustaloup approximation is stronger dependent on range of angular frequency between ωl and ωh, than approximation order, described by N. The order of approximation N, equal 8 assures the good accuracy of approximation. The improving of this order does not significantly improve this accuracy, but it increases a computational complexity of approximant. Decreasing the width range of angular frequency from ωl to ωh improves the accuracy of approximation in the sense of considered cost functions. Method presented in the paper can be applied to effective tuning of Oustaloup approximant for elementary non integer order integrator, independently on its gain (describing for example an integral and derivative actions in FO PID controller). The approach presented in this paper was applied to Charef approximation – see [8]. It can be also applied to discrete PSE and CFE approximations. This problem is planned to be considered by authors. As an another area of further investigations will be formulating analytical conditions directly associating with the cost functions (10) and (11) with integrator order and approximation parameters: N , ωl and ωh.
Acknowledgements This paper was partially supported by the AGH (Poland) – project no 11.11.120.815 and partially supported by the AGH (Poland) – project no 11.11.120.817.
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