Discretisation of different non-integer order system approximations Paweł Piatek ˛ ∗ , Marta Zagórowska∗ , Jerzy Baranowski∗ , Waldemar Bauer∗ and Tomasz Dziwi´nski∗ ∗ AGH
University of Science and Technology, Department of Automatics and Biomedical Engineering, Al. Mickiewicza 30, 30-059 Kraków Email:
[email protected],
[email protected],
[email protected],
[email protected],
[email protected]
Abstract—The goal of this paper is to compare popular methods of approximation on non-integer order systems and propose strategies of increasing their robustness to numerical errors of their coefficients. It is shown, that popular methods during discretisation lead to instability.
I.
I NTRODUCTION
Nowadays, non-integer controllers are a widely researched problem. Although non-integer calculus was known since XIX, first the rapid development of computational power permitted in-depth analysis. Therefore, there are more and more methods and approaches to that problem. One of questions that is of great importance is the design of non-integer order controllers (see [1]–[4]) and the approximation (see [5], [6]). There are some popular methods of realisation of noninteger order systems in the form of integer order transfer functions. There are however certain issues with their discretisation and subsequent implementation. In this paper two popular discretisation methods are considered and approximation is based on method proposed by Oustaloup in [7]. The authors presented both the approximation and discretisation methods. They used zero order hold (ZOH) equivalent method, Tustin’s approximation and Al-Alaoui operator as tool to discretise the approximated system. At the end, there are some conclusions and plans for future work. II.
P OPULAR APPROXIMATION METHODS
Oustaloup filter approximation to a fractional-order differentiator G(s) = sα is a widely used one in applications [8]. A generalized Oustaloup filter can be designed as Gt (s)
= K
N Y s + ωi0 i=1
s + ωi
(1)
where: ωi0 ωi K ωu
= ωb ωu(2i−1−α)/N = ωb ωu(2i−1+α)/N = ωhα s ωh = ωb
(2) (3) (4) (5)
(ωb , ωh ) is the frequency range and N is the order of the approximation. As it can be seen its representation takes form of a product of a series of stable first order linear systems.
Another type of approximation was analysed in [6]. This approach is based on so called ”distribution function”. This method is best suited for approximation of the following type of systems 1 F (s) = (6) 1 + (τ0 s)m where m ∈ (0, 1). The following approximation can be used
F (s) ≈ FN (s) =
2N −1 X i=1
G(τi ) 1 + τi s
(7)
where: G(τ ) =
1 2π
sin((1 − m)π) τ cosh m log − cos((1 − m)π) τ0 τi = τ0 · λN −i
(8)
(9)
Parameter λ is obtained through the following optimisation problem λ = arg min max ||FN (jω)| − |F (jω)|| (10) ω∈[ωb ,ωh ]
This approach represents the approximation as a sum of stable first order lags. III.
D ISCRETISATION
In order to implement the approximated fractional system one needs a discrete model. Analysis included two methods of dicretisation - ZOH equivalent model, Tustin’s approximation and Al-Alaoui operator. ZOH equivalent discretisation takes form −1 −1 Gc (s) G(z) = (1 − z )Z L (11) s where Gc (s) is the continuous transfer function, L−1 is the inverse Laplace transform, Z is the Z-transform with discretisation step T . This is the most popular method, which behaves properly for moderately small discretisation steps. Tustin’s approximation is based on the trapezoidal rule of integration and uses the following substitution s≈
2 z−1 · T z+1
(12)
Approximation order 5 -0.0000363078 -0.0091201084 -2.2908676528 -575.4399373372 -14457543.99373372 -
so G(z) = Gc (s)
(13)
Tustin’s approximation guarantees that stable systems are discretised as stable (in infinite numerical precision). Al-Alaoui operator is obtained by interpolating the trapezoidal and the rectangular integration rules. Al-Alaoui operator transforms s to z using substitution (see [9]): s≈
2 z−1 T (1 − a) + (1 + a)z
(14)
where a ∈ [0, 1] is a constant(in all numerical experiments a = 0.75) and T is sample period. The above procedure is equivalent to interpolating directly the bilinear operator (Tustin), and the backward difference operator.
TABLE I.
IV.
Approximation order 20 -0.0000024547 -0.0000097724 -0.0000389045 -0.0001548817 -0.0006165950 -0.0024547089 -0.0097723722 -0.0389045145 -0.1548816619 -0.6165950019 -2.4547089157 -9.7723722096 -38.9045144994 -154.8816618912 -616.5950018615 -2454.7089156850 -9772.3722095581 -38904.5144994280 -154881.6618912476 -616595.0018614809
P OLES OF APPROXIMATION OF ORDER 5 AND 20
ANALYSIS
In this section the following example is analysed: Pole−Zero Map
G(s) = sα .
1
(15)
1 0.8
A. Oustaloup approximation
1
1
1
1
1
1
Systematic analysis shows that for approximation order equal or less than 5 the discrete approximation behaves properly. In figure (1), there are poles × and zeros ◦ of Oustaloup approximation. Pole−Zero Map
Imaginary Axis (seconds−1)
0.6 0.4 1 0.2 7e+05 0
6e+05
5e+05
4e+05
−5
−4
3e+05
2e+05
1e+05
−0.2 −0.4 1
1 1
1
1
1
1
1 −0.6
0.8 1 −0.8
1
0.6 Imaginary Axis (seconds−1)
1 0.4
−1 −7
1
1
1 −3
1 −2
Real Axis (seconds−1)
1
1 −1
0 5
x 10
0.2 1.4e+05 0
1.2e+05
1e+05
8e+04
6e+04
4e+04
2e+04
Fig. 2.
Poles of approximation for order N = 20
−0.2 −0.4
For both approximation orders, it can be seen that the poles and zeros are widely spread on the real axis. One can suppose that this dispersion will have some influence on discretisation. The results from the next subsection prove this assumption true.
1
−0.6 1 −0.8 1 −1 −15
1
1
−10
1
1
1
−5
Real Axis (seconds−1)
Fig. 1.
−6
0 4
x 10
Poles of approximation for order N = 5
It is clearly visible, that the approximation is stable as all poles are in the left complex half-plane. However, the figure shows most of them close to the origin. Table I presents the numerical values of poles. The same thing can be observed in higher order approximation. For N = 20, the numerical values are in right hand side of table I and the plot of poles and zeros is in figure (2). The continuous integer order system obtained from Oustaloup approximation is stable.
B. Discretisation As mentioned above, three types of discretisation are considered - ZOH equivalent model, Tustin approximation and Al-Alaoui operator. There are some differences in these approaches. The discretisation with Tustin method and Al-Alaoui operator guarantees the stability of the discrete system. For approximation order N = 5 it is clearly visible in figure 3 and figure 5. Moreover, the authors used two equivalent form of transfer function to discretise the system. The first one uses the direct (s) form of transfer function G(s) = N D(s) where N (s) and
zero pole unit circle
1
0.6
0.6
0.4
0.4
0.2 2
0 −0.2
0.2
−0.6
−0.6
−0.8
−0.8
−1
−1 0 Real Part
0.5
1
Poles of Tustin discretisation Oustapoup approximation for N = 5
−1
Fig. 4.
2
−0.2 −0.4
−0.5
2
0
−0.4
−1
Fig. 3.
0.8
Imaginary Part
Imaginary Part
0.8
zero pole unit circle
1
−0.5
0 Real Part
0.5
1
Poles of ZOH discretisation of Oustaloup approximation for N = 5 Oustaloup approximation, Al−Alaoui operator, 5−order
D(s) are polynomials. The second one, however, uses the transfer function as series of transfer functions of first order. In case of approximation order smaller or equal to 5, there is no difference in discretisation connected with these two approaches. Table II and table III presents the poles of two resulting discretisations.
TABLE II.
zero pole unit circle
0.8 0.6 0.4 Imaginary Part
N =5 series of transfer functions of first order 0.999999636922011 0.999908803074694 0.977350755579337 -0.484163787883313 -0.997236499866286
1
direct form of transfer function 0.999999636948703 0.999908803047891 0.977350755579445 -0.997236499866285 -0.484163787883313
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
P OLES OF DISCRETE SYSTEM . T USTIN APPROXIMATION WAS USED WITH N = 5 Fig. 5.
−0.5
0
0.5 Real Part
1
1.5
Poles of discrete system. Al-Alaoui operator was used with N = 5
N =5 series of transfer functions of first order 0.999998903522645 0.999724630219861 0.934012554025306 -0.320055347639005 0.428104057515693
TABLE III.
direct form of transfer function 0.999998903543115 0.999724630199327 0.934012554025371 0.320055347639004 -0.428104057515693
P OLES OF DISCRETE SYSTEM . A L -A LAOUI OPERATOR APPROXIMATION WAS USED WITH N = 5
For ZOH approximation of order N = 5 the same situation occurs, even though the discretisation method does not guarantee that - the discretised system is stable for direct and series form of transfer function. The poles lie in the unit circle, which can be seen in figure 4. The table IV presents the poles of both discretisations. A problem occurs for higher order approximation. Tustin method and Al-Alaoui operator should provide stability for every N . However, for N = 20 this is not the case. Approximation of series of transfer functions provides a stable discrete system with poles in table V and table VI(left hand side).
The direct approach, however, results in unstable system, presented in figure 6 and figure 8. Most of poles lie inside the unit circle, but some of them form an additional ”circle” around point (1, 0). In case of ZOH approximation, the resulting system is even ”more unstable” - the spread of poles around point (1, 0) is larger than in case of Tustin approximation and Al-Alaoui operator (see figure 7. The same is visible in table (VII), where numerical values are presented. V.
C ONCLUSION
The poles of discrete order systems that cause the numerical instability are located near the point (1, 0) almost on the boundary of unit circle. Numerical limitations require for example rounding of calculated values or of transfer function coefficients. Therefore, there is a possibility that rounding error ”pushes” the poles out of unit circle causing instability.
N =5 series of transfer functions of first order 0.999999636922011 0.999908803074757 0.977351734851963 0.003168809310851 0
TABLE IV.
N = 20 direct form of transfer function 0 0.999999637199247 0.999908802796435 0.977351734853051 0.003168809310851
Series of transfer functions of first order 0.999999967640635 0.999999871175057 0.999999487138800 0.999991871741087 0.999967641367280 0.999487322668693 0.997961175976391 0.991917681722918 0.968357388688670 0.881831207415650 0.622619379230273 0.159507646529784 0.215027148987375 0.368170690900175 -0.215027148987375 -0.368170690900175 -0,412903397310894 -0.424603207327706 -0.427572578839676 -0.428320397431475
P OLES OF DISCRETE SYSTEM . ZOH APPROXIMATION WAS USED WITH N = 5
N = 20 Series of transfer functions of first order 0.999999975452911 0.999999902276283 0.999999610954931 0.999998451184580 0.999993834068991 0.999975453212119 0.999902281052634 0.999611030518350 0.998452381869428 0.993853001025606 0.975750537699695 0.906828796312646 0.674309088876416 0.127136290639267 -0.510161096886249 -0.849324346772391 -0.959889182674448 -0.989771002015487 -0.997420713738027 -0.999351486314265
TABLE V.
Direct form of transfer function -0.999351487101443 -0.997420712689764 -0.989771002293298 -0.959889182657533 -0.849324346772579 -0.510161096886253 1.075726620635316 + 0.022314663152380j 1.075726620635316 - 0.022314663152380j 1.052249584080806 + 0.060309802747964j 1.052249584080806 - 0.060309802747964j 1.012420776470067 + 0.080619746035397j 1.012420776470067 - 0.080619746035397j 0.967576960371940 + 0.078190582323937j 0.967576960371940 - 0.078190582323937j 0.928407085161366 + 0.054343651537072j 0.928407085161366 - 0.054343651537072j 0.900801606245107 + 0.016719866638254j 0.900801606245107 - 0.016719866638254j 0.674309078575385 0.127136290639266
TABLE VI.
N = 20
That problem appears in this work, that is in discretisation of non-integer order systems approximations. The authors presented the results of discretising an Oustaloup approximation with methods: Tustin transformation, ZOH equivalent and AlAlaoui operator. Those methods were chosen to verify if they preserve the stability of approximating continuous system.
TABLE VII.
zero pole unit circle
Imaginary Part
0.5
P OLES OF DISCRETE SYSTEM . A L -A LAOUI OPERATOR WAS USED WITH N = 20
Series of transfer functions of first order 0.999999975452911 0.999999902276283 0.999999610954931 0.999998451184581 0.999993834068991 0.999975453212121 0.999902281052712 0.999611030523255 0.998452382178561 0.993853020440873 0.975751740509824 0.906899425418300 0.677703674394852 0.212499292738529 0.002099722641157 0.000000000021844 0.000000000000000 0.000000000000000 0 0
P OLES OF DISCRETE SYSTEM . T USTIN APPROXIMATION WAS USED WITH N = 20
1
Direct form of transfer function 1.086952736394901 - 0.026013988511432j 1.086952736394901 + 0.026013988511432j 1.059181140213900 - 0.069902977158210j 1.059181140213900 + 0.069902977158210j 1.012676659882425 - 0.092592956116126j 1.012676659882425 + 0.092592956116126j 0.961071353370407 - 0.088651516583313j 0.961071353370407 + 0.088651516583313j 0.916750796377530 - 0.060073321165506j 0.916750796377530 + 0.060073321165506j 0.883058696729180 - 0.014935825652994j 0.883058696729180 + 0.014935825652994j 0.622619373754632 -0.412903397310894 -0.427572569410924 -0.424603209890053 -0.412903397137895 -0.368170690903462 -0.215027148987355 0.159507646529791
Direct form of transfer function 0 0 1.111586562239917 + 0.033256089305533j 1.111586562239917 - 0.033256089305533j 1.075685248074987 + 0.088562093147388j 1.075685248074987 - 0.088562093147388j 1.017267596380368 + 0.115758108548299j 1.017267596380368 - 0.115758108548299j 0.954946722804788 + 0.109941597447535j 0.954946722804788 - 0.109941597447535j 0.903952990337114 + 0.076558048962737j 0.903952990337114 - 0.076558048962737j 0.873779507558468 + 0.026814434469236j 0.873779507558468 - 0.026814434469236j 0.677703526876975 0.212499292738484 0.002099722641157 0.000000000021844 0 0
P OLES OF DISCRETE SYSTEM . ZOH APPROXIMATION WAS USED WITH N = 20
Is is shown that for approximation order less or equal than 5 both methods are providing correct results. However, for approximation order N = 20 even the Tustin method and Al-Alaoui operator fails and the resulting discretisation is unstable. The problems with implementation of high order approximations is substantial. It leads to situations, where proper approximation might be impossible, as discrete form will not be implementable. Methods of increasing robustness help, but they introduce additional design difficulties. New promising method is currently being developed by authors ( [5]) and is free of at least part of problems.
0
−0.5
−1 −1
−0.5
0 Real Part
0.5
1
Fig. 6. Poles of discrete system. Tustin approximation was used with N = 20
In this work, the authors propose three ways to avoid problem with numerical instability: •
Use discrete forms of system more resistant to numerical errors, in particular use a serial connection of transfer functions of the second order (so called SOS
calculation. Due to the very ”low-level” programming language, implementation of more complex calculations in FPGA is time-consuming.
zero pole unit circle
1
ACKNOWLEDGMENT Work realised in the scope of project titled ”Design and application of non-integer order subsystems in control systems”. Project was financed by National Science Centre on the base of decision no. DEC-2013/09/D/ST7/03960.
Imaginary Part
0.5
5
0
R EFERENCES −0.5
[1]
−1
[2] −1
−0.5
0 Real Part
0.5
1
[3] Fig. 7. Poles of discrete system. ZOH approximation was used with N = 20 Oustaloup approximation, Al−Alaoui operator, 20−order 1
[4]
zero pole unit circle
0.8 0.6
[5]
Imaginary Part
0.4 0.2 0
[6]
−0.2 −0.4 −0.6
[7]
−0.8 −1 −1
−0.5
0
0.5 Real Part
1
1.5
Fig. 8. Poles of discrete system. Al-Alaoui operator was used with N = 20
– Second Order Section). •
”Push” the poles back in the unit circle. The poles, which as a result of rounding errors ”float” outside the area of stability, can be ”slidden” back inside the unit circle. In such a situation, however, smaller or larger changes in the dynamic properties of the discrete system in relation to the original are possible.
•
Increase the number of bits in words used for storing coefficients of transfer function. The operation tis not possible in every computing system. Typically, processors and microprocessors have several predefined data types. Within these types, one can freely convert from one type to another, but an attempt to use an unforeseen type in the design of the processor is complex and cumbersome. Changing the length of the word is easier in computing systems based on FPGA. These systems, due to their design, allow a very flexible design of variable types and manners of
[8]
[9]
J. Baranowski, M. Długosz, and W. Mitkowski, “Parametric optimization of nonlinear system controller,” in Materiały XIII sympozjum Podstawowe Problemy Energoelektroniki, Elektromechaniki i Mechatroniki, Wisła, 14–17 grudnia 2009, pp. 206–211. J. Baranowski, W. Mitkowski, and P. Skruch, “Stability regions of time delay controller for lc ladder network,” in Materiały XXXII Mi˛edzynarodowej konferencji z podstaw elektrotechniki i teorii obwodów IC-SPETO, Ustro´n, 20–23.05 2009, pp. 103–104, extended version on CD. J. V. Burke, A. S. Lewis, and M. L. Overton, “A nonsmooth, nonconvex optimization approach to robust stabilization by static output feedback and low-order controllers,” in Proceedings of 4th IFAC Symposium on Robust Control Design, Milan, Italy, 2003, pp. 175–181. S. Joo and J. Seo, “Design and analysis of the nonlinear feedback linearizing control for an electromagnetic suspension system,” Control Systems Technology, IEEE Transactions on, vol. 5, no. 1, pp. 135–144, Jan 1996. P. Bania and J. Baranowski, “Laguerre polynomial approximation of fractional order linear systems,” in Advances in the Theory and Applications of Non-integer Order Systems: 5th Conference on Non-integer Order Calculus and Its Applications, Cracow, Poland, W. Mitkowski, J. Kacprzyk, and J. Baranowski, Eds. Springer, 2013, pp. 171–182. A. Djouambi, A. Charef, and A. V. Besancon, “Approximation and synthesis of non integer order systems,” in 2nd IFAC Workshop on Fractional Differentiation and its Applications, FDA06, Porto : Portugal (2006), July 2006. A. Oustaloup, F. Levron, B. Mathieu, and F. M. Nanot, “Frequencyband complex noninteger differentiator: characterization and synthesis,” Circuits and Systems I: Fundamental Theory and Applications, IEEE Transactions on, vol. 47, no. 1, pp. 25–39, 2000. C. A. Monje, Y. Chen, B. M. Vinagre, D. Xue, and V. Feliu, Fractionalorder systems and controls. Fundamentals and applications, ser. Advances in Industrial Control. London: Springer-Verlag, 2010. M. A. Al-Alaoui, “Al-alaoui operator and the α-approximation for discretization of analog systems,” Electrical Engineering, no. 1, pp. 143– 146.
