On Fractional-order Calculus Applications to Industrial Control Problems Aleksei Tepljakov, Eduard Petlenkov and Juri Belikov Department of Computer Control Tallinn University of Technology Ehitajate tee 5, 19086, Tallinn, Estonia
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[email protected] Abstract— Fractional-order calculus (FOC) offers novel methods for control system engineering. In this paper, we discuss the applications of fractional-order calculus to industrial control problems. We introduce fractional-order controllers, which are generalizations of the classical ones, namely the fractional PID controller and the fractional-order lead-lag compensator. Tuning and control optimization are also discussed with illustrative examples, based on system modeling and simulation in FOMCON – a new fractional-calculus based modeling and control toolbox for MATLAB.
follows. In Section II we briefly introduce the basic concepts of fractional-order calculus. In Section III we present two fractional-order controllers derived from conventional controllers. Several issues are addressed, such as controller tuning and implementation. A more detailed tuning procedure for the fractional PID controller is given in Section IV. Two illustrative examples are also provided in Section V. Finally, in Section VI conclusions are drawn. II. I NTRODUCTION TO F RACTIONAL - ORDER C ALCULUS
I. I NTRODUCTION The concept of the differential operator D = d/ dx is a fundamental tool of modern calculus. Given a suitable function f the n-th derivative is well defined as D n f (x) = dn f (x)/ dxn , where n is a positive integer. However, what happens if we extend this concept to a situation, when the order of differentiation is fractional? For example, what if n were 12 ? This was the very same question L’Hôpital addressed to Leibniz in a letter in 1695. Since then the concept of fractional calculus has drawn the attention of many famous mathematicians. Fractional-order calculus was not particularly popular until the recent years when benefits stemming from using its concepts became evident in various scientific fields. Recent findings support the notion that fractional-order calculus should be employed where more accurate modeling and robust control are concerned. In general, fractional-order calculus may be useful when modeling any system which has memory and/or hereditary properties [1]. In the field of control system design fractional calculus is used to obtain more accurate models, develop new control strategies and enhance the characteristics of control systems. In order to facilitate fractional-order system modeling and controller design a set of tools is required. Since using MATLAB in control system engineering is standard practice, several MATLAB toolboxes have been developed for this particular set of tasks. Among them are CRONE [2], developed by the CRONE team, NINTEGER, developed by Duarte Valério [3] and FOMCON [4], developed by the authors of this paper. Further we will focus on fractional-order control and discuss its use in industrial applications. The paper is organized as
Fractional calculus is a generalization of integration and differentiation to non-integer order operator a D α t , where a and t denote the limits of the operation [5]. There exist several definitions of the fractional differintegral. A commonly used definition is the Riemann-Liouville differintegral [6] m Zt d f (τ ) 1 α a Dt f (t) = α−m+1 dτ (1) Γ (m − α) d t (t − τ ) a
for m − 1 < α < m, m ∈ N, where Γ (·) is Euler’s gamma function. Consider also the Grünwald-Letnikov definition t−a [X h ] 1 j α α (−1) f (t − jh) , a Dt f (t) = lim α h→0 h j j=0
(2)
where [·] denotes the integer part. The Laplace transform of an α-th derivative with α ∈ R+ of f (t), assuming zero initial conditions, is given by (3) L D α f (t) = sα F (s) , R ∞ −st where F (s) = 0 e f (t) d t. Thus a fractional order differential equation can be expressed in transfer function form bm sβm + bm−1 sβm−1 + · · · + b0 sβ0 G (s) = . (4) an sαn + an−1 sαn−1 + · · · + a0 sα0 A system given by (4) is said to be of commensurate order if all the orders of the fractional operator s are integer multiples of a base order q such that αk , βk = kq, q ∈ R+ , 0 < q < 1. For more information on fractional-order calculus and its applications the interested reader is referred to the books [1], [6], [7].
III. OVERVIEW OF F RACTIONAL - ORDER C ONTROLLERS In this section an overview of two fractional-order controllers is provided. These controllers are derived from their classical variants and since generalization to the fractional order is applied to the differentiation operators, these controllers can be tuned to meet the given specifications more accurately and thus provide enhanced performance of the control loop. A. Fractional PID Controller PID controllers are ubiquitous in the industry. In process control more than 95% of the control loops are of PI/PID type [8]. This is probably the main motivation for using the fractional PID controller in industrial process control since it is expected to improve control loop performance. The fractional-order PID controller was first introduced by Podlubny in [9]. This generalized controller is called the PIλ Dµ controller and has an integrator of order λ and a differentiator of order µ. Recent researches illustrate the benefits of using the PIλ Dµ controller in place of the classical PID controller [10], [11]. The fractional PID controller transfer function has the following form Ki (5) Gc (s) = Kp + λ + Kd sµ . s Obviously, with λ = µ = 1 the result is the classical integer-order PID controller. With more freedom in tuning the controller, the four-point PID diagram can now be seen as a PID controller plane, which is conveyed in Fig. 1.
where α is the fractional order of the controller and parameters λ, x are such that λ1 = ωz is the zero frequency and 1 0 α xλ = ωp is the pole frequency when α > 0, and k = Kc x is the full controller gain. The contribution of parameter α is such, that the lower its value, the longer the distance between the zero and pole and vice versa so that the contribution of phase at a certain frequency stands still. This makes the controller more flexible and allows a more robust approach to the design. Tuning and auto-tuning techniques are discussed in [14]. It can be seen, that equation (6) is an implicit form of a fractional-order transfer function. In order to obtain an explicit form, a method was proposed in [15], which resulted in very good frequency fitting. However, in practice an integerorder approximation would be beneficial for discretization and direct implementation. Two methods can be proposed: • •
Frequency response fitting to obtain an integer-order model; Approximating the implicit fractional-order model by an integer-order model.
The former was used in [6], [14]. Here, we propose a method, derived from a regular Newton process for iterative approximation of the α-th root described in [16], [17]. Given the following relationships 1
(H(s)) α − (G(s)) = 0,
H(s) = (G(s))α ,
(7)
where G(s) corresponds to the implicit transfer function Gc (s) in (6), and defining q = 1/α, m = q/2, in each k0 iteration, starting from H0 (s) = 1 an approximated rational function is obtained in the form Hi (s) = Hi−1 (s)
(q − m)(Hi−1 (s))2 + (q + m)G(s) . (8) (q + m)(Hi−1 (s))2 + (q − m)G(s)
The necessary number of iterations required to ensure accuracy will most likely lead to a model of a very high order. However, we can apply a method to reduce the model order described in [18]. For an example see Section V of the present paper. Fig. 1.
The PIλ Dµ controller plane
Several methods of tuning fractional PID controllers were proposed over the years [6], [12], [13]. However, there is still no well-established, general tuning algorithm. Nevertheless, optimization techniques may be applied to the tuning problem instead with respect to necessary design specifications, see Section IV for further information. B. Fractional Lead-Lag Compensator The lead-lag compensator is a well-known type of feedback controller widely used in practice. Coupled with concepts from fractional-order calculus a more robust controller could be obtained. The fractional lead-lag compensator is described by the following transfer function: α λs + 1 0 , (6) Gc (s) = k xλs + 1
Note that this method works for the case, when α < 1, i.e. for the lead compensator. When α > 1, that is for approximating a lag compensator by a rational transfer function, the controller should be composed as follows: G(s) = k 0 · GC (s) · GK (s),
(9)
where GC is the Carlson approximation such that GC = α GK (s) and GK (s) = Gck(s) corresponds to the implicit 0 transfer function in (6). IV. C ONTROLLER T UNING AND O PTIMIZATION In this section we address the problem of fractional PID optimization. For PID controller design one may use the FPID Optimization Tool, which is a part of the FOMCON toolbox. It can be called from the MATLAB command line by typing fpid_optim. The corresponding graphical user
Fig. 2.
Fractional PID optimization tool
interface is presented in Fig. 2. Further we briefly describe the optimization tool.
Performance specifications include gain and phase margin settings, required sensitivity and complementary sensitivity function constraints and means for limiting the control signal. Several optimization algorithms are available, most notably the Nelder-Mead algorithm which was found especially suitable for the task. Also it does not require the MATLAB Optimization toolbox to be installed, so if it is not available the user will still be able to use this algorithm. Thus, because it is possible to select the parameters to optimize and also individually pick the necessary performance specification settings, a flexible tool is obtained. It should be noted, that optimization to full specifications may take a lot of time and a feasible optimal solution cannot be guaranteed. V. I LLUSTRATIVE EXAMPLES A. Example 1 Consider a dynamic model of a heating furnace in [13]: G1 (s) =
1 . 14994s1.31 + 6009.5s0.97 + 1.69
We shall design a fractional-order PID controller for this plant using the optimization tool.
1.1 1 0.9 0.8 Amplitude
The objective of optimization is to find a suitable paramKp Ki Kd λ µ consisting of the eter set θ = fractional PID controller gains and integrator/differentiator exponents subject to parameter constraints and performance specifications using a performance metric given by one of the following: Rt 2 • Integral square error ISE = e (t) dt, 0R t • Integral absolute error IAE = e(t) dt, 0 Rt 2 • Integral time-square error IT SE = te(t) dt, 0R t dt. • Integral time-absolute error IT AE = t e(t) 0
1.2
0.7 0.6 0.5 0.4 0.3 0.2
Integer−order controller Optimized fractional−order controller
0.1 0
0
25
Fig. 3.
50
75
100 125 Times [s]
150
175
200
225
Fractional-order PID optimization results
First, an approximate FOPDT model is found in form GF OP DT (s) =
0.588586 −11.2571s e 1 + 4801.86s
and the corresponding conventional PID controller parameters are obtained with the Ziegler-Nichols tuning method such that Kp = 802.915, Ki = 24.3806, Kd = 6.09515 using the iopid_tune tool, also found in FOMCON toolbox. Next, gains are fixed and fractional PID exponents λ and µ are tuned with an initial setting of λ = 0.5, µ = 0.6. The performance metric is IAE. Gain margin is set to Gm = 10 dB and phase margin ϕ = 45◦ . Also, control signal value is limited within a range Clim = [0.001; 1000]. The achieved performance improvement after 50 iterations using the Nelder-Mead algorithm is depicted in Fig. 3. The obtained exponent values are λ = 0.03 and µ = 0.523. Openloop phase margin is ϕm = 77.57◦ . B. Example 2 In this example we will study the implementation method for the fractional lead compensator discussed in Section III.
Consider a model of a position servo given by 2 G(s) = . s(0.5s + 1)
VI. C ONCLUSIONS
In [6] a fractional lead compensator was designed for this plant. The chosen parameters were such that k 0 = 10, x = 0.005, λ = 0.6404, α = 0.5. The resulting controller was obtained in the form of an integer-order transfer function using a MATLAB function invfreqs, available in the Signal Processing toolbox. In [15] a similar method was used to obtain a fractional-order transfer function. In the following, we will use the Carlson method and model order reduction. To obtain the initial approximation of this fractional lead compensator, the following statements can be entered into the MATLAB command prompt: s = tf(’s’); G = (0.6404*s+1)/(0.0032*s+1); Gc = 10*carlapp(0.5,3,G); The model can be reduced to a 4-th order model by using the balred function of the Control System toolbox: Gc = balred(Gc,4); Next, the original lead compensator frequency response is obtained for comparison: w = logspace(-5,5,1000); r = frlc(10,0.005,0.6404,0.5,w); A transfer function is also obtained by using the invfreqs command: [b,a] = invfreqs(r,w,4,4,[],6,0.001); Gc1 = tf(b,a); The comparison of frequency responses of the resulting models is depicted in Fig. 4. It can be seen from the Bode diagram, that the method proposed in this paper provides an excellent frequency fitting result. Bode Diagram 50 45
Magnitude (dB)
40 35 30 25 20 15 10
Original controller response Phase (deg)
Carlson method with model reduction invfreqs() method
30
0 −4 10
−3
10
−2
10
−1
10
0
10 Frequency (rad/sec)
1
10
2
10
3
10
4
10
Fig. 4. Comparison of frequency responses of different fractional lead controller rational approximations
In this paper, we presented two important generalized controllers, conventional variants of which are widely used in industrial applications. It is expected that the new controllers will become ubiquitous once necessary methods for their implementation and tuning are developed and tested. An optimization tool for the fractional PID controller was described in this paper, as well as an efficient method for obtaining a rational approximation of a fractional lead-lag compensator. Illustrative examples were also provided. R EFERENCES [1] I. Podlubny, Fractional differential equations, ser. Mathematics in science and engineering. Academic Press, 1999. [2] A. Oustaloup, P. Melchior, P. Lanusse, O. Cois, and F. Dancla, “The CRONE toolbox for Matlab,” in Proc. IEEE Int. Symp. ComputerAided Control System Design CACSD 2000, 2000, pp. 190–195. [3] D. Valério. (2005) Toolbox ninteger for MatLab, v. 2.3. [Online]. Available: http://web.ist.utl.pt/duarte.valerio/ninteger/ninteger.htm [4] A. Tepljakov, E. Petlenkov, and J. Belikov. (2011) FOMCON toolbox. [Online]. Available: http://www.fomcon.net/ [5] Y. Q. Chen, I. Petras, and D. Xue, “Fractional order control - a tutorial,” in Proc. ACC ’09. American Control Conference, 2009, pp. 1397–1411. [6] C. A. Monje, Y. Chen, B. Vinagre, D. Xue, and V. Feliu, Fractionalorder Systems and Controls: Fundamentals and Applications, ser. Advances in Industrial Control. Springer Verlag, 2010. [7] K. Miller and B. Ross, An introduction to the fractional calculus and fractional differential equations. Wiley, 1993. [8] K. Åström and T. Hägglund, Advanced PID control. The Instrumentation, Systems, and Automation Society (ISA), 2006. [9] I. Podlubny, “Fractional-order systems and PIλ Dµ -controllers,” vol. 44, no. 1, pp. 208–214, 1999. [10] Y. Luo and Y. Chen, “Fractional-order [proportional derivative] controller for robust motion control: Tuning procedure and validation,” in Proc. ACC ’09. American Control Conference, 2009, pp. 1412–1417. ˇ [11] M. Cech and M. Schlegel, “The fractional-order pid controller outperforms the classical one,” in Process control 2006. Pardubice Technical University, 2006, pp. 1–6. [12] D. Xue and Y. Chen, “A comparative introduction of four fractional order controllers,” in Proceedings of the 4th World Congress on Intelligent Control and Automation, 2002, pp. 3228–3235. [13] C. Zhao, D. Xue, and Y. Chen, “A fractional order pid tuning algorithm for a class of fractional order plants,” in Proc. IEEE Int Mechatronics and Automation Conf, vol. 1, 2005, pp. 216–221. [14] C. A. Monje, B. M. Vinagre, A. J. Calderon, V. Feliu, and Y. Q. Chen, “Auto-tuning of fractional lead-lag compensators,” in Proceedings of the 16th IFAC World Congress, 2005. [15] A. Tepljakov, “Fractional-order calculus based identification and control of linear dynamic systems,” Master’s thesis, Tallinn University of Technology, 2011. [16] G. Carlson and C. Halijak, “Approximation of fractional capacitors (1/s)1/n by a regular newton process,” vol. 11, no. 2, pp. 210–213, 1964. [17] B. M. Vinagre, I. Podlubny, A. Hernández, and V. Feliu, “Some approximations of fractional order operators used in control theory and applications,” Fractional Calculus & Applied Analysis, vol. 3, pp. 945–950, 2000. [18] A. Varga, “Balancing free square-root algorithm for computing singular perturbation approximations,” in Proc. 30th IEEE Conf. Decision and Control, 1991, pp. 1062–1065.
