Computers and Mathematics with Applications

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1 Computers and Mathematics with Applications 59 (2010) Contents lists available at ScienceDirect Computers and Mathematics with Applications journal homepage: Control of a heat diffusion system through a fractional order nonlinear algorithm Isabel S. Jesus, J.A. Tenreiro Machado, Ramiro S. Barbosa Institute of Engineering of Porto, Rua Dr. António Bernardino de Almeida, 431, Porto, Portugal a r t i c l e i n f o a b s t r a c t Keywords: Fractional calculus Nonlinear control Heat diffusion systems The application of the FC concepts has increased significantly in different fields of science and engineering, because FC captures properties that classical integer order models neglect. This paper studies a heat diffusion system, that is described through the fractional operator s 0.5, under the control of a fractional nonlinear algorithm. The tuning of the algorithm follows the optimization of performance control indices. The results demonstrate the good performance of the proposed controller Elsevier Ltd. All rights reserved. 1. Introduction Fractional calculus (FC) is a generalization of integration and differentiation to a non-integer order α C, being the fundamental operator a D α t, where a and t are the limits of the operation [1 4]. During the last few years, FC has been used increasingly to model the constitutive behavior of materials and physical systems exhibiting hereditary and memory properties [5,6]. This is the main advantage of fractional derivatives in comparison with classical integer models, where these effects are simply neglected. It is well known that the fractional order operator s 0.5 appears in several types of problems. The transmission lines [7], heat flow [8,9] or the diffusion of neutrons in a nuclear reactor are examples where the half-operator is the fundamental element. On the other hand, diffusion is one of the three fundamental partial differential equations of mathematical physics [10 13]. In this paper we investigate the heat diffusion system in the perspective of applying the FC theory. A nonlinear controller with a fractional order model is presented and compared with other algorithms, namely the fractional PID controller. The fractional PI β D α controller involves an integrator of order β R + and a differentiator of order α R + [6,8,14,15]. Bearing these ideas in mind, the paper is organized as follows. Section 2 gives the fundamentals of fractional order control systems. Section 3 introduces the heat diffusion system and describes its simulation. Section 4 points out a control strategy for the heat system and discusses the results. Finally, Section 5 draws the main conclusions and addresses perspectives towards future developments. 2. Fractional order control systems Fractional controllers are characterized by differential equations that have, in the dynamical system and/or in the control algorithm, an integral and/or a derivative of fractional order. Due to the fact that these operators are defined by irrational continuous transfer functions, in the Laplace domain, or infinite dimensional discrete transfer functions, in the Z domain, we often encounter evaluation problems in the simulations. Therefore, when analyzing fractional systems, we usually adopt continuous or discrete integer order approximations of fractional order operators. Corresponding author. addresses: isj@isep.ipp.pt (I.S. Jesus), jtm@isep.ipp.pt (J.A.T. Machado), rsb@isep.ipp.pt (R.S. Barbosa) /$ see front matter 2009 Elsevier Ltd. All rights reserved. doi: /j.camwa

2 1688 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 1. Nonlinear structure control. Fig. 2. Step responses of the FNC (with α = 0.8) and the ILC (with α = 1.0) closed-loop systems for = 1.0. The mathematical definition of a fractional derivative and integral has been the subject of several different approaches [1,4,14]. One commonly used definition is given by the Riemann Liouville expression (α > 0 and n 1 < α < n): d n ad α t f (t) = 1 Ɣ (n α) dt n t a f (τ) dτ (1) α n+1 (t τ) where f (t) is the applied function and Ɣ(x) is the Gamma function of x. Another widely used definition is given by the Grünwald Letnikov approach (α R): ad α t f (t) = lim h 0 1 h α ( ) α ( 1) k f (t kh) k [ t a h ] k=0 ( α Ɣ (α + 1) = k) Ɣ (k + 1) Ɣ (α k + 1) where h is the time increment and [x] means the integer part of x. The memory effect of these operators is demonstrated by (1) and (2), where the convolution integral in (1) and the infinite series in (2), reveal the unlimited memory of these operators, ideal for modelling hereditary and memory properties in physical systems and materials. An alternative definition to (1) and (2), which reveals useful for the analysis of fractional order control systems, is given by the Laplace transform method. Considering vanishing initial conditions, the fractional differintegration is defined in the Laplace domain, F(s) = L{f (t)}, as: L { ad α t f (t)} = s α F (s), α R. In this paper we adopt discrete integer order approximations to the fundamental element s α (α R) of a fractional order control (FOC) strategy. The usual approach for obtaining discrete equivalents of continuous operators of type s α adopts the Euler, Tustin and Al-Alaoui generating functions [2,3,16]. It is well known that rational-type approximations frequently converge faster than polynomial-type approximations and have a wider domain of convergence in the complex domain. Thus, by using the Euler operator ω(z 1 ) = (1 z 1 )/T c, and (2a) (2b) (3)

3 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 3. The FNC parameters {K p, K i, K d, K, Ψ } versus α for = {0.5, 1.0, 2.0}. performing a power series expansion of [ω(z 1 )] α = [(1 z 1 )/T c ] α gives the discretization formula corresponding to the Grünwald Letnikov definition (2): D α ( z 1) ( ) 1 z 1 α = = h α (k) z k (4) T k=0

4 1690 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 4. Step responses of the NLC closed-loop system for = 1.0 and α = {0.2, 0.4, 0.6, 0.8}. Fig. 5. Step responses ( 1 ) of the FNC closed-loop system with {K p, K i, K d, α, K, Ψ } {0.01, 0.42, , 0.8, 0.07, 1.90}, {K p, K i, K d, α, K, Ψ } {0.01, 0.28, 55.80, 0.8, 0.04, 2.40} and {K p, K i, K d, α, K, Ψ } {0.01, 0.44, , 0.8, 0.04, 1.60}, for = {0.5, 1.0, 2.0}, respectively. ( ) α ( ) 1 h α k α 1 (k) =. T k (5) A rational-type approximation can be obtained by applying the Padé approximation method [17] to the impulse response sequence (5) h α (k), yielding the discrete transfer function: H ( z 1) = b 0 + b 1 z b m z m 1 + a 1 z a n z n = h (k) z k (6) k=0 where m n and the coefficients a k and b k are determined by fitting the first m + n + 1 values of h α (k) into the impulse response h(k) of the desired approximation H(z 1 ). Thus, we obtain an approximation that matches the desired impulse response h α (k) for the first m + n + 1 values of k. Note that the above Padé approximation is obtained by considering the Euler operator, but the determination process will be exactly the same for other types of discretization schemes. At low frequencies the Padé fraction behaves simply as a proportional gain. Therefore, the total proportional gain consists, in fact, of two components.

5 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 6. Step responses ( 1 ) of the FNC closed-loop system with {K p, K i, K d, α, K, Ψ } {0.01, 0.28, 55.80, 0.8, 0.04, 2.40} (i. e., tuned for = 1.0) when = {0.5, 1.0, 2.0}. 3. Heat diffusion The heat diffusion is governed by a linear unidimensional partial differential equation (PDE) of the form: c t = k 2 c c (7) x 2 where k c is the diffusivity, t is the time, c is the temperature and x is the space coordinate. The system (7) involves the solution of a PDE of parabolic type for which the standard theory guarantees the existence of a unique solution [10,11]. For the case of a planar perfectly isolated surface we usually apply a constant temperature C 0 at x = 0 and we analyze the heat diffusion along the horizontal coordinate x. Under these conditions, the heat diffusion phenomenon is described by a non-integer order model: C (x, s) = C 0 s s e x k (8) where x is the space coordinate and C 0 is the boundary condition. In our study, the simulation of the heat diffusion is performed by adopting the Crank Nicholson implicit numerical integration based on the discrete approximation to differentiation as [6,12]: rc [j + 1, i + 1] + (2 + r) c [j + 1, i] rc [j + 1, i 1] = rc [j, i + 1] + (2 r) c [j, i] + c [j, i 1] (9) where r = k c t( x 2 ) 1, { x, t} and {i, j} are the increments and the integration indices for space and time, respectively [8,18]. 4. Control strategy This section studies a new control strategy for the heat diffusion system. In fact, in a previous work [9] it was analyzed the closed-loop system with a classical PID controller: de (t) g (t) = K p e (t) + K i e (t) dt + K d. (10) dt In the sequel we will denote (10) as the integer linear controller (ILC). However, the poor results of the ILC [19,20] indicated that it might not be the most adequate for the control of the heat system. In fact, the inherent fractional dynamics of the system lead us to consider other configurations. In this perspective, we propose the use of fractional order schemes tuned by the minimization of the integral square error (ISE) index. In this line of thought, we develop the nonlinear controller with a fractional order algorithm (FNC), represented in Fig. 1. The closed-loop system consists in the PID α controller and the nonlinearity described by Eqs. (11) and (12), respectively: d α e (t) m (t) = K p e (t) + K i e (t) dt + K d dt α (11) { K m (t) Ψ if m (t) 0 n (t) = K m (t) Ψ (12) if m (t) < 0

6 1692 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 7. ISE ( 1 2 ) versus α for = {0.5, 1.0, 2.0}. where the symbol e represents the error, α the order of the fractional derivative term, 0 α 1, and the constants K p, K i and K d are the proportional, the integral and the derivative gains, respectively. The fractional derivative term s α in (11) is implemented through a 4th order Padé discrete rational transfer function of type (6), with a sampling period of T = 0.1 s. In expression (12) m is the output of fractional algorithm, K > 0 is a constant and the exponent Ψ is a real number. The nonlinearity (12) can be considered as a generalization of the standard variable structure controller (VSC) [21 26]. In fact, in the simplest form, a VSC consists in a saturation-like function which is a special case of (12); therefore, expression (12) gives an extra degree of freedom in the controller design though the tuning of the parameters (K, Ψ ). In general nonlinearities must be avoided, but the truth is that VSCs demonstrated good robustness, leading to linear-like responses. In this line of thought, in the sequel we will verify that the quasi-linear response will be a characteristic of the control algorithm (11) (12). The controller is tuned by the minimization of an integral performance index. For that purpose, we analyze the indices that measure the response error, namely the ISE criteria: ISE = [r (t) c (t)] 2 dt. (13) 0 We can use other performance criteria such as the integral time square error (ITSE), the integral absolute error (IAE) or the integral time absolute error (ITAE); however, in the present case the ISE criterion had produced the best results and is adopted in the study [9]. For the numerical experiments is applied a step reference input R(s) = /s and the output is the temperature c(t) measured at x = 3.0 m. The heat system is simulated with a controller having sampling period T = 0.1 s. Fig. 2 compares the step responses of the closed-loop system with the FNC tuned in the ISE perspective, for the best fractional order α = 0.8 and the ILC (without nonlinearity). This study confirms that the fractional order controller reveals best results than the corresponding of integer order. In fact, for the FNC we obtain the overshoot (ov) {ISE, ov(%), α} = {14.35, 7.94, 0.8} and for the ILC {ISE, ov(%), α} = {18.48, 15.55, 1.0}. Fig. 3 illustrates the variation of the fractional order control parameters {K p, K i, K d, K, Ψ } as function of the order s derivative α, when minimizing the ISE criterion for the FNC. Fig. 4 shows the step responses of the closed-loop system, for the FNC tuned in the ISE perspective, and for 0 α 1. The FNC parameters {K p, K i, K d, α, K, Ψ } corresponding to the minimization of the ISE, lead to the values {K p, K i, K d, α, K, Ψ } {0.01, 0.28, 55.8, 0.8, 0.04, 2.40} for the best case. The step responses reveal a large diminishing of the ov and the rise time (t r ) when compared with the linear integer PID where we get {ov(%), t r } {68.56%, 12.0} [9], showing a good transient response and a zero steady-state error. In order to analyze the system dynamics we evaluate the response of the FNC system for different input systems amplitudes, namely, = {0.5, 1.0, 2.0}, when considering two different cases study. In the first case, the controller parameters are tuned (minimization of the ISE index) for each different input amplitude. Fig. 3 depicts the parameters values as function of α. For the different reference amplitudes = {0.5, 1.0, 2.0} the tuning leads to the controller parameters: {K p, K i, K d, α, K, Ψ } {0.01, 0.42, , 0.8, 0.07, 1.90}, {K p, K i, K d, α, K, Ψ } {0.01, 0.28, 55.80, 0.8, 0.04, 2.40}, {K p, K i, K d, α, K, Ψ } {0.01, 0.44, 225.6, 0.8, 0.04, 1.60}, respectively.

7 I.S. Jesus et al. / Computers and Mathematics with Applications 59 (2010) Fig. 8. Parameters t s, t p, t r, ov(%) ( 1 ) versus 0.0 α 1.0 for the step responses of the closed-loop system, when = {0.5, 1.0, 2.0}. Fig. 5 shows, the step response of the closed-loop system ( 1 ), for R 0 = {0.5, 1.0, 2.0}. We can verify that this controller reveals good characteristics and that the system output does not change significantly with the variation of input amplitude. On the other hand, when we compare the controller parameters for these three distinct input amplitudes we verify that they lead almost to similar time response. In a second set of experiments, we adopt the same controller parameters, found previously for = 1.0, namely {K p, K i, K d, α, K, Ψ } {0.01, 0.28, 55.80, 0.8, 0.04, 2.40}, and we analyze the output response for the three distinct input amplitudes = {0.5, 1.0, 2.0}. Fig. 6 depicts the three step responses of the closed-loop system ( 1 ), for the FNC. The results in this second case are inferior to those attained in the first case. In this line of thought, in the rest of the study the controller parameters are tuned separately. Fig. 7 depicts the ISE (with a scale factor of 1 ) for 0.0 α 1.0, when R R 2 0 = {0.5, 1.0, 2.0}. We verify the existence 0 of a minimum for α 0.8 for the ISE. Fig. 8 shows the variation of the settling time t s, the peak time t p, the rise time t r, and the percent overshoot ov(%) (with a scale factor of ( 1 ) versus α, for the closed-loop response tuned through the minimization of the ISE indices. Again, we verify a smooth variation with α with good results for 0.8 α 0.9 and an abrupt variation for α = Conclusion This paper studied the control of a heat diffusion system, using a fractional order and a nonlinear algorithm. The results reveal the superior performance of the proposed controller. Moreover, the fractional order model of the controller is more flexible and gives the possibility of adjusting more carefully the closed-loop system characteristics than the classical PID controller. In conclusion, the results demonstrate the effectiveness of the FNC when used for the control of fractional systems.

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