PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 69 9 controller for the optimally reduced model G r (s) and let us see if the designed controller still works for the original system. The integer order PID controller to be designed is in the following form: The optimum ITAE criterion-based PID tuning formula [43] can be used Based on this tuning algorithm, a PID controller can be designed for G r (s) as follows: L=0.63; T=3.5014; K=0.9951; N=10; Ti=T+0.5*L; Kp=(0.7303+0.5307*T/L)*Ti/(K*(T+L)); Td=(0.5*L*T)/(T+0.5*L); [Kp,Ti,Td] Gc=Kp*(1+1/Ti/s+Td*s/(Td/N*s+1)) The parameters of the PID controller are then K p =3.4160,T i =3.8164,T d = 0.2890, and the PID controller can be written as Fig. 3. Step response comparison of the optimum FOLPD and the original model. (12) (13) (14) G c (s) = 1.086s 2 + 3.442s + 0.8951 0.0289s 2 + s K p = (0.7303 + 0.5307T/L)(T + 0.5L) K(T + L) , T i = T + 0.5L, T d = 0.5LT T + 0.5L . G c (s) = K p 1 + 1 T i s + T d s T d /Ns + 1 . 1070 D INGY PID controller is shown in Fig. 4. A satisfactory performance can be clearly observed. Therefore, we believe, the method presented in this paper can be 0 2 4 6 8 10 12 14 16 18 0 0.2 0.4 0.6 0.8 1 1.2 Step Response Time (sec) Amplitude 5 Concluding Remarks H 2 -norm. Relevant MATLAB codes useful for practical applica tions are also given in the appendix. Through illustrations,we show that the tion is simple and effective. It is also demonstrated approximation method is effective in designing integer order controllers for FO-LTI systems in general form. Finally, we would like to remark that the so-called pseudo-rational ap- proximation is essentially by cascading irrational transfer function (a time delay) and a rational transfer function. Since a delay element is also infinite system involving time delay. Although it might not fully make physical sense, the pseudo-rational approximation proposed in this paper will find its prac- systems, as illustrated in Example 3. ¨ U and YANGQUAN CHEN Xue and Chen Finally, the step response of the original FO-LTI with the above -designed used for integer-order controller design for general FO-LTI systems. Fig. 4. Step response of fractional-order plant model under the PID controller. to arbitrary FO-LTI pseudo-rational approxima- that this suboptimum dimensional, it makes sense to approximate a general fractional-order LTI tical applications in designing an integer-order controller for fractional-order systems with suboptimummation In this paper, we presented a procedure to achieve pseudo-rational approxi- PSEUDO-RATIONAL APPROXIMATIONS TO FO-LTI SYSTEMS 7111 We acknowledge that this paper is a modified version of a paper published in the Proceedings of IDETC/CIE 2005 (Paper# DETC2005-84743). We would like to thank the ASME for granting us permission in written form to pub- lish a modified version of IDETC/CIE 2005 (Paper# DETC2005-84743)as Professors Machado, Sabatier, and Agrawal (Springer). Appendix 1 MATLAB functions for optimum fractional model reduction a chapter in the book entitled editedby AdvancesinFractional Theoret-Calculus: icalDevelopmentsandApplicationsinPhysicsandin Engineering Acknowledgment • ousta fod.m Outstaloup’s rational approximation to fractional differ- entiator, with the syntax G=ousta fod( function G=ousta_fod(r,N,w_L,w_H) mu=w_H/w_L; k=-N:N; w_kp=(mu).^((k+N+0.5-0.5*r)/(2*N+1) )*w_L; w_k=(mu).^((k+N+0.5+0.5*r)/(2*N+1) )*w_L; K=(mu)^(-r/2)*prod(w_k./w_kp); G=tf(zpk(-w_kp’,-w_k’,K)); function G_r=opt_app(G,nn,nd,key,G0) GS=tf(G); num=GS.num{1}; den=GS.den{1}; Td=totaldelay(GS); GS.ioDelay=0; GS.InputDelay=0; GS.OutputDelay=0; if nargin<5, n0=[1,1]; for i=1:nd-2, n0=conv(n0,[1,1]); end G0=tf(n0,conv([1,1],n0)); end beta=G0.num{1}(nd+1-nn:nd+1); alph=G0.den{1}; Tau=1.5*Td; x=[beta(1:nn),alph(2:nd+1)]; if abs(Tau)<1e-5, Tau=0.5; end if key==1, x=[x,Tau]; end dc=dcgain(GS); y=opt_fun(x,GS,key,nn,nd,dc); x=fminsearch(’opt_fun’,x,[],GS,key,nn,nd,dc); alph=[1,x(nn+1:nn+nd)]; beta=x(1:nn+1); if key==0, Td=0; end beta(nn+1)=alph(end)*dc; if key==1, Tau=x(end)+Td; else, Tau=0; end G_r=tf(beta,alph,’ioDelay’,Tau); r,N ,ω L ,ω H ) • opt app.m Optimal model reduction function, and the pseudo-rational transfer function model G r , i.e., the transfer function with a possible delay term, can be obtained. G r =opt app(G,r,d,key,G 0 ), where key indicates whether a time delay is r equired in the reduced order model. G 0 is the initial reduced order model, optional. 1272 D INGY • opt fun.m internal function used by opt app, function y=opt_fun(x,G,key,nn,nd,dc) ff0=1e10; alph=[1,x(nn+1:nn+nd)]; beta=x(1:nn+1); beta(end)=alph(end)*dc; g=tf(beta,alph); if key==1, tau=x(end); if tau<=0, tau=eps; end [nP,dP]=pade(tau,3); gP=tf(nP,dP); else, gP=1; end G_e=G-g*gP; G_e.num{1}=[0,G_e.num{1}(1:end-1)]; [y,ierr]=geth2(G_e); if ierr==1, y=10*ff0; else, ff0=y; end • get2h.m internal function to evaluate the H 2 norm of a rational transfer function model. function [v,ierr]=geth2(G) G=tf(G); num=G.num{1}; den=G.den{1}; ierr=0; n=length(den); if abs(num(1))>eps disp(’System not strictly proper’); ierr=1; return else, a1=den; b1=num(2:end); end for k=1:n-1 if (a1(k+1)<=eps), ierr=1; v=0; return else, aa=a1(k)/a1(k+1); bb=b1(k)/a1(k+1); v=v+bb*bb/aa; k1=k+2; for i=k1:2:n-1 a1(i)=a1(i)-aa*a1(i+1); b1(i)=b1(i)-bb*a1(i+1); end, end, end v=sqrt(0.5*v); ¨ U a ndYANGQUAN CHEN Xue and Chen References 1. 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Wang FS, Juang WS, Chan CT (1995) Optimal tuning of PID controllers for single and cascade control loops. Chem. Eng. Comm. 132:15–34. LINEAR DIFFERENTIAL EQUATIONS OF FRACTIONAL ORDER Blanca Bonilla 1 , Margarita Rivero 2 , and Juan J. Trujillo 1 1 Departamento de An´alisis Matem´atico, Universidad de la Laguna. 2 Departamento de Matem´atica Fundamental, Universidad de la Laguna. 38271 La Laguna-Tenerife, Abstract This manuscript presents the basic general theory for sequential linear fractional dif- D α a + (a ∈ R, 0 <α≤ 1) L nα (y)= D nα a+ + n−1 k=0 a k (x)D kα a+ (y)=y (nα + n−1 k=0 a k (x)y (kα = f(x). (1) where {a k (x)} n−1 k=0 are continuous real functions defined in [a, b] ⊂ R and D α a+ = D α a+ D kα a+ = D α a+ D (k−1)α a+ . (2) We also consider the case where f(x) is a continuous real function in (a, b] ⊂ R and f(a)=o(x α−1 ). λx α , which we will call α- exponential. This function is the product of a Mittag-Leffler function and a power function. This function allows us to directly obtain the general solution to homo- geneous and non-homogeneous linear fractional differential equations with constant coefficients. This method is a variation of the usual one for the ordinary case. Keywords 1 Introduction Questions as to what we mean by, and where we could apply, the fractional calculus operators have fascinated us all ever since 1695 when the so-called fractional calculus was conceptualised in connection with the infinitesimal 38271 La Laguna-Tenerife, Spain;E-mail: BBonilla@ull.es;JTrujill@ull.es Spain; MRivero@ull.es ferential equations, involving the well known Riemann Liouville fractional operators, − We then introduce the Mittag-Leffler-type function e Fractional differential equations, Caputo, RiemannLiouville, linear. − © 2007 Springer. 77 in Physics and Engineering, 77– 91. J. Sabatier et al. (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications E-mail: 2 78 calculus, see [15] and [13]. A rigourous and encyclopedic study of fractional operators can be found in [17]. It is known that the classical calculus provides a powerful tool for ex- plaining and modelling many important dynamic processes in most applied sciences. But experiments and reality teach us that there are many com- plex systems in nature and society with anomalous dynamics, such as charge transport in amorphous semiconductors, the spread of contaminants in under- ground water, relaxation in viscoelastic materials like polymers, the diffusion of pollution in the atmosphere, and many more. In most of the above-mentioned cases, this kind of anomalous process has a complex macroscopic behaviour, the dynamics of which cannot be charac- terised by classical derivative models. Nevertheless, a heuristic solution to the corresponding models of some of those processes can be frequently obtained using tools from statistical physics. For such an explanation, one must use some generalised concepts from classical physics such as fractional Brownian motion, the continuous time random walk (CTRW) method involving L´evy stable distributions (instead of Gaussian distributions), the generalised cen- tral limit theorem (instead of the classical central limit theorem), and non- Markovian distributions which means non-local distributions (instead of the classical Markovian ones). From this approach it is also important to note that the anomalous behaviour of many complex processes includes multi-scaling in the time and space variables. The above-mentioned tools have been used extensively during last 30 years. But the connection between these statistical models and certain fractional differential equations involving the fractional integral and derivative operators been formally established during the last 15 years; (see, for instance, [10], [9] [19], [14]). We could ask what are the useful properties of these frac- processes? From the point of view of the authors and from known experi- mental results, most of the processes associated with complex systems have Perhaps this is one of the reasons why these fractional calculus operators lose the above-mentioned useful properties of the ordinary derivative D. This manuscript is organised as follows. Sections 2 and 3 presents some fractional operators and their main properties and introduce some types of new direct method for solving the homogeneous and non-homogeneous case with constant coefficients, using the α-exponential function and certain frac- tional Green functions, including some illustrative examples. Bonilla,Rivero,andTrujillo (RiemannLiouville, Caputo, Liouville or Weyl, Riesz, etc.; see [17]) has only − ourselves, tional calculus operators,which help in the modelling of so many anomalous non-local dynamics involving longmemory in time, and the fractional inte- gral and fractional-derivative operators do have some of those characteristics. Mittag-Leffler functions. In section 4 we develop a general theory for sequen- tial linear fractional differential equations, while in section 5 we introduce a [...]... coherent fractional derivatives using as starting point the Diaz and Osler integral formulation for the fractional differences [4] The framework we proposed was based on the following steps: 1 Also with INESC-ID, R Alves Redol, 9, 2º, Lisbon 93 J Sabatier et al (eds.), Advances in Fractional Calculus: Theoretical Developments and Applications in Physics and Engineering, 93 –112 © 2007 Springer 94... mere substitution of for N in (8) and (12) In the first case, this leads easily to ( +1) c1f(t) = 2 ih f(z+w) Cc -w h +1 -w h + 2 +1 w h w h + 2 +1 dw (31 ) The integrand has infinite poles at every nh, with n Z The integration path must consist of infinite lines above and below the real axis closing at the infinite The easiest situation is obtained by considering two straight lines near the real axis,... problem We are mainly interested in obtaining a generalisation of a well-known formulation The paper outline is as follows In section 2 we present the type 1 and type 2 centred differences and their integral representations Centred derivative definitions similar to Grünwald–Letnikov ones are presented in section 3 and their integral representations obtained generalising the Riesz potentials In section 4... References 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Carpinteri A, Mainardi F (eds.) (1997) Fractals and Fractional Calculus in Continuum Mechanics, CIAM Courses and Lectures 37 8 Springer, New York Erdélyi A, Magnus W, Oberhettinger F, Tricomi FG (19 53) Higher Transcendental Functions, Vol I McGraw-Hill, New York Gorenflo R, Kilbas AA, Rogosin SV (1998) On the generalized Mittag-Leffler Type functions, Int Trans... uniqueness theorems for nonlinear fractional differential equations, Demostratio Math 3( 33) :5 83 602 Kilbas AA, Pierantozzi T, Vázquez L, Trujillo JJ (2004) On solution of fractional evolution equation, J Phys A: Math Gen 37 :1– 13 Kilbas AA, Srivastava HM, Trujillo JJ (20 03) Fractional differential equations: an emergent field in applied and mathematical sciences, in: Factorization, Singular Operators and... diffusion: A fractional dynamic approach, Phys Rep 1 (33 9)1–77 Miller KS, Ross B (19 93) An Introduction to the Fractional Calculus and Fractional Differential Equations Wiley, New York Mittag-Leffler MG (19 03) Sur la nouvelle fonction, Compt Rend Acad Sci Paris, 137 :554–558 Oldham KB, Spanier J (1974) The Fractional Calculus Academic Press, New York Podlubny I (1999) Fractional differential equations, in: Mathematics...LINEAR FRACTIONAL DIFFERENTIAL EQUATIONS 79 3 We must point out that the Laplace or Fourier transform can only be used for the case where the fractional derivative involved in the fractional α differential equation is D0+ , and never when we want to use the more general α fractional derivative Da+ (a < 0), as must be done when the initial conditions of the corresponding model are given in the origin... always find, in a way similar to the ordinary case, a fundamental set of solutions for Eq (30 ) in some interval V ⊂ [a, b] Usually, the general solution to a non-homogeneous linear sequential fractional differential equation Lnα (y) = f (x) (35 ) will be given as in the following proposition: Proposition 3 If yp (x) is a particular solution to (35 ) and yh (x) is a general solution to the corresponding homogeneous... /2+k+1) (33 ) Definition 8 For the second case, we define the type 2 fractional centred derivative given by Dc2f(t) = lim h 0 ( +1) + h - (-1)k f(t-kh+h/2) [( +1)/2-k+1] [( -1)/2+k+1] (34 ) Formulae (33 ) and (34 ) generalise the positive integer order centred derivatives to the fractional case, although there should be an extra factor (-1) /2 in the first case RIESZ POTENTIALS AS CENTRED DERIVATIVES 1 03 and... (-1)( +1)/2 in the second case that we removed It is a simple task to obtain the derivative analogues to (28), (29), and (30 ): { } = Dc + f(t) (35 ) { } = - Dc + f(t) (36 ) { } = Dc + f(t) (37 ) Dc1 Dc1f(t) 1 and Dc2 Dc2f(t) 1 while Dc2 Dc1f(t) 2 again with + > -1 4.2 Integral formulae To obtain the integral formulae for the centred fractional derivatives, we follow the same procedure used in the integer . differentiator/integrator. J. Franklin Inst. 34 0(5) :34 9 36 2. 32 . Chen YQ, Vinagre BM (20 03) A new IIR-type digital fractional order differentiator. Signal Processing, 83( 11) November: 235 9– 236 5. 33 . Al-Alaoui. 42(4):220–2 23. 35 . Al-Alaoui MA (1997) Filling the gap between the bilinear and the back-ward difference transforms: an interactive design approach. Int. J. Electr. Eng. Educ. 34 (4) :33 1 33 7. 36 DETC2005-847 43) as Professors Machado, Sabatier, and Agrawal (Springer). Appendix 1 MATLAB functions for optimum fractional model reduction a chapter in the book entitled editedby Advances in Fractional