SOME APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS USED IN CONTROL THEORY AND APPLICATIONS B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu Abstract In this paper, starting from the formulation of some possible models of fractional-order systems, several approximations are discussed. For continuous models, some methods for obtaining an approximated rational function using evaluation, interpolation and curve fitting techniques are studied. For discrete models, approximations using Lubich s formula, the trapezoidal rule, and the application of continued fractions expansion technique to integro-differential operators formulated in the Z domain are studied. The methods are compared, in both the time and the frequency domains, using an illustrative example. Mathematics Subject Classification: A (main), 9C, 9C, 9C8 Key Words and Phrases: fractional calculus, fractional-order systems, fractional-order controllers, integer-order approximations. Introduction At least since the sixties some researchers have been interested in obtaining approximated integer order models for fractional order systems, or finite dimensional models for infinite dimensional systems. Most of these researchers worked in electrochemistry where the problem is (roughly speaking) to build an equivalent electrical circuit for processes in which diffusion is present. Even earlier, other authors Partially supported by FEDER Research Grant IFD9--C- and by VEGA Reserach Grant /98/.
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu used fractional-order models for modeling physical systems, in which the presence of viscoelasticity was important, or in the characterization of certain class of real materials. However, applications of fractional-order models in control theory are relatively new comparing to mentioned applications. As far as the authors of this paper know, except in old papers of Carlson and Halijak [, ], only in the last two decades the possibility of using fractional order controller has been considered (see, for example, [,,,,, 8,, 9,,,, 8]). In all the cases, the important point for the purposes of this paper, is that the equivalent circuit or the fractional-order controller in its practically realizable form is a finite dimensional integer-order system resulting from the approximation of a fractional-order or infinite-dimensional system. In this paper the Riemann-Liouville definition of fractional integration and differentiation is used [,,, ]. For the case of < α < and f(t) being a causal function of t, that is, f(t) = for t <, the fractional integral is defined as: D α f(t) = Γ(α) t and the expression for the fractional order derivative is: D α d f(t) = Γ( α) dt (t τ) α f(τ) dτ, () t (t τ) α f(τ) dτ () Of particular interest for the purposes of this paper are the expressions of the former operators in the Laplace domain. The Laplace transform of the Riemann- Liouville fractional integral is: { D α f (t) } = s α F, () and for the fractional derivative of order < α < it is: {D α f (t)} = s α F [ D α f (t) ] t= () A fractional-order control system can be described by a fractional differential equation of the form [, ]: a n D α n y(t) + a n D α n y(t) +... + a D α y(t) = b m D β m u(t) + b m D β m u(t) +... + b D β u(t) or by a continuous transfer function of the form [, ]: G = b ms β m + b m s β m +... + b s β a n s α n + an s α n +... + a s α ()
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS For obtaining discrete models of fractional order systems, it is necessary to use discrete approximations of the fractional integrals and derivatives. By doing so in the expression given above for G a general expression for the discrete transfer function of the fractional system, G(z), can be obtained, of the form: G(z) = b ( ( m w z )) β m ( ( + b m w z )) β m ( ( +... + b w z )) β a n (w (z )) α n + a n (w (z )) α n +... + a (w (z )) α () where ( w ( z )) denotes the discrete equivalent of the Laplace operator s, expressed as a function of the complex variable z or the shift operator z. As can be seen in the former expressions, a fractional-order system has an irrational continuous transfer function in the Laplace domain or, in the Z domain, an infinite dimensional discrete transfer function. In other words, a fractionalorder system has an unlimited memory, being the integer order systems particular cases of this general case in which the memory is limited. It is obvious that only in the case of integer order it is possible to realize a transfer function exactly by using conventional lumped elements (resistances, inductances, and capacitors, in the case of analog realizations), or procedures (finite order difference equations or digital filters in the case of discrete realizations). Because of this, and taking into account that the final step for applying a fractional controller demands a realizable form of it, in this paper some continuous and discrete integer-order approximations of fractional-order operators are considered.. Integer-Order Continuous Models of Fractional Order Systems The problem of obtaining a continuous realizable model for a fractional order controller can be viewed as a problem of obtaining a rational approximation of the irrational transfer function, modeling the fractional controller. Among other mathematical methods, two of them are particularly interesting for this purpose, from a control theory point of view: the continued fraction expansion method used for evaluation of functions, and the rational approximation method used in interpolation of functions. On the other hand, the use of frequency identification or curve fitting methods for obtaining rational approximations to the irrational frequency responses, characterizing fractional-order systems, is proposed. In this section, the general form of the methods and some specially interesting applications of them are described.
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu Approximations using continued fraction expansions and interpolation techniques It is well known that the continued fraction expansions (CFE) is a method for evaluation of functions, that frequently converges much more rapidly than power series expansions, and converges in a much larger domain in the complex plane []. The result of such approximation for an irrational function, G, can be expressed in the form: b G a + a + = a + b a + b a + b a +... b a + b a +... () where a i s and b is are rational functions of the variable s, or are constant. The application of the method yields a rational function, Ĝ, which is an approximation of the irrational function G. On the other hand, for interpolation purposes, rational functions are sometimes superior to polynomials. This is, roughly speaking, due to their ability to model functions with poles. (As it can be seen later, branch points can be considered as accumulations of interlaced poles and zeros). These techniques are based on the approximations of an irrational function, G, by a rational function defined by the quotient of two polynomials in the variable s: G R i(i+)...(i+m) = P µ Q ν = p + p s +... + p µ s µ q + q s +... + q ν s ν (8) m + = µ + ν + passing through the points (s i, G(s i )),..., (s i+m, G(s i+m )). General CFE method for approximation of fractional integro-differential operators. In general [], a rational approximation of the function G = s α, < α < (the fractional integral operator in the Laplace domain) can be obtained by performing the CFE of the functions: G h = G l = ( + st ) α (9) ( + α () s) where G h is the approximation for high frequencies (ωt >> ), and G l the approximation for low frequencies (ω << ).
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS Carlson s method. The method proposed by Carlson in [], derived from a regular Newton process used for iterative approximation of the α-th root, can be considered as belonging to this group. The starting point of the method is the statement of the following relationships: (H) /α (G) = ; H = (G) α () Defining α = /q, m = q/, in each iteration, starting from the initial value H =, an approximated rational function is obtained in the form: H i = H i (q m) (H i ) + (q + m)g (q + m) (H i ) + (q m)g () Matsuda s method. The method suggested in [] is based on the approximation of an irrational function by a rational one, obtained by CFE and fitting the original function in a set of logarithmically spaced points. Assuming that the selected points are s k, k =,,,..., the approximation takes on the form: H = a + s s a + s s a + s s a +, () where a i = v i (s i ), v = H, v i+ = s s i v i a i () Approximations using curve fitting or identification techniques In general, any available method for frequency domain identification can be applied in order to obtain a rational function, whose frequency response fits the frequency response of the original irrational transfer function. For example, this may be minimization of the cost function of the ISE form, that is J = W (ω) G(ω) Ĝ(ω) dω where W (ω) is a weighting function, G(ω) is the original frequency response, and Ĝ(ω) is the frequency response of the approximated rational function. Oustaloup s method. The method [, 8, ] is based on the approximation of a function of the form: H = s µ, µ R + ()
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu by a rational function: N Ĥ = C k= N using the following set of synthesis formulas: + s/ω k + s/ω k () ω = α. ω u ; ω = α. ω u ; ω k+ ω k = η > ; ω k ω k ω k+ ω k = ω k+ ω k = αη > ; () = α > ; N = log (ω N /ω ) ; µ = log α log (αη) log (αη) with ω u being the unit gain frequency and the central frequency of a band of frequencies geometrically distributed around it. That is, ω u = ω h ω b, ω h, ω b are the high and low transitional frequencies. Chareff s method. This method, proposed in [], which is very close to Oustaloup s method, is based on approximation of a function of the form H = by a quotient of polynomials in s in a factorized form: ) ( + s zi ( + s p T ) α (8) n Ĥ = i= n ) (9) ( + s pi i= where the coefficients are computed for obtaining a maximum deviation from the original magnitude response in the frequency domain of y db. Defining a = y/( α), b = y/α, ab = y/α( α) () the poles and zeros of the approximated rational function are obtained by applying the following formulae: p = p T b, pi = p (ab) i, z i = ap (ab) i () The number of poles and zeros is related to the desired bandwidth and the error criteria used by the expression: ( ) N = log ωmáx p + () log (ab)
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS. Discrete Models In general, if a function f(t) is approximated by a grid function, f(nh), where h the grid size, the approximation for its fractional derivative of order α can be expressed as []: y h (nh) = h α ( ω ( ζ )) α fh (nh) () where ζ is the shift operator, and ω ( ζ ) is a generating function. This generating function and its expansion determine both the form of the approximation and its coefficients. It is worth mentioning that, in general, the case of controller realization is not equivalent to the cases of simulation or numerical evaluation of the fractional integral and differential operators. In the case of controller realization it is necessary to take into account some important considerations. First of all, the value of h, the step when dealing with numerical evaluation, is the value of the sample period T, and it is limited by the characteristics of the microprocessor-based system, used for the controller implementation, in two ways: (i) each microprocessor-based system has its own minimum value for the sample period, and (ii) it is necessary to perform all the computations required by the control law between two samples. Due to this last reason, it is very important to obtain good approximations with a minimal set of parameters. On the other hand, when the number of parameters in the approximation increases, it increases the amount of the required memory too. It is also important to have discrete equivalents or approximations with poles and zeros, that is, in a rational form. In the following, the notation normally used in control theory is adopted, that is: T, the sample period, is used instead of h, and z, the complex variable resulting from the application of the Z transform to the functions y(nt ), f(nt ) considered as sequences, is used instead of ζ. Discrete Approximations using Numerical Integration and Power Series Expansion Using the generating function corresponding to the backward fractional difference rule, ω ( z ) = ( z ), and performing the power series expansion (PSE) of ( z ) α, the Grünwald Letnikov formula for the fractional derivative of order α is obtained: ( ) α α T f(nt ) = T α ( ) k f((n k)t ) () k k=
8 B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu Performing the PSE of the function ( z ) α leads to the formula given by Lubich for the fractional integral of order α []: α T f(nt ) = T α ( ) α ( ) k f((n k)t ) () k k= In any case, the resulting transfer function, approximating the fractional-order operators, can be obtained by applying the relationship: Y (z) = T α PSE { ( z ) ±α } F (z) () where T is the sample period, Y (z) is the Z transform of the output sequence y(nt ), F (z) is the Z transform of the input sequence f(nt ), and PSE{u} denotes the expression, which results from the power series expansion of the function u. Doing so gives: D ±α (z) = Y (z) { ( F (z) = T α PSE z ) } ±α () where D ±α (z) denotes the discrete equivalent of the fractional-order operator, considered as processes. Another possibility for the approximation is the use of the trapezoidal rule, that is, the use of the generating function ω(z ) = z + z (8) It is known that the forward difference rule is not suitable for applications to causal problems [, ]. It should be mentioned that, at least for control purposes, it is not very important to have a closed-form formula for the coefficients, because they are usually pre-computed and stored in the memory of the microprocessor. In such a case, the most important is to have a limited number of coefficients because of the limited available memory of the microprocessor system. Discrete approximations using numerical integration and continued fraction expansion The approximations, considered in the previous section, lead to discrete transfer functions in the form of polynomials, and this is not convenient, at least from the control point of view. On the other hand, it can be recalled that the CFE leads to approximations in rational form, and often converges much more rapidly than PSE and has a wider domain of convergence in the complex plane, and,
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS 9 consequently, a smaller set of coefficients will be necessary for obtaining a good approximation. In view of these reasons, a method for obtaining discrete equivalents of the fractional-order operators, which combines the well known advantages of the trapezoidal rule in the control theory and the advantages of the CFE, is proposed here. The method implies: the use of the generating function ω(z ) = z + z where z is the complex variable, and z is the shifting operator, and the continued fraction expansion (CFE) of ( ω(z ) ) ±α = ( ) ±α z + z for obtaining the coefficients and the form of the approximation. The resulting discrete transfer function, approximating fractional-order operators, can be expressed as: { ( D ±α (z) = Y (z) F (z) = T α CFE ) ±α } z + z = T α P p(z ) Q q (z ) where T is the sample period, CFE{u} denotes the function resulting from applying the continued fraction expansion to the function u, Y (z) is the Z transform of the output sequence y(nt ), F (z) is the Z transform of the input sequence f(nt ), p and q are the orders of the approximation, and P and Q are polynomials of degrees p and q, correspondingly, in the variable z. p,q (9) Other approximations In the paper [9] some others continuous approximations have been studied, which, in fact, are particular cases of the methods considered here. Some methods are not considered in the present paper, because they can be used only for particular values of fractional order (e.g., [] for α =.). On the other hand, it is necessary to mention the diffusive realizations proposed in [] and [8]. While the starting point in these works is quite different
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu from the starting point of the methods considered in the present paper, the resulting approximations can be viewed, in the Laplace domain, as rational approximations of the fractional-order operators. Furthermore, these approximations exhibit a common feature, which we observe in all good rational approximations: they have poles and zeros interlaced on the negative real axis of the s plane, and the distance between successive poles and zeros decreases as the approximation is improved by increasing the degree of the numerator and denominator polynomials. Probably, this fact was noted for the first time in [], where the following idea appeared: a dense interlacing of simple poles and zeros along a line in the s plane is, in some way, equivalent to a branch cut; and s α, < α <, viewed as an operator, has a branch cut along the negative real axis for arguments of s on ( π, π) but is otherwise free of poles and zeros.. Illustrative Example: Fractional Integrator of Order α =. For comparing the methods, the approximations of the fractional integrator of order., and the corresponding step and frequency responses have been obtained. For simplicity, the approximations obtained are (except for the one derived from Lubich s formula) of low order ( or depending on the particular requirements of the methods), and necessary scale factors have been applied in order to have the same central frequency and db at ω = rad/s. Approximated rational functions General CFE method. Performing the CFE of the function (9), with T =, α =., we obtain: H =.s +.s +.8s +.s +.899 s +.s +.8s +.s +.8 Performing the CFE of the function (), with T =, α =., we obtain: H = s + s +.s +.8s +. 9s + s +.s +.s +. Carlson s method. Starting from: H = ( s ) /, H =, after two iterations we obtain: H = s + s + s + 8s + 9 9s + 8s + s + s +
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS Matsuda s method. With G = ( s ) /, finitial =, f final =, f k ={,.8,.,.,,.8,.,., }, we obtain: H =.89s +.8s +.8s +.99s + s + s +.8s +.8s +.8 Least-squares method. Using the MATLAB invfreqs function we obtain: H =.99s +.99s +.s +.9s +.89 s + 8.s +.s +.9s +.8 Using a method similar to that proposed in [9], we obtain: H =.s +.s +.s +.s +.9 s + 8.s + 9.s +.s +. Chareff s method. For y = db, p T =, ω [, ], the obtained approximation is: H =.s +.8s +.s + 9.9s +.998 s + 9.8s +.8s +.8s +.9s +. Oustaloup s method. Using the Oustaloup s method with: ω h =, ω b =, from which we have α = η =.9, the obtained approximation is: H 8 = s +.9s + 8.s + 8s + 98.s + s + 98.s + 8s + 8.s +.9s + Discrete approximation using backward rule and PSE. This approximation can be viewed as the Z version of Lubich s formula with the necessary use of the short-memory principle []. With T =. and coefficients, the resulting discrete transfer function is: ( ) H 9 (z) =.. z ( ) k. z k k k= Discrete approximation using backward rule and CFE. With T =. we obtain: H (z) =.z.z +.9z.9z +. z.z +.8z.88z +.
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu Phase (deg); Magnitude (db) H Phase (deg); Magnitude (db) H Phase (deg); Magnitude (db) H Phase (deg); Magnitude (db) H Figure : Bode plots of H, H, H, and H. Phase (deg); Magnitude (db) H Phase (deg); Magnitude (db) H Phase (deg); Magnitude (db) 8 H Phase (deg); Magnitude (db) H 8 Figure : Bode plots of H, H, H, and H 8.
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS H H H H Figure : Step responses of H, H, H, and H. H H H H 8 Figure : Step responses of H, H, H, and H 8.
B. M. Vinagre, I. Podlubny, A. Hernández, V. Feliu Phase(deg); Magnitude (db) H 9 (z) H 9 (z) Time (sec) Phase(deg); Magnitude (db) H (z) H (z) Phase(deg); Magnitude (db) H (z) H (z) Figure : Bode plots and step responses of H 9 (z), H (z), H (z). Discrete approximation using trapezoidal rule and CFE. With T =. we obtain: H (z) =.z.9z +.88z.9z. z.8z +.9z.9z +.9 The figures - shows the step responses, compared with the exact step response, and the frequency responses (Bode plots) of the obtained approximations.. Controller Realization In general, there are two possibilities for realizing a controller: a hardware realization based on the use of a physical device, or a software (or digital) realization based on a program, which will run on a computer or microprocessor. In electronics, hardware realizations imply the use of electronic devices or circuits, implementing the required function as an admittance or impedance function. Analog realizations. For hardware electronic realizations, the starting point is the admittance or impedance function. For realizing such functions, at least two
APPROXIMATIONS OF FRACTIONAL ORDER OPERATORS ways can be used: producing a microelectronic specific device that, for construction, has the required admittance or impedance (see, for example, []), or realizing an approximate rational function by using a finite lumped-element network, in a ladder, tree, cascaded, or lattice topology. Since this paper deals with rational approximations of the fractional-order operators, only the last way is discussed below. In order to be realizable by a finite lumped-element network, the rational approximation must be a positive real rational function. We see that the continuous approximations given in Section, fulfill this requirement for being rational functions with poles and zeros interlaced on the negative real axis of the s plane. In view of this, they can be realized as the driving point impedance or admittance of RL or RC networks, that is, as passive RL or RC filters. Furthermore, these realizations can be complemented by using active electronic devices, such as operational amplifiers. That is, a more flexible way for realizing the required functions can be obtained by considering the possibility of using active filters. In laboratory conditions, we have successfully tested an active filter, which realizes the fractional integrator of order. by using a lattice RC network, obtained from a former version of the Carlson s method []. Digital realizations. For digital realizations, a finite difference equation is needed. This equation can be obtained by numerical approximation of by performing the inverse Z transform of a discrete transfer function. For this, the discrete approximations given in Section or discrete equivalents of the continuous approximations can be used. Furthermore, by considering the system described by a discrete transfer function as a IIR digital filter, useful considerations well known in digital filter realization can be taken into account in order to choose a best structure for realizing a discrete approximation.. Conclusion In this paper some approximations of fractional-order operators have been considered, in order to use them for implementing fractional-order controllers. These approximations have been compared, both in frequency domain and in time domain, for a fractional integrator of order α =.. From the obtained results a general conclusion can be made: both in continuous and discrete domains it is possible to use rational approximations for these operators in order to have suitable realizable forms in control applications (with some limitation regarding the frequency bandwidth or the time interval, correspondingly). Furthermore, it has been shown that very good continuous approximations can be obtained by using the methods proposed by Carlson, Oustaloup, Matsuda, and Chareff, or least-square methods for fitting the original frequency response. For
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