Introduction to fractional linear systems. Part 1 : Continuous-time case M.D.Ortigueira Abstract: In the paper, the class of continuous-time linear systems is enlarged with the inclusion of fractional linear systems. These are systems described by fractional differential equations. It is shown how to compute the impulse, step, and frequency responses from the transfer function. The theory is supported by definitions of fractional derivative and integral, generalisations of the usual. An introduction to fractal signals as outputs of fractional differintegrators is presented. It is shown how to define a stationary fractal. 1 Introduction Fractional calculus is nearly 300 years old. In fact, in a letter to Leibnitz, Bernoulli put him a question about the meaning of a non-integer derivative order. It was the beginning of a discussion on the theme that involved other mathematicians such as L‘ Hhpital, Euler and Fourier [ 1-31, However we can trace the beginning of the fractional calculus in the works of Liouville and Abel. Abel solved an integral equation representing an operation of fractional integration. Liouville made several attempts and presented a formula for fractional integration where T(p) is the gamma function. With the term (- 1)p omitted, we call this the Liouville fractional integral. Liouville developed ideas on this theme and presented a generalisation of the notion of incremental ratio to define a fractional derivative. This idea was discussed again by Grunwald (1867) and Letnikov (1 868). Riemann reached an expression similar to eqn. 1 for the fractional integral. Holmgren (1 86Y66 and Letnikov 1868/74) discussed this problem when looking for the solution of differential equations, putting in a correct statement thc fractional differentiation as inverse operation of the fractional inte- gration. Hadamard proposed a method of fractional differ- entiation based on the differentiation of the Taylor’s series associated with the function. Weyl (1917) defined a frac- tional integration suitable to periodic functions, and Marchaud (1 927) presented a form of differentiation based on finite differences. More recently, the unified formulation of integration and differentiation (differ- integration) based on Cauchy’s integral [4-61 has gained great popularity. C IEE 2000 IEE Proceedings online no. 20000272 DOI: 10.1049/ip-vis:20000272 Paper received 9th September 1999 The author is with the lnstituto Superior Tecnico and UNINOVA, Ca~npus da FCT da UNL, Quinta da Tort-e 2825 - 114 Monte da Caparica, Portugal and also with INESC, K. Alves Redol, 9, 2 , 1000 Lisbon, Portugal 62 Applications to physics and engineering are not recent: application to viscosity dates back to the 1930s [7]. The work of Mandelbrot [8, 91 in the field of fractals had great influence and attracted attention to fractional calculus. During the last 20 years, application domains of fractional calculus have increased significantly: seismic analysis [7], dynamics of motor and premotor neurones of the oculo- motor systems [lo], viscous damping [ 1 1, 121, electric fractal networks [13], l/f noise [14, 151: fractional order sinusoidal oscillators [ 161, and, more recently, control [ 17- 191 and robotics [20]. However, there is no publication with a coherent presentation of fractional linear system theory. Most elementary books on signals and systems consider only the integer derivative order case and treat the corresponding systems, studying their impulse, stcp and frequency responses and their transfer function. It is not such a simple matter, if one substitutes fractional deriva- tives for the common derivatives. The objective of this paper is to treat the fractional continuous-time linear system as is done with usual systems. Attempts have been made to create a formal framework for the study of fractional linear systems, but without the desired generality, coherence and usefulness of the final results [2, 11, 21-23]. To our knowledge, the approach we propose here is original, although it has an ‘already seen’ character. This is because we are dealing with very well known concepts. We merely generalise them to the fractional case. We intend to make a first contribution for a correct understanding of some experimental results [ 14, 22, 231 and to create a new way into modelling, simulation, and estimation in real fractional systems. To begin a study of fractional systems, we need to define fractional derivative. The most obvious approach to the fractional derivative is the Griinwald-Letnikov method, a generalisation of the usual definition based on the incre- mental ratio [ 1-31. However, it is very hard to manipulate and obtain new results. This motivates us to adopt a definition based on the Laplace transforms. Consider first the Laplace transform (LT) case. Essen- tially, we are looking for the Laplace inverse transform of 3, d(’)(t), with 2 being any positive real number. General- ising the well known property of the Laplace transform, the convolution of a Laplace transformable signal .Y( t) with a(’)(t) has s’X(s) as the LT and is defined as the 2 order derivative of x(t). If r < 0, the definition remains valid and we are performing a (fractional) integration. For this IEE Pi.oc.- ICs. huge Sigiinl Procex, Ell. 147, .Yo. I, Fehi.ilni.1, 2000