Gs () = 1 RC s 1 + a 6 ■ a , where Ca is the capacitance in Fs1 a@ and a1 1. The IR of this system has only fractional component. It is interesting to note that when a 1= the situation is reversed. In figure 1 we depict the step responses of a RC circuit with . RC a = 1s - a B. Stability Consider a given term of (31) corresponding to one pseudo-pole p: gt pe t up p () = 1 11 pt 1/a /aa + sin() r -+ ar #3 a 2 cos ar () f() 0 22eu t f u - d ·( ). ut We can extract some conclusions [81]: 1) The fractional part is always bounded, for bounded. Therefore, () #3 0 22eu A t u a upcos ar 2 a-+p - 120 (40) utd t , This term exists whenever a ! 1 and does not contribute to unstability. 2) As mentioned previously, the integer part only exists if - ra 1 arg p # ra. In this case, we have () ■ ■ three situations [81]: (/ ) arg p arg p () 1 2ar - the exponential increases without bound - unstable system; (/ ) () 2 2ar - the exponential decreases to zero - stable system; Step Response 1 0.8 0.6 0.4 0.2 C. General Non Commensurate Case The non commensurate case cannot be dealt as we discussed above, unless we know the pseudo-poles. In this case, we can use (31). In the general case it is possible to obtain the IR in the form of a fractional McLaurin series through the recursive application of the general initial value theorem [86], [87]. This theorem relates the asymptotic behavior of a causal signal, (),gt as t 0 " Gg tLv = + to the asymptotic behavior of its LT, () () v = Re s " 3. Theorem III.1 (The initial-value theorem). Assume a = 0.2 a = 0.4 a = 0.6 a = 0.8 a = 1 02468 10 12 14 16 18 20 t Figure 1. Step responses of the RC circuit (example III.1) for ., a== (from below). 02kk ,, 44 12 5g IEEE CIRCUITS AND SYSTEMS MAGAZINE that g(t) is a causal signal such that in some neighborhood of the origin is a regular distribution corresponding to an integrable function and its LT is G(s) with region of convergence defined by () a real number b 2 1Re s 2 0. Also, assume that there is such that li ()+mgt tb t 0 " finite complex value. Then () lim+ " t 0 gt t b = lim v " 3 vv C b+ 1 () () G b + 1 . (41) SECOND QUARTER 2022 exists and is a [( )], as a (39) () =- the exponential oscillates sinusoidally - critically stable system. arg p ar 2(/ ) The above considerations allow us to conclude that the behavior of stable systems can be integer, fractional, or mixed: ■ Classical integer order systems have impulse responses corresponding to linear combinations of exponentials that, in general, go to zero very fast. They are short memory systems. ■ In fractional systems without poles, the exponential component disappears. These are long memory systems. All pseudo-poles have arguments with absolute values greater than /, ra where a11 . ■ Mixed systems have both components. Some pseudo-poles have arguments with absolute values larger than /.2ra ^h Remark III.5. The three rules above are usually known a 2 0 and any p C! . In fact, it is a simple matter to verify that function /( ) uu cospp222ar aa-+ arg p^h is as Matignon's theorem, that can be put in the following way: TF (10) is stable iff all pseudo-poles pk ^h verify () k 2ar/2 . The above procedure for studying the stability demands knowing the pseudo-poles. In the integer order case, there are several criteria to evaluate the stability of a given linear system without the knowledge of the poles. One of the most important is the RouthHurwitz criterion that gives information on the number of poles on the right hand half complex plane [83]. The generalization of this criterion for the fractional case was proposed in [84] and is very similar to the integer order case [23]. A very interesting alternative is the Mikhailov criterion [85] that is formulated in the frequency domain. re(t)

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