The regularized tempered derivative is defined by f Df t = m a ,f () # =ft-3 () ()/xfax dx, C() N () ()() -1 mm m! t mG e mx a-- -1 x -a (115) that generalizes the causal expression (114) to real orders and where N a= 6@ For stability reasons, we con. sider always m ! R .0 + B. On the Tempered LS We introduced in (14) the notion of a tempered LS through its TF. After having the derivatives defined in the previous sub-section we can write the corresponding differential equation. Let x(t) and y(t) be two functions assumed almost everywhere continuous, with bounded variation, and of exponential order. Therefore, they have LT with non empty regions of convergence. We define a tempered fractional LS with input x(t) and output y(t) as the one following the differential equation N 00 k k aD yt = k m a ,f () k where ta ,,f,, ! kN and bk = f01 M R k ,, = 01 M // (116) k== bD xt k c b k k ,f (), k ,, ,, , are real valued constant coefficients. The parameters a k and kb are the derivative orders that, without loss of generality, we assume to form strictly increasing sequences of positive real numbers. The exponential coefficients ,, ,, ,kN m = 01 f k and ,, ,, ,kM are real c = 01 f k numbers. The differential equation (116) is very general in the sense that we can use forward, backward or both types of derivatives. However, for most practical applications, where we deal with causal systems and, therefore, the use of the forward tempered GL or L derivatives is more appropriate. For stability reasons, the parameters ,, ,, ,kN m = 01 f k and ,, ,, ,kM must be positive. c = 01 f k The TF (14) corresponding to (116) poses difficulties for an analytic manipulation. Therefore, just consider the commensurate case defined with ab !a== ,. kk kk N0 Furthermore, this case is only manageable if ,, ,, . mc m== = kk 0 k 123 f The TF becomes: V. Conclusions M Gs () = N or, equivalently () SECOND QUARTER 2022 / / M = 0 % % N Gs K k=1 k=1 6 6 () () sp sz ++m m a a k k @ @ , (118) k=0 k=0 as bs k k () () + + m m , ka (117) ka The fractional continuous-time linear systems were presented. Two classes were introduced, namely, the fractional ARMA and the Tempered systems. For both classes we showed how to handle the standard tools, like impulse response, transfer function, and frequency response. We considered also the stability and the initial-condition problem. Additionally, for backward compatibility with classic systems, suitable fractional derivatives were introduced having IEEE CIRCUITS AND SYSTEMS MAGAZINE 53 This expression shows clearly the presence of two factors with different characteristics described by exponential and fractional power functions [116]. Let us compute the impulse response merely for the lead controller. We can write () Cs = x e a =+ + a s sa + + The inverse LT gives () ct = x e x t that can be written as () ct = CC - () () 0#t aa 1 e aa x a -at ut ue utd f(). - -- () 1 -- -cmau 1 1 x (123) x o a x a `s x a j ^sah-a a where pk and zk 12= f are the pseudo-poles and -zeroes and K0 is a constant. The corresponding differk ,, , ential equation can be written as () N // (119) k 00 aD yt = ka k gt0 == m00 (). k bD xt k ka m If we denote by () the IR of this system, corresponding to m0 0 ,= and use the shift property of the LT, we conclude that the IR of (119) is given by () gt eg t t = -m0 0(). (120) This shows that the tempering procedure leads to an increase in the stability domain of a system. Example IV.1. The fractional lead ()a+ compensator used in Control to increase the phase of a system around a chosen frequency and the lag ()acompensator, used to increase the static gain of a plant, are defined by the TF [23], [114], [115] Cs () = ax + ` x + !a sa sa j ,, ,. + a ! R 21 (121) M . a;;EE (122) - at aa--11 -at CC() () -a f() ) te t a f t ()

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