**3.1 Fractional-order PID (FOPID) controller**

The PIDs are linear and in particular symmetric and they have difficulties in the presence of non-linearities. We can solve this problem by using a fractional-order PID (FOPID) controller. A FOPID controller is presented below [7-9]:

$$\text{G(s)} = \text{K}\_P + \frac{\text{K}\_I}{\text{S}^\alpha} + \text{K}\_D \text{S}^\beta = \frac{\text{K}\_P \text{S}^\alpha + \text{K}\_I + \text{K}\_D \text{S}^\alpha \text{S}^\beta}{\text{S}^\alpha} \tag{14}$$

Figure.7 describes the possibilities a FOPID for the different controllers.

Fig. 7. Generalization of the FOPID controller: from point to plane.

There are several methods to calculate the fractional order derivative and integrator of a fractional order PID controller. For this purpose we present a real order calculus according to the Riemann-Liouville definition.

PID Control Theory 223

0 0 ( , ) ( , ), min : *<sup>n</sup> at n n at*

 

*k*

(21)

(22)

*<sup>g</sup> <sup>d</sup> <sup>f</sup> <sup>t</sup> <sup>g</sup> <sup>t</sup>* (23)

*ssa*

 

*s a* (26)

(24)

0 ( ) (0, ) ( 1)

<sup>0</sup> [ ( ) ( ) ] [ ( )] [ ( )] *<sup>t</sup>*

1 1 <sup>1</sup> [ ( , ] [ ][ ] ( ) ( )

<sup>1</sup> [ ( , )] [ ( , ] () () *n n*

<sup>1</sup> [ (0, )] [ ] *at Ea e <sup>t</sup>*

Approximation of Fractional Order Derivative and Integral There are many different ways of finding such approximations but unfortunately it is not possible to say that one of them is the best, because even though some are better than others in regard to certain characteristics, the relative merits of each approximation depend on the differentiation order, on whether one is more interested in an accurate frequency behaviour or in accurate time responses, on how large admissible transfer functions may be, and other factors such like these. For that reason this section shall present several alternatives and conclude with a comparison of

 

*dt s sa s sa* 

(25)

*k*

*ssa* 

*k at E a* *<sup>d</sup> De D D e En a E a n k Nk*

*E a <sup>t</sup>*

*f t* 

*at Ea t e <sup>t</sup>*

For negative orders, applying the convolution theorem (23) and (19) we obtain

 

*t t n n d s E a En a*

For positive orders, applying the Laplace transform and we have:

*t*

For positive orders, the same definition gives

 

which is the series development of eat. Finally, the Laplace transform of Et is:

The Convolution theorem:

And when ν = 0 , we find:

them.

**3.3 Approximation of fractional order** 

If *v* = 0 , we have:

*tt t t n*

<sup>1</sup> [ ( , )] ( )

*dt*

#### **3.2 Fractional calculus**

Fractional calculus is a branch of mathematics dealing with real number powers of differential or integral operators. It generalizes the common concepts of derivative and integral. Among all the different definitions, the definition which has been proposed by Riemann and Liouville is the most usual one [9,10]. The definition is as follows:

$$\,\_{\alpha}D\_{\chi}^{-n}f(\mathbf{x}) = \int\_{c}^{\chi} \frac{(\chi - t)^{n-1}}{(n-1)!} f(t)dt, \qquad n \in \mathfrak{R} \tag{15}$$

The general definition of D is given by (2):

$$\begin{aligned} \,\_{c}D\_{\mathbf{x}}\,^{v}f(\mathbf{x}) = \begin{cases} \,\_{c}^{\mathbf{x}} (\mathbf{x}-\mathbf{t})^{-v-1} \\ \,\_{c}^{\mathbf{f}} \,\_{c}^{\mathbf{f}}f(\mathbf{x}) = \begin{cases} f(\mathbf{x}) \, dt \\ f(\mathbf{x}) \end{cases} & \quad \mathrm{if}\,\mathbf{v} = \mathbf{0} \\ \,\_{c}^{\mathbf{u}} \,\_{c}^{\mathbf{u}}D\_{\mathbf{x}}\,^{\mathbf{v}-n}f(\mathbf{x}) \mathbf{I} & \quad \mathrm{if}\,\mathbf{v} > \mathbf{0} \end{cases} \end{aligned} \tag{16}$$

$$n = \min\left\{K \in \mathfrak{R}, K > v\right\},$$

Where Γ(·) is the well-known Euler's gamma function.

Function *Fs s* ( )

$$F(\mathbf{s}) = \mathbf{s}^{\nu} \tag{17}$$

Function (17) is not only the simplest fractional order transfer function hat may appear but is also very important for applications, as shall be seen subsequently. For that reason, we analyse its time and frequency responses.

Time responses of (17)

The derivatives of the exponential function are given by

$$\begin{array}{ll} \text{ao } D^{\nu} \, \_t e^{at} = E\_t(-\nu \, \_t \mathbf{a}) , t > 0 \end{array} \tag{18}$$

For negative orders, from definition (16) we have:

$$\_0D\_t^{-\nu}e^{at} = \frac{1}{\Gamma(\nu)} \int\_0^t (t-\xi)^{\nu-1} e^{a\xi} d\xi, \nu \in \mathfrak{R}^+ \tag{19}$$

By means of the substitution x = t −ξ , in the first place, and of the substitution ax = y, in the second place, we obtain

$$\begin{split} \, \_0D\_t^{-\nu} e^{at} &= -\frac{1}{\Gamma(\nu)} \int\_t^0 \mathbf{x}^{\nu-1} e^{a(t-\mathbf{x})} d\mathbf{x} = \frac{e^{at}}{\Gamma(\nu)} \int\_0^t \mathbf{x}^{\nu-1} e^{-a\mathbf{x}} d\mathbf{x} = \\ \, \_0\frac{e^{at}}{\Gamma(\nu)} \int\_0^{at} \underbrace{\binom{y}{a}}\_0 \mathbf{x}^{\nu-1} e^{-y} \frac{dy}{a} &= \frac{e^{at}}{\Gamma(\nu) a^{\nu}} \int\_0^{at} \mathbf{y}^{\nu-1} e^{-y} d\mathbf{y} = \mathbf{E}\_t(\nu, a) \end{split} \tag{20}$$

Fractional calculus is a branch of mathematics dealing with real number powers of differential or integral operators. It generalizes the common concepts of derivative and integral. Among all the different definitions, the definition which has been proposed by

<sup>1</sup> ( ) ( ) () , ( 1)!

<sup>1</sup> ( ) () , 0 ( )

*x t f t dt ifv*

[ ( )] 0

() () 0

*n K Kv* min ,

Function (17) is not only the simplest fractional order transfer function hat may appear but is also very important for applications, as shall be seen subsequently. For that reason, we

<sup>0</sup> ( , ), 0 *at D e E at t t*

*<sup>t</sup> at <sup>a</sup> D e t ed <sup>t</sup>*

By means of the substitution x = t −ξ , in the first place, and of the substitution ax = y,

<sup>0</sup> 1 1 ( ) <sup>0</sup> <sup>0</sup>

*at at at at y y*

0 0

*a a a*

1 1

 

*e e y dy <sup>e</sup> y e dy E a*

( ) ( )

*at <sup>t</sup> at at x ax <sup>t</sup> <sup>t</sup>*

*<sup>e</sup> D e x e dx x e dx*

( ) (,) ( ) ( )

 

1

 

 

 

(19)

 

*t*

<sup>1</sup> () , ( )

 

<sup>0</sup> <sup>0</sup>

1

 

 

 *D D f x ifv* 

(15)

(16)

(17)

(18)

(20)

*x t D fx f t dt n n*

*x v*

 

*n n c x*

*D fx fx ifv*

Riemann and Liouville is the most usual one [9,10]. The definition is as follows:

*<sup>x</sup> <sup>n</sup> <sup>n</sup> c x c*

*c*

 

**3.2 Fractional calculus** 

The general definition of D is given by (2):

*<sup>v</sup> c x*

Where Γ(·) is the well-known Euler's gamma function.

The derivatives of the exponential function are given by

For negative orders, from definition (16) we have:

Function *Fs s* ( )

analyse its time and frequency responses.

Time responses of (17)

in the second place, we obtain

  For positive orders, the same definition gives

$$\,\_0D\_t^{\ \nu}e^{dt} = \mathbf{D}^{\mu}{}\_0\mathbf{D}^{\nu-n}\_t e^{dt} = \frac{d^n}{dt^n}\mathbf{E}\_t\{n-\nu, a\} = \mathbf{E}\_t\{-\nu, a\}, \nu \in \mathfrak{R}^+ \land n = \min\left\{k \in N : k > \nu\right\}$$

If *v* = 0 , we have:

$$E\_t(0, a) = \sum\_{k=0}^{+\infty} \frac{(at)^k}{\Gamma(k+1)}\tag{21}$$

which is the series development of eat. Finally, the Laplace transform of Et is:

$$\ell[E\_t(\nu, a)] = \frac{1}{s^{\nu}(s - a)}\tag{22}$$

The Convolution theorem:

$$\ell[\int\_0^t f(t-\tau)g(\tau)d\tau] = \ell[f(t)]\ell[g(t)]\tag{23}$$

For negative orders, applying the convolution theorem (23) and (19) we obtain

$$\ell[E\_t(\nu, a] = \frac{1}{\Gamma(\nu)} \ell[t^{\nu - 1}] \ell[e^{at}] = \frac{1}{s^\nu(s - a)} \tag{24}$$

For positive orders, applying the Laplace transform and we have:

$$\ell[\mathcal{E}\_t(-\nu, a)] = \ell[\frac{d^\nu}{dt^\nu}\mathcal{E}\_t(\nu - \nu, a) = \frac{s^\nu}{s^{\nu-\nu}(s-a)} = \frac{1}{s^{-\nu}(s-a)}\tag{25}$$

And when ν = 0 , we find:

$$\ell[E\_t(0, a)] = \ell[e^{at}] = \frac{1}{s - a} \tag{26}$$

#### **3.3 Approximation of fractional order**

Approximation of Fractional Order Derivative and Integral There are many different ways of finding such approximations but unfortunately it is not possible to say that one of them is the best, because even though some are better than others in regard to certain characteristics, the relative merits of each approximation depend on the differentiation order, on whether one is more interested in an accurate frequency behaviour or in accurate time responses, on how large admissible transfer functions may be, and other factors such like these. For that reason this section shall present several alternatives and conclude with a comparison of them.

PID Control Theory 225

poles and zeros is selected at first and the desired performance of this approximation depends on the order N. Simple approximation can be provided with lower order N, but it

, , , [0,1] *v n s ss v n n*

arg ( ) arg( ) arg( )

Now there are several complex numbers *z* with different arguments such that *z* = *jν*; by

arg ( ) /2 *F j* 

*F*( ) 20lo *j* g 20 log ( ) *dB* 

 

 

Thus the Bode and Nichols plots of *F(s)* = *s*ν are those shown in Figure 8 and Figure 9:

*Fj j j*

 

 

 

   

> 

 

 **for real orders** 

 

 

, the role of the zeros and the poles is swapped. The number of

  usually is separated as (34) and only the first term


(34)

(35)

(36)

(37)

For negative values of

*s* not satisfactory. The fractional order

needs to be approximated.

**3.4 Bode and Nichols plots of s<sup>ν</sup>**

The frequency response of *s*

The gain in decibel shall be

Fig. 8. Bode diagrams

can cause ripples in both gain and phase characteristics. When |

is: choosing the one with a lower argument in interval [0; 2π[ , we will obtain:

( )( )

( )

*Fj j Fj j*

Approximations are available both in the s-domain and in the z-domain. The former shall henceforth be called continuous approximations or approximations in the frequency domain; the latter, discrete approximations, or approximations in the time domain.

There are 32 approximation methods for fractional order derivative and integral, we present here Crone approximation method [10, 11].

#### **3.3.1 Crone approximation method**

The Crone methodology provides a continuous approximation, based on a recursive distribution of zeros and poles. Such a distribution, alternating zeros and poles at wellchosen intervals, allows building a transfer function with a gain nearly linear on the logarithm of the frequency and a phase nearly constant being possible for the values of the slope of the gain and of the phase for any value of *ν* [12-14].

The functions we are dealing with in this section provide integer-order frequency-domain approximation of transfer functions involving fractional powers of s.

For the frequency-domain transfer function C(s) which is given by:

$$\mathbf{C}(\mathbf{s}) = \mathbf{K}\mathbf{s}^{\upsilon} \qquad \qquad \nu \in \mathfrak{R} \tag{27}$$

One of the well-known continuous approximation approaches is called Crone. Crone is a French acronym which means 'robust fractional order control'. This approximation implements a recursive distribution of N zeros and N poles leading to a transfer function as (28).

$$C(s) = K \prod\_{n=1}^{N} \frac{1 + \frac{s}{o o\_{2n}}}{1 + \frac{s}{o o\_{pn}}} \tag{28}$$

Where K' is an adjusted gain so that both (26) and (27) have unit gain at 1 rad/s. Zeros and poles have to be found over a frequency domain [ , *l h* ] where the approximation is valid, they are given for a positive v, by (29), (30) and (31).

$$
\alpha\_{z1} = \alpha\_l \sqrt{\eta} \tag{29}
$$

$$
\alpha\_{\rm pnt} = \alpha\_{\rm 2,n-1} \alpha \qquad \qquad n = 1 \ldots N \tag{30}
$$

$$
\alpha\_{2n} = \alpha\_{p, n-1} \eta \qquad \qquad n = 2...N \tag{31}
$$

Where α and η can be calculated thanks to (32) and (33).

$$\alpha = \left(\frac{\alpha\_{\parallel}}{\alpha\_{\parallel}}\right)^{\frac{\upsilon}{N}}\tag{32}$$

$$\eta = \left(\frac{\alpha \eta\_l}{\alpha \eta}\right)^{\frac{1-\upsilon}{N}} \tag{33}$$

Approximations are available both in the s-domain and in the z-domain. The former shall henceforth be called continuous approximations or approximations in the frequency

There are 32 approximation methods for fractional order derivative and integral, we present

The Crone methodology provides a continuous approximation, based on a recursive distribution of zeros and poles. Such a distribution, alternating zeros and poles at wellchosen intervals, allows building a transfer function with a gain nearly linear on the logarithm of the frequency and a phase nearly constant being possible for the values of the

The functions we are dealing with in this section provide integer-order frequency-domain

One of the well-known continuous approximation approaches is called Crone. Crone is a French acronym which means 'robust fractional order control'. This approximation implements a recursive distribution of N zeros and N poles leading to a transfer function

'

*n*

Where K' is an adjusted gain so that both (26) and (27) have unit gain at 1 rad/s. Zeros and

1

1

1 *<sup>N</sup> zn*

, 1 1...

, 1 2...

*v <sup>N</sup> <sup>h</sup> l* 

1 *v <sup>N</sup> <sup>h</sup> l* 

 

  *pn*

*s*

*s* 

> *l h*

(27)

(28)

] where the approximation is valid,

(29)

(32)

(33)

*n N* (30)

*n N* (31)

domain; the latter, discrete approximations, or approximations in the time domain.

here Crone approximation method [10, 11].

slope of the gain and of the phase for any value of *ν* [12-14].

poles have to be found over a frequency domain [ ,

they are given for a positive v, by (29), (30) and (31).

*pn z n* 

*zn p n* 

Where α and η can be calculated thanks to (32) and (33).

approximation of transfer functions involving fractional powers of s. For the frequency-domain transfer function C(s) which is given by:

( ) *<sup>v</sup> C s Ks*

( )

*Cs K*

*z l* <sup>1</sup> 

> 

 

**3.3.1 Crone approximation method** 

as (28).

For negative values of , the role of the zeros and the poles is swapped. The number of poles and zeros is selected at first and the desired performance of this approximation depends on the order N. Simple approximation can be provided with lower order N, but it can cause ripples in both gain and phase characteristics. When | |>1, the approximation is not satisfactory. The fractional order usually is separated as (34) and only the first term *s*needs to be approximated.

$$\mathbf{s}^{\mathcal{V}} = \mathbf{s}^{\rho}\mathbf{s}^{\mathcal{V}}, \quad \mathbf{v} = \mathbf{n} + \rho, \ \mathbf{n} \in \mathfrak{R}, \ \ \rho \in [0, 1] \tag{34}$$

#### **3.4 Bode and Nichols plots of s<sup>ν</sup> for real orders**

The frequency response of *s* is:

$$\begin{aligned} \left| F(jo) = (jo)^{\nu} \\ \left| F(joo) \right| = \left| j^{\nu} o^{\nu} \right| = \left| oo^{\nu} \right| = o^{\nu} \end{aligned} \tag{35}$$
 
$$\begin{aligned} \arg \left[ F(joo) \right] = \arg(j^{\nu} o^{\nu}) = \arg(j^{\nu}) \end{aligned} $$

Now there are several complex numbers *z* with different arguments such that *z* = *jν*; by choosing the one with a lower argument in interval [0; 2π[ , we will obtain:

$$\arg\left[F(j\rho)\right] = \nu \pi \not> 2\tag{36}$$

The gain in decibel shall be

$$\left| F(j\alpha) \right| = \text{20}\log \alpha^{\nu} = \text{20}\nu \log \alpha \qquad \text{(dB)}\tag{37}$$

Thus the Bode and Nichols plots of *F(s)* = *s*ν are those shown in Figure 8 and Figure 9:

Fig. 8. Bode diagrams

PID Control Theory 227

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[2] Barbosa, Ramiro S.; Machado, J. A. Tenreiro; FERREIRA, Isabel M, A fractional calculus

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[4] D. Maiti, A. Acharya M. Chakraborty, A. Konar, R. Janarthanan, "Tuning PID and

[5] Z. Yongpeng , Sh. Leang-San , M. A.Cajetan , and A. Warsame, '' Digital PID controller

[6] Pierre, D.A. and J.W. Pierre, "Digital Controller Design-Alternative Emulation

[7] D. Xue, C. N. Zhao and Y. Q. Chen, "Fractional order PID control of a DC-motor with an

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perspective of PID tuning. In Proceedings of ASME 2003 design engineering technical conferences and Computers and information in engineering conference.

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Table 2. Approximation of 1/ Sv for different v values

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Dec 2004, pp. 483-495.

3187, June 2006.

305-321.

pp. 457-462

**6. References** 

Fig. 9. Nichols diagrams
