**3. Variable, fractional-order PID controller synthesis**

In the synthesis of the classical PID controller there are three parameters to evaluate. Namely, *K*,*KI*,*KD* known as the proportional, integral and differential gains. In the fractional-order PID controllers there are two additional parameters: the differentiation order *ν*ð Þ *kh* ∈ <sup>þ</sup> and the integration one �*μ*ð Þ *kh* ∈ þ. In the variable, fractionalorder PID controller the mentioned orders are generalized to functions. This means that there are three constant coefficients and two discrete variable functions to find

$$K\_P, K\_I, K\_D, \nu(kh), \mu(kh) \tag{29}$$

In the rejection of the external disturbation one can assume that **r**ð Þ¼ *kh* **0** so Eq. (29) simplifies to

$$\mathbf{e}(kh) = -[\mathbf{1}\_k + \mathbf{G}\_o(kh)\mathbf{C}(kh)\mathbf{H}(kh)]^{-1}\mathbf{H}(kh)\mathbf{d}(kh) \tag{30}$$

Usually the sensor matrix **H**ð Þ *kh* is treated as constant, by assumption that sensors do not introduce its own dynamics to the system. Hence, **H**ð Þ¼ *kh* **H** ¼ const. It may be assumed that **H** ¼ *h*0**1***<sup>k</sup>* or further, for *h*<sup>0</sup> ¼ 1, formula (30) takes a form

$$\mathbf{e}(kh) = -\left[\mathbf{1}\_k + \mathbf{G}\_o(kh)\mathbf{C}(kh)\right]^{-1}\mathbf{d}(kh) \tag{31}$$

The optimal parameters (29) are evaluated due to the assumed optymality criterion. The most popular is so called ISE one (Integral of the Squared Error) or in the discrete-system case: Sum of the Squared Error (SSE).

$$\text{SSE}[K\_P, K\_I, K\_D, \nu(kh), \mu(kh)] = \sum\_{i=0}^{k\_m \text{ax}} e(ih)^2 h = \mathbf{e}(kh)^T \mathbf{e}(kh) h \tag{32}$$

Substitution of (31) into (32) gives

$$\begin{aligned} &\text{SSE}[K\_P, K\_I, K\_D, \nu(kh), \mu(kh)] \\ &= \mathbf{d}(kh)^T [\mathbf{1}\_k + \mathbf{G}\_o(kh)\mathbf{C}(kh)]^{-T} [\mathbf{1}\_k + \mathbf{G}\_o(kh)\mathbf{C}(kh)]^{-1} \mathbf{d}(kh) \end{aligned} \tag{33}$$

In the proposed VFOPID controller synthesis method with partially intuitive and supported by closed-loop systems synthesis experience the classical optimisation due to the performance criterion (32) is performed. The pre-defined differentiation and integration order functions orders are as follows

$$
\boldsymbol{\nu}(kh) \ge \mathbf{0} \tag{34}
$$

and in the convention proposed above as

which may be expressed as

**Figure 1.** *Closed-loop system.*

**e**ð Þ*k* , respectively. Then, denoting.

tion used makes it possible.

lowing relations

where

**6**

• **r**ð Þ *kh* - a reference signal vector,

• **d**ð Þ *kh* - an external disturbance signal vector,

**<sup>u</sup>**ð Þ¼ *kh KP***1***k***e**ð Þþ *kh KD*0**G**½ � *<sup>ν</sup>C*ð Þ*<sup>k</sup>*

**<sup>u</sup>**ð Þ¼ *kh KP***1***<sup>k</sup>* <sup>þ</sup> *KD*0**G**½ � *<sup>ν</sup>C*ð Þ*<sup>k</sup>*

**<sup>C</sup>**ð Þ¼ *kh KP***1***<sup>k</sup>* <sup>þ</sup> *KD*0**G**½ � *<sup>ν</sup>C*ð Þ*<sup>k</sup>*

one gets a VFOPID controller transfer function-like description

To simplify the description one assumes a sensor matrix as

**<sup>y</sup>**ð Þ¼ *kh* ½ � **<sup>1</sup>***<sup>k</sup>* <sup>þ</sup> **<sup>G</sup>***o*ð Þ *kh* **<sup>C</sup>**ð Þ *kh* **<sup>H</sup>**ð Þ *kh* �<sup>1</sup>

þ½ � **<sup>1</sup>***<sup>k</sup>* <sup>þ</sup> **<sup>G</sup>***o*ð Þ *kh* **<sup>C</sup>**ð Þ *kh* **<sup>H</sup>**ð Þ *kh* �<sup>1</sup>

*<sup>k</sup>* **<sup>e</sup>**ð Þþ *kh KI*0**G**½ � �*μC*ð Þ*<sup>k</sup>*

*<sup>k</sup>* <sup>þ</sup> *KI*0**G**½ � �*μC*ð Þ*<sup>k</sup>*

**u**ð Þ¼ *kh* **C**ð Þ *kh* **e**ð Þ *kh* (25)

**H**ð Þ¼ *kh* **1***<sup>k</sup>* (26)

**G***o*ð Þ *kh* **C**ð Þ *kh* **r**ð Þ *kh*

**d**ð Þ *kh*

*<sup>k</sup>* <sup>þ</sup> *KI*0**G**½ � �*μC*ð Þ*<sup>k</sup> k*

h i

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

where *νC*ð Þ*k* , *μC*ð Þ*k* ≥0 and controlling and error signals are denoted as **u**ð Þ*k* and

*Remark 2.2*. The plant may be described by classical integer order, fractional or even variable, fractional - order difference equations. The matrix - vector descrip-

The closed-loop system is presented in **Figure 1** from which one gets the fol-

*<sup>k</sup>* **e**ð Þ *kh* (22)

**e**ð Þ *kh* (23)

*<sup>k</sup>* (24)

(27)

*Control Based on PID Framework - The Mutual Promotion of Control and Identification…*

$$\nu(kh) = \begin{cases} \nu\_1(kh) & \text{for} \\ \nu\_2(kh) & \text{for} \\ & \vdots \\ \nu\_N(kh) & \text{for} \quad k \in [k\_{NN-1}, k\_N N) \\ & \text{for} \\ \mathbf{0} & \text{for} \quad k \in [k\_{NN}, +\infty) \end{cases} \tag{35}$$

and

$$
\mu(kh) \le 0 \tag{36}
$$

The plant is discretized with the sampling time *h* ¼ 0*:*5 and a VFOPID controller

*ν*1ð Þ¼ *kh* 1 for *k* ¼ 0

0 for *k*∈½ Þ 1, þ∞

�1 for *k*∈ ð Þ 10, þ∞

(39)

(40)

*ν*ð Þ¼ *kh*

(

*DOI: http://dx.doi.org/10.5772/intechopen.95232*

the optimal parameters are as follows

• *KP* ¼ 1*:*000

• *Ki* ¼ 0*:*514

• *KD* ¼ 0*:*890

−1.5

*VFOPID controller order functions: ν*ð Þ *kh (in black) and μ*ð Þ *kh (in red).*

ν*I*

**Figure 2.**

**9**

(kh)(bk

*−* .), ν

*F*

(kh)(bk

*−* o), μ*I*

(kh)(red

*−* .), μ

*F*

(kh))(red

*−*

o)(red)

−1

−0.5

0

0.5

1

1.5

(

*Variable, Fractional-Order PID Controller Synthesis Novelty Method*

*<sup>μ</sup>*ð Þ¼ *kh <sup>μ</sup>*1ð Þ¼� *kh* <sup>1</sup> <sup>þ</sup> *<sup>d</sup>*1*ed*2ð Þ *kh*�1*<sup>h</sup>* for *<sup>k</sup>* <sup>¼</sup> ½ � 0, 10

and controller gains *KP*, *kI*,*KD* and order function parameters *d*1, *d*2.

Hence, there are 5 parameters to evaluate. Due to the performance index (33)

0 5 10 15 20 25

kh

is applied

$$\mu(kh) = \begin{cases} \mu\_1(kh) & \text{for} \\ \mu\_2(kh) & \text{for} \\ & \vdots \\ \mu\_N(kh) & \text{for} \\ -\mathbf{1} & \text{for} \end{cases} \tag{37}$$

Every function *νi*ð Þ *kh* for *i* ¼ 1, 2, ⋯, *N* and *μi*ð Þ *kh* for *i* ¼ 1, 2, ⋯, *M* is characterized by a sets of parameters *cij* and *dij*, respectively.

In the classical closed-loop system with PID controller there is introduced the integration part preserving the steady - state error signal tending to zero. So, in (38) there is a constant order �1 for *k*≥½ Þ *KMM*, þ∞ .

Now, for initially assumed order functions one applies the following algorithm based on well known Gauss method.


*Remark 3.1*. Algorithm described above can be applied also to the classical discrete PID controller with three parameters.
