**4. Numerical example**

One considers a closed-loop system depicted in **Figure 1**. A plant is described by a transfer function

$$G\_o(s) = \frac{b\_0}{s^2 + a\_1s + a\_0} \tag{38}$$

where


*Variable, Fractional-Order PID Controller Synthesis Novelty Method DOI: http://dx.doi.org/10.5772/intechopen.95232*

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

$$\nu(kh) = \begin{cases} \nu\_1(kh) = 1 & \text{for } \qquad k = 0 \\\\ 0 & \text{for } \quad k \in [1, +\infty) \end{cases} \tag{39}$$

$$\mu(kh) = \begin{cases} \mu\_1(kh) = -1 + d\_1 e^{d\_1(kh - 1h)} & \text{for} \quad k = [0, 10] \\\\ -1 & \text{for} \quad k \in (10, +\infty) \end{cases} \tag{40}$$

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) the optimal parameters are as follows

• *KP* ¼ 1*:*000

*ν*ð Þ¼ *kh*

*μ*ð Þ¼ *kh*

ized by a sets of parameters *cij* and *dij*, respectively.

value alongside the first variable (eg. K\_P),

4. If the SSE value is satisfactory stop else return to step 2.

One considers a closed-loop system depicted in **Figure 1**.

*Go*ðÞ¼ *s*

3.Repeat step 2 for the next parameter,

crete PID controller with three parameters.

A plant is described by a transfer function

**4. Numerical example**

where

• *a*<sup>1</sup> ¼ 0*:*5

• *a*<sup>0</sup> ¼ 0*:*1

• *b*<sup>0</sup> ¼ *a*<sup>0</sup>

**8**

there is a constant order �1 for *k*≥½ Þ *KMM*, þ∞ .

based on well known Gauss method.

and

8 >>>>>><

>>>>>>:

8 >>>>>><

>>>>>>:

*ν*1ð Þ *kh* for *k*∈½ Þ 0, *kN*<sup>1</sup> *ν*2ð Þ *kh* for *k*∈½ Þ *kN*1, *kN*<sup>2</sup>

*νN*ð Þ *kh* for *k*∈½ Þ *kNN*�1, *kNN* 0 for *k*∈½ Þ *kNN*, þ∞

*μ*1ð Þ *kh* for *k*∈½ Þ 0, *kM*<sup>1</sup> *μ*2ð Þ *kh* for *k*∈½ Þ *kM*1, *kM*<sup>2</sup>

*μN*ð Þ *kh* for *k*∈½ Þ *kMM*�1, *kMM* �1 for *k*∈½ Þ *kMM*, þ∞

⋮

Every function *νi*ð Þ *kh* for *i* ¼ 1, 2, ⋯, *N* and *μi*ð Þ *kh* for *i* ¼ 1, 2, ⋯, *M* is character-

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)

Now, for initially assumed order functions one applies the following algorithm

2.Applying the classical Gauss algorithm find a minimal SSE performance index

*Remark 3.1*. Algorithm described above can be applied also to the classical dis-

*b*0 *s*<sup>2</sup> þ *a*1*s* þ *a*<sup>0</sup>

1.Chose a starting set of coefficients *KP*, *K* � *I*, *KD*, *c*11, ⋯ and *d*11, ⋯,

*μ*ð Þ *kh* ≤0 (36)

(35)

(37)

(38)

⋮

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


**Figure 2.** *VFOPID controller order functions: ν*ð Þ *kh (in black) and μ*ð Þ *kh (in red).*

0 5 10 15

kh

0 5 10 15

kh

*The closed-loop system response with the VFOPIS (in red) and IOPID controllers (in blue).*

0

0

*The closed-loop controlling signals.*

0.5

u*I*

**Figure 5.**

**11**

(kh)(bk),

 u

*F*

(kh)(rd)

1

1.5

2

2.5

0.2

0.4

y*o*(kh)(bk),

**Figure 4.**

y*cI*(kh)(

b), y*cF*

(kh)(rd)

0.6

0.8

1

1.2

1.4

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

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

**Figure 3.** *VFOPID (in red) and IOPID (in black) controller unit step response.*


The VFOPID controller order functions are plotted in **Figure 2** whereas the PID and VFOPID controllers unite step responses are given in **Figure 3**.

The achieved VFOPID controller synthesis result is compared with the classical discrete-time PID controller optimized due to criterion (30). The optimal parameters are


**Figure 4** contains the closed - loop systems with PID (in blue) and VFOPID (in red) controllers unit step responses. There is included a plant unit step response of the plant (in black.)

In **Figure 5** the controlling signals are presented (PID - in black, VFOPID - in red). The controlling signals have typical shapes: first differentiation action and

*Variable, Fractional-Order PID Controller Synthesis Novelty Method DOI: http://dx.doi.org/10.5772/intechopen.95232*

**Figure 4.** *The closed-loop system response with the VFOPIS (in red) and IOPID controllers (in blue).*

**Figure 5.** *The closed-loop controlling signals.*

• *d*<sup>1</sup> ¼ �0*:*35

**Figure 3.**

0

5

10

15

v

(kh)(black),

 v

*F*

(kh)

*I* 20

25

30

35

40

45

50

• *d*<sup>2</sup> ¼ �0*:*5

parameters are

• *KP* ¼ 1*:*00

• *Ki* ¼ 0*:*81

• *KD* ¼ 0*:*90

the plant (in black.)

**10**

The VFOPID controller order functions are plotted in **Figure 2** whereas the PID

0 5 10 15

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

kh

The achieved VFOPID controller synthesis result is compared with the classical

**Figure 4** contains the closed - loop systems with PID (in blue) and VFOPID (in red) controllers unit step responses. There is included a plant unit step response of

In **Figure 5** the controlling signals are presented (PID - in black, VFOPID - in red). The controlling signals have typical shapes: first differentiation action and

and VFOPID controllers unite step responses are given in **Figure 3**.

*VFOPID (in red) and IOPID (in black) controller unit step response.*

discrete-time PID controller optimized due to criterion (30). The optimal

finally the classical integration preserving zero steady - state closed - loop system error.

*Remark 4.1*. In the Numerical example proposed here the VFOPID and the classical PID controllers maximal control signal values are the same reaching assumed bounding value max ½ � *uI*ð Þ *kl* , max ½ �¼ *uF*ð Þ *kh* 2.

*Remark* 4.2. In the Numerical example

$$\begin{aligned} \text{SSE}[K\_P, K\_I, K\_D, \mathbf{1}, -\mathbf{1})] &= \mathbf{1}.\mathbf{3}\mathbf{3}\mathbf{1}\mathbf{2} \\ \text{SSE}[K\_P, K\_I, K\_D, \nu(kh), \mu(kh)] &= \mathbf{1}.\mathbf{2}\mathbf{8}\mathbf{9}\mathbf{9} \end{aligned} \tag{41}$$
