**4. Conclusion**

26 Introduction to PID Controllers – Theory, Tuning and Application to Frontier Areas

**M1**

**GC(j0,5c) GC(j0,35c)**

Fig. 17. a) Open-loop Nyquist plots; b) closed-loop step responses of the flow valve, required

0.5

Controlled variable y(t)

1

1.5

0.5

Controlled variable y(t)

1

1.5

5. Using the PID controller, the first identified point GC(j0,35c) (Design No. 1) was moved into the gain crossover LC1(j0,35c)=1e-j127 located on the unit circle M1; this verifies achieving the phase margin M1=180-127=53 (dashed line in Fig. 17a). Achieved performance in terms of the closed-loop step response in Fig. 17b is max1\*=29,6%, ts1\*=81,73[s] (dashed line). The second identified point GC(j0,5c) (Design No. 2) was

line in Fig. 17a). Achieved performance in terms of the closed-loop step response parameters max2\*=19,7%, ts2\*=82,44[s] (dotted line in Fig. 17b) meets the required specification. Frequency characteristics LC1(j), LC2(j) begin near the negative real halfaxis of the complex plane, because both open-loops contain a 2nd order integrator.

All data necessary to design two PID controllers of all three plants GA(s), GB(s) and GC(s) along with specified and achieved performance measure values are summarized in Tab. 10 where max and ts in the last two columns marked with "\*" indicate closed-loop performance

Model max;τ<sup>s</sup> c[rad/s] ts[s] B-par. <sup>M</sup> n/<sup>c</sup> G(jn) GR(jn) max\* ts

parametres and PID controller tunings for GA(s), GB(s) and GC(s)

GA(s) 30%;12 173,22 0,0693 Fig. 11 50 0,5 0,43e-j120 2,31e-j10 29,7% 0,0584 GA(s) 5%;12 173,22 0,0693 Fig. 11 70 0,8 0,19e-j165 5,20ej55 4,89% 0,0605 GB(s) 30%;12 0,3521 34,08 Fig. 11 55+45,9 0,35 1,03e-j23 0,97e-j56 18,6% 24,78 GB(s) 5%;12 0,3521 34,08 Fig. 11 70+26,2 0,2 1,09e-j13 0,92e-j71 0,15% 28,69 GC(s) 30%;20 0,2407 83,09 Fig. 16 53+10,1 0,35 12,7e-j122 0,08ej5,8 29,6% 81,73 GC(s) 20%;20 0,2407 83,09 Fig. 16 62+14,5 0,5 8,10e-j129 0,12e-j28 19,7% 82,44 Table 10. Summary of required and achieved performance measure values, identification

achieving the phase margin M2=180-118=62 (dotted

Closed-loop step response of the flow valve

**max1\*=29,6%, ts1\*=81,73[s]** 

Closed-loop step response of the flow valve

**M2=62, n2=0,5<sup>c</sup>**

**M1=53, n1=0,35<sup>c</sup>**

**max2\*=19,7%, ts2\*=82,44[s]**

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>0</sup>

Time [s]

<sup>0</sup> <sup>20</sup> <sup>40</sup> <sup>60</sup> <sup>80</sup> <sup>100</sup> <sup>120</sup> <sup>140</sup> <sup>160</sup> <sup>180</sup> <sup>200</sup> <sup>0</sup>

Time [s]

\*[s]

performance max1=30%, max2=20% and τs=20

Real Axis


**LC2(j0,5c)**

**GC(j) 62 <sup>53</sup>**

**LC2(j)**

Open-loop Nyquist plots, M1=53, n1=0,35c; M2=62, n2=0,5<sup>c</sup>

moved into LC2(j0,5c)=1e-j118

complying with the required one.

**Discussion** 

Imaginary Axis


**LC1(j0,35c)** 

**LC1(j)** 

The proposed new engineering method based on the sine-wave identification of the plant provides successful PID controller tuning. The main contribution has been construction of empirical charts to transform engineering time-domain performance specifications (maximum overshoot and settling time) into frequency domain performance measures (phase margin). The method is applicable for shaping the closed-loop response of the process variable using various combinations of excitation signal frequencies and required phase margins. Using B-parabolas, it is possible to achieve optimal time responses of processes with various types of dynamics and improve their performance. When applying digital PID controller, it is recommended to set the sampling period Ts from the interval

$$T\_s \in \left\langle \frac{0,2}{\alpha\_c}, \frac{0,6}{\alpha\_c} \right\rangle \tag{46}$$

where c is the critical frequency of the controlled plant (Wittenmark, 2001).

By applying appropriate PID controller design methods including the above presented 51+3 tuning rules for prescribed performance, it is possible to achieve cost-effective control of industrial processes. The presented advanced sine-wave design method offers one possible way to turn the unfavourable statistical ratio between properly tuned and all implemented PID controllers in industrial control loops.

#### **5. Acknowledgment**

This research work has been supported by the Scientific Grant Agency of the Ministry of Education of the Slovak Republic, Grant No. 1/1241/12.
