**Discussion**

Inspection of Fig. 14a, 15a and 16a reveals, that increasing results in decreasing of the maximum overshoot max, narrowing of the B-parabolas of relative settling times τs=f(M,n) for each identification level n/c, and consequently settling time increasing. Consider e.g. the B0,95 parabolas in Fig. 14b, Fig. 15b and Fig. 16b: if M=70 and =4, relative settling time is τs=30, for =8 it grows to τs=40, and for =12 even to τs=45. If a 10% maximum overshoot is acceptable, then the standard interaction PID controller can be used with no need to use a setpoint filter; however a larger settling time is to be expected.

Procedure 1 is used to specify the performance in terms of (M,n) from (max,ts) using pertinent B-parabolas in Fig. 14 – 16. Procedure 2 is used for plant identification and PID controller design.

#### **Example 3**

Using the sine-wave method, design ideal PID controller for the flow valve modelled by the transfer function GC(s) (system with integrator and time delay)

$$\mathbf{G}\_{\mathbb{C}}(\mathbf{s}) = \frac{\mathbf{K}\_{\mathbb{C}}e^{-D\_{\mathbb{C}}s}}{s(T\_{\mathbb{C}}s+1)} = \frac{\mathbf{1}, \mathbf{3}e^{-2.1s}}{s(7, 51s+1)}\tag{45}$$

Control objective is to provide the maximum overshoots of the closed-loop step response max1=30%, max2=20% and a maximum relative settling time τs=20.

#### **Solution**


Inspection of Fig. 14a, 15a and 16a reveals, that increasing results in decreasing of the maximum overshoot max, narrowing of the B-parabolas of relative settling times τs=f(M,n) for each identification level n/c, and consequently settling time increasing. Consider e.g. the B0,95 parabolas in Fig. 14b, Fig. 15b and Fig. 16b: if M=70 and =4, relative settling time is τs=30, for =8 it grows to τs=40, and for =12 even to τs=45. If a 10% maximum overshoot is acceptable, then the standard interaction PID controller can be used with no need to use a

Procedure 1 is used to specify the performance in terms of (M,n) from (max,ts) using pertinent B-parabolas in Fig. 14 – 16. Procedure 2 is used for plant identification and PID

Using the sine-wave method, design ideal PID controller for the flow valve modelled by the

2,1 1,3 ( ) ( 1) (7,51 1) *D s <sup>C</sup> <sup>s</sup> <sup>C</sup>*

Control objective is to provide the maximum overshoots of the closed-loop step response

1. Critical frequency of the plant identified by the Rotach test is c=0,2407[rad/s]. Then,

2. For GC(s) the time delay/time constant ratio is DC/TC=2,1/7,51=0,28<1, hence, the influence of the time constant prevails - GC(s) is a so-called "lag-dominant system" with integrator, therefore B-parabolas are to be chosen carefully. From one side, due to time delay it would be desirable to choose B-parabolas from Fig. 14, Fig. 15 or Fig. 16 with the lowest identification level n/c=0,2. However, the minima of B0,2 parabolas in Fig. 14b (for =4), Fig. 15b (for =8) and Fig. 16b (for =12) indicate the smallest achievable relative settling time τs=36,5 (for =4), τs=33 (for =8) and τs=34 (for =12),

and GC(j0,5c)=8,10e-j129

M2=M2+(180/)n2DC=62+14,5=76,5 was supplied into the PID

frequency response GC(j) (solid line) in Fig. 17a, verifying correctness of the sine-wave

M1=M1+(180/)n1DC=53+10,1=63,1 into the controller design algorithm. The second performance specification (max2;τs)=(20%,20) is achievable using the B0,5 parabolas in Fig. 16 for =12 and n/c=0,5 and parametres (M2;n2)=(62;0,5c) (Design No. 2). To reject the influence of DC, instead of M2=62 the augmented open-

4. The first performance specification (max1;τs)=(30%;20) can be provided using the B0,35 parabolas for =12 (Fig. 16b) at the level n/c=0,35 and for parameters (M1;n1)= =(53;0,35c) (Design No. 1), supplying the augmented open-loop phase margin

*sT s s s* 

(45)

are located on the plant

*C K e <sup>e</sup> G s*

setpoint filter; however a larger settling time is to be expected.

transfer function GC(s) (system with integrator and time delay)

*C*

max1=30%, max2=20% and a maximum relative settling time τs=20.

the required settling time is ts=τs/c=20/0,2407[s]=83,09[s].

which do not satisfy the required value τs=20.

3. Identified points GC(j0,35c)=12,7e-j122

type identification.

loop phase margin ´

controller design algorithm.

**Discussion** 

controller design.

**Example 3** 

**Solution** 

´

Fig. 14. B-parabolas: a) ηmax=f(M,n); b) τs=cts=f(M,n) for systems with integrator, =4

Fig. 15. B-parabolas: a) ηmax=f(M,n); b) τs=cts=f(M,n) for systems with integrator, =8

Fig. 16. B-parabolas: a) ηmax=f(M,n); b) τs=cts=f(M,n) for systems with integrator, =12

PID Controller Design for Specified Performance 27

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

02 06

*, , T ,*

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

This research work has been supported by the Scientific Grant Agency of the Ministry of

Åström, K.J. & Hägglund, T. (1995). *PID Controllers: Theory, Design and Tuning* (2nd Edition), Instrument Society of America, Research Triangle Park, ISBN 1-55617-516-7 Åström, K.J. & Hägglund, T. (2000). Benchmark Systems for PID Control. *IFAC Workshop on* 

Bakošová, M. & Fikar, M. (2008). *Riadenie procesov (Process Control),* Slovak University of Technology in Bratislava, ISBN 978-80-227-2841-6, Slovak Republic (in Slovak) Balátě, J. (2004). *Automatické řízení (Automatic Control)* (2nd Edition), BEN - technická

Bucz, Š.; Marič, L.; Harsányi, L. & Veselý, V. (2010). A Simple Robust PID Controller Design

Bucz, Š.; Marič, L.; Harsányi, L. & Veselý, V. (2010). A Simple Tuning Method of PID

Method Based on Sine Wave Identification of the Uncertain Plant. *Journal of Electrical Engineering,* Bratislava, Vol. 61, No. 3, (2010), pp. 164-170, ISSN 1335-3632

Controllers with Prespecified Performance Requirements. *9th International Conference Control of Power Systems 2010,* High Tatras, Slovak Republic, May 18-20,

*Digital Control PID'00,* pp. 181-182, Terrassa, Spain, April, 2000

literatúra, ISBN 80-7300-148-9, Praha, Czech Republic (in Czech)

*c c*

(46)

*s*

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

PID controllers in industrial control loops.

Education of the Slovak Republic, Grant No. 1/1241/12.

**5. Acknowledgment** 

**6. References** 

2010

**4. Conclusion** 

interval

Fig. 17. a) Open-loop Nyquist plots; b) closed-loop step responses of the flow valve, required performance max1=30%, max2=20% and τs=20

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 moved into LC2(j0,5c)=1e-j118 achieving the phase margin M2=180-118=62 (dotted 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.

#### **Discussion**

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 complying with the required one.


Table 10. Summary of required and achieved performance measure values, identification parametres and PID controller tunings for GA(s), GB(s) and GC(s)
