**2.5.2 Tuning formulae for FOPDT models**

FOPDT models (7a) are used for chemical processes, thermal systems, manufacturing processes etc. Corresponding P, PI and PID coefficients are calculated using formulae in Tab. 3.


Table 3. PID tuning rules based on FOPDT model; 1=K1D1/T1 is the normed process gain

Formulae No. 11 – 13 represent the time-domain (or reaction curve) Ziegler-Nichols method (Ziegler & Nichols, 1942) and usually give higher open-loop gains than the frequencydomain version. Algorithms by Chien-Hrones-Reswick provide different settings for setpoint regulation and disturbance rejection for two representative maximum overshoot values.

#### **2.5.3 Tuning formulae for IPDT and FOLIPDT models**

While dynamics of slow industrial processes (polymer production, heat exchangers) can be described by IPDT model (7b), electromechanic subsystems of turning machines and servodrives are typical examples for using FOLIPDT model (7c).


Table 4. Tuning rules based on IPDT and FOLIPDT model parameters

FOPDT models (7a) are used for chemical processes, thermal systems, manufacturing processes etc. Corresponding P, PI and PID coefficients are calculated using formulae in Tab. 3.

ratio (δDR=1:4) 12. PI 0,9/<sup>1</sup> 3D1 -

15. PID 0,95/<sup>1</sup> 2,38D1 0,42D1 D1/T1(0,1;1) 16. PI 0,7/<sup>1</sup> 2,33D1 - max=20%, 17. PID 1,2/<sup>1</sup> 2D1 0,42D1 D1/T1(0,1;1)

19. PID 0,6/<sup>1</sup> D1 0,5D1 D1/T1(0,1;1) 20. PI 0,6/<sup>1</sup> D1 - max=20%, 21. PID 0,95/<sup>1</sup> 1,36D1 0,47D1 D1/T1(0,1;1)

23. PID 2/K1 T1+D1 min(D1/3;T1/6) Fast loop Table 3. PID tuning rules based on FOPDT model; 1=K1D1/T1 is the normed process gain Formulae No. 11 – 13 represent the time-domain (or reaction curve) Ziegler-Nichols method (Ziegler & Nichols, 1942) and usually give higher open-loop gains than the frequencydomain version. Algorithms by Chien-Hrones-Reswick provide different settings for setpoint regulation and disturbance rejection for two representative maximum overshoot

While dynamics of slow industrial processes (polymer production, heat exchangers) can be described by IPDT model (7b), electromechanic subsystems of turning machines and

24. (Haalman, 1965), IPDT P 0,66/(K2D2) - - Ms=1,9 25. (Ziegler & Nichols, 1942), IPDT PI 0,9/(K2D2) 3,33D2 - δDR=1:4 26. (Ford, 1953), IPDT PID 1,48/(K2D2) 2D2 0,37D2 δDR=1:2,7 27. (Coon, 1956), FOLIPDT P x3/[K3(D3+T3)] - - δDR=1:4 28. (Haalman, 1965), FOLIPDT PD 0,66/(K3D3) - T3 Ms=1,9

*D s*

;

*FOLIPDT K e G s*

roller K Ti Td Performance

P 1/<sup>1</sup> - - Quarter decay

PI 0,6/<sup>1</sup> 4D1 - max=0%,

PI 0,35/<sup>1</sup> 1,17D1 - max=0%,

PID 2/K1 T1+D1 max(D1/3;T1/6) Slow loop

roller K Ti Td

3 3 3 ( ) <sup>1</sup>

(7)

Performance

*D s*

 

*sTs*

*s* 

; <sup>2</sup> <sup>2</sup> ( )

*IPDT K e G s*

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

Cont-

13. PID 1,2/<sup>1</sup> 2D1 0,5D1

**2.5.3 Tuning formulae for IPDT and FOLIPDT models** 

No. Design method, year, model Cont-

servodrives are typical examples for using FOLIPDT model (7c).

Table 4. Tuning rules based on IPDT and FOLIPDT model parameters

*D s*

*T s* 

*FOPDT K e G s*

No. Design method, year, control purpose

11. (Ziegler & Nichols,

(Chien et al., 1952), Regulator tuning

(Chien et al., 1952), Servo tuning

22. (ControlSoft Inc., 2005)

1942)

14.

18.

values.

**2.5.2 Tuning formulae for FOPDT models** 

According to Haalman (rules No. 24 and 28), controller transfer function GR(s)=L(s)/G(s), where L(s)=0,66e-Ds/(Ds) is the ideal loop transfer function guaranteeing maximum closedloop sensitivity Ms=1,9 to disturbance d(t), (see subsection 2.8.1). For various G(s), various controller structures are obtained. The gain K in rule No. 27 depends on the normed time delay 3=D3/T3 of the FOLIPDT model; for corresponding couples hold: (3;x3)={(0,02;5), (0,053;4); (0,11;3); (0,25;2,2); (0,43;1,7); (1;1,3); (4;1,1)}. Due to integrator contained in IPDT and FOLIPDT models, I-term in the controller structure is needed just to achieve zero steady-state error e() under steady-state disturbance d().

#### **2.5.4 Tuning formulae for SOPDT plant models**

Flexible systems in wood processing industry, automotive industry, robotis, shocks and vibrations damping are often modelled by SOSPTD models with transfer functions

$$\mathbf{G}\_{\rm SOPDT}(\mathbf{s}) = \frac{K\_4 e^{-D\_4 s}}{(T\_4 s + 1)(T\_5 s + 1)}; \quad \mathbf{G}\_{\rm SOPDT}(\mathbf{s}) = \frac{K\_6 e^{-D\_6 s}}{T\_6^2 s^2 + 2\xi\_6 T\_6 s + 1} \tag{8}$$

For SOPDT model (8b), the relative damping 6(0;1) indicates oscillatory step response.

Fig. 5. Step response of SOPDT model: a) non-oscillatory, b) oscillatory

If 6>1, SOPDT model (8a) is used; its parameters are found from the non-oscillatory step response in Fig. 5a using the following relations

$$T\_{4,5} = \frac{1}{2} \left( C\_2 \pm \sqrt{C\_2^2 - 4C\_1^2} \right);\ D\_4 = \frac{t\_{0,33}}{0,516} - \frac{t\_{0,7}}{1,067};\ \ C\_1 = \frac{\left( t\_{0,33} - t\_{0,7} \right)}{1,259};\ \ C\_2 = \frac{S}{y(\infty)} - D\_4 \tag{9}$$

where S=K4(T4+T5+D4) denotes the area above the step response of y(t), and y() is its steady-state value. Parameters of the SOPDT model (8b) can be found from evaluation of 2-4 periods of step response oscillations (Fig. 5b) using following rules (Vítečková, 1998)

$$\xi\_6 = \frac{-\ln\frac{a\_{i+1}}{a\_i}}{\sqrt{\pi^2 + \ln^2\frac{a\_{i+1}}{a\_i}}} ; \ T\_6 = \frac{\sqrt{1 - \xi\_6^2}}{\pi N} (t\_{N+1} - t\_1) ; \ D\_6 = \frac{1}{N} \left[ \sum\_{i=1}^N t\_i - \frac{N+1}{2} (t\_{N+1} - t\_1) \right] \tag{10}$$

Quality of identification improves with increasing number of read-off amplitudes N. If N>2 several values 6, T6 and D6 are obtained and their average is taken for further calculations. Tab. 5 summarizes useful tuning formulae for both oscillatory and non-oscillatory systems with SOPDT model properties.

PID Controller Design for Specified Performance 11

leading to simple tuning rules for PID controller (4a) (No. 42 – 44 in Tab. 7). Tuning rules No. 45 and 46 for PID controller (4c) show that settling time ts increases with growing

No. Method, year K Ti Td Tf Performance

43. 1,371/K1 4,12T110,90 0,55T1 - Minimum ISTE 44. 1,701/K1 4,52T111,13 0,50T1 - Minimum IST2E

46. 2,0217/K1 4,65T1 0,2366T1 0,0696T1 ts=0,8T1: 1=0,5

Using tuning methods shown in Tab. 2 – 7, achieved performance is a priori given by the chosen metod (e.g. a quarter decay ratio if using Ziegler-Nichols methods No. 11 – 13 in Tab. 3), or guaranteed performance however not specified by the designer (e.g. in Chen method No. 33 in Tab. 5, a gain margin GM=1,96, a phase margin M=44,1, and a maximum

These methods provide tuning rules are based on a single tuning parameter that enables to

Most frequent parameters for tuning PID controllers are following performance measures

Ms and Mt: maximum peaks of sensitivity S(j) and complementary sensitivity T(j)

If a controller GR(j) guarantees that S(j) or T(j) do not overrun prespecified values Ms

over 0,), then the Nyquist plot L(j) of the open-loop L(s)=G(s)GR(s) avoids the

If L(j) avoids entering the circles corresponding to MS or MT, a safe distance from the point CS is kept (Fig. 6a). Typical S(j) and T(j) plots for properly designed controller are plotted in Fig. 6b. The disturbance d(t) is sufficiently rejected if Ms(1,2;2). The reference w(t) is properly tracked by the process output y(t) if Mt(1,3;2,5). With further increasing of

; ( ) sup ( ) sup 1 () *<sup>t</sup> L j M Tj*

*T*

 *L j*

, 2 <sup>1</sup>

*<sup>M</sup> <sup>R</sup>*

*t*

*t*

*<sup>M</sup>* (14)

(13)

2 <sup>2</sup> , 0 1 *t*

*t <sup>M</sup> C j <sup>M</sup>* 

*T*

respective circle MS or MT , each given by the their center and radius as follows

*s*

*M* ;

1,371/K1 2,42T111,18 0,60T1 - Minimum ISE

10,3662/K1 0,3874T1 0,0435T1 0,0134T1 ts=0,1T1: 1=0,1

normed time delay 1=D1/T1 of the FOPDT model (12).

Table 7. Tuning rules for unstable FOPDT model

peak of the sensitivity to disturbance d(t) Ms=1,5).

**2.8 PID controller design for specified performance** 

M and GM: phase and gain margins, respectively,

<sup>1</sup> sup ( ) sup 1 () *M Sj <sup>s</sup>*

*CS* 1, 0*<sup>j</sup>* , <sup>1</sup>

 *L j*

*S*

*R*

systematically affect closed-loop performance by step response shaping.

**2.8.1 Performance measures used as a PID tuning parameter** 

42. (Visoli, 2001), Regulator tuning

45. (Chandrashekar et al., 2002)

(Åström & Hägglund, 1995):

magnitudes, respectively,

or Mt, respectively, defined by

: required closed-loop time constant.

Mt the closed-loop tends to be oscillatory.


Table 5. Tuning rules based on SOPDT model parameters

#### **2.6 PID controller design based on optimization techniques**

Optimal PID controller tuning can be found by minimizing the performance index

$$I(K, T\_i, T\_d) = \bigcap\_{0}^{\infty} \left[ t^n c(K, T\_i, T\_{d'}, t) \right]^2 dt \tag{11}$$

Its particular cases are known as integral square error (ISE) for n=0; integral squared time weighed error (ISTE) for n=1, and integral squared time-squared weighed error (IST2E) for n=2. Some tuning formulae for PID controller in form (4a) are shown in Tab. 6. Settling time ts in rules No. 40 and 41 is affected by D2.


Table 6. Tuning rules based on minimizing performance indices

#### **2.7 PID controller setting for unstable FOPDT models**

Minimization of performance indices can be applied also for unstable FOPDT models

$$\mathcal{G}\_{FOPDT\\_LIS} \text{(s)} = \frac{K\_1 e^{-D\_1s}}{T\_1s - 1} \tag{12}$$

roller K Ti Td Performance for

4 5 4 5 *T T T T*

4 5 4 5 *T T T T*

2

1,47310,970/K1 0,897T110,753 0,550T110,948 Minimum ISE

1,52410,735/K1 0,885T110,641 0,552T110,851 Minimum ISE

0,9588/[K2D2] 3,0425D2 0,3912D2 ts=D2

1

(12)

*D s*

1 ( ) <sup>1</sup>

*T s* 

(11)

Closed-loop step response overshoot max=10%

Overdamped plants; T5>T4

[GM=2, M=45]: x6=1,571 [GM=5, M=72]: x6=0,628

Underdamped plants (0,5<61)

[GM;M;Ms]=[3,14;61,4;1]: x6=1,0 [GM;M;Ms]=[1,96;44,1;1,5]: x6=1,6

max=0%: x4=0,368 max=30%: x4=0,801

max=0%: x6=0,736 max=30%: x6=1,602

T4+T5

T4+T5

26T6

26T6

26T6

Optimal PID controller tuning can be found by minimizing the performance index

0 ( , , ) ( , , ,) *<sup>n</sup> i d i d I K T T t e K T T t dt* 

No. Method, year, model K Ti Td Performance

35. 1,46810,970/K1 1,062T110,725 0,443T110,939 Minimum ISTE 36. 1,53110,960/K1 1,030T110,746 0,413T110,933 Minimum IST2E

38. 1,51510,730/K1 1,045T110,598 0,444T110,847 Minimum ISTE 39. 1,59210,705/K1 1,045T110,597 0,414T110,850 Minimum IST2E

41. 0,3144/[K2D2] 11,1637D2 0,1453D2 ts=5D2

Minimization of performance indices can be applied also for unstable FOPDT models

*FOPDT US K e G s*

<sup>1</sup> \_

Table 6. Tuning rules based on minimizing performance indices

**2.7 PID controller setting for unstable FOPDT models** 

Its particular cases are known as integral square error (ISE) for n=0; integral squared time weighed error (ISTE) for n=1, and integral squared time-squared weighed error (IST2E) for n=2. Some tuning formulae for PID controller in form (4a) are shown in Tab. 6. Settling time

No. Method, year

29. (Suyama,

30. Vítečková, (1999), Vítečková et al., (2000)

32. (Wang &

33. (Chen

Cont-

4 4 2 *T T K D* 

> 4 4 *T T*

*K D* 

6 6 *x T K D* 

6 6 *x T K D* 

6 6 *x T K D* 

**2.6 PID controller design based on optimization techniques** 

Table 5. Tuning rules based on SOPDT model parameters

PID 4 5 4

*x*

1992) PID 4 5

31. PID <sup>666</sup>

Shao, 1999) PID <sup>666</sup>

et al., 1999) PID <sup>666</sup>

ts in rules No. 40 and 41 is affected by D2.

34. (Zhuang & Atherton, 1993), FOPDT model,

37. (Zhuang & Atherton, 1993), FOPDT model,

40. (Wang a Cluett, 1997), IPDT model

10,1;1

11,1;2

leading to simple tuning rules for PID controller (4a) (No. 42 – 44 in Tab. 7). Tuning rules No. 45 and 46 for PID controller (4c) show that settling time ts increases with growing normed time delay 1=D1/T1 of the FOPDT model (12).


Table 7. Tuning rules for unstable FOPDT model

Using tuning methods shown in Tab. 2 – 7, achieved performance is a priori given by the chosen metod (e.g. a quarter decay ratio if using Ziegler-Nichols methods No. 11 – 13 in Tab. 3), or guaranteed performance however not specified by the designer (e.g. in Chen method No. 33 in Tab. 5, a gain margin GM=1,96, a phase margin M=44,1, and a maximum peak of the sensitivity to disturbance d(t) Ms=1,5).

#### **2.8 PID controller design for specified performance**

These methods provide tuning rules are based on a single tuning parameter that enables to systematically affect closed-loop performance by step response shaping.

#### **2.8.1 Performance measures used as a PID tuning parameter**

Most frequent parameters for tuning PID controllers are following performance measures (Åström & Hägglund, 1995):


If a controller GR(j) guarantees that S(j) or T(j) do not overrun prespecified values Ms or Mt, respectively, defined by

$$M\_s = \sup\_{\phi} \left| S(j\phi) \right| = \sup\_{\phi} \left| \frac{1}{1 + L(j\phi)} \right|; \quad M\_l = \sup\_{\phi} \left| T(j\phi) \right| = \sup\_{\phi} \left| \frac{L(j\phi)}{1 + L(j\phi)} \right| \tag{13}$$

over 0,), then the Nyquist plot L(j) of the open-loop L(s)=G(s)GR(s) avoids the respective circle MS or MT , each given by the their center and radius as follows

$$\mathbf{C}\_{S} = \begin{bmatrix} -1 \text{ } j0 \end{bmatrix}, \ R\_{S} = \frac{1}{M\_{s}}; \quad \mathbf{C}\_{T} = \left\lfloor -\frac{M\_{l}^{2}}{M\_{l}^{2} - 1} \right\rfloor \\ \text{j. } R\_{T} = -\frac{M\_{l}}{\left\lfloor 1 - M\_{l}^{2} \right\rfloor} \tag{14}$$

If L(j) avoids entering the circles corresponding to MS or MT, a safe distance from the point CS is kept (Fig. 6a). Typical S(j) and T(j) plots for properly designed controller are plotted in Fig. 6b. The disturbance d(t) is sufficiently rejected if Ms(1,2;2). The reference w(t) is properly tracked by the process output y(t) if Mt(1,3;2,5). With further increasing of Mt the closed-loop tends to be oscillatory.

PID Controller Design for Specified Performance 13

Kcsin<sup>M</sup>

*M Gj M*

> *K G*

( ) 1 *t Mt t*

cos *c M M*

1 1 1 1 *T D* 0,5 *K D* 

> 

11 1

*TD T*2 *K D*

0,91 64,55 38 ;71 1,53 88,46 12 ;38 *M M M M*

 

*for for*

1.18 (0) <sup>100</sup> (0) *M T <sup>t</sup> T*

the controller has the integral part then T(0)=T(=0)=1.

original PID controller design method is presented.

 

 

1 1

K Ti Td

(1 cos ) sin *c M M*

> 2 arg *Mt*

*d G d*

*Mt*

<sup>2</sup> <sup>1</sup> *<sup>c</sup> M M*

1 2

; <sup>2</sup> <sup>2</sup> max <sup>100</sup> *<sup>t</sup> b M <sup>e</sup>*

<sup>3</sup> 1,3;1,5 *s t*

*a t for M* 

; \* \*

*tg tg* 

11 1 1 1

*TD T*2 *T D* 

 

 

2

 <sup>2</sup> <sup>1</sup> 4

*T D* 1 1


*T*

(1 cos ) 4 sin *c M*

1 arg

*<sup>c</sup> M M*

1 1 2 *T D T D*

4

 

(17)

*a a*

, *<sup>s</sup>*

 

*t*

(18)

*tg tg* 

*d G d*

*T*

2

 

*M*

*Mt Mt*

> 

*T*

 

<sup>2</sup>

*Mt*

*T*

1 1

Table 8. PID design formulae for specified performance based on tuning parameters M, GM,

Phase margin M is the most wide-spread performance measure in PID controller design. Maximum overshoot max and settling time ts of the closed-loop step response are related

valid for 2nd order closed-loop with relative damping (0,25;0,65) where a\* is the gain

[%]; \*

(Hudzovič, 1982); (Grabbe et al., 1959-61) are general for any order of the closed-loop T(s); if

The engineering practice is persistently demanding for PID controller design methods simultaneously guaranteeing several performance criteria, especially maximum overshoot ηmax and settling time ts. However, we ask the question: how to suitably transform the above-mentioned engineering requirements into frequency domain specifications applicable for PID controller coefficients tuning? The response can be found in Section 3 where a novel

2

2

No.

47.

49.

50.

51.

Mt and

Design method,

(Hang & Åström,

Non-model 2

FOPDT

FOPDT

year, model

1988), Non-model

48. (Rotach, 1994),

(Wojsznis et al., 1999), FOPDT

1989),

2002),

max

(Morari & Zafiriou,

(Chen & Seborg,

**2.8.3 Performance evaluation** 

with M according to Reinisch relations

max

crossover frequency (Hudzovič, 1982). Relations

Fig. 6. a) Definition and geometrical interpretation of M and GM in the complex plane; b) Sensitivity and complementary sensitivity magnitudes S(j), T(j) and performance measures Ms, Mt

From Fig. 6a results, that increasing open-loop phase margin M causes moving the gain crossover L(ja\*) lying on the unit circle M1 away from the critical point (-1,j0). Increasing open-loop gain margin GM causes moving the phase crossover L(j<sup>f</sup> \*) away from (-1,j0). Therefore, parameters M or GM given by

$$\text{Cov} = 180^\circ + \text{arg}\, L(o\_a^\circ) \; ; \quad G\_M = \frac{1}{\left| L(jo\_f^\circ) \right|} \tag{15}$$

are frequently used performance measures, their typical values are M(20;90), GM(2;5). Relations between them are given by following inequalities

$$\phi\_M \ge 2 \arcsin\left(\frac{1}{2M\_s}\right); \quad \phi\_M \ge 2 \arcsin\left(\frac{1}{2M\_l}\right); \quad G\_M \ge \frac{M\_s}{M\_s - 1}; \quad G\_M \ge 1 + \frac{1}{M\_l} \tag{16}$$

The point at which the Nyquist plot L(j) touches the MT circle defines the closed-loop resonance frequency Mt.

#### **2.8.2 Tuning formulae with performance specification**

Table 8 shows open formulae for PID controller design. The coefficients tuning is carried out with respect to closed-loop performance specification. Rules No. 47 – 49 consider tuning of ideal PID controller (4a). To apply the Rotach method, knowledge of the plant magnitude G(j) is supposed as well as of the roll-off of argG() at =Mt, where the maximum peak Mt of the complementary sensitivity is required. Method No. 50 is based on so-called -tuning, with the resulting closed-loop expressed as a 1st order system with time constant ; this rule considers a real PID controller (4b) with filtering constant in the derivative part Tf=Td/N=0,5D1/(1+D1) where is to be chosen to meet following conditions: >0,25D1; >0,25T1 (Morari & Zafiriou, 1989). The -tuning technique is used also in the rule No. 51 to design interaction PI controller.

1

S(j)

T(j)

0

1

0

Fig. 6. a) Definition and geometrical interpretation of M and GM in the complex plane; b) Sensitivity and complementary sensitivity magnitudes S(j), T(j) and performance

Re

**L(j<sup>f</sup> \*) M**

> **argL(<sup>a</sup> \*)**

**0** 

Im

**1 G**

open-loop gain margin GM causes moving the phase crossover L(j<sup>f</sup>

*L*

\* *<sup>M</sup>* 180 arg ( )

From Fig. 6a results, that increasing open-loop phase margin M causes moving the gain crossover L(ja\*) lying on the unit circle M1 away from the critical point (-1,j0). Increasing

are frequently used performance measures, their typical values are M(20;90), GM(2;5).

The point at which the Nyquist plot L(j) touches the MT circle defines the closed-loop

Table 8 shows open formulae for PID controller design. The coefficients tuning is carried out with respect to closed-loop performance specification. Rules No. 47 – 49 consider tuning of ideal PID controller (4a). To apply the Rotach method, knowledge of the plant magnitude G(j) is supposed as well as of the roll-off of argG() at =Mt, where the maximum peak Mt of the complementary sensitivity is required. Method No. 50 is based on so-called -tuning, with the resulting closed-loop expressed as a 1st order system with time constant ; this rule considers a real PID controller (4b) with filtering constant in the derivative part Tf=Td/N=0,5D1/(1+D1) where is to be chosen to meet following conditions: >0,25D1; >0,25T1 (Morari & Zafiriou, 1989). The -tuning technique is used also in the rule No. 51 to

<sup>1</sup> 2arcsin <sup>2</sup> *<sup>M</sup> Mt*

 

*<sup>a</sup>* ; \*

*G*

1 ( ) *<sup>M</sup> f*

*L j*

; 1 *<sup>s</sup> <sup>M</sup> s*

*<sup>M</sup> <sup>G</sup> M* ; \*) away from (-1,j0).

Mt

Mt

1

*M* (16)

*t*

(15)

Ms

Ms

*<sup>M</sup>* 1

*G*

measures Ms, Mt

**MT**

**MS**

**RT**

resonance frequency Mt.

design interaction PI controller.

Therefore, parameters M or GM given by

**<sup>M</sup>**

**-1** 

**RS**

**CT**

**L(j)** 

<sup>1</sup> 2arcsin <sup>2</sup> *<sup>M</sup> Ms*

 

**M1**

**L(j<sup>a</sup> \*)** 

Relations between them are given by following inequalities

;

**2.8.2 Tuning formulae with performance specification** 



#### **2.8.3 Performance evaluation**

Phase margin M is the most wide-spread performance measure in PID controller design. Maximum overshoot max and settling time ts of the closed-loop step response are related with M according to Reinisch relations

$$\eta\_{\text{max}} = \begin{cases} -0.91 \phi\_M + 64, 55 & \text{for } \phi\_M \in \{38^\circ; 71^\circ\} \\ -1, 53 \phi\_M + 88, 46 & \text{for } \phi\_M \in \{12^\circ; 38^\circ\} \end{cases}; \quad \eta\_{\text{max}} = 100e^{-2\pi b^2 M\_i}; \; t\_s \in \left(\frac{\pi}{o\_a^\*}, \frac{4\pi}{o\_a^\*}\right) \tag{17}$$

valid for 2nd order closed-loop with relative damping (0,25;0,65) where a\* is the gain crossover frequency (Hudzovič, 1982). Relations

$$
\eta\_{\text{max}} \le 100 \frac{1.18 M\_t - \left| T(0) \right|}{\left| T(0) \right|} \text{ [\%]}; \ t\_s \approx \frac{3}{o\_a^\*} \text{ } for \ M\_t \in \{1, 3; 1, 5\} \tag{18}
$$

(Hudzovič, 1982); (Grabbe et al., 1959-61) are general for any order of the closed-loop T(s); if the controller has the integral part then T(0)=T(=0)=1.

The engineering practice is persistently demanding for PID controller design methods simultaneously guaranteeing several performance criteria, especially maximum overshoot ηmax and settling time ts. However, we ask the question: how to suitably transform the above-mentioned engineering requirements into frequency domain specifications applicable for PID controller coefficients tuning? The response can be found in Section 3 where a novel original PID controller design method is presented.

PID Controller Design for Specified Performance 15

The output sinusoid amplitude Yn can be affected by the amplitude Un of the excitation sinusoid generated by the sine wave generator; it is recommended to use Un=37%umax. Identified plant parameters are represented by the triple n,Yn(n)/Un(n),φ(n). In the SW position "4", identification is performed in the open-loop. Hence, this method is applicable only for stable plants. The excitation frequency n is to be adjusted prior to identification and taken from the empirical interval (29) (Bucz et al., 2010a, 2010b, 2011).

In the control loop in Fig. 7, let us switch SW in "5"and put the PID controller into manual mode. The closed-loop characteristic equation 1+L(j)=1+G(j)GR(j)=0 at the gain crossover frequency a\* can be broken down into the amplitude and phase conditions as

> *a Ra* ; \* \* arg ( ) arg ( ) 180 *G G*

where M is the required phase margin, L(jn) is the open-loop transfer function. Denote =argGR(a\*). We are searching for K, Ti and Td of the ideal PID controller (4a). Comparing

\* \* \* 1 cos sin () () *d a i a a a*

frequency transfer functions of the PID controller in parallel and polar forms

coefficients of PID controller can be obtained from the complex equation

\*

(20a). The complex equation (22) is solved as a set of two real equations

\* cos ( ) *<sup>a</sup>*

A positive solution of (24) yields the rule for calculating the derivative time Td

2

1 1

 

\* \*

<sup>2</sup> <sup>4</sup> *<sup>d</sup> a a tg tg <sup>T</sup>* 

 

; \*

*G j*

*K jK T <sup>j</sup> <sup>T</sup> Gj Gj*

at =a\* using the substitution GR(ja\*)=1/G(ja\*) resulting from the amplitude condition

where (23a) is a general rule for calculation of the controller gain K. Using (23a) and the ratio of integration and derivative times =Ti/Td in (23b), a quadratic equation in Td is obtained

<sup>2</sup> 2\* \* <sup>1</sup> <sup>0</sup> *T T tg d a da*

 ; \* 180 ar *<sup>M</sup>* <sup>g</sup> ( ) 

 

*K T*

*a Ra* 

; ( ) ( ) ( ) cos ( ) sin *<sup>j</sup> G j G j e G j jG j RR R R* 

> \* \* 1 sin ( ) *d a i a a*

*T G j*

 

 

 

 

*<sup>M</sup>* (20)

 

, (22)

(23)

(24)

*G <sup>a</sup>* (25)

 

(21)

**3.2 Sine-wave method tuning rules** 

\* \* ( ) ( )1 *Gj G j*

 

*i*

 

*K*

<sup>1</sup> ( ) *R d*

*G j K jK T <sup>T</sup>*

after some manipulations

follows
