**Discussion**

When choosing M=40 on the B-parabola corresponding to the excitation level 5=n5/c=0,8 (further denoted as B0,8 parabola), maximum overshoot max=40% and relative settling time τs10 are expected. Point corresponding to these parameters is located on the left (falling) portion of B0,8 yielding oscillatory step response (see response in Fig. 10c). If the phase margin increases up to M=60, the relative settling time decreases up to the point on the right (rising) portion of the B0,8 parabola; the corresponding step response in Fig. 10c is weakly-aperiodic. For the phase margin M=80 the B0,8 parabola indicates a zero maximum overshoot, the relative settling time τs=20 corresponds to the position on the B0,8 parabola with aperiodic step response (Fig. 10c). If the maximum overshoot max=20% is acceptable then M=53 yields the least possible relative settling time τs=6,5 on the given level 5=0,8 ("at the bottom" of B0,8) (Bucz et al., 2011), (Bucz, 2011).

#### **Procedure 2. PID controller design using the sine-wave engineering method**


#### **Example 1**

Using the sine-wave method, ideal PID controller (4a) is to be designed for the operating amplifier modelled by the transfer function GA(s)

$$G\_A(\mathbf{s}) = \frac{1}{\left(T\_A \mathbf{s} + 1\right)^3} = \frac{1}{\left(0, 01s + 1\right)^3} \tag{38}$$

The controller has to be designed for two values of the maximum overshoot of the closedloop step response max1=30% (Design No. 1) and max2=5% (Design No. 2) and maximum relative settling time τs=12 in both cases.

#### **Solution**


9. Calculate the excitation frequency n according to the relation n=c using the critical

When choosing M=40 on the B-parabola corresponding to the excitation level 5=n5/c=0,8 (further denoted as B0,8 parabola), maximum overshoot max=40% and relative settling time τs10 are expected. Point corresponding to these parameters is located on the left (falling) portion of B0,8 yielding oscillatory step response (see response in Fig. 10c). If the phase margin increases up to M=60, the relative settling time decreases up to the point on the right (rising) portion of the B0,8 parabola; the corresponding step response in Fig. 10c is weakly-aperiodic. For the phase margin M=80 the B0,8 parabola indicates a zero maximum overshoot, the relative settling time τs=20 corresponds to the position on the B0,8 parabola with aperiodic step response (Fig. 10c). If the maximum overshoot max=20% is acceptable then M=53 yields the least possible relative settling time τs=6,5 on the given level 5=0,8 ("at the bottom" of B0,8) (Bucz et al., 2011), (Bucz, 2011).

frequency c (from Step 1) and the chosen excitation level (from Step 4).

**Procedure 2. PID controller design using the sine-wave engineering method** 

Procedure 1. The switch SW is in position "4".

Step 4, if not, change (n;M) and repeat Steps 1-3.

the controller by switching SW into position "5".

amplifier modelled by the transfer function GA(s)

relative settling time τs=12 in both cases.

rules in Tab. 9 to calculate PID controller parameters.

*G s*

1. From the required values (ηmax,ts) specify the couple (n;M) using Procedure 1.

2. Identify the plant using the sinusoidal excitation signal with frequency n specified in

3. Specify =argG(n), andG(jn). Calculate the controller argument by substituting and M into (27c); if is within the range shown in the last column of Tab. 9, go to

4. Substitute the identified values =argG(n), G(jn) and specified M into the tuning

5. Adjust the resulting PID controller values, switch into automatic mode and complete

Using the sine-wave method, ideal PID controller (4a) is to be designed for the operating

1 1 ( ) ( 1) (0,01 1) *<sup>A</sup>*

The controller has to be designed for two values of the maximum overshoot of the closedloop step response max1=30% (Design No. 1) and max2=5% (Design No. 2) and maximum

1. Critical frequency of the plant identified by the Rotach test is c=173,216[rad/s] (the process is "fast"). The prescribed settling time is ts=τs/c=12/173,216[s]=69,3[ms]. 2. For the Design No. 1 (max1;τs)=(30%;12), a suitable choice is (M1;n1)=(50;0,5c) resulting from the B0,5 parabola in Fig. 11. The performance in Design No. 2 (max2;τs)=(5%;12) can be achieved for (M2;n2)=(70;0,8c) chosen from the B0,8 parabola

3. Identified points for the Designs No. 1 and No. 2 are GA(j0,5c)=0,43e-j120

*T s <sup>s</sup>*

*A*

3 3

, respectively. According to Fig. 12a, both points are located in the

(38)

and

**Discussion** 

**Example 1** 

**Solution** 

in Fig. 11.

GA(j0,8c)=0,19e-j165

Quadrant II of the complex plane, on the Nyquist plot GA(j) (solid line) which verifies the identification.


Fig. 12. a) Open-loop Nyquist plots; b) closed-loop step responses of the operational amplifier, required performance max1=30%, max2=5% and τs=12

#### **3.4.2 Systems with time delay**

The sine-wave method is applicable also for plants with time delay considered as difficultto-control systems. It is a well-known fact, that the time delay D turns the phase at each frequency n0,) by nD with respect to the delay-free system. For time delayed plants, phase condition of the sine-wave method (20b) is extended by additional phase φD=-nD

$$\left(\boldsymbol{\phi}^{\prime} + \boldsymbol{\varphi}\_{\rm D}\right) + \boldsymbol{\Theta} = -180^{\circ} + \boldsymbol{\phi}\_{\rm M} \tag{39}$$

where φ´ is the phase of the delay-free system and

$$
\varphi = \stackrel{\circ}{\varphi} + \varphi\_{\text{D}} \tag{40}
$$

is the identified phase of the plant including the time delay. The added phase φD=-nD can be associated with the required phase margin M

$$
\rho^{\tilde{\cdot}} + \Theta = -180^{\circ} + \left(\phi\_M + \alpha\_n D\right) \tag{41}
$$

The only modification in using the PID tuning rules in Tab. 9 is that increased required phase margin is to be specified (Bucz, 2011)

$$
\dot{\phi\_M} = \phi\_M + o\_n D \tag{42}
$$

PID Controller Design for Specified Performance 23

to the PID design algorithm is ´

3. Identified points GB(j0,35c)=1,03e-j23

gain crossover LB1(j0,35c)=1e-j125

the required performance specification.

Open-loop Nyquist plots, M1=55, n1=0,35c; M2=70, n2=0,2<sup>c</sup>

**LB2(j)**

required performance max1=30%, max2=5% and τs=12

**LB2(j0,2c)**

Real Axis


j=1...8, k=1...6 and three various ratios Ti/Td: =4, 8 and 12).

**3.4.3 Systems with 1st order integrator** 

Imaginary Axis

**70 55**

**LB1(j0,35c)**


open-loop gain crossover LB2(j0,2c)=1e-j110

=45,9 compared with M1=55 in case of delay-free system.

**M1**

**GB(j0,35c) GB(j0,2c)** 

n2DB(180/)=0,2cDB(180/)=0,2.0,3521.6,5.180/=26,2, hence the phase supplied

M2=70 for a delay-free system). The required performance (max1;τs)=(30%;12) (Design No. 1) can be achieved by choosing (M1;n1)=(55;0,35c) from the B0,35 parabolas in

algorithm was increased by n1DB(180/)=0,35cDB(180/)=0,35.0,3521.6,5.180/=

the Quadrant I of the complex plane at the beginning of the frequency response GB(j) (solid line). The point GB(j0,2c) (Design No. 2) was shifted by the PID controller to the

the same location in the complex plane as LA2 in Fig. 12a, however at a considerably lower frequency n2B=0,2.0,3521=0,07[rad/s] compared to n2A=0,8.173,216= =138,6[rad/s] (ts2\_B\*=28,69[s] is almost 500 times larger than ts2\_A\*=0,0584[s] which demonstrates the key role of the excitation frequency n in achieving required closedloop dynamics). The identified point GB(j0,35c) (Design No. 1) was moved into the

(dashed line in Fig. 13a).

4. Achieved performances (max1\*=18,6%, ts1\*=24,78[s], dashed line), (max2\*=0,15%, ts2\*=28,69[s], dotted line) in terms of closed-loop step responses in Fig. 13b comply with

Fig. 13. a) Open-loop Nyquist plots; b) closed-loop step responses of the distillation column,

0.5

Controlled variable y(t)

1

1.5

0.5

Controlled variable y(t)

**max1\*=18,6%, ts1 L \*=24,78[s] B1(j)**

1

1.5

**GB(j)** 

By testing the sine-wave method on benchmark systems with 1st order integrator, the B-parabolas in Fig. 14 – 16 were obtained (for Cartesian product Mj×nk of sets (31) and (32),

and GB(j0,2c)=1,09e-j13

M2=M2+n2DB(180/)=70+26,2=96,2 (instead of

M1=55+45,9 supplied into the design

(dotted line in Fig. 13a). Note that LB2 has

Closed-loop step response of the distillation column

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

Time [s]

<sup>0</sup> <sup>10</sup> <sup>20</sup> <sup>30</sup> <sup>40</sup> <sup>50</sup> <sup>60</sup> <sup>70</sup> <sup>80</sup> <sup>90</sup> <sup>100</sup> <sup>0</sup>

**M2=70; n2=0,2<sup>c</sup>**

**M1=55; n1=0,35<sup>c</sup>**

**max2\*=0,15%, ts2\*=28,69[s]** 

Time [s]

Closed-loop step response of the distillation column

in Fig. 13a are located in

and the controller working angle Θ is computed using the relation

$$
\dot{\Theta} = -180^\circ - \dot{\varphi}^\cdot + \left(\phi\_M + \alpha\_\hbar D\right) \tag{43}
$$

The phase delay nD increases with increasing frequency of the sinusoidal signal n. To lessen the impact of time delay on closed-loop dynamics, it is recommended to use the smallest possible added phase φD=-nD.

#### **Discussion**

Time delay D can easily be specified during critical frequency identification as the time D=Ty-Tu, that elapses since the start of the test at time Tu until time Ty, when the system output starts responding to the excitation signal u(t). A small added phase φD=-nD due to time delay can be secured by choosing the smallest possible n attenuating effect of D in (43) and subsequently in the PID controller design.

Therefore, when designing PID controller for time delayed systems according to Procedure 1, in Step 4 it is recommended to choose the lowest possible excitation level on the performance B-parabolas (most frequently n/c=0,2 resp. 0,35) and corresponding couples of B-parabolas in Fig. 11. Procedure 2 is used for plant identification and PID controller design. M is specified from the given couple (max;ts) using the chosen couple of Bparabolas, however its increased value M´ given by (42) is to be supplied in the design algorithm thus minimizing effect of the time delay on closed-loop dynamics.

#### **Example 2**

Using the sine-wave method, ideal PID controllers (4a) are to be designed for the distillation column modelled by the transfer function GB(s)

$$\text{G}\_{B}(\text{s}) = \frac{K\_{B}e^{-D\_{B}s}}{T\_{B}s+1} = \frac{1,11e^{-6,5s}}{3,25s+1} \tag{44}$$

Control objectives are the same as in Example 1.

#### **Solution**


´

The only modification in using the PID tuning rules in Tab. 9 is that increased required

To lessen the impact of time delay on closed-loop dynamics, it is recommended to use the

Time delay D can easily be specified during critical frequency identification as the time D=Ty-Tu, that elapses since the start of the test at time Tu until time Ty, when the system output starts responding to the excitation signal u(t). A small added phase φD=-nD due to time delay can be secured by choosing the smallest possible n attenuating effect of D in (43)

Therefore, when designing PID controller for time delayed systems according to Procedure 1, in Step 4 it is recommended to choose the lowest possible excitation level on the performance B-parabolas (most frequently n/c=0,2 resp. 0,35) and corresponding couples of B-parabolas in Fig. 11. Procedure 2 is used for plant identification and PID controller design. M is specified from the given couple (max;ts) using the chosen couple of Bparabolas, however its increased value M´ given by (42) is to be supplied in the design

Using the sine-wave method, ideal PID controllers (4a) are to be designed for the distillation

6,5 1,11 ( ) 1 3,25 1 *D sB <sup>s</sup> <sup>B</sup>*

1. Critical frequency of the plant is c=0,3521[rad/s]. Based on comparison of critical frequencies, GB(s) is 500-times slower than GA(s). Required settling time is ts=τs/c=

2. Because DB/TB=2>1, the plant is a so-called "dead-time dominant system". Due to a large the time delay, it is necessary to choose the lowest possible excitation frequency n to minimize the added phase nDB in (43). Hence, for the required performance (max2;τs)=(5%;12) (Design No. 2) we choose the B0,2 parabolas in Fig. 11 at the lowest possible level n/c=0,2 to find (M2;n2)=(70;0,2c). The added phase is

*Ts s* 

*B Ke e G s*

algorithm thus minimizing effect of the time delay on closed-loop dynamics.

*B*

The phase delay nD increases with increasing frequency of the sinusoidal signal n.

 

´

180 *M nD* (41)

*<sup>M</sup> M nD* (42)

180 *M nD* (43)

(44)

and the controller working angle Θ is computed using the relation

´ 

phase margin is to be specified (Bucz, 2011)

smallest possible added phase φD=-nD.

and subsequently in the PID controller design.

column modelled by the transfer function GB(s)

Control objectives are the same as in Example 1.

=12/0,3521[s]=34,08[s].

**Discussion** 

**Example 2** 

**Solution** 

n2DB(180/)=0,2cDB(180/)=0,2.0,3521.6,5.180/=26,2, hence the phase supplied to the PID design algorithm is ´ M2=M2+n2DB(180/)=70+26,2=96,2 (instead of M2=70 for a delay-free system). The required performance (max1;τs)=(30%;12) (Design No. 1) can be achieved by choosing (M1;n1)=(55;0,35c) from the B0,35 parabolas in Fig. 11 (i.e. n/c=0,35). The phase margin ´ M1=55+45,9 supplied into the design algorithm was increased by n1DB(180/)=0,35cDB(180/)=0,35.0,3521.6,5.180/= =45,9 compared with M1=55 in case of delay-free system.


Fig. 13. a) Open-loop Nyquist plots; b) closed-loop step responses of the distillation column, required performance max1=30%, max2=5% and τs=12
