**3. Advanced PID controller design method based on sine-wave identification**

The presented method is applicable for linear stable SISO systems even with unknown mathematical model. The control objective is to provide required maximum overshoot max and settling time ts of the process variable y(t). The method enables the designer to prescribe max and ts within following ranges (Bucz et al., 2010b, 2010c), (Bucz, 2011)


where c is the plant critical frequency. The PID controller design provides guaranteed phase margin M. The tuning rule parameter is a suitably chosen point of the plant frequency response obtained by a sine-wave signal with excitation frequency n. The designed controller then moves this point into the gain crossover with the required phase margin M. With respect to engineering requirements, the pair (n;M) is specified on the closed-loop step response in terms of ηmax and ts according to parabolic dependencies in Fig. 11 and Fig. 14-16. A multipurpose loop for the proposed sine-wave method is in Fig. 7.

Fig. 7. Multipurpose loop for identification and control using the sine-wave method

#### **3.1 Plant identification by a sinusoidal excitation input**

By switching SW into "4", the loop in Fig. 7 opens; a stable plant with unknown model G(s) is excited by a persistent sinusoid u(t)=Unsin(nt) (Fig. 8a) where Un denotes the amplitude and n excitation frequency. The plant output y(t)=Ynsin(nt+) is also a persistent sinusoid with the same frequency n, amplitude Yn and phase shift with respect to the input excitation sinusoid (Fig. 8b). From the stored records of y(t) and u(t) it is possible to read-off the amplitude Yn and phase shift n and thus to identify a particular point of the plant frequency response G(j) under excitation frequency n with coordinates G≡G(jn)

$$\mathbf{G}(jco\_n) = \left| \mathbf{G}(jco\_n) \right| e^{j \arg \mathbf{G}(o\_n)} = \frac{Y\_n(oo\_n)}{U\_n(oo\_n)} e^{j \arg \mathbf{G}(o\_n)} \tag{19}$$

where =argG(n). The point G(jn) can be plotted in the complex plane (Fig. 8c).

Fig. 8. Time responses of a) u(t); b) y(t), and c) location of G(jn) in the complex plane

**3. Advanced PID controller design method based on sine-wave identification**  The presented method is applicable for linear stable SISO systems even with unknown mathematical model. The control objective is to provide required maximum overshoot max and settling time ts of the process variable y(t). The method enables the designer to prescribe

where c is the plant critical frequency. The PID controller design provides guaranteed phase margin M. The tuning rule parameter is a suitably chosen point of the plant frequency response obtained by a sine-wave signal with excitation frequency n. The designed controller then moves this point into the gain crossover with the required phase margin M. With respect to engineering requirements, the pair (n;M) is specified on the closed-loop step response in terms of ηmax and ts according to parabolic dependencies in Fig. 11 and Fig. 14-16. A multipurpose loop for the proposed sine-wave method is in Fig. 7.

Fig. 7. Multipurpose loop for identification and control using the sine-wave method


Sine-wave generator

PID controller

frequency response G(j) under excitation frequency n with coordinates G≡G(jn)

*<sup>Y</sup> Gj Gj e <sup>e</sup>*

where =argG(n). The point G(jn) can be plotted in the complex plane (Fig. 8c).

Fig. 8. Time responses of a) u(t); b) y(t), and c) location of G(jn) in the complex plane

Tn

 

y(t)

t

By switching SW into "4", the loop in Fig. 7 opens; a stable plant with unknown model G(s) is excited by a persistent sinusoid u(t)=Unsin(nt) (Fig. 8a) where Un denotes the amplitude and n excitation frequency. The plant output y(t)=Ynsin(nt+) is also a persistent sinusoid with the same frequency n, amplitude Yn and phase shift with respect to the input excitation sinusoid (Fig. 8b). From the stored records of y(t) and u(t) it is possible to read-off the amplitude Yn and phase shift n and thus to identify a particular point of the plant

w(t) e(t) u(t) y(t)

arg ( ) arg ( ) ( ) () () ( ) *n n j G n n j G n n*

*n n*

t

Im Re

n n Y U

G(jn)

(19)

SW

<sup>5</sup>G(s)

4

*U*

Yn

**3.1 Plant identification by a sinusoidal excitation input** 

Tn=2/<sup>n</sup>

u(t)

Un

max and ts within following ranges (Bucz et al., 2010b, 2010c), (Bucz, 2011) max0%; 90% and ts6,5/c; 45/c for systems without integrator, max9,5%; 90% and ts11,5/c; 45/c for systems with integrator,

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).

#### **3.2 Sine-wave method tuning rules**

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 follows

$$\left| \left| \mathbf{G} (j o\_a^\*) \right| \mathbf{G}\_R (j o\_a^\*) \right| = 1; \quad \arg \mathbf{G} (o\_a^\*) + \arg \mathbf{G}\_R (o\_a^\*) = -180^\circ + \phi\_M \tag{20}$$

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

$$\mathbf{G}\_R(j\rho o) = \mathbf{K} + j\mathbf{K} \left[ T\_d o o - \frac{\mathbf{1}}{T\_i o o} \right]; \quad \mathbf{G}\_R(j\rho o) = \left| \mathbf{G}\_R(j\rho o) \right| e^{j\Theta} = \left| \mathbf{G}\_R(j\rho o) \right| \cos \Theta + j \left| \mathbf{G}\_R(j\rho o) \right| \sin \Theta \tag{21}$$

coefficients of PID controller can be obtained from the complex equation

$$K + jK \left[ T\_d o\_a^\star - \frac{1}{T\_i o\_a^\star} \right] = \frac{\cos \Theta}{\left| \mathbf{G} (j o\_a^\star) \right|} + j \frac{\sin \Theta}{\left| \mathbf{G} (j o\_a^\star) \right|} \tag{22}$$

at =a\* using the substitution GR(ja\*)=1/G(ja\*) resulting from the amplitude condition (20a). The complex equation (22) is solved as a set of two real equations

$$K = \frac{\cos \Theta}{\left| \overline{\mathbf{G}(j o o\_a^\*)} \right|} ; \quad K \left| T\_d o\_a^\* - \frac{1}{T\_i o\_a^\*} \right| = \frac{\sin \Theta}{\left| \overline{\mathbf{G}(j o o\_a^\*)} \right|} \tag{23}$$

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 after some manipulations

$$T\_d^2 \left(\boldsymbol{\alpha}\_a^\*\right)^2 - T\_d \boldsymbol{\alpha}\_a^\* \text{tg}\boldsymbol{\Theta} - \frac{1}{\beta} = 0\tag{24}$$

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

$$T\_d = \frac{t\text{g}\Theta}{2\alpha\_a^\*} + \frac{1}{\alpha\_a^\*} \sqrt{\frac{t\text{g}^2\Theta}{4} + \frac{1}{\beta}}; \quad \Theta = -180^\circ + \phi\_M - \arg\text{G}(\phi\_a^\*) \tag{25}$$

where =argGR(a\*) is found from the phase condition (20b). Thus, using the PID controller with coefficients {K;Ti=Td;Td}, the identified point G(jn) of the plant frequency response with coordinates (19) can be moved on the unit circle M1 into the gain crossover LA≡L(ja\*); the required phase margin M is guaranteed if the following identity holds between the excitation and amplitude crossover frequencies n and a\*, respectively

$$
\alpha\_a^\* = \alpha\_n \tag{26}
$$

PID Controller Design for Specified Performance 17

The working range (30) can be interpreted by means of an imaginary transparent triangular ruler turned as in Fig. 9b; its segments to the left and right of the axis of symmetry represent the PD and PI working ranges, respectively. Put this ruler on Fig. 9a, the middle of the hypotenuse on the complex plane origin and turn it so that its axis of symmetry merges with the ray (0,G). Thus, the ruler determines in the complex plane the cross-hatched area representing the full working range of the PID controller argument. The controller type is chosen depending on the situation of the ray (0,LA) forming the angle M with the negative real halfaxis: situation of the ray (0,LA) in the left-hand-sector suggests PD controller, and in the right-hand sector the PI controller. The case when the phase margin M is achievable using both PI or PID controller is shown in Fig. 9b (Bucz et al., 2010b, 2011), (Bucz, 2011).

 

Im

**0** 

**G n**

**G(jn) L(jn)** 

**-1 1** 

Re

**G(j)** 

**PI**

**LA**

*Mj* 20 ,30 ,40 ,50 ,60 ,70 ,80 ,90 , j=1...8 (31)

 *c cc cc c* , k=1...6 (32)

**L(j)** 

Fig. 9. a) Graphical interpretation of M, a\* and shifting G into LA at a\*=n; b) controller structure selection with respect to location of G and LA using the "triangle ruler" rule

Re

**3.4 Evaluation of closed-loop performance under the sine-wave type PID controller**  This subsection answers the following question: how to transform required the maximum overshoot max and settling time ts into the couple of frequency-domain parametres (n,M)

Looking for appropriate transformation : (max,ts)(n,M) we have considered typical

 

Elements of (32) divided by the plant critical frequency c determine the set of so-called

split into 5 equal sections n=0,15c; let us generate the set of excitation frequencies

*nk* 0,2 ;0,35 ;0,5 ;0,65 ;0,8 ;0,95

 

needed for identification and PID controller coefficients tuning (Bucz, 2011)?

**G(j)**

**3.4.1 Systems without integrator** 

**LA**

**L(j)** 

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

**M1**

**a \***

phase margins M given by the set

excitation levels

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

**M1**

*PID PI PD* 90 ,0 0 , 90 90 , 90 (30)

**n**

Im

**0** 

**G n**

**PD** 

**G(jn) L(jn)** 

**-1 1** 

Thus

$$\left| \mathbf{G} (j o o\_a^\*) \right| = \left| \mathbf{G} (j o o\_n) \right|; \quad \arg \mathbf{G} (o o\_a^\*) = \arg \mathbf{G} (o o\_n) = \boldsymbol{\varphi}; \quad \boldsymbol{\Theta} = -180^\circ + \phi\_M - \boldsymbol{\varphi} \tag{27}$$

and coordinates of the gain crossover LA are

$$L\_A \equiv L(j o o\_a^\* \equiv j o o\_n) = \left\lceil \left| L(j o o\_n) \right| \arg L(o o\_n) \right\rceil = \left\lceil \left| 1 \right| \left. -180^\circ + \phi\_M \right\rceil \right\rceil \tag{28}$$

Substituting (27a) and (27b) into (23a) and (23b), respectively, and (26) into (25a), tuning rules in Table 9 are obtained (Bucz et al., 2010a, 2010b, 2010c, 2011), (Bucz, 2011). Resulting PID controller coefficients guarantee required phase margin M for =4.


Table 9. PI, PD and PID controller tuning rules according to the sine-wave method

Note that PI controller tuning rules were derived for Td=0, and PD tuning rules for Ti in (21a). The excitation frequency is taken from the interval (Bucz et al., 2011), (Bucz, 2011)

$$a o\_{\rm in} \in \{0, 2o\_{\rm c}; 0, 95o\_{\rm c}\} \tag{29}$$

obtained empirically by testing the sine-wave method on benchmark examples (Åström & Hägglund, 2000). Shifting the point G(jn)=G(jn)ej into the gain crossover LA(jn) on the unit circle M1 is depicted in Fig. 9a.

#### **3.3 Controller structure selection using the "triangle ruler" rule**

The argument Θ appearing in tuning rules in Tab. 9 indicates, what angle is to be contributed to the identified phase φ by the controller at n to obtain the resulting open-loop phase (-180°+M) needed to provide the required phase margin M. The working range of PID controller argument is the union of PI and PD controllers phase ranges symmetric with respect to 0

where =argGR(a\*) is found from the phase condition (20b). Thus, using the PID controller with coefficients {K;Ti=Td;Td}, the identified point G(jn) of the plant frequency response with coordinates (19) can be moved on the unit circle M1 into the gain crossover LA≡L(ja\*); the required phase margin M is guaranteed if the following identity holds between the

> \* *a n*

*a n* 

Substituting (27a) and (27b) into (23a) and (23b), respectively, and (26) into (25a), tuning rules in Table 9 are obtained (Bucz et al., 2010a, 2010b, 2010c, 2011), (Bucz, 2011). Resulting

> 1 *n tg*

<sup>1</sup>

*Td*

Note that PI controller tuning rules were derived for Td=0, and PD tuning rules for Ti in (21a). The excitation frequency is taken from the interval (Bucz et al., 2011), (Bucz, 2011)

0,2 ;0,95

 

*n cc* 

obtained empirically by testing the sine-wave method on benchmark examples (Åström &

The argument Θ appearing in tuning rules in Tab. 9 indicates, what angle is to be contributed to the identified phase φ by the controller at n to obtain the resulting open-loop phase (-180°+M) needed to provide the required phase margin M. The working range of PID controller argument is the union of PI and PD controllers phase ranges symmetric with

Table 9. PI, PD and PID controller tuning rules according to the sine-wave method

 ;

 

> *n tg*

2 4 *n n tg tg* 

*<sup>M</sup>* (28)

;0

<sup>2</sup> 1 1

 

> 

(29)

into the gain crossover LA(jn) on the

;

 ; \* arg ( ) arg ( ) *G G* 

\* *L Lj j Lj L A an n n* ( ) ( ) ,arg ( ) 1 , 180

 

PID controller coefficients guarantee required phase margin M for =4.

cos ( ) *G j <sup>n</sup>* 

cos ( ) *G j <sup>n</sup>* 

cos ( ) *G j <sup>n</sup>* 

**3.3 Controller structure selection using the "triangle ruler" rule** 

roller K Ti Td

(26)

*<sup>M</sup>* (27)

Range of ; =180+M

2 

0; 2 

2 2 

 

 180 

excitation and amplitude crossover frequencies n and a\*, respectively

\* () () *Gj Gj* 

and coordinates of the gain crossover LA are

*a n* 

Cont-

Hägglund, 2000). Shifting the point G(jn)=G(jn)ej

Thus

No. Design

52. Sine-wave

53. Sine-wave

54. Sine-wave

respect to 0

method, year

method, 2010 PI

method, 2010 PD

method, 2010 PID

unit circle M1 is depicted in Fig. 9a.

$$
\Theta\_{\rm PID} \in \Theta\_{\rm PI} \cup \Theta\_{\rm PD} = \left( -90^{\circ}, 0^{\circ} \right) \cup \left( 0^{\circ}, +90^{\circ} \right) = \left( -90^{\circ}, +90^{\circ} \right) \tag{30}
$$

The working range (30) can be interpreted by means of an imaginary transparent triangular ruler turned as in Fig. 9b; its segments to the left and right of the axis of symmetry represent the PD and PI working ranges, respectively. Put this ruler on Fig. 9a, the middle of the hypotenuse on the complex plane origin and turn it so that its axis of symmetry merges with the ray (0,G). Thus, the ruler determines in the complex plane the cross-hatched area representing the full working range of the PID controller argument. The controller type is chosen depending on the situation of the ray (0,LA) forming the angle M with the negative real halfaxis: situation of the ray (0,LA) in the left-hand-sector suggests PD controller, and in the right-hand sector the PI controller. The case when the phase margin M is achievable using both PI or PID controller is shown in Fig. 9b (Bucz et al., 2010b, 2011), (Bucz, 2011).

Fig. 9. a) Graphical interpretation of M, a\* and shifting G into LA at a\*=n; b) controller structure selection with respect to location of G and LA using the "triangle ruler" rule

#### **3.4 Evaluation of closed-loop performance under the sine-wave type PID controller**

This subsection answers the following question: how to transform required the maximum overshoot max and settling time ts into the couple of frequency-domain parametres (n,M) needed for identification and PID controller coefficients tuning (Bucz, 2011)?

#### **3.4.1 Systems without integrator**

Looking for appropriate transformation : (max,ts)(n,M) we have considered typical phase margins M given by the set

$$\left(\phi\_{M\circ}\right) = \left(20^{\circ}, 50^{\circ}, 40^{\circ}, 50^{\circ}, 60^{\circ}, 70^{\circ}, 80^{\circ}, 90^{\circ}\right), \text{ j=1...8} \tag{31}$$

split into 5 equal sections n=0,15c; let us generate the set of excitation frequencies

$$\left\{ \left. \alpha\_{nk} \right\} = \left\{ 0, 2a\_{\odot}; 0, 35a\_{\odot}; 0, 5a\_{\odot}; 0, 65a\_{\odot}; 0, 8a\_{\odot}; 0, 95a\_{\odot} \right\}, \text{ k} = 1...6 \tag{32}$$

Elements of (32) divided by the plant critical frequency c determine the set of so-called excitation levels

PID Controller Design for Specified Performance 19

Considering (26) resulting from the assumptions of the engineering method, the settling

*n*

(35)

(36)

(37)

similar to (17c) (Hudzovič, 1989), is the curve factor of the step response. In (17c) valid for a 2nd order closed-loop,is from the interval (1;4) and depends on the relative damping (Hudzovič, 1989). In case of the proposed sine-wave method, varies in a considerably broader interval (0,5;16) found empirically, and strongly depends on M, i.e. =f(M) at the given excitation frequency n. To examine closed-loop settling times of plants with various

> *s sc t*

Substituting n=c into (35), the following relation for the relative settling time is obtained

where ts is related to the critical frequency c. By substituting c in (37) its left-hand side is constant for the given plant, independent of n. Fig. 11b depicts (37b) empirically evaluated for different excitation frequencies nk; it is evident that at every excitation level k with increasing phase margin M the relative settling time τs first decreases and after achieving its minimum s\_min it increases again. Empirical dependencies in Fig. 11 were approximated by quadratic regression curves and called B-parabolas. B-parabolas are a useful design tool to carry out the transformation :(max,ts)(n,M) that enables choosing appropriate values of phase margin and excitation frequencies M and n, respectively, to provide performance specified in terms of maximum overshoot max and settling time ts (Bucz et al., 2011). Note

*<sup>s</sup>*

that pairs of B-parabolas at the same level (Fig. 11a, Fig. 11b) are always to be used. **Procedure 1. Specification of M and n from max and ts from B-parabolas prior to** 

2. From the required settling time ts calculate the relative settling time τs=cts. 3. On the vertical axis of the plot in Fig. 11b find the value of τs calculated in Step 2.

6. Find M from Step 5 on the horizontal axis of the plot in Fig. 11a.

multipurpose loop in Fig. 7 (SW in position "3").

4. Choose the excitation level (e.g. 5=n5/c=0,8).

with the chosen excitation level found in Step 4.

performance measures ηmax and ts are satisfied.

chosen excitation level found in Step 4.

1. Set the PID controller into manual mode. Find the plant critical frequency c using the

5. For τs, find the corresponding phase margin M on the parabola τs=f(M,n) with the

7. For M, find the corresponding maximum overshoot ηmax on the parabola ηmax=f(M,n)

8. If the found ηmax is inappropriate, repeat Steps 4 to 7 for other parabolas τs=f(M,n) and ηmax=f(M,n) corresponding to other levels k=nk/c (related with the choice 5=n5/c=0,8 for k=0,2;0,35;0,50;0,65;0,95, k=1...4,6). Repeat until both the required

*s*

*t*

dynamics, it is advantageous to define the relative settling time (Bucz et al., 2011)

*s c t*

time can be expressed by the relation

**designing the controller** 

$$\left\{\sigma\_{k} = \alpha\_{nk} / \alpha\_{\mathcal{C}}\right\} \Rightarrow \left\{\sigma\_{k}\right\} = \left\{0, 2; 0, 35; 0, 5; 0, 65; 0, 8; 0, 95\right\}, \text{ k} = 1...6 \tag{33}$$

Fig. 10 shows closed-loop step responses under PID controllers designed for the plant

$$G\_{1}(s) = \frac{1}{(s+1)(0.5s+1)(0.25s+1)(0.125s+1)}\tag{34}$$

for three different phase margins M=40,60,80 each on three excitation levels 1=n1/c=0,2; 3=n3/c=0,5 and 5=n5/c=0,8. Qualitative effect of nk and Mj on closed-loop step response is demonstrated.

Fig. 10. Closed-loop step responses of G1(s) under PID controllers designed for various <sup>M</sup> and <sup>n</sup>

Achieving ts and ηmax was tested by designing PID controller for a vast set of benchmark examples (Åström & Hägglund, 2000) at excitation frequencies and phase margins expressed by a Cartesian product Mj×nk of (31) and (32) for j=1...8, k=1...6. Acquired dependencies ηmax=f(M,n) and ts=(M,n) are plotted in Fig. 11 (Bucz et al., 2010b, 2011).

Fig. 11. Dependencies: a) ηmax=f(M,n); b) τs=cts=f(M,n) for nk×Mj, j=1...8, k=1...6 (relative settling time τs is ts weighed by the critical frequency c of the plant)

<sup>1</sup> ( ) ( 1)(0,5 1)(0,25 1)(0,125 1)

for three different phase margins M=40,60,80 each on three excitation levels 1=n1/c=0,2; 3=n3/c=0,5 and 5=n5/c=0,8. Qualitative effect of nk and Mj on

Closed loop step responses, <sup>n</sup>

Fig. 10. Closed-loop step responses of G1(s) under PID controllers designed for various <sup>M</sup>

Fig. 11. Dependencies: a) ηmax=f(M,n); b) τs=cts=f(M,n) for nk×Mj, j=1...8, k=1...6

Relative settling time s=ct

s [-]

Dependencies max=f(M,n), for systems without integrator, =4 Dependencies τs=f(M,n), for systems without integrator, =4

(relative settling time τs is ts weighed by the critical frequency c of the plant)

Achieving ts and ηmax was tested by designing PID controller for a vast set of benchmark examples (Åström & Hägglund, 2000) at excitation frequencies and phase margins expressed by a Cartesian product Mj×nk of (31) and (32) for j=1...8, k=1...6. Acquired dependencies ηmax=f(M,n) and ts=(M,n) are plotted in Fig. 11 (Bucz et al., 2010b, 2011).

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time [s]

*ss s s*

Fig. 10 shows closed-loop step responses under PID controllers designed for the plant

*<sup>k</sup>* 0,2;0,35;0,5;0,65;0,8;0,95 , k=1...6 (33)

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

<sup>45</sup> max <sup>M</sup> <sup>n</sup>

n =0,2<sup>c</sup> n =0,35<sup>c</sup> n =0,5<sup>c</sup> n =0,65<sup>c</sup> n =0,8<sup>c</sup> n =0,95<sup>c</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>5</sup>

Phase margin M [°]

Controlled variable y(t)

=0,5<sup>c</sup>

M=40° M=60° M=80° (34)

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time [s]

Closed loop step responses, <sup>n</sup>

=0,8<sup>c</sup>

M=40° M=60° M=80°

<sup>0</sup> <sup>5</sup> <sup>10</sup> <sup>15</sup> <sup>20</sup> <sup>0</sup>

Time [s]

<sup>90</sup> max <sup>M</sup> <sup>n</sup>

n =0,2<sup>c</sup> n =0,35<sup>c</sup> n =0,5<sup>c</sup> n =0,65<sup>c</sup> n =0,8<sup>c</sup> n =0,95 <sup>c</sup>

Closed loop step responses, <sup>n</sup>

and <sup>n</sup>

Maximum overshoot max [%]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Controlled variable y(t)

 *k nk c* 

1

=0,2<sup>c</sup>

M=40° M=60° M=80°

<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>0</sup>

Phase margin M [°]

0.2 0.4 0.6 0.8 1 1.2 1.4 1.6

Controlled variable y(t)

closed-loop step response is demonstrated.

*G s*

Considering (26) resulting from the assumptions of the engineering method, the settling time can be expressed by the relation

$$t\_s = \frac{\mathcal{Y}\pi}{o\nu\_n} \tag{35}$$

similar to (17c) (Hudzovič, 1989), is the curve factor of the step response. In (17c) valid for a 2nd order closed-loop,is from the interval (1;4) and depends on the relative damping (Hudzovič, 1989). In case of the proposed sine-wave method, varies in a considerably broader interval (0,5;16) found empirically, and strongly depends on M, i.e. =f(M) at the given excitation frequency n. To examine closed-loop settling times of plants with various dynamics, it is advantageous to define the relative settling time (Bucz et al., 2011)

$$
\boldsymbol{\tau}\_{\rm s} = \mathbf{t}\_{\rm s} \boldsymbol{o}\_{\rm c} \tag{36}
$$

Substituting n=c into (35), the following relation for the relative settling time is obtained

$$
\pi \, t\_s a\_{\ell} = \frac{\pi}{\sigma} \gamma \implies \tau\_s = \frac{\pi}{\sigma} \gamma \tag{37}
$$

where ts is related to the critical frequency c. By substituting c in (37) its left-hand side is constant for the given plant, independent of n. Fig. 11b depicts (37b) empirically evaluated for different excitation frequencies nk; it is evident that at every excitation level k with increasing phase margin M the relative settling time τs first decreases and after achieving its minimum s\_min it increases again. Empirical dependencies in Fig. 11 were approximated by quadratic regression curves and called B-parabolas. B-parabolas are a useful design tool to carry out the transformation :(max,ts)(n,M) that enables choosing appropriate values of phase margin and excitation frequencies M and n, respectively, to provide performance specified in terms of maximum overshoot max and settling time ts (Bucz et al., 2011). Note that pairs of B-parabolas at the same level (Fig. 11a, Fig. 11b) are always to be used.

#### **Procedure 1. Specification of M and n from max and ts from B-parabolas prior to designing the controller**


PID Controller Design for Specified Performance 21

4. Using the PID controller designed for (M1;n1)=(50;0,5c), the point GA(j0,5c) is moved

5. Achieved performance according to the closed-loop step response in Fig. 12b (dashed line) is max1\*=29,7%, ts1\*=58,4[ms]. Performance in terms of max2\*=4,89%, ts2\*=60,5[ms] identified from the closed-loop step response in Fig. 12b (dotted line) fulfils the

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

phase margin M1=180-130=50 (dashed line in Fig. 12a). The point GA(j0,8c) has been

0.5

Controlled variable y(t)

Fig. 12. a) Open-loop Nyquist plots; b) closed-loop step responses of the operational

 ´ 

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

> 

> > ´

is the identified phase of the plant including the time delay. The added phase φD=-nD can

 

 

*D M* 180 (39)

*<sup>D</sup>* (40)

**M1**

yielding the phase margin M2=180-110=70 (dotted line in Fig. 12a).

1

1.5

0.5

Controlled variable y(t)

1

1.5

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

Closed-loop step response of the operational amplifier

<sup>0</sup> 0.05 0.1 0.15 0.2 0.25 0.3 <sup>0</sup>

Closed-loop step response of the operational amplifier

on the unit circle M1, which verifies achieving the

by the PID controller designed for (M2;n2)=(80;0,8c)

Time [s]

**M2=70, n2=0,8<sup>c</sup>**

**M1=50, n1=0,5<sup>c</sup>**

**max1\*=29,7%, ts1\*=58,4[ms]** 

**max2\*=4,89%, ts2\*=60,5[ms]** 

Time [s]

Real Axis

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

be associated with the required phase margin M


**LA2(j0,8c)**

Open-loop Nyquist plots, M1=50, n1=0,5c; M2=70, n2=0,8<sup>c</sup>

into the gain crossover LA1(j0,5c)=1e-j130

moved into LA2(j0,8c)=1e-j110

performance requirements.

**LA2(j)**

**GA(j)** 

**GA(j0,5c) GA(j0,8c)**

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

Imaginary Axis

**70 50**

**LA1(j)** 

the identification.

**LA1(j0,5c)** 

**3.4.2 Systems with time delay** 


9. Calculate the excitation frequency n according to the relation n=c using the critical frequency c (from Step 1) and the chosen excitation level (from Step 4).
