3.2.2. Power

Utilising the location of critical infrastructure (such as electricity substations) and comparing against predicted flood outlines can help infer power outages associated with flood events. Substations located within the CAFRA and SWMP flood model outlines within central Bristol, St Philips and Ashton were analysed. Specific information relating to the criticality of individual major substations cannot be disclosed in greater detail due to security reasons, but the number of potential substations affected and the percentage over the total studied are given in Table 3 for the three Average Water Depth (AWD) categories established. In the most critical case (AWD > 1.60 m), the increase in severe flooding occurrences rises from 2 to 76 when increasing the return period from T20 to T1000, meaning a 17.2% increase over the total number of 11 kV substations. These results are also displayed in Figure 11.

**Table 3.** Number of substations affected resulting from the electrical sector analysis, and the percentage over the total of substations studied, according to different return periods, type of substations and

**Figure 11.** Map of substations potentially affected by flooding for various return periods in the 2115 future case scenarios. The points are sized according to the three categories of flood event magnitude established and coloured according to the year. This map shows the evolution of the flooding depending on the return period.


*3.4. Coincident Bands Evaluation Methodology*

When the optimization algorithms employed in *PF2* and *PF3* selected the bands, different bands were identified for each test image. Thus, the six HS test images were evaluated employing the same selected bands, i.e., using the coincident and non-coincident selected bands in each test image obtained in the leave-one-patient-out cross-validation for the SVM training and classification. The procedure was as follows:

This process was repeated until reaching the possible six coincidences. However, in our case, a maximum of three levels of coincidence were obtained.


**3. Optical sensing of heavy metal ions**

#### **3.1 Spectrophotometric/ Fluorometric methods**

Wide range of materials including organic molecules, nanomaterials, hybrid of earlier two, and quantum dots are used to sense HMIs by generating either color or fluorescence signal. Simultaneous signal generation for multiple HMIs is more often observed in electrochemical sensing. Whereas in case of optical methods, generally, signal will be generated exclusively due to target HMI. Hence, below are the sections categorized w.r.t. type of the material used for sensing. Lead i.e. Pb is considered as an example to explain the commonly followed strategies for the optical sensing of HMIs.

#### *3.1.1 Cyclodextrins*

In the year 1996 Czarnik et al. proposed a bench mark work for the fluorescence sensing of Pb2+ ions using a small molecule, heteroatom containing ligands. On complexation with Pb2+ ions proposed ligand exhibited enhanced fluorescence signal for about 15-fold [63]. Cyclodextrin molecule in binary solvent system exhibited 20-fold enhancement in the fluorescence signal after the addition of Pb2+ ions [64]. Similar cyclodextrin molecules are proposed by Chen et al. and Hayashita et al. for fluorometric sensing of the HMI as shown in **Figure 3** [65, 66].

#### *3.1.2 Organic molecules*

Color resulting dibromo-p-methyl-methylsulfonazo molecule was explored by Li et al. for spectrophotometric detection of the Pb2+ions [67]. Inspired by this strategy Meng et al. worked out similar organic molecule for the colorimetric sensing of HMI [68].

#### **Figure 3.**

*(A) Representation of the structure of the ligand. (B) Image of the fluorescence response of the proposed ligand. (C) Binding mechanism of the Pb2+ ions with the proposed ligand (reprinted with permission from [65] copyright 2002 American Chemical Society).*

#### *3.1.3 Rhodamines*

Rhodamine family molecules and its derivatives have been utilized extensively for the optical sensing of not only Pb2+ ions but most of the HMIs. Majority of such works result the optical signal through spirolactum ring opening or forming mechanism after interaction between the rhodamine and HMI. Yoon et al. reported rhodamine B derivative for both fluorometric and colorimetric sensing of Pb2+ ion in methyl cyanide medium [69].

#### *3.1.4 Calixarenes*

Calixarene family molecules are another category contributed significantly for the HMIs sensing. Calixarene structures generally from a dative bond with HMI through the functional groups to result an optical signal. Calix [4]arene derivatives are reported for the Pb2+ ion sensing can be observed as a regular method [70]. Switchable i.e. on–off–on fluorescent sensor observed due to the binding capacity of the calixarene molecule with two different cations i.e. Pb2+ and K+ as shown in **Figure 4**. Observed phenomenon is due to the interaction of the K<sup>+</sup> ions with the molecular orbitals whereas that is absent in the Pb2+ ions case [71].

Apart from the above-mentioned categories there exist another variety of organic and bioorganic molecule extensively studied for the HMI. To name a few polymers based, peptide based, DNAzyme based sensors.

#### *3.1.5 Carbon quantum dots*

Carbon quantum dots (CQDs): Again, inherent functional groups on its surface and their deceptive role in anchoring desired molecules, nanoparticles etc.

*Electrochemical and Optical Methods for the Quantification of Lead and Other Heavy Metal… DOI: http://dx.doi.org/10.5772/intechopen.95085*

#### **Figure 4.**

*Representation of the switchable fluorescence of the calix [4] arene derivative and its binding mechanism with Pb2+ and K+ (reprinted with permission from [71] copyright 2004 American Chemical Society).*

made CQDs as emerging optosensors for HMIs. It is prerequisite to understand the mechanism of sensing of HMIs using CQDs before considering the individual articles on the same. Interaction between the light (electromagnetic radiation) of suitable wavelength and CQDs leads to the generation of charge carriers i.e. due to the excitation of electron from valence band to the conduction band. Generated charge carriers are utilized in the sensing of HMIs depending on the interaction of HMI with the ligand on the CQDs surface. Photo induced charge transfer, fluorescence enhancing/quenching, inner filter effects, phosphorescence etc. [72]. Sodium citrate and polyacrylamide sodium citrate resulted blue fluorescent CQDs which was quenched upon the addition of Pb2+ ion. By adding pyrophosphoric acid to the quenched solution, fluorescence was regenerated. This fluorescence off–on method was able to detect the Pb2+ of the order of 4.6 nM [73]. CQDs synthesized from the green approach using *Lantana camara* berries were exhibited sensitive and wide linear range up to 200 nM Pb2+ ion concentration [74]. CQDs synthesized from chocolate source also exhibited nanomolar HMI detection [75]. Household sugar was used to prepare the CQDs and found to be sensitive and selective for the naked eye sensing of Pb2+ ions in water [76].

#### *3.1.6 Nanoparticles*

Nanoparticles forms an interesting domain for the HMIs sensing. Though there exist a wide range of metal nanoparticles majority of the work has been done on Ag and noble metals Pt, Au nanoparticles in spite of their costly affair. Surface plasmon resonance is the corner stone of the optical sensing of HMIs using Ag, Au, and Pt. Hupp et al. used 11-mercaptoundecanoic acid capped Au nanoparticles as color generating agent for the detection of Pb2+, Cd2+, and Hg2+ ions [77]. Thomas et al. proposed quite a different approach wherein they started with Au3+/Ag+ ions and gallic acid solution. After the addition of Pb2+ Au/Ag nanoparticles are formed to result pink or red color respectively [78]. Non-aggregation-based sensor is developed by Huang et al. Originally Gold nanoparticles surrounded by thiosulphate and

2-mercaptoethanol in a solution. After the addition of Pb2+ due to the formation of Pb-Au alloy gold nanoparticles dissolved into solution and hence Surface plasmon resonance is decreased [79]. Opposite to that, i.e. aggregation based, sensing method was developed by Su et al. using glutathione modified Au nanoparticles. Upon the addition of HMI Au nanoparticles aggregate to result a color change from red to blue [80]. Theme of this work is summarized in the **Figure 5**.

Above discussed category of organic molecules, bioorganic molecules, nanoparticles have also been extensively used for the sensing of remaining HMIs. Functional moieties in the sensing molecules are different for different HMIs. But, conceptual wise it will be a repetition if the discussion is extended for the other HMIs. Ha Na Kim et al. has reviewed all the above discussed type of materials except CQDs for HMIs with an exhaustive literature survey and detailed discussion [81]. One can get more details and literature on CQDs for HMIs sensing in the review article presented by Pooja Devi et al. [72].

**Figure 5.**

*(A) Scheme represents the preparation of glutathione capped Au nanoparticles. (B) Uv–Vis spectra, inset shows the images, (C) TEM images of Au nanoparticles in the absence (a) and presence of Pb2+ ions (b). (D) Images of the Au nanoparticle in the presence of various metal cations and Pb2+ ions (reprinted with permission from [80] copyright 2010 American Chemical Society).*


Figure 4. Photograph of the experimental two-mass resonance model.

192 PID Control for Industrial Processes

In this chapter, classical PID speed and current controllers are used to suppress the resonance vibrations for the two-mass resonance model. In general, the PI controller, which consists of a proportional controller and an integral controller that are placed in parallel to determine the speed error, is used as the angular speed controller. However, because the resonance system has a complex structure, it is difficult to suppress the vibrations using the classic PI controller alone. Therefore, the I-PD controller is used in this chapter and a first-order lag element is also used to increase the degrees of freedom for the controller design. Additionally, a simple PI controller is

### 3. Controller design using coefficient diagram method

#### 3.1. Coefficient diagram method

First, this section explains the coefficient diagram method (CDM) for design of the proposed controller, which is required to suppress the resonance vibrations for the two-mass resonance model. The CDM is an algebraic approach that is used to design the characteristic polynomial directly in the parameter space. In the CDM, a coefficient diagram (CD) is used to perform the controller design. The CD provides the ability to analyze the time response, stability and robustness qualities of the controller using a diagram. In the CD, the vertical axis shows the coefficient of the characteristic polynomial (ai), the stability indices (γi), and the equivalent time constant (τ) logarithmically, while the horizontal axis shows the order i values that correspond to the coefficient. Here, the characteristic polynomial is as shown in Eq. (12).

$$P(\mathbf{s}) = a\_n \mathbf{s}^n + a\_{n-1} \mathbf{s}^{n-1} + \dots + a\_1 \mathbf{s} + a\_0 = \sum\_{i=0}^n a\_i \mathbf{s}^i \tag{12}$$

In the CDM, the stability indices (γi), which are defined in Eq. (13) below, are indicators of the stability of the control system.

$$\gamma\_i = \frac{a\_i^2}{a\_{i+1}a\_{i-1}}, \quad i = 1 \| i - 1 \tag{13}$$

The equivalent time constant τ, which represents the transient response characteristic, is expressed using the following equation:

$$
\pi = \frac{a\_1}{a\_0} \tag{14}
$$

design of the CD for the control system is performed using the differential evolution (DE)

In the proposed design method, the reference value of the equivalent time constant τref is first specified. Then, the coefficients of the characteristic polynomial are calculated using the six controller gains with random initial settings. Each of the coefficients is determined using the

a5 ¼ JMpu þ JMpuKap þ KapKd þ KepuT þ JMpuKaiT þ JMpuTτeω2

a0 ¼ KaiKiω2

The stability indices are then computed using these calculated coefficients and the evaluation function F in Eq. (25) is calculated using the terms from Eqs. (26)–(30). In this case, the evaluation function F consists of an evaluation to match with the set reference value of the equivalent time constant, an evaluation to reduce the change in the next stability index, and an evaluation to match the standard forms of the stability indices γs,i. The weights w1 to w5 of the evaluation functions are set to have values of (w1, w2, w3, w4, w5) = (100, 2, 10, 1, 4), respectively.  The DE method is an optimization search method [12, 13].

a4 ¼ Kepu þ JMpuKai þ KaiKd þ KapKp þ JMpuTω2

r þ KapKi þ KaiKp þ JMpuKapω2

a þ KaiKi þ JMpuKaiω2

a1 ¼ KapKiω2

are subsequently repeated to obtain the optimal controller gains by the DE method.

F ¼ 1

f 1 ¼

f 2 ¼ X2

f 3 ¼

f 4 ¼ X5

f 5 ¼ X6

q

i¼1

r

i¼3

i¼4

r

q

r

a2 ¼ Kepuω2

following equations:

a3 ¼ JMpuω2

The coefficient ai can then be calculated using τ and the stability indices γi as shown in Eq. (15).

$$a\_{i+1} = \frac{a\_0 \pi^i}{\mathcal{V}\_{i-1} \mathcal{V}\_{i-2}^2 \cdots \mathcal{V}\_2^{i-2} \mathcal{V}\_1^{i-1}} \tag{15}$$

In the CDM, use of the standard values of the stability indices is recommended, and these values are listed as follows:

$$
\gamma\_{n-1} = \dots = \gamma\_3 = \gamma\_2 = \text{2.0}, \quad \gamma\_1 = \text{2.5} \tag{16}
$$

This form is called "the standard form of the stability indices."

#### 3.2. Design method for the controller gains using the CDM

The CDM is a very simple and effective method for controller design. However, in higher order systems, it is difficult to complete the design by trial and error alone. In this work, the design of the CD for the control system is performed using the differential evolution (DE) method. The DE method is an optimization search method [12, 13].

3. Controller design using coefficient diagram method

to the coefficient. Here, the characteristic polynomial is as shown in Eq. (12).

n þ an�1s

γi ¼ a2 i aiþ1ai�1

P sð Þ¼ ans

First, this section explains the coefficient diagram method (CDM) for design of the proposed controller, which is required to suppress the resonance vibrations for the two-mass resonance model. The CDM is an algebraic approach that is used to design the characteristic polynomial directly in the parameter space. In the CDM, a coefficient diagram (CD) is used to perform the controller design. The CD provides the ability to analyze the time response, stability and robustness qualities of the controller using a diagram. In the CD, the vertical axis shows the coefficient of the characteristic polynomial (ai), the stability indices (γi), and the equivalent time constant (τ) logarithmically, while the horizontal axis shows the order i values that correspond

In the CDM, the stability indices (γi), which are defined in Eq. (13) below, are indicators of the

The equivalent time constant τ, which represents the transient response characteristic, is

τ ¼ a1 a0

The coefficient ai can then be calculated using τ and the stability indices γi as shown in Eq. (15).

In the CDM, use of the standard values of the stability indices is recommended, and these

The CDM is a very simple and effective method for controller design. However, in higher order systems, it is difficult to complete the design by trial and error alone. In this work, the

i�2⋯γi�2

2 γi�1 1

γn�1 ¼ ⋯ ¼ γ3 ¼ γ2 ¼ 2:0, γ1 ¼ 2:5 (16)

aiþ1 ¼ a0τi γi�1γ2

This form is called "the standard form of the stability indices."

3.2. Design method for the controller gains using the CDM

n�1 þ ⋯ þ a1s þ a0 ¼ Xn

i¼0 ais

, i ¼ 1n~ � 1 (13)

i (12)

(14)

(15)

3.1. Coefficient diagram method

194 PID Control for Industrial Processes

stability of the control system.

values are listed as follows:

expressed using the following equation:

In the proposed design method, the reference value of the equivalent time constant τref is first specified. Then, the coefficients of the characteristic polynomial are calculated using the six controller gains with random initial settings. Each of the coefficients is determined using the following equations:

$$a\_{\overline{\tau}} = f\_{Mpu} T \tau\_e \tag{17}$$

$$a\_{\theta} = f\_{Mpu} \tau\_{\epsilon} + f\_{Mpu} T + f\_{Mpu} K\_{ap} T \tag{18}$$

$$a\_5 = f\_{Mpu} + f\_{Mpu} K\_{ap} + K\_{ap} K\_d + K\_{epu} T + f\_{Mpu} K\_{u\bar{t}} T + f\_{Mpu} T \tau\_\epsilon \omega\_r^2 \tag{19}$$

$$\mathbf{a}\_{4} = \mathbf{K}\_{\rm cpu} + \mathbf{J}\_{\rm Mpu}\mathbf{K}\_{\rm ui} + \mathbf{K}\_{\rm id}\mathbf{K}\_{\rm d} + \mathbf{K}\_{\rm ap}\mathbf{K}\_{\rm p} + \mathbf{J}\_{\rm Mpu}T\boldsymbol{\omega}\_{r}^{2} + \mathbf{J}\_{\rm Mpu}\boldsymbol{\tau}\_{\boldsymbol{\epsilon}}\boldsymbol{\omega}\_{r}^{2} + \mathbf{J}\_{\rm Mpu}\mathbf{K}\_{\rm ap}T\boldsymbol{\omega}\_{r}^{2} \tag{20}$$

$$\boldsymbol{\sigma}\_{3} = \boldsymbol{I}\_{\text{Mpu}} \boldsymbol{\omega}\_{r}^{2} + \boldsymbol{K}\_{\text{ap}} \boldsymbol{K}\_{i} + \boldsymbol{K}\_{\text{ati}} \boldsymbol{K}\_{p} + \boldsymbol{I}\_{\text{Mpu}} \boldsymbol{K}\_{\text{up}} \boldsymbol{\omega}\_{r}^{2} + \boldsymbol{K}\_{\text{up}} \boldsymbol{K}\_{\text{d}} \boldsymbol{\omega}\_{a}^{2} + \boldsymbol{K}\_{\text{cpu}} \boldsymbol{T} \boldsymbol{\omega}\_{a}^{2} + \boldsymbol{I}\_{\text{Mpu}} \boldsymbol{K}\_{\text{ati}} \boldsymbol{T} \boldsymbol{\omega}\_{r}^{2} \tag{21}$$

$$\sigma\_2 = K\_{\rm cpu} \omega\_a^2 + K\_{\rm ai} K\_i + J\_{\rm Mpu} K\_{\rm ai} \omega\_r^2 + K\_{\rm ai} K\_d \omega\_a^2 + K\_{\rm ap} K\_p \omega\_a^2 \tag{22}$$

$$
\omega\_1 = K\_{ap} K\_i \omega\_a^2 + K\_{di} K\_p \omega\_a^2 \tag{23}
$$

$$a\_0 = K\_{di} K\_i \omega\_a^2 \tag{24}$$

The stability indices are then computed using these calculated coefficients and the evaluation function F in Eq. (25) is calculated using the terms from Eqs. (26)–(30). In this case, the evaluation function F consists of an evaluation to match with the set reference value of the equivalent time constant, an evaluation to reduce the change in the next stability index, and an evaluation to match the standard forms of the stability indices γs,i. The weights w1 to w5 of the evaluation functions are set to have values of (w1, w2, w3, w4, w5) = (100, 2, 10, 1, 4), respectively. These steps are subsequently repeated to obtain the optimal controller gains by the DE method.

$$F = \frac{1}{w\_1 f\_1 + w\_1 f\_1 + w\_1 f\_1 + w\_1 f\_1 + w\_1 f\_1} \tag{25}$$

$$f\_1 = \sqrt{\left(\tau\_{n\!f} - \tau\right)^2} \tag{26}$$

$$f\_2 = \sum\_{i=1}^{2} \sqrt{\left(\gamma\_{s,i} - \gamma\_i\right)^2} \tag{27}$$

$$f\_3 = \sqrt{\left(\gamma\_{s,3} - \gamma\_3\right)^2} \tag{28}$$

$$f\_4 = \sum\_{i=3}^{5} \sqrt{\left(\gamma\_i - \gamma\_{i+1}\right)^2} \tag{29}$$

$$f\_5 = \sum\_{i=4}^6 \sqrt{\left(\gamma\_{s,i} - \gamma\_i\right)^2} \tag{30}$$

### 3.3. Simulation results

Figure 8 shows an example of the simulation results in the form of the angular speed step responses for the control input. The resonance vibrations can be seen in this figure.

Figure 9 shows the frequency response characteristics of the two-mass resonance model from the control input uc to the motor angular speed ωM. The peak point of the mechanical resonance can be observed in this figure. It is therefore essential to construct the controller design method such that it reduces this resonance peak gain.

Table 2 shows the controller gains that were designed using the proposed method, where the reference time constant τref is set to 0.05. Figure 10 shows the designed CD. In the figure, each coefficient is multiplied by 100n , where n is the order of the characteristic polynomial. The figure shows that the form of the diagram is very smooth and that the convex shape is appropriately upward. The designed stability indices are shown in Figure 11. The results in this figure confirm that the stability indices nearly fit the standard form of these indices, and the fluctuations in the numerical values of adjacent indices are also small.

Figure 8. Angular speeds of the step responses of the two-mass model.

Figure 12 shows the simulation results, where the speed reference command input ωref is changed from 0 to 30 rad/s at t = 0 s, and the disturbance torque input is changed from 0 to 20% of the rated torque at t = 0.25 s. The figure indicates that the wave provides a good reference-following performance and illustrates the validity of the vibration suppression characteristics and the disturbance response simultaneously. Additionally, the gain characteristic that was derived using the proposed method over the range from the reference speed ωref to

Gain name Value Gain name Value Gain name Value Kp 2.792 102 Ki 9.007 103 Kd 3.522 T 0.4368 Kap 1.834 Kai 96.53

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Table 2. Controller gain results when designed using the proposed CDM method.

Figure 10. Designed coefficient diagram.

Figure 11. Designed and standard form stability indices.

Figure 9. Frequency responses of the two-mass resonance model from uc to ωM.

Table 2. Controller gain results when designed using the proposed CDM method.

Figure 10. Designed coefficient diagram.

3.3. Simulation results

196 PID Control for Industrial Processes

coefficient is multiplied by 100n

Figure 8 shows an example of the simulation results in the form of the angular speed step

Figure 9 shows the frequency response characteristics of the two-mass resonance model from the control input uc to the motor angular speed ωM. The peak point of the mechanical resonance can be observed in this figure. It is therefore essential to construct the controller design

Table 2 shows the controller gains that were designed using the proposed method, where the reference time constant τref is set to 0.05. Figure 10 shows the designed CD. In the figure, each

figure shows that the form of the diagram is very smooth and that the convex shape is appropriately upward. The designed stability indices are shown in Figure 11. The results in this figure confirm that the stability indices nearly fit the standard form of these indices, and

, where n is the order of the characteristic polynomial. The

responses for the control input. The resonance vibrations can be seen in this figure.

the fluctuations in the numerical values of adjacent indices are also small.

method such that it reduces this resonance peak gain.

Figure 8. Angular speeds of the step responses of the two-mass model.

Figure 9. Frequency responses of the two-mass resonance model from uc to ωM.

Figure 11. Designed and standard form stability indices.

Figure 12 shows the simulation results, where the speed reference command input ωref is changed from 0 to 30 rad/s at t = 0 s, and the disturbance torque input is changed from 0 to 20% of the rated torque at t = 0.25 s. The figure indicates that the wave provides a good reference-following performance and illustrates the validity of the vibration suppression characteristics and the disturbance response simultaneously. Additionally, the gain characteristic that was derived using the proposed method over the range from the reference speed ωref to

4.1. Frit

closed-loop system used to implement C(r).

either the model parameters or the state equations.

Figure 14. Typical closed-loop system.

Figure 15. Measurement of the initial data.

4.2. Vibration suppression controller design method by FRIT

Figure 16. Reference model showing the input of the fictitious reference signal.

Figure 14 shows a typical control system, in which G is the transfer function of the object to be controlled, r is the reference signal, r represents the controller gains, C(r) represents the controller, u is the control input parameter, and y is the output parameter. In this case, the mathematical model of G is not known in advance and is not required for this method.

Initially, as shown in Figure 15, a single-shot experiment is performed using the initial controller gains r0, and the control input u0 and output y0 are measured. Then, the reference model M(s), which matches the desired response, is determined. A fictitious reference signal is then generated using the controller, the control input u0, and the output y0, as shown in Eq. (31) below. This means that the initial data u0 and y0 can be obtained using any value of r if ~rð Þ r is input to the

~rð Þ¼ r Cð Þ r �1

The optimal controller gains that are required to achieve yM = y0 are then determined, as shown in Figure 16, using an optimization search method. Finally, these controller gains then represent the best available solutions that allow the desired control system response to be obtained. Therefore, the controller design process can be performed without any prior information about

Figure 17 shows a simplified form of the proposed vibration suppression control system, where Cω1, Cω2, Cω3, and Ci are the controllers. While the FRIT method generally uses one

u0 þ y0 (31)

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Figure 12. Simulation results when using the proposed method.

Figure 13. Gain characteristics when using the proposed method over the range from ωref to ωM.

the motor angular speed ωM is illustrated in Figure 13. The effectiveness of the proposed method is confirmed by the characteristic shown in this figure because the resonance peak is reduced considerably. Therefore, the results for the proposed method show that it is effective as a design method for the vibration suppression controller for the two-mass resonance system.

### 4. Controller design using fictitious reference iterative tuning (FRIT)

In this section, an off-line tuning method for the vibration suppression-type speed and current controller gains for the two-mass system is proposed based on the FRIT method; this method uses only single-shot experimental input-output data and does not use either the model parameters or the state equation of the two-mass resonance model. While most FRIT designs only use one state variable, this method uses specific multiple state variables to design the controller gains when using the FRIT method [6, 7, 14–16].

### 4.1. Frit

Figure 14 shows a typical control system, in which G is the transfer function of the object to be controlled, r is the reference signal, r represents the controller gains, C(r) represents the controller, u is the control input parameter, and y is the output parameter. In this case, the mathematical model of G is not known in advance and is not required for this method.

Initially, as shown in Figure 15, a single-shot experiment is performed using the initial controller gains r0, and the control input u0 and output y0 are measured. Then, the reference model M(s), which matches the desired response, is determined. A fictitious reference signal is then generated using the controller, the control input u0, and the output y0, as shown in Eq. (31) below. This means that the initial data u0 and y0 can be obtained using any value of r if ~rð Þ r is input to the closed-loop system used to implement C(r).

$$\tilde{r}(\mathfrak{p}) = \mathbb{C}(\mathfrak{p})^{-1}u\_0 + y\_0 \tag{31}$$

The optimal controller gains that are required to achieve yM = y0 are then determined, as shown in Figure 16, using an optimization search method. Finally, these controller gains then represent the best available solutions that allow the desired control system response to be obtained. Therefore, the controller design process can be performed without any prior information about either the model parameters or the state equations.

#### 4.2. Vibration suppression controller design method by FRIT

Figure 17 shows a simplified form of the proposed vibration suppression control system, where Cω1, Cω2, Cω3, and Ci are the controllers. While the FRIT method generally uses one

Figure 14. Typical closed-loop system.

the motor angular speed ωM is illustrated in Figure 13. The effectiveness of the proposed method is confirmed by the characteristic shown in this figure because the resonance peak is reduced considerably. Therefore, the results for the proposed method show that it is effective as a design

method for the vibration suppression controller for the two-mass resonance system.

Figure 13. Gain characteristics when using the proposed method over the range from ωref to ωM.

controller gains when using the FRIT method [6, 7, 14–16].

Figure 12. Simulation results when using the proposed method.

198 PID Control for Industrial Processes

4. Controller design using fictitious reference iterative tuning (FRIT)

In this section, an off-line tuning method for the vibration suppression-type speed and current controller gains for the two-mass system is proposed based on the FRIT method; this method uses only single-shot experimental input-output data and does not use either the model parameters or the state equation of the two-mass resonance model. While most FRIT designs only use one state variable, this method uses specific multiple state variables to design the

Figure 15. Measurement of the initial data.

Figure 16. Reference model showing the input of the fictitious reference signal.

Figure 17. Simplified proposed control system.

control input and one state variable for the initial experimental data, the proposed method uses one control input, uc, plus two state variables, ωM and ia. The controller gain vector r is given as follows:

$$\mathbf{p} = \begin{bmatrix} \mathbf{K}\_p & \mathbf{K}\_i & \mathbf{K}\_d & \mathbf{T} \mathbf{K}\_{ap} & \mathbf{K}\_{ai} \end{bmatrix}^T \tag{32}$$

Therefore, the fictitious reference signal ω~ refð Þ r can be calculated using the following equation without any need for the two-mass resonance model.

$$
\tilde{\omega}\_{n\circlearrowright}(\mathfrak{p}) = \left(1 + \mathsf{C}\_{a1}^{-1} \mathsf{C}\_{a2}\right) \omega\_{\mathsf{M}0} + \mathsf{C}\_{a1}^{-1} \mathsf{C}\_{a3}^{-1} \mathbf{i}\_{a0} + \mathsf{C}\_{a1}^{-1} \mathsf{C}\_{a3}^{-1} \mathsf{C}\_{i}^{-1} \mathbf{u}\_{\mathfrak{d}0} \tag{33}
$$

Here, uc0, ωM0 and ia0 represent the initial experimental data. The reference model M(s), as shown in Eq. (34), is then used depending on the purpose of the system, where the time constant τ is a reference model parameter.

$$M(\mathbf{s}) = \frac{1}{\left(\pi \mathbf{s} + 1\right)^3} \tag{34}$$

Here, τ is calculated using the following equation with the 99% response time parameter T99.

$$
\pi = \frac{T\_{99}}{4.4 \times 3^{0.6}} \tag{35}
$$

input and output waves of the voltage step response using the method that was proposed in [20]. Additionally, both the resonance and anti-resonance points can be found in these figures. Figure 20 shows the experimental results obtained when using the general PI speed and current controller for comparison with the effects of the proposed control system. Figure 21 shows the gain characteristics for the frequency responses shown in Figure 20, which relate ωref to ωM and ωL, where ωref is 30 rad/s. These characteristics show that the peak gain of the

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Figure 19. Calculated frequency responses to dc voltage input (left: uc to ωM, right: uc to ωL).

Figure 20. Experimental results obtained using conventional PI controller.

Figure 18. Angular speeds of the step responses (DC voltage input).

The differential evolution method is then used to search for the optimal gains. The performance index function F is then defined as shown in Eq. (36) below using ωM0 and yM ¼ M sð Þω~ refð Þ r .

$$F(\mathbf{p}) = \left\| M(\mathbf{s})\tilde{\omega}\_{r\circ}(\mathbf{p}) - \omega\_{M0} \right\|\_{2} \tag{36}$$

#### 4.3. Experimental results

Figure 18 shows an example of the experimental results in the form of the angular speed step response of the control input. The resonance vibrations can again be observed in a similar manner to the case of the simulation results shown in Figure 8 above.

Figure 19 shows the gain characteristics of the frequency responses from the experimental results shown in Figure 18, which relate the input voltage to the motor angular speed ωM and the load angular speed ωL. From these characteristics, the peak resonance vibrations at approximately 300 rad/s are also observed. These results were calculated based on the experimental

Figure 18. Angular speeds of the step responses (DC voltage input).

control input and one state variable for the initial experimental data, the proposed method uses one control input, uc, plus two state variables, ωM and ia. The controller gain vector r is

r ¼ Kp Ki Kd TKap Kai

Therefore, the fictitious reference signal ω~ refð Þ r can be calculated using the following equation

Here, uc0, ωM0 and ia0 represent the initial experimental data. The reference model M(s), as shown in Eq. (34), is then used depending on the purpose of the system, where the time

M sð Þ¼ 1

Here, τ is calculated using the following equation with the 99% response time parameter T99.

τ ¼ T99

The differential evolution method is then used to search for the optimal gains. The performance index function F is then defined as shown in Eq. (36) below using ωM0 and

Fð Þ¼ r M sð Þω~ refð Þ� r ωM0

Figure 18 shows an example of the experimental results in the form of the angular speed step response of the control input. The resonance vibrations can again be observed in a similar

Figure 19 shows the gain characteristics of the frequency responses from the experimental results shown in Figure 18, which relate the input voltage to the motor angular speed ωM and the load angular speed ωL. From these characteristics, the peak resonance vibrations at approximately 300 rad/s are also observed. These results were calculated based on the experimental

manner to the case of the simulation results shown in Figure 8 above.

ω1C�1

ω3 ia0 þ C�1

without any need for the two-mass resonance model.

constant τ is a reference model parameter.

Figure 17. Simplified proposed control system.

ω~ refð Þ¼ r 1 þ C�1

ω1Cω2 ωM0 þ C�1

T (32)

ω1C�1 ω3C�1

ð Þ τs þ 1 3 (34)

4:4 � 30:6 (35)

2 (36)

i uc0 (33)

given as follows:

200 PID Control for Industrial Processes

yM ¼ M sð Þω~ refð Þ r .

4.3. Experimental results

Figure 19. Calculated frequency responses to dc voltage input (left: uc to ωM, right: uc to ωL).

input and output waves of the voltage step response using the method that was proposed in [20]. Additionally, both the resonance and anti-resonance points can be found in these figures.

Figure 20 shows the experimental results obtained when using the general PI speed and current controller for comparison with the effects of the proposed control system. Figure 21 shows the gain characteristics for the frequency responses shown in Figure 20, which relate ωref to ωM and ωL, where ωref is 30 rad/s. These characteristics show that the peak gain of the

Figure 20. Experimental results obtained using conventional PI controller.

Figure 21. Frequency responses of conventional PI speed control system (left: ωref to ωM; right: ωref to ωL).

resonance is not greatly attenuated. Figures 22–24 show the initial experimental waves for ωM0, ia0 and uc0, respectively, that were obtained using the values of the initial controller gain r0, which are listed as follows:

$$
\boldsymbol{\rho}\_0 = \begin{bmatrix} 0.1 & \Re & 0.0001 & 0.001 & 1 & 10 \end{bmatrix}^T \tag{37}
$$

shows the experimental results that were obtained for ωM, ωL, and ia when the proposed FRIT method was used. Here, ωref is stepped from 0 to 30 rad/s when t is 0 s. The figure shows the good response of the proposed vibration suppression speed controller. Figure 27 shows the experimental results, where ωref is stepped from 30 to 50 rad/s when t is 0 s and the disturbance torque is increased from 0 to 10% of the rated torque when t is 0.5 s. As these figures show, good waves were observed in terms of their reference-following performance and disturbance

Gain name Value Gain name Value Gain name Value Kp 1.24 Ki 17.0 Kd 6.53 103 T 8.59 104 Kap 2.37 Kai 135

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Table 3. Controller gain results when designed using the proposed FRIT method.

response.

Figure 23. Initial ia0 data.

Figure 24. Initial uc0 data.

Both the initial rise and the oscillation can be observed in these figures.

Table 3 shows the results for the controller gains determined using the proposed FRIT design method with searching by the DE method, where T99 was set at 0.2 s. Figure 25 shows a comparison of the experimental results obtained using the controller gains that were designed using the proposed method with the simulated results for the reference output yM. The results in the figure show that the proposed off-line tuning method works very well, despite the fact that the design was performed using the initial one-shot experimental data alone. Figure 26

Figure 22. Initial ωM0 data.

Figure 23. Initial ia0 data.

resonance is not greatly attenuated. Figures 22–24 show the initial experimental waves for ωM0, ia0 and uc0, respectively, that were obtained using the values of the initial controller gain

Figure 21. Frequency responses of conventional PI speed control system (left: ωref to ωM; right: ωref to ωL).

Table 3 shows the results for the controller gains determined using the proposed FRIT design method with searching by the DE method, where T99 was set at 0.2 s. Figure 25 shows a comparison of the experimental results obtained using the controller gains that were designed using the proposed method with the simulated results for the reference output yM. The results in the figure show that the proposed off-line tuning method works very well, despite the fact that the design was performed using the initial one-shot experimental data alone. Figure 26

Both the initial rise and the oscillation can be observed in these figures.

r0 ¼ ½ � 0:1 30 0:0001 0:001 1 10 T (37)

r0, which are listed as follows:

202 PID Control for Industrial Processes

Figure 22. Initial ωM0 data.

Figure 24. Initial uc0 data.

Table 3. Controller gain results when designed using the proposed FRIT method.

shows the experimental results that were obtained for ωM, ωL, and ia when the proposed FRIT method was used. Here, ωref is stepped from 0 to 30 rad/s when t is 0 s. The figure shows the good response of the proposed vibration suppression speed controller. Figure 27 shows the experimental results, where ωref is stepped from 30 to 50 rad/s when t is 0 s and the disturbance torque is increased from 0 to 10% of the rated torque when t is 0.5 s. As these figures show, good waves were observed in terms of their reference-following performance and disturbance response.

Figure 27. Experimental results for speed step response and disturbance response.

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Figure 28. Frequency responses of the proposed control system (left: ωref to ωM; right: ωref to ωL).

Figure 29. Experimental responses of ωL to various values of T99.

Figure 25. Initial experimental results for ωM0 and simulation results for yM.

Figure 26. Experimental results for ωM, ωL, and ia obtained when the proposed FRIT method.

Figure 28 shows the gain characteristics of the frequency responses of the proposed control system, where these characteristics are shown from the perspectives of ωref relative to ωM and ωL. The resonance vibration suppression effect can be observed in these figures. Therefore, the effectiveness of the proposed control system and the design method based on use of the FRIT method can be confirmed. Additionally, Figure 29 shows the experimental results (ωL) that were obtained for various values of the speed reference time parameter, where T99 = 0.15, 0.175, 0.2, 0.25, 0.3, and 0.35. As shown in this figure, the response times change satisfactorily and the proposed design method for the controller gains can thus also be used to design the response times arbitrarily.

Figure 27. Experimental results for speed step response and disturbance response.

Figure 28. Frequency responses of the proposed control system (left: ωref to ωM; right: ωref to ωL).

Figure 29. Experimental responses of ωL to various values of T99.

Figure 28 shows the gain characteristics of the frequency responses of the proposed control system, where these characteristics are shown from the perspectives of ωref relative to ωM and ωL. The resonance vibration suppression effect can be observed in these figures. Therefore, the effectiveness of the proposed control system and the design method based on use of the FRIT method can be confirmed. Additionally, Figure 29 shows the experimental results (ωL) that were obtained for various values of the speed reference time parameter, where T99 = 0.15, 0.175, 0.2, 0.25, 0.3, and 0.35. As shown in this figure, the response times change satisfactorily and the proposed design method for the controller gains can thus also be used to design the

Figure 26. Experimental results for ωM, ωL, and ia obtained when the proposed FRIT method.

Figure 25. Initial experimental results for ωM0 and simulation results for yM.

204 PID Control for Industrial Processes

response times arbitrarily.

### 5. Conclusion

This chapter has proposed two design methods for the controller gains required for vibration suppression control in a two-mass resonance system. The proposed controller consists of a modified-IPD speed controller and a PI current controller. The proposed controller design methods are based on application of the coefficient diagram method (CDM) and application of the fictitious reference iterative tuning (FRIT) method. Both methods use the motor side variables only, including the motor's angular speed and the armature current. The CDM method uses the coefficient of the characteristic polynomial of the control system and can determine the control performance based on the shape of the coefficient diagram and the stability indices. In this chapter, the fitting performances for the standard form of the stability indices and the coefficient values were used to determine the controller gains, which were designed using the differential evolution method. The effectiveness of the proposed CDM was confirmed by the simulation results. The FRIT method can be used to design the controller without knowledge of the model state equations and their parameters. Furthermore, a fictitious signal that was calculated using the initial experimental data for multi-state variables was also proposed in this chapter. The effectiveness of the proposed FRIT method was confirmed using the experimental results. Consequently, the CDM and the FRIT method were shown to produce the same design performance. The CDM is useful for controller design when the mathematical model and the object parameters are known. The FRIT method is effective when the mathematical model is unknown but the initial experimental data can be observed.

[5] Ikeda H, Hanamoto T, Tsuji T, Tomizuka M. Design of vibration suppression controller for 3-inertia systems using Taguchi method. Proc. of SPEEDAM 2006, Mechatronic Sys-

PID Controller Design Methods for Multi-Mass Resonance System

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[6] Matsui Y, Ayano H, Nakano K. A controller tuning for 2-mass system using closed-loop transient data, Proc. of 11th IFAC International Workshop, THS6T2. 1; 2013. pp. 564-569

[7] Azuma T, Watanabe S. A design of PID controllers using FRIT-PSO. Proc. of 8th ICST;

[8] Manabe S, Kim Y-C. Recent development of coefficient diagram method. Proc. of 3rd

[9] Hanamoto T, Takenouchi T, Ikeda H. Vibration suppression control of 3-mass resonance system using particle swarm optimization for Design of Coefficient Diagram Method.

[10] H. Ikeda, T. Hanamoto. Vibration suppression control for multi-mass system using CDM designed by group-based-PSO. Proc of ICEE 2012, P-EM-12; 2012. pp. 1160-1165

[11] Ikeda H, Hanamoto T. Fuzzy controller of three-inertia resonance system designed by differential evolution. Journal of International Conference on Electrical Machines and Sys-

[12] Ikeda H, Hanamoto T. Fuzzy controller of multi-inertia resonance system designed by

[13] Ikeda H, Hanamoto T. Design of m-IPD controller of multi-inertia system using differen-

[14] Ikeda H, Hanamoto T. Design of vibration suppression controller for 2-inertia system by fictitious reference iterative tuning. Proc of ICEE 2015, ICEE15A-123; 2015. p. 6

[15] Ikeda H, Ajishi H, Hanamoto T. Application of fictitious reference iterative tuning to vibration suppression controller for 2-inertia resonance system. Proc of IECON 2015, TS-

[16] Ikeda H, Kogura Y, Kim K, Era R, Hanamoto T. Vibration suppression controller tuning method for 2-mass resonance system using FRIT, Proc of ICEMS 2016, DS4G-3-2; 2016. p. 6

[17] Ikeda H. Vibration Suppression Controller of Multi-Mass Resonance System Using Fuzzy Controller. In: Chapter 19 of Modern Fuzzy Control Systems and Its Applications. Rijeka:

[20] Matsui Y, Kimura T, Nakao K. Frequency response estimation for mechanical systems using closed-loop step response data. IEEJ Transactions on Electronics, Information and

differential evolution. Proc of ICEMS 2013, MC-1883; 2013. pp. 2291-2295

tial evolution. Proc of IPEC 2014, 21J1–2; 2014. pp. 2476-2482

tems; 2006. pp. S10–19-S10–25

Asian Control Conference; 2000. pp. 2055-2060

Journal of Japan AEM. 2011:S16-S20

48, YF-008451. 2015. pp. 1825-1830

Systems. 2011;131(5):751-757

InTech; 2017. pp. 399-417. ISBN 978-953-51-3390-2

[18] Myway Plus Corporation, https://www.mayway.co.jp/, Yokohama, Japan

[19] The Mathworks, https://www.mathworks.com, Massachusetts, U.S.A

2014. pp. 459-464

tems. 2014:184-189

### Author details

### Hidehiro Ikeda

Address all correspondence to<EMAIL_ADDRESS>
Department of Electrical Engineering, Nishi-Nippon Institute of Technology, Japan

### References

[5] Ikeda H, Hanamoto T, Tsuji T, Tomizuka M. Design of vibration suppression controller for 3-inertia systems using Taguchi method. Proc. of SPEEDAM 2006, Mechatronic Systems; 2006. pp. S10–19-S10–25

5. Conclusion

206 PID Control for Industrial Processes

Author details

Hidehiro Ikeda

References

1999;46(1):162-168

Address all correspondence to<EMAIL_ADDRESS>
tions. 1999;119-D(4):544-545 (in Japanese)

Department of Electrical Engineering, Nishi-Nippon Institute of Technology, Japan

compensation. Proceedings of ICCAS 2008; 2008:1367-1372

[1] Hori Y, Sawada H, Chun Y. Slow resonance ratio control for vibration suppression and disturbance rejection in torsional system. IEEE Transactions on Industrial Electronics.

[2] Yoon K-H et al. Hybrid robust controller design for a two mass system with disturbance

[3] Ikeda H, Hanamoto T, Tsuji T, Tanaka Y. Position control of 2-mass systems with speed minor loop designed by pole placement method. IEEJ Transactions on Industry Applica-

[4] Ikeda H, Hanamoto T, Tsuji T. Design of multi-inertia digital speed control system using

Taguchi method. Proc. of ICEM 2008; 2008. pp. 1-6. Paper ID 1167, PB.3.9

This chapter has proposed two design methods for the controller gains required for vibration suppression control in a two-mass resonance system. The proposed controller consists of a modified-IPD speed controller and a PI current controller. The proposed controller design methods are based on application of the coefficient diagram method (CDM) and application of the fictitious reference iterative tuning (FRIT) method. Both methods use the motor side variables only, including the motor's angular speed and the armature current. The CDM method uses the coefficient of the characteristic polynomial of the control system and can determine the control performance based on the shape of the coefficient diagram and the stability indices. In this chapter, the fitting performances for the standard form of the stability indices and the coefficient values were used to determine the controller gains, which were designed using the differential evolution method. The effectiveness of the proposed CDM was confirmed by the simulation results. The FRIT method can be used to design the controller without knowledge of the model state equations and their parameters. Furthermore, a fictitious signal that was calculated using the initial experimental data for multi-state variables was also proposed in this chapter. The effectiveness of the proposed FRIT method was confirmed using the experimental results. Consequently, the CDM and the FRIT method were shown to produce the same design performance. The CDM is useful for controller design when the mathematical model and the object parameters are known. The FRIT method is effective when the mathematical model is unknown but the initial experimental data can be observed.

## *Edited by Mohammad Shamsuzzoha*

*PID Control for Industrial Processes* presents a clear, multidimensional representation of proportional–integral–derivative (PID) control for both students and specialists working in the area of PID control. It mainly focuses on the theory and application of PID control in industrial processes. It incorporates recent developments in PID control technology in industrial practice. Emphasis has been given to finding the best possible approach to develop a simple and optimal solution for industrial users. This book includes several chapters that cover a broad range of topics and priority has been given to subjects that cover real-world examples and case studies. The book is focused on approaches for controller tuning, i.e., method bases on open-loop plant tests and closed-loop experiments.

Published in London, UK © 2018 IntechOpen © AlexandrVedmed / iStock

PID Control for Industrial Processes

PID Control

for Industrial Processes