# Actuator Location and Voltages Optimization for Shape Control of Smart Beams Using Genetic Algorithms

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Formulation of the Problem

#### 2.1. Strains and Electrical Fields

_{y}, as:

_{x}(x,y,z) ≈ zθ

_{y}(x,t), u

_{y}(x,y,z) ≈ 0, u

_{z}(x,y,z) ≈ w(x,t)

_{y}is the rotation of the beam cross section about the positive y−axis. Assuming small deformation, the strain−displacement relation can be expressed as:

_{k}−th piezoelectric layer is given by

_{k}= −[B

_{ϕ}]ϕ

^{Pk}

^{Pk},ϕ

^{Pk}are the thickness and the electric voltage of the p

_{k}−th piezoelectric layer. It should be noted that such formulation gives one electric degree of freedom per layer per element of the electric field.

#### 2.2. Constitutive Equations

_{11}, Q

_{55}are the transformed plane stress−reduced stiffness coefficients, e

_{31}, e

_{32}are the transformed piezoelectric moduli given in [15] and is the electric permittivity.

#### 2.3. Finite Element Formulation

_{c}} is the concentrated force vector, {f

_{S}} is the surface force vector, {F

_{b}} is the body force vector, {q} is the surface charge vector, S

_{1}is the surface area where external force is acting, and S

_{2}is the surface area of piezoelectric layer where applied electric charge is acting.

_{y}, per node and one additional degree of freedom, ϕ, per piezoelectric layer. Using standard discretization techniques,

_{e}= {w

_{1}, θ

_{y}

_{1}, w

_{2}, θ

_{y}

_{2}}

^{T}, [N

_{w}] is a cubic shape function and [N

_{θ}] is a quadratic shape function. These shape functions lead to a shear−locking free element and their explicit expressions are given in Ref. [9]. The strain field is given by:

_{e}is the derivative operator between the corresponding strain and the generalized nodal displacements. The electric voltage vector of the e

^{th}element can be expressed as:

^{th}element.

_{e}, the elastic stiffness matrix [K

_{uu}]

_{e}, the electromechanical coupling matrix [K

_{uϕ}]

_{e}, the permittivity matrix [K

_{ϕϕ}]

_{e}, the surface electric charge density {F

_{Q}}

_{e}and the mechanical load vector {F

_{m}}

_{e}are given in the Appendix.

_{uu}]{X} = {F

_{m}} − {F

_{el}}

_{uu}] ∈

**R**

^{N×N}is the global stiffness matrix, {X} ∈

**R**

^{N×1}is the nodal displacement vector, {F

_{m}} ∈

**R**

^{N×1}is the mechanical force vector and {F

_{el}} = [K

_{uϕ}]{ϕ} is the electrical force vector due to the actuation.

## 3. Optimal Shape Control

#### 3.1. The Fitness Function

_{1}, as given by Equation (14), which is the sum of all the squared difference of the transverse displacements between the desired (pre−defined) and the achieved (calculated) shape at all nodes.

_{y}rotations . Therefore, a fitness function based on the following cost function is used in the aforementioned paper:

_{2}is very restricted for bending problems such as the ones studied here. Nevertheless, in this work, the two above fitness functions will be used for comparison reasons. Results obtained by using f

_{1}as the fitness function will show improvements over f

_{2}. In following, a general symbol f is used to denote any fitness function.

#### 3.2. Design Optimization Problems

_{i}for a given number and position of actuators, which minimizes the cost function f under the constraint:

_{min}≤ ϕ

_{i}≤ ϕ

_{max}

_{i}is the actuation voltage of the i

^{th}actuator and ϕ

_{min}and ϕ

_{max}the lower and upper saturation voltages. In this work, this problem is solved by genetic algorithms. The MatLab software package was used for the development of an algorithm to optimize actuator placement and voltage for a given cost function and for given number of actuators and beam dimensions and properties. The computer code developed makes no assumption of linearity between the displacements and the electric voltages, thus, it can be used for non−linear models as well.

#### 3.3. Genetic Algorithm and Great Deluge

_{i}* is computed based on the existing value V

_{i}and the following equation:

_{i}* = V

_{i}+ (2 × rand()) × step

_{k}

_{i}* gets a value out of that range, an adjustment occurs so as the value to become valid again. In particular, when the value violates either the lower or the upper limit it is modified so as to be spaced at the same distance from the limit that is violated but in the feasible range of values.

#### 3.3.1. Chromosome Encoding

#### 3.3.2. GA Implementation Issues

## 4. Numerical Results

_{1}= 150 GPa, v

_{12}= 0.3, G

_{13}= 7.1 GPa. The piezoelectric material has the following properties: E

_{1}= 63 GPa, v

_{12}=0.3, G

_{13}= 24.2 GPa, e

_{31}= 17.584 C/m

^{2}and other entries in the piezoelectric stress matrix are zero. For comparison reason, in all the following examples the fitness value is scaled as [9]:

#### 4.1. The Voltage Problem

^{d}(x) = 0 and the aim is to calculate the actuator voltages required to induce this desired shape. The Voltage Problem is studied in two cases. In the first case, it is assumed that all the elements have a piezoelectric material layer bonded on its upper surface and the specified group is activated by the user. This is the same situation of [9]. In the second case, each element is considered as having no patch or being fully covered with piezoelectric material. A special assembly procedure was used to account for the piezoelectric actuator patches instead of complete layers of piezoelectric material throughout the structures. It is noted that in the former case, the stiffness characteristics of the beam remain constant throughout the shape control procedure, while, in the latter, they change depending on the number of actuators used. More precisely, the actuator patches contribute less to the beam stiffness than the continuous actuator layers in the first case. In addition, it is pointed out that for five actuator groups, the two cases of the problem (case 1 and case 2) are identical to that of [9].

#### 4.2. The Location and Voltage Problem

_{2}was 31.45, the minimum value was 27.35, the average value was 29.35, the standard deviation was 0.97, and the range of values was 4.1. Similar results were obtained for other instances of the problem demonstrating that the approach is fairly robust giving consistently good results.

Number of used Actuator Groups | Method | Voltage of the Actuator Groups (V) | Fitness f_{2} | |||||
---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | ||||

432.61 | 0.00 | 389.20 | 0.00 | 0.00 | 17.86 | |||

Present (case 1) | 432.60 | 389.21 | 17.86 | |||||

Present (case 2) | 438.87 | 400.55 | 17.76 | |||||

(1,2,3) | [9] | 324.83 | 217.58 | 272.11 | 0.00 | 0.00 | 18.99 | |

Present (case 1) | 324.91 | 217.41 | 272.20 | 18.99 | ||||

Present (case 2) | 325.35 | 216.01 | 277.59 | 18.89 | ||||

(1,2,3,4) | [9] | 320.15 | 243.17 | 169.88 | 122.44 | 0.00 | 22.14 | |

Present (case 1) | 320.20 | 243.30 | 169.51 | 122.57 | 22.14 | |||

Present (case 2) | 320.33 | 243.39 | 169.21 | 123.79 | 22.03 | |||

(1,2,3,4,5) | [9] | 320.54 | 240.47 | 178.09 | 98.57 | 41.94 | 23.41 | |

Present (case 1) | 321.06 | 241.18 | 173.58 | 107.45 | 30.37 | 23.61 | ||

Present (case 2) | 320.18 | 243.09 | 171.98 | 107.91 | 31.20 | 23.56 |

#### 4.2.1. Clamped−Free Beam

^{d}(x) = 0. Table 2 shows the optimal solutions for placement of the actuators and the corresponding optimal voltages for various numbers of actuators. The genetic algorithms were run using the following parameters: Generations = 2,000, Population = 100, EliteCount = 2. Marginally better results can be obtained in some cases by further fine−tuning of GA parameters like EliteCount and mutation rate. It should be noted that for a small number of actuators (8, 12), the GA was terminated before 2000 generations. We observe that for a small number of actuators (8, 12) the optimal actuation voltages are close to the upper saturation limit and the optimal positions are closed to the clamped end. A graphical presentation of these results is given in Figure 3.

**Table 2.**Optimal Location and voltages of actuators within the 30 finite element mesh for Clamped−Free Beam.

Number of Actuators in Use | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|

8 | 12 | 18 | 24 | 30 | ||||||

f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | |

1 | 240.00 | 240.00 | 72.88 | 170.6295 | 211.79 | 178.82 | 150.63 | 172.21 | 196.49 | 171.86 |

2 | 0 | 240.00 | 205.33 | 166.2474 | 137.64 | 240.00 | 181.99 | 164.26 | 135.51 | 165.25 |

3 | 240.00 | 240.00 | 57.59 | 239.5262 | 224.25 | 0 | 233.47 | 160.70 | 200.80 | 159.43 |

4 | 240.00 | 0 | 227.76 | 0 | 0 | 237.75 | 142.78 | 152.82 | 106.08 | 154.16 |

5 | 240.00 | 0 | 240.00 | 235.2183 | 236.74 | 150.84 | 62.97 | 148.71 | 172.72 | 147.67 |

6 | 240.00 | 240.00 | 240.00 | 240.00 | 157.88 | 141.92 | 153.98 | 141.55 | 134.09 | 142.17 |

7 | 240.00 | 240.00 | 240.00 | 0 | 144.00 | 136.04 | 133.83 | 137.12 | 175.95 | 136.62 |

8 | 240.00 | 0 | 0 | 240.00 | 101.20 | 131.17 | 202.06 | 129.91 | 77.37 | 130.25 |

9 | 240.00 | 0 | 240.00 | 0 | 111.33 | 124.13 | 82.88 | 125.06 | 133.95 | 125.01 |

10 | 0 | 240.00 | 0 | 240.00 | 154.75 | 179.14 | 155.75 | 118.98 | 147.23 | 118.82 |

11 | 0 | 240.00 | 0 | 240.00 | 124.40 | 0 | 20.06 | 113.21 | 89.92 | 112.97 |

12 | 0 | 0 | 240.00 | 0 | 138.96 | 220.14 | 142.41 | 107.06 | 88.79 | 107.72 |

13 | 0 | 240.00 | 240.00 | 0 | 0 | 0 | 215.45 | 101.73 | 163.93 | 101.21 |

14 | 0 | 0 | 0 | 240.00 | 232.81 | 149.81 | 0 | 95.90 | 54.65 | 95.94 |

15 | 0 | 0 | 0 | 0 | 0 | 89.29 | 0 | 89.52 | 66.10 | 89.88 |

16 | 0 | 0 | 240.00 | 0 | 0 | 84.59 | 223.23 | 84.19 | 137.71 | 84.19 |

17 | 0 | 0 | 0 | 240.00 | 180.02 | 116.37 | 54.80 | 78.72 | 16.61 | 78.05 |

18 | 0 | 0 | 0 | 0 | 126.31 | 0 | 50.74 | 71.97 | 126.12 | 72.91 |

19 | 0 | 0 | 0 | 240.00 | 0 | 167.23 | 135.11 | 99.19 | 72.24 | 66.33 |

20 | 0 | 0 | 0 | 0 | 118.10 | 0 | 0 | 0 | 17.67 | 61.24 |

21 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 87.00 | 79.27 | 54.98 |

22 | 0 | 0 | 240.00 | 0 | 0 | 132.40 | 133.63 | 72.41 | 27.59 | 49.26 |

23 | 0 | 0 | 0 | 0 | 194.29 | 0 | 28.23 | 0 | 60.20 | 43.54 |

24 | 0 | 0 | 0 | 0 | 0 | 124.34 | 36.36 | 77.41 | 55.35 | 37.67 |

25 | 0 | 0 | 0 | 0 | 0 | 0 | 0.12 | 0 | 21.19 | 31.62 |

26 | 0 | 0 | 0 | 0 | 0 | 0 | 79.99 | 68.89 | 3.47 | 26.56 |

27 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 31.47 | 20.06 |

28 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 27.63 | 14.41 |

29 | 0 | 0 | 0 | 0 | 122.02 | 0 | 17.03 | 0 | 0.53 | 8.90 |

30 | 0 | 0 | 0 | 186.51 | 1.87 | 46.15 | 2.03 | 22.71 | 0.38 | 2.59 |

Fitness | 22.00 | 16.11 | 25.60 | 19.09 | 30.65 | 20.88 | 31.29 | 23.12 | 33.66 | 26.77 |

_{1}is smaller than the one obtained using f

_{2}(e.g., 7.536e−6m compared to 1.093e−5m for 8 actuators). Hence, it can be concluded that the deflection controlled by fitness f

_{1}is closer to the desired shape than the one controlled by f

_{2}.

**Figure 3.**The centerline of the cantilever smart beam under the action of various numbers of actuators for X

^{d}(x) = 0 with the optimal location of the actuators and the optimal values of actuation voltages.

#### 4.2.2. Clamped−Clamped Beam

^{d}(x) = 0. The optimal values of voltages for the most efficient combinations of number of actuators to shape control of the beam are presented in Table 3. The GA runs using the following parameters: Generations = 3,000, Population = 150, EliteCount = 0. It should be noted that, for a small number of actuators (8, 12), the GA was terminated before 1,500 generations. We observe that for a small number of actuators (8, 12) the optimal positions of the actuators are close to the clamped ends where the optimal actuation voltages are close to the upper saturation limit and at the middle of the beam where the optimal actuation voltages are close to the lower saturation limit. A graphical presentation of these results is given in Figure 4. By comparing the curves in Figure 4, it can be seen that the deflection controlled by fitness f

_{1}is closer to the desired shape than the one controlled by f

_{2}(the maximum displacement from 3.67e−6m is reduced to 1.72e−6m for 12 actuators). Again, the results indicate that increasing the number of actuators has a beneficial effect on controlling the shape of the beam.

## 5. Conclusions

**Table 3.**Optimal Location and voltages of actuators within the 30 finite element mesh for Clamped−Clamped Beam.

Number of Elements | Number of Actuators in Use | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

8 | 12 | 18 | 24 | 30 | ||||||

f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | f_{1} | f_{2} | |

1 | 240.00 | 240.00 | 239.98 | 240.00 | 240.00 | 240.00 | 240.00 | 235.43 | 239.95 | 211.92 |

2 | 240.00 | 240.00 | 0 | 0 | 240.00 | 240.00 | 240.00 | 205.09 | 196.47 | 180.26 |

3 | 0 | 0 | 239.97 | 0 | 240.00 | 213.31 | 240.00 | 174.19 | 141.02 | 149.30 |

4 | 0 | 0 | 0 | 0 | 240.00 | 157.75 | 99.60 | 143.31 | 33.89 | 118.34 |

5 | 0 | 0 | 0 | 240.00 | 0 | 169.21 | 70.087 | 112.43 | 72.03 | 87.38 |

6 | 0 | 0 | 0 | 0 | 0 | 0 | 3.87 | 108.23 | 72.32 | 56.42 |

7 | 0 | 0 | 0 | 0 | 157.35 | 91.23 | 93.21 | 0 | −51.29 | 25.46 |

8 | 0 | 0 | 0 | −168.77 | 65.58 | 0 | 46.64 | 46.47 | 48.25 | −5.51 |

9 | 0 | 0 | 0 | 0 | −172.58 | 0 | −59.20 | −11.14 | −162.20 | −36.47 |

10 | 0 | −240.00 | 0 | 0 | 0 | 0 | −240.00 | −42.00 | 1.25 | −67.42 |

11 | −240.00 | 0 | 0 | 0 | −240.00 | 131.61 | 0 | −72.92 | −151.25 | −98.38 |

12 | 0 | 0 | −239.96 | −240.00 | 0 | 121.53 | 0 | −103.80 | −144.01 | −129.35 |

13 | 0 | 0 | 239.97 | −240.00 | −240.00 | 156.44 | −240.00 | −134.69 | −230.12 | −160.31 |

14 | −240.00 | 0 | 239.98 | 0 | −240.00 | 191.35 | −240.00 | −165.58 | −239.98 | −191.27 |

15 | 0 | 0 | 239.99 | −240.00 | 0 | 197.23 | −240.00 | −167.45 | −239.93 | −193.21 |

16 | −240.00 | −240.00 | 239.98 | −240.00 | −240.00 | 174.10 | −240.00 | −140.27 | −222.46 | −166.13 |

17 | 0 | −240.00 | 0 | 0 | −240.00 | 150.97 | −191.31 | −158.49 | −186.16 | −139.05 |

18 | 0 | 0 | 0 | 0 | −192.47 | 230.14 | 0 | 0 | −106.64 | −111.97 |

19 | 0 | 0 | 239.93 | 0 | −199.99 | 0 | −143.09 | −125.64 | −115.69 | −84.89 |

20 | 0 | −240.00 | 0 | −240.00 | 0 | 0 | −96.15 | 0 | −54.88 | −57.80 |

21 | 0 | 0 | 0 | 0 | 0 | −184.05 | 13.02 | 0 | −67.16 | −30.72 |

22 | 0 | 0 | 0 | 0 | 0 | 0 | −46.17 | 0 | −66.87 | −3.64 |

23 | 0 | 0 | 0 | 0 | 129.42 | 0 | 14.11 | 97.55 | 61.63 | 23.44 |

24 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 5.15 | 50.52 |

25 | 0 | 0 | 0 | 240.00 | 0 | 98.34 | 139.55 | 144.69 | 45.88 | 77.61 |

26 | 0 | 0 | 0 | 0 | 0 | 0 | 0 | 131.25 | 95.75 | 104.68 |

27 | 0 | 240.00 | 239.93 | 240.00 | 0 | 0 | 240.00 | 158.41 | 100.36 | 131.77 |

28 | 240.00 | 240.00 | 239.97 | 240.00 | 240.00 | 240.00 | 0 | 185.55 | 102.98 | 158.85 |

29 | 240.00 | 0 | 239.96 | 0 | 240.00 | 0 | 240.00 | 214.49 | 206.31 | 185.93 |

30 | 240.00 | 0 | 239.98 | 240.00 | 185.24 | 211.60 | 240.00 | 240.00 | 239.91 | 213.64 |

Fitness | 22.24 | 16.92 | 24.31 | 18.17 | 28.46 | 20.79 | 30.26 | 22.37 | 31.90 | 24.97 |

**Figure 4.**The centerline of the clamped−clamped smart beam under the action of various numbers of actuators for X

^{d}(x) = 0 with the optimal values of actuation voltages.

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## Appendix

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## Share and Cite

**MDPI and ACS Style**

Foutsitzi, G.A.; Gogos, C.G.; Hadjigeorgiou, E.P.; Stavroulakis, G.E. Actuator Location and Voltages Optimization for Shape Control of Smart Beams Using Genetic Algorithms. *Actuators* **2013**, *2*, 111-128.
https://doi.org/10.3390/act2040111

**AMA Style**

Foutsitzi GA, Gogos CG, Hadjigeorgiou EP, Stavroulakis GE. Actuator Location and Voltages Optimization for Shape Control of Smart Beams Using Genetic Algorithms. *Actuators*. 2013; 2(4):111-128.
https://doi.org/10.3390/act2040111

**Chicago/Turabian Style**

Foutsitzi, Georgia A., Christos G. Gogos, Evangelos P. Hadjigeorgiou, and Georgios E. Stavroulakis. 2013. "Actuator Location and Voltages Optimization for Shape Control of Smart Beams Using Genetic Algorithms" *Actuators* 2, no. 4: 111-128.
https://doi.org/10.3390/act2040111