# Optimal Placement and Active Control Methods for Integrating Smart Material in Dynamic Suppression Structures

^{1}

^{2}

^{*}

## Abstract

**:**

_{infinity}controller, optimization is carried out to determine the best controller weights, which will dampen the oscillation. Challenging issues arise in the design of control techniques for piezoelectric smart structures. Piezoelectric materials have been investigated for use in distributed parameter systems (for example airplane wings, intelligent bridges, etc.) to provide active control efficiently and affordably. Still, no full suppression of the oscillation with this approach has been achieved so far. The controller’s order is then decreased using optimization techniques. Piezoelectric actuators are positioned optimally according to an enhanced optimization method. The outcomes demonstrate that the actuator optimization strategies used in the piezoelectric smart single flexible manipulator system have increased observability in addition to good vibration suppression results.

## 1. Introduction

_{∞}) control methods. For example, the appropriate position could be selected by modeling the equipment or simulating the process [22]. Many important researchers have dealt with the problem of control in smart construction [23,24,25]. In this paper, the problem of topological optimization of structures is presented [25], while in our thesis the modeling of the vector and the complete suppression of oscillations are presented in detail.

- -
- Modeling of intelligent constructs execution of control in oscillation suppression.
- -
- Uncertainties in dynamic loading.
- -
- Measurement noise.
- -
- Appropriate selection of weights for complete suppression of oscillations.
- -
- Using various choice places to stifle oscillations.
- -
- Results in the frequency domain as well as the time-space domain.
- -
- Introduction of the uncertainties in the construction’s mathematical model.

## 2. Modeling

_{m}is the global external loading mechanical vector, K is the global stiffness matrix, M is the global mass matrix, and D is the viscous damping matrix. It is difficult to specify how structural damping is determined because there are so many variables involved. To keep things simple, the structural damping matrix D can be analyzed as either linearly combined mass or stiffness (Rayleigh damping), which is D = αM + βK, or as mass proportional. Here α and β are calculated in terms of the first and second normal mode of vibration, α and β are 0.0005, and f

_{e}is the global control force vector resulting from electromechanical coupling effects. Rotations w

_{i}and transversal deflections ψ

_{i}make up the unrelated variable q(t), or

_{e}(t) = F

_{e}× u(t) as, where (of size 2n × n) is the piezoelectric force for a unit put on the appropriate actuator,

_{m}(t). Then,

_{1}(t) x

_{3}(t) … x

_{n−1}(t)]T = Cx(t).

## 3. Controller Synthesis

_{u}(F(jω), Κs(jω)) ≤ β, ∀ω is transmuted into the difficulty of discovering transfer function matrices D(ω) ∈ H and G(ω) ∈ H, such that,

_{∞}synthesis and frequently generates good outcomes. The initial point is the maximum value of μ in terms of the scaled singular value, where:

- K-step. Create a controller for the scaled issue. $\underset{K}{\mathrm{min}}\Vert DN\left(K\right){D}^{-1}{\Vert}_{\infty}$ with fixed D(s).
- D-step. Find D(jω) to minimalize at each frequency $\overline{\sigma}\left(DN{D}^{-1}\left(\mathrm{j}\omega \right)\right)$ with fixed N.
- Fit the degree of each factor of D(jω) to a stable and the lowest phase transfer function D(s) and move to Step 1.

## 4. Results

#### 4.1. Results in Simulation and Analysis of the Smart Structural Control

^{−1},

_{de}(transfer function disturbance to error) is

_{de}= J × (I − HBKC)

^{−1}H × G.

_{ne}(transfer function noise to error) is

_{ne}= J × (I − HBKC)

^{−1}HBK.

_{du}(transfer function disturbance to control) is

_{du}= (I − KCHB)

^{−1}KCH × G.

_{nu}(transfer function noise to control) is

_{nu}= (I − KCHB)

^{−1}K.

^{−1}H × Gd + J × (I − HBKC)

^{−1}HBKn

^{−1}KCH × Gd + (I − KCHB)

^{−1}Kn

_{zw}(s) = P

_{zw}(s) + P

_{zu}(s)K(s) (I − P

_{yu}(s)K(s))

^{−1}P

_{y}

_{w}(s)

_{zw}w = F(P, K)w.

_{w}= W

_{e}Jx= W

_{e}JHv = W

_{e}JH(GW

_{d}d

_{w}+ Bu) = W

_{e}JHGW

_{d}d

_{w}+ W

_{e}JHBu,

_{w}= W

_{u}u,

_{n}= Cx + W

_{n}n

_{w}= CHv + W

_{n}n

_{w}= CH(GW

_{d}d

_{w}+ Bu) + W

_{n}n

_{w}=

_{d}d

_{w}+ CHBu + W

_{n}n

_{w}.

_{ij}’s. We achieve this utilizing Equation (18) and noticing that:

_{d}d

_{w}, n = W

_{n}n

_{w}, e

_{w}= W

_{e}e, u

_{w}= W

_{u}u.

_{K}x

_{K}(t) + D

_{K}y(t).

_{0}(I + k

_{p}I

_{2n×2n}δ

_{K})

_{0}(I + m

_{p}I

_{2n×2n}δ

_{M}).

_{0}(I + k

_{p}I

_{2n×2n}δ

_{K}) + M

_{0}(I + m

_{p}I

_{2n×2n}δ

_{M})]=

_{0}+ 0.0005[K

_{0}k

_{p}I

_{2n×2n}δ

_{K}+ M

_{0}m

_{p}I

_{2n×2n}δ

_{M}].

_{0}(I + d

_{p}I

_{2n×2n}δ

_{D}).

_{p}and k

_{p}are employed to scale the proportion value and the zero subscript represents nominal values.

_{n×m}the norm is determined via $\u2551\mathrm{A}{\u2551}_{\infty}=\underset{1\le \mathrm{j}\le \mathrm{m}}{\mathrm{max}}{\displaystyle \sum}_{\mathrm{j}=1}^{\mathrm{n}}\left|{\mathrm{a}}_{\mathrm{ij}}\right|$).

_{1}, and E

_{2}are used to extract:

_{1}and E

_{2}are:

_{u}, and q

_{u}(which will be employed as extra inputs/outputs correspondingly) and use the auxiliary signals α, β, and γ.

_{w}to q

_{u}):

_{d}replaced by G

_{u}, i.e.,

#### 4.2. Results for the Open Loop (Initial Condition without Control)

_{o}(s) = C(sI − A)

^{−1}G.

^{0.007}×

^{−0.013}. This means some preconditioning would be beneficial to sensitive calculations (like pole placement). A solution to this issue is to stabilize the system matrix. Matlab delivers the routine [36,37,38] [T, S] = balance(A) which creates a diagonal alteration matrix T whose elements are integer powers of 2, and matrix B such that

^{−1}.

^{−1}x⇒x = Tz,

#### 4.3. Results with LQR Control

_{L}[42].

^{−1}is the beam’s transfer function.

#### 4.4. Results with Hinfinity Control

_{infinity}control, where the matrixes have been obtained after optimization. Nominal performance is depicted in Figure 24, Figure 25, Figure 26, Figure 27 and Figure 28. The resulting controller is order 36. The max singular value for this controller is 0.074. In Figure 24b the performance of the controller is significant since it appears that there is a significant improvement in the error noise for frequencies above 1000 Hz. Moreover, Figure 24a shows the noteworthy enhancement of the effect of disturbances on the error up to the frequency of 1000 Hz.

## 5. Discussion

_{∞}) techniques is used to synthesize controllers for stabilization with assured performance. By expressing the control problem as a mathematical optimization problem and then identifying the controller that resolves this optimization, a control designer can employ Hinfinity techniques. In comparison to classical control techniques, H

_{∞}techniques have the advantage of being easily adaptable to problems involving multivariate systems with cross-coupling between channels. Hinfinity techniques’ drawbacks include the level of mathematical sophistication required for successful application and the requirement for a passably accurate model of the system to be controlled. Keep in mind that the resultant controller may not always be the best and is just the optimum solution with regard to the required cost function. The closed-loop impact of a perturbation may be reduced using H

_{∞}methods. Depending on how the problem is phrased, the impact will either be assessed in terms of stability or performance. The issue of control in intelligent structures has been addressed by a number of significant researchers [23,24,25]. While the modeling of the vector and the thorough suppression of oscillations are provided in detail in our research, the challenge of topological optimization of structures is covered in this study [25].

- On the modeling of uncertainty in smart constructions.
- In the creation of advanced control techniques.
- In the complete suppression of vibrations under dynamic loading.
- Analytical explanation of the equations used in programming.
- Advanced programming techniques have been used to make the simulations.
- The model has been worked both in simulation and in advanced programming.
- It is not possible in one article to present both the modeling and the experimental results in such detail. For this reason, they will be presented in future research papers.

## 6. Conclusions

- -
- Modeling of intelligent constructs execution of control in oscillation suppression.
- -
- Using various choice places to stifle oscillations.
- -
- Results in the frequency domain as well as the time-space domain.
- -
- Introduction of the uncertainties in the construction’s mathematical model.
- -
- The integration of smart structures using methods for optimal placement and active control.
- -
- Uncertainties in dynamic loading.
- -
- Measurement noise, appropriate selection of weights for complete suppression of oscillations.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Nomenclature

M | Mass Matrix | ψ_{i}(t) | Displacement deflection |

K | Stiffness Matrix | x(t) | The state vector of our system |

D | Viscous damping Matrix | y(t) | Output vector of our system |

fe(t) | piezoelectric force | d_{31} | Piezoelectric constant |

n | Number of nodes in finite element formulation | cp | Piezoelectric constant |

u(t) | Control voltages of actuators | K(s) | Hinfinity Controller of the system |

Fe | Matrix with piezoelectric constant | Kl_{Q} | LQR controller of the system |

w_{i}(t) | Rotation deflection | P(s) | Augment Plant of the smart system |

μ | Singular value | e(t) | The error of the system |

d(t) | Disturbances of the system | n(t) | Noise of the system |

A, B, G, H | Matrices of our system | D, G-K | D-K interaction in the frequency domain |

Tde, Tne, Tdu, Tnu | The transfer function disturbance error, noise error, disturbance control, noise control | W_{e} | The error Weight for Hinfinity control |

W_{n} | The noise Weight for Hinfinity control | W_{u} | The control Weight for Hinfinity control |

W_{d} | The disturbance Weight for Hinfinity control | N | The transfer function for the smart system |

Δ | The Uncertainty of the system | δ_{M t} | The Uncertainty terms for the mass matrix |

δκ | The Uncertainty terms for the stiffness matrix | kp, mp | Numerical constant from zero to one |

J | Matrix which is utilized to select states that we are concerned with controlling | Q, R | The weight vectors for LQR control |

κ(jω) | Frequency-dependent condition number | F | Fractional transformation |

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**Figure 2.**(The second case) the actuators have been placed at the end of the beam and concentrated in positions 3 and 4.

**Figure 3.**The actuators have been placed alternately, i.e., in positions 2 and 4 (the red color indicates these positions, the blue color is the beam without piezoelectric patches and the white color is the remaining two positions 1 and 3, without actuator).

**Figure 4.**The actuators have been placed at the end of the beam and concentrated in positions 3 and 4 (the red color indicates these positions, the blue color is the beam without piezoelectric patches and the white color is the remaining two positions 1 and 2, without actuator).

**Figure 18.**Responses for unit white noise for every node of the structure. The curves almost coincide; thus, the different curves cannot be clearly distinguished in the graphs.

**Figure 24.**H

_{∞}control: max singular value plots for the closed loop unweighted system’s error (

**a**) disturbance to error, (

**b**) noise to error.

**Figure 27.**Rotation for the second place of piezoelectric patches; Blue line—no control; Green line—Hinf.

Parameters | Values |
---|---|

L, for Beam length | 1.00 m |

W, for Beam width | 0.08 m |

h, for Beam thickness | 0.02 m |

ρ, for Beam density | 1600 kg/m^{3} |

E, for Young’s modulus of the Beam | 1.5 × 10^{11} N/m^{2} |

bs, ba, for Pzt thickness | 0.002 m |

d_{31} the Piezoelectric constant | 280 × 10^{−12} m/V |

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**MDPI and ACS Style**

Moutsopoulou, A.; Stavroulakis, G.E.; Petousis, M.; Pouliezos, A.; Vidakis, N.
Optimal Placement and Active Control Methods for Integrating Smart Material in Dynamic Suppression Structures. *Vibration* **2023**, *6*, 975-1003.
https://doi.org/10.3390/vibration6040058

**AMA Style**

Moutsopoulou A, Stavroulakis GE, Petousis M, Pouliezos A, Vidakis N.
Optimal Placement and Active Control Methods for Integrating Smart Material in Dynamic Suppression Structures. *Vibration*. 2023; 6(4):975-1003.
https://doi.org/10.3390/vibration6040058

**Chicago/Turabian Style**

Moutsopoulou, Amalia, Georgios E. Stavroulakis, Markos Petousis, Anastasios Pouliezos, and Nectarios Vidakis.
2023. "Optimal Placement and Active Control Methods for Integrating Smart Material in Dynamic Suppression Structures" *Vibration* 6, no. 4: 975-1003.
https://doi.org/10.3390/vibration6040058