# Robust Control and Active Vibration Suppression in Dynamics of Smart Systems

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## Abstract

**:**

## 1. Introduction

_{infinity}(H

_{inf}) controllers for active structures, the numerical simulation demonstrates that sufficient vibration suppression can be achieved by using the suggested general methods in a tutorial manner for the case of a piezoelectric smart structure [12,13,14]. The novelty of the work is that it calculates an H

_{infinity}controller with very good results in the frequency domain and the state space, even for different values of the mass and stiffness matrix, considering the uncertainty of the modeling; additionally, good results were acquired with a reduced order H

_{infinity}controller. No similar work achieves vibration suppression if there are different values of the mass and the stiffness matrix.

## 2. Materials and Methods

_{e}is the global control force vector produced by electromechanical coupling effects, and f

_{m}is the global external loading vector for a beam structure used in this work.

_{i}and rotations ψ

_{i}constitute the independent variable q(t), i.e.,

**w**and

**f**

_{m}are upward positive.

_{e}(t) as Bu(t) we write it as ${F}_{e}^{*}u$ where ${F}_{e}^{*}$ (of size 2n × n) indicates the voltages on the actuators. The ${F}_{e}^{*}$ (of size 2n × n) matrix also denotes the piezoelectric force for a unit mounted on its corresponding actuator. Lastly, the disturbance vector is designed by the following equation d(t) = f

_{m}(t) [15]. Then,

^{T}= Cx(t)

_{S}and b

_{A}, individually. The electromechanical properties of the beam of interest depicted in Figure 1a,b are listed in Table 1.

#### 2.1. Frequency Domain

#### 2.2. Design Objectives

- Small control effort.
- Attenuation of disturbances with acceptable transient characteristics (overshoot, settling time).
- Strength of closed loop system (plant + controller).

- 4.
- The above criteria (1)–(3) should be satisfied even when noise exists in the modeling procedure.

#### 2.3. System Specifications

_{s}(s)y(s)

_{zw}(s) with K

_{s}(s) the controller of our system,

_{zw}(s)=P

_{zw}(s)+P

_{zu}(s)K

_{s}(s)(I − P

_{yu}(s)K

_{s}(s))

^{−1}P

_{yw}(s)

_{u}(N, Δ)w = [N

_{22}+ N

_{21}Δ(I − N

_{11}Δ)

^{−1}N

_{12}]w = Fw

- I.
- If M is internally stable, the system is presumably stable;
- II.
- If the system performs about average;
- III.
- If and only if, the system (M, Δ) is robustly stable,

_{Δ}in the criterion, for the structured uncertainty set Δ. This condition is known as the generalized small gain theorem [12,13,14].

- IV.
- The system (N, Δ) exhibits robust performance if and only if,

_{p}is fully complex and has the same structure as Δ and dimensions corresponding to (w, z). Unfortunately, only bounds on μ can be estimated [19,20].

#### 2.4. Controller Synthesis

_{s}such that μ(Φ

_{u}(F(jω)), K

_{s}(jω)) ≤ β, $\forall $ω, is transformed into the problem of finding transfer function matrices D(ω) $\u03f5\forall \mathsf{\Delta}$ and G(ω) ϵ Γ, such that,

_{inf}synthesis and μ-analysis and often produces positive results. An upper limit on μ in terms of the scaled single value serves as the starting point,

_{s}or D (while maintaining the other constant) [9].

## 3. Results and Discussion

_{m}(t) was produced from the wind velocity data.

_{u}= 1.2.

_{infinity}control (close loop with PZTvoltages)in the schematic with the blue line. The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices. In Figure 7 with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices A and B of our system for the open loop that means without PZT voltages. Also in Figure 8 with green, red, light blue and petrol line we can see the displacement of the free end of the beam with different prices of matrices M and K of our system for the open loop that means without PZT voltages. Figure 7 and Figure 8 in the last graph show the changes when the PZT material properties change. The smart piezoelectric structure almost has no vibrations, and it maintains equilibrium even when the key system matrices (A, B, M, and K) have different prices. The initial parameters are the mass, the damping, and the stiffness matrix. In Figure 7 and Figure 8 these parameters change for the open and the closed loop—this means without PZT material and with PZT material. This work focuses on a specific PZT material with its properties shown in Table 1. Figure 7 (last graph) shows the changes when the PZT material properties change.

_{infinity}controller is 24 in order. Numerous scientists have proposed algorithms for order reduction as a result of the fact that the order of the controller, which is equal to the order of the system, is substantially higher than the order of conventional controllers such as PI and LQR. The following process will use the most widely used of these algorithms, known as Hifoo [21], which has been implemented in the Matlab environment. The main issue is to calculate a reduced-order n < 24 controller that preserves the performance of the H

_{infinity}criterion and the behavior of a full-order controller of the given system. As a mechanical input to this controller, 10 KN is taken at the free end of the structure. In Figure 9 we can see the beam-free end displacement with and without control, using Hifoo recovery time 0.05 sec (0.03 with H

_{infinity}), the steady-state error of the order of 10

^{−5}m (10

^{−6}with H

_{infinity}) maximum elevation 2.1 × 10

^{−4}(0.3 × 10

^{−4}with H

_{infinity}) and vibration suppression at 90% (98% with H

_{infinity}). In Figure 10, we can see the voltages within the piezoelectric limits of 30 volts.

## 4. Conclusions

_{infinity}-based controller is designed to suppress the vibration of the smart piezoelectric structure under dynamical loading. The robustness of the H

_{infinity}controller to parametric uncertainty in vibration suppuration problems is shown. The benefit of robust control and active vibration suppression in the dynamics of smart structures is amply illustrated by this work. H

_{infinity}control has certain advantages for the analysis of robust control systems. Unfortunately, relatively complicated modeling and resulting controllers lead to restricted practical applications. These drawbacks will be gradually eliminated due to the availability of cheaper and more powerful electronic components for control implementation. Future research will be focused on experimental verification in this direction.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 7.**Results for matrices A and B with and without H

_{in}

_{f}, controlling for sinusoidal external inputs, at various prices.

**Figure 8.**Results for matrices M (mass) and K (stiffness) at various costs, both with and without H

_{inf}control of the external sinusoidal inputs.

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

Beam length, L | 0.8 m |

Beam width, W | 0.07 m |

Beam thickness, h | 0.0095 m |

Beam density, ρ | 1600 kg/m^{3} |

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

Piezoelectric constant, d_{31} | 254 × 10^{−12} m/V |

Nominal stability (NS) ⇔ | N internally stable |

Nominal performance (NP) ⇔ | ║N_{22}(jω)║_{∞} < 1, ∀ω and NS |

Robust stability (RS) ⇔ | F = Φu(N, Δ) stable ∀Δ, ║Δ║_{∞} < 1 and NS |

Robust performance (RP) ⇔ | ║F║_{∞} < 1, ∀Δ, ║Δ║_{∞} < 1 and NS |

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**MDPI and ACS Style**

Moutsopoulou, A.; Stavroulakis, G.E.; Pouliezos, A.; Petousis, M.; Vidakis, N. Robust Control and Active Vibration Suppression in Dynamics of Smart Systems. *Inventions* **2023**, *8*, 47.
https://doi.org/10.3390/inventions8010047

**AMA Style**

Moutsopoulou A, Stavroulakis GE, Pouliezos A, Petousis M, Vidakis N. Robust Control and Active Vibration Suppression in Dynamics of Smart Systems. *Inventions*. 2023; 8(1):47.
https://doi.org/10.3390/inventions8010047

**Chicago/Turabian Style**

Moutsopoulou, Amalia, Georgios E. Stavroulakis, Anastasios Pouliezos, Markos Petousis, and Nectarios Vidakis. 2023. "Robust Control and Active Vibration Suppression in Dynamics of Smart Systems" *Inventions* 8, no. 1: 47.
https://doi.org/10.3390/inventions8010047