#
Robust H_{∞} Control of STMDs Used in Structural Systems by Hardware in the Loop Simulation Method

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Motion Equations of Building Models

#### 2.2. Robust Control Design

#### 2.2.1. Defining the System and Model Reduction

#### 2.2.2. Control Design

#### 2.2.3. Selection of Frequency Shaping Filters and ${\mathrm{H}}_{\infty}$ Control in Solution Mixed Sensitivity Structure

#### 2.3. Application of the Controller to the Semi-Active System

## 3. Results and Discussions

#### 3.1. Performance Analysis and Results with the HILS Method

#### 3.1.1. Introduction of the Experimental Setup

#### 3.1.2. Determination of Parameters

_{1},…,Wn

_{10}), 1.0108, 3.0097, 4.9414, 6.7628, 8.4331, 9.9149, 11.1753, 12.1861, 12.9247, and 13.3745 Hz. respectively. As the excitation forces, two different acceleration excitations were applied for 20 s at an amplitude of 0.02*g m/${s}^{2}$. Here g is the acceleration of gravity.

#### 3.2. Application of the Robust ${H}_{\infty}$ Controller

#### 3.2.1. Time Responses

#### 3.2.2. Frequency Analysis

#### 3.3. Structural Vibration Performance Evaluations

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Nomenclature

MDOF | Multi degree of freedom |

SDOF | Single degree of freedom |

TMD | Tuned mass damper |

STMD | Semi-active tuned mass damper |

MR | Magnetorheological |

HILS | Hardware in the loop simulation |

RTHS | Real-time hybrid simulation |

FOM | Full order model |

ROM | Reduced-order model |

RMS | Root mean square |

PSD | Power spectral density |

M_{s} | Mass matrix of the structural system |

C_{s} | Stiffness matrix of the structural system |

K_{s} | Damping matrix of the structural system |

L | The seismic input vector |

${\mathrm{H}}_{\mathrm{s}}$ | The placement of control units |

$\ddot{\mathrm{x}}\left(\mathrm{t}\right)$ | Acceleration vector |

$\dot{\mathrm{x}}\left(\mathrm{t}\right)$ | Velocity vector |

$\mathrm{x}\left(\mathrm{t}\right)$ | Displacement vector |

f(t) | The damping force of MR damper |

${\ddot{\mathrm{x}}}_{\mathrm{g}}\left(\mathrm{t}\right)$ | The earthquake ground acceleration |

${\mathrm{x}}_{\mathrm{i}}$ | i th floor displacement |

H_{∞} | H infinity control |

${\mathrm{A}}_{\mathrm{f}},{\text{}\mathrm{B}}_{\mathrm{f}},{\text{}\mathrm{C}}_{\mathrm{f}}$ | State-space matrices for the full order model |

${\mathrm{A}}_{\mathrm{r}},{\text{}\mathrm{B}}_{\mathrm{r}},{\text{}\mathrm{C}}_{\mathrm{r}}$ | State-space matrices for the reduced-order model |

$\mathsf{\Phi}$ | Modal transformation vector |

${\mathrm{x}}_{\mathrm{f}}$/${\mathrm{x}}_{r}$ | The state vectors of the full-order/reduced-order system model |

${\mathrm{y}}_{\mathrm{f}}$/${\mathrm{y}}_{r}$ | The output vectors of the full-order/reduced-order system model |

P_{f} (s) | Full order model of the system |

P_{r} (s) | Reduced-order model of the system |

C_{y} | Locations of the measurements of the system |

ƞ | Modal space of the system |

w | The input excitation |

K | Controller of the system |

u | The control signal of the system |

y | The measured response of the system |

z_{1,} z_{2} | They are regulating outputs of frequency shaping filters. |

n | The noise of the measurement |

${\mathrm{G}}_{\mathrm{zw}}$ | The transfer function of the mixed sensitivity structure |

${\mathsf{\xi}}_{\mathrm{nm}}$ | The damping ratio of last controlled mode |

${\mathsf{\xi}}_{\mathrm{dm}}$ | The damping ratio of first uncontrolled mode |

${\mathsf{\omega}}_{\mathrm{nm}}$ | Frequency of the last controlled mode |

${\mathsf{\omega}}_{\mathrm{dm}}$ | Frequency of the first uncontrolled mode |

${\mathrm{V}}_{\mathrm{min}}$ | Minimum voltage in the MR damper |

${\mathrm{V}}_{\mathrm{max}}$ | The maximum voltage in the MR damper |

${\mathrm{f}}_{\mathrm{d}}$ | The force necessary for system |

${\mathrm{f}}_{\mathrm{c}}$ | The system measures force |

W_{T}, W_{M} | Filters |

S(s) | Sensitivity transfer function |

T(s) | The complementary sensitivity transfer function |

$\Delta $ | Additive uncertainty of the system |

$\overline{\mathsf{\sigma}}$ | Maximum singular value of the S(s) |

$\gamma $ | Positive design parameter |

G(s) | The augmented system structure |

${\mathrm{x}}_{\mathrm{G}}$ | The state vector of the augmented system model |

${\mathrm{x}}_{\mathrm{K}}$ | The state vector of the controller |

${\mathrm{G}}_{\mathrm{MR}}$ | The MR damper controller gain |

${\mathrm{f}}_{\mathrm{opt}}$ | Optimum frequency ratio of TMD |

${\mathsf{\xi}}_{\mathrm{opt}}$ | The optimum damping ratio of TMD |

μ | Mass ratio |

W_{ni} | The natural frequency of the system model |

${\dot{\mathrm{x}}}_{\mathrm{v}}$ | The velocity of the floor to which the MR damper was connected in the system |

g | The acceleration of gravity |

J_{n} | Performance indices of the system |

h_{i} | Distance between floors |

d_{i} | Displacement between floors |

${\ddot{\mathrm{x}}}_{\mathrm{i}}{}^{\mathrm{max}}$(t) | Absolute acceleration without the controller |

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**Figure 4.**The Excitation-1 and the Excitation-2 forces, acceleration (left), and frequency changes (right).

**Figure 8.**The ${\mathrm{TMD}}_{0.0300}$and the $\mathrm{STMDh}{\infty}_{0.0300}$ the displacements (top) and accelerations (bottom) under the Excitation-1 force.

**Figure 9.**The maximum displacement (left-top), maximum acceleration (left-bottom), displacement RMS (right-top), and acceleration RMS (right-bottom) values of all floors under the Excitation-1 force.

**Figure 10.**The ${\mathrm{TMD}}_{0.0300}$and the $\mathrm{STMDh}{\infty}_{0.0300}$, the displacements (top) and accelerations (bottom) under the Excitation-2 force.

**Figure 11.**The maximum displacement (left-top), maximum acceleration (left-bottom), displacement RMS (right-top), and acceleration RMS (right-bottom) values of all floors under the Excitation-2 force.

**Figure 13.**The ${\mathrm{TMD}}_{0.0300}$ damping forces and the MR damper forces in $\mathrm{STMDh}{\infty}_{0.0300}$: Excitation-1 (top); Excitation-2 (bottom).

**Figure 14.**Displacement power spectrum density (PSD) curves of the system first (left) and 10th (right) floors under the Excitation-1 (top) and Excitation-2 forces (bottom).

**Figure 15.**Acceleration PSD curves of the system first (left) and 10th (right) floors under the Excitation-1 (top) and Excitation-2 forces (bottom).

Mass Ratio | Abbreviations of the TMD | Abbreviations of the STMD | G Values |
---|---|---|---|

$\mathsf{\mu}=0.0300$ | ${\mathrm{TMD}}_{0.0300}$ | $\mathrm{STMDh}{\infty}_{0.0300}$ | 0.09 |

$\mathsf{\mu}=0.0250$ | ${\mathrm{TMD}}_{0.0250}$ | $\mathrm{STMDh}{\infty}_{0.0250}$ | 0.05 |

$\mathsf{\mu}=0.0220$ | ${\mathrm{TMD}}_{0.0220}$ | $\mathrm{STMDh}{\infty}_{0.0220}$ | 0.01 |

$\mathsf{\mu}=0.0200$ | ${\mathrm{TMD}}_{0.0200}$ | $\mathrm{STMDh}{\infty}_{0.0200}$ | 0.01 |

Abbreviations | Excitation-1 | Excitation-2 | ||||||
---|---|---|---|---|---|---|---|---|

MR Damper Forces | MR Damper Voltage | MR Damper Forces | MR Damper Voltage | |||||

Maximum | RMS | Maximum | RMS | Maximum | RMS | Maximum | RMS | |

$\mathrm{STMDh}{\infty}_{0.0200}$ | 1910.60 | 328.51 | 10 | 2.52 | 1906.78 | 344.02 | 10 | 3.82 |

$\mathrm{STMDh}{\infty}_{0.0220}$ | 1840.27 | 330.10 | 10 | 2.58 | 1948.94 | 340.36 | 10 | 3.67 |

$\mathrm{STMDh}{\infty}_{0.0250}$ | 1847.94 | 348.74 | 10 | 3.57 | 1886.71 | 348.59 | 10 | 4.92 |

$\mathrm{STMDh}{\infty}_{0.0300}$ | 1883.10 | 359.30 | 10 | 4.06 | 1821.12 | 355.17 | 10 | 4.84 |

Abbreviations | Excitation-1 | Excitation-2 | ||
---|---|---|---|---|

Maximum | RMS | Maximum | RMS | |

${\mathrm{TMD}}_{0.0200}$ | 1307.03 | 490.19 | 1288.10 | 487.57 |

${\mathrm{TMD}}_{0.0220}$ | 1387.54 | 518.96 | 1369.70 | 516.06 |

${\mathrm{TMD}}_{0.0250}$ | 1497.22 | 559.94 | 1478.27 | 556.62 |

${\mathrm{TMD}}_{0.0300}$ | 1662.28 | 623.44 | 1652.36 | 619.38 |

Abbreviations (TMD and STMD) | Maximum Relative Displacement [m] | Maximum Displacement [m] | ||
---|---|---|---|---|

Excitation-1 | Excitation-2 | Excitation-1 | Excitation-2 | |

${\mathrm{TMD}}_{0.0200}$ | 0.01636 | 0.01663 | 0.01699 | 0.01655 |

${\mathrm{TMD}}_{0.0220}$ | 0.01507 | 0.01538 | 0.01577 | 0.01531 |

${\mathrm{TMD}}_{0.0250}$ | 0.01356 | 0.01385 | 0.01423 | 0.01374 |

${\mathrm{TMD}}_{0.0300}$ | 0.01164 | 0.01197 | 0.01227 | 0.01184 |

$\mathrm{STMDh}{\infty}_{0.0200}$ | 0.02499 | 0.02543 | 0.02548 | 0.02492 |

$\mathrm{STMDh}{\infty}_{0.0220}$ | 0.02324 | 0.02369 | 0.02355 | 0.02309 |

$\mathrm{STMDh}{\infty}_{0.0250}$ | 0.02114 | 0.02085 | 0.02111 | 0.02062 |

$\mathrm{STMDh}{\infty}_{0.0300}$ | 0.01743 | 0.01808 | 0.01826 | 0.01778 |

**Table 5.**Evaluations of the $\mathrm{STMDh}\infty $ and the TMD according to the performance indices.

Abbreviations (TMD and STMD) | Excitation-1 | Excitation-2 | ||||||
---|---|---|---|---|---|---|---|---|

${\mathbf{J}}_{1}$ | ${\mathbf{J}}_{2}$ | ${\mathbf{J}}_{3}$ | ${\mathbf{J}}_{4}$ | ${\mathbf{J}}_{1}$ | ${\mathbf{J}}_{2}$ | ${\mathbf{J}}_{3}$ | ${\mathbf{J}}_{4}$ | |

${\mathrm{TMD}}_{0.0200}$ | 0.29636 | 0.30228 | 0.24416 | 0.24497 | 0.34919 | 0.79590 | 0.24969 | 0.76976 |

${\mathrm{TMD}}_{0.0220}$ | 0.29636 | 0.30228 | 0.23536 | 0.23620 | 0.34919 | 0.79590 | 0.24108 | 0.76841 |

${\mathrm{TMD}}_{0.0250}$ | 0.27220 | 0.30571 | 0.22414 | 0.22501 | 0.32287 | 0.79296 | 0.23010 | 0.76671 |

${\mathrm{TMD}}_{0.0300}$ | 0.25577 | 0.30862 | 0.20920 | 0.21013 | 0.30297 | 0.78950 | 0.21548 | 0.76446 |

$\mathrm{STMDh}{\infty}_{0.0200}$ | 0.27896 | 0.36409 | 0.14706 | 0.15492 | 0.32710 | 0.76680 | 0.15859 | 0.76293 |

$\mathrm{STMDh}{\infty}_{0.0220}$ | 0.26825 | 0.36426 | 0.13966 | 0.14817 | 0.31405 | 0.76188 | 0.15037 | 0.76195 |

$\mathrm{STMDh}{\infty}_{0.0250}$ | 0.25257 | 0.36240 | 0.13123 | 0.14094 | 0.29688 | 0.76144 | 0.14378 | 0.76151 |

$\mathrm{STMDh}{\infty}_{0.0300}$ | 0.23578 | 0.36444 | 0.12197 | 0.13310 | 0.27372 | 0.76429 | 0.13249 | 0.76131 |

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**MDPI and ACS Style**

Aggumus, H.; Guclu, R.
Robust H_{∞} Control of STMDs Used in Structural Systems by Hardware in the Loop Simulation Method. *Actuators* **2020**, *9*, 55.
https://doi.org/10.3390/act9030055

**AMA Style**

Aggumus H, Guclu R.
Robust H_{∞} Control of STMDs Used in Structural Systems by Hardware in the Loop Simulation Method. *Actuators*. 2020; 9(3):55.
https://doi.org/10.3390/act9030055

**Chicago/Turabian Style**

Aggumus, Huseyin, and Rahmi Guclu.
2020. "Robust H_{∞} Control of STMDs Used in Structural Systems by Hardware in the Loop Simulation Method" *Actuators* 9, no. 3: 55.
https://doi.org/10.3390/act9030055