# Multi-Mode Vibration Suppression in MIMO Systems by Extending the Zero Placement Input Shaping Technique: Applications to a 3-DOF Piezoelectric Tube Actuator

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## Abstract

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## 1. Introduction

## 2. Preliminaries on Multiple Input Shaping and Single Output Input Shaping Control

## 3. A Novel Multi-Mode MIMO Shaping Control

^{th}output state vector.

**B**shown in Equation (8) in the problem solving would allow us to design a different shaper for each of the inputs. The newly formed

**P**matrix shown in Equation (11), which can be constructed from

**B**and sub-system poles, would ensure that the designed shapers will suppress all modes of vibrations for all outputs. The shaper amplitudes vector $\mathit{a}$ can be obtained from solvin $\mathit{a}={\mathit{P}}^{\u2020}\mathit{W}$.

**P**matrix in Equation (11) becomes as shown in Equation (12). Where $[({w}_{1},{\zeta}_{1}),({w}_{2},{\zeta}_{2})]$ and $[({w}_{3},{\zeta}_{3}),({w}_{4},{\zeta}_{4})]$ are the structural frequencies and damping ratios for the first and second output respectively. ${s}_{1,2}=-{\zeta}_{1,2}{w}_{1,2}\mp j{w}_{d1,2}$ are system poles for the first output and $\text{}{s}_{3,4}=-{\zeta}_{3,4}{w}_{3,4}\mp j{w}_{d3,4}$ are system poles for the second output.

**P**matrix in Equation (11) allows us to design a compensator for any MIMO system with multiple modes of vibration. However, as the number of inputs, outputs or modes of vibration increase, the P matrix size becomes larger, and solving for shaper impulses becomes more challenging. The determination of this complexity is one of the perspective works. In the next section, we will apply the proposed generalized MIMO input shaping technique to a three-input three-output system with three modes of vibrations. The multi-mode in this context refers to the dominant resonance frequencies for each output of the MIMO system. The three-input three-output with three modes of vibration is not three SISO systems but rather it is, in our approach, three Multi Input Single Output (MISO) systems. Each of the outputs is coupled with all inputs through the ${\mathit{B}}_{j}$ matrix [23]. We will explain the MIMO system simplification in detail.

## 4. Applications to a 3-DOF Piezoelectric Actuator

#### 4.1. Presentation of the Experimental Setup

- A piezoelectric actuator with a tubular structure, capable of deflecting along the x-axis, y-axis or z-axis when a voltage is applied to ux, uy or uz respectively. The piezotube (PT230.94 from PIceramic Company, Lederhosen, Germany) is 30 mm in length and has a 3.2 mm external diameter and can tolerate ±200 V voltages range.
- Three inductive sensors (ECL202 from IBS company, Eindhoven, Netherlands) that are used to measure the displacements x, y and z. The sensors are tuned to have 40 nm of resolution, ±250 μm of measurement range and 15 kHz of bandwidth. Notice that the sensors are only used to characterize the oscillations of the actuator and to verify the performances of the control technique, they are not used to make a feedback control.
- A computer and a dSPACE data acquisition board, which are used to manage the different signals (voltages, reference input and measured output) and to implement the input shaping controller. The sampling time is set to 50 μs, which is largely sufficient to consider the dynamics of the actuator in our case.
- Three high-voltage (HV) amplifiers used to amplify the control signals ux, uz and uz from the dSPACE board before supplying the piezoactuator. The amplifiers can provide up to ±200 V.

#### 4.2. Characterization and Modeling of the 3-DOF Piezoelectric Tube Actuator

#### 4.3. Vibration Feedforward Controller Design

**B**(shown in Equation (10)) and the resultant poles from the reduced transfer functions are required for the formulation of the

**P**matrix (shown in its generic form in Equation (11)). Since our system under test is a three-input three-output system, we have nine transfer functions and nine poles with their conjugates, as listed below:

S1x = −326.7763621 + 5736.019955i; | S1y = −238.7149106 + 5769.089347i; | S1z = −632.18957547 + 4379.5395930i; | ||

S1xc = −326.7763621 − 5736.019955i; | S1yc = −238.7149106 − 5769.089347i; | S1zc = −632.18957547 − 4379.5395930i; | ||

S2x = −302.1598020 + 5803.751059i; | S2y = −184.4922971 + 7461.789129i; | S2z = −96.57235885 + 7522.7038840i; | ||

S2xc = −302.1598020 − 5803.751059i; | S2yc = −184.4922971 − 7461.789129i; | S2zc = −96.57235885 − 7522.7038840i; | ||

S3x = −35.84556158 + 1222.586399i; | S3y = −3.945002797 + 456.1052076i; | S3z = −39.356673163 + 15967.113359i; | ||

S3xc = −35.84556158 − 1222.586399i; | S3yc = −3.945002797 − 456.1052076i; | S3zc = −39.356673163 − 15967.113359i; |

**T**between them (which is the same for all shapers) has to be selected such that it is the minimum value

**T**to make all impulse amplitudes for all shapers positive. The goal is to make T as small as possible so the delay caused to the shaped (compensated) input is reduced. Solving Equation (7) yields an infinite number of solutions as a function of T. The desired one is the smallest value of T which satisfies the positive impulse condition. A simple MATLAB code was used to extract all of this information as shown in Figure 3.

**T**, if we design the shapers with four impulses each then the first value to make all impulse amplitudes positive is T = 0.00025 s (Figure 3, right-hand side) and as a result the calculated amplitude vector is:

**T**for these four-impulse input shapers that satisfy the positive amplitudes condition are marked using square dots in Figure 3 (right-hand side): 0.300, 0.500, 0.550, 0.600 and 0.650 ms.

**T**to make all impulse amplitudes positive is T = 0.0005 s. For this T, the resultant impulse amplitudes vector is:

**T**to satisfy the positive amplitudes condition for the three-impulse shapers are marked using square dots in Figure 3-left: 0.500, 0.550 and 0.600 ms.

#### 4.4. Simulation Results and Discussion

**T**value that would make all shaper impulse amplitudes positive and this can be considered as a limitation for this approach. For five-impulse shapers, the designed compensator was not able to find

**T**that satisfies the positive impulse amplitudes condition.

#### 4.5. Experimental Results

## 5. Conclusion and Perspectives

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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**Figure 1.**Feedforward control of badly damped vibrations. (

**a**) control of SIMO (single input multiple outputs) systems [22]; (

**b**) generalized control for MIMO (multiple inputs multiple outputs) systems.

**Figure 3.**Shapers’ impulse amplitudes versus T for three-impulses (

**left**) and four-impulses (

**right**) schemes.

When Exciting X Only | When Exciting Y Only | When Exciting Z Only | |
---|---|---|---|

Un-comp./3-imp./4-imp.(X) | 12.68/09.25/04.45 | 35.00/21.53/18.85 | 11.20/04.39/02.75 |

Un-comp./3-imp./4-imp.(Y) | 19.13/09.92/06.57 | 17.00/09.00/04.00 | 08.00/05.00/04.00 |

Un-comp./3-imp./4-imp.(Z) | 10.00/02.00/07.00 | 30.00/24.00/13.00 | 30.00/02.50/08.00 |

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Al Hamidi, Y.; Rakotondrabe, M.
Multi-Mode Vibration Suppression in MIMO Systems by Extending the Zero Placement Input Shaping Technique: Applications to a 3-DOF Piezoelectric Tube Actuator. *Actuators* **2016**, *5*, 13.
https://doi.org/10.3390/act5020013

**AMA Style**

Al Hamidi Y, Rakotondrabe M.
Multi-Mode Vibration Suppression in MIMO Systems by Extending the Zero Placement Input Shaping Technique: Applications to a 3-DOF Piezoelectric Tube Actuator. *Actuators*. 2016; 5(2):13.
https://doi.org/10.3390/act5020013

**Chicago/Turabian Style**

Al Hamidi, Yasser, and Micky Rakotondrabe.
2016. "Multi-Mode Vibration Suppression in MIMO Systems by Extending the Zero Placement Input Shaping Technique: Applications to a 3-DOF Piezoelectric Tube Actuator" *Actuators* 5, no. 2: 13.
https://doi.org/10.3390/act5020013