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Research Papers

# Robust and Optimal Output-Feedback Control for Interval State-Space Model: Application to a Two-Degrees-of-Freedom Piezoelectric Tube Actuator

[+] Author and Article Information
Mounir Hammouche, Philippe Lutz

Automatic Control and MicroMechatronic
Systems Department,
FEMTO-ST Institute,
Université Bourgogne Franche-Comté, CNRS,
24, rue Alain Savary,
Besançon 25000, France

Micky Rakotondrabe

Automatic Control and MicroMechatronic
Systems Department,
FEMTO-ST Institute,
Université Bourgogne Franche-Comté, CNRS,
24, rue Alain Savary,
Besançon 25000, France
e-mail: mrakoton@femto-st.fr

1Corresponding author.

Contributed by the Dynamic Systems Division of ASME for publication in the JOURNAL OF DYNAMIC SYSTEMS, MEASUREMENT,AND CONTROL. Manuscript received January 24, 2018; final manuscript received July 18, 2018; published online October 10, 2018. Assoc. Editor: Dumitru I. Caruntu.

J. Dyn. Sys., Meas., Control 141(2), 021008 (Oct 10, 2018) (14 pages) Paper No: DS-18-1039; doi: 10.1115/1.4040977 History: Received January 24, 2018; Revised July 18, 2018

## Abstract

The problem of robust and optimal output feedback design for interval state-space systems is addressed in this paper. Indeed, an algorithm based on set inversion via interval analysis (SIVIA) combined with interval eigenvalues computation and eigenvalues clustering techniques is proposed to seek for a set of robust gains. This recursive SIVIA-based algorithm allows to approximate with subpaving the set solutions $[K]$ that satisfy the inclusion of the eigenvalues of the closed-loop system in a desired region in the complex plane. Moreover, the LQ tracker design is employed to find from the set solutions $[K]$ the optimal solution that minimizes the inputs/outputs energy and ensures the best behaviors of the closed-loop system. Finally, the effectiveness of the algorithm is illustrated by a real experimentation on a piezoelectric tube actuator.

###### FIGURES IN THIS ARTICLE
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## Figures

Fig. 1

Region Ω of desired eigenvalues for the closed-loop

Fig. 2

Output-feedback with integral compensator

Fig. 3

General framework for the design of optimal and robust controllers

Fig. 4

Presentation of the experimental setup

Fig. 5

Structure and operation of the piezoelectric tube actuator

Fig. 6

The control design of a MIMO piezoelectric tube actuator using output-feedback with integral compensator

Fig. 7

Resulting subpaving [Ky] and [Ki]

Fig. 8

Step response of piezoelectric tube for the closed-loop system (Simulation using matlab)

Fig. 9

The test of the robustness of the closed-loop system using Monte Carlo techniques

Fig. 10

Step response of piezoelectric tube for the closed-loop system (experimental test)

Fig. 11

Open- and closed-loop frequency responses. (a), (b), (c), and (d) for Gxx, Gxy, Gyx, and Gyy, respectively.

Fig. 12

Deflection response of piezoelectric tube

Fig. 13

Helix trajectory tracking

Fig. 14

Step response of piezoelectric tube actuator with different gains values

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