Signal shaper with a distributed delay: Spectral analysis and design

doi:10.1016/j.automatica.2013.08.029

Automatica

Volume 49, Issue 11, November 2013, Pages 3484-3489

https://doi.org/10.1016/j.automatica.2013.08.029 Get rights and content

Abstract

Input shapers with time delays have proved useful in many applications related to controls for various oscillatory devices, for example flexible manipulators and cranes. In the paper, a novel approach for designing a zero-vibration signal shaper based on equally distributed delay is proposed. The parameter assessment of the shaper is based on the spectral approach. Various characteristics of the shaper are analyzed and compared with the classical zero-vibration shaper with a lumped delay. The analysis shows that the novel shaper is a slower, but more robust alternative to the classical shaper. Besides, the discrete implementation of the shaper is proposed and tested. It includes zero placement based parameter adjustment with the objective to preserve full compensation of the oscillatory mode by the discrete algorithm.

Introduction

Input shaping is a control technique for reducing vibrations in computer controlled flexible machines by smoothing the reference command. Significant filtering features of simple time-delay shapers were first reported by Smith, 1957, Smith, 1958 and applications for effective manipulation and control of flexible systems were immediately proposed. In the 1990s, Singer, Seering and Singhose revisited the concept of delay-based signal shapers, see e.g. Singer and Seering (1990), and Singhose, Seering, and Singer (1994). They developed alternative methodology and time-domain formulas for the Smith’s posicast (Smith, 1957) resulting in the concept of zero-vibration (ZV) shaper. These results were followed by modifications with improved robustness, particularly the zero-vibration-derivative (ZVD) shaper and extra insensitive (EI) shaper (Singhose, Crain, & Seering, 1997). Particular simple shapers are the starting point for multi-modes shapers tuned to two or more selected flexible modes (Cole, 2011, Hyde and Seering, 1991, Singhose and Sung, 2009, Tuttle and Seering, 1994). Next to the shaper design and analysis in continuous time domain, an extensive research has been devoted to discrete signal shapers, see e.g. Hyde and Seering (1991), Tuttle and Seering (1994), Magee and Book (1993), Baumgart and Pao (2007), Cole (2011) and Cole and Wongratanaphisan (2011). Robustness analysis of classical shapers is discussed, e.g. in Singer and Seering (1990), Gurleyuk and Cinal (2007), Vaughan, Yano, and Singhose (2008) and Hurák, Hromčík, and Spiller (2007). Input shapers have proved most useful in many projects related to controls for flexible devices like reference tracking for flexible manipulators and cranes (Kim & Singhose, 2010), vibration suppression of industrial robots (Park et al., 2006, Pereira et al., 2009), orientation and pointing of solar panels of satellites (Singhose, Derezinski, & Singer, 1996), to name a few. Next to the signal shaping, various command profiles, such as trapezoidal, S-curve, and even more complex functions can be used to smooth the rapid changes in the reference or input signals of flexible systems, see e.g. Meckl, Arestides, and Woods (1998). However, as shown in Singhose, Eloundou, and Lawrence (2010) by applying the theory of signal shapers, the input shaping is considerably faster and more efficient technique in reducing the vibrations compared to the command smoothing.

In this paper, the novel zero-vibration shaper with a distributed delay is introduced as the main result. After the preliminary Section 2, the novel shaper is proposed in Section 3. Next, its spectral features are analyzed, resulting in the parameter design procedure. Consequently, comparison of the classical ZV shaper and the novel shaper is performed. In Section 4, the discretization issues of the shaper are addressed. Residual vibration characteristics are given in Section 5 and the results summary can be found in Section 6. Let us remark that the preliminary results have been presented in the brief conference paper (Vyhlídal, Kučera, & Hromčík, 2012).

Section snippets

Zero vibration shaper

Consider the objective to compensate a single oscillatory mode of a system $G (s) = \frac{y (s)}{v (s)} = \frac{N (s)}{M (s)}$ where $v, y$ are system input and output, respectively. Let the undesirable oscillatory mode be represented by the natural frequency $ω_{0}$ and the damping ratio $ζ$ , determining the complex conjugate couple of system poles $r_{1, 2} = - β \pm j Ω, β = ω_{0} ζ, Ω = ω_{0} \sqrt{1 - ζ^{2}}$ . An example of a system under consideration is as follows $G (s) = \frac{ω_{0}^{2}}{s^{2} + 2 ζ ω_{0} s + ω_{0}^{2}} .$ For the oscillatory mode compensation purpose, we can use the classical ZV

Main result—zero vibration shaper with an equally distributed delay

Substituting the lumped delay in the ZV shaper (2) by the delay (7), the distributed zero-vibration (DZV) shaper can be defined as $v (t) = B w (t) + \frac{(1 - B)}{ϑ} \int_{0}^{ϑ} w (t - η) d η .$ Consequently, its transfer function is given by $S_{D Z V} (s) = B + (1 - B) \frac{1 - e^{- s ϑ}}{s ϑ} .$ Let us remark that various types of delay distribution can be considered instead of (7). However, the use of more complex distribution functions may result in difficulties regarding their implementation, as shown e.g. in Verriest (1999).

Discrete implementation of the DZV shaper

For the purpose of the DZV shaper implementation, we propose its discretization scheme. As the preliminary step, consider the state space implementation of the DZV shaper transfer function (9) as follows $\dot{x} (t) = w (t) - w (t - ϑ)$ $v (t) = B w (t) + \frac{1 - B}{ϑ} x (t)$ where $x \in R$ is the state.

With the objective to keep the discretized algorithm as simple as possible, we apply the Euler explicit discretization scheme to (19), which results in $x (k + 1) = x (k) + Δ t (w (k) - \sum_{q = 0}^{N} α_{q} w (k - d + q)),$ where $k$ is discrete time and $Δ t$ is the

Residual vibration characteristics

Robustness of the connection of a shaper and a system with respect to the ratio between the nominal (design) natural frequency $ω_{n}$ and the true natural frequency $ω_{0}$ can be visualized by the sensitivity curves (Singer and Seering, 1990, Singhose and Sung, 2009). Based on the comparison of step responses of the system (1) given by $h_{G} (t) = L^{- 1} {\frac{G (ζ, ω_{0}, s)}{s}}$ , and the shaper–system connection given by $h_{S G} (t) = L^{- 1} {\frac{S_{.} (s) G (ζ, ω_{0}, s)}{s}}$ , the residual vibration sensitivity function can be defined as $U (ζ, ω_{0}) = max (h_{S}$

Conclusions

As the main contribution, the novel DZV signal shaper with an equally distributed delay and its spectrum based design have been introduced. Besides, various features of the shaper have been analyzed and compared with the classical ZV shaper. On the one hand, the DZV shaper provides slower performance in vibration suppression compared to the ZV shaper. On the other hand, it provides smoother command and better robustness at the higher frequency range. Analogously to DZV, more advanced shaper

Acknowledgments

The research benefited from discussions with professors Pavel Zítek, Nejat Olgac and Erik Verriest. We also thank to the anonymous reviewers for many valuable comments and suggestions.

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Citation Excerpt :
Moreover, the ZV shaper constraints at the certain frequency of single-mode system. Similar methods have been employed in DZV [26,30]. To expand the frequency control range, the DZV shaper designed with the equally distributed density function delays.
Dielectric elastomer actuators (DEAs) are an emerging type of soft actuator. When electrostatically actuated the DEA with a step signal, the inherent vibrations may limit their motion accuracy in practical applications. Moreover, increasing the applied voltage reduces the natural frequency of the dielectric elastomer structure, as the electrostatic stress decreases the tension in the DE device. This paper proposes a robust input shaper for a DEA system to eliminate vibration under various applied voltage. To extend the frequency control range of the shaper, an improved zero distribution method is proposed. The zeros are symmetrically distributed in the radial direction of the designed zero of a distributed zero-vibration (DZV) shaper. Simulations and experiments are performed to explain the robust control effects of the proposed design method. The step responses of the DEA are conducted at 2, 3 and 4 kV, the vibration attenuations of 27.66, 4.867 and 9.803 dB are achieved, respectively. Meanwhile, the overshoots are eliminated from the step responses. The designed method can pave the way to improve the control accuracy of soft material devices under varying operating conditions.

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Tomáš Vyhlídal was born in Dačice, Czech Republic, in 1974. He graduated in Automatic Control and Engineering Informatics in 1998 and received Ph.D. in Control and Systems Engineering in 2003, both from the Faculty of Mechanical Engineering (FME), Czech Technical University in Prague (CTU). Since 2000, he has been with the Dept. of Instrumentation and Control Eng., FME-CTU, and with the Centre for Applied Cybernetics. Since 2012, he has been Professor of Systems and Control Engineering at FME-CTU. His research interests include spectral analysis and control design of time-delay systems, algebraic control design, mathematical modeling and applied control theory. He is a member of editorial board of Kybernetika Journal.

Vladimír Kučera was born in Prague, Czech Republic, in 1985. He graduated in Cybernetics and Measurements at the Faculty of Electrical Engineering, Czech Technical University in Prague (CTU), in 2011. His master thesis focused on vibration control of large blended wing body aircraft. He is currently Ph.D. student of Control and Systems Engineering at the Faculty of Mechanical Engineering, CTU. He focuses on applications of delay based controllers and compensators in vibration control of mechanical systems.

Martin Hromčík was born in Ledeč nad Sázavou, Czech Republic, in 1975. He graduated in Cybernetics and Measurements in 1999 and received Ph.D. in Control Engineering and Robotics in 2005, both from the Faculty of Electrical Engineering (FEE), Czech Technical University in Prague (CTU). Since 2000, he has been with the Dept. of Control Eng., FEE-CTU. He became Assistant Professor of Control Engineering at FEE-CTU in 2009. His research interests cover flexible systems and controls, robust control, and flight control systems. He has been a member of the IFAC Technical Committee for Control Design since 2005, and of the Technical Committee for Aerospace since 2012.

^☆: The presented research has been supported by the Ministry of Education of the Czech Republic under the project KONTAKT II - LH12066. The material in this paper was partially presented at the 10th IFAC Workshop on Time Delay Systems, June 22–24, 2012, Boston, Massachusetts, USA. This paper was recommended for publication in revised form by Associate Editor Fen Wu under the direction of Editor Roberto Tempo.

¹: Tel.: +420 224352877.

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Brief paperSignal shaper with a distributed delay: Spectral analysis and design☆

Abstract

Introduction

Section snippets

Zero vibration shaper

Main result—zero vibration shaper with an equally distributed delay

Discrete implementation of the DZV shaper

Residual vibration characteristics

Conclusions

Acknowledgments

Automatica

Automatica

Control Engineering Practice

Automatica

Journal of Sound and Vibration

Optimal FIR input shaper designs for motion control with zero residual vibration

Transactions of the ASME. Journal of Dynamic Systems, Measurement and Control

Robust three-impulse sequence input shaper design

Journal of Vibration and Control

Trends in the stability properties of CLSS controllers: a root-locus analysis

IEEE Transactions on Control System Technology

Minimization of l2 norm of the error signal in posicast input command shaping: a polynomial approach

International Journal of Robust and Nonlinear Control

Design of learning input shaping technique for residual vibration suppression in an industrial robot

IEEE/ASME Transaction on Mechatronics

Preshaping command input to reduce system vibration

ASME. Journal of Dynamic Systems, Measurement and Control

Brief paper
Signal shaper with a distributed delay: Spectral analysis and design☆