Designing unity magnitude input shaping by using PWM technique

Sırrı Sunay GürleyükKaraelmas University, Engineering Faculty, Electric & Electronics Engineering Department, 67100 Zonguldak, Turkey

a r t i c l e i n f o

Article history:Received 8 May 2006Accepted 16 September 2010Available online 14 October 2010

Keywords:Vibration reductionInput shapingRobust control and unity magnitude shaper

a b s t r a c t

This paper proposes a method for reducing vibrations in flexible systems by using the input shaping. Anew input shaper technique based on Pulse Width Modulation (PWM) is used to shape the input.

Since unity magnitude (UM) input shaping is easy to implement and provides less control duration, ithas been preferred for many application areas. An analytical solution to time locations of the UM shaperrequires very complex mathematical expressions due to involving in dependent constraint equations.This paper presents a simple PWM based input shaping method. The proposed technique allows design-ing UM shapers without analytical solution especially for the shapers having higher order impulsesequence. The method requires only the estimated value of system natural frequency and damping ratio.Desired numbers of impulses can be obtained by comparing reference signal amplitude and carrier wavefrequency of a PWM. Analysis and experimental results demonstrate the effectiveness of the proposedmethod.

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1. Introduction

Flexible systems such as robot arms, cranes, actuators and stepmotors require vibration-damping techniques to maintain accuratepositioning. Problems occur due to the precision requirements asdifficulty in system modeling and non-minimum phase character-istics of the system [1]. For a flexible system, a control mechanismcan be classified as feedback and feedforward control schemes[2,3]. Feedback techniques use the measurements of the motionby locating sensors, thus a robust controller to the parameteruncertainty can be put into practice. However calibration of sensoris a problem due to the assembling difficulties for many systems[4]. Feedforward techniques are designed through considerationof the system properties. It does not require any sensor informa-tion and account for changing in the system once the input isdeveloped [5]. A feedforward technique known as input shapinghas been proposed for control of vibration in oscillatory structures[6,7]. Shaped input in this method is produced by convolving a se-quence of impulses with a desired system command [8]. Since in-put shaping introduces a time lag in the input command, shortersequence of impulses is desired for many fast systems [9,10].

Different solution techniques have been proposed since inputshaping first introduced [6]. The robustness for the input shapingto the modeling errors has been accepted as the main goal for mostof the control designers [8–10]. Zero Vibration (ZV), Zero VibrationDerivative (ZVD) and Extra Insensitive (EI) shapers are developedto force the residual vibration to zero with a wide insensitivity

curve or high robustness [7]. Sahinkaya proposes a method to pro-duce desired output by inverse dynamics using high order polyno-mials where a continuous and differentiable function is introducedto define the desired motion and then the input is shaped using in-verse dynamic analysis [11]. Alici defines an acceleration trajectoryconsisting of a cycloid. It was applied swing free transportation ofobjects with a robot manipulator [12]. Alici et al. have also devel-oped a robust motion design technique for flexible jointed manip-ulation systems using a ramp function and a sequence of impulses[13]. Piazzi and Visioli presents an approach for motion planningbased on system inversion using a polynomial function as the de-sired output [14].

For an input shaper, the constraint equations are used to deter-mine the required positive or negative values and time locations ofthe impulse amplitudes [8,15]. Negative impulses can significantlyreduce the move time. But, they cause the overcurrent problem inmost applications [16]. Negative but unity amplitude input shapersatisfies faster duration than positive shaping methods like ZV,ZVD, and EI [4,7,15–18]. Constraint equations are time dependentand cannot be solved analytically for UM shaper except the un-damped systems, f = 0 [3,12]. Numerical solution or graphicalapproximation is used to define the time locations of impulse se-quence [4,15,17]. Basic UM shaper is formed with an input shapercontaining two positive impulses and one negative impulse inunity amplitude [17]. It is simple and easy to apply, however, theinsensitivity of this shaper is weak to the natural frequencyvariations. Application areas generally require more robustness.Impulse sequence which has more impulses can increase therobustness of input shaper [18]. However, adding new impulsesincreases the move time and makes difficulty finding any relation

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Mechatronics 21 (2011) 125–131

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