Original articles
Minimal modeling methodology to characterize non-linear damping in an electromechanical system

https://doi.org/10.1016/j.matcom.2015.05.007Get rights and content

Abstract

This paper presents a minimal modeling methodology for capturing highly non-linear dynamics in an electromechanical cart system. A theoretical foundation for the method is given including a proof of identifiability and a numerical example to demonstrate the theory. The second order differential equation model describing the cart system is reformulated in terms of integrals to enable a fast method for identification of both constant and time varying parameters. The model is identified based on a single experimental proportional step response and is validated on a proportional–derivative (PD) controlled step input for a range of gains. Two models with constant damping and time varying non-linear damping were considered. The fitting accuracy for each model was tested on three separate data sets corresponding to three proportional gains. The three data sets gave similar non-linear damping models and in all cases the non-linear model gave smaller fitting errors than the linear model. For the PD control responses, the non-linear model reduced the mean absolute prediction error by a factor of three. The non-linear model also provided significantly better PD control design. These results demonstrate the ability of the proposed method to accurately capture significant non-linearities in the data. Computationally, the proposed algorithm is shown to be significantly faster than standard non-linear regression.

Introduction

Many systems have non-linear oscillatory behavior, for example pitch control of an aircraft  [21], building response to seismic load  [3], [23], robotic arm  [6], [20], and car suspension control  [1], [25]. A common approach to analyzing vibration responses in various applications is to assume linear damping and break the responses into various modal frequencies. For example, modal analysis is commonly used for assessing damage of a building after an earthquake  [29], [13], and for characterizing stability of power systems  [17], [16], [28]. Another common approach is Prony’s Analysis  [16], [24], [2], which is similar in concept to the Fourier Transformation, and extracts a series of damped complex exponentials from the signal. Other methods have involved Kalman filter based designs,  [31], non-linear regression analysis  [19], [22], and statistical methods such as the Mean-Likelihood estimator  [14], to estimate the damping co-efficient.

In electro-mechanical systems, significant non-linear damping exists, including static friction and gear backlash  [32]. The usual approach to capturing this phenomena is to have a pre-assumed non-linear representation, for example, the Bingham viscoplastic model, assumes that the force of the damper is a given non-linear function of the damper velocity  [26], and the Bouc–Wen model includes a model describing hysteric behavior of the damper  [30]. There are also several models of gear backlash  [15], [5]. Thus, the concept is to start with complex model structures first, and then fit it to the data. If required, more complex models of damping can be used including finite elements  [18], [12].

This research presents a different philosophy by starting with initially simplified structures for the model then using correlations between the time varying parameters and measured data to “bootstrap” a more complex and accurate non-linear model. In this paper a general non-linear modeling approach to damping in second-order systems is presented. The approach is based on a time-varying damping formulation which is proven in theory and applied to an electromechanical cart system. The method identifies a general damping profile under continuity constraints so avoids any potentially incorrect assumptions on the precise nature of the non-linear damping present. Furthermore, this approach allows an integral based method for parameter identification to be developed that linearizes the optimization problem. This identification method has been validated in biomedical applications which tend to involve coupled first order systems  [11], [8], [27], [9], and rocket roll dynamics which are overdamped and first order  [10]. The extension to second order systems is not straightforward as dynamics can be significantly oscillatory and initial conditions involve derivatives. This paper includes the extension of this method to both second order and non-linear parameters, and provides a proof of the theoretical convergence and identifiability of this nonlinear modeling approach. Although, this paper is primarily on the non-linear characterization and comparison with linear models, this extended integral method is important in terms of minimizing computational requirements for the non-linear identification. The second order non-linear model is identified on a closed-loop response and is shown to accurately capture the system non-linear dynamics over a wide range of inputs. The non-linear model is shown to be significantly more accurate than a model with linear damping, and thus provides better control system prediction and design.

Section snippets

Spring–mass–damper with time-varying damping

The differential equation for a linear spring is defined: mÿ+cÿ+ky=u(t) where yy(t) is the displacement (m), u(t) the input force (N), m the mass (kg), c the damping factor (kg/s), and k the stiffness of the spring N/m. For the case of free vibration where u(t)=0, without loss of generality it can be assumed that m=1, and thus Eq. (1) is written in the form: ÿ+cẏ+ky=0initial conditionsy(0)=y0,y(0)=dy0.

Eqs. (2), (3) describe the transient response of a spring–mass–damper which is useful

Theoretical analysis of model

There is no guarantee that the damping profile obtained from Fig. 2 is unique or physically realistic. For example, consider the following linear time varying damping profile: ÿtrue+Ftrue(t)ẏtrue+Kytrue=0Ftrue(t)=(0.9+0.7t),K=30ytrue(0)=y0=2,ytrue(0)=dy0=1.

Fig. 3 shows simulated data from Eq. (30). Choose k intervals [Ti1,Ti], i=1,,k and for a given ϵ>0, define a piecewise model approximation: Fmodel,k(t)={Ftrue(t)+ϵ(ΔT)2ytrue(t),t[0,T1],Ftrue(t)+2(1)i1ϵ(ΔT)2ytrue(t)t[Ti1,Ti],i=2,,

Setup and data acquisition

A schematic of the cart system is shown in Fig. 7. The cart can move back and forward along a support rail. The cart is tethered and has two wires, one for the commanded voltage and another for feedback of the cart position which is achieved using an encoder. The origin is set at the middle of the track. The cart system is connected to a space system with Control Desk to allow real-time access to changing control gains, for data acquisition and viewing the signals. A

Conclusion

A general modeling methodology for non-linear damping in second order systems is developed. The method using a piecewise constant model of damping and relates the identified values to the absolute value of velocity by a decaying exponential function. An analytical proof shows that this approach is uniquely identifiable under the constraint that the derivative of the damping is bounded. The exponential decay of the damping versus speed reveals the increased friction or “stiction” that occurs at

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      The steps of this method are evaluated by means of a Matlab Program and the integrals in the following equations are calculated using the trapezium method. This methodology has been used in the identification of damping in an electromechanical system [15,24] with constant stiffness. In this work, a new formulation is created to estimate the damping model in gear system with time varying stiffness.

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