Automatica 46 (2010) 369–374
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Brief paper
Modeling and identification of systems with backlashI Jozef Vörös ∗ Institute of Control and Industrial Informatics, Faculty of Electrical Engineering and Information Technology, Slovak Technical University, Ilkovicova 3, 812 19 Bratislava, Slovakia
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Article history: Received 8 October 2008 Received in revised form 12 August 2009 Accepted 2 November 2009 Available online 27 November 2009 Keywords: Backlash Modeling Identification
abstract A new analytic form of backlash characteristic description is introduced, which uses appropriate switching functions and their complements. The backlash parameters in the model equation are separated; hence their estimation can be solved as a quasi-linear problem using an iterative method with internal variable estimation. Also, the identification of cascaded systems consisting of an input backlash followed by a linear dynamic system is presented. Simulation studies of backlash and cascaded systems identification are included. © 2009 Elsevier Ltd. All rights reserved.
1. Introduction One of the most important nonlinearities that limits the performance of control systems in many applications is the so-called backlash (Kalaš et al., 1985). The backlash (see Fig. 1 can be classified as a hard (i.e. non-differentiable) and dynamic nonlinearity. Backlash is present in every mechanical system where a driving member (motor) is not directly connected with the driven member (load). This is the case in many driven mechanical systems, notably those with gears, e.g. x–y positioning tables, industrial robots, overhead crane mechanisms. Backlash is particularly common in actuators, such as mechanical connections, hydraulic servovalves and electric servomotors. It is well known that this kind of nonlinearity may often cause delays, oscillations and inaccuracy, which severely limit the performance of control systems (Barreiro & Banos, 2006; Hägglund, 2007; Nordin, Galic, & Gutman, 1997; Tao & Kokotovic, 1993, 1995, 1997). Therefore, the compensation of backlash has attracted research efforts for several decades (Nordin & Gutman, 2002; Selmic & Lewis, 2001). In most applications the backlash parameters are either poorly known or completely unknown; therefore the identification of backlash is fundamental for its compensation and implementation of the corresponding inverse. Unfortunately, there are only few contributions in the literature on the identification of systems with hard nonlinearities (Bai, 2002; Tao and Canudas de Wit, 1997;
I This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Antonio Vicino under the direction of Editor Torsten Söderström. ∗ Tel.: +421 35 6410 572; fax: +421 2 654 29 521. E-mail address:
[email protected].
0005-1098/$ – see front matter © 2009 Elsevier Ltd. All rights reserved. doi:10.1016/j.automatica.2009.11.005
Vörös, 1997), and even fewer on backlash identification (Cerone & Regruto, 2007; Giri, Rochdi, Chaoui, & Brouri, 2008; Sun, Liu, & Sano, 1999). A three-stage identification approach is applied to the identification of cascade systems with input backlash followed by a linear dynamic system in Sun et al. (1999). First, the parameters of the transfer function model are identified by using the blind identification method. Secondly, the unknown internal variable is restored by applying the input estimation method. Finally, the two backlash parameters are identified by using the input signal and the restored internal variable. A two-stage parameter bounding procedure for linear systems with input backlash in presence of bounded output errors was published in Cerone and Regruto (2007). In the first stage, a set of square wave inputs with different amplitudes is applied to the system to bind the parameters of the backlash from steady-state operating conditions. In the second stage, the system is excited with a pseudo-random binary signal to evaluate bounds on the parameters of the linear dynamic block. In Giri et al. (2008), an approach is presented that is based on a specific system parameterization (model rescaling) and an appropriate periodic input signal. In this paper, an identification method based on a new mathematical model for backlash is proposed. First, an analytic description of this hard dynamic nonlinearity is introduced, which uses appropriate switching functions and their complements. This multivalued nonlinearity was never before described in analytic form by one output equation. The backlash parameters in the model equation are separated; hence their estimation can be solved as a quasi-linear problem using an iterative method with internal variable estimation, similarly as in Vörös (2002). Then the identification of cascaded systems consisting of an input backlash followed by a linear dynamic system is presented. This means that the cascade of two dynamic subsystems is identified. Application of a decomposition technique leads to a system
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The constant part, i.e., the third line in (1), can be covered using the sum of the complements to f1 (t ) and f2 (t ). Now the backlash can be modeled by one difference equation as x(t ) = mL u(t )f1 (t ) + mL cL f1 (t ) + mR u(t )f2 (t )
− mR cR f2 (t ) + x(t − 1)[1 − f1 (t )][1 − f2 (t )].
(7)
The input/output relation (7) is identical to that of (1). The slopes of straight lines mL and mR may be simultaneously positive or negative, while cL and cR must be positive. This model allows the upward and downward line slopes to be different provided that the intersection of the two lines is not in the region of practical interest. 3. Backlash parameter estimation Fig. 1. Backlash characteristic.
description, which is again quasi-linear, and the parameters can be estimated iteratively based on available measured inputs and outputs. Finally, simulation studies of backlash and cascaded systems with input backlash identification are included.
The proposed new model can be used for estimation of the backlash parameters. Defining the following vector of data
ϕ(t ) = [u(t )f1 (t ), f1 (t ), u(t )f2 (t ), −f2 (t )]T and the vector of parameters
2. Backlash
θ = [mL , mL cL , mR , mR cR ]T = [m1 , c1 , m2 , c2 ]T ,
The backlash characteristic with input u(t ) and output x(t ) is described by two straight lines, upward (right) and downward (left) sides of backlash, connected with horizontal line segments (Cerone & Regruto, 2007). The backlash nonlinearity is shown in Fig. 1, and the mathematical model for the discrete-time case is given by
where
mL [u(t ) + cL ] mR [u(t ) − cR ] x(t − 1)
( x(t ) =
u(t ) ≤ zL u(t ) ≥ zR zL ≤ u(t ) ≤ zR
(1)
where mL , mR , cL > 0, cR > 0 are constant parameters characterizing the backlash and zL = zR =
x(t − 1) mL x(t − 1) mR
− cL
(2)
+ cR
(3)
h(s) =
0, 1,
mL = m1 ,
cL = c1 /m1 ,
mR = m2 ,
cR = c2 /m2 ,
(9)
(10)
the backlash model can be written in the form x(t ) − x(t − 1)[1 − f1 (t )][1 − f2 (t )] = ϕ T (t ) θ .
(11)
As the variables f1 (t ) and f2 (t ) in (8) are unmeasurable and must be estimated, an iterative parameter estimation process has to be considered similarly as in Vörös (2002). Assigning the estimated variables in the s-th step as s
f1 (t ) = h{[s mL u(t ) + s mL s cL − x(t − 1)]/s mL }
(12)
s
f2 (t ) = h{[x(t − 1) − s mR u(t ) + s mR s cR ]/s mR },
(13)
the error to be minimized in the estimation procedure is s +1
e(t ) = x(t ) − x(t − 1)[1 − s f1 (t )][1 − s f2 (t )] − s ϕ T (t ) s+1 θ , (14)
are the u-axis values of the intersections of the two lines, with slopes mL , mR , with the horizontal inner segment containing x(t − 1). It can be seen that the backlash is a first-order nonlinear dynamic system. It is evident that (1) is not well suited for the estimation of the backlash parameters and it is desirable to find an analytical model which is at least quasi-linear and has separated parameters. One way to simplify the backlash description is by the use of an appropriate switching function. In Vörös (2002), the following function was proposed:
(8)
if s > 0 if s ≤ 0
(4)
switching between two sets of values, i.e., (−∞, s) and (s, ∞). The complementary function to h(s) is simply [1 − h(s)]. To describe three branches of (1) in one equation, two applications of the switching function and its complement are needed. The following variable, based on (2), can be defined for the description of the left side of backlash, i.e., the first line in (1): f1 (t ) = h[u(t ) − zL ] = h{[mL u(t ) + mL cL − x(t − 1)]/mL };
(5)
and similarly, based on (3), the following variable can be defined for the description of the right side of backlash, i.e., the second line in (1): f2 (t ) = h[zR − u(t )] = h{[x(t − 1) − mR u(t ) + mR cR ]/mR }.
(6)
where ϕ(t ) is the data vector with the corresponding estimates of variables f1 (t ) and f2 (t ) according to (12) and (13) and s+1 θ is the (s + 1)-th estimate of the parameter vector. If the input is persistently excited (with respect to the dead zone) the steps in the iterative procedure may be stated as follows. s
(a) Minimizing the mean squares criterion based on (14), the estimates of parameters s+1 θ are computed using s ϕ(t ) with the s-th estimates of variables s f1 (t ) and s f2 (t ). (b) Using (12) and (13), the estimates of s+1 ϕ(t ) are evaluated by means of the recent estimates of the corresponding parameters. (c) If the criterion is met the procedure ends; else it continues by repeating steps (a) and (b). In the first iteration, nonzero initial values of the backlash parameters have to be considered for evaluation of 1 ϕ(t ) to begin the iterative algorithm. In the simplest case, mL = mR = 1 for a backlash with positive slopes or mL = mR = −1 for a backlash with negative slopes, while cL and cR are chosen sufficiently small. The convergence of the above algorithm with estimation of internal variables f1 (t ) and f2 (t ) cannot be exactly proved. However, as the corresponding linear segments of the nonlinearity are always included into the computation, the convergence of the iterative algorithm is good.
J. Vörös / Automatica 46 (2010) 369–374
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Assigning the estimated variables in the s-th step according to (12) and (13), and the estimate of internal variable x(t ) as s
x(t ) = s m1 u(t )s f1 (t ) + s c1 s f1 (t ) + s m2 u(t )s f2 (t ) − s c2 s f2 (t )
+ s x(t − 1)[1 − s f1 (t )][1 − s f2 (t )], Fig. 2. Cascade system with backlash.
the error to be minimized in the estimation procedure is now s+1
4. Cascade systems with input backlash In many real control systems, the backlash appears in a cascade connection with a linear dynamic system. One of the possible cases is a cascade system, where the backlash is followed by a linear dynamic system, as shown in Fig. 2. The linear dynamic system can be described by the difference equation model B(q−1 )y(t ) = A(q−1 )x(t ),
(15)
where x(t ) and y(t ) are the inputs and outputs, respectively, and A(q−1 ) and B(q−1 ) are scalar polynomials in the unit delay operator q−1 : A(q−1 ) = a1 q−1 + · · · + am q−m ,
(16)
B(q−1 ) = 1 + b1 q−1 + · · · + bn q−n .
(17)
The output equation of this cascade system can be constructed from (7) and (15). However, a direct substitution of (7) into (15) would lead to a very complex expression; therefore the so-called key term separation principle can be applied (Vörös, 1995). In this connection of two systems we can assume that a1 = 1 and substitute (7) only for the separated variable x(t − 1), leading to the following equation: y(t ) = mL u(t − 1)f1 (t − 1) + mL cL f1 (t − 1)
+mR u(t − 1)f2 (t − 1) − mR cR f2 (t − 1) +x(t − 2)[1 − f1 (t − 1)][1 − f2 (t − 1)] m n X X + ai x ( t − i ) − bj y(t − j), i=2
(18)
j =1
where the parameters of both the backlash and the linear system are separated and the equation is quasi-linear as the variables f1 (t ) and f2 (t ) depend on the backlash parameters. The parameter estimation for a cascade system with input backlash can be performed similarly, as in the previous case. Defining the vector of data
Φ (t ) = [u(t − 1)f1 (t − 1), f1 (t − 1), u(t − 1)f2 (t − 1), − f2 (t − 1), x(t − 2), . . . , x(t − m),
− y(t − 1), . . . , −y(t − n)]T
(23)
(19)
e(t ) = y(t ) − s x(t − 2)[1 − s f1 (t − 1)][1 − s f2 (t − 1)]
−s Φ T (t ) s+1 Θ ,
(24)
where s Φ (t ) is the data vector with the corresponding estimates of variables f1 (t ) and f2 (t ) according to (12) and (13) and the internal variable x(t ) according to (23), while s+1 Θ is the (s + 1)-th estimate of the parameter vector. The steps in the iterative procedure are stated as follows (for persistently excited input). (a) Minimizing the mean squares criterion based on (24), the estimates of parameters s+1 Θ are computed using s Φ (t ) with the s-th estimates of variables s f1 (t ), s f2 (t ) and s x(t ). (b) Using (12), (13) and (23), the estimates of s+1 Φ (t ) are evaluated by means of the recent estimates of the corresponding parameters. (c) If the criterion is met the procedure ends; else it continues by repeating steps (a) and (b). In the first iteration, nonzero initial values of the backlash parameters have to be considered for evaluation of 1 Φ (t ) to begin the iterative algorithm, while the initial values of the linear system parameters can be chosen as zero. This identification method enables simultaneous estimation of both the backlash parameters and the parameters of linear dynamic system compared to the three-stage identification approach proposed in Sun et al. (1999). In contrast to the method in Cerone and Regruto (2007), only one set of input/output data is needed to perform the identification of a cascaded system with input backlash. 5. Simulation studies The methods for the identification of backlash systems were implemented and tested in MATLAB. Several cases were simulated and the estimations of parameters were carried out on the basis of input and output records. The performance of the proposed methods is illustrated in the following examples.
5.1. Backlash
and the parameter vector
Θ = [m1 , c1 , m2 , c2 , a2 , . . . , am , b1 , . . . , bn ]T ,
(20)
where again mL = m1 ,
cL = c1 /m1 ,
mR = m2 ,
cR = c2 /m2 ,
(21)
the model of a cascade system with backlash can be written in the following form: y(t ) − x(t − 2)[1 − f1 (t − 1)][1 − f2 (t − 1)] = Φ T (t ) Θ .
(22)
As the variables f1 (t ), f2 (t ) and the internal variable x(t ), i.e. the backlash output, are unmeasurable and must be estimated, an iterative parameter estimation process with internal variable estimation has to be considered, similarly as in the previous section.
Example 1. The backlash with unequal slopes shown in Fig. 3 was simulated with the following parameters: mL = 1.3, cL = 0.5, mR = 2.5, cR = 0.7. The identification was performed on the basis of 800 samples of uniformly distributed random inputs with |u(t )| < 1.0 and simulated outputs. Relatively high normally distributed random noise with zero mean and signal-to-noise ratio SNR = 10 (the square root of the ratio of output and noise variances) was added to the outputs. The iterative estimation algorithm was applied with initial values mL = mR = 1 and cL = cR = 0.001 for the first estimate of f1 (t ) and f2 (t ). The process of parameter estimation is shown in Fig. 4 (the top-down order of parameters is mR , mL , cR , cL ). The estimates converge to the values of real parameters after 7 iterations.
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Fig. 6. Parameter estimates — Example 2. Fig. 3. Characteristic — Example 1.
Fig. 4. Parameter estimates — Example 1.
Fig. 7. Backlash characteristic — Example 3.
Fig. 5. Characteristic — Example 2.
Fig. 8. Parameter estimates — Example 3.
Example 2. The backlash with relatively narrow unequal slopes shown in Fig. 5 was simulated with the following parameters: mL = 1.7, cL = 0.1, mR = 1.4, cR = 0.15, i.e., the deadzones were relatively small. The parameter estimation was performed under the same conditions as in Example 1. The process of parameter estimation is shown in Fig. 6 (the top-down order of parameters is mL , mR , cR , cL ). The estimates converge to the values of real parameters after 5 iterations. As the simulation studies show, the convergence of the proposed identification algorithm is good, despite the relatively high level of additive noise. This is because the nonlinearity actually has linear segments, and the switching functions separate these segments. The main estimation problem is with the ‘‘hard nonlinear element’’ of backlash, i.e., the generally asymmetric deadzones. In Example 2, the deadzones are smaller, therefore the number of iterations is less than in Example 1. 5.2. Cascade systems with input backlash Example 3. The cascade system with an input backlash (Fig. 7 characterized by the parameters mL = 1.3, cL = 0.6, mR = 2.0,
cR = 0.7 and followed by the linear dynamic system described by the difference equation y(t ) = x(t − 1) + 0.5x(t − 2) + 0.2y(t − 1) − 0.35y(t − 2) was considered. The identification was performed on the basis of 1500 samples of uniformly distributed random inputs with |u(t )| < 1.0 and simulated outputs. Normally distributed random noise with zero mean and SNR = 25 was added to the outputs. The iterative estimation algorithm was applied with initial values mL = mR = 1 and cL = cR = 0.001 for the first estimate of f1 (t ) and f2 (t ), while the initial values of linear system parameters were chosen as zero. The process of parameter estimation is shown in Fig. 8 (the top-down order of parameters is mR , mL , cR , cL , a2 , b2 , b1 ). The estimates converge to the values of real parameters after 9 iterations. Example 4. The cascade system with a narrow input backlash (Fig. 9) characterized by the parameters mL = 0.24, cL = 0.035, mR = 0.26, cR = 0.07 and followed by the same linear dynamic system was considered. The identification was performed under the same conditions as in Example 3. The process of parameter estimation is shown in Fig. 10 (the top-down order of parameters
J. Vörös / Automatica 46 (2010) 369–374
Fig. 9. Backlash characteristic — Example 4.
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Fig. 11. Backlash characteristic — Example 5.
Fig. 12. Parameter estimates — Example 5.
Fig. 10. Parameter estimates — Example 4.
is a2 , b2 , mR , mL , cR , cL , b1 ). The estimates converge to the values of real parameters after 4 iterations. Example 5. This example shows that the proposed method can also deal with negative slopes of backlash, and that the choice of parameter initial values is not too restrictive. A cascade system with the same linear dynamic system was considered, but the input backlash was with negative and equal slopes mL = mR = −1.3 and with cL = cR = 0.2 (see Fig. 11). The parameter estimation was performed under the same conditions as in Example 3. The process of parameter estimation is shown in Fig. 12 (the top-down order of parameters is a2 , b2 , cR = cL , b1 , mR = mL ). Despite the fact that the initial values of mL and mR were chosen positive, the iterative process converged after 5 iterations. As the simulation studies show, the convergence rate of the identification algorithm for cascade systems with input backlash is relatively high. The main estimation problem is again with the asymmetric deadzones (compare Examples 3 and 4) and naturally with the connection of nonlinear dynamic and linear dynamic systems. The choice of initial values for the backlash parameters is not crucial (see Example 5), while those of the linear system may be chose as zero. 6. Conclusions The identification of systems with unknown backlash is still an open theoretical problem of major relevance to control applications. In this paper, a new analytic form of backlash characteristic was introduced, where this three-branch nonlinearity is described by one output equation with separated parameters. The new model
was applied to the identification of generally asymmetric backlash systems and to cascaded systems consisting of an input backlash followed by a linear dynamic system. For both cases, iterative parameter estimation algorithms were proposed and their feasibility was shown in simulation studies. Although no convergence proof of the identification methods with internal variable estimation is available, testing of the proposed algorithms showed very good results. The convergence proof of the proposed algorithms is a very challenging and difficult task and worth further research, because up to now, no convergence proofs are available for cascade model parametric identification methods which explicitly work with internal variables. Compared to the previously published approaches, the proposed model for cascaded systems enables simultaneous estimation of both the input backlash parameters and the parameters of a cascaded linear dynamic system on the basis of one set of input/output data. The presented backlash identification method is proposed for systems with ‘normal’ operation conditions, i.e., eventual collision situations are not considered. Finally, note that the technique presented can be easily extended to more complex cascaded systems with backlash. Acknowledgement The author gratefully acknowledges financial support from the Slovak Scientific Grant Agency (VEGA). References Bai, E. W. (2002). Identification of linear systems with hard input nonlinearities of known structure. Automatica, 38(5), 853–860. Barreiro, A., & Banos, A. (2006). Input–output stability of systems with backlash. Automatica, 42, 1017–1024. Cerone, V., & Regruto, D. (2007). Bounding the parameters of linear systems with input backlash. IEEE Transactions on Automatic Control, 52(3), 531–536. Giri, F., Rochdi, Y., Chaoui, F. Z., & Brouri, A. (2008). Identification of Hammerstein systems in presence of hysteresis-backlash and hysteresis-relay nonlinearities. Automatica, 44(3), 767–775. Hägglund, T. (2007). Automatic on-line estimation of backlash in control loops. Journal of Process Control, 17(6), 489–499.
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Jozef Vörös received his Ph.D. degree in Control Theory from the Institute of Technical Cybernetics of the Slovak Academy of Sciences, Bratislava, in 1983. Since 1992 he has been with the Faculty of Electrical Engineering and Information Technology at the Slovak Technical University in Bratislava, where he is acting as a senior research scientist in the Institute of Control and Industrial Informatics. His research interests include the analysis and identification of nonlinear systems. He is also interested in the area of mobile robot path planning using quadtree and octree representations.
