Proceedings of the 13th WSEAS International Conference on SYSTEMS
Identification of Systems with Friction via Distributions using the Modified Friction LuGre Model RADU ZGLIMBEA, VIRGINIA FINCA, EMILIAN GREABAN, MARIN CONSTANTIN Department of Automation University of Craiova Bd. Decebal, no.5, Craiova ROMANIA
[email protected] Abstract: - This paper presents the identification procedures based on distributions theory to continuous time systems with friction using the modified LuGre friction model. Mainly it applies the results of the identification procedures based on distributions theory to continuous time systems with friction. There are defined the so called generalized friction dynamic systems (GFDS) as a closed loop structure around a smooth system with discontinuous feedback loops representing friction reaction vectors. Both GFDS with static friction models (SFM) and dynamic friction models (DFM), also the modified friction LuGre model is analyzed. The identification problem is formulated as a condition of vanishing the existence relation of the system. Then, this relation is represented by functionals using techniques from distribution theory based on testing function from a finite dimensional fundamental space. The advantage of representing information by distributions are pointed out when special evolutions as sliding mode, or limit cycle can appear.The proposed method is a batch on-line identification because identification results are obtained during the system evolution after some time intervals but not in any time moment. This method does not require the derivatives of measured signals for its implementation. Some experimental results are presented to illuminate its advantages and practical use. Keywords: - Identification; Distribution theory; Friction; the modified friction LuGre model.
1 Introduction There are many mechanical systems, from machine tool positioning to tracking in navigation, where the so-called friction forces influence the motion. Friction models contain some specific nonlinearity such as stiction, hysteretic, Stribeck effect, stick-slip, depending on velocity [1], [2], [3], [4]. The values of model parameters can change during the system evolution or are influenced by some other causes as external temperature, quality of materials etc. A large variety of friction models, as Coulomb friction model, Dahl model [5], [6], LuGre model [19], exponential model [7], bristle model [8], state variable model [9], there are accepted in literature [1]. Ignoring friction in controlling such systems can lead to tracking errors, limit cycles, undesired stick-slip motion [2]. Such adaptive strategies involve identification procedures of the controlled system, including identification of the model friction parameters. Unfortunately friction models are nonlinear, involving a discontinuous dependence with respect to velocity. Because of this, many techniques, as identification based on time-discretized models fail to offer good results.
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A survey of models, analysis tools and compensation methods for the control of machines with friction is presented in [9]. A distribution-based approach of mechanical systems with friction identification is developed in [10]. This extends the results on continuous time system identification based on distribution theory, reported in [11], for linear systems, or in [12], for nonlinear systems. To perform continuous time domain identification the system differential equations is transformed to an algebraic system that reveals the unknown parameters, [13]. This can be done also by using some modulating functions to generate functionals to avoid the direct computation of the input-output data derivatives [14], [15], [16]. Through this, is intended to demonstrate the effectiveness of the distribution based identification technique from [10], for friction mechanical systems. As the identification results are obtained during the system evolution after some time intervals but not in any time moment, proposed self-tuning method is a batch on-line. The paper is organized as follows: After introduction in the first section, Section 2 is dedicated to the generalized friction dynamic systems GFDS, as presented in [10]. Section 3, presents the main steps
ISBN: 978-960-474-097-0
Proceedings of the 13th WSEAS International Conference on SYSTEMS
the of continuous time system identification based on distributions. Section 4 illustrates application for a friction mechanical system identification using the modified friction LuGre model . Section 5 presents experimental results, again conclusions are resumed in Section 6.
2 Generalized Friction Dynamic Systems A generalised dynamic friction system (GFDS) is a system characterised by the state equation of the form x f ( x, u , r1 ,.., ri ,.., rp ) (1)
where x(t ) X R n , t t0 , is the state vector and u (t ) U R q , t t0 is the input vector. The vectors ri (t ) R i R n , t t0 , i 1: p (2) are called friction reaction vectors. They depend on x and u through a specific operator i {} , called friction operator, ri i {x, u}, i 1: p (3) There are two categories of friction models: static friction models (SFM) and dynamic friction models.(DFM). For SFM, the operator (3) is a non dynamic mapping ri Fi ( x, u ) : X × U R i R mi , i 1: p (4) with a specific structure as follows. For any i 1: p , there are two functions
vi i ( x, u ) : X × U Vi R mi , i 1: p , (5) which determines the so called generalized velocity vector vi , and ai i ( x, u ) : X × U A i R mi , i 1: p , (6) expressing the so called active component of the velocity vector vi . In SFM, the non dynamic mapping (4) can be expressed as a function of vi , and ai only, that means, ri i (vi , ai ) Fi ( x, u ), i 1: p (7) The structure of a GDFS with SFM is illustrated in Fig.1. u
u
x f ( x, u , ri )
ri
ri
vi i (vi , ai )
ai
u h ( x, u )
x
i ( x)
zi ai vi zi vi
(10)
ri ci zi
(11)
(9) There are many types of DFM but we shall present a simplified Dhal model [2], [5], for mi 1 , characterized by a first order nonlinear dynamic system
where
the
velocity
vi
is
vi i ( x) R . (12) This system can be expressed as a single differential equation with respect to the friction reaction variable ri ci [1 ( ai / ci ) sgn(vi ) ri ] vi (13)
3 Continuous Time System Identification Based on Distributions This section presents the main results on continuous time system identification based on distribution, as have been presented in [11]. Let n be the fundamental space from distribution theory [17] of the real testing functions, : , t (t ) , having continuous derivatives at least up to the order n , with compact support T for any of the above derivative. The linear space n is organized as a topological space considering a specific norm [17]. A distribution is a linear, continuous real functional on n , F : n , F () . Let q : , t q(t ) be a function that admits a Riemann integral on any compact interval T from . Using this function, a unique distribution Fq : n , Fq () can be build by the relation Fq () q (t ) (t ) dt , n .
The m-order derivative of a distribution is a new distribution, Fq( m ) n/ uniquely defined by the
i 1: p
Fig.1, The feedback structure of a GDFS with SFM
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(8)
Considering, at least, q C0 ( ) , the following important equivalence take place [18], Fq () 0, n q (t ) 0, t (14)
ri Fi ( x, u ),
u
ri hi ( zi , x, u ) zi fi ( zi , x, u )
y
x
i ( x, u )
For dynamic friction models, (DFM) the operator (3) is a dynamic system characterized by an additional state vector zi , expressing internal changes in some surfaces of relative movements. The friction reaction vector ri is the output of a dynamic system
relations,
580
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Proceedings of the 13th WSEAS International Conference on SYSTEMS
Fq( m ) () (1) m Fq (( m ) ), n
(15)
When q C ( ) , then m
Fq( m ) () Fq( m ) () q ( m ) (t )(t )dt
that means the k-order derivative of a distribution generated by a function q C m ( ) equals to the distribution generated by q ( m ) , the k-order time derivative of the function q . If q Cm ( ) , from (13), (16) one can write, n Fq( m ) () q ( m ) (t )(t )dt (1) m q (t ) ( m ) (t ) dt (16)
p
where wi and v represent a sum of the derivatives of some known, possible nonlinear, functions , , with respect to the input and output variables, j 0
wi j i 1[ ij (u , y )]( ni ) , i 1: p ,
(20)
v j 0 1[ 0j (u , y )]( n0 ) .
(21)
j
where parameters pi , ni , p0 , n are given integer numbers. In [11] the existence and uniqueness conditions for a problem of distribution based continuous time system identification are presented. Suppose that it is possible to record the functions (u, y) in an the time interval T , called observation time interval or just time window. The restriction of the functions (u , y ) to the time interval j
j 0
T is denoted by (uT , yT ) respectively. If no confusion would appear, then we may drop the subscript T . An identification problem means to determine the parameter , given the priori information on the model structure FC , (17), and a set of observed input output pairs (uT , yT ) , (uT , yT , FC ) in a such a way that, q /(u , y ) (t ) 0, t (22) T
T
This condition involves, q /(u , y ) (t ) 0, t , (u , y )
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j
R
j i 1 ( 1) ni p
j
[ R
j i
j
(t )]( ni ) (t )dt
which determines the row vector, FwT () [ Fw1 (),..., Fwi (),..., Fwp ()] p .
Fv () j 1 [ 0j (t )]
( n0j
p0
)
(t )dt
j
j
(24)
(25)
(26)
R
q /( u , y ) FC (u , y, ) i 1 wi v wT v , (19)
p
p
p
It represents a family of models with a given structure in constant parameters. A special case is the model (17) expressing a linear relation in the parameters
j
Fwi () j i 1 [ ij (t )]( ni ) (t )dt
j 0 1 (1) n0 [ 0j (t )]( n0 ) (t )dt
whose expression depends on a vector of parameters [1 ... i ... p ]T . (18)
p
the functions (20), (21),
R
Let us consider a dynamical continuous time system expressed by a differential operator, q /( u , y ) FC (u , y , ) (17)
j i
for any input-output pair (u , y ) observed to that system. Let us consider two families of regular distributions, Fwi , i 1: p , and Fv () created based on
(23)
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Any input-output pair ( u, y ) observed from the system (27) is described by a pair of regular distribution ( Fw , Fv ) for any n , [11]. The problem of the system (17) parameter identification can be represented now by distributions. For example, the regular distribution generated by the continuous function q /( u , y ) from (17), is related to the parameter vector , n as
(27)
Fqθ () Fqθ /( u , y ) () i 1 Fwi ()i Fv () p
FwT () Fy () If a triple (u*, y*, *) is a realization of the model (17), then the identity (39) takes place, Fqθ* () Fqθ* /( u *, y *) () 0, n (28) and vice versa, if an input-output pair (u*, y*) of the family of models (27), with unknown parameter , generates a distribution Fqθ () Fqθ /( u *, y *) () i 1 Fwi () i Fv () (29) p
which satisfies Fqθ () Fqθ /( u *, y *) () 0, n * , (30) As has p components it is enough a chose (utilize) a finite number N p of fundamental function i , i 1: N
and to build an algebraic equation,
Fw Fv (31) where Fw is an ( N p ) matrix of real numbers Fw [ FwT (1 );...; FwT (k );...; FwT ( N )]T
(32)
where k-th row F (k ) is given by (25). The symbol T w
Fv denotes an N -column real vector built from (26), Fv [ Fv ( 1 ),..., Fv ( k ),..., Fv ( N )]T .
(33)
When only the restriction ( uT ,yT ) of the pair ( u , y ) on the time interval T is available, any i must have
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Proceedings of the 13th WSEAS International Conference on SYSTEMS
for its k derivative (km ) (t ), m 1: n the same compact support Tk ,
where z v
0 v
(41)
z
g (v)
v supp{(km ) (t )} Ti [tak , tbk ] T , m 1: n, k 1: N (34) (42) and g (v) Fs ( Fc Fs ) 1 v Below there are some simple testing functions with 0 known as dynamic friction parameters and is k n , k k k k k (t ) k k (ta , tb ) k (t , ta , tb ) (35) a stiffness parameter, respectively Fs , Fc and known as static friction parameters, where Fc is the n k k k sin k [ (t tb ) / (tb ta ) ], k , nk n (36) (t , tak , tbk ) Coulomb friction, Fs is the static friction and is a k k 0, t (, ta ] [tb , ) viscous friction coefficient, again z is the internal where i is a scaling factor and i normalizes the state of the modified friction model. area The state x of (1) has three components, x1 x , tbk k k k k k k x (t , t ) 1 / (t , t , t ), t t . (37) 2 v and x3 z , so k
a
b
tak
k
a
b
a
b
If r rank (Fw ) p , then a unique solution is obtained. (FwT Fw ) 1 FwT Fv * (38)
4 Application for a Friction Mechanical System Identification Using the Modified of Friction LuGre Model To combine the elements of GFS with the theory of identification based on distributions, as presented in [10], let us consider a simple mechanical system consisting of a mass m , moving on a surface with bristle. The system is excited by a force u , while the (unmeasurable) friction force f resists the motion (Fig. 2). The model for this system is: mv u F (39)
v
c
m
s
s
s
F F v v v v F v F vu F v u v vu u F except a set of points of a zero measure. Denoting mF F 1 m ; 2 ; 3 0 ; 4 c ; 5 c ; Fs Fs m 0 0 0 Fc Fc 6 ; 7 ; 8 ; 9 and Fs Fs Fs Fs
2
0
10
0
c
2
s
c
0
s
s
0
the parameters that have to be identified, Fs (44) is expressed as the operator (19), where p 10 . The distribution image (29) of this differential operator, evaluated for a testing function k on the
F the zone of friction
Tk ta k , tb k T , contains the elements given by (24), (25), (26) of the form
time Fig.2, A simple mechanical system with friction The friction force F is given of the modified LuGre model: F 0z v (40)
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0
c
0
s
v
u
x1 x2 1 (43) x2 u 0 x3 x2 m 0 x2 x3 x2 x3 x Fs ( Fc Fs ) 2 x2 1 where x1 x is position, x2 v represents velocity, again x3 z represents the displacement of bristles which is unmeasureable. The system can be described by the following differential equation (44): mF F m mv v v vv vv v v v vv F F F
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interval
Fwi k
tb k
w (t ) i
k (t ) dt
where,
ta k
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Proceedings of the 13th WSEAS International Conference on SYSTEMS
tb k
Fw1 k v(t ) k (t ) dt Fw2 k
ta k
tb k
0.1s with 0.01s 0.01s 1 initial state x(0) [0 0 0] and only m , 0 , Fc , , Fs as parameters necessary for identification. Ten testing functions k on Tk, as (35), with nk=10 and T1=[1,1.4]; T2=[1.4,1.8]; T3=[1.8,2.2]; T4=[2.2,2.6]; T5=[2.6,3]; T6=[3,3.4]; T7=[3.4,3.8]; T8=[3.8,4.2]; T9=[4.2,4.6]; T10=[4.6,5] are utilized. through function of transfer
v(t ) k (t ) dt
ta k
tb k
Fw3 k
v(t ) (t )dt k
ta k tb k
tb k
k
k
Fw4 k v t v(t ) k (t )dt
ta
tb k
Fw5 k
v t (t )dt 2
k
ta
6
Measured signal States: x=[x1 x2 x3] x(0)=[0 0 0]
v(t ) v(t ) k (t ) dt
ta k
Fw k
tb k
6
v(t ) v(t ) k (t )dt
ta k
v(t ) v(t ) v(t )
(t )dt
2
(t ) k (t )dt
tb k
v(t ) v(t )
k
(t )dt
ta k
n tio ra le ce ac
v(t ) v
2
x3
tb k
ta k
Fw8 k
k
ta k
x2
7
tb k
x1
Fw k
2
-2
tb k
v (t ) 2
ta
k (t ) dt
Time (sec.)
k
tb k
tb k
ta k
ta k
-6 0
Fw k v(t ) u (t ) k (t ) dt v(t ) u (t ) k (t ) dt tb k
tb k
ta k
ta k
10
v(t ) v(t ) u (t )
k
4
6
8
10
Fig.3, Measured variables for the system with the modified friction LuGre model and initial state x(0)=[0 0 0]
9
Fw k v(t ) u (t ) k (t )dt
2
(t )dt
The real and identified parameter values are respectively:
tb k
Fv k u (t ) k (t )dt .
ta k
For the evaluation of these integrals only input-output pairs ( v, u ), respectively ( v, u ) and the module of v are necessary. Otherwise also the speed v and the acceleration v will be measured, for to measure acceleration will use an accelerator. Integrals are utilized to build the system (31), (32), (33), whose solution is (38).
5 Experimental Results To implement distribution based identification methods, an experimental platform (DBI) has been developed. It allows creating testing functions with settable parameters, automatically to create and solve the system (31). The input-output data for identification are obtained from an external source (a data file) or internally by simulation. Many examples and types of friction systems have been implemented for identification but, because of limited space in this paper, only one example is analysed, based on the application presented in section 4. In this example, the measured signals, as indicated in Fig.3., are generated by a step input u (t ) 1(t ) passed
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m: 0 : Fc : : Fs :
0.2000000000000 300.00000000000 0.3300000000000 0.0300000000000 0.2900000000000
0.1999919109469 299.99937408757 0.3299672364812 0.0300235552826 0.2899840840260
For the same input but with x(0)=[1 1 1] and T1=[1,1.3]; T2=[1.3,1.6]; T3=[1.6,1.9]; T4=[1.9,2.2]; T5=[2.2,2.5]; T6=[2,5.2.8]; T7=[2.8,3.1]; T8=[3.1,3.4]; T9=[3.4,3.7]; T10=[3.7,4], the identification results are:
m : 0.2000000000000 0.1999999236194 0 : 300.00000000000 299.99993551165 Fc : 0.3300000000000 0.3300031565991 : 0.0300000000000 0.0299974467911 Fs : 0.2900000000000 0.2899997173826 The measured variables for this case are illustrated in Fig.4.
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Proceedings of the 13th WSEAS International Conference on SYSTEMS
6
Measured signal States: x=[x1 x2 x3] x(0)=[1 1 1]
2
x
2
3
le ce
x
1
ac
x
-2
tio ra
Time (sec.) 2
4
6
n
-6 0
8
10
Fig.4, Measured variables for the system with the modified friction LuGre model and initial state x(0)=[1 1 1]
6 Conclusions The above results illustrates the advantages of distribution based identification for systems with discontinuities on the right side. Description by functionals allows to enlarge the area of systems to which identification procedures can be applied. This paper is development of a paper “Identification of systems with friction via distributions”.
References: [1] B. Armstrong-Helouvry, Control of machines with friction, Boston MA, Kluwer, 1991 [2] C. Canudas de Wit, H. Olsson, K. J. Åström, P. Lischinsky, "A New Model for Control of Systems with Friction", IEEE Trans. Automat. Contr., vol.40, pp. 419–425, March 1999. [3] H. Du and N. Satish, "Identification of Friction at Low Velocities Using Wavelets Basis Function Network", Proc. of American Control Conference, Philadelphia, June 1998, pp. 1918– 1922. [4] S.J. Kim. and I.J. Ha, “A Frequency-Domain Approach to Identification of Mechanical Systems with Friction,” IEEE Trans. Automat. Contr., vol.46, pp. 888–893, June. 2001. [5] P. Dahl, “A solid friction model,” Aerospace Corp., El Segundo, CA,Tech. Rep. TOR0158(3107-18)-1, 1968. [6] B. Armstrong-Helouvry, P. Dupont, and C. Canudas de wit, , "A Survey of Models, Analysis Tools, and Compensation Methods for the Control of Machines with Friction," Automatica, Vol. 30, No. 7, pp. 1083-1138, 1994. [7] Canudas de wit, C., Noel, P., Aubin, A., and Brogliato, B., "Adaptive Friction Compensation in Robot Manipulators: Low Velocities," Int. Journal of Robotics Research, Vol. 10, No. 3, pp. 189-199, 1991.
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[8] Haessig, D. A. and Friedland, B., "On the Modeling and Simulation of Friction," ASME Joumal of Qynamic Systems, Measurement, and Control, Vol. 113, pp. 354-362, Sept. 1991. [9] B. Helouvry, P. Dupont, and C. de Wit, “A survey of models, analysis tools and compensation methods for the control of machines with friction”, Automatica, vol. 30, no. 7, pp. 1083–1138, 1994. [10] S. J. Kim and I. J. Ha, “On the existence of Carathèodory Solutions in mechanical systems with friction,” IEEE Trans. Automat. Contr., vol. 44, pp. 2086–2089, Nov. 1999. [11] C. Marin, "System Identification Based on Distribution Theory". Proceedings of the IASTED International Conference Applied Simulation and Modelling (ASM 2002), Crete, June, 2002 [12] C. Marin, E. Petre, D. Selisteanu, D. Sendrescu, "Identification of Nonlinear Plants via Distributions. Application for Wastewater Biodegradation Process," The 15th Int. Conference on Control Systems and Computer Science-CSCS15, May 25-27,2005, Bucharest. [13] N. K.Sinh, G. P. Rao, Identification of continuous–time systems. Dordreht: Kluwer Academic Press. [14] C. Marin, System Identification by the method of weighted moments, 12th Int. Conference on Control Systems and Computer Science, CSCS 12, Bucharest. [15] T. Bastogne, H. Garnier, P. Sibille, "PMF subspace method for continuous-time model identification deterministic study," Proc. Of IFAC Symp.SYSID'97, Fukuoka, July, 16651670. [16] A. E. Pearson, A. E. Lee, "On the identification of polynomial input-output differential systems," IEEE Trans.Aut.Contr.,Vol. AC30, No8, 778782. [17] W. Kecs, P. Teodorescu, Introduction to distribution theory with technical applications, Editura Tehnica, Bucharest 1975. [18] V. Barbu, Differential equations. Editura Junimea, Iasi 1985. [19] D. P. Liu , "Research on Parameter Identification of Friction Model for Servo Systems Based on Genetic Algorithms", Proceedings of the Fourth International Conference on Machine Learning and Cybernetics, Guangzhou, 18-21 August 2005, pp. 1116-1120.
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