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Control design of nonlinear mechanical systems with friction

Takashi Nakakuki and Katsutoshi Tamura1

Abstract—This paper presents a solid analysis and control design of nonlinear mechanical systems with friction.Utilizing the Filippov solution concept,it is shown that the class of nonsmooth systems can not be asymptotically stabilized via any Lipschitz continuous static control law.Then,a sliding mode controller which makes the nonsmooth mechanical system asymptotically stable is derived.The validity of the controller design is veri?ed by a simulation example for an electronic throttle control system.

I.I NTRODUCTION

There are several mechanical systems with nonsmooth-ness in applications.Nonsmooth nature in mechanical sys-tems is caused by several phenomena such as friction, backlash,hysteresis of dynamics,nonsmooth controller or switching structure of hybrid systems.While nonsmooth systems have many problems to be resolved,there are few theoretical tools to deal with them.In a mathematical sense,they are described by differential equations with nonsmooth right hand side.Also,in a practical sense, control performance degradation may appear such as limit cycle(chattering phenomenon),dead zone and so on.The systems with friction have important and interesting nature from the practical point of view.There are two kinds of friction such as static and dynamic friction.In either case, it is essentially that friction causes the nonsmoothness or discontinuity at zero-velocity.

In nonsmooth analysis,the Filippov solution concept[1] and related Lyapunov stability theory[2],[3]are useful. In[2],the stability of a harmonic oscillator with Coulomb friction was shown by employing nonsmooth analysis.Pa-pers of the controller design for mechanical systems with friction include[3],[4],[5],[6].In[3],a sliding mode controller design of robot manipulators was presented,while the algorithm was a little complicated.[4]showed that for a Coulomb friction oscillator any continuous algorithm does not admit asymptotic stability.On the basis of this fact, [5]proposed a switched chattering controller for electrical servo-motors.Recently,[6]proposed a Lyapunov-based friction compensator for a hydraulic actuator.

In this paper,we consider generic second order nonlinear mechanical systems with friction.It is proven that the class of nonsmooth systems does not admit asymptotic stability via any Lipschitz continuous static control law.Then,a sliding mode controller is proposed by employing rigorous nonsmooth analysis to make the system asymptotically stable.After illustrating our results by a simple example, 1Department of Mechanical Engineering,Graduate Division of Science and Technology,Sophia University,7-1,Kioicho,Chiyoda-ku,Tokyo, Japan,1028554;e-mail:t-nakaku@sophia.ac.jp a simulation for an electronic throttle control system is presented.

II.P RELIMINARIES

In this section,we present some important de?nitions and theorems for nonsmooth analysis.Consider the vector autonomous differential equation

˙x=f(x),x(t0)=x0(1) where f:R n→R n is de?ned almost everywhere and is measurable and essentially locally bounded in an open region D?R n.Under these conditions,the existence of Filippov solution is assured(see[1]).

De?nition2.1(Filippov solution):A vector function x(t)is called a solution of(1)on[t0,t1]if x(t)is absolutely continuous on[t0,t1],and for almost all t∈[t0,t1]

˙x∈K[f](x)(2) where

K[f](x)=

δ>0

μN=0

ˉco f(B(x,δ)?N).(3)

μN=0

denotes the intersection over all sets N of Lebesgue measure zero,B(x,δ)={y∈R n| y?x <δ}andˉco denotes the convex closure.

In the nonsmooth analysis,the behavior of the solution on nonsmooth surface is speci?cally important.Next two theorems by Filippov[1]are useful to investigate them. Theorem2.1(Lemma3in[1]):Let the regions G?and G+in the space R n be separated by a smooth surface S(x). Suppose that the vector function f(x)is bounded and its limiting values f?and f+exist when S(x)is approached from G?and G+.Let f?N and f+N be the projections of the vectors f?and f+on the normal to the surface S(x) directed from G?and G+.Let the vector function x(t)be absolutely continuous and for t1≤t≤t2let x(t)∈S(x), f?N≥0,f+N≤0,f?N?f+N>0.In order for x(t)to be a solution of equation(1),it is necessary and suf?cient that for almost all t∈[t1,t2]

˙x=αf++(1?α)f?,α=

f?N

f?N?f+N

.(4) Theorem2.2(Lemma8in[1]):If f+N≤0for t1

43rd IEEE Conference on Decision and Control

December 14-17, 2004

Atlantis, Paradise Island, Bahamas

ThA07.2

Under the concept of Filippov solution,some related Lyapunov stability results have been proposed in[2]and [3].

Theorem2.3(Chain rule[2]):Let x(t)be a Filippov solution of(1)on an interval containing t and V:R n→R be a Lipschitz and regular function(See[7]for de?nition). Then,V(x(t))is absolutely continuous,(d/dt)V(x(t)) exists almost everywhere and

V(x(t))∈a.e.˙?V(x)(5) where

˙?V(x)=

ξ∈?V(x(t))ξT K[f](x(t)).(6)

Remark2.1:In Theorem2.3,we used Clarke’s general-ized gradient?V(See[7]).

Theorem2.4(Lyapunov’s theorem generalized,[3]):If

1)V:R n→R,V(0)=0,and V(x)>0,?x=0and

2)x(t)and V(x(t))are absolutely continuous on[t0,∞) with

V(x(t))

III.A S TABILIZING C ONTROLLER D ESIGN Consider the nonsmooth mechanical system with friction

˙x1=x2

˙x2=g(x1,x2)?μ(x2,t)+h(x1,x2)u(8) where g:D→R and h:D→R are Lipschitz,with g(0,0)=0and h(x1,x2)=0for all x1,x2∈D,and D?R2is a domain containing the origin x=[x1,x2]T=0.We assume that both g and h are known.u∈R is the control input.The frictionμ(x2,t)which has the discontinuity at x2=0is illustrated in Fig.1,where narrow part represents the dynamic friction and the discontinuous part the static friction.We?rst start with the following theorem.

Fig.1.friction modelμ(x2,t):On x2=0the friction force can be represented as a fuction of t.The complex friction phenomena such as Stribeck effect may exist around x2=0.

Theorem3.1:Consider system(8).Any Lipschitz con-tinuous static control lawξ(x1,x2)withξ(0,0)=0can not achieve the asymptotic stability of the origin.

Proof:We will show that there exists a nontrivial equilibrium set including the origin.The closed loop system with u=ξ(x1,x2)is given by

˙x1

˙x2

?μ(x2,t)+φ(x1,x2)

(9) whereφ(x1,x2)=g(x1,x2)+h(x1,x2)ξ(x1,x2),andφis Lipschitz.From(9)the switching surface is given by ?={x∈D|S(x)=x2=0}.Let the regions G+and G?be de?ned by

G+={x∈D|S(x)>0}

G?={x∈D|S(x)<0}.(10) Since the normal vector N s for S(x)=0is N s=[0,1]T, at a point x∈?we obtain

f+N=?γf+φ(x1,0)

f?N=γf+φ(x1,0).

(11) Let the set S I be

S I={x∈?

f+N≤0,f?N≥0,f?N?f+N>0}.(12) Then,from(11)we have the following.

φ(x1,0)≤γf→f+N≤0

φ(x1,0)≥?γf→f?N≥0

f?N?f+N=2γf>0,?x∈?

(13) which implies

S I=

|φ(x

1,0)|≤γf

.(14) Sinceφis Lipschitz continuous andφ(0)=0,for given γf,there existsδ>0such that

|x1|<δ?|φ(x1,0)|<γf.(15) Therefore,there exists a nonempty subset S I?S I such that

S I=

1|<δ

.(16) From Theorem2.1,any solution on S I is dominated by the differential equation

˙x=αf++(1?α)f?(17) where

α=

f?N

f?N+N

=γf

+φ(x1,x2)

2γf

.(18) Calculating the right hand side of(17),we obtain˙x=0on S I,which implies that S I is the nontrivial equilibrium set including the origin.

Theorem3.1implies that a discontinuous control law should be applied in order to achieve the asymptotic sta-bility of the system.

Theorem 3.2:Consider system (8).Assume that there

exists a positive constant ˉγf such that ˉγ

f >|μ(x 2,t )|for all x ∈D and t ∈R +.Then,the followin

g control law asymptotically stabilizes the origin of system (8):

u =

h (x 1,x 2)

{?g (x 1,x 2)+ˉγf sgn (αx 1?x 2)+αx 2}(19)

where α<0is a design parameter which is the slope of the switching surface in R 2.

Proof:The closed loop system is given by ˙x 1

˙x 2

=f (x,t )=

x 2

?μ(x 2,t )+αx 2+ˉγf sgn (αx 1?x 2)

.(20)

First,we investigate the equilibrium point of the system.Let the set S 0be

S 0={x ∈D |0∈K [f ](x,t )}.

(21)

On the x 1-axis,K [f ](x,t )is given by

K [f ](x,t )=K 0

?μ(0,t )+ˉγf sgn (αx 1)

(x,t )

=???????

0?γf

SGN (0)+ˉγf SGN (αx 1) 0,if x 1=0

0?γf SGN (0)+ˉγf SGN (0)

0,if x 1=0

(22)

where SGN (·)is the set-valued sign function de?ned by

SGN (x )=?

{1},

if x >0{?1},if x <0[?1,1],if x =0.(23)Thus,S 0={0}.Since it can be shown that x =0is the Filippov solution with ˙x =0,x =0is the equilibrium point.

We now prove the asymptotic stability of (20).Sliding mode control consists of two modes,that is,sliding mode and reaching mode.

·Stability of sliding mode

The switching surface of the controller is given by

?c ={x ∈D |S (x )=αx 1?x 2=0}.

(24)

We ?rst show that ?c is the sliding surface.At a point x ∈?c ,f +and f ?are given by

f += x 2

?lim S (x )→0+μ(x 2,t )+αx 2+ˉγf

f ?= x 2

?lim S (x )→0?

μ(x 2,t )+αx 2?ˉγf .

(25)

Since the normal vector N s for S (x )=0is N s =[α,?1]T

and ˉγf >|μ(x 2,t )|for all x ∈D and t ∈R +,we have

f +

N =lim

S (x )→0+μ(x 2,t )?ˉγf <0f ?N

lim

S (x )→0?

μ(x 2,t )+ˉγf >0

(26)

for all x ∈D and t ∈R +.Hence,from Theorem 2.2,we

conclude that ?c is actually the sliding surface.Further-more,from Theorem 2.1,it follows that the trajectory on ?c is the Filippov solution,which is dominated by

˙x =αf ++(1?α)f ?

(27)where

α=f ?

N f ?N ?f +

=μ(x 2,t )+ˉγ

f 2ˉγf .(28)

Substituting (25)into (27),we obtain

˙x 1

=αx 1

˙x 2

=αx 2.

(29)

Since α<0,it follows that x →0as t →∞.

·Behavior of reaching mode

Let V (S (x ))be a positive de?nite,Lipschitz and regular function de?ned by

V (S (x ))=|S (x )|=|αx 1?x 2|.

(30)

The generalized gradient of V is given by

?V (S (x ))= α

SGN (S (x )).

(31)

Also,K [f ](x,t )is given by

K [f ](x,t )=

x 2

?K [μ(x 2,t )]+αx 2+ˉγf SGN (S (x ))

.(32)

Thus,

˙?V (S (x ))= ξ∈?V (S (x ))

ξT K [f ](x,t )=

????K [μ(x 2,t )]?ˉγ

f ,if S (x )>0?K [μ(x 2,t )]?ˉγf ,if S (x )<00,if S (x )=0

.(33)

Since ˉγf >|μ(x 2,t )|for all x ∈D and t ∈R +,for any

v ∈˙?V

(S (x )),there exists a positive constant =ˉγf ?sup x ∈D

t ∈R +

|μ(x 2,t )|>0

(34)

such that

v ≤? <0,if S (x )=0.

(35)

Thus,from Theorem 2.3and 2.4,the trajectory starting from x (0)∈D reaches the switching surface ?c in ?nite time.In the following example,we will examine two con-trollers associated with Theorem 3.1and 3.2.

Example 3.1:Consider system (9)with μ(x 2,t )=sgn (x 2)

˙x 1

˙x 2 =

x 2?sgn (x 2)+φ(x )

.(36)

-2-1.5-1-0.50

0.5

11.52x x 1

Fig.2.

trajectories via the controller associated with Theorem

3.1

-2-1.5-1-0.500.5

11.52x x 1

Fig.3.trajectory via the controller associated with Theorem 3.2

(1)Case when φ(x )=?a 0x 1?a 1x 2:

This controller is associated with Theorem 3.1.The equi-librium set is given by

S 0= q 0 ?q ∈ ?γf a 0,γf a 0 = ?1a 0,1a 0 .(37)

In Fig.2,the trajectories starting from some initial condi-tions are shown where a 0=a 1=1.Every trajectory is

captured on S 0.

(2)Case when φ(x )=αx 2+ˉγf sgn (αx 1?x 2):

This controller is associated with Theorem 3.2.The equi-librium point is given by

S 0={0}.

(38)

In Fig.3,the trajectory starting from x (0)=(1,1)is shown where α=?1and ˉγf =2.The trajectory reaches S (x )=0in ?nite time,then it converges to the origin on the surface.

IV.S IMULATION

In this section,we will apply our controller to an elec-tronic throttle control (ETC)system.ETC system is an equipment of engine control systems,which adjusts the quantity of air ?owing into the engine.Since the opening angle of valve plate is decided depending on the required quantity of air to achieve the desirable speed or torque in vehicle,the high performance control system of the valve plate is required.However ETC system has discontinuity and nonlinearity such as friction,spring and backlash.These facts motivate researchers to tackle the control problem,and some robust controllers including switching mechanism have been proposed [8],[9],[10].The ETC system is constructed by DC motor,some gears,valve plate and spring.Consider the model of ETC system as shown in Fig.4.Whole system is constructed by two systems.One is actuator system and another one is throttle system.actuator system:

The relation between the input voltage v [V]and armature current i [A]is given by

+Ri +K v ωm =v (39)where L [H]is the armature inductance,R [?]is the ar-mature resistance,K v [V ·sec /rad]is the back electromotive force constant and ωmt [rad /sec]is the angular velocity of motor.The relation between the armature current i and the torque generated by the motor is given by

τmt =K i i

(40)

where K i [Nm /A]is torque constant.Since the rotating angle of motor θmt [rad]becomes the rotating angle of valve plate through the gear ratio K g ,we obtain the following relations.

θ=K g θmt ω=K g ωmt τ=K g τmt

(41)throttle system:

The throttle’s dynamic equation is given by

J ¨θ

=τapp ?τf (˙θ,τapp )(42)

Fig.4.

Electronic Throttle Control System

p o s i t i o n : θ, θ [d e g ]

time : t [sec]

-10-8-6-4-20246

i n p u t v o l t a g e : v [v ]

time : t [sec]

Fig.5.

position θ,θm and input voltage v

where J [kg ·m 2]is the inertia of valve plate and τapp =τ?τS (θ).τS (θ)is nonlinear spring torque:

τS (θ)=m (θ?θ0)+γs sgn(θ?θ0),

(43)

τf (˙θ,

τapp )is static and dynamic frictional torque.In this simulation,we employ the bristle model [11]as the friction model.

Now assume that we want to design the model following controller.The reference model is given by

¨θ

m +λ1˙θm +λ0θm =λ0r.(44)

where λ0,λ1are some constants and r is the reference

position.Ignoring the armature inductance L and putting the error signal e =θm ?θ,the error system is given by ˙x 1

˙x 2 = x 2?λ0x 1?λ1x 2+β(θ,˙θ,r )+μ(t )?1J

(45)

τ=K g K i

R v ?K v K i R

˙θ(46)where [x 1x 2]=[e ˙e

],β(θ,˙θ,r )=?λ0θ?λ1˙θ+λ0r +(1/J )τs ,and μ(t )=(1/J )τf .Setting τ=J (β(θ,˙θ,

)?u ),-0.05

0.05

0.1

0.15

0.2

time : t [sec]

p o s i t i o n e r r o r : e [d e g

]

0123v e l o c i t y e r r o r : e [d e g / s e c ]

time : t [sec]

Fig.6.

position error e and velocity error ˙

e we have

˙x 1˙x 2 = x 2

g (x 1,x 2)+μ(t )+h (x 1,x 2)u

(47)

where g (x 1,x 2)=?λ0x 1?λ1x 2and h (x 1,x 2)≡1.Then,the control law is given by

u =?g (x 1,x 2)+ˉγ

f sgn(αx 1?x 2)+αx 2v =RJ K

g K i

(β(θ,˙θ,r )?u )+αx 2.(48)

As shown in Fig.5and Fig.6,the controller realizes the

good performance,and the tracking error asymptotically converges to 0as t tends to in?nity in spite of uncertain and complex friction phenomena.

V.C ONCLUSION

We have presented a stabilizing controller design for nonsmooth mechanical systems with friction.The approach to design the asymptotic stabilizing controller is based on the Filippov solution concept and related Lyapunov stability theory for nonsmooth systems.For the class of nonsmooth systems,it was shown that any Lipschitz continuous con-troller can not achieve the asymptotic stability because of the existence of the nontrivial equilibrium set.A sliding

mode controller which makes the nonsmooth mechanical system asymptotically stable was derived.

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