Handling actuator non-smooth nonlinearities using variable structure control

14th World Congress ofIFAC

HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

Copyright © 1999 IFAC 14th Triennial \',lorld Congress,

F-2c-22-2 Beijing~

P.R. China

HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING VARIABLE STRUCTURE CONTROL Cem Hatipoglu t and Ulnit Ozguller t t AlliedSi,gnal Truck Brake Systems

901 C1.evel(lnd Street Elyria] Off 44035, [lSA [email protected]

t The Ohio State University 2015 Neil Ave. 205 Dreese Labs, ColurrLb'us, OH 43210, USA Dzguner~2~osll.edu

Ab~tract: This paper deals with robust variable structure control design for continuous-time linear systems involving non-smooth actuator nonlinearities. Also examined in this paper are globa.l boundedncss, ~tabilizability to origin, non-zero set point regulation and arbitrary reference tracking topics when discontinuous controllers are employed "vithin the class of systems studied. i\.dvantages of full-state feedback and robust observer design issues to overcome difficulties int.roduced into control due to t.he existence of non-smooth nonlineaI'ities are addressed and the benefits of not having to identify the actual nonlinearity to achieve robust performance within the framev~rork of sliding mode controllers are clarified. Copyright ~t:J 19991J~~4C:

KCY\\-1ords: Nonlinearity, Nonlinear control, Input non linearity, Variable structure control: Tracking control.

1. INTRODUCTION

nonlinearity to an extent that can vary from a exto (;none" depending on the completeness of the information available about the nonlinearity. If t.he non-smooth nonlineal'ity can Le identified perfectly~ a pre-filtering process can be utilized by means of the exact-pre-inverse function so as to annihilate its effects ,-vhich jn return yields an identical response frorll input to output to that of the system without the nonlinearity. A vaila.bility of the exact kno\vledge of the nonlinearity is relatively unrea1i~tic of an assumption since even though such identification may be possible, it would be costly, unit dependent and possibly tiule-varying. Nevertheless, the properties of the non-smoothness are likely to change from mildly to drastically (in time) resulting in nlismatches bcact'~

~vlany physical systerIls involve an actuation assernbly that has some sort of non-smooth char~ acteristics and a plant ,vhich is either linear or nonlincar with all the nonlinearities embedded in the plant being ~mooth. In such a system the nonsmooth nonlincarity can be put in a separate block \vhich pre-shapes the designed control input that effectively drives the smooth plant. Handling such systen1.s \vithin the frarne\.vork of available control theory varies significantly in nature. The rnajar Ineans of cornpensation for uncertainties at the input channel a.re the adaptive schemes (Tao, 1996), fuzzy logic controllers and variable structure control ~ehernes (Drakllnov et.al., 1997). Being alternatives for one another in many aspects of design and analysis phases, all of those methods derIlonstratc to yield desired tracking and regulation perforIIlance under certain assumptions which differ in each approach.

If the exact Ilre-inverse exists for a non-smooth nonlinearity at the input channel of a system, a pre-filtering process can compensate for the anomalies caused by the non-snlooth nuture of the

t\vcen the fixed pre-inverse-nonlinearity model and the nonlinearity itself. The objective of the classification of non-smooth

nonlinearitics in a standard forIll as sUlnnlari~cd in (Hatipoglu~ 1997, 1998) "ras to handle the sarIle control problem utilizing variable structure control la\vs v,.-hich ",·ouId solve the regulation and tracking problenls in systelIls \vith non-slTlooth nonlinearities ,"vithout ha.ving to identify the parameters of the nonlinearity -not even the type of it- ex-

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HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

plicitly. The objective is similar to that of (Tao; 1996), ho\vever, the assumptions and the approach rClnain significantly different.

Inverse Definitions: For (], non-smooth nonlinearity given in the form yr =: c/J( [I), the exact pre-inver-se is defined as c/>;1 (.) /·;atisjying ep( ~b; 1 CLI)) == [Ion E C; IR For a non-smooth nonlincarity given in the form 17 == q,(f]) , the exact posl-invel'sf:: is defined a,s q),~1 (.) satisfying 4>.-;1 (ep(U)) = [T on E ~ ffi A pseudo-pre-2:nverse r/J;;pl (-) of Q non-sm.ooth nonlincarity ( rP;pl eCI)) ~ [/2. J

Pseudo-pre-inverse can also be referred to as ;'inverse in the sliding Illode sense~' because it states that you can find a pre-filtcring function that guarantees to achieve a signal of equal or greater rnagnitude at the output of the nonIinearity. In the case of input nonlinearities ",rhere sliding Inode type control is used 1 employnlent of such a preinverse filter would by-pa..~s the effects of the nonsmoothness at the input.

Stability Definitions: The 111,otions of a systent is called LagT(Lnge .r.;table if starting from any initial conditions x(t :;::::: 0) ::=: xo,~ there exists a ;3(x o ) E 1R such that Jlx(t)Jj < p(x o ) Lagrange stability describes the boundedness of all state trajeetories in lRn . The motions of a system is called uniformly ultirnately bounded with a bound 0if there exists positive constants X, J, t} E lR fUT any "in"itiaZ condition x(O) :::::: X o EEC !Rn su.ch that for all t > tl (x o ) ~ IxI ~ x and hence y(t) S (T. If, E == mn then: the rnotions of the systern is called globally uniformly 'Ultimately bounded g.'u.u.b .. l~niform ultimate boundedness imply that after a finite tinle transient, motions of the system ~~ill be contained in a x neighborhood, and output will be contajned in a t7 neighborhood. 1

x=

_Ax + bu;

y = ex

for all v E JR s.t. 'vi > v* with a linea'f' output feedback law, v == ~ky(t) if there exists a single p()~r;itive definite rnatrix P that satisfies; PC.4 - bkc)

2. NON-Sl'vIOOTH ACTU~~TORS

Consider for no,v the following SISO linear

14th World Congress of IFAC

~Y8tem

(1)

"\-vhere x E 1R n, U E lR and Y E JR. This systern is globally uniforruly asyrnptotically stabilizabJe to the origin \vith output feedback u =:: -ky = -kcx if K ;;::: {k E 1R : (A - bkc) is Hurv..~itz} is nonempty.

Theorelll 1 The 'nl.otions of the system (1) 'which is controllable and observable with a non~

+ (A

sim'li.ltaneously for

- bkc)TP

k =::

ke an(l

<

k ===

-El

k u , (3£

(3)

> 0).

Proof is similar in nature to the simultaneous Lyapunov function generation problem, (I{halil, 1992) Consider the Lyapunov function candidate V(x) ::::: x T Px where P is a sYlnmetric positive definite lIlatrix. Consider ~~T(X,t) :;: : : xT(PA + ~4T P)x-2x T Pbr:P(kY1 t) For all v E lR s.t. lvl > v· for some finite v*, the sector condition (2) is satisfied. Define the time varying gain by 1

u(t) == 4J( 'v(t) ,t) ==

-~(t)y(t)

(4)

'-v-'" -ky(t)

Note that. for all ky > v* i kg :5 ~(t) ::; k u \vhich yields (x, t) == x T[P(.A. ~ bl\.(t)c) + (AbK.(t)C)T P]x ,vhich in return reduces the problem into the stabilit.y of the time varying system = c.4. - bK.(t)C)x for K(t) ta.king values in the convex region kl S ~(t) :::; k u for all {x E IRn : kcx:::; v*}. There exists an E > 0 such that t r (x ~ t) < -cx T X \iK(t) E [k.e, kuJ if and only if it (x, t) < -€X T X is satisfied for K.(t) == ke and tt(t) :::::: k u since l? (x, t) is linear in K,(t) and attains a maximum at either of the end-points. Therefore if there exists a positive definite IIlatrix P that satisfies (3) at k£ and kt" any the same P will satisfy simultaneously satisfy the same inequality for any gain that lies in the convex region [ki , k u ]. 'Therefore, \Iv E lR "\vith Ivl > v*, motions of (1) can be concluded to exhibit stable dynamics in the Lyapunov sense \vith u === rj:;( -ky(t)) type output feedback. Consider the salne la\v for \ky(t) I :; v* < 00. T'he worst case is to have an unstable system for that interval. Then, the unstable modes v.,rill be reflected to the output slnce (..4 , c) is observable resulting in an instability at the output causing y to gro\\,'" accordingly. Eventually K(t)y(t) v.rill have to beCOlne larger thall v* and the state trajectories can be proven to be stable in the Lyapunov sense once again. For a continuous time systeln, an infinite frequency switehing bet\veen stable and unstable structures might occur resulting in the bordering line Yb ::; 'l~ to become an equilibriuln point of the nonlinear systenl. The equilibrium point Yo of the stahle system for jkyl > v* has to satisfy IkYol ~ v* because of (3). Then, the nonlinear s:ystem is said to be g.u.u.b ..

1t\/

x

1t.

1t

Rernark: The inequality (2) is a less restrictive condition than classically defined sector condition. lVe do not seek asynlptotic stabilizability around the origin, therefore, "ve do not require the nOusmooth nonlinearity to be trapped globally by linear functions. This condition only requires 1(', .)

smooth actuator nonlinearity of type u ;::;; cjJ(v, t) can be g. u. u. b. if ther'e exists k l , k'iJ, E IR 1vith < k l < k u s'uclt that [ki, k 1L J E JC, 1)* > 0 and

o

(2)

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Copyright 1999 IFAC

ISBN: 0 08 043248 4

HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

14th World Congress of IFAC

to be upper and lOfver bounded by linear functions every\vhere except on an interval around the origin of finite Lebesgue measure. The other difference is that~ \-vc have very little constraint on the non-smooth nonlinearity. It is not required that 46(0, t) == 0 and also it is not required that r/J(., .) be rnelnoryless. Actually the ~ector eondition (2) can be replaced \vith a more generalized form utilizing deadzone type bounding functions. IVloreovcr, the conclusions are valid for any lllonotonic, llnbounding nOll-Rlnooth nonlinearity Vilith or ,~,rithout dynamics.

Phase ponrait for Linear and Di5(;:onUnuous Otrtpu( Feedbar:::k

,:

Theorem 2 Consider the linear SISO system (1) 10here (A, b) is controllable and (A, c) is observable. Assume that this system has a nonsrnooth actuator nonlinearity at its input channel. If the motions of this system is g.u.u.b. with linea1~ output feedback v = -ky and stable in the LyaIJunov sense for jvf > v"") then the same nonlinear system, ca.n be stabilized to the origin by means of an, additionally injected discontinuous te rrn, i. f:. V

=== - ky +1.L';

w == -

v~/~Sign (5 (.))

(5)

_ ___..::::=-.L--_ _----l-

_1O'-----L--~---L

Figure 1: The phase portraits for the linear output feedback and discontinuous feedback injection.

Example:

=

for some large enough J;V > 0 and sorne manz-

(6) l"\he lower bound is significant. Since a pseudopre-inverse exists for unbounding nonlinearities, by making Ifl/"j large, lcP(~l/~ t)1 can be made as large as desired. As a result, ,vith the addition of the discontinuous terrn, output may be steered to zero on the designed manifold S (.) if it is picked such that S(-) explicitly depends on the control input, and on the rnanifold the dynarnics stably take the output to zero. The motion of the trajectories are bounded by the linear output feedback term. and hence, bounded discontinuous control is ca.~able of dOlninating SS. Since zero dynaulics of the underlying plant are assulned stable, regulating the output to zero will yield stable internal dynanlics.

Consider :1;2;

X2

==

Xl -

X2

+ u;

1 3 ba.cklash ( v, -, r) ~ 1) ~ 2 ~

fold S(·) if the 'u.nderlying plant zero dyna.mics are stable.

I\ote that (5) ma.kes the domain of the inputs to the non-smooth nonlinearity JR. - ( - rv', fff) c 1Ft R.ecall that a pseudo-pre-inverse exists for the nonSITlooth nonJjnearity r,&(. ~ .) meaning that by making llf large ~ one can shrink the image space as \vell. If l~i is picked larger than v* (which is achieved \vith w == 0) l then it is guaranteed to hav~ t.he. non~smooth nonlinearity operate exclusivel.y in the region R = {v E IR : vet) 2 v*}. If T-l.7 is picked to ensure this, 1rV :::::: TV + 'Y for some ""'l > 0 \vill ensure a discont.inuous t,erm at the output of the non-smooth nonlincarity as well due to lIlonotonici ty of t.he signal and existence of a sector condition as in (2). This condition yields upper and lower bounds for the effective magnitude of the discontinuous term at the output of the non-smooth nonlinearity,

...L_____ _.........

-6

(7)

y

= Xl,

\vhere Xl and X2 are the state variables, u and v are the control inputs as seen by the plant and as designed respectively, y is the output. Kote that . 4 is not HUTlvitz, hovt='ever, (A, b) js controllable 1 C4, c) is observable. Consider a fixed output feedback laVv~ 'U ::::: -ky. 'Then, (.4 ~ bkc) is Hur\vitz for all k E K v/here lC. ~ {k E lR : k > I}. The linear system v..rith k = 3 is asymptotically stable, For this second order system Aizerman's conj ccture holds true. Therefore, the linear systern \vith the non-smooth biased backlash type actuator nonIinearity as given in (7) can be concluded to be g.u.u.b. and stable in the Lyapunov sense for large enough v with k == 3 for IvJ > v* ~ ~ which is obtained froll1 the intersection of keY line with the backlash nonlinearity's directional graph. By nleans of a discontinuous term with l~,"r =:: + E for any € > 0, origin can be luade a globally attractive stable point (See Figure 1).

£

2.1. Decp.ntralized Non-smooth J'tlonlinearities Corollary

1 Consider the decentralized Jt..fIMO

system,

x ~ . 4 x + Bu; y ~ Cx;

'U:=

cIt(v, t)

(8)

tJ)here x E IRn, 'l1.,y E IRP a.nd f:P(V,t) diag (4)1 (VI) , 92 (V2 ), ... , rP p ( V p ) ) • Assume 10 r the linear part that CA, B) is controllable J ( ...4, C) is obser"vable) and (A, B, C) is tlecentTal-ized output feedback stabilizable with u ~ -k y u.~here I{ is diagonal. Then; the nontincar system with the decentr-alized non-smooth actuator nnnlinearities

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HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

cPi (., .) at the input chf1.nne[8 can be 1lnifo1'mly ultimately bounded in ffi n with a linear output feedback law u =: - K y for some diagonal K if the condition (9) he;

v; ~ tPi (Vi, t)Vi ~ k u;

(9)

Ui

'Vi E m. with IVi I ~ v; for some vt > 0 and all i == 1" .. , p is satisfied and if there exists P > 0 satisfying

for all fOT

peA - Bk(i) C)

+ (A

- Bk(i)C)Tp

<

-EI(lO)

at the ver-tices of the convex hypcrcube

11 = {k

E

lR.P : k1i

::;

ki

:s;

k ni }

(11)

namely k( i) for i = 1, ... , Then, the nnnlinear system can be stab~'lizcd to the origin with decent'Tnll:zed discontinuous o'u.tput injection if the underlying plant has sta.ble zero dyna:rnics~ 2P ,

Simultaneous Lyapunov function generation problenl satisfying the negative definiteness property at all vertices can be transformed into a minimax problem, (Khalil~ 1996). This method assumes that at least one of the matrices on one of the vertices of the convex hypercube is Hurwitz. Starting from there, a minjrnization process can be elllployed for the maximum eigenvalue of the respeetive rnatrices, siInul tancously. By Incans of cutting-plane algorithm \vithln nonlinear programming frame,\\rork and the use of linear matrix inequalities (L!vIIs) mininlUX problelu can be solved yielding the corresponding P > o. l

2.2. Finite Dom,ain Results ~1'\.1l the results reported (as summarized in TheoreIns 1 ~ 2 ~ and Corollary 1) require a linear control input term to assure global stability of the given system. If the system itself is stable (Le ..4 is Hur,vitz), then purely discontinuous, bounded control input of sufficient magnitude Vl-rould stabilize the nonlinear system around the origin. HOv\rever, unstable plants can not be stabilized globally l\dth hounded input~~ therefore~ for such systelns if linear term is to be neglected, only "Finite Donlain Results" can be achieved, On the other hand, if a pseudo-pre-inversc does not exist for a non-smooth actuator Ilonlinearity (corresponding to a "bounding" non-smoothness) the sector conditions Illay not be satisfied globally for all 1) > v:+' for some finite v*. Actually these tVlO problerns possess similarities in nature: The first corresponds to a user bounded r the other to a system bounded control iIlput~ Nevertheless, the effective control input available to drive the linear system is bounded in either case.

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that for a nonsingular .14 matrix~ controllability t.o the origin irnplies reachability. Let Yd be the desired output set-point for the actual output to follow. Define error as: E y == Y - ,Yd For the linear system a feedback la\v 1L == - kEy results in the following closed loop system dynamics: X

::=:

(A - bkc)x - bk-r'Yd,

y

== ex

(12)

which exhibits the same stability properties around x eq =- (.A. - bkc)-lbk·.-,tYd as does the linear system (1) with u == -ky around X eq == O. Then~ the equilibriuln point for y becornes Yeq ::::: cX eq == c(.4. - bkc)-lbk'Yd. For Yeq to be identical to Yd, ,should be picked as "'Y' = c(A~b~c)-1tJk for snnle output feedback gain k that stabilizes the unforced system. The observation is that if
:i :::::

(A - bkc)x,

fj ::::: ex

(13)

for the linear SISO system. K ote tha.t (x ~ y) to be g.u.u.b. around the origin is equivalent to (x, y) to be g. 'U. u. b. around (x eq , Y d) \vith the nOIl-Slnooth nonlinearity at its input channel. By Theorerll 2 the nlotions of the nonlinear system that can be g.u. u. u. around y == Yd by means of a linear output feedback v =: -~n,(y - -rYd) can be globally stabilized to y ::::: Yd by means of a discontinuous term injection based on the error as follo",~s; v = -k(y - ~(Yd) - lllSign(y - Yd)

(14)

'-.".--'

v·,rhere TV is to be picked exactly the same way as in the controllability to the origin case. Proof is sirnilar to the proof of Theorem 2 with the transformed states. In the case that plant pararneters a.re not kno,vn exactly, l' can not be determined explicitly~ It is alVv"ays possible to lump the uncprtainty into the non-sITlooth eharacteristics and simply pick { = 1. .t\s long as the non-SIlloothness satisfies the condition (2) the statements still hold true. Decentralized 1vlIlVIO case \vhose controllability and stability to the origin "'..ere studied in Section 2.1 can be sho~rn to exhibit similar properties for non-zero output regulation if reachability assumption is also imposed onto the the underlying linear plant.

2.4. Advantages of Full State Feedback

2.3. Output Regulation to a Non-zero Set-point Should the matching conditions be nlet, it is shov{u that a variable structure observer scheme yields in asyu1.ptotically convergent error dynamics making full state information available to the controller.

Consider the SISO linear system (1). Assume that (..4., b) is controllable and (A, c) is observablc~ .A.dditionally assume that (A, c) is reaehable. Note

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HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

\\Then state feedback is availa.ble either explicitly or by means of a robust observer, given a reference signal Yd and all its derivatives, one can define a sliding nlanifold: S

==

E,~P)

+ I'l:p-l £~P-l) + ... + J£l£Y +

x;oE y

(15)

=

where f,y Y - Yd is the tracking error, and the polynolnial sP + Kp_l~,;p-l + ... + ~lS + 1'\:'0 ==: 0 is asymptotically stable and p + 1 is the relative degree of the system. Since the relative degree of the systerll is p+ 1 1 S is dependent on t.he control input explicitly and a discontinuous control input can enforce SS < 0 which forces the systern trajectories onto the manifold a = {x E IR/~ : S(x):::= D} exhibiting asymptotically stable dynamics while steering the error term f.. y to zero. ExarIlple: Consider the following linear system with a backlash type non-smooth nonlinearity at its input Xl == X2; X2 == X3; y:=: Xl , Xs

-=

-0.1'7:'] -

a2 X 2 -

a3X3

+ bu

(16)

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ahove ean be forced to exhibit sliding mode like behavior yielding e X2 =: 0 if /!,2 is picked such that .£2 > lC X3 l resulting the other equivalent control of [Sign[Sign(ey)]cqJef} == 7;-. Similarly if £3 dominates the uncertainty in the equation governing the motions of C X3 stemming frOlll the imprecisely knO""ill input nonlinearity, eJ~3 can also be forced to go to zero in the sliding mode sense. The equivalent controls can be implemented by means of appropriately selected low-pass filters considering the frequeney response characteristics of the systenl in hand. If the gains and the filters are selected appropriately, the errors eX], e X2 and e X3 go to zero sequentially and all in finite time. :v1oreover, the equivalent control obtained from the last equation can be used to estinlate the non-srnooth nonlinearity characteristics since:

[Sign ([Sign[Sign (c y)]eq ]eq )Jeq == b(ttt( v) -~(v) .(18) -<.4.ssume that the reference signal to be tracked is given by ret). Then, design a sliding manifold, S

where x = [;T:'1 X2 X3]T is the sta.te vector, u is the input seen by the linear system, v is the user generated control input, y is the output. Clearly the relative degree of this system is 3. Let the reference signal be given by Yd(t)~ ..~ssume that full state feedback is available. Then the manifold: S:::::: Ey + K'l Ey + tl'2Ey \~lith f y == 11 - Yd, and a control input v == -kEy - OSign(S) \vould guarantee reachability to S == 0 in the sliding rllode sense, and to Cy == 0 asyrllptotically if n is picked according to n » Ikl- IICyll and if (since cP has a pseudopre-inverse) it is made sure that v(t),p(v(t)) ~ 'v(t)f!Sign(S(t)) and if > 1a ll ~ Ilxlll + 1~2 - a2 ~ • HX2J1 + jK:l - a31 . Jl x 3jj + ly~3) + KIYa + K2Ydf is satisfied. In the case of non-available full-stat.e feedback, consider the follcl\ving variable structure observer, (IIaskara., 1908):

n

l

X2

+ i\Sign(y -

X2

X3

+

.i

X3

y)

(17)

£zSign([Sign(y ~ Y)]eq)

-Q,lX] ~ a2X'2 - asxs + btP(v) + ... . ~ . + £3 Sig n ([Sign([Sign(y - Y)]eq) ]eq)

Y Xl ¥.l"here (·Jeq denotes the equivalent control of its argulllcnt. Let e denote the estinlation error. Then, the error dynamics become: eXl == e X2 - i\Sign(e.y); ex:;:! ~ e. Xa f 2 Sign([Sign(c y )]eq); ex:~ := -ale X1 - a2eX2 a;ie X3 + beifJ - e3 SignC[Sign[Sign( ey ) JeqJeg).

:=:

(x3(t)-f(t)+K:l (xz(t)-i-(t))+K2(y(t)-r(t))(19)

and a discontinuous control input, U :::: -ky(t) j'-'fSign(S), ~~here the linear term serves for global stabilization. Assume that .1'I-{ is picked such that M > v* and that SS < 0 is satisfied for the given I'eference signal. Then, S(t) is found as 5 :::: -aliI - (0,2 - #1:2).1;2 - (a3 - K,l)Xg + b(j)( v, t) - (r(3) + Kir + /(,21") + '1' (eX] ~ e X2 , €X3' et/J) ",~here 'lr(e X1 ' ex~, e X3 , e4J) ----t 0 in finite time as the estirnates of the states converge to their actual counterparts. In the design, if rp( v, t) dominates the right hand side in a discontinuous Inannel', SS < 0 can be achieved. If the nonlinearity f/1(v, t) has a pseudopre inverse, then, picking Ivl large enough can 111ake ljJ( v, t) magnitude-"vise as large as desired by the definition of pseudo-pre inverse. Therefore, tracking error dynamics \vill eventually satisfy S==:O in Equation (19) . Figure (2) displays the st.abilizability to the origin results on a manifold S == e X3 + 4e z :2 + 3e X1 starting from an arbitrary initial condition ,vith a backlash type Ilonlinearity at the input channel. _~f 4 can not sta.bilize the system globally. The sole linear output feedback with k :=: 3 though exhibi ts bounded traject.ories, yields oscillatory response when a non-smooth backlash type nonlinearity is present at the input.

=

1"~igure (3) displays the simulation results for tracking a sinusoidal reference signal of rei) :=: 4S-in(O.25t) in the existence of a backlash nonlinearity at, the input. The control la,v u :::: -3(y r)-.lfSign(S) with S == (X3-f)+4(X2-r)+3(y-r) is used v,...-here .i\1 == 20 ensures convergence to the sliding manifold and the S == 0 ensures stable asyrnptotically deeaying response in sliding mode

If Cl is picked large enough, such that £1 > fe :r2 I dominating the first equation, eX] e X1 < 0 can be enforced and the manifold e X1 == 0 can be reached in finite time. 'Then the equivalent control associated \vith this equation yields [Sign(cy)]eg = Then~ the second equation in the error dynaulics

7;-.

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HANDLING ACTUATOR NON-SMOOTH NONLINEARITIES USING ...

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nacking a

SQli::!: Aclual statE'S; Oe:sfl&d: Observer e:stjtnaM:5

sinusoidEl.~

wrth Observer

1

\ I I

-WQL----=::-L---1...--~.L----6.l....-----~---:------:"'::-------:'12

15

10

2C

25

30

40

35

50

45

timl9'ls9!.:)

tirn'illsgc)

; 20 - /

I

-'~

The SI!d1r,g

mal"li'fo~d

20r--------,----------,----.----.----~ 10

~-IO -20

~ Or"'"---~-~------:--~------~~

'/

"

Ij/·; ~-~L

J"". -

\\'

-30

",

,"

.

.. '

-40 Ci

-40 0

10

...L5~------l..10~------l15~-----.J2l.-.a~~2:'-:-5~-------:-L:-30~----:-.35;---.-----;4';::-D~---;4-;-;-5-----;;5'O time (sec'

Figure 2:

stabilizability to the origin problem:

Figure 3: T'he tracking simulations: The output yet)

The state traject.ories, control input and the sliding manifold.

tracks the reference signal r(t) effectively.

~rhe

as y (t) ----t 1~ (t). Linear term helps bound the trajec tories g10 bally.

3. CONCLUSIONS Existence of input llonlincarities often cause undesirable oscillations or stictioIl at a non-de~igned manifold ~~ith conventional output feedback la'\vs. Input nonlinearitics have been sho"\vn to be very \vell suited for variable structure eontroJ as the

nonlinearity input (and hence its output) can be nlade discontinuous and rejection of the uncertainty at t,he input may be attained by means of rich enough control authority. A modified sector condition is required to conclude giohal attractiveness of the origin along with the existence of Cl. sirnultaneously satisfied negati ve definite condition. Robust observer design subject is also investigated and concluded to be feasible by lllca-ns of equivalent control based \'se observers. Generalized tracking probleln is solved by coupled ob-

server/ controller design. Some simulation exa111pIes are provided to display the performance of the \'SCr. The conservativene.ss of the approach is significantly narro\ve-d by means of robust observer coupling and large control authority.

4~

S.

REFERENCES

"'T.

u.

Drakunov, D. Hanchin, C. Su, Ozgiiner(1997). "Nonlinear control of a rod-less pneUlllatic servo-actuator or slid-

ing modes vs coulolnb friction" 1 A'll,tomo.tica, \7"01 33, Ko 7, pp. 1401-1406. i. IIaskara, (T. <5 zgiiner, \1.1. Utkin (1998) . "On sliding mode observers via. equivalent- control approac.h~', to appear in Int. J. of et/l. C. Hatipoglu and t. Ozgiiner(1997). £(Co'rLt1~oller Design for a Class of Systems with Inherent

Right Hand Side Discontinuities Proc. of 19i CDe, San Diego, CA, pp. 4024-4025. C. Hatipoglu and U. Ozgiiner(199S). URobust Control of Systems Involving }-lon~sm.ooth JJ

,

Nonlinearities using lvfulti-Layer Sliding Man'lfolds JJ , Proc. of '98 ACe, Plliladelphia~ PA. H.I{. l{halil(1992). ftlonlinear Systems, :rvlaclvlillan Publishing Cornpany. G. Tao, P. Kokotovic(lg96)~ Adaptive Control of Sy.~tems un:th Actuator and Sensor Nonlinearities, vViley-Interscience Publication.

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ISBN: 0 08 043248 4