Adaptive Control of Plants with Unknown Output Dead-zones

Copyright © IFAC 12th Triennial World Congress, Sydney, Australia, 1993

ADAPTIVE CONTROL OF PLANTS WITH UNKNOWN OUTPUT DEAD-ZONES Gang Tao* and P.V. Kokotovic** *Deparrment of EE, University of Virginia. C/Illr/ottesville, VA 22903, USA **Department of ECE. University ofCalifomia , Santa Barbara, CA 93106, USA

Abstract . For plants with unknown dead-zones at output we propose a new controller structure with an adaptive dead-zone inverse. We consider two control problems: one when only the dead-zone is unknown and the other when the whole plant is unknown. We allow the dead-zone to be with unequal slopes . The proposed controller structure results in a linear parametrization which is crucial for developing adaptive update laws. We prove the signal boundedness for the closed-loop system, and for a special case we establish asymptotic tracking. Simulation results show that our adaptive control schemes improve system performance. Key Words. Adaptive dead-zone inverse; adaptive control ; control system design; nonlinear systems; stability.

1

Introduction

y

z

Control system components, such as hydraulic servovalves, electric servomotors or mechanical connections, often have dead-zone characteristics (Truxall , 1958; Recker et al. , 1991 ; Tao and Kokotovic, 1992), which are usually poorly known and often severely limit system performance. The design of controllers for plants with dead-zones is of a major practical interest . Recently in Recker et al. (1991) and Tao and Kokotovic (1992) we have developed adaptive control schemes for plants with an input dead-zone. In thi s paper , we ; deal with plants consist of a linear part and an unknown deadzone at its output. Our control designs will be based on a new controller structure with an adaptive dead-zone inverse, which is capable of achieving exact tracking when the plant is known and can be updated by a linear adaptive law when the plant is unknown.

y

Figure 1: The plant model.

2

Problem Statement

We consider the following discrete-time plant with a linear part G(D) and a dead-zone nonlinearity DZ(-) at its output (see Figure 1):

y(t)

The paper is organized as follows. In Section 2 we formulate the control problem . In Section 3 we present the new controller structure for plants with output dead-zones, which employs a dead-zone inverse and results in a linear parametrization for the closed-loop system. In Section 4 we design two adaptive controllers with robust update laws: one for plants with known linear parts but unknown dead-zones and the other for plants with unknown linear parts as well as unknown dead-zones. Our adaptive control schemes allow the slopes of the dead-zone to be unequal and ensure the closed-loop signal boundedness. The tracking performance is characterized in a mean sense by a bounded error due to the unobservability of the dead-zone input. For a special case when the dead-zone vanishes and the nonlinearity becomes a piecewise linear element with unequal slopes, our adaptive controllers guarantee asymptotic tracking. In Section 5 we present simulations results to show the significantly improved system performance by our adaptive controllers.

= DZ( z(t)) , z(t) = G(D)[u](t)

(2 .1)

where u(t) is the applied input , y(t) is the measured output, and G(D) = kp ~:!~l with Zo(D) and Po(D) being monic polynomials and kp being a constant. The dead-zone characteristic y(t) = DZ(z(t)) is described mr( z(t) - b,) for z(t) > b, . by: y(t) = 0 for b, ::; z(t) ::; b, m,(z(t) - bt) for z(t) < b, . Our objective is to design a control u(t) for the plant (2.1) with an unknown dead-zone DZ(·) and a non-accessible z(t) to achieve stabilization and tracking of a bounded reference signal Ym(t) by y(t). To design adaptive controllers achieving the stated objective, we make the following assumptions: (AI) G(D) is minimum phase; (A2) the relative degree n° of G(D) is known ; (A3) the degree n of Po(D) is known; (A4) the sign of kp is known ; (A5) ~. 2 mo, ~ 2 mo for some known constant mo > 0, and b, 2 0, b, 2 O.

1

73

In view of (A4) and (A5) we represent kp by the dead-zone slopes rn" rnl and assume: (A4') kp = l. Unlike the input dead-zone problem (Recker et al., 1991 ; Tao and Kokotovic , 1992) where an adaptive right dead-zone inverse which precedes the dead-zone is capable of cancelling its effects, the output dead-zone problem can not be solved by using a right dead-zone inverse because the dead-zone input z(t) is neither accessible for control nor available for measurement. Instead , we will develop an adaptive left deadzone inverse to deal with the output dead-zone.

3

and Kokotovic, 1992) where a right dead-zone inverse results in the zero tracking error. We also show how to eliminate the dead-zone effect by using an initialization of the deadzone Inverse. We then address the tracking problem for the general case. Example 3.1 Consider a first-order plant with a known linear part z(t+ 1) = az(t)+u(t), and a known dead-zone DZ(·) whose parameters are rnr = rnl = 1, br = -bl = b > O. If z(t) were available for measurement, a stabilizing controller would be u(t) = -az(t) + zm(t + 1), where zm(t) is defined as: zm(t) = (~,Ym(t) + br)xrm(t) + (~Ym(t) + bl)Xlm(t). Because z(t) is not available we use i(t) = (....Ly(t) + br )X,( t) + (;!;;y( t) + bl)XI( t) as its estimate and the c';;ntroller u(t) = -ai(t) + zm(t + 1). Let Ym(t) = r > 0 be a constant. Then, zm(t) = r + b, i(t) = (y(t) + b)X,(t) + (y(t) - b)X/(t) . There are two cases of interest: (i) z(t) > b or z(t) < -b; (ii) -b ~ z(t) ~ b. For (i), y(t) = z(t) - b > 0 or y(t) = z(t) + b < O. It follows that u(t) = -az(t) + r + b, z(t + 1) = r + b > b, and y(t + 1) = r . This implies that y( r) = r for all T ~ t + 1. Therefore the exact tracking is achieved. For (ii), y(t) = 0, and)t follows that u(t) = r + band z(t + 1) = az(t) + r + b. If a = 1 or lal > 1 and z(O) # ~, then Iz(tl)1 > b for some finite tt, so it becomes (i). If lal < 1 or a = -1 or lal > 1 and z(O) = ~ then it may happen that -b ~ z(t) ~ b so that y(t) = 0, for all t ~ O. However, for lal < 1 or a = -1, u(t) = r + b + il, with u > lalb - r, results in z(t + 1) > b, and for lal > 1 and z(O) = ~, u(t) = r + b + il, with il # 0, results in Iz(t + tdl > b for some finite t 1 • Hence it becomes (i) again. This shows that a bounded variation of u(t) from the nominal control law can eliminate the effect of the unobservability of the dead-zone input so that asymptotic tracking is achieved. Making z(to) > b or z(t o) < -b is to initialize the dead-zone inverse at to.

Dead-zone Inverse Control

For the output dead-zone problem, we need a new adaptive controller. We first develop such a controller structure, and specialize it for different cases of the output dead-zone problem. We then show that this controller can achieve exact tracking in the presence of an output dead-zone.

3.1

A New Controller Structure

Let us first parametrize the dead-zone DZ(-). Denoting X[Xj as the indicator function of X: X[Xj = 1 if X is true and X[Xj = 0 if X is untrue, from the measured output y(t), we introduce. three indicator functions: Xr(t) = X[y(t) > 0], XI(t) = X[y(t) < 0], and Xo(t) = X[y(t) = 0], and express the dead-zone (2.2) as: z(t) = (~,y(t) + b')Xr(t) + (;!;;y(t) + bl)XI(t) + do(t)xo(t) , where do(t) E [bl, br ], and the term do(t)Xo(t) represents the unobservable part of z(t). Then, using two vector filters: a(D) = (D-n+l,··,D-1jT, b(D) (D-n+t,···,D-l,ljT, and two rectified signals from the measured y(t): y,(t) = y(t)Xr(t), YI(t) = y(t)XI(t) , we define five regressors:

wu(t) = a(D)[u](t), wy,(t) = b(D)[Yr](t), wr(t) b(D)[Xrj(t), Wyl(t) = b(D)[Yd(t), WI(t) = b(D)[xd(t). For a given reference Ym(t), we al so introduce two In dicator functions: X,m(t) = X[Ym(t) > OJ and Xlm(t) = X[Ym(t) < 0]' and define two rectified signals from Ym(t): Ymr(t) = Ym(t)Xrm(t) , Yml(t) = Ym(t)Xlm(t). Our adaptive controller structure is: u(t) = B~w"(t) +Om,Ym,(t

For the general case, let us assume that the linear part G(D) and the dead-zone DZ(·) are known but the signal z( t) is not accessible and consider the controller:

(3.3)

+ B~rwyr(t) + B; w,(t) + B~WYI(t) + oTWI(t) where Wi(t) = b(D)[z](t) is a new regressor.

+ nO) + B,mX,m(t + nO) + BmlYml(t + nO) (3.1)

Lemma 3.1 The exael tracking is achieved, that is, y(t) = Ym(t),joral/t ~ t o+n+no-l and some to ~ 0 iji(r) = z(r) jor T = to , to + 1" .. , to + n - 1, to + n, .. " to + n + n* - 2.

where Bll E Rn-I, By"B" By/, BI E Rn , Bm" Orm, Bml, Olm E R to be updated from an adaptive law. The nominal values of t.he controller parameters are defined from: B:T a(D)Po(D) + O;Tb(D)Zo(D) = Po(D) - Zo(D)Dn', 0;, = ~, B; , 0; = b,O; , 0;1 = ;!;;O;, 0i = bIO; , 0;", =

The condition that i( T) = z( T) for T = to, t o+ 1, . .. , to+n1, to+n,' .. , to+n+n° -2 initializes the dead-zone inverse for the exact tracking. The dead-zone inverse characterization of the controller (3.1) is dynamic because the inverted Ym(t) and y( t) are applied as a part of the control u( t) to the plant to cancel the dead-zone effect.

~r' O;m = br) O~l = ~ , Oim = btooT ooT ooT ooT 0° 0° 0° oo)T , W I'th 0° = (ooT U ' IT! r , yl' I , mT' rrn' ml' Im w(t) = (w[(t),w;,(t) ,W, (t),w~(t),wT(t),y,(t + nO) ,x, (t + nO) , YI(t + nO), XI(t + nO)jT , we express the plant (2.1) as (3.2)

4

where d1(t+nO) = do(t+nO)Xo(t+no)+O;Tb(D)[doXo](t) , due to the unobservability of the ·dead-zone input , is bounded for all t.

3.2

Adaptive Designs

In this section we develop adaptive control schemes for the plant (2.1) with an unknown dead-zone DZ(·). In Section 4.1 we consider the case when the linear part G(D) of the plant is known, and in Section 4.2 we design an adaptive controller for both G(D) and DZ( ·) unknown. In Section 4.3 we present stability and tracking performance of our adaptive control schemes. In Section 4.4 we apply our designs to plants with a piecewise linear element, that is, a degenerated dead-zone with b, = b/ = 0, to prove that in this case the asymptotic tracking is achieved.

Exact Tracking

In this subsection, we first use an example to show that if the dead-zone is present, b, # 0, bl # O. then the unobservable "disturbance" do(t) may be nonzero so that a tracking error may remain . y(1) - Ym(t) -I 0, even if the estimated deadzone parameters are equal to the true ones. This is different from the input dead-zon e problem (Recker et al.. 1991; Tao

74

4.1

Design with G(D) Known

We have shown that our adaptive control schemes for plants with unknown output dead-zones of unequal slopes ensure the closed-loop signal boundedness. We have not yet shown that the tracking error e( t) = y( I) - Ym (t) converges to zero for z( I) not accessible. In Section 5 we will study the tracking performance by simulations which show that with our adaptive control schemes the performance of the closed-loop system is significantly improved.

When C( D) is known , that is, 0: and 0; are known, we define four known signals: wm,(t) = O;TWy,(t) + Ym,(t + nO), w,m(t) = O;Tw,(I) + X,m(t + nO) , Wml(t) = O;TWyl(t) + Yml(t + nO) , Wlm(t) = O;TWI(t) + Xlm(t + nO) . Then the controller (3.1) with nominal parameters is u(t) = O:Twu(t) + O;",wm,(t) + O;mw,m(t) + O;"IWml(t) + 0imWlm(t). With 0;", = ~" (};"I = ~ , O;m = be, Oim = bl unknown, we use the adaptive controller:

u(l)

= O~TuJu(t) + Om,wmr(t) + Ormwrm(l) + Omlwml(l) + OlmWlm(t)

4.4 (4.1 )

When the dead-zone parameters br = bl = 0, (2.1) represents a plant with a piecewise linear element at its output. Adaptive control of such plants can be solved as an application of our above designs and analysis. If we do not known in a priori that br = bl = 0, then we still use the controller (4.1) or (3.1) for C(D) known or unknown. In this case the unobservable part dolt) equals zero for all I 2: 0 so that in addition to signal boundedness we also achieve asymptotic tracking.

where (}m" (}rm , Om/' Olm are adaptively updated. Using (3.2), and introducing wd(l) = (O;TWyr(t) + Yr(1 + no) , O;Twr(t) + Xr(1 + no) , O;TwYI(I) + YI(t + no) , O;TWI(t) + XI(I + no»T , 0:; = (O;"r,O;m,O;"I,Oimf , we express u(l) as: u(l) = O:Twu(t) + O;/wd(l) + dl(1 + nO). Let Od(t) be the estimate of 0:; , define the estimation error: (d(l) = 0I(I-l)wd(t-nO)-u(l - nO)+O:Twu(l - nO). Asour update law for Od(I) , we choose the gradient-type algorithm (Astrom and Wittenmark , 1989; Egardt , 1979; Goodwin and Sin, 1984; IQannou and Tsakalis, 1986; Kreisselmeier and Anderson, 1986; Landau, 1990; Praly, 1990):

Od(t) =Od(t - 1) where 0< Id

IdWd(~~(:td(l)

- fd(l)

Corollary 4.1 The controller (4-1) (respectively, (3.1)) applied to Ihe planl (2.1) with CID) known (respectively, C(D) unknown), m" ml unknown and br = bl = 0 guaranlees that all closed-loop signals are bounded and limt_oo y( I) - Ym( I) =

(4.2)

O.

VI

+ wI(1 - nO)Wd(t - no) + 82 (1) with (Od(1 - 1) - Od(t - nOWwd(1 - nO), and fd(l) is a signal for robustness with respect to d l (t) . The switching-omodification (Ioannou and Tsakalis, 1986) is: 8(1)

< 1, md(l) =

Since dolt) = 0 for all I 2: 0, we can set fd(t) = f(l) = 0 in (4.2) and (4.3) and still have the closed-loop signal boundedness and asymptotic tracking. It is the new parametrization of our controller (4 .1) or (3.1), that leads to these desired properties of the developed adaptive control schemes for plants with a piecewise linear element.

=

f ( I) d

=

{o-o for IIOd(1 - 1) 112 > 2Md > 2110:;112 0 otherWIse

where 0 < 0-0 < ~(1 -Id). Although not shown in (4.2) we use projection to ensure that Omr(l) 2: mo , Oml(l) 2: mo , O,m(l) 2: 0, Olm(t) <:: O.

4.2

5

Design with G(D) Unknown

Ott) = O(t - 1) _

Iw(1 - nO){(t) - f(l) 1 + wT(1 - no)w(t - no)

Example 5.1 We consider the plant (2.1) with C(D) = _1_ and DZ(-) = DZ(m"ml,b"b l ;·), where al = 1.7 and ~:a~ 0.01, ml = 0.013 , b = 21, bl = -27. With 0: = 0 we have 0; = al = 1.7. The dead-zone DZ(·) is unknown to any of the following simulated control schemes: (a) fixed linear controller, no dead-zone inverse: u(t) = O;y(t) + 100Ym(t + 1) which achieves asymptotic tracking for ml = mr = 0.01, br = bl = 0; (b) adaptive linear controller, no dead-zone inverse: u(t) = O,(I)y(t) + (}m(I)Ym(t + 1) which is updated for 0,(1), Om(t) with 0,(0) = 3.1, Om(O) = 95 ; (c) fixed linear controller, adaptive dead-zone inverse: the adaptive controller (4.1) with Omr(O) = 95, Orm(O) = 23.5, Oml(O) = SO, Olm(O) = -22 .3, while the nominal values are O;"r = 100, O;m = 23.5, 0;"1 = 76.92, 0im = -27; (d) adaptive linear controller, adaptive dead-zone inverse: the adaptive controller (3.1) with 0(0) = (294.5,70.5, 24S, 69.13, 95, 23.5, SO, -22 .3f, while the nominal controller parameter vector is 0° = (170,35.7, 130.78,45.9,100,21,76.92, 27f. Typical responses of these control schemes are shown in Figure 2 for Ym(l) = 10sinO.09421, I = 0.8, 0-0 = 0.05, which show that the control schemes (a) and (b) which ignore the effects of the dead-zone result in large tracking errors while our adaptive control schemes (c) and (d) which take into account the effects of the parameter uncertainty of the deadzone lead to asymptotic tracking.

r

(4.3)

where 0 < I < 1, and f(l) is similar to fd(t). We also use projection to ensure that (}mr(t) 2: mo, Oml(l) 2: mo, (}rm(t) 2: 0 and (}Im(t) <:: O.

Stability

The bounded input-output stability and tracking performance of the adaptive systems with these two adaptive controllers are given by the following theorem. Theorem 4.1 All signals in the closed-loop syslem consisling of the planl (2.1) , conlroller (4.l) or (3.1) and updale law (4.2) or (4.3) are bounded, and there exisl conslants 00 > 0, 130 > 0 such Ihal tl+t2

tl+t2

I: e (t) <:: 2

t:=t}

for no

= 2n° + n

-

Simulations

In this section we present simulation results to show that our adaptive control schemes lead to significant improvements of the system performance.

For both CID) and DZ(·) unknown , the adaptive controller structure is in (3.1). To develop an adaptive law for updating the controller parameter vector O( I) = (()~ (I), (}~r( 1),0; (t J..(}~(t) , oT( I), 0mr( I), Orm( t), (};I( I), Olm (I»T we define the estimatIon error as: {(I) = 0 (I l)w(1 - nO) - u(1 - nO). From (3.2), this estimation error becomes t(l) = (0(1-1)oo)Tw(t - nO) + dl(I) , which suggests the following update law for 0(1):

4.3

Adaptive Control of Plants with a Piecewise Linear Element

00

I:

d~(t)

+ 130

(4.4)

Example 5.2 We study the responses of the adaptive deadzone inverse controller (4.1) to different reference Ym(t) when the linear part CID) = D'-~D+5 is known and the dead-zone

t=tl-nO

2 and any II 2: no , t2 2: O.

75

(a) fiJlCd linear c ontroller. no dead -tooe inverse for G(D) Icno wn

0.5

'-\ H r' HHrlrlHr-' rlHH r l H I-'

~:V~r r-A

-5 -10 -15

(.) ym(I)' 10 .ib(0.09421)

-I

o

lOO

200

)00

400

500

600

700

800

900

-1.5'---~--~--~--~--~-----~--~--~---'

o

1000

lOO

200

JOO

400

500

600

700

800

900

1000

(b) ym(t) = 10 sp(ain(O.03141t» (square wave)

(b) adaptive linear coollolla. DO dad-loonc inverse (or G(D) unknown

-2

10r---__

--~--~--~--

__- -__- -__- -__- -__- - _ ,

(d) adaptive linear cOQl1ollcr. adaptive dud -tooe inverse fa G(D) unknovm

Figure 3: Tracking errors for different Ym(t).

Figure 2: Tracking errors with different control schemes _

References

DZ(-) with rnT = 0.01 , rnl = 0.013 , bT = 21, bl = -27 is unknown. In this case 0: = -3, 0; = (15, -4f. The tracking errors for difference Ym(t) and with the same parameters and initial estiamtes as that in Example 3.1, which converge to very small values, are shown in Figure

Astrom, K. J . and B. Wittenmark (1989). Adaptive Control. Addison- Wesley.

3.

Stability of Adaptive Controllers. Egardt, B_ (1979). Springer-Verlag, Berlin.

6

Goodwin, G. C. and K. S. Sin (1984). Adaptive Filtering Prediction and Control. Prentice-Hall, Englewood Cliffs, New Jersey.

Conclusions

In this paper we have solved the adaptive control problem with a dead-zone at the plant output. We proposed two adaptive control schemes: one for plants with a known linear part but an unknown dead-zone and the other for plants with both the linear part and dead-zone unknown. Our adaptive controller structures consist of a linear feedforward part and a nonlinear feedback part incorporated with an adaptive dead-zone inverse , which resu lt in a linear error model convenient for parameter estimation. We have showed that for output dead-zones wi t h unequal slopes all signals in the closed-loop system are bounded for any bounded reference signal and a mean generalized tracking error ,is of the order of the error due to the unobservability of the dead -zone input . The asymptotic tracking is achieved when the deadzone degenerates as a piecewise linear element. Extensive simulations indicate that our adaptive designs significantly improve system performance.

Ioannou, P. A. and K. Tsakalis (1986). Robust discrete time adaptive control. in Adaptive and Learning Systems: Theory and Applications, Plenum Press, edited by K. S. Narendra.

Acknowledgements

Tao, G_ and P. V. Kokotovic (1992). Adaptive control of plants with unknown dead-zones. Proc. of the 1992 ACC, pp. 2710-2714, Chicago, IL.

Kreisselmeier, G. and B. D. O. Anderson (1986) . Robust model reference adaptive control. IEEE Trans. on Automatic Control, vol. 31, no. 2, pp. 127- 133. Landau , I. D_ (1990). System Identification and Control Design_ Prentice-Hall, Englewood Cliffs, New Jersey. Praly, L. (1990). Almost exact modelling assumption in adaptive control. Int . J. 0Jntrol, vo!. 51, no. 3, pp. 643- 668. Recker, D., P. Kokotovic, D. Rhode and J. Winkelman (1991). Adaptive nonlinear control of systems containing a dead-zone. Proc . of the 30th IEEE CDC, pp. 21112115, Brighton, England.

This work was supported by National Science Foundation under Grant ECS 87-15811 , by the Air Force Office of Scientific Research under Grant AFOSR 90-0011 and by a Ford Motor Company grant.

Truxal, J . G. (1958). Control Engineers' Handbook. McGraw-Hill, New York _

76