Robustness of model reference adaptive control systems with magnified leakage term

WA5

-

1O:OO

Promdlngr of the 32nd Confenme

on Declrlon and Contml

San Antonlo, Tear * D e c " r 1993

Robustness

of

Model Reference Adaptive Control Systems

with Magnified Leakage Term

A-Cheng Wu', Li-Chen Fu',' and men-Fa Hsu' *Department of Electrical Engineering

'Department of Computer Science and Information Engineering National Taiwan University

Taipei, Taiwan, Republic of China

Abstract

This paper investigates the robustness of model reference adaptive control(MRAC) of a class of uncertain dynamical sys- tem using state information. The controller structure used here is quite standard in the literature of adaptive control, and the parameter update law adopted here incorporates some magnified leakage term like the well-known o-modification[ll]. But unlike MRAC of systems with only 1/0 measurement, the scheme pre- sented here can be robust to arbitrarily high level of matched un- certainties(besides its robustness to some degree of mismatched uncertainties). It is also worthwhile to point out that such scheme differs from those proposed in the literature of (adaptive) robust control is that the strong robustness is obtained under no high-gain condition and hence there is no phenomenon for the possibility of saturating the input channel. Furthermore, the present adaptive controller remains to have asymptotic tracking capability in the ideal case, which is hardly obtained by conven- tional (adaptive) robust controller under practical design.

1

Introduction

The control of uncertain systems has attracted the attention of system theorists for a long time. By this we mean that the system or, to be more precise, the model of the system, contains uncertain elements which may be nonlinear and/or time-varying and are unknown or imperfectly known. Furthermore, the con- trol input may be contaminated by measurement error. Under such a circumstance, one tries to design a control such that the plant has certain satisfactory properties.

Recently, robustness of uncertain systems with matched un-

certainties were proved by Chen and Leitmann [2], where the matched uncertainty with arbitrarily large strength is shown sta- bilizable by a fixed robust control law. They also showed that in the absence of matching assumptions the proposed control law remains a stabilizing one provided the uncertainty is small enough. While in the adaptive literature, Coreless and Leitman- n [4] first construct a class of saturation-type adaptive robust controllers with state measurement error. The control gain is

tuned adaptively to deal with the unknown uncertainty bound. In Chen [l] and Wu et al. [8], adaptive control algorithms uti- lizing concave function are presented to overcome the unknown

U.S. Government work not protected

by U.S. copyright

uncertainty bound. In the absence of input uncertainty, two new classes of adaptive robust control schemes were developed by Chen [3]. Both uncertain dynamics and measurement noise are considered but at the price that the tracking error(state)

~ ( t ) does not converge to the origin even in the absence of those non-idealities.

In this paper, we present a model reference adaptive control (MRAC) scheme which is robust to different types of uncertain- ties. The proposed adaptive controller adopted the standard structure [6] with a parameter update law augmented by some magnified leakage term similar to the well-known a-modification

[ll]. This kind of modified update law is not new to the literature of MRAC with only 110 information, i.e., state is not accessi- ble. Therefore, the present scheme is shown to similarly possess the robustness to both matched and mismatched system uncer- tainties, which can be nonlinear and time-varying unlike [7], and

the capability of asymptotic convergence when in the ideal case. Furthermore, the residual set of the tracking error(state) is such that the magnitude of the limiting time-average of the squared error is of the order of the levels of uncertainties and some design constant uo inside the leakage term. But, here what is better for the present scheme is its robustness to arbitrarily high level of

matched uncertainties, which is hardly obtained in the literature of MRAC[6][11]. It is, however, worthwhile to point out that the proceeding feature is obtained under no high-gain condition like what has usually been assumed explicitly or implicitly[l][3][8].

As a result, the control force does not have any indication to be large for possible saturation problem.

The layout of this paper is as follows : The first section is the introduction. The problem of the adaptive tracking control of the uncertain plants is formulated in section 2. The design and analysis of robust model reference adaptive control system is presented in section 3. Section 4 is the simulation. Section 5

is the conclusion.

2

Problem Formulation

Let the class of plants to be controlled here be described as

where x E R" is the state vector, U E

R"

is the vector of control inputs, Ao(., .) : R" x R+ 4 R"'" and B(.,.) :

R"

x R+ + WXm

are C'(continuous1y differentiable) with respect to both I and t , and for any control U = u(z,t) which is also

C'

with respect to both x and t the solution trajectory exists. Now suppose the control task is to drive the system (2.1) to follow the following reference model :

where x, E R" denotes the reference state, r E Rm denotes the vector of reference exogenous inputs, and (A,,,, B,,,) is a stabi- lizable pair such that A,,, itself is a Hurwitz matrix. Then, the problem to be solved here is clearly how to design the vector of plant inputs U such that the control task can be fulfilled, namely,

x(t) + xm(t) as closely possible when t + 00 This problem is generally hard t.o solve and can be made more tractable if the following assumptions about (2.1) are satisfied : The system (2.1) can be rearranged in the following form :

wherep1,p2 E R+,plAAI(x,t) and plABI(x,t)uaregen- erally termed as mismatched uncertainties whereas p2BmA A4x, t) is generally called matched uncertainties, and B' E P x " is not known a priori so that B,B*z represents the parametric uncertainty (in contrast with the former non- parametric uncertainties with levels yl and p2, respective- ly).

The upper bound on IlO'lll is assumed to be known apriori

as Oms=( II.lI1 denotes the one-norm of the argument vector or the induced one-norm of the argument matrix). The nonlinear time-varying Uncertainties in (2.3) satisfy the following cone-bounded conditions :

for some gi 2 0, i = 1 , 2 , .

. .

,5, where

1)

.

11

denotes either the matrix norm or the Euclidean vector norm depending on its argument.

This new form (2.3) with the system properties (2.4) implies that the system behavior of the original plant in fact is domi- nated by that of a linear plant :

provided p1 and pz are small, or equivalently that (2.5) is the nominal model of the original plant (2.1).

Remarks :

It should not be difficult to rewrite (2.1) into the form

(2.3). In general, we can first linearize (2.1) around the

origin to obtain

~ = A I + B u

where A = ~ J z . = o , t = o and B = -1==o,t=o. SUP- pose that (A, 8 ) is a controllable pair, then there exists a

matrix IC E PXn such that A - BK = A, is a Hurwitz matrix. If we further define B, = B, then the linearized system is indeed (2.5) with 0' = IC.

In thesecond, let Ao(x,t)z-Az = jllAA1(2,t)+pZBmAAz ( q t ) and Bo(z,t)

-

B = plABl(x,t), we readily obtain

(2.3). It is noteworthy that AAl(z,t),AAZ(s, t ) , and AB1

(x, t ) can be arbitarily fast time-varying with the inequal- ities (2.4) being satisfied.

(2) One should note that BmPz is actually one kind of matched uncertainty. Therefore, the expressions @.I)? (2.3), (2.5) implicitly imply that (A,

+

B,,,B')s represents the maxi- mum parameterizable information available from Ao(E, t)

so that p2 (level of matched uncertainty) can normally be kept small.

Based on the expression (2.3) of the plant, we can derive the error system conforming the control task by subtracting (2.2) from (2.3), namely,

1 = A,e

+

B,(u

+

B'x - r

+

pzAA?(z, t))

+pi(AAi(z,t)

+

A B i ( 2 , t ) ~ ) (2.6)

A

where e = x

-

xm will be referred t o as error state. Now, we can restate the problem in the context of robust model refer- ence adaptive control as follows : devise a continuous control law U = u(b,x,t), where

b

is an estimate of B', as well as a pa- rameter update law

4

= O ( b , x , t) so that e ( t ) approaches zero asymptotically in the ideal case, i.e., p1 = pz = 0, and e ( t ) con- verges to a small residue set asymptotically when p1 and pz are not zero but generally small, i.e., the control has to be robust to the existence of nonparametric uncertainties. Furthermore, the limiting time average of the squared error satisfies

where IC is an increasing function with p 1 , p 2 , and uo separately,

and K(O,O,O) = 0.

3

Robust Model Reference

Adaptive Control

Under the problem formulation in the previous section, it is quite natural to devise the control law as :

using certainty equivalence principle [6]. By this design, we em- phasize that this is really a continuous control so long as the parameter update law is well specified and the reference inputs are themselves continuous.

Now, let the parameter update law be chosen as :

4 =

.

-(e - e*)

d - = @ ( 6 , x , t ) = -BlfPexT

-

du _(3.2)

where P E

R n x n

is a positive definite matrix satisfying the Lya- punov equation $(A;P+PA,) =

-&

for some positive definite matrix

&,

and U = d i u g ( u l , ~ ~ ~ , u , ) with

=

{

uo(l+ 11e1111x11) otherwise dt

0 if ~ l l i i l l l

I

nemor,i = 1 , 2 , . . . , n

7

(3.3)

where

0,

denotes the i-th column of

e.

To investigate the effectiveness of this adaptive control, we adopt the popular Lyapunov analysis, i.e., by constructing a Lya- punov function V(e,

4)

as follows :

V(e,

4) = -

2

Y

eTPe

+

T ~ ( J + ) ) (3-4)

and then evaluating its time derivative along the solution trajec- tories of (2.6), (3.2). The investigation will first be performed in the ideal case in which the error system (2.6) will become

(3.5)

d = Ame

-

Bm4x

50 that the time derivative of (3.4) becomes :

1

V(e,

4)

= [;e'(AfP

+

PA,)e - eTPBm4x +Tr[dT(-B;PexT - du)]

= -eTQe - Tr[4*64 n

5

-PIIeIIz - CUi(IIPiIIz - IIdiIIlIIerIIi), _(3.6) where q = A,i,(Q) and 0; denotes the i-th column of 0'. It is

clear from (3.3) that

i=l

ui(IIdiI12 -

IIjiIIlIIef

111)

2

:lldiIIi(IIe^iIIi - nIlerIIi) _(3.7)

and, hence, (3.6) satisfies V

5

-qllellz. Then, boundedness of all signals is clear, and the asymptotic convergence of the error

state to the origin can be concluded via Barbdat's lemma [6]. But in the presence of nonparametric uncertainties, (2.6) be- comes

e =

Arne

-

Bmdz

+

pzBmAAZ(x, t )

where K1 = suptzo

-

k l ( x , p i , p p , Om,,, t) and ( , represent the i-th column of

C.

From (3.9), there exist 71,72

>

0 such that

v I

-71v

+

72v;

+

v i , (3.16) where

to derive the following

(3.20)

where Kp = suptzo ICz(*, p l , p2,uo, t ) and Nz is the subset of N

such that Nl and Nz are disjointed and NI U N Z = N. For convenience of presentation, we define

to further reduce (3.20) to

Now, we can conclude from (3.16) and (3.21) that the value of V

defined in (3.9) will eventually fall into a residual interval [0,

&I,

where & = max[R1, Rz] with Rz =

(q)',

and hence

that the global stability of the adaptive system is established. Now, we are in a position to present the main results of this paper in the following theorem :

Main T h e o r e m : There exist pi

>

0 such that for all p1 E

[O,p;) and under the adaptive control law (3.1)-(3.3), all sig- nals in the closed-loop systems (2.6) remain globally uniformly bounded. In addition, the state tracking error will converge to 135

the residual set

1 to+T

Re

= {e

'

Flo

lle(T)lIZdT

5

1{3(k1,P2rcO)} (3.22) where K3 is a positive function of the order p l , p z , and oo, i.e., Z(3 is an increasing function with itl, iiz, and 00 separately, and

1C3(O,0,O) = 0. Moreover, in the absence of uncertainties, the state tracking error will converge to zero asymptotically.

0

Proof : Please refer to [lo].

with rl = 10, andrz = -10. Simulation plots are depicted in Fig. 1 where satisfactory tracking performance is shown.

Another physical example in (31 for the control of pendulum motion is given as : E ( t ) = A z ( t )

+

B u ( t )

+

(f

-

A z ( t ) ) with z ( t ) = [zl(t),z2(t)lT, where

1

A = [ ! ) 2 ! 3 ] . B = [ ; ] J = [ -sZn(z)-q(t)cos(z) x2 k m ( t ) = A z , ( t ) Remarks :

The precondition for the global stability of the adaptive system is that p1 must be small enough, which implies that the mismatched uncertainties have to be small. How- ever, there is no constraint on the strength of the matched uncertainties. Therefore, the matched uncertainties can not destroy the global stability of the closed-loop system while the parameter update law is turned on. Neverthe- less, the larger the matched uncertainties are, the larger

pz and, hence, r are and so that the tracking performance degrades because the size of the residual set grows.

and q ( t ) = random[-l,l]. Since the controller in [3] is only for regulation, we choose the reference model to be

(2) The magnified leakage term in adaptation law (3.2) pulls the drifted parameters back can generally avoid the prob- lem of instability, which is very similar to the case of

M-

RAC systems with only 1/0 measurements. Notwithstand- ing this, it may lead to conservative bounds associated with (3.20) and, perhaps, the adaptive system may have fluctu- ating transient. Since the bound obtained by Lyapunov function analysis is usually not as tight as the one the adaptive system actually has, we claim that the transient behavior of the proposed scheme is anticipated to be good as will be observed in the illustrative simulated examples in the next section.

4

Simulation Examples

To illustrate effectiveness of the scheme proposed in this pa- per, we consider the following system with some nonlinear un- certain dynamics :

where pi = 0.01 and pz = 10. A simple reference model is chosen as:

and the Lyapunov matrix P to be 5 1 P = [ i

r ]

The control force used in the scheme in [3](Fig.(2.d)) is ap- parently larger than the one used in the scheme proposed here (Fig.(Z.c)) since the former is witch high gain nature. In Fig.(2.a) and Fig.(2.b), the trajectories of the states resulting from the present scheme are depicted in solid line whereas those resulting from the scheme in [3] are in dashed line. Although applying the smaller control force, the present scheme in this paper seems to yield slightly faster transient response.

5

Conclusion

This paper presents an MRAC for a class of dynamical sys- tems with nonlinear timevarying uncertainties. Under the as-

sumption of state accessibility, the robustness against input, mis- matched, and arbitrarily high level of matched uncertainties is shown. Further, the present control retains the capability of

asymptotic convergence of the tracking error (state) in the ideal case. But in the presence of uncertainties, the residual tracking error is such that the limiting time-average of its square is of the order of the level of uncertainties and the design constant 00. S-

ince no high gain concept is adopted in the presented scheme, the control force is generally more acceptable than those generated by the schemes in [1][3][8], which is supported by the numerical simulation provided in the end. Ongoing research will concen- trate on the development of the observer-based adaptive robust control scheme for a more practical environment for implemen- tation.

6

Reference

11) Chen,

Y.

H.,

"Modified Adaptive Robust Control System Design", I d .

J.

Control, Vol. 49, No. 6, pp. 1869-1882,

1989.

[2] Chen, Y. H., "Robustness of Uncertain Systems in the Ab- sence of Matching Assumptions", Int. J. Control, Vol. 45,

NO. 5, pp. 1527-1542, 19S7.

[3] Chen, Y. H., "Adaptive Robust Control of Uncertain Sys- tems with Measurement Noise", Automatica, Vol. 28, No.

4, pp. 715-728, 1992.

[4] Coreless, M. and G. Leitmann, "Adaptive Control of Sys- tems Containing Uncertain Functions and Unknown Func- tions with Uncertain Bounds", J. Optimization Theory Ap- plications, 41, pp. 155-16S, 1983.

[5] Fu, L. C., "A New Robust MRAC Using Variable Struc-

ture Design for Relative-Degree-Two Plants", Automati- ca, Vol. 25, 1992.

[S] Narendra, K. S. and A. M. Annaswamy, "Stable Adaptive Systems", PrenticeHall, 1989.

(71 Maciejowski, J. M., "Multivariable Feedback Design", Ad- dison-Wesiey, 1989.

[SI Wu, A. C., L. C. Fu, and C. F. Hsu, "On the Robust Opti- mal Control for Robot Manipulators", International Sym- posium on Implicit and Nonlinear Systems, Ft. Worth, Texas, Dec., 1992.

(91 Wu, A. C., L. C. Fu, and C. F. Hsu, "A New Decentral- ized Model Reference Adaptive Control for a Class of In- terconnected Dynamic Systems Using Variable Structure Design", International Journal of Adaptive Control and Signal Processing, In press, 1992.

[lo] Wu, A. C., L. C. Fu, and C. F. Hsu, "Robustness of Model Reference Adaptive Control Systems with Magnified Leak- age Terms", EE Report, NTU, Taipei, Taiwan, R.O.C., 1992.

[ l l ] Ioannou, P.A. and K.S. Tsakalis, "A Robust Direct Adap- tive Controller", IEEE Trans. Autom. Contr., Vol. AC-31, No. 11, pp. 1033-1043, 1986. 0

9,

2

c C

Control input u l xl-solid line, xml-dashed line

I

0

-I

Y

0

Control input u2 x2-solid line, xm2-dashed line

Control input U o G o u , r Control input U in [3] 0 Y 0 0 I

&

CD + 0 x l x2 137