310 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4. NO. 3, JUNE 1988

Stabilization of Uncertain Systems Subject to Hard Bounds on Control with Application to a Robot

Manipulator MEHREZ HACHED, S. MEHDI MADANI-ESFAHANI, AND STANISLAW H. ZAK

Abstract-This paper examines the problem of estimating the region of stability for uncertain systems with bounded controllers and the “sliding mode” requirement. A new type of controller for a class of linear time- invariant systems subject to uncertainties is proposed, A transformation for decoupling the “fast” and “slow” states is utilized to investigate the stability domain estimates of the system. The results are then applied to a two-joint planar manipulator and illustrated by computer simulation results.

I. INTRODUCTION

URING recent years many approaches were developed to D control robotic manipulators ([5], [6], [8], and 1111 among many others). One of the most attractive methods is based on the theory of Variable Structure Systems (VSS) developed by Utkin 131. In this approach one seeks a hypersurface in the state space towards which all motions of the system converge and stay thereafter. Once on this hypersurface, the system is insensitive to a class of uncertain- ties and disturbances. However, only few authors considered the control problem of robotic manipulators with hard control bounds (151, [ I l l , 1121).

In this paper we consider a class of linear time-invariant multivariable control systems in the presence of uncertainties. A variable structure controller with hard constraints on the control action is proposed. We utilize a transformation for decoupling the “fast” and “slow” states. Such a decomposi- tion, also appearing in [I] and 121, simplifies the investigation of the stability of the system. We then find an estimate of the region where the Lyapunov derivative of the “fast” subsystem is negative under two different assumptions: i) the uncertain- ties in the system satisfy the matching condition, and ii) the uncertainties are bounded by a positive constant. We then find an estimate of the region where the Lyapunov derivative of the “slow” subsystem is negative under the assumption that the motion of the system when constrained to an m-dimensional hyperplane is stable. The results found in the investigation of both the “slow” and “fast” subsystems are then used to obtain an estimate of the region of attraction of the overall system.

Manuscript received November 3 , 1986; revised November 22, 1987. Part of the material in this paper was presented at the 25th Annual Allerton Conference on Communication, Control, and Computing, Monticello, IL, October 1987.

The authors are with the School of Electrical Engineering, F’urdue University, West Lafayette, IN 47907.

IEEE Log Number 8820156.

Next we apply the developed theory to second- and third- order linear time-invariant systems. Finally, we illustrate the considerations on a two-degree-of-freedom manipulator. We find an estimate of the region of stability of the two-link manipulator when the Coriolis and centripetal terms are present in the system and when they are ignored.

11. PROBLEM STATEMENT AND PRELIMINARY RESULTS We consider systems modeled by differential equations of

the form

i=Ax( t )+Bu( t )+f (x) , t E [to, 03) (1)

wherex E an, A E !Anxn, B E R“”‘”, U E Rm, and f(*): R n + R n is a nonlinear vector function representing the perturbation in the system ( f ( 0 ) = 0).

condition; i.e., there exists g(.): R n + R m such that Assumption I : The function f( .) satisfies the matching

f ( x ) =Bg(x).

Assumption 2: The function g(.) satisfies the Lipschitz condition; i.e.,

Ilg(x)ll 5 Lllxll, L>O.

Assumption 3: Matrix B is of full rank, i.e., rank B = m.

Taking into account Assumption 1, the system (1) can be represented as follows:

X(t)=Ax(t)+ B(u(t)+g(x)). (2)

Let

y = S x = a ( x ) (3)

where S E W m x n and rank S = m. We also assume that det (SB) # 0. Our goal is to find a linear transformation of the form

[:] = [?] (4)

where

det [?] # O

and MIB = 0. Note that in the new coordinates the original

0882-4967/88/0600-03 10$01 .OO @ 1988 IEEE

HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS

system has the form

31 1

the matrix S to the form

SV=[O i Z,]

where V E R n x n , and det V # 0. Hence This particular form, labeled as the special normal form, introduced and analyzed in [l] , is of great importance in our further considerations.

The following lemma gives an alternative procedure for finding the transformation (4).

Lemma I: Given two matrices S E R where rank S = rank B = m, and det SB # 0.

Then there exists a matrix M I E R ( n - m ) x n such that

,I and B E R ,I

MB & [ 71 B = [ SOB] and

det 171 #O. L A

Proof: By assumption, rank S = m. Therefore, we can always find a matrix VI E W ( n - m ) x n such that

- - det I : I f O .

L J

Then

[ :] B = [:A] .

We now form a matrix

Premultiplying (7) by Mz yields

Therefore, the transformation matrix M has the form

(7)

(9)

Remark 1 A simple procedure for finding the matrix VI such that

- - det I : I + O

L - J

is as follows. Using elementary Column operations transform

[:I * S = [ O i Z,]V-l=[O i Z,]

Thus a possible choice of the matrix VI is the first n - m rows of V-I.

In most engineering problems, the designer is faced with hard bounds on the control action. Let us assume that

IuiI<ai, i = l , . . e , m , a i > O (10)

and

k A min [ai; i = 1, . . e , m ] .

We propose the following controller satisfying constraints (10):

- k ( S B ) - l ~ ( x ) (1 1)

9 U ( X ) # O

. (X) = 0 U = [ Il(SB)- ' 4x ) l l

{ f i l I I f i I I s k l ,

where a(x) = y = Sx. Applying the control law (1 1) to the system (5) yields

2 = DllZ + DIZY

We will label the subsystem described by the equation

z =Dl12 + D n y (13)

as the "slow" subsystem, and the subsystem described by

as the "fast" subsystem. Note that the control law (1 1) is of a bang-bang type. This

may lead to chattering and hence premature wear of mechani- cal parts of a system. In order to reduce chattering one may apply a boundary-layer version of (1 1) of the form

where the constant E is a design parameter.

Stability Estimates Theorem 1 gives a sufficient condition for the stability of the

system (1,2). Before proving this theorem we need the following lemmas.

312 lEEE JOURNAL OF ROBOTICS AND AUTOMATION. VOL. 4. NO. 3, JUNE 1988

k - t Lemma 2: For the subsystem (14) there exists a Lyapunov function whose Lyapunov derivative is negative in the region

k llzll ad

Fig. 1. The region where r'/ < 0; shaded area.

Proof: Let us choose the following Lyapunov function Thus the region where v-(y) < 0 is candidate for the subsy-stem (14):

L

The Lyapunov derivative is given by

which gives A straightforward computation yields

W ) 5 l I D 2 l II IlYll ll4 + 11~2211 IlY1l2 k dllzll +ellvll -;CO.

-~

We assumed that 1 1 g(x) I( 5 L llxll. In the new coordinates Remark 2

The region where the Lyapunov derivative of the "fast" subsystem (14) is negative is also a region that contains points in the state space satisfying a sufficient condition for sliding, aTci < 0 ( o r y T j < 0).

The Lemma below gives the region where the Lyapunov derivative of the "slow" subsystem (13) is negative. We assume that the matrix DII is stable, i.e., Re ( X ; ( ~ I I ) ) < 0, i = 1, . * * , n - m.

Lemma 3: For the subsystem (13) there exists a Lyapunov I1 [a function whose derivative is negative in the region

x=M-1 [;I where

M = [ T ] .

Thus

llt?(z, u>ll = Ilg(x)ll 5 Lllxll = L M-'

5 LIIM-'II 11 [;] 11 IlYll < 4 I Z I I

S LIIM-'II (11z11+ I I Y I I ) . (16) where

The region where the Lyapunov derivative vf is negative is depicted in Fig. 1. Taking into account (16) we obtain

min X(Q) max X(P)C

CY=

and

+LlI(sB)ll IlYII llM-'ll (1141 + IlYll). Proofi The motion of the system (12) when restricted to

HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS 313

i=D11 Z. (18)

Equation (18) represents the dynamics of the reduced order system. It is obtained by setting y = 0. It can also be obtained by using the equivalent control method ([3] and [9]).

The assumption that Dll is stable will lead to the following Lyapunov matrix equation:

PDl,+DT,P= -2Q (19)

where both P and Q are symmetric positive-definite matrices. Let us choose the following Lyapunov function candidate

for the subsystem (13):

where V,(s) =zTPz.

The Lyapunov derivative is given by

V ( Z ) = ZTPZ + ZTPZ

= z T ( P D1 I + D T, P)z + z TP Dl2y + y TDT,P~.

Taking into account (19) yields and

V,(Z) = - ~ z ' Q z + z T P D12y +Y'DT,Pz. (20)

llzll Fig. 2. The region where Vs < 0; shaded area.

By the Rayleigh's principle [4] For the case d = 0, we define

2 min X(Q)llzl12 5 z'Qz 9 max X ( Q ) I I Z I / ~ . dl=-

Hence (20) becomes 2+a

which gives

0

Using Lemma 2 and Lemma 3 yields

2kllYll 2ad (2 + a d )

.~ l l Z l l - a(2 + a d ) *

Using Lemma 1 and Lemma 2 from Pate1 and Toda [ 101 we conclude that the region where ~ J z ) is negative is maximized when Q = I. Hence

For the overall system (12) to be asymptotically stable l?z, U) should satisfy the inequality V(Z, y ) < 0 in a neighborhood of the origin. Thus the system (12) will be stable if

1 max X(P)C

a=

The region where the Lyapunov derivative v, is negative is (22) depicted in Fig. 2.

For (9 to be the region of asymptotic stability of system (12), we prove that T i s negative for all llyll < Ilyoll. We define Theorem I : The system (12) is stable in the region

IlYll i2 I l Y o l l - E (23)

where E is a positive constant less than (\yo (1. Taking into

1 (2 , y): V(Z, y " j dl IIyo(12, y E Rrn, z E W-"

IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 3, JUNE 1988 3 14

account (23), (22) becomes

This quadratic form is negative for all values of E < [ [yo 1 1 . Hence a region of attraction of system (12) is given by

l lY l l

t

/

Fig. 3. The region of stability with sliding of the system (12); shaded area.

Note that when e = d = 0, then (22) implies that T(z, y ) < 0. where

d' = 1) 41 1) Remark 3 e' = 11D2211

The constant JJyo)J is the 1Jy)J coordinate of the intersection point of the line dll z 1 1 + e Ily 11 - (k/a) = 0 and the line JIy 11

and

= aIIzII.

Remark 4

The constant dl was chosen to assure that the hypersurface T(z, y) = 0, is always tangent to the set IIy 11 = [ [yo 1 1 . Hence the region of attraction for the system (12) is the largest V

described by the equation T(z, y) = 0. The region of stability of the overall system (12) depends on

the parameters of the system. Hence the region of stability with sliding also depends on the parameters of the system. The boundary of this region is the greatest level set of V(z, y )

Proof: We choose the following Lyapunov function candidate for the subsystem (14)

1

2 W 9 = - Y T Y . contour which we can fit on the origin side of the surface

The Lyapunov derivative is given by

~ ~ ( ~ 1 = y ~ y

k l l y ' 1 2 +yr(SB)g(z, y). =Y 'D21 z + Y 'Dzzy - contained in the region C = (R f l N (Fig. 3), where II(SB)-'y II

k a

<-, y E W", z E W"-"

Taking into account (24) yields

s I I 0 2 1 I I II Y II I I zll + II 0 2 2 II II Y II

Hence the region where the Lyapunov derivative of the "fast" subsystem is negative is

Lemma 4: If the perturbation in the system (2) satisfies the which gives

d'llzll +e'll.Yll + Y < O . (25) Igi I<ai, i = l , e . . , m

condition

K Theorem 2: If the perturbation in the system (2) satisfies Ilg(x)ll .:P, O<P<- (24) condition (24), then the system (12) is stable in the region

aII(SB)II 1

then for the subsystem (13) there exists a Lyapunov function @ ' = ( z , y ) : V'(z, Y ) < ~ d,' llyd \ I 2 , y E Wm, z E W n - m

whose Lyapunov derivative is negative in the region I d'Il~ll+e'Ilull+Y<o

HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS 3 15

where Example 2 Let us consider a third-order system modeled by the 1

2 V’(Z, y)=- d;yTY+(l -d ; )zTPz following equations:

and

- f fY d ‘ + cye’ llYoll’=-

d‘=-, 2 Let 2 + a d ’

Proof: To prove Theorem 2, we replace k / a by -7, d by d‘ , e by e ‘ , and dl by d{ and follow the same arguments as in the proof of Theorem 1 .

In the following we implement our approach to two linear H

~ = S X = X I + x z + x ~ .

Using transformation (4) with

M = [ i -! -!] systems of second and third order.

Example I the system (31) can be represented as

Let us consider the system -1/2 0 -112 [;I=[ -312 - 1 1 / 2 1 [;] -3/2 0 112 [;:I=[: :][::]+[:I (26)

and y = a(x) = XI + x2. Using the transformation (4) with L -

- 2 1 M=l 1 1 1

we can represent (26) as follows:

Thus d = 113, e = 8/3, a = 1/3, a = 217. If we use the control law (1 1) with k = 10. we obtain the

following region of attraction for the system (27):

1 1 - ~ I ~ ~ Y + ( ~ - ~ I ) z ~ P z < - di(lyo)1’ 2 2

where

and dl = 0.954 P= 3/2

11 yo 1 1 =7.826.

In the original coordinates, (28) becomes

1.977X: + 0.545~: - 1 . 7 7 2 ~ 1 ~ 2 < 29.21.

To eliminate the xIx2 term in (29) we define

X I = 0.9X+ 0.43 Y

x Z = - 0.43X+ 0.9 Y.

Hence in the new coordinates XY, (29) becomes

x2 Y’ (3.5)’ (15.5)’

+-= 1.

Stability regions are depicted in Fig. 4.

Thus

d = I( 4 1 11 = 3/2

e = 11DZ211 = 1/2.

If we use the control law (11) with k = 10, we obtain the following region of attraction for the system (3 1):

1 1 2 2 - dlyTY+(l -d l )zTPz<- dl IIy011’ (33)

where

di=0.81

2 - 2 .=[ -2 3 1

and

IIyo I( = 3.746. (30)

In the original coordinates, (33) becomes 0.78~: + 2.1 lx: + 2 .11~ ; + 2 . 3 2 ~ ~ ~ ~ - 0 . 7 2 ~ ~ ~ ~ - 2 . 6 2 ~ ~ ~ ’ < 5.68.

316 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 3, JUNE 1988

111. APPLICATION TO A ROBOTIC MANIPULATOR

Many approaches to the control of manipulators are based on various nonlinear compensation schemes to eliminate the interactions. The major disadvantages of such approaches are the requirements of detailed models, accurate load forecasting, and costly implementation. Another approach is to eliminate nonlinearities or interactions at the mechanical design stage. This approach is costly and only deals with certain aspects of the problem.

A third approach to control robotic manipulators is the use of variable structure control (VSC), [5], [6]. The major advantage of using a variable structure controller is the insensitivity of the closed-loop system to the parameter variations and disturbances once in the sliding mode. Also, the inherent nonlinear interactions can be rejected as disturbances.

Morgan and Ozguner [5], and Young [6] designed variable structure controllers for robotic manipulators without examin- ing the region of stability of the system. The controllers were designed so that the constraints on the input torques were not exceeded. In this paper, we will use the theory developed in the previous sections to design a variable structure controller for a two-joint manipulator as shown in Fig. 5. We will then find an estimate of the region of attraction of the system.

Description of the Model

region of attraction

Fig, 4. Region of asymptotic stability of the system (26); and the region of stability with sliding.

R

The model of the manipulator is the same as in [7]

Fig. 5 . A planar two-link manipulator

matrix form as f o l l o ~ ~ :

] [Bg:] [ ::] = [ A21(e2) A22 Ail(&) A ~ z ( & )

1 1: +m&+m2 -+m21112 cos O 2 + I 1 + I ,

4

COS e 2 + ~ 2 + - 4

A122(&)4: + A I I ~ ( & ) ~ I ~ GI(61’ 0 2 )

m21112 sin 82 + (thez)( - mzlllz sin 6 2 ) ’ ’ ] + [ G Z ( ~ I , 0 2 1 1

( 3 5 )

represents the velocity vector of the two-joint manipulator, and uj’s are the applied torques.

where 13 = (e,, 1 9 ~ ) ~ is the position vector, e = (81,

The individual terms of (35) are defined as follows ([7],

1 1

2

1 + [ cos O1 + mzll cos O1 +- m212 cos (0, +e,) g

1 PI) :

COS (e, +e,) m2d2 t- 2

where

mi17 m,R2 I . = - + - i = l , 2 I 1 2 4 ’

(34)

AI1(O2) = 5/3mI2 + m12 cos (e,)

mP 2

AlZ2(e2) = -- sin (0,)

= - m12 sin (0,) and g is the gravitational constant. The values of the model parameters used are m, = m2 = 1 kg, l1 = l2 = 0.5 m, R = m12

2 AzII(&)=- sin (62) 0.05 m, and g = 9.8 m/s2. Equation (34) can be put in the

317 HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS

ml 3m 2 2

ml 2

the gravitational terms, and L a positive constant. This fact will allow us to use the approach from the previous sections to find an estimate of the stability domain for the system (36). We can now represent the system (36) as

w e 1 , e 2 ) = - g COS (e1+e2)+-gicos (e,)

c2(el, e,) =- g cos (e, + e2) . k = A ~ ( t ) + B(x)u( t ) + f ( x ) (39)

We define the state vectorxT A ( O l , O 2 , 8 1 , 8 ~ ) = (XI, x2, x3, x4) . The following state equations are obtained: where

where

(37)

0 0 1 0

A = [ ' 0 0 0 0 '1 0 0 0 0

and

and

We then define the vector We use the transformation

[:I = [ Y ]

M1=[0 1 0 O J

f ( x ) A where

1 0 0 0

where M,B=O

and

and the matching condition is satisfied. For the system (36), f ( x ) does not satisfy Assumption 2 or

the requirement (24). Moreover B(x) is not a constant matrix. However, we have the following inequality for

L t Y = d x ) = s x and Z A (21, Z z I T = ( X I , X d T , then in the new coordinates the system (39) becomes

(41) where p is a positive constant representing the upper bound on

318

where

and

D H = [ -b" -pol 10 0

O L 2 = [ 0 1 0 1

D21= [ -b" -pol D22=[ 0 1 0 1

10 0

Since the matrix B is not constant, we define the quantity a (see Lemma 2)

a A sup Il(SB)-'(z2)ll =7.3 22

and it can be verified that SB(z2) is always nonsingular.

Remark 5

we found L = 0.817 and = 73.70.

Controller Design Although no actuator dynamics are included in the manipu-

lator dynamics, one must place practical limits on any control input. The maximum torques applied in the control of system (41) are the same as used in [5] and [6]. They are

For this paricular two-joint manipulator (shown in Fig. 5 ) ,

Iu l I<360N m

(u21 < 182 N * m. (43)

For the system (41), we apply the control law (1 l), where the saturation part is

where

(45)

and kl and k2 have to be chosen so that the maximum torques applied will not be exceeded. In our case, kl = 360 and k2 = 182, thus

k=min [k,, k2] = 182.

Stability Estimates In what follows we examine the region of asymptotic

stability for two different cases.

Case I : We will assume that the model of the kinetics of the two-joint manipulator shown in Fig. 5 includes the Coriolis and centripetal terms, i.e., L # 0.

IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4. NO. 3. JUNE 1988

Case 2: We will assume that the model of the kinetics of the two-joint manipulator includes only the inertial and the gravitational terms, i.e., L = 0, orAI22 = AI12 = A211 = 0. For most arms, a model of kinetics which includes only the inertial and gravity terms is quite satisfactory [8].

For Case 1 we first consider the region where the Lyapunov derivative of the fast subsystem is negative. From (15) we have

The Lyapunov derivative is given by

Taking into account (42) yields

G(f(y) 5 IlYll (ID21 II llzll + 1 1 0 2 2 II llrll

Upon substituting 11 (SB(z2)) - 11 by a, and the other terms by their values we obtain

vf 5 IlYll [lOllzll + 1011Yll + 16.411Yl12 - 17.56+ 16.411~11~1.

The Lyapunov derivative of the fast subsystem is negative in the region

16.411~11~ + lOllzll + 16.411y112 + lOllyll - 17.56<0

which gives

Thus the Lyapunov derivative of the "fast" subsystem of the two-joint manipulator is negative in the region described by (46). Obviously vf < 0 in the region

l lY l l+ llzll 5 0.815. (47)

From Lemma 3, the Lyapunov derivative of the "slow" subsystem is negative in the region llyll < cr IIzII. For the system (41) we have a = 1. Thus the Lyapunov derivative of the "slow" subsystem is negative in the region

l l ~ l l ~ l l ~ l l ~ (48)

From Theorem 1 we conclude that the region of asymptotic stability of the overall system (41) is

where V(z, y ) is given by (21)

HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS

and

Fig. 6. Positions of both links for a set point regulation problem.

0.1 0 p= [ 0 0.11

yo=o.4.

The boundary of the stability domain with sliding is the greatest level set of V(z, y) contained in the region C = (R fl N where

N={(z,y): I(zIJ +llyll <0.815,y E W’, z E W’}.

For Case 2, we assumed that L is negligible compared to p . Hence (42) becomes

Ilf(z, u)ll < P . (49)

Inequality (49) is the same as (24). This enables us to use the results of Theorem 2.

From Lemma 4, the Lyapunov derivative of the “fast” subsystem is negative in the region

d‘llzll +e’Ilyll +r<O. For the two-joint manipulator we have

d’ =e’= 10 and y= - 17.56

and hence

lOllzll+ lOllyll- 17.56<0. (50)

Using Theorem 2, we conclude that the system (41) is asymptotically stable in the region

1 d,’yTY+(l-d,’)ZTPz

3 1 <z d ; \]yo) (I2, y E W’, z E Elz

where

d,’ =2/3

and

319

Time (sec)

IIyoI(’=O.878.

Referring to Fig. 3, the stability region with sliding is the greatest level set of V(z, y) contained in the region C = CR n N where

N = ((2, y ) : lOllzll + lOllyll< 17.56, z E W’, y E W’}. In the original coordinates the region CR is

CR= {0.37x;+0.37x;+3.4~ 1 0 - 3 4

+3.4x 10-3x~+0.2x,x3+0.2x2x4<0.256)

and

N={10(x;+x;)1/2+ ~ O ( ( X ~ + O . ~ X ~ ) ~

+ (X2 + 0.1 X4)’) ’’’ < 17.56).

IV. SIMULATION RESULTS

In this Section we present the results of the computer simulations for the two-link manipulator (Fig. 5) . We simulate the case where the centripetal and Coriolis terms are ignored. Furthermore, we only consider the set point regulation problem.

We set the desired positions and velocities of both links to zero. Hence the control law (1 1) has to drive the system from any initial condition that belongs to the region of stability of the system to the origin.

For the simulation shown in Figs. 6, 7, and 8, we took xl(t0) = 0.6, x2(t0) = 0.45, andx3(tO) = Xq(t0) = 0. Figs. 6 and 7 show the time responses, while Fig. 8 shows the phase plane portraits.

320 IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4. NO. 3. JUNE 1988

h

U

VI \ D

L ” pl

0

c 0

L

r. .e

._

- . M 1 ~ 0 .OO ,062 , lP.5 ,188 ,254 ,313 ,375 .q3E . S O

6.61

3.13

1.97

,811

s

3

r.

.e C

0 .-.

9 . s ~

2.15-

-.157-

-7.08

-9.38 I- -11.7 0.00 1 ,062 I ,254 ,313

Time (sec) Time (sec)

Fig. 7. Velocities of both links for a set point regulation problem.

6.761 n

-.we -.WO - e o 0 8 -.y6 - . 9 Y -.m -.30’)

Position of joint one (rad)

Fig. 8. Phase plane trajectories

Position of joint tu0 (rad)

for a set point regulation problem.

Note that the controller (44) results in undesirable robot chattering. To eliminate this problem, a continuous version of (44) was implemented

boundary-layer-type control. As can be seen from these figures, chattering was significantly reduced.

V. CONCLUSIONS

k( SB(x2)) - 1 a(x) The main objective of this paper was to find the region of attraction of a linear time-invariant variable structure system subject to uncertainties and hard bounds on the control action. A transformation which resulted in the decoupling of the

9 if lI(SB(xz))-la(x)ll O.5- “s1ow”and “fast” dynamics of the system was utilized. This transformation simplified the investigation of the stability of the overall system. The regions where the Lyapunov deriva- tives of the “slow” and “fast” subsystems were negative were investigated and the results were used to determine an estimate of the region of asymptotic stability of the whole system.

9

R( SB(X2)) - 1 a( x)

if II(SB(X2))-“J(x)ll > 0.5 Il(Wx2))- Ia(x)ll

U = [

- E

(51)

Figs. 9 and 10 depict time responses and phase-plane portraits of the system driven by the controller (51). In Figs. 11 and 12, we compare the action of a bang-bang-type control and a

HACHED et al.: STABILIZATION OF UNCERTAIN SYSTEMS

3.62802i

32 1

I \

8 92759

7 8116'l

A

U

6 69569 U L "

5 57974

C

c) C 4 L16380

0 4

Y

% 0 3 39785

3 e +

2 23190 4

a, 3

1 11595

0 00000 - 6

3.62802

\

" - ,748755

3

? -2.93714

0

% -5.12553 - 7 + g-7.31392-

a, 3

0

.+ 7

4

4

-9.50231 -

-11.6907,' 0.00000 ,062500 ,125000 ,187500 .250000 ,312500 ,375000 237500 .500000

-9.50231i11

" - ,748755 - 3

?-2.93714 - 0

0

.+ 7

7 + g-7.31392-

a, 3

4

4

-11.6907d

Time (sec) Time (sec)

Fig. 9. Velocities of both links for a set point regulation problem with the control (51).

DO -. 5253'47 - . L)Y)695-. 3760q2- .301389-. 226737 -. 1&?084 - .O771)31- ,002778

aJ 3

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IEEE JOURNAL OF ROBOTICS AND AUTOMATION, VOL. 4, NO. 3 , JUNE 1988

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Fig 12. Control action, boundary-layer type (t = 0.5)

The theory developed was first applied to second- and third- order linear systems. We then analyzed a two-degree-of- freedom manipulator. We were able to find an estimate of the region of attraction of the system under hard bounds on the input torques.

ACKNOWLEDGMENT

The authors wish to gratefully acknowledge the reviewers for their constructive remarks.

REFERENCES [ I ] K. K. D. Young, P. V . KokotoviC, and V. I. Utkin, “A singular

perturbation analysis of high gain feedback systems,” IEEE Trans. Automat. Contr., vol. AC-22, no. 6, pp. 931-938, 1977.

121 P. V. KokotoviC, “A Ricatti equation for block-diagonalization of ill- conditioned systems,” IEEE Trans. Automat. Contr., vol. AC-20, no. 6, pp. 812-814, 1975.

[3] V. I. Utkin, Sliding Mode and Their Application in Variable Structure Systems.

[4] J. N. Franklin, Matrix Theory. Englewood Cliffs, NJ: Prentice- Hall, 1968.

[SI R. G . Morgan and U . Ozgiiner, “A decentralized variable structure control algorithm for robotic manipulator,” IEEE J . Robotics Automat., vol. RA-I, no. 1, pp. 57-65, Mar. 1985. K. K. D. Young, “Controller design for a manipulator using theory of variable structure systems,” IEEE Trans. Systems, Man, Cybern., vol. SMC-8, no. 2 , pp. 101-109, Feb. 1978.

[7] J . M. Hollerbach, “Dynamic scaling of manipulator trajectories,” Trans. ASME, vol. 106, pp. 102-106, Mar. 1984.

[8] W. E. Snyder, Industrial Robots: Computer Interfacing and Control. Englewood Cliffs, NJ: Prentice-Hall, 1985.

Moscow, USSR: MIR. 1978.

[6]

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191

[lo]

V. I. Uthn, “Variable structure systems with sliding modes,” IEEE Trans. Automat. Contr., vol. AC-22, no. 2, pp. 212-222, 1977. R. V. Patel, and M. Toda, “Quantitative measures of robustness for multivariable systems, in Proc. J.A.C.C., San Francisco, CA, 1980, pp. TP8-A. M. W. Spong, J. S. Thorp, and J. M. Kleinwaks, “Robust micropro- cessor control of robot manipulators,” Automatica, vol. 23, no. 3, pp. 373-379, May 1987. A. Weinreb and A. E. Bryson, “Optimal control of systems with hard control bounds,” IEEE Trans. Automat. Contr., vol. AC-30, no. 11, pp. 1135-1138, 1985.

S. Mehdi Madani-Esfahani was born in Ahwaz, Iran, in 1954. He received the B.S. degree in electrical engineering from the Tehran University, Tehran, in 1977, and the M.S. degree in electrical engineering from Purdue University, West Lafay- ette, IN, in 1979. In 1985 he joined the Department of Electrical Engineering, Purdue University as a Ph.D. student and a research assistant, where he is currently pursuing the Ph.D. degree in control sciences.

From 1979 to 1985, he worked as a Control and Electronics Engineer in Automation Engineering Company of Iran. From 1986 to 1987, he was a Teaching Assistant in the Mathematics Department at Purdue University. His current research interests are in nonlinear systems and control, stability theory, variable structure control systems, and robust control of robotic manipulator.

[ l l ]

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manipulators.

Mehrez Hached was born in Sfax, Tunisia, on December 16, 1962. He received the B.S. degree from the Pennsylvania State University, University Park, in 1985, and the M.S. degree from Purdue University, West Lafayette, IN, in 1986, both in electrical engineering.

He is currently a Ph.D. student and research assistant in the Department of Electrical Engineer- ing, Purdue University. His research interests in- clude robust control; variable structure control; stability theory, and their application to robotic mary areas of research

systems control, and col

Stanislaw H. Zak received the Ph.D. degree from the Technical University of Warsaw, Poland, in 1977.

He was a faculty member in the Institute of Control and Industrial Electronics, Technical Uni- versity of Warsaw, from 1977 to 1980. From 1980 until 1983, he was a visiting assistant professor in the Department of Electrical Engineering, Univer- sity of Minnesota, Minneapolis. In August 1983, he joined the School of Electrical Engineering at Purdue University, West Lafayette, IN. His pri-

are systems theory, time delay systems, nonlinear mputer algebra.