2000 Tomei - Robust adaptive friction compensation for tracking control of robot manipulators.pdf

2164 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000
where a = 8 T (1)K 01
e 8(1)k p and b = 8 T (1)K 01
e 8 0 (1)k p. Note
that a> 0. Substituting (41) into (42) and after some manipulation,
we obtain
"(1) = a
2b
sin 1 0
2+a 2+a 1 2: (43)
The bending angle can also be calculated from the flexible coordinate p
(1) = (8 0 (1)) T p = b(s 1x 2 0 c 1x 2 ) 0 c1 2 (44)
where c =(8 0 (1)) T K 01
e 8 0 (1)k p . Substituting (41) into (44) yields
(1) = 00:5b sin 1 0 0:5b"(1) 0 c1 2: (45)
From (43) and (45), it is simple to derive
(1) = 0
b
2+a
sin 1 0
2c + ac 0 b2
2+a
1 2: (46)
Note that ac 0 b 2 0. By substituting (25) into (46), we obtain
b
(1) = 0
2+a + sin 1 0
1 (47)
2+a +
where =2c + ac 0 b 2 . Clearly, is positive. Substituting (47) into
(43) derives
"(1) =
a + ac 0 b2
2+a + sin 1 0 2b
1: (48)
2+a +
Substituting (47) and (48) into (40) yields a constraint equation only
on 1:
4+2a + 0 b cos 1 2+a + 0 2b
1 + sin 1
2+a +
2(2 + a + )
a + ac
0
0 b2
sin 21 =0: (49)
4(2 + a + )
It can be proved that this equation has such a unique solution: 1 =0
[12]. Therefore, we conclude that (1) = 0 from (47) and "(1) =
0 from (48), which implies that both 1 1 and 1 2 are zero. From
(36) and (37), we conclude that the position errors 1x 1 ; 1x 1 ) and
(1x 2 ; 1x 2 ) are zero.
REFERENCES
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[8] J. K. Mills, “Multi-manipulator control for fixtureless assembly of elastically
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objects,” Ph.D. dissertation, The Chinese Univ. Hong Kong, Shatin,
Hong Kong, 1997.
[13] D. Sun, Y. H. Liu, and J. K. Mills, “Cooperative control of a two-manipulator
system handling a general flexible object,” in Proc. IEEE/RSJ
Conf. Intell. Robots Syst., 1997, pp. 5–10.
[14] T. J. Tarn et al., “Design if dynamic control of two cooperating robot
arms closed chain formulation,” in Proc. IEEE Int. Conf. Robot Automat.,
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[15] I. D. Walker, R. A. Freeman, and S. I. Marcus, “Analysis of motion and
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[16] J. T. Wen and K. Kreutz-Delgado, “Motion and force for multiple robotic
manipulators,” Automatica, vol. 28, no. 4, pp. 729–743, 1992.
[17] T. Yukawa et al., “Stability of control system in handling of a flexible
object by rigid arm robots,” in Proc. IEEE Int. Conf. Robot. Automat.,
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[18] Y. F. Zheng and M. Z. Chen, “Trajectory planning for two manipulators
to deform flexible objects,” in Proc. IEEE Int. Conf. Robot. Automat.,
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Robust Adaptive Friction Compensation for Tracking
Control of Robot Manipulators
Patrizio Tomei
Abstract—The tracking problem is considered for robot manipulators
with unknown parameters and dynamic friction, in the presence
of bounded disturbances and/or modeling uncertainties. The authors
design a robust adaptive control algorithm which guarantees arbitrary
disturbance attenuation. If the disturbances belong to , asymptotic
tracking is also achieved.
Index Terms—Disturbance attenuation, dynamic friction, robotic manipulators,
robust adaptive control.
I. INTRODUCTION
High-performance position tracking control of mechanical systems
cannot be performed adequately if friction phenomena are not properly
taken into account. Moreover, a more accurate representation of friction
should allow the control gains (and therefore the feedback control
component) to be decreased, since the friction disturbance terms
could be compensated by the feedforward component of the control
algorithm. The previous statement understands that friction is exactly
known, and this, in turn, implies that the friction parameters are known
Manuscript received November 1, 1999. Recommended by Associate Editor,
G. Tao. This work was supported in part by MURST and in part by ASI.
The author is with the Dipartimento di Ingegneria Elettronica, Università di
Roma ‘Tor Vergata’ 00133 Roma, Italy.
Publisher Item Identifier S 0018-9286(00)04217-3.
0018–9286/00$10.00 © 2000 IEEE

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000 2165
and the friction state variables are available from measurements. Unfortunately,
this is rarely the case, so that adaptive (or robust) techniques
are needed to deal with uncertain parameters and state observers are
needed to estimate the friction unmeasured variables.
For the tracking control of robot manipulators, other parameter uncertainties
related to the mechanical structure have to be taken into account.
By considering only linear viscous friction effects, many adaptive
controls for robot manipulators have been proposed in the literature
(see [1]–[3] for a survey). Since time-varying disturbances are not
considered in all of these schemes, they may not work satisfactorly in
the presence of disturbance torques and/or modeling uncertainties. On
the other hand, the robust and the H1 controllers developed in [4]–[7]
allow for unstructured time-varying disturbances but they are unable to
guarantee asymptotic tracking (even though the disturbances vanish).
Moreover, disturbance attenuation is achieved by enforcing the feedback
control component to counteract the perturbations due to exogeneous
disturbances and modeling uncertainties. To retain both the
advantages of robust and adaptive controls, in [8] a robust adaptive
controller was designed which tolerates time-varying parameters and
disturbances, and guarantees asymptotic tracking when parameters are
constant and disturbances are vanishing. However, the transient performance
depends upon disturbance bounds and cannot be arbitrarily
improved. An adaptive control for mechanical systems with nonlinear
friction was proposed in [9], while in [10] the friction was treated as a
nonlinearly parametrized function. In both papers time-varying disturbances
are not allowed and transient performance are not guaranteed.
A robust adaptive controller with transient performance and arbitrary
disturbance attenuation has been proposed in [11], where only instantaneous
friction is taken into account, without modeling dynamical effects.
As far as dynamically modeled friction is concerned, the problem
of its compensation in a DC motor along with a new dynamic model
was treated in [12], assuming that all parameters were known. A further
extension of this result to the case in which the friction force depends
linearly on only one unknown parameter was given in [13] for the
stabilization problem. A discontinuous tracking control law has been
recently proposed in [14] for adaptive friction compensation in robot
manipulators, where more uncertainty is allowed in the friction terms.
However, none of the previous works takes into account the action of
bounded time-varying disturbances (with unknown bound), due to exogeneous
disturbance forces or to bounded modeling uncertainty, as well
as transient performance bounds which are considered in this paper.
We present a robust adaptive tracking controller which guarantees
arbitrary attenuation on the joint position and joint velocity tracking
errors of the effects of bounded disturbances. The proposed controller
achieves asymptotic tracking when disturbances belong to L2. Dynamic
friction forces depending on unknown parameters are taken into
account, which are not available from measurements.
II. MAIN RESULT
We consider a robot manipulator consisting of n +1links interconnected
by n joints whose dynamic model is
B(q; )q + C(q; _q; )_q + h(q; ) +F0z
+ F1 _z + F2 _q = u + d(t)
j _q j j
_z j = 0f0j zj +_qj; 1 j n (1)
g j (_q j )
in which the vector q = [q1; 111;q n ] T represents the joint relative
displacements, the vector z = [z1; 111;z n ] T takes into account the
friction dynamics (see [12]), the vector 2 R m is the vector of the
kinematic and dynamic parameters of robot and actuators which enters
linearly in the robot equations, the vector u denotes generalized forces
(forces or torque) applied at the joints, B(q; ) is the symmetric positive
definite inertia matrix, and C(q; _q; )_q represents Coriolis and
centripetal forces. The vector z represents the average deflection between
the contact surfaces during the stiction phase. The friction force
F = F0z + F1 _z + F2 _q consists of three terms: the viscous friction
F2 _q, the stiffness force F0z, and the damping force F1 _z, with F i diagonal
positive semidefinite matrices (F i =diag[f ij ]). The disturbance
forces due to exogeneous disturbances are grouped into the vector d(t),
which is assumed to be bounded; d(t) may also contain bounded unstructured
modeling uncertainties. The vector is assumed to be unknown
and to belong to a known compact set, which for the sake of
simplicity is supposed to be a closed ball centered at N (the nominal
value of ). As stated in [12], by measuring the steady-state friction
force when the velocity _q is constant, the functions g j and the matrix
F0 can be determined and, therefore, they are supposed to be known.
The case in which they are not known will be addressed in Remark 2.4.
The functions g j are always positive and decrease monotonically from
g j (0) as _q j increases. As shown in [12], the z j variables are always
bounded and, in particular, if jz j(0)j g j(0) then jz j(t)j g j(0),
for any t 0. The matrices F1 and F2 are supposed to be unknown
and to belong to a known compact set; the nominal values are denoted
by F1N and F2N . The choice of C(q; _q; ) is not unique; we choose
the elements of C as follows [1], [15]:
C ij = 1 2
_q T @B ij
@q + n
k=1
@B ik
@q j
0 @B jk
q i
_q k (2)
where i; j = 1; 111;n, so that the matrix _ B(q; ) 0 2C(q; _q; ) is
skew-symmetric. We assume that only the position q(t) and the velocity
_q(t) are available from measurements. For the robot model (1),
we consider the adaptive tracking problem formulated in the following
definition.
Definition 2.1: The robust adaptive tracking problem with arbitrary
disturbance attenuation is said to be globally solvable for
the robot manipulator (1) if, given any smooth bounded reference
trajectory q r (t) with bounded derivatives _q r (t) and q r (t), a parametrized
adaptive control law (k > 0)u = u(q; _q; q r; _q r; q r; ^; k);
_^ = (q; _q; q r ; _q r ; q r ; ^; k), exists such that for the closed-loop
system: i) (boundedness) k^(t)k; kq(t)k, and k _q(t)k are bounded
8t 0; ii) (arbitrary disturbance attenuation) the following inequality
holds 8t >t0 and 8t0 0:
2
t
q( ) 0 q r( )
d
t
_q( ) 0 _q r( )
2(t0)+ 1 t
k 1(M )(t 0 t0) + kd( )k 2 d (3)
t
where 2 is a nonnegative constant depending on initial conditions
q(t0)0q r (t0); _q(t0)0 _q r (t0);z(t0); ^(t0) and 1 is a positive constant
depending only on a known constant M ; and iii) (asymptotic tracking)
if d(t) 2 L2 \ L1, then
lim
t!1 q(t) 0 q r(t)
_q(t) 0 _q r(t)
=0:
The following result shows how an adaptive tracking control which
solves the problem given in the previous definition may be actually
designed.
Theorem 2.1: The robust adaptive tracking problem with arbitrary
disturbance attenuation is globally solvable for the robot system (1).

2166 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000
Proof: Let x 1 = q; x 2 = _q, so that (1) is written as
_x 1 = x 2
B(x 1;)_x 2 = 0C(x 1;x 2;)x 2 0 h(x 1;) 0 F 0z
0 (F 1 + F 2)x 2 + F 18(x 2)z + u + d(t)
_z = 08(x 2 )z + x 2 (4)
where 8(x 2 )=diag[f 0j (jx 2j j)=(g j (x 2j ))] = 4 diag[ j (x 2j )]. Perform
the change of coordinates ~x 1 = x 1 0 q r; ~x 2 = x 2 0 x 3 2 with
x 3 2 = 0K 1 (x 1 0 q r )+ _q r and K 1 a symmetric positive definite matrix
(K 1 > 0). Let 0 a (m + n) 2 n matrix such that 0C(x 1 ;x 2 ;)
(0K 1~x 1 +_q r)0h(x 1;)0(F 1 +F 2) x 2 +B(x 1;)K 1 (x 2 0 _q r)0
B(x 1 ;)q r = 0 T (x 1 ;x 2 ;q r ; _q r ; q r ) in which = [ T ;f 11 +
f 21 ; 111;f 1n + f 2n ] T 4 = N + a with f ij the jth diagonal element of
matrix F i . Let aM be the (known) largest value of k a k. We obtain
_~x 1 = 0K 1 ~x 1 +~x 2
B(x 1 ;) _~x 2 + C(x 1 ;x 2 ;)~x 2 = 0F 0 z + F 1 8(x 2 )z
+0 T (x 1 ;x 2 ;q r ; _q r ; q r ) + d(t)+u
_z = 08(x 2)z + x 2: (5)
Since we know a bound for the z-variables kz(t)k k[g 1 (0); 111;
g n (0)]k = 4 z M ; 8t 0, we define (k >0;K 2 > 0)
u = 00 T ( N + ^ a )+F 0 ^z 0 (F 1N + 1F ^ 1 )8^z
0 k 4 ~x2 0 k 4 0T 0~x 2 0 K 2~x 2 0 ~x 1 + v(t) (6)
with (k^z(0)k z M)
_^z =Proj(08^z + x 2 0 F 0 ~x 2 + F 1N 8~x 2 ; ^z) (7)
where the dynamics of the estimates ^ a and 1F ^ 1 and the control v(t)
are yet to be defined, while Proj(y; ^) is the smooth projection algorithm
introduced in [16], which in this case is given by: Proj(y; ^) =y,
if p(^) 0; Proj(y; ^) =y, ifp(^) 0 and p^(^)y 0;,
Proj(y; ^) =[I 0 (p(^)p^(^) T p^(^))=(kp^(^)k 2 )]y; if p(^) > 0
with
and p^(^)y >0;
p(^) =(^ T ^ 0
2
M)=( 2 +2 M );p^(^) =(dp(^))=(d^)
and an arbitrary positive real. The projection algorithm
is such that, if
_^ = Proj(y; ^) and k^(0)k M then:
k^(t)k M + ; 8t 0; Proj(y; ^) is Lipschitz and continuous;
kProj(y; ^)k kyk; ~ T Proj(y; ^) ~ T y. From (5) and (6),
we obtain ( ~ = 0 ^)
_~x 1 = 0K 1~x 1 +~x 2
B _~x 2 + C ~x 2 =0 T ~ a 0 ~x 1 0 k 4 0 k 4 0T 0 0 K 2 ~x 2
0 [F 0 + F 1N 8+1F 18]~z + 8 T (x 2; ^z) ~ b + d + v
(8)
where b = [1f 11 ; 111; 1f 1n ] T and 8 T (x 2 ; ^z) ~ b = (1F 1 0
1F ^ 1 )8(x 2 )^z. Let bM be the largest (known) value of k b k. Consider
the function
W = 1 2 ~xT 1 ~x 1 +~x T 2 B(x 1;)~x 2 +~z T ~z : (9)
Its time derivative, in view of (8), is such that
_W = 0~x T 1 K 1 ~x 1 +~x T 1 ~x 2 + 1 2 _ ~xT 2 B~x 2 +~x T 2 0C ~x 2 +0 T a
~
0 ~x 1 0 k 4 ~x 2 0 k 4 0T 0~x 2 0 K 2 ~x 2 0 F 0 ~z + F 1N 8~z
+1F 1 8~z + 8 T ~ b + d + v +~z T (08z + x 2 0 _^z) (10)
which, recalling property d) of Proj and the skew-symmetry of _ B02C,
implies
_W 0~x T 1 K 1~x 1 0 ~x T 2 K 2~x 2 + kdk2
+ 1
k k ~ T ~ a a
+~x T 2 [8 T b
~ + v(t)] + ~x T 2 1F 18~z 0 ~z T 8~z: (11)
Since ~x T 2 1F 1 8~z 0 ~z T n
8~z = [~x j=1 2j ~z j j 1f 1j 0~z 2 j j ] =
n
j=1 j (~x 2j ~z j 1f 1j 0~z 2 j )= n j=1 j[0((1=2)~x 2j 1f 1j 0 ~z j ) 2
+(1=4)~x 2 2j1f1j] 2 from (11), we obtain
Now, choose
_W 0~x T 1 K 1~x 1 0 ~x T 2 K 2~x 2 + kdk2
k
+~x T 2 [8 T ~ b + v(t)] + 1 4
n
j=1
+ k~ a k 2
k
j ~x 2 2j1f 2 1j: (12)
v = 0 k 4 8T (x 2; ^z)8(x 2; ^z)~x 2 0 8T (x 2 ; ~x 2 ) ^ c
4
0 k 16 8T (x 2 ; ~x 2 )8(x 2 ; ~x 2 )~x 2 (13)
where ^ c is an estimate of c =[1f11; 2 111; 1f1n] 2 T , which belongs
n
to a closed ball of known radius cM . Since j=1 j ~x 2 2j1f 2 1j =
~x T 2 8 T (x 2 ; ~x 2 ) c , from (12), (13) we have
_W 0~x T 1 K 1 ~x 1 0 ~x T 2 K 2 ~x 2 + kdk2
k
+ 1 k k~ a k 2 + k ~ b k 2 + k ~ c k 2 : (14)
For the dynamics of the estimates we use a gradient algorithm with
projection as given in the following:
_^ a = Proj(0(x 1 ;x 2 ;q r ; _q r ; q r )~x 2 ; ^ a ); k^ a (0)k aM
_^ b = Proj(8(x 2 ; ^z)~x 2 ; ^ b ); k^ b (0)k bM
_^ c = Proj(8(x 2 ; ~x 2 )~x 2 ; ^ c ); k^ c (0)k cM : (15)
The projections guarantee the boundedness of ^z; ^ a; ^ b ; ^ c and, consequently
(since z(t) is bounded), of ~z; ~ a ; ~ b ; ~ c . By virtue of (9) and
(14) it follows that ~x 1 and ~x 2 are bounded so that property i) of Definition
2.1 is proved. Integrating (14), we obtain
t
t
~x T 1 K 1 ~x 1 +~x T 2 K 2 ~x 2 d
t
W (t 0 )+ 1 kdk 2 d
k
t
+ 1 k sup (k ~ a k 2 + k ~ b k 2 + k ~ c k 2 )(t 0 t 0 ): (16)
2[t ;t]
By virtue of property a) of the operator Proj, we have, 8t
0; k^ a (t)k aM + ; k^ b (t)k bM + ; k^ c (t)k cM +
so that, with a proper redefinition of k, (3) in Definition 2.1 easily
follows. As far as property iii) is concerned, consider the function
V = W + 1 2 ~ T a ~ a + ~ T b ~ b + ~ T c ~ c (17)
whose time derivative, taking (10), (13), (15) and property d) of Proj
into account, is such that
_V 0~x T 1 K 1 ~x 1 0 ~x T 2 K 2 ~x 2 + 1 k kd(t)k2 : (18)
From (18), since d(t) 2 L 2 , we obtain lim t!1 t 0 (~xT 1 K 1 ~x 1 +
~x T 2 K 2 ~x 2 ) d < 1. Since by (5), _~x 1 and _~x 2 are bounded, by the
Barbalat lemma (see [17]), property iii) in Definition 2.1 follows.
Remark 2.1: By virtue of the disturbance attenuation terms in (6)
and (13), the adaptation dynamics for ^ a; ^ b , and ^ c may be switched

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000 2167
Fig. 1.
Tracking and observer error, reference, and torque.
off at any time without compromising the stability of the closed loop
system, as apparent from (9) and (14), and preserving properties i) and
ii) in Definition 2.1.
Remark 2.2: If, in addition to the L 2 bound represented by the inequality
(3) in Definition II.1, an L1 bound of this kind is required
q(t) 0 q r (t)
_q(t) 0 _q r
4(t 0)+ p 1 3( M )
(t) k
+ p 1
sup kd( )k; 8t
k
t 0
2[t ;t]
K 1 and K 2 should be chosen as: K 1 = kK 1 ;K 2 = kK 2 with K 1 >
0; K 2 > 0.
Remark 2.3: The proposed robust adaptive control may be easily
modified to deal with time-varying parameters, following the design
given in [11].
Remark 2.4: The only parameters which are a priori assumed to be
known are the entries f oj of matrix F 0 . Suppose that the actual value
of F 0 (denoted by F 3 0
) is different than its nominal value F 0. We can
write (1) as
B q + C _q + h + F 0z + F 1 _z + F 2 _q = u + d(t)
_z j = 0f 0j
_z 3 j = 0f 3 0j
k _q j k
g j (_q z j + _q j ; 1
j
j n (19)
)
k _q j k
g j (_q j ) z3 j + _q j; 1 j n (20)
where d(t) =d(t)+F 0 z0F 3 0
z 3 . Since F 0 z0F 3 0
z 3 is a time-varying
bounded vector, we can treat d(t) as a time-varying bounded disturbance
(even though it is state-dependent), so that (19) is still in the
form (1), with d(t) in place of d(t). Moreover, the nominal friction
dynamics given by the second equation in (19) contains the known coefficients
f 0j, while the true friction dynamics (20) is not considered.
The control law (6), (7), (13), (15) is therefore robust with respect to
uncertainties on F 0. The same reasoning may be applied for uncertainties
on the g j functions.
III. SIMULATION RESULTS
Some simulations have been carried out with reference to the current-controlled
DC motor illustrated in [13]: B q + f 0z + f 1
_z + f 2 _q = u + d(t); _z = 0(f 0j _qj)=(a 0 + a 1e 0(_q=! )
)z + _q;
in which B is the total inertia (motor plus load), (f 0 z + f 1 _z + f 2 _q)
is the friction torque, q is the motor shaft angular position and u is
the DC motor torque. The nominal known values of parameters and
disturbances are: B N =0:0025 kg/m 2 , f 0N =260Nm/rad, f 1N =
0:7 Nms/rad, f 2N =0;a 0N =0:285 Nm, a 1N =0:05 Nm, ! 0N =
0:01 rad/s, d N (t) = 0. Following the procedure given in Section II,
we designed the control law
u = 00 T ( N + ^ a )+f 0 ^z 0 (f 1N + ^ f b
0j _qj
) ^z 0 k ~x 2
g(_q) 4
0 k 4 0T 0~x 2 0 k 2 ~x 2 0 ~x 1 + v
v = 0 k 4 82 ^z 2 ~x 2 0
_^ a = Proj(0~x 2 ; ^ a );
_^ c = Proj
8~x 2 2; ^ c
1 4 8^z ^ c 0 k 16 2 ~x 3 2
_^z = Proj(08^z + _q 0 f 0 ~x 2 0
_^ b = Proj(8^z~x 2 ; ^ b )
f 1N 8~x 2 ; ^z)

2168 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 45, NO. 6, JUNE 2000
Fig. 2.
Estimates of parameter deviations.
with ~x 2 = _q 0 _q r + k 1(q 0 q r); g(_q) = a 0N + a 1N e 0(_q=! ) ;
0 T = [k 1 (_q 0 _q r ) 0 q r ; 0 _q]; T N = [B N ;f 1N + f 2N ];
8 = (f 0N j _qj)=(g(_q)). In the first set of simulations we assumed
that d(t) = 0 and that the nominal values of f 0;a 0;a 1;! 0
coincided with their actual values, while the actual values of the
other parameters were chosen as: B = 0:0022 kg/m 2 , f 1 = 0:6
Nms/rad, f 2 = 0:018 Nms/rad. The initial conditions of the motor
and of the controller were set equal to zero while the following gain
constants were chosen: k = 0:1;k 1 = 5;k 2 = 25. The reference
position was (as in [13]): q r (t) = 5:6 sin(0:4t) sin(0:02t). The
results of the simulations are illustrated in Figs. 1 and 2, which show
a satisfactory behavior of the closed-loop system. A second set of
simulations was then performed with actual values of f 0 ;a 0 ;a 1 ;! 0
different from their nominal values, precisely: f 0 = 280 Nm/rad,
a 0 =0:3 Nm, a 1 =0:055 Nm, ! 0 =0:012 rad/s. Moreover, a torque
disturbance d(t) =0:1 sin(100t) was added to perturb the system.
The initial conditions and the controller gains were left unchanged.
In the corresponding results there are no appreciable differences with
respect to Figs. 1 and 2, except in the input torque which has a small
oscillation at the same frequency of the disturbance.
IV. CONCLUSION
An adaptive tracking controller has been designed for robots subject
to dynamic friction, which is able to guarantee arbitrary disturbance
attenuation on the joint position and joint velocity tracking errors of
the effects of modeling uncertainties and unknown bounded disturbances.
The unknown parameters are supposed to belong to a known
compact set while the bounds of the disturbances may be unknown.
Even though the friction is dynamically modeled, only positions and
velocities of the joints must be available from measurements, while
the friction variables are estimated by means of a suitable observer.
Three separate control components may be distinguished: disturbance
attenuation terms, adaptation terms, and an observer for the friction
dynamics. The adaptation part is needed to guarantee asymptotic
tracking for L 2 disturbances. However, the adaptation dynamics may
be stopped (i.e., the parameters estimates may be frozen) without
affecting the stability of the closed-loop system and still ensuring arbitrary
disturbance attenuation.
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Robust Stability of Uncertain Time-Delay Systems
Yun-Ping Huang and Kemin Zhou
Abstract—This paper considers the robust stability and control of uncertain
time-delay systems. Sufficient stability conditions are derived by
using the small theorem. It is then shown that most existing results in the
literature are much more conservative than this condition. Furthermore,
robust control of uncertain delay systems can be studied by combining this
stability criterion and the standard synthesis techniques.
Index Terms—Robust control, robust stability, structured singular value,
uncertain delay systems.
I. INTRODUCTION
It is well known that checking the stability of a feedback system with
time delays is in general a nontrivial question because of the infinite
dimensional nature of the system [21]. Many computational methods
have been proposed over the years; see, e.g., [1] and references therein.
A relatively easier problem is to check if the system is stable independent
of delay [7], [8]. Many sufficient conditions for delay-independent
stability have been derived in recent years. Unfortunately, most of the
criteria in the published literature are actually more conservative than
the criteria obtained in [23] using small gain theorem. The exact condition
for delay-independent stability was derived in [2] using structured
singular value. Many of the recent research papers have focused on the
delay-dependent stability; see, [4], [9]–[12], [14]–[16], [19], and [17]
for an extended list of references. An advantage of these methods is
that they may be applied to synthesis problems. Unfortunately, these
methods can be extremely conservative, as we shall show in this paper.
Manuscript received March 15, 1999; revised December 9, 1999. Recommended
by Associate Editor, J. Chen.
The authors are with the Department of Electrical and Computer Engineering,
Louisiana State University, Baton Rouge, LA 70803 USA (e-mail:
kemin@ee.lsu.edu).
Publisher Item Identifier S 0018-9286(00)09998-0.
A less conservative computational method has also been proposed recently
in [5] using discretized LMI approach. This motivates us to find
less conservative stability criteria that at the same time can be used
for synthesis problems. We shall derive in this paper some sufficient
stability conditions using standard robust control techniques. The advantage
of our stability conditions is that all standard robust control
techniques such as H1 control and synthesis can be applied directly.
This paper is organized as follows: Section II gives the problem formulation.
Section III contains the main results. Some concluding remarks
are given in Section IV.
II. PROBLEM FORMULATION
Consider the following uncertain time-delay system:
_x(t) =Ax(t) + A ix(t 0 i) (1)
i=1
where i 2 [ i ; i ], i =1; 2; 111;m, are uncertain constant delays
such that i 0 and i 6= j for i 6= j. We are interested in the
stability question of this uncertain time-delay system. In the case when
i =0and i = 1, the question is the well-known delay-independent
stability problem. A complete solution has been obtained in [2].
For convenience, we shall define
m
h i := i 0 i: (2)
We shall denote D as the delay operator such that
D (t) =(t 0 )
for any scalar function (t).
We shall also assume that there is a B i 2 R n2r and a C i 2 R r 2n
such that
A i = B i C i :
In particular, B i and C i can be chosen to have full rank so that r i =
rank(A i ). Of course, results presented in this paper do not necessarily
require these factorization be full rank. For example, one can always
use a trivial factorization: B i = A i and C i = I. However, results may
become computationally more difficult to apply when r i > rank(A i ).
In the subsequent development, we shall also use the following definition:
B =[B1 B2 111 B m ] ; C =
C1
C2
. .
C m
D = diagfh1I r ;h2I r ; 111;h m I r g: (3)
To derive an explicit stability condition for the uncertain delay
system, we shall need the notion of structured singular value [20],
[24]. Let n c = r1 + r2 + 111+ rm. Define
1 := fdiag(1I r ;2I r ; 111; m I r ): i 2 Cg :
The structured singular value of a matrix M 2 C n 2n with respect to
a block structure 1 is defined to be 1(M )=0if there is no 1 2 1
such that det(I 0 1M )=0and
otherwise.
1 (M )=
min
121 max jij: det(I 0 1M )=0
i
01
0018–9286/00$10.00 © 2000 IEEE