Applied Nonlinear Control - Chapter 8: Adaptive Control - Nguyễn Tân Tiến

n-parametric uncertainties can be present. These 
include 
- high-frequency un-modeled dynamics, such as actuator 
dynamics or structural vibrations 
- low-frequency un-modeled dynamics, such as Coulomb 
friction and stiction 
- measurement- noise 
- computation round-off error and sampling delay 
Since adaptive controllers are designed to control real physical 
systems and such non-parametric uncertainties are 
unavoidable, it is important to ask the following questions 
concerning the non-parametric uncertainties: 
- what effects can they have on adaptive control systems ? 
- when are adaptive control systems sensitive to them ? 
- how can adaptive control systems be made insensitive to 
them ? 
While precise answers to such questions are difficult to obtain, 
because adaptive control systems are nonlinear systems, some 
Applied Nonlinear Control Nguyen Tan Tien - 2002.6 
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 
___________________________________________________________________________________________________________ 
Chapter 8 Adaptive Control 
46 
qualitative answers can improve our understanding of adaptive 
control system behavior in practical applications. 
Parameter drift 
When the signal v is persistently exciting, both simulations 
and analysis indicate that the adaptive control systems have 
some robustness with respect to non-parametric uncertainties. 
However, when the signals are not persistently exciting, even 
small uncertainties may lead to severe problems for adaptive 
controllers. The following example illustrates this situation. 
Example 8.7 Rohrs’s example_________________________ 
Consider the plant described by the following nominal model 
p
p
ap
k
pH +=)(0 
The reference model has the following SPR function 
3
3)( +=+= pap
k
pM
m
m 
The real plant, however, is assumed to have the transfer 
function relation 
u
ppp
y
22930
229
1
2
2 +++= 
This means that the real plant is of third order while the 
nominal plant is of only first order. The un-modeled dynamics 
are thus to seen to be )22930/(229 2 ++ pp , which are high-
frequency but lightly-damped poles at )15( j±− . Beside the 
un-modeled dynamics, it is assumed that there is some 
measurement noise )(tm in the adaptive system. The whole 
adaptive control system is shown in Fig. 8.18. The 
measurement noise is assumed to be )1.16sin(5.0)( ttn = . 
)(tym
)(tr
u
22930
229
1
2
2 +++ ppp
3
3
+p )(te
1y
reference model
nominal un-modeled
yaˆ
raˆ
)(tn
Fig. 8.18 Adaptive control with un-modeled dynamics and 
measurement noise 
Corresponding to the reference input 2)( =tr , the results of 
adaptive control system are shown in Fig. 8.19. It is seen that 
the output )(ty initially converges to the vicinity of 2=y , 
then operates with a small oscillatory error related to the 
measurement noise, and finally diverges to infinity. 
Fig. 8.19 Instability and parameter drift 
_________________________________________________________________________________________ 
In view of the global tracking convergence proven in the 
absence of non-parametric uncertainties and the small amount 
of non-parametric uncertainties present in the above example, 
the observed instability can seem quite surprising. 
Dead-zone 
8.7 On-line Parameter Estimation 
Few basic methods of on-line estimation are studied. 
Continuous-time formulation is used. 
8.7.1 Linear parameterization model 
The essence of parameter estimation is to extract parameter 
information from available data concerning the system. The 
quite general model for parameter estimation applications is in 
the linear parameterization form 
aWy )()( tt = (8.77) 
where 
nR∈y : known “output” of the system 
mR∈a : unknown parameters to be estimated 
mnRt ×∈)(W : known signal matrix 
(8.77) is simply a linear equation in terms of the unknown a. 
Model (8.77), although simple, is actually quite general. Any 
linear system can be rewritten in this form after filtering both 
side of the system dynamics equation through an 
exponentially stable filter of proper order, as seen in the 
following example. 
Example 8.9 Filtering linear dynamics__________________ 
Consider the first-order dynamics 
ubyay 11 +−=& (8.78) 
Assume that 11,ba in model are unknown, and that the output 
y and the input u are available. The above model cannot be 
directly used for estimation, because the derivative of 
y appears in the above equation (noting that numerically 
differentiating y is usually undesirable because of noise 
consideration). To eliminate y& in the above equation, let us 
filter (multiply) both side of the equation by )/(1 fp λ+ 
(where p is the Laplace operator and fλ is a known positive 
constant). Rearranging, this leads to the form 
11)()( buayty fff +−= λ (8.78) 
where )/( ff pyy λ+= and )/( ff puu λ+= with subscript 
f denoting filtered quantities. Note that, as a result of the 
filtering operation, the only unknown quantities in (8.79) are 
the parameters )( 1af −λ and 1b . 
The above filtering introduces a d.c. gain of fλ/1 , i.e., the 
magnitudes of fy and fu are smaller than those of y and 
u by a factor of fλ at low frequencies. Since smaller signals 
may lead to slower estimation, we may multiply both side of 
(8.79) by a constant number, i.e., fλ . 
Applied Nonlinear Control Nguyen Tan Tien - 2002.6 
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 
___________________________________________________________________________________________________________ 
Chapter 8 Adaptive Control 
47 
Generally, for a linear SISO system, its dynamics can be 
described by 
upBypA )()( = (8.80) 
where 
nn
n ppapaapA ++++= −− 1110)( K 
1
110)(
−−+++= nn papbbpB K 
Divide both sides of (8.80) by a known monic polynomial of 
order n, leading to 
u
pA
pB
pA
pApAy
)(
)(
)(
)()(
00
0 −= (8.81) 
where 
nn
n pppA ++++= −− 11100 ααα K 
has known coefficients. In view of the fact that 
1
1111000 )()()()()(
−−− −++−+−=− nnn papaapApA ααα K
(8.81) can be rewritten in the form 
)(ty T wθ= (8.82) 
where 
[ ]Tnnn bbaaa 10111100 −−− −−−= LL αααθ 
Tnn
A
up
A
u
A
yp
A
py
A
y



=
−−
0
1
00
1
00
LLw 
Note that w can be computed on-line based on the available 
values of y and u . 
_________________________________________________________________________________________ 
Example 8.9 Linear parametrization of robot dynamics_____ 
lc1
l1
l2
lc2
I2, m2
I1, m1
q2,τ2
q1,τ1
Fig. 6.2 A two-link robot 
Consider the nonlinear dynamics of a two-link robot 


=

+



 −−−=




2
1
2
1
2
1
1
212
2
1
2221
1211
0 τ
τ
g
g
q
q
qh
qhqhqh
q
q
HH
HH
&
&
&
&&&
&&
&&
 (6.9) 
where, 
[ ]Tqqq 21= : joint angles 
[ ]T21 τττ = : joint inputs (torques) 
2221
2
2
2
121
2
1111 )cos2( IqllllmIlmH ccc +++++= 
2
2
2222122112 cos IlmqclmHH c ++== 
2
2
2222 IlmH c += 
2212 sin qllmh c= 
]cos)cos([cos 1121221111 qlqqlgmqglmg cc +++= 
)cos( 21222 qqglmg c += 
Let us define ,21 ma = ,222 clma = ,21113 clmIa += and 
2
2224 clmIa += . Then each term on the left-hand side of (6.9) 
is linear terms of the equivalent inertia parameters 
[ ]Taaaa 4321=a . Specially, 
212
2
114311 cos2 qlalaaaH +++= 
422 aH = 
42122112 cos aqlaHH +== 
Thus we can write 
aqqqYτ ),,(1 &&&= (8.83) 
This linear parametrization property actually applies to any 
mechanical system, including multiple-link robots. 
Relation (8.83) cannot be directly used for parameter 
estimation, because of the present of the un-measurable joint 
acceleration q&& . To avoid this, we can use the above filtering 
technique. Specially, let )(tw be the impulse response of a 
stable, proper filter. Then, convolving both sides of (6.9), 
yields 
∫∫ ++−=− tt drrtwdrrrtw 00 ])[()()( GqCqHτ &&& (8.84) 
Using partial integration, the first term on the right hand side 
of (8.84) can be rewritten as 
∫
∫∫
−−
−−=
−−=−
t
ttt
drrtwrtw
ww
drw
dr
drtwdrrtw
0
000
])()([
)0(])0([)0()()0(
][)()(
qH-qH
qqHqqH
qHqHqH
&&&&
&&
&&&&
This means that the equation (8.48) can be rewritten as 
aqqWy ),()( &=t (8.85) 
where y is the filtered torque and W is the filtered version of 
1Y . Thus the matrix W can be computed from available 
measurements of q and q& . The filtered torque y can also be 
computed because the torque signals issued by the computer 
are known. 
_________________________________________________________________________________________ 
8.7.2 Prediction-error-based estimation model 
8.7.3 The gradient estimator 
8.7.4 The standard least-squares estimator 
8.7.5 Least-squares with exponential forgetting 
Applied Nonlinear Control Nguyen Tan Tien - 2002.6 
_________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________________ 
___________________________________________________________________________________________________________ 
Chapter 8 Adaptive Control 
48 
8.7.6 Bounded gain forgetting 
8.7.7 Concluding remarks and implementation issues 
8.8 Composite Adaptation