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Robust Nonlinear Predictive Control with Modeling
Uncertainties and Unknown Disturbance for Single-Link
Flexible Joint Robot
Adel Merabet and Jason Gu
Department of Electrical & Computer Engineering
Dalhousie University
Halifax, Nova Scotia B3J 2X4, Canada
{Adel.Merabet & Jason.Gu}@dal.ca
Abstract - A robust nonlinear predicti ve controller for single-
link flexible joint robot is presented in this paper. The objective
is to track some predefined profiles for angular displacement of
the link. The prediction model, used in the controller design, is
carried out via Taylor series expansion. The uncertainties and
error modeling of the system are taken into account by this
controller. In order to deal with them, a disturbance observer is
designed from the predictive control law. Simulation results show
the high performance of the proposed control scheme.
Index Terms – Flexible joint robot, predictive control.
I. INTRODUCTION
The flexible joint robot is modeled by a nonlinear
representation to describe its dynamic behavior. However, this
mathematical model is only an approximation of the real
system. The simplified representation of the system behavior
contains model inaccuracies such as parametric uncertainties,
unmodeled dynamics, and external disturbances [9, 11].
Because of these inaccuracies, the controller design must take
into account their effects in order to improve the performance
of the closed loop system.
Model based predictive control (MPC) has received a
great deal of attention and is considered by many to be one of
the most promising methods in control engineering. The
predictive control strategy belongs to the optimal control
methods. The difference is that the cost function to be
optimized is defined over a future horizon. In traditional
optimal control law, the online computation burden is heavy to
solve the optimization problem [1, 5, 11], which is
unacceptable for systems characterized by fast dynamics like
robotics. To overcome this advantage, several works, about
nonlinear predictive control, have been done as in [2, 10],
where the model output prediction for error tracking is
obtained by expanding the output signal and the reference
signal. Then, the optimization of the predictive tracking errors
is used to derive the offline control laws. This method applied
for mechanical systems with good performance [4, 6, 8]. In
addition of that, to solve the problem of modeling
inaccuracies, one of the methods is to design an observer to
deal with these uncertainties. A nonlinear disturbance observer
have been developed in [3], and applied in case of mechanical
system with good performance for disturbance rejection [4].
The aim of this paper is to design a nonlinear predictive
controller for the single-link flexible joint robot with a
disturbance observer to deal with uncertainties and unmodeled
quantities. It is organized as follows. Section II describes the
mathematical model of the robot. Section III derives the
nonlinear predictive controller in detail. The objective is to
track the angular displacement of the link. Section IV presents
the design of the disturbance observer to take into account the
mismatched model and external disturbance. Section V gives
the stability analysis of the closed-loop system. Section VI
presents simulation results for tracking problem of the robot
using the proposed controller. Section VII concludes the
paper.
II. MATHEMATICAL MODEL OF THE ROBOT
Fig. 1 Single-link flexible joint robot [12]
Consider the single-link flexible joint robot shown in Fig.
1. For simplicity, the viscous damping is neglected in system
modeling. The equations of motion are [9, 12]
uqqkqJ
qqkqMglqI
=−+
=−++
)(
0)()sin(
122
2111
(1)
where, q1 is the link angular displacement, q2 is the motor
angular position, I is the link inertial, J is the rotor inertia, k is
the stiffness , M is the link mass, g is the gravity constant, and
l is the center of mass. The control u is the torque delivered by
the motor.
978-1-4244-2114-5/08/$25.00 © 2008 IEEE. 1516
Proceedings of the 7th
World Congress on Intelligent Control and Automation
June 25 - 27, 2008, Chongqing, China
In order to built the state space model, the angles and
velocities, which are assumed to be known by measurement,
are taken as state variables
;;;; 24231211 qxqxqxqx ==== (2)
Then, the system (1) is written as
u
J
xx
J
k
x
xx
xx
I
k
x
I
Mgl
x
xx
1
)(
)()sin(
314
43
3112
21
+−=
=
−−−=
=
(3)
The state space model is given under the form
uxgxfx )()( 1
+=
(4)
with
»
»
»
»
»
¼
º
«
«
«
«
«
¬
ª
=
»
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
«
¬
ª
−
−−−
=
J
xg
xx
J
k
x
xx
I
k
x
I
Mgl
x
xf
1
0
0
0
)(;
)(
)()sin(
)( 1
31
4
311
2
The output to be controlled is the link’s angle
1
)( xxhy == (5)
III. NONLINEAR PREDICTIVE CONTROL
The predictive control algorithm belongs to the optimal
control. The difference is that the cost function is defined over
a future horizon. It can be defined by the simple quadratic
form
³+=ℑ
T
dte
0
2
)(
2
1
ττ
(6)
)()()( TtyTtyTte ref +−+=+ is the tracking error at the next step
(t+T).
T is the prediction time, y(t+T) a T-step ahead prediction of
the system output and yref(t+T) the future reference trajectory.
The control weighting term is not included in the performance
index (2); rather, a weighting can be achieved by the
predictive time [2, 4].
The prediction output can be carried out from the Taylor
series expansion
)()(
!
)(
!
...)(
!2
)()()(
1
2
2
tuxhLL
r
T
xhL
r
T
xhL
T
xhTLxhTty
r
fg
r
r
f
r
ff
−
+
++++=+ (7)
r is the relative degree defined to be the number of times of
output differentiation until the control input appears.
The Lie derivative of h with respect to f, denoted Lfh, is
defined as
)()(
1
xf
x
h
xf
x
h
hL i
n
ii
f¦
=∂
∂
=
∂
∂
= (8)
Iteratively, we have
)( 1hLLhL k
ff
k
f
−
=for k=1,…,n;
and )(
1
1xg
x
hL
hLL f
fg ∂
∂
=
Lie derivatives of the link angle output are
huLLhLy
hLy
hLy
hLy
hy
fgf
f
f
f
34)4(
3
2
1
+=
=
=
=
=
(9)
where,
IJ
k
hLL
x
I
Mgl
J
k
I
k
xx
I
k
I
k
x
I
Mgl
xx
I
Mgl
hL
xx
I
k
xx
I
Mgl
hL
xx
I
k
x
I
Mgl
hL
xhL
xh
fg
f
f
f
f
=
¸
¹
·
¨
©
§++−
+
¸
¹
·
¨
©
§++=
−−−=
−−−=
=
=
3
131
2
2
21
4
4221
3
311
2
2
1
1
)cos()(
)cos()sin(
)()cos(
)()sin(
(10)
The relative degree of the system is r = 4.
Then,
)()(
!4
)(
!4
)(
!3
)(
!2
)()()(
3
4
4
4
3
3
2
2
tuxhLL
T
xhL
T
xhL
T
xhL
T
xhTLxhTty
fgf
fff
+
++++=+ (11)
Similarly, the prediction of the reference may be expanded in
Taylor series expansion
)(
!4
)(
!3
)(
!2
)()()(
)4(
4
32
ty
T
ty
T
ty
T
tyTtyTty
ref
refrefrefrefref ++++=+
(12)
1517
Then, the predicted error is given by
()
)(Y)(Y
)()()(
rtt
TtyTtyTte ref
−Τ=
+−+=+ (13)
where,
;
2462
1
432
»
¼
º
«
¬
ª
=Τ TTT
T
;
)(
0
0
0
0
)(
)(
)(
)(
)(
)(Y
34
3
2
)4( »
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
+
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
=
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
=
thuLLhL
hL
hL
hL
h
ty
ty
ty
ty
ty
t
fgf
f
f
f
T
refrefrefrefref tytytytytyt ])()()()()([)( )4(
=
r
Y
The cost function (6) is rewritten as
()()
)(Y)(Y)(Y)(Y
2
1
rr tttt T−∏−=ℑ (14)
where,
³ΤΤ=∏
T
Td
0
τ
The optimal control is obtained by putting 0=∂∂ℑ u
()
)4(4
4
13 )()( refffg yhLhLLtu −+ΚΜ−= − (15)
with
»
»
»
»
»
¼
º
«
«
«
«
«
¬
ª
−
−
−
−
=
reff
reff
reff
ref
yhL
yhL
yhL
yh
3
24
M
The elements of K are carried out from the matrix
[]
»
¼
º
«
¬
ª
=
=Κ
TTTT
kkkk
*576
2557
*168
2557
*72
2557
*60
2557
234
4321
IV. DISTURBANCE OBSERVER DESIGN
The disturbance variable d contains the uncertainties,
unmodeled quantities, and external disturbances. The equation
of the link is rewritten, with the disturbance variable, as
dqqkqMglqI =−++ )()sin( 2111
(16)
Then, the state space model (4) becomes
dxguxgxfx )()()( 21 ++=
(17)
with
T
I
xg ]00
1
0[)(
2=
The output derivatives are given in matrix form by
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
+
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
+
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
=
»
»
»
»
»
»
¼
º
«
«
«
«
«
«
¬
ª
=
hdLL
hdLL
thuLLhL
hL
hL
hL
h
ty
ty
ty
ty
ty
t
fg
fg
fg
f
f
f
f
334
3
2
)4(
2
2
1
0
0
0
)(
0
0
0
0
)(
)(
)(
)(
)(
)(Y
(18)
where,
2
1
2
2
3
3
2
)cos(
1
2
2
I
k
x
I
Mgl
g
x
hL
hLL
I
g
x
hL
hLL
f
fg
f
fg
−−=
∂
∂
=
=
∂
∂
=
Then, the optimal control (15) has the form
{
}
dhLLkdhLLyhLhLLu fgfgrefffg ˆˆ
)( 221 3
3)4(4
4
13 ++−+ΚΜ−= − (19)
where,
d
ˆis the estimated disturbance.
An initial disturbance observer is given by [3]
()
)()(
ˆˆ 12 tugxfxLdLgd −−+−=
(20)
where
L is a gain vector to be designed.
()
{
}
dhLLkdhLLyhL
hLLLgdLgxfxLd
fgfgreff
fg
ˆˆ
)(
ˆ
)(
ˆ
22
1
3
3)4(4
4
13
12
++−+ΚΜ
+−−= −
(21)
L can be chosen as
¸
¸
¹
·
¨
¨
©
§
∂
∂
+
∂
∂
=x
hL
k
x
hL
pL ff
3
3 (22)
p is a constant
The choice of the gain L will be explained in the section
about the stability analysis.
The Lie derivatives used in (21) can be defined in function of
L as
()
ykyp
t
x
x
hL
k
t
x
x
hL
pxL
hLkLf
p
f
x
hL
hL
hLLkLg
p
g
x
hL
hLL
Lg
p
g
x
hL
hLL
ff
f
f
f
fg
f
fg
f
fg
3
)4(
3
3
2
3
3
4
322
3
3
11
3
3
1
1
1
22
1
+=
¸
¸
¹
·
¨
¨
©
§
∂
∂
∂
∂
+
∂
∂
∂
∂
=
−=
∂
∂
=
−=
∂
∂
=
=
∂
∂
=
(23)
Substituting (23) in (21), and after simplification, we get
)()(
)()()(
ˆ
12
34
)4()4(
refref
refrefref
yypkyypk
yypkyypkyypd
−+−
+−+−+−=
(24)
1518
Then, by integrating (24), the equation of the disturbance
observer is given by
dtyypkyypk
yypkyypkyypd
refref
refrefref
³−+−
+−+−+−=
)()(
)()()(
ˆ
12
34
)3()3(
(25)
The observer contains an integral action, which allows the
elimination of the steady state error and enhances the
robustness of the control scheme with respect to model
uncertainties and disturbances rejection.
V. STABILITY ANALYSIS
One of the main issues in control task is to guarantee the
stability of the closed loop system under the derived optimal
control law and the disturbance observer.
Substituting the control optimal (19) into the last equation
of (18), the dynamic error of the system is given by
(
)
)()()()()()( 22 3
3
1234
4tehLLkhLLtektektektekte dfgfgyyyyy +=++++ (26)
where,
)()()( tytyte refy −= is the output tracking error.
)(
ˆ
)()( tdtdted−= is the disturbance error.
In order to guarantee the stability, the disturbance error
must tend to zero. Then, it is easily to verify that the system
dynamic (26) is Hurwitz.
Since, in general, there is no prior information about the
derivative of the disturbance d, it is reasonable to suppose that
0=d
(27)
this implies that the disturbance varies slowly relative to the
observer dynamics [4].
Then, from (17) and (20), the error dynamic of the disturbance
observer is given by
0)()( 2=+ teLgte dd
(28)
It can be shown that th e obser ver is exponentially stable by
choosing
0
2>Lg (29)
Then,
0)(lim =
∞→ te d
t
(30)
In general, it is not easy to select the nonlinear function L.
However, for a single-link flexible joint robot, with the help of
its characteristics and from its dynamic error (26), the function
L is chosen as shown above in (22), such that the observer
given by (20) is asymptotically stable.
From the definition (22), the choice of p depends of robot’s
characteristics and prediction time. It is taken with respect of
the condition
0)cos( 3
2
1
2>
¸
¹
·
¨
©
§+−− I
k
I
k
x
I
Mgl
p (31)
VI. SIMUL ATION RESULTS
In this section, simulations are conducted to test the
performance of the proposed control strategy. The robot
parameters are
M
=0.2 kg,
L
=0.02m,
I
=1.35×10-4 kg-m2,
k
=7.47 N-m/rad, and
J
=2.16×10-3 kg-m2. The controller
parameters are chosen by trial and error, with respect to
condition (31), in order to get an acceptable tracking.
First, the robot is controlled by the predictive control law
without disturbance observer. The information about the
disturbance (Fig. 2) is not included in the controller
computation. The system is run with the nominal values of the
robot parameters. The tracking performance of the angular
displacement to a smooth step reference is shown in Fig. 3.
The result shows an acceptable performance. However, a
steady error occurs with the disturbance variations. Then, the
disturbance observer is taken into account while carrying out
the control law. The tracking performance in case of a smooth
step reference and sinusoidal reference are shown in Fig. 4
and Fig. 5 respectively. Fig. 6 and Fig. 7 give the estimation
of the disturbance for both references respectively. As shown
in results, the tracking performance is achieved successfully
and the effect of disturbance is well rejected.
Fig. 2 External disturbance applied to the robot
Fig. 3 a. Angular position of the link with smooth step reference under NPC
controller without disturbance observation.
b. Tracking error.
Disturbance
Angular position [rad]
Position error [rad]
Time [s]
Time [s]
Time [s]
1519
Fig. 4 a. Angular position of the link with smooth step reference
b. Tracking error (
T
= 10-3s and
p
=10-9)
Fig. 5 a. Angular position of the link with sinusoidal reference
b. Tracking error (
T
= 10-3s and
p
=10-9)
Fig. 6 External disturbance estimation for smooth step reference
Fig. 7 External disturbance estimation for sinusoidal reference
Then, in case of mismatched model, the robot parameters
are varied 50 %. These variations are not taken into account
when the control law is carried out. The nominal values of
parameters are used in the control law computation. The same
values of prediction time and observer gain p, as above, are
used in this simulation.
The tracking performances for angular displacement with
smooth step and sinusoidal references are shown in Fig. 8 and
Fig. 9 respectively. These results show that the robustness to
parameters variations and external disturbance is successfully
achieved by this control strategy. The disturbance observer
takes care about the unmodeled quantities, uncertainties and
unknown external disturbances.
Fig. 8 a. Angular position of the link with smooth step reference in case
of mismatched model
b. Tracking error
Position error [rad] Angular position [rad]
Time [s]
Position error [rad] Angular position [rad]
Time [s]
Time [s]
Time [s]
Disturbance
Disturbance
Time [s]
Time [s]
Time [s]
Time [s]
Position error [rad] Angular position [rad]
1520
Fig. 9 a. Angular position of the link with sinusoidal reference in case of
mismatched model
b. Tracking error
V. CONCLUSIONS
In this paper, a nonlinear predictive control method with a
disturbance observer is applied to single-link flexible joint
robot. The controller is robust with respect to modeling errors,
very effective in disturbance rejection, and gives no steady
state error caused by either parameters uncertainties or
external disturbance.
The asymptotic stability of the closed loop system is
guaranteed. The controller parameters are easy to choose
under certain conditions. Results show that the single-link
flexible-joint robot under nonlinear predictive control has
good output tracking performance.
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Time [s]
Time [s]
Position error [rad] Angular position [rad]
1521
