Zhang / Front Inform Technol Electron Eng 2016 17(12):1331-1343 1333
which the parameters are self-tuned by the adap-
tive laws, is first constructed to approximate the un-
known dynamics. The SMC controller is designed
based on the DFLS. The stability of the tracking er-
ror and the reaching condition of the sliding mode are
then validated using the Lyapunov stability theory.
This paper is an extension of Zhang et al. (2015).
The contributions of the current work are listed as
follows:
1. The analysis and discussion of the disturbance
are elaborated to further clarify the significance of
the discrete model considered in this paper. The as-
sumption of the disturbance is revised, which further
relaxes the model condition and makes the method
adapt to more general applications.
2. The adaptive DFLS design is extended in de-
tail and the parameters of the forward and backward
parts are clearly designed.
3. The main results achieve more strict and rig-
orous improvements. First, in the DFLS design for
approximation, relations of the estimated function,
its optimal approximation, and the absolute error
are clearly clarified. The point is very important
for the rigorous controller design (Theorem 1). Sec-
ond, the parameter design of the controller and the
adaptive law are improved. Under the same tun-
ing condition (α
i
> 0, 0 <β
i
< 1), the controller
parameters (k
i,1
, k
i,2
) have been simplified. Again,
the chattering problem is considered with analysis
and discussion. Chattering depends on the absolute
approximation error. Fortunately, this value is very
small due to the strong approximation ability of the
DFLS. Therefore, the proposed design method has
nearly no chattering.
4. An application to a robotic arm with two
degrees of freedom is added via simulation.
2 Problem formulation
Consider a class of discrete nonlinear systems:
y
(r)
k
= f(x
k
, u
k
)+d(x
k
), (1)
where y
k
=[y
1,k
,y
2,k
,...,y
m,k
]
T
denotes the
output vector, r =[r
1
,r
2
,...,r
m
]
T
with each
element r
i
denoting a subsystem’s relative de-
gree (the total relative degree n =
m
i=1
r
i
),
and u
k
=[u
1,k
,u
2,k
,...,u
m,k
]
T
denotes the
control input vector. The system state vec-
tor is x
k
=[y
T
1,k
, y
T
2,k
,...,y
T
m,k
]
T
with y
i,k
=
[y
i,k
,y
i,k+1
,...,y
i,k+r
i
−1
]
T
for i =1, 2,...,m.Con-
sequently, y
(r)
k
=[y
(r
1
)
1,k
,y
(r
2
)
2,k
,...,y
(r
m
)
m,k
]
T
denotes its
forward difference vector with y
(r
i
)
i,k
= y
i,k+r
i
(i =
1, 2,...,m). Additionally,
f(x
k
, u
k
)=[f
1
(x
k
,u
1,k
),f
2
(x
k
,u
2,k
),
...,f
m
(x
k
,u
m,k
)]
T
is defined as the nonlinear dynamic vector, whose
components f
i
(x
k
,u
i,k
) ∈ L
2
(R)(i =1, 2,...,m)
are unknown. Moreover,
d(x
k
)=[d
1
(x
k
),d
2
(x
k
),...,d
m
(x
k
)]
T
represents the unknown disturbance or unmodeled
dynamics.
The disturbance d(x
k
) in the controlled sys-
tem (1), which has relation with only the system
state x
k
, is certainly unknown and is not required
to be bounded. It is only a type of interference to
the state equation, related with the state x
k
itself
and affecting the movement of the system state x
k
.
Therefore, if the adaptive mechanism has the abil-
ity of tracking it, this kind of effect can be treated.
In this study, the disturbance d(x
k
) is considered
as part of the unknown dynamics, which can be ap-
proximated by the DFLS online. Then the SMC
controller is not required to consider that the distur-
bance is matched or unmatched, or compensated by
the switching signal (the nonlinear part of the SMC).
The subscript i represents every branch of the
subsystem and the subscript k denotes the discrete
time instant. If
¯
y
k
=[¯y
1,k
, ¯y
2,k
,...,¯y
m,k
]
T
repre-
sents the trajectory to be tracked and comprises the
following vectors:
¯y
i,k
=[¯y
i,k
, ¯y
i,k+1
,...,¯y
i,k+r
i
−1
]
T
,i=1, 2,...,m,
the tracked mathematic model is described by
¯y
(r
i
)
i,k
= −
r
i
−1
j=0
a
j
¯y
i,k+j
+ r
i
(k), (2)
where ¯y
(r
i
)
i,k
=¯y
i,k+r
i
(i =1, 2,...,m) with a
j
(j =
0, 1,...,r
i
− 1) being Hurwitz coefficients and r
i
(k)
the reference signal. The control problem is to design
a controller for system (1) such that the tracking
error is
e
k
=[e
T
1,k
, e
T
2,k
,...,e
T
m,k
]
T
, (3)
where each element
e
i,k
= y
i,k
− ¯y
i,k
=[e
i,k
,e
i,k+1
,...,e
i,k+r
i
−1
]
T
(4)