SEKE 2012 Proceedings - Knowledge Systems Institute

scribed by the equation:
where
η il (t) =
∑N i
x ′ (t) = η il (ξ(t))(A il x(t)+B il u i (t)),
l=1
∏ n
ρ=1 μ Mρ l
∏ (t)
∑N i
n
ρ=1 μ , 0 ≤ η il ≤ 1, η il (t) =1,
Mρ l (t)
∑ Ni
l=1
l=1
and μ M l
ρ (t) denotes the membership function of the fuzzy
state variable x ρ that belongs to the fuzzy set M l ρ.
2.2 Switching Logic For σ(t)
Let X be the universe of discourse. Assume that X
is partitioned to m parts, i.e. there are X i ∈ R 1×n ,i =
1, 2,...,msuch that X = X 1 ∪ X 2 ∪···∪X m . We also
assume that X i is associated with subsystem i. The switching
is based on the Region Rule, which is defined as the
following:
[Region Rule]
If ξ 1 is Ω i1 and ξ 2 is Ω i2 and ... and ξ p is Ω ip
Then [Local Plant Rule]
where Ω ij are crisp sets, X i =Ω i1 × Ω i2 ×···×Ω ip , and
{ 1, ξj ∈ Ω
Ω ij (ξ j )=
ij ;
0, otherwise.
Thus at time t, if the state variable x has value x(t) =
(x 1 (t),x 2 (t),...,x p (t),...,x n (t)) such that x 1 (t) ∈ Ω i1
and x 2 (t) ∈ Ω i2 and ... and x p (t) ∈ Ω ip , then σ(t) =i.
If we regard that x(∈ X i ) as a state, then our switching
is actually a Finite State Machine (FSM) based switching.
2.3 A Running Example
We employ the the differential-drive two-wheeled mobile
robots (TWMR) as the example to illustrate our
method. Based on the design control, TWMR can move
on a reference trajectory. Figure 1 pictures the movement.
In the figure, variable y represents hight of the rear axle and
the variable θ specifies the angle of the robot orientation in
a reference frame. Both are the functions of time t.
In order to simply and effectively control the nonlinear
dynamics, the authors in paper[4] introduced the switched
T-S fuzzy model. Based on the values d of the premise variable,
the premise variable space are partitioned into three
regions. In each region, the local nonlinear dynamic is represented
by a T-S fuzzy model. The switched T-S fuzzy
model is described as follows:
[Region Rule 1]: If d