Page 15 - IJOCTA-15-3

Improving the performance of a chaotic nonlinear system of fractional-order...
part of the control system satisfies the sliding The equivalent control law u eq (t) is obtained
mode condition because: as follows:

˙
S i S i = −α i S i (x) Sign(S i (x)) < 0, S i (x) ̸= 0 u eq (t) = − (−x 2 − x 1 x 3 + γx 3 ) (30)
(23)
Now, considering the Lyapunov candidate
(b) Linear continuous feedback:
function as a positive definite function, we have
(x) = −β i S i (x) where (β i >
By choosing u i N
the following:
0) as the discontinuous function, the sufficient
condition for sliding mode according to the fol-
1
2
lowing equation is satisfied. In more general cases, V = S > 0 (31)
2
the discontinuous part of the control system can
be considered as a combination of the first and For asymptotic stability, the derivative of this
second cases. function must be negative definite:
˙
˙
˙
2
S i S i = −β i S (x) < 0 (24) V = SS < 0 (32)
i
This is a proposed candidate for the switching
(c) Combination of the relay with a constant
control law to satisfy Equation (32), and based on
gain and linear continuous feedback
Equation (23), it is given by:
In general, the discontinuous control compo-
nent can be considered as a combination of the
first and second modes: u N = −β Sign(S(x)) − α S(x) (33)
where α and β are positive real values. By com-
(
−α i Sign(S i (x)) − β i S i (x), S i (x) ̸= 0, α i > 0, β i > 0
= bining Equations (30) and (33) and substituting
u i N
0, S i (x) = 0
into Equation (20), the general sliding mode con-
(25)
trol law for regulating the state variables of a typ-
With this choice, the sufficient condition for
ical BLDC system is obtained:
the discontinuous part of the control is satisfied
because
u(t) = x 2 + x 1 x 3 − γx 3 − β Sign(S(x)) − α S(x)
(34)
˙
2
S i S i = −α i S i (x) Sign(S i (x)) − β i S (x) < 0,
i
S i (x) ̸= 0 4. Simulation and comparative analysis
(26)
4.1. Numerical implementation of the
Now, we denote the location of the control in-
proposed controller
put in the fractional-order chaotic system:
Considering the parameter vector in the form of
 homogeneous orders (µ, γ, σ) = (1, 20, 5.46),
q 1
t
 0 D x 1 = −x 1 + x 2 x 3 we model the system as q = q = q = 0.995
q 2
D x 2 = −x 2 − x 1 x 3 + γx 3 + u(t) (27) 1 2 3
0 t
q 3
 D x 3 = −σ(x 3 − x 2 ) and T sim = 50 sec. We set the time
0 t
constant h = 0.005 and initial conditions as
which u(t) is the same as the combined control in- (x 1 (0) , x 2 (0) ,x 3 (0)) = (5, 5, 5). With the
put. Since the sliding surface equation is defined switching gain k = 5 and based on the sliding
based on the state variables according to Equation mode control input, we design a relation (32) and
(19), the system converges to the desired state on implement it in MATLAB software. By plotting
the sliding surface. the phase trajectories ( ˙ x − x) for the reference
The following equations hold on the sliding currents (d − q) of the electric motor and the ro-
surface: tor angular speed (ω) in both uncontrolled and
controlled states, we illustrate the system behav-

S = 0 ior in a single diagram. Similarly, we plot the
˙
S = 0 (28) previously separately drawn parametric diagrams
for uncontrolled and controlled states. 28
To obtain the equivalent control law, it suf- Figure 7 shows that the state variables and
˙
fices to substitute S = 0:
their rates of change have converged well to the
zero set point. This indicates that the FO-SMC
˙
q 2
S = 0 D x 2 = −x 2 − x 1 x 3 + γx 3 + u eq (t) = 0 has effectively regulated the state variables of the
t
(29) BLDC system to the desired zero set point.
387