A Novel Second Order Sliding Mode Control Algorithm for Velocity Control Permanent Magnet Synchronous Motor - Intelligent Computing for Sustainable Energy and Environment - page 216

Because the error equation(4)is first order corresponding to the state equation(2), the
chattering phenomenon is eliminated by the second order sliding mode control
[15]-[16].
In order to achieve the good dynamic performance and improve the speed tracking
precision, an integral manifold is designed as follows:
S = k 1 e ω + k 2 e ω dt

(5)

where k 1 and k 2 is the proportion gain and the integral gain, and k 1 > 0, k 2 > 0.
Theorem1: The speed error system 3 can converge to zero in finite time, while the
integral manifold is chosen as 5, and the control law is designed as follows:

i q = i eq + i sw (6)

with

1
k 1

+ k 1 ˙

i eq = ( k 2 e ω + k 1 βω ω ∗ ) (7)

α

1
k 1 α

i eq = ( k 2 e ω + k 1 βω + k 1 ˙ ω ∗ ) (8)

where λ 1 , λ 2 , γ 1 and γ 2 are designed parameters, and λ 1 > 0, λ 2 = λ 21 + λ 22 , λ 21 >

k 1 δ γ 2 > 0.

Proof: When the Lyapunov function V < 0, the sliding mode control exists and the error
system can converge. So the Lyapunov function can be defined as [17]:

and λ 22 > 0, γ 1 > 0,

V = 1

2 γ 1 S T S + 1

S T S +

λ 1 S (9)

2

Take the time derivative of (9), and obtain:

V = S T (

γ 1 S + S + λ 1 sgn( S )) (10)

According to (5), the S and S are rewritten as below:

S = k 1 e ω + k 2 e ω

(11)

+ ˙

S = k 1 ( ¨ i q +

ω ∗− α β ω ˙ δ )+ k 2 e ω (12)

The following equations is obtained from the (6), (7), (8) and (12):

ω + ˙

S = k 1 ( ¨

ω ∗− α( i eq + i sw )+β ˙ δ)+ k 2 e ω (13)

1
k 1 α

i eq =

˙

+ k 1 ¨ ω ∗ ( k 2 e ω + k 1 β ω ) (14)

1
k 1 α

λ 2 sign ( S )+( γ 2 S )) i sw = ( λ 1 sign ( S )+ γ 1 S + (15)

Hence:

− γ 2 S + k 1 ˙

S =

− λ 2 sgn( S ) − λ 1 sgn( S ) − γ 1 S δ (16)