# Systems, Man & Cybernetics - January 2016 - 28

```Z di
]] d = - R i d +
dt
Ld
[ di
q
R
]
= - L iq q
\ dt

Lq
L d ~i q +
Ld
L q ~i d +

ud
Ld
uq }m ,
Lq - Lq ~

Te = 1.5P [}i q + (L d - L q) i d i q],

(1)

objective function given by (3) has multiple local optima as
the reality PMSM is a typical dynamic nonlinear system.

(2)

Parallel Immune Cooperative
Dynamic PSO Learning Algorithm

where ~ is the electrical angular velocity, ud, uq, id, and iq
are dq-axis stator voltage and current, and P is the number
of pole pairs. R, Ld, Lq, and } are the motor winding resistance, dq-axis inductances, and magnet flux, and Te is
electromagnetic torque, respectively, while those parameters are needed to be estimated.
Optimization-Based Parameter Estimator Modeling
The parameter estimation problem can be ascribed to a
system optimization problem for a system with known
model structure but unknown parameters. The parameter
identification problem can be treated as a gray-box modelbased parameter optimization task without needing a lot
of prior knowledge of reality systems, and the issue of
parameter cross-coupling intersystems can be adaptively
decoupled by using a bioinspired optimization algorithm.
The idea is to compare the real system output with the
ideal mathematical model output through optimization
based on an objective function. The schematic can be
roughly illustrated as in Figure 1. The problem of parameter identification could be expressed as a minimum optimization problem with some restrictions. Based on (1) and
Figure 1, the objective function f (it ) is built and shown as
in (3), which follows:
n

1
f (it ) = n | w 1 u d0 (k) - ut d0 (k) + w 2 u q0 (k) - ut q0 (k)
k=1
+ w 3 u d (k) - ut d (k) + w 4 u q (k) - ut q (k) ,

(3)

where n is the length of samples, w1, w2, w3, w4 are the
weight coefficients satisfy, 0 < w i < 1 , and the variables
with '^' signify that they are calculated voltages through
the estimated parameters and measured currents. The

ud
uq
did
dt
diq
dt

= -
= -

R
Ld
R
Lq

id +
iq -

Lq
Ld
Ld
Lq

ωiq +
ωid +

f (θ^ )

ud
Ld
uq
Lq

-

ψm
Lq

ω

y^

-
+

∧
θ
Optimization-Based
Parameters Estimation
Figure 1. a parameter identification model based on

PICDPSO-m.

28

Vid (t + 1) = zVid + c 1 * rand 1 () (Pbest id (t) - X id (t))
+ c 2 * rand 2 () (gBest d (t) - X id (t)),

(4)

X id (t + 1) = X id (t) + Vid (t + 1) ,

(5)

where c 1 and c 2 are the acceleration coefficients, z is
the inertia weight, rand 1 and rand 2 are two uniformly d istr ibuted ra ndom nu mber s generated w ith i n
[0, 1], Pbest id represents the best position found by
the ith particle up to now and gBest d is the best position in the entire population, and d represents the
dimension of solved problem. The symbol t is the
iteration generation.
Principle of the PICDPSO-M Algorithm
The PICDPSO-M architecture is composed of one memory and several subpopulations, which integrate the
cooperative evolution mechanism, immune evolutionary mechanism, and swarms social behavior based on
multicore parallel framework. The model of PICDPSO-M
is as shown in Figure 2. In Figure 2, the memory conserves searched information selected from the different subpopulation, where PopM (M > 0) represents the
Mth normal subpopulation of the whole population.
The key operator of PICDPSO-M is shown in the following sections.
Dynamic Velocity Modification Strategy for PSO
In order to enhance the dynamics of the particles, a
dynamic velocity modification strategy is designed for
PSO. The details are as follows:

y

PMSM

Basic PSO Algorithm
In a d dimensional solution space, each particle i is
composed of two vectors, where the velocity vector
is Vi = {Vi1, Vi2, f, Vid} and the position vector is X i =
{X i1, X i2, f, X id} . The searching procedure can be given
as follows:

IEEE SyStEmS, man, & CybErnEtICS magazInE Janu ar y 2016

Vid (t + 1) = zVid + c 1 * rand 1 () (Pbest id (t) - X id (t))
+ c 2 * rand 2 () (gBest d (t) - X id (t))
min .
(7)
+ (x max
id - x id ) Gaussian (0, 1),
X id (t + 1) = X id (t) + Vid (t + 1),

(8)

where Gaussian (0, 1) is a random number of a Gaussian
distribution with a 0 mean and a 1 standard deviation,
which can provide momentum for particles with few
activities. The term will be activated when (growth factors) Gf i < m (m is a n adaptive threshold value as

```

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