or transfer function 2. However, the same procedure is used for parameter identification of
the d axis or transfer function 3.
4. RMRAC gains adaptation algorithm
The gradient algorithm used to obtain the control law gains is given by
˙ θ = −σP θ − P ξ ε,
(7)
m 2
with
˙
m = δ 0 m + δ 1 up + yp + 1 , m(0) > δ 1
δ , δ 1 ≥ 1,
(8)
0
and
ξ = Wm( s)I w,
(9)
w = [ w 1 w 2 yp up] ,
(10)
w 1 , w 2 are auxiliary vectors, δ 0, δ 1 are positive constants and δ 0 satisfies δ 0 + δ 2 ≤ min( p 0, q 0), q 0 ∈ + is such that the Wm( s − q 0) poles and the (F − q 0I) eigenvalues are stable and δ 2 is a positive constant. The sigma modification σ in 7 is given by
⎧
⎪
⎪
⎨
0
if
θ < M 0
θ
σ =
−
⎪ σ
1
if M
,
(11)
⎪ 0
0 ≤
θ < 2 M 0
⎩
M 0
σ 0
if
θ ≥ 2 M 0
where M 0 > θ∗
and σ 0 > 2 μ− 2/ R 2, R, μ ∈
+ are design parameters. In this case, the
parameters used in the implementation of the gradient algorithm are
⎧
⎪
⎪ δ
⎪ 0 = 0.7
⎪
⎨ δ 1 = 1
δ
⎪
,
(12)
⎪ 2 = 1
⎪
⎪
⎩ σ 0 = 0.1
M 0 = 10
More details of design of the gradient algorithm can be seen in (Ioannou & Tsakalis, 1986). As
defined in (Lozano-Leal et al., 1990), the modified error in 7 is given by
ε = e 1 + θ ξ − Wmθ w,
(13)
or
ε = φ ξ + μη.
(14)
When the ideal values of gains are identified and the plant model is well known, the plant can
be obtained by equation analysis of MRC algorithm described in the next section.
150
Electric Machines and Drives
r
1
+
up
y
å
G s
p
p ( )
θ 4
+ + + θ T a( s)
1
θ Λ( s)
4
θ T a( s)
2
θ Λ( s)
4
θ 3
θ 4
Fig. 4. MRC structure.
5. MRC analysis
The Model Reference Control (MRC) shown in Fig. 4 can be understood as a particular case
of RMRAC structure, which is presented in Fig. 3. This occurs after the convergence of the
controller gains when the gradient algorithm changes to the steady-state. It is important to
note that this analysis is only valid when the plant model is well known and free of unmodeled
dynamics and parametric variations.
To allow the analysis of the MRC structure, the plant and reference model must satisfy some
assumptions as verified in (Ioannou & Sun, 1996). These suppositions, which are also valid for
RMRAC, are given as follow:
Plant Assumptions:
P1. Zp( s) is a monic Hurwitz polynomial of degree mp;
P2. An upper bound n of the degree np of Rp( s);
P3. The relative degree n∗ = np − mp of Gp( s);
P4. The signal of the high frequency gain kp is known.
Reference Model Assumptions:
M1. Zm( s), Rm( s) are monic Hurwitz polynomials of degree qm, pm, respectively, where pm ≤ n;
M2. The relative degree n∗m = pm − qm of Wm( s) is the same as that of Gp( s), i.e., n∗m = n∗.
In Fig. 4 the feedback control law is
θ α ( s)
θ α ( s)
u
1
2
p = θ
yp + 1 r,
(15)
4 Λ ( s) up + θ 4 Λ ( s) yp + θ 3
θ 4
θ 4
and
α ( s) Δ
= αn− 2 ( s) = sn− 2, sn− 3, . . . , s, 1
for n ≥ 2,
(16)
α ( s) Δ
= 0
for n = 1,
θ 3, θ 4 ∈ 1; θ , θ ∈ n− 1 are constant parameters to be designed and Λ( s) is an arbitrary 1
2
monic Hurwitz polynomial of degree n − 1 that contains Zm( s) as a factor, i.e.,
Λ ( s) = Λ0 ( s) Zm ( s) ,
(17)
A RMRAC Parameter Identification Algorithm Applied to Induction Machines
151
which implies that Λ0( s) is monic, Hurwitz and of degree n 0 = n − 1 − qm. The controller parameter vector
θ = θ θ θ θ
∈ 2 n,
(18)
1
2
3
4
is given so that the closed loop plant from r to yp is equal to Wm( s). The I/O properties of the closed-loop plant shown in Fig. 4 are described by the transfer function equation
yp = Gc( s) r,
(19)
where
Gc( s) =
kpZpΛ2
,
(20)
Λ θ Λ − θ α R
α + θ Λ
4
1
p − kp Zp θ 2
3
Now, the objective is to choose the controller gains so that the poles are stable and the
closed-loop transfer function Gc( s) = Wm( s), i.e.,
kpZpΛ2
=
Z
k
m .
(21)
Λ θ Λ − θ α
m
R
α + θ Λ
R
4
1
p − kp Zp θ 2
3
m
Thus, considering a system free of unmodeled dynamics, the plant coefficients can be known
by the MRC structure, i.e., kp, Zp( s) and Rp( s) are given by 21 when the controller gains θ 1 , θ 2 , θ 3 and θ 4 are known and Wm( s) is previously defined.
6. Parameter identification using RMRAC
The proposed parameter estimation method is executed in three steps, described as follows:
6.1 First step: Convergence of controller gains vector
The proposed parameter identification method is shown in Fig. 2. In this figure the parameter
identification of q axis is shown, but the same procedure is performed for parameter
identification of d axis, one procedure at a time.
A Persistent Excitant (PE) reference current i∗sq is applied at q axis of SPIM at standstill rotor.
The current isq is measured and controlled by the RMRAC structure while isd stays at null value. The controller structure is detailed in Fig. 3. When e 1 goes to zero, the controller gains go to an ideal value. Subsequently, the gradient algorithm is put in steady-state and the system
looks like the MRC structure given by Fig. 4. Therefore, the transfer function coefficients can
be found using equation 21.
6.2 Second step: Estimation of kpi, h0 i, a1 i and a0 i
This step consists of the determination of the Linear-Time-Invariant (LTI) model of the
induction motor. The machine is at standstill and the transfer functions given in 2 and 3 can
be generalized as follows
i
Z
si =
pi ( s)
s + h
k
= k
0 i
,
(22)
v
pi R
pi s 2 + sa
si
pi ( s)
1 i + a 0 i
where
kpi = Lri , h
, a
,
(23)
¯ σ
0 i = Rri
1 i = pi and a 0 i = RsiRri
i
Lri
¯ σi
152
Electric Machines and Drives
The reference model given by 5 is rewritten as
Z
s + z
W
m
0
m ( s) = km
= k
,
(24)
R
m
m
s 2 + p 1 s + p 0
and from the plant and reference model assumptions results
mp = 1, np = 2, n∗ = 1,
(25)
qm = 1, pm = 2, n∗m = 1,
The upper bound n is chosen equal to np because the plant model is considered well known and with n = np only one solution is guaranteed for the controller gains. Thus, the filters are given by
Λ ( s) = Zm ( s) = s + z 0,
α (
(26)
s) = z 0,
Assuming the complete convergence of controller gains, the plant coefficients are obtained
combining the equations 22, 24 and 26 in 21 and are given by
⎧
⎪
⎪
⎪ k
⎪ pi = kmθ 4 i,
⎪
⎪
⎪
⎨ h 0 i = z 0 θ
θ
4 i − θ
,
1 i
4i
⎪
(27)
⎪
⎪
⎪ a
⎪ 1 i = p 1 + kmθ 3 i,
⎪
⎪
⎩ a 0 i = p 0 + kmz 0 θ + θ
2 i
3 i
.
6.3 Third step: Rsi, Rri, Lsi, Lri and Lmi calculation Combining the equations 4, 23 and using the values obtained in 27 after the convergence of
the controller gains, we obtain the parameters of the induction motor:
⎧
⎪
⎪
⎪ ˆ
⎪ R
,
⎪ si = a 0 i
⎪
k
⎪
pih 0 i
⎪
⎪
⎪
⎪
⎪ ˆ
− ˆ
⎪ R
R
⎨ ri = a 1 i
k
si,
pi
⎪
ˆ
(28)
⎪
R
⎪
ri
⎪ ˆ L
,
⎪ si = ˆ Lri =
⎪
⎪
h 0 i
⎪
⎪
⎪
⎪
⎪
ˆ ˆ
⎪
R R
⎩ ˆ L
si ri
mi =
ˆ L 2 −
.
si
a 0 i
In the numerical solution it-is considered that stator and rotor inductances have the same
values in each winding.
7. Simulation results
Simulations have been performed to evaluate the proposed method. The machine model given
by 1 was discretized by Euller technique under frequency of fs = 5 kHz. The SPIM was
performed with a square wave reference of current and standstill rotor. The SPIM used is
a four-pole, 368 W, 1610 rpm, 220 V/3.4 A. The parameters of this motor obtained from classical no-load and locked rotor tests are given in Table 1.
