INTRODUCTION
AC motors are categorized into two types, synchronous motor and induction
motor. The class of synchronous motors is further comprised of PMSM and
wound synchronous motors. Since the rotor of PMSM is made of magnetic
materials, PMSM has a lower moment of inertia than an induction motor,
indicating a higher response speed under the same load torque. While a
single smaller inverter is only required under the same output power,
PMSM has an improved performance without the problem of temperature rise
and loss of rotor due to absence of excitation current. Therefore, PMSM
has been widely applied to highperformance Servo drive systems.
In recent years, Vector Control theory (Senjyu et al., 1996; Santisteban
and Stephan, 2001) is wellproven in Permanent Magnet Synchronous Motor
(PMSM), so Fuzzy ProportionalIntegral (PI) controllers are widely applied
thanks to simple construction and good performance (Cerruto et al.,
1997; Kazemian, 2005). However, PI controllers shall be adjusted according
to the running state and load torque likely varies in this context. So,
traditional PI controllers cannot achieve good output performance under
a wide range of operating conditions. To this end, an adaptive controller
enables PMSM to obtain good speed control since it can overcome the influence
of changes of motor parameters or load. In fact, Model Reference Adaptive
Control (MRAC) is a perfect adaptive control method (Islam et al.,
1990; Lin and Liaw, 1993; Astron and Wittenmark, 1995; Tsai and Tzou,
1997; Kao et al., 1997), since it can be treated as an adaptive
servo system. If the desired output performance specifications are represented
by model reference, the parameters of controller could be adjusted through
the output of model reference and output error of actual system. In such
a case, the output response of PMSM is forced to track the output response
of model reference, irrespective of the change of motor parameters or
load. Using the method of MRAC and stabilization theory of Lyapunov, an
adaptive speed controller of PMSM is designed in such a manner that actual
output response of PMSM is insensitive to the change of motor parameters
and load. So, inferior speed control of traditional PI controllers could
be improved. Generally speaking, PMSM controller has a longproven infrastructure
without complex computation, of which the SmalltheBetter (STB) output
features of PMSM include: Overshoot, Rise Time and Settling Time. In previous
designs of controllers, only individual quality characteristics were considered
without overall output planning of multiquality characteristics (Dolinar
et al., 1991; Salvatore and Stasi, 1994; Sozer et al., 1997;
Perng et al., 2000). To provide a further insight into the overall
output performance of PMSM, this study employed Desirability Function
(Wu, 2005) to integrate three quality characteristics (Overshoot, Rise
Time and Settling Time) into a single quality indicator. Then, the parameters
of adaptive speed controller of PMSM were optimized with Genetic Algorithm
(GA), thereby achieving the objective of improving speed control and robustness
design of PMSM.
MATHEMATICAL MODELS OF PMSM
The permanent magnet synchronous motor (PMSM) in this study has a permanent
magnet rotor and sinusoidal stator windings with a spacing of 120°.
Under a synchronous reference coordinate system, the voltage equation
of PMSM is expressed by Eq. 1:
Where:
R_{a}, L_{a} 
= 
Motor’s Armature Resistance and Inductance 
v_{da}, v_{qa} 
= 
Daxis and qaxis Armature Voltage 
i_{da}, i_{qa} 
= 
Daxis and qaxis Armature currents 
e_{da}, e_{qa} 
= 
Electromotive Force (EMF) daxis and qaxis Armature Coil 
P 
= 
Differential Operators (d/dt) 
ω_{re} 
= 
Rotor Electrical Radian Speed 
φ_{fa} 
= 
Field Flux Linkages 
Equation 1 is rearranged into:
It is observed from Eq. 2 that daxis and qaxis Armature
Current i_{da}, i_{qa} is controllable, but e_{qa}
= ω_{re}xφ_{fa} is uncontrollable, since e_{qa}
is EMF generated by Field Flux Linkages of Permanent Magnet. PMSM’s torque
T_{e} is:
where, p_{n} is number of pole pairs. It can be seen from above
deduction process that, after change of coordinate, PMSM allows to decouple
the electrical equation and then obtain 2axis current i_{da},
i_{qa}. Given a constant φ_{fa}, it is only required
to control armature current i_{qa} in order to control the torque
of PMSM.
It is learnt from Eq. 3 that, if the position of magnetic
pole is measured, twophase dand qaxis armature voltage v_{da},
v_{qa} can be converted into threephase output voltage (v_{ua},
v_{va}, v_{wa}) using Eq. 4.
Thus, it should be possible to control motors easily with v_{da}
and v_{qa} under d, q coordinate. It is also learnt from Eq.
1 that, there exists mutual interference of dqaxis of PMSM. For
the purpose of control, noninterference control is discussed below. Let
v_{da}, v_{qa}:
Substituting Eq. 5 into Eq. 1 to
obtain Eq. 6, with the relationship shown by Fig.
1:
It is learnt from Eq. 6 and Fig. 1
that, the current of two axes generates no interference and can be controlled
individually. Figure 2 depicts a vector control framework
of common PMSM.

Fig. 1: 
Block diagram of PMSM with noninterference control 
MODEL REFERENCE ADAPTIVE CONTROL (MRAC)
MRAC system comprises a Model Reference, a Plant, an adjustable controller
and an adjustment mechanism (adaptive control rule), with the infrastructure
shown in Fig. 3.
Assuming the devices and models are expressed separately with FirstOrder
Differential Equation:
where, x_{m} represents the output signal of Model Reference,
r represents the input reference signal and the values of a_{m}
and b_{m} are selected according to the desired output response.
In general, a_{m}>0, b_{m}>0 and a_{p}(t),
b_{p}(t) are timevarying parameters of devices.
It is clearly observed from Fig. 3 that, the MRAC system
is actually based on original plant feedback control system, along with the
Model Reference and an adjustment mechanism (adaptive control rule) for automatic
adjustment of control parameters. The parameters of Model Reference can be permanently
stored into computer memory, with the output response x_{m} (the signal
of rotational speed) expressed as the desired output performance specification.
According to the actual output of closedloop system and output error of Model
Reference, the adjustment mechanism is used to regulate the control parameters
to ensure that the output response of system can meet the performance requirements
and also make the controlled system insensitive to the changes of its parameters.
So, MRAC is a convenient control method for plant control systems with inherent
disturbance.
The parameters of PMSM are difficult to regulate for a MRAC system with
PMSM as a plant. So, an adjustable controller must be provided to ensure
that the controlled output of PMSM corresponds to the output response
of Model Reference. It is assumed that adjustable feedforward controller
g(t) and feedback controller f(t) is separately mounted into the feedforward
and feedback paths of controlled vector control PMSM system, an adjustment
system is shaped as shown in Fig. 4, wherein r is input
reference signal. The following state equation of an adjustment system
could be deducted form Fig. 4:

Fig. 3: 
MRAC system infrastructure 

Fig. 4: 
Structural diagram of firstorder adjustment system 
The purpose of MRAC is to make the adjustment system’s dynamic output
response to the reference input command r correspond to that of Model
Reference. It is found from Eq. 8 and 9
that, the gain f(t) and g(t) of controller in Eq. 9
shall be regulated to:
In other words, output x_{p} of controlled system will be the
same of output x_{m} of Model Reference. It is assumed that, due
to temperature rise and wear of plants, most of MRAC systems have a parameter
drift process, which is much slower than the response speed of Model Reference
and plant as well as the regulation process of controller. Thus, during
the regulation process of control parameters f(t) and g(t), a_{p}(t)
and b_{p}(t) are treated as constants for a simplified design.
The state equation of adjustment system can thus be rewritten from Eq.
9 into:
Let the error of output state between Model Reference and adjustment
system as e(t), namely:
By subtracting Eq. 8 from Eq. 12,
the following differential equation for error state could be obtained:
Where, let:
φ(t) and ψ(t) represent the error of Model Reference parameters
and plant parameters. Thus, Eq. 14 can be rewritten
as:
It can be seen from Eq. 17 that, parameter error φ(t),
ψ(t) and output state error e must meet the abovespecified differential
equation for error state. To make the differential equation for error
state more stable, an adaptive control rule shall be designed to regulate
the control parameters f(t) and g(t), such that the output of controlled
system could satisfy the output performance of model reference. In this
paper, Lyapunov stabilization theory is used to ensure the stability of
control system. And adaptive control rule is planned using Lyapunov stabilization
theory (Åstron and Wittenmark, 1995). Three error variables, i.e.,
e, φ and ψ are contained in the abovespecified differential
equation for error state. So, the stability analysis of entire error system
shall cover e, φ and ψ. According to Lyapunov stabilization
theory, the following secondary Lyapunov function is selected:
where, λ_{1}>0, λ_{2}>0. As the parameter
of controlled system b_{p}>0, V(e,φ,ψ)>0 if e,
φ and ψ are not equal to 0. Then, the derived function of time
can be obtained from Eq. 18:
If substituting Eq. 14 into Eq. 19:
Let:
Equation 22 is then simplified as:
.
Thus, it shall be possible to maintain a stable control system by following
the adaptive control rule in Eq. 21 and 22
and Lyapunov stabilization theory. Since control gains f(t) and g(t),
not φ(t) and ψ(t), are adjustable parameters in actual MRAC
system, the adaptive control rule of f(t) and g(t) could be derived from
Eq. 15, 16, 21
and 22:
Thus, a stable system could be guaranteed if the adaptive control rule
in Eq. 24 and 25 is observed based
on Lyapunov stabilization theory, namely, consistent output of the system
and Model Reference could be achieved if t →∞ and e(t)→0.
According to the adaptive control rule in Eq. 24 and
25, the infrastructure of MRAC system for vector control
PMSM is shown in Fig. 5.

Fig. 5: 
Infrastructure of MRAC for PMSM 
DESIRABILITY FUNCTION
As mentioned earlier, the output response of PMSM shall comprise Maximum
overshoot M_{o}, rise time R_{t} and settling time S_{t},
of which M_{o} is often used to measure the relative stability
of control system and represented by the percentage of specific order
response value:
where, y_{max }is maximum specific order response value, y_{ss}
is stable value of specific order response. In addition, R_{t}
is generally the period when the specific order response value increases
from 1090%. The performance characteristic index S_{t} is maintained
within the 2% when the specific order response is reduced. It is clearly
seen that the smaller performance response characteristics represent more
stable output quality of PMSM. In other words, the quality of PMSM is
characterized by SmalltheBetter of multiqualitye. In previous designs
of controllers, only individual quality characteristics were compared
without consideration of interaction amongst control parameters and Tradeoff
of multiple quality characteristics. So, the relationship between control
parameters and overall multiquality characteristics output is difficult
to measure. In 1965, Harrington developed a mathematic conversion method
with desirability function, which was used to resolve the multiquality
problem by converting it into single response. The Individual Desirability
d was defined using Eq. 27:
where, d represents desirability function, z_{i} represents
ith actual response, USL and LSL represents separately upper and lower
limit of quality specifications and n is called Deviation Importance.
Then Overall Desirability (D) is obtained from average of individual desirability,
thus converting optimum parameter into maximum overall desirability. According
to desirability function concept designed by Harrington, the response
value shifts within the tolerance far away from Midpoint of specification;
this design, however, is not suitable for practical applications. For
this reason, Wu (2005) made a rectification, with the SmallertheBetter
desirability function as:
where, T represents the target value of quality characteristic, n is
set depending upon the characteristics of individual response. A bigger
value means a quicker slowdown of desirability when it is far away from
T. To obtain optimum output response of PMSM, the quality response output
d_{slt}(s,l,t = 1, 2, 3) of PMSM is given with different
load conditions: without load, semiload (1.2 Ntm) and 3/4 load (1.8
Ntm) under low, middle and high speed (300, 900 and 1800 rpm). Twenty
seven quality response items (d_{111} − d_{333}),
target (T) of quality response, upper limit (USL) and desirability d_{slt}(z)
are listed in Table 1. In this study, let n = 1, T =
0, USL is set properly as the upper limit according to speed and load
conditions. USL is set as 0.4 sec under low speed and reduced to 0.25
sec under middle and high speed. The desirability of quality response
0≤d_{slt}(z)≤1 is shown in Table
1 and d_{slt}(z) = d_{slt}(T) = d_{slt}(z)_{max}
= 1 in the case of z = t, or otherwise z = USL d_{slt}(z)
= d_{slt}(z)_{min} = 0. In addition, d_{sl2}
= 0.1 for R_{t} in Table 1. All d_{slt}
are converted by Eq. 29 into singleresponse overall
desirability D, 0≤D≤1. A bigger D indicates better speed
response and robustness of PMSM.
Table 1: 
Desirability function of quality characteristics under different
speed/load conditions 

PARAMETERIZATION AND SIMULATION
Parameter control in MRAC is a crucial factor to design the robust response
of PMSM. The parameters of PMSM are shown in Table 2.
Based upon MRAC of PMSM as discussed in section 3, parameters λ_{1}
and λ_{2} in Eq. 24 and 25
are taken as control variables. And overall quality desirability D in
Eq. 29 is taken as an output variable. Finally, GA
is used to find optimum parameters λ_{1}, λ_{2}
and desirability D,enabling to design an optimal MRAC parameter control
model. This helps to improve the robustness of PMSM under different speed
and load conditions, in addition to the consistency of output between
PMSM and Model Reference.
Table 2: 
Parameters of PMSM 

Genetic Algorithms (GA), proposed by Holland (1992) has a solution process
similar to biological evolution. That’s to say, the survival pressure
and threats represent the adaptive function of target issues, which could
guide the searching for a solution. After continuous evolution, a highly
adaptive value could be maintained as the solution. Every chromosome in
GA represents a possible solution. Firstly, possible solutions of fixed
quantity may be generated during searching process, of which every possible
solution is an individual for the solution space. Genes are generally
represented by binary strings. The evolution process of population within
the solution space is finished through Selection, Crossover and Mutation.
GA evaluates each of those solutions and decides on a fitness level for
each solution set (Goldberg, 1989; Holland, 1992). In GA optimization,
the factors involved are the size of initial population, the crossover
probability and the mutation probability. The optimal plasma cleaning
parameter design through the procedures of executing GA is simplified
as follows.
Step 1: 
Encoding:GA permits encoding of
the variables for the searching space. The MRAC parameters: λ_{1}
and λ_{2} will vary within a known range and encoding
as a binary string. We use 10 bit encoding which resulted in a 20
bit chromosome for each parameter. Figure 6 show
the encoding and mapping of two parameters. 
Step 2: 
Creation of a random initial population:Set
the GA’s operating conditions: the generation size was set to 500,
the size of the initial population was set to 100, the crossover and
mutation probabilities were set to 0.75 and 0.03, respectively. 
Step 3: 
Scoring each member of the current population by computing the individual’s
fitness value. The GA algorithm meeting a given fitness function is
expressed as: 
Step 4: 
Selection of members: The members
of the new population are selected based on their fitness. Elite children
are the individuals in the current generation with lower fitness values.
Roulette wheel selection is employed in this algorithm. These individuals
automatically survive to the next generation. 
Table 3: 
Desirability value searched by GA 


Fig. 6: 
Encoding λ_{1} and λ_{2} using a 20 bit
string 

Fig. 7: 
Speed simulation response diagram of unloading motor under speed
command of 300 rpm 
Step 5: 
Crossover and mutation: Production of children
from their parents. Dependent on the crossover rate, crossover of
the bits from each chosen chromosome occurs at a random position,
where there is an interchange between the two parts. Proceed through
the chosen chromosome bits and flip them in dependence to the mutation
rate. Replace the current population with the children, to form the
next generation. 
Step 6: 
Step 3, 4, 5 are repeated until a stopping criteria is met. 
After 30 calculations with GA, the maximum desirability is 0.99399, as
shown in Table 3. The optimum parameter is: λ_{1}
= 2 and λ_{2} = 2, D = 0.979. To validate the feasibility
of the aforementioned MRAC method, Matlab/Simalink software was used according
to the control structure in Fig. 6. Optimum parameters
λ_{1}, λ_{2} obtained from GA were taken as
control objects and PCbased simulation was made when input command r
is 300, 900 and 1800 rpm, with the results shown in Fig.
715.

Fig. 8: 
Speed simulation response diagram of unloading motor under speed
command of 900 rpm 

Fig. 9: 
Speed simulation response diagram of unloading motor under speed
command of 1800 rpm 
While GA method is used to find optimum MRAC parameters λ_{1},
λ_{2} and desirability value D of PMSM, an optimal MRAC parameter
control model is designed. The response simulation from Fig.
715 shows that, under any speedload conditions, PMSM is proved to
have stronger robustness in addition to the consistency of output between
PMSM system and Model Reference.

Fig. 10: 
Speed simulation response diagram of semiload (1.2 Nt m) motor
under speed command of 300 rpm 

Fig. 11: 
Speed simulation response diagram of semiload (1.2 Nt m) motor
under speed command of 900 rpm 

Fig. 12: 
Speed simulation response diagram of semiload (1.2 Nt m) motor
under speed command of 1800 rpm 

Fig. 13: 
Speed simulation response diagram of 3/4 load (1.8 Nt m) motor under
speed command of 300 rpm 

Fig. 14: 
Speed simulation response diagram of 3/4 load (1.8 Nt m) motor under
speed command of 900 rpm 

Fig. 15: 
Speed simulation response diagram of 3/4 load (1.8 Nt m) motor under
speed command of 1800 rpm 
EXPERIMENTAL SYSTEMS AND RESULTS
Introduction to infrastructure of experimental system: Signals
were previously processed through analog devices. With recent rapid development
of digital systems, there is a growing trend of signal processing with
much quicker computers or specialpurpose digital processors. In such
cases, this paper enables PCbased control of PMSM in order to obtain
more stable operating performance within a shorter sampling period. As
shown in Fig. 16, the control system of PMSM mainly
comprises: (1) PC processor; (2) motor control interface card; (3) inverter
power drive circuit.
Experimental results: To validate the feasibility of the aforementioned
MRAC method, a system integrating hardware and software is developed through
PCbased motor controller. And, optimum parameters λ_{1},
λ_{2} obtained from GA are taken as control objects. The
field test is made when input command r is 300, 900 and 1800 rpm, with
the simulation and actual response results compared in Fig. 1725.

Fig. 16: 
PMSM drive control system 

Fig. 17: 
Comparison diagram of speed simulation and actual response for unloading
motor under speed command of 300 rpm 
While GA is used to search optimum MRAC parameters λ_{1},
λ_{2} of PMSM, the actual response diagram and simulation
response diagrams are compared and analyzed. Figure 17,
20 and 23 show the comparison of
simulation and test results in the case of lowspeed unloading, semiload
(1.2 N m) and 3/4 load (1.8 Nm). But, overshoot and delay are observed
due to the following reasons:
• 
Floatingpoint calculating ability and control program of PC are
impossible to realize optimum calculation time. 
• 
Initial angle of rotor may affect seriously the performance of PMSM
drive system, so high priority of concern shall be paid to calibration
of the rotor’s initial angle. 

Fig. 18: 
Comparison diagram of speed simulation and actual response for unloading
motor under speed command of 900 rpm 

Fig. 19: 
Comparison diagram of speed simulation and actual response for unloading
motor under speed command of 1800 rpm 

Fig. 20: 
Comparison diagram of speed simulation and actual response for semiload
(1.2 Ntm) motor under speed command of 300 rpm 

Fig. 21: 
Comparison diagram of speed simulation and actual response for semiload
(1.2 Ntm) motor under speed command of 900 rpm 

Fig. 22: 
Comparison diagram of speed simulation and actual response for semiload
(1.2 Ntm) motor under speed command of 1800 rpm 

Fig. 23: 
Comparison diagram of speed simulation and actual response for 3/4
load (1.8 Ntm) motor under speed command of 300 rpm 

Fig. 24: 
Comparison diagram of speed simulation and actual response for 3/4
load (1.8 Ntm) motor under speed command of 900 rpm 

Fig. 25: 
Comparison diagram of speed simulation and actual response for 3/4
load (1.8 Ntm) motor under speed command of 1800 rpm 
But, the phenomenon in low speed occurred in Fig. 18,
19, 21, 22, 24
and 25 have been improved considerably in middle speed
(900 rpm), high speed (1800 rpm) and different loads. It is proved that
MRAC in this paper presents excellent speed control and robustness against
PMSM parameter or load change. The experimental results in low speed are
acceptable despite of limited deviation from the ideal simulation results.
To improve overshoot and delay during lowspeed operation, one solution
is to improve the digital signal processor in cooperation with calibration
of the initial angle.
CONCLUSIONS
This study has developed a design method for adaptive speed controller
(MRAC) of PMSM. And, an adaptive control rule is designed with Lyapunov
stabilization theory such that the control parameters could be regulated
properly to be suitable for a PMSM speed control system according to the
output state error of the system and Model Reference. By taking overall
output quality of PMSM into account based on the Desirability Function,
it is possible to design optimum MRAC parameters with GA, thereby serving
the purpose of improved speed control and robustness design of PMSM. A
PMSM speed control system with hardware and software can be developed
through PCBased motor controller. The PCbased simulation and experimental
results show that, MRAC in this study presents excellent speed control
and robustness against the PMSM parameter or load change.
On a whole, this study takes the overall output quality of multiresponse
characteristic of controller into consideration while designing synchronous
motor and integrates the multiresponse characteristic through desirability
function to achieve the single quality characteristic index. Then it uses
GA to design the optimal parameter standards for λ_{1} and
λ_{2} in order to enhance the stability and output quality
of the speed control of synchronous motor. The optimal parameters λ_{1}
and λ_{2} obtained from GA are applied in the simulation
result control comparison and actual testing of the adaptive speed control
of PMSM reference model and conventional PI controller. The results indicate
that the adaptive speed control of parameter design model of multiresponse
characteristic proposed in this study could increase the tracing and regulating
abilities of high, medium and low rational speeds of the motor. In other
words, the results can contribute to the parameters or load interference
of synchronous motor. Therefore, the adaptive control strategy proposed
herein can generate better speed response result.