An Improved Adaptive Tracking Controller of Permanent Magnet Synchronous Motor

and Applied Analysis 3


Introduction
Permanent magnet synchronous motors have had a great deal of attention for industrial applications in the recent years [1][2][3][4][5][6].Due to the compact size, high speed, high efficiency, high power density, and low inertia, PMSMs are widely used in many fields of industry.However, the control and the stabilization for PMSMs still have some challenges as their high nonlinearity and even chaotic behavior.
In the field of power electronics, chaos in motor drive systems occurs when parameters fall into a certain area as defined by Kuroe and Hayashi [7].Then, the existence of chaos has been found in several types of motor driver systems [7][8][9].Chaotic phenomenon in PMSMs is completely studied by Li et al. [10], and this study points out that the chaotic oscillations appear when the parameters lie in a certain area.Since the undesired chaotic oscillations can affect the stabilization of a motor drive system negatively, causing the drive system's collapse, it has been more and more critical to control and eliminate the chaos.Until now, despite various chaos control methods developed for PMSMs such as feedback linearization [11], sliding mode control [12], quasisliding mode control [13,14], adaptive backstepping control [15], and time-delay feedback control [16], there are some shortcomings which still exist.Most of the methods require the exact mathematical models to calculate the control laws.This leads to implemental difficulties in practical systems where mathematical models can be dynamic due to undesired uncertainties.Moreover, in the adaptive backstepping method, a single system parameter is considered as an unknown and constant parameter, which can be seen as a restriction.The time-delay feedback control encounters some problems; for example, the control object is not an equilibrium or unstable periodic orbit.Furthermore, determination of delay time for this controller is quite difficult.
In the past two decades, neural networks (NNs) and fuzzy logic (FL) have been widely used to model and control highly uncertain, nonlinear, and complex systems [17][18][19][20][21][22].Incorporating the estimation abilities of NNs (or FL) into adaptive control method, direct and indirect adaptive control methods were developed [23].In the indirect adaptive control method, the control input () usually appears in the form of with unknown model of PMSM, but some disadvantages still remain.The self-organizing fuzzy sliding mode control needs much knowledge of specialist to set up the fuzzy rules.The fuzzy guaranteed cost control may have the awful performance when the reference point is not the origin.In the adaptive fuzzy control method via backstepping, the tracking error is impossible to converge to zero.Therefore, in comparison with previous methods, the proposed control shows the improvements in controlling chaotic PMSM.With the proposed controller, chaotic oscillations in a PMSM are successfully suppressed and the motor speed is forced to follow the desired trajectory and the tracking error converges to zero asymptotically.Finally, the simulations are carried out to illustrate the effectiveness and robustness of the proposed controller.
The rest of this paper is organized as follows.The dynamics of a PMSM and the formulation of the chaos control problem are outlined in Section 2. The design of the adaptive controller based on a fuzzy neural network is described in Section 3. In Section 4, simulation results are given to confirm the validity of the proposed method.Finally, the conclusion is offered in Section 5.

Dynamic Model of Chaotic PMSM.
The dynamic model of a PMSM with the smooth air gap can be described as follows [10]: where ,   , and   are state variables, which denote angle speed and - axis currents, respectively.The state  can be directly measured, while states   and   can be calculated by using - transformation. and  are system parameters.T , ũ , and ũ stand for the load torque and - axis voltages, respectively.
In system (1), when the external inputs are set to zero, namely, T = ũ = ũ = 0, the system becomes an unforced system [10] as The theories of bifurcation and chaos have been widely used to study the stability of PMSM drive systems in [10].The study showed that a PMSM produces chaotic oscillations when system parameters  and  fall into a certain area.For example, the system in (2) displays chaos when the system parameters are set as  = 5.45 and  = 20, and the initial states are given as [ (0)   (0)   (0)]  = [1 − 1 0]  .The typical chaotic attractor of a PMSM is exhibited in Figure 1 and the bifurcation diagrams of the quadrature current   versus the parameters,  and , are presented in Figure 2, respectively.

Problem Formulation.
Since the chaotic oscillations can destroy the stability of the PMSM drive system, we propose an adaptive controller, a PMSM, to suppress chaos and achieve the speed tracking control.Let us consider the PMSM drive system in (2).We add a control input  to the second differential equation as the manipulated variable, which is desirable for real applications.And for simplicity, the following notations are introduced as  1 = ,  2 =   , and  3 =   .In this manner, the system in (2) with uncertainties can be rewritten as follows: where   (, ) ∈ ,  = 1, 2, 3, is uncertainty applied to the PMSM due to parameter perturbation and external uncertainties. and  are unknown system parameters and are also located within the chaotic area.
Assumption 1.The uncertainty,   (, ) ∈ ,  = 1, 2, 3, is bounded.In order to force the speed  1 of PMSM to follow the desired trajectory   () ∈ , the system in (3) is expressed in the standard form of the single input, single output (SISO) system with output () =  1 as where With the control signal  inserted into the system above, the SISO system in (4) has the relative degree  = 2.By using Lie derivatives, we take the transformation as which leads to where The control goal is to design a controller that can suppress chaos and allow the output () ∈  to track the desired trajectory   () ∈ .Based on linearization feedback control method [33], the ideal control law  * () is given to reach the control goal as where V() ∈  is the linearization input and can be computed as where  is a positive factor.  () and   () can be calculated according to the following equations: where  0 () is the tracking error and  is chosen to ensure that Δ() =  +  is a Hurwitz polynomial.
In order to make (9) proper and use () to determine the property of the Lyapunov function candidate, the following assumptions are needed.

Assumption 2. 𝑏(𝑥) is lower bounded by a known positive constant
Assumption 3. Desired trajectory   () is continuously differentiable and bounded up to the second-order.ẏ  () and ÿ  () are measurable.Substituting ( 9) into (7), one can get Using ( 14) and ( 11) leads to The error dynamics can be obtained by substituting ( 13) into (15) as Since  is assumed to be positive, as mentioned in (10), and  satisfies the Hurwitz polynomial Δ() =  + , the equation in (16) expresses that both   () and therefore  0 () converge to zero exponentially.For this reason, the controlled system is stable and the perfect tracking is achieved.
Moreover, due to the relative degree  = 2 and the order of the system  = 3, the zero dynamics is considered.It is possible to find a function ℎ 3 () such that (ℎ 3 ()/)() = 0, and then we define the state  3 = ℎ 3 () =  3 to obtain the additional state equation as When  1 =  2 = 0, (17) can be rewritten as ż 3 = − 3 +  3 .This equation expresses the stable zero dynamics of the system.Because the zero dynamics is stable, the system is a minimum phase system.Thus, the state variable  3 is also stable when both state variables  1 and  2 are stable.
However, since , , and   ,  = 1, 2, 3, are unknown, () and () cannot be determined exactly.This leads to the fact that the ideal control law in (9) cannot perform.In order to overcome this problem, we use fuzzy neural networks to estimate () and ().

Description of Fuzzy Neural Networks.
In this section, a fuzzy neural network (FNN), which is used to estimate unknown functions () and (), is described.The FNN incorporates the advantages of a fuzzy logic system and a neural network; that is, the FNN possesses the learning ability of a neural network and the human thinking of a fuzzy logic system [25].The basic structure of a fuzzy logic system consists of fuzzification, rulebase, fuzzy inference, and defuzzification.The fuzzification is the process of mapping inputs, state variables  1 ,  2 , and  3 , to membership values in the input universes of discourse.The rulebase consists of nine antecedent-consequent linguistic rules (IF-THEN rules) in which the th rule is described in the form of where The fuzzy inference is the process of mapping membership values from the input windows, through the rulebase, to the output window.The fuzzy inference employs product inference for mapping.The defuzzification is the procedure of mapping from a set of inferred fuzzy signals contained within a fuzzy output window to a crisp signal.Using the center-average defuzzification techniques, the outputs of a fuzzy logic system can be represented as where ] is a fuzzy basic vector where each element   (),  = 1, 2, . . ., 9, is defined as The fuzzy logic system can be expressed by a neural network, which is known as a fuzzy neural network [25,26].As shown in Figure 3, the fuzzy neural network has four layers, including the input layer, membership layer, rule layer, and output layer.At the input layer, each node is an input representing a state variable.At the membership layer, the Gaussian functions are used as membership functions to calculate the membership values.At the rule layer, each node stands for an element   (),  = 1, 2, . . ., 9, of the fuzzy basis vector () and performs a fuzzy rule.The links between the rule layer and output layer are fully connected by the components of weighting vectors   and   .At the output layer, two outputs represent the value of â() and b().

Adaptive Fuzzy Neural Controller Design
Because () and (), as described in (8), cannot be calculated explicitly, the ideal control law ( 9) is unable to be implemented.In order to overcome this impediment, a neural network, as shown in Figure 3, is proposed to estimate () and ().Let â(, ) and b(, ) be the estimations of () and (), respectively.Then, following the certainty equivalent approach, the fuzzy neural controller   () based on the ideal control law (9) can be obtained as However, the control law in ( 21) may face the singularity problem when b(, ) closes to zero or even receives the zero value in some point in time initially, leading to possible large values for control signal   ().In such situation, the closedloop controlled system may lose controllability.To avoid this problem, we replace the control law in (21) with where  is a nonzero constant.The constant  is introduced to guarantee that the term b2 (, ) +  is always nonzero, and therefore the singularity problem can be avoided.The estimations â(, ) and b(, ) are calculated by a fuzzy neural network as where   () and   () are weighting vectors at the output layer of the neural network in Figure 3, while () is the fuzzy basic vector defined in (20).In the adaptive mechanism,   () and   () are online tuned so that â(, ) and b(, ) converge to () and (), respectively, and reach their optimal values.The achieved optimal weighting vectors where Θ  and Θ  are sets of acceptable values of vectors   () and   (), respectively, and Ω is a compact set of state variable .In this paper, we assume that the used fuzzy neural network does not violate the estimation property on the compact set Ω, and the compact set Ω is large enough so that state variables remain within Ω under the control action.
In adaptive mechanism, the adaptive laws for   () and   () are chosen as where   and   are positive-definite weighting matrices.
For the ideal situation, when   () and   (), respectively, approach  *  and  *  , â(, ) and b(, ), respectively, approach () and ().However, there exist the unavoidable estimation errors because â(, ) and b(, ) are estimated by a neural network which has a finite number of units in the hidden layer.Consequently, â(, ) and b(, ) cannot converge to () and () even when   () and   () converge to  *  and  *  , respectively.Let   () and   () be the estimation errors; then, the exact models of () and () can be expressed by The differences between the estimation models and exact models can be described by where W () =   () −  *  and W () =   () −  *  are parameter errors.
We suppose that the estimation errors of the neural network are bounded, and they can be expressed in the following assumption.
Assumption 4. The estimation errors are upper bounded by some known constants   > 0 and   > 0 over the compact set Ω ⊂  3 ; that is, sup The estimation errors are unavoidable, and sometimes they can break down the stability of the close-loop controlled system.In order to keep the system robustness, a compensatory controller   () is used as an additional controller to compensate for the estimation errors.The compensatory controller   () is designed as where   () = (/( b2 (, ) + ))(−â(, ) + V()).Therefore, the controller () has two control terms: the fuzzy neural controller   () and the compensatory controller   ().The overall scheme of the controller is illustrated in Figure 4 and the total control signal is given as Theorem 5. Consider the system in (3) and that the desired trajectory   () satisfies Assumption 3. If Assumptions 1-4 hold, then under the effect of controller (30) with the adaptive laws (25), chaos in the PMSM can be suppressed and its speed can asymptotically track the desired trajectory successfully.

Numerical Simulations
Numerical results are given in this section to verify the proposed method.The system parameters and initial conditions are maintained as above; that is,  = 5.45,  = 20, and [ 1 (0)  2 (0)  3 (0)]  = [1 − 1 0]  .First, the system without control action and uncertainties is considered.The simulation result points out that the state response falls into chaotic oscillations, as displayed in Figure 5.
Second, proposed controller is used to suppress chaos in the PMSM and track the desired speed under the effect of uncertainties.The uncertainties were chosen as  1 = 1 + cos(),  2 = −1, and  3 = sin( 1 ) for simulation while the desired trajectory   () = sin(), which satisfies Assumption 3, is also chosen for this simulation.On the other hand, the control parameters are chosen as follows: The results, as shown in Figures 6, 7, and 8, illustrate that the chaotic oscillations are removed and the speed of PMSM follows the desired trajectory perfectly while the tracking error converges to zero asymptotically.Abstract and Applied Analysis 11

Conclusion
In this paper, based on a fuzzy neural network, a new adaptive controller has been developed to suppress chaos and track the desired speed in a chaotic permanent magnet synchronous motor drive system.Derived from Lyapunov function, the stability of the system is ensured and the controller guarantees the perfect tracking performance where the tracking error converges to the origin even if uncertainties are applied to the system.The robustness and simple neural network structure can allow the controller to be feasibly applied to practical systems where the uncertainties are present.Simulation results are given to illustrate the effectiveness and robustness of the proposed method.

Figure 3 :
Figure 3: Structure of a fuzzy neural network.

Figure 4 :
Figure 4: Overall scheme of the adaptive controller.
1 ,   2 ,   3 ,    , and    are fuzzy sets which are denoted by the membership functions    1 ,    2 ,    3 ,     , and     , respectively.â() ∈  and b() ∈  are outputs of the fuzzy logic system, which stand for the estimations of () and (), respectively.   1 ,    2 , and    3 use Gaussian functions to calculate their values, while     and     are fuzzy singletons.