Supervisory Genetic Evolution Control for Induction Machine
Using Fuzzy Design Technique
Rong-Jong Wai*, Jeng-Dao Lee, and Li-Jung Chang
Department of Electrical Engineering, Yuan Ze University, Chung Li 320, Taiwan, R.O.C.
*E-mail:
AbstractThis study presents a supervisory genetic evolution control (SGEC) system for achieving high-precision position tracking performance of an indirect field-oriented induction motor (IM) drive. Based on fuzzy inference and genetic algorithm (GA) methodologies, a newly design GA control law is developed first for dominating the main control task. However, the stability of the GA control can not be ensured when huge unpredictable uncertainties occur in practical applications. Thus, a supervisory control is designed within the GA control so that the states of the control system are stabilized around a predetermined bound region. In addition, the effectiveness of the proposed control scheme is verified by numerical simulation and experimental results, and its advantages are indicated in comparison with a feedback control system.
I. Introduction
Nowadays, the field-oriented control technique has been widely used in industry for high-performance IM drive [1, 2], where the knowledge of synchronous angular velocity is often required in the phase transformation for achieving the favorable decoupling control. However, the performance is sensitive to the variations of motor parameters, especially the rotor time-constant, which varies with the temperature and the saturation of the magnetizing inductance. Recently, much attention has been given to the possibility of identifying the changes in motor parameters of an IM while the drive is in normal operation. Some researchers have proposed various IM drives with rotor-resistance or rotor time-constant identification to produce better control performance [3-5]. However, the control performance of the IM is still influenced by the uncertainties, such as mechanical parameter variation, external disturbance, unstructured uncertainty due to nonideal field orientation in transient state, and unmodelled dynamics, etc. In the control fields, the acquirement of the uncertainty information is an important research topic. From a practical point of view, however, it is usually very difficult to get the complete information of uncertainties. Therefore, the motivation of this study is to design a suitable control scheme to confront the uncertainties existing in practical applications of an indirect field-oriented IM drive.
To deal with the mentioned uncertainties, much research has been done in recent years to apply various approaches to attenuate the effect of uncertainties. On the basic aspect, the conventional proportional-integral-derivative (PID)-type controllers are widely used in industry due to their simple control structure, ease of design, and inexpensive cost. However, the PID-type controller can not provide perfect control performance if the controlled plant is highly nonlinear and uncertain [6, 7]. On the other hand, computed torque or inverse dynamics technique is a special application of feedback linearization of nonlinear systems [8, 9]. The computed torque controller is utilized to linearize the nonlinear equation by cancellation of some, or all, nonlinear terms such that a linear feedback controller is designed to achieve the desired closed-loop performance. However, since the computed torque approach is based on perfect cancellation of the nonlinear dynamics, the objection to the real-time use of such control scheme is the lack of knowledge of uncertainties.
Genetic algorithm (GA), which uses the concept of Darwin’s theory, has been widely introduced to deal with nonlinear control difficulties and to solve complicated optimization problems [10-17]. Darwin’s theory basically stressed the fact that the existence of all living things is based on the rule of “survival of the fittest”. In the theory of evolution, different possible solutions to a problem are selected first to a population of binary strings encoding the parameter space. The selected solutions undergo a parallel global search process of reproduction, crossover, and mutation to create a new generation with highest fitness function [10]. This process of production of a new generation and its evaluation is repeated until there is satisfactory convergence within a predefined fitness grade. Since the GA simultaneously evaluates many points in the parameter space, it is more likely to converge toward the global solution [16]. Recently, this underlying GA-based global optimization technique has been applied in several fuzzy logic control applications [18-22]. In view of the previous research results, the favorable control or optimization performance can reach their destination owing to the powerful global searching capability of GA. However, the role of GA control is usually used as a minor compensatory tuner in the open literatures because the stability of the GA-based control scheme can not be guaranteed until now. The aim of this study is to design an on-line GA control scheme as a major controller, moreover, the stability of this strategy can be ensured with the aid of supervisory control during the whole control process.
II. Indirect Field-Orientation Induction Motor Drive
The IM used in this drive system is a three-phase Y-connected four-pole 800W 60Hz 130V/5.6A type. Moreover, the drive system is a ramp comparison current-controlled pulse width modulated (PWM) voltage source inverter (VSI). The current-controlled VSI is implemented by isolated gate bipolar transistor (IGBT) switching components with a switching frequency of 15kHz. For the position control system, the braking machine is driven by a current source drive to provide braking torque. An inertia varying mechanism is coupled to the rotor of the IM. The mechanical equation of an IM drive can be represented as
(1)
where J is the moment of inertia; B is the damping coefficient; is the rotor position; represents the external load disturbance; denotes the electric torque. With the implementation of field-oriented control [1, 2], the electric torque can be simplified as
(2)
with the torque constant is defined as
(3)
Substituting (2) into (1) as follows can represent the mechanical dynamic of the IM drive system:
(4)
where; ; ,and is the control effort. Dynamic modeling based on measurements [23] is applied to find the drive model off-line at the nominal condition. The results are (on a scale of 50(rad/s)/V)
(5)
The overbar symbol represents the system parameters in nominal conditions.
III. Supervisory Genetic Evolution Control
With the field-oriented method, the dynamic behavior of the IM is rather similar to that of a separately excited dc motor. The decoupled relationship is obtained by means of a proper selection of state coordinates under the hypothesis that the synchronous angular velocity is precise. Therefore, the rotor speed is asymptotically decoupled from rotor flux, and the speed is linearly related to torque current after the slip angular velocity can be obtained precisely. However, the control performance of the IM is still influenced by the uncertainties of the plant, such as mechanical parameter uncertainty, external load disturbance, unstructured uncertainty due to nonideal field orientation in transient state, and unmodelled dynamics in practical applications. Therefore, a SGEC scheme is designed in the sense of fuzzy inference and GA methodologies to increase the robustness of the indirect field-oriented IM drive for high-performance applications.
Consider the parameters in the nominal condition without external load disturbance, rewriting (4) as follows can represent the nominal model of the IM drive system:
(6)
where and are the nominal values of and , respectively. Consider (6) parametric variation, external load disturbance and unpredicted uncertainties for the actual IM drive system
(7)
where and denote the uncertainties introduced by system parameters J and B; represents the unstructured uncertainty due to nonideal field orientation in transient state, and the unmodelled dynamics in practical applications; is called the lumped uncertainty and is defined as
(8)
Here the bound of the lumped uncertainty is assumed to be given; that is,
(9)
where is a given positive constant. The control problem is to find a control law so that the rotor position can track any desired commands. To achieve this control objective, define a tracking error as and its derivative , in which represents a reference trajectory specified by a reference model. The control law for a SGEC system is assumed to take the following form:
(10)
where is a GA control that is a main tracking controller, and is a supervisory control that is designed so that the states of the control system are stabilized around a predetermined bound region. The overall scheme of the SGEC strategy is depicted in Fig. 1 and the detailed descriptions of each control part are exhibited in the following subsection.
A. GA Control
In the GA controller, the tracking error () and its derivative() are chosen as the input signals, and is the output signal. In this study, the spirit of fuzzy inference mechanism is utilized to design this GA controller. It can divide three main parts: GA membership region, quantization number/levels and GA lookup table introducing in the following paragraphs.
GA Membership Region
The membership regions and denote some area, where the tracking error and its derivative maybe varied in practical applications. The selection of membership regions usually depends on the expert’s experience and various applications.
Quantization Number/Levels
According to the quantization number and , the tracking error and its derivative can be separated into several different levels. Note that, the selection of and has a great influence with higher or lower accuracyof system performance. If the selection of quantization number is too large, it will cause heavy computation load, and the learning speed of the GA controller will be reduced. On the contrary, if the selection of quantization number is too small, it may cause the chattering efforts in the controlled system, even to be unstable. In the study, the quantization functions are denoted as and , and each of them has nine levels, which are composed of NE (Negative Extend), NB (Negative Big), NM (Negative Medium), NS (Negative Small), ZE (Zero), PS (Positive Small), PM (Positive Medium), PB (Positive Big), and PE (Positive Extend).
Fig. 1. Block diagram of SGEC system.
Table I. GA Lookup Table
/ PE / PB / PM / PS / ZE / NS / NM / NB / NE
NE / / / / / / / / /
NB / + / / / / / / / /
NM / + / + / / / / / / /
NS / + / + / + / / / / / /
ZE / + / + / + / + / / / / /
PS / + / + / + / + / + / / / /
PM / + / + / + / + / + / + / / /
PB / + / + / + / + / + / + / + / /
PE / + / + / + / + / + / + / + / + /
GA Lookup Table
When the input signals are passed through the quantization number/levels step, the GA lookup table shown in Table I will be constructed on line with the genetic evolution mechanism: reproduction, crossover and mutation introduced later. Note that, based on the fuzzy inference mechanism, the sign of the associated control efforts are predefined in Table I such that it has more possibility to search optimal control efforts. In this study, each control effort in the GA lookup table can be represented as a chromosome and can be expressed via binary string representation as
with (11)
where is one chromosome that has l-bits binary string, and denotes a gene. On the other words, the values of the are converted into their binary equivalent values. In order to evaluate the fitness grade of each chromosome, a fitness function is chosen as
(12)
where is the evaluated tracking error induced by the original control effort, and denotes the evaluated tracking error-change at two continuous iterations via a new chromosome in area of the GA lookup table. During the on-line searching process, the chromosome with a highest fitness grade will be saved on the GA lookup table. The basic genetic operation used in this study is summarized as follows:
Reproduction
This reproduction procedure is used to decide which chromosomes would be selected into the mating pool for further genetic operations. First, an initial chromosome, named as mother chromosome, is taken as a control input of the IM drive. According to the running result, it will produce new tracking error and its derivative such that a corresponding chromosome, named as father chromosome, can be detected via GA lookup table. Both of them are selected to the crossover operation.
Crossover
The crossover operation combines the features of two parent chromosomes to form one offspring by swapping corresponding segments of the parents. In this study, the crossover operation is performed with one crossover rate defined as
(13)
where denote rounding the element to its nearest integer. If the selection of the crossover rate is bigger, then the offspring has more characteristics in the father chromosome. This operation is repeated until there is satisfactory convergence within a predefined fitness grade.
Mutation
In order to avoid chromosome trapping in local optimal point, every gene is subject to random change with probability of the pre-assigned mutation rate, , at each iteration. In the binary string case, mutation operators just to change the bit form 0 to 1 or vice versa. According to the corresponding fitness value, the mutation rate can be represented as
(14)
where is a given upper bound of mutation number. As time goes by, the fitness grade will gradually increase, and the crossover and mutation operators also tend to settle. This evolution procedure progresses until the fitness grade reaches the desired specification. Thus, the output of the GA controller can be represented as
(15)
However, the stability of the GA control can not be ensured when huge unpredictable uncertainties occur in practical applications. Therefore, the auxiliary design of a supervisory control is necessary for the condition of divergence of states to pull the states back to the predetermined bound region and guarantee the system stability.
B. Supervisory Control
Assume that the lumped uncertainty is available, there exists an ideal control law as follows such that the favorable control performance can be ensured:
(16)
where is a given positive constant vector and is a tracking error vector. From (7), (10) and (16), an error equation is then obtained as follows:
(17)
where is a stable matrix and . Define a Lyapunov function as
(18)
where Pis a symmetric positive definite matrix which satisfies the following Lyapunov equation:
(19)
and is selected by the designer. Take the derivative of the Lyapunov function and use (17) and (19), then
(20)
To satisfy , the supervisory control is designed as follows [8, 9]:
(21)
where sgn() is a sign function; is an absolute function; I is an index function and is defined as
(22)
in which is a positive constant designed by the user. Substitute (21) into (20) and consider the case, then
(23)
Using the designed supervisory control as shown in eqn. 28, the inequality can be obtained for non-zero value of the tracking error vector Ewhen . As a result, the stability of the SGEC system can be guaranteed. The effectiveness of the proposed control scheme is verified by the following simulation and experimental results.
IV. Simulation and Experimental Results
The simulation of the proposed SGEC system is implemented via the “Matlab” package based on the scheme shown in Fig. 1, and its control parameters are given as follows:
, , , ,
, , , (24)
All the parameters in the proposed control system are chosen to achieve the best transient control performance in both simulation and experimentation considering the requirement of stability. In order to let the GA controller have the self-organizing property, the initial GA lookup table in this study is set at a null table. The effect due to the inaccurate selection of the initialized population can be retrieved by the on-line searching methodology. The parameter searching process remains continually active for the duration of the simulation and experiments runs.
A second-order transfer function with rise time 0.5s is chosen as the reference model for the periodic step command:
(25)
where and are the damping ratio (set at one for critical damping) and undamped natural frequency. Moreover, two simulation cases including parameter variations and external load disturbance in the shaft due to periodic commands are addressed as Case 1: , , occurring at 5.5s; Case 2: , , occurring at 5.5s. The control objective is to make the rotor position follow the periodic step reference trajectory under the occurrence of uncertainties.
Fig. 2. Simulated responses of feedback control system at Case 1 and Case 2.
In the simulation, first the ideal control law in (16) without considering lumped uncertainty (), which is called a feedback control system, is demonstrated for comparison. The simulated results of feedback control system for periodic step command at Case 1 and Case 2 are depicted in Fig. 2. From the simulation results, favorable tracking response shown in the beginning of Fig. 2(a) only can be obtained at the nominal condition, and poor tracking responses are resulted owing to parameter variations and external load disturbance. Though a large control gain may solve the problem of delay or degenerate tracking responses, it will result in impractical large control efforts. Therefore, the control gains are difficult to determine due to the unknown uncertainties in practical applications, and are ordinarily chosen as a compromise between the stability and control performance. Now, the SGEC system is considered under the same simulated cases as the feedback control system. The simulated results of SGEC system for periodic step command at Case 1 and Case 2 are depicted in Fig. 3. The tracking errors converge quickly, and the robust tracking performance of the proposed control scheme can be obtained under the occurrence of uncertainties.
Fig. 3. Simulated responses of SGEC system at Case 1 and Case 2.
Fig. 4. Experimental results of feedback control system at external disturbance condition and parameter variation condition.