Nonlinear Torque Control of BLAC Motor

5th IFAC Symposium on Mechatronic Systems Marriott Boston Cambridge Cambridge, MA, USA, Sept 13-15, 2010

Nonlinear Torque Control of BLAC Motor Karel Jezernik, Robert Horvat, Milan Čurkovič University of Maribor, Faculty of Electrical Engineering and Computer Science, Maribor, Slovenia (e-mail: [email protected], [email protected], [email protected])} Abstract: This study presents the implementation of a hybrid control strategy applied to a brushless AC

(BLAC) motor drive. Hybrid control is a general approach for control of a switching-based Hybrid Systems (HS). This class of HS includes a continuous process controlled by a discrete controller with a finite number of states. The overall stability of the system is shown using Lyapunov technique. The Lyapunov functions used contain a term penalizing incremental energy of control error, torque and stator current, enhancing the stability. The closed-loop system with the proposed control low provides good transient response and good regulation the BLAC motor control. A new logical FPGA torque controller based on Lyapunov theory are developed, analyzed and experimentally verified. Keywords: Brushless motors, Discrete-event systems, Lyapunov methods, Motor control, Stability analysis quarantines a sufficient stability region in the state space for the system against large-signal disturbances. In this approach, Lyapunov direct method which is the most important tool for nonlinear control system design is used.

1. INTRODUCTION The Brushless AC (BLAC) motor is now widely used industrial drive applications due to its many advantages such as high power factor, high torque density, high efficiency, small size, and weight. A position sensor is required to enable high quality control and performance of a BLAC motor.

Lyapunov theory has for long time been an important tool in linear as well as nonlinear control. However, its use within nonlinear control has been changed by the difficulties to find the Lyapunov function for a given system. If one can be found, the system is known to be stable, but the task of finding such a function has often been left to the imagination and experience of the designer (Branicky at al.).

The nonlinear and complex dynamic interactions of AC machine, coupled with the fact that some important qualities are not measurable, cause considerable difficulties in designing high performance AC drive control. Additionally, the uncertainties arising from imperfect knowledge of system inputs and inaccuracies in the mathematical modeling itself, contribute to performance degradation of the feedback control system. Over the years it has been generally accepted that field orientation, in one of its many forms, is the most promising control method for a high dynamic performance AC drive. The direct torque control (DTC) is basically an advanced scalar control method and has found acceptance in replacement of conventional volts/Hz control (Boldea at al.).

In using the direct method, the idea is to construct a scalarenergy-like function (Lyapunov function) for the system and to examine the function's time variation. The closed-loop system with the proposed control law not only guarantees a sufficient stability region, but also provides good transient response and good regulation. The aim of this paper is the design of a control law for a BLACM to achieve good torque control in steady state and transient operating conditions. The feedback system is globally asymptotically stable in the sense of the Lyapunov stability theory. Therefore we are interested in an extension of the Lyapunov function concept. This concept used such a scalar function which contains a logical discontinuous input switching function penalizing the torque, i.e. current control error, enhancing the stability.

In spite of AC drives widespread popularity no rigorous stability proof for field oriented control or DTC control was available in the literature. In this paper we provide some answers to these questions (Buja at al.). This paper introduces recently developed hybrid based approach for modelling of discrete event systems in the field of power electronics and motion control. Power electronic circuits are hybrid dynamic systems. Because of the ON and OFF switching of power electronic devices, the operation of power electronic circuits can be described by a set of discrete states with associated continuous dynamics. A hybrid structure arises when a logical control unit governs such a system by using logic decision (Buisson at al.).

2. BLAC MOTOR CONTROL The BLAC motor combines many of the advantages of the permanent excited AC motor and the synchronous motor. The BLAC motor needs low reactive current, so similar to the DC motor, the current is proportional to the torque. The shaft torque is therefore easy to estimate by detecting threephase current of BLAC motor. The BLAC motor is the combination of a permanent excited synchronous motor and a

In this paper a hybrid control law based on Lyapunov stability theory, without linearization, is proposed that strictly 978-3-902661-76-0/10/$20.00 © 2010 IFAC

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three-phase inverter. The inverter in Fig. 1 has to replace the commutator of a conventional DC motor.

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Hybrid controllers can be used to obtain improved closedloop performance, beyond what can be achieved by using either classical linear or smooth nonlinear controllers. Motivation for hybrid control is the following: If the hybrid controller is appropriately defined, then the hybrid closed loop can reflect, to some degree, multiple performance properties associated with the closed-loop properties provided by each individual input function. This statement can be expressed in a slightly different way: motivation for use of hybrid control is that the performance of a hybrid closed loop can exceed the performance that can be achieved by any fixed feedback controller without switching. The results that we subsequently present illustrate the realization of this potential (Cassandras at al.).

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Fig. 1. Basic circuit of voltage source inverter. The BLAC motor dynamics are governed as electrical current with d ik 1 = ( uk − Rs ik − ek ) ; k = 1, 2,3 , dt Ls

(1)

and developed torque of BLAC motor as

∑ k =1 ek ik . = 3

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In this paper, hybrid controllers are represented by the block diagram in Fig. 3. That is, the hybrid control architecture consists of a family of discontinuous input functions and a supervisor; at each instant, the supervisor selects a particular input function from the family. The supervisor controls the input function selection via particular voltage vector Vi as well as when the selection changes. Such hybrid controllers have often been called logic-based switching controllers and are the most widely studied class of hybrid controllers.

(2)

Variables uk, ik, ek are motor phase voltages, currents and electromotive forces. Rs is resistance and Ls is inductance of motor windings. The mechanical equation is dω 1 = (Te − TL ) , (3) dt J where ω is shaft speed, Te and TL are electromechanical and load torque, respectively.

The commutation of a BLAC motor depends on the position of the rotor. The angle between the magnetomotive forces of stator and rotor is fixed to 90◦ , so the motor produces maximum torque and needs low reactive current.

Decision maker

Strategy Supervisor

Voltage vector 1

We performed a simple speed control of BLAC motor. Fig. 2 shows the architecture of speed / current regulation of BLAC motor implemented on FPGA circuit. The considered control task is a tracking of three-phase current reference signal, which amplitude is determined by output of PI speed controller. Drivers are made to incorporate every external device. A "signal select" output is implemented on FPGA for monitoring and external supervisor activity (Bomar).

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Fig. 3. Hybrid control architecture. The supervisor selects a input function to be active by specification of control Lyapunov function. The supervisor, as well as the family of input functions, may include dynamics and delays or other memory elements. The supervisor can be time driven, in which case switching is scheduled according to a clock as in a digital controller; or it can be event driven, in which case switching occurs according to a state partition or a state-dependent switching condition. This structure for the supervisor allows transitions

3. HYBRID SYSTEM FOCUS The most common class of control architectures has, historically, been based on linear feedback. In the case of classical nonlinear controllers, there has been substantial growth in recent years in control design methods that are based on feedback linearization, passivity, adaptation, and control Lyapunov functions.

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between input function selections according to an automata model, common hybrid control architecture. It is important that the supervisor does not switch between inputs functions infinitely often in a finite time period; this can be achieved in a number of ways. Design of the complete hybrid controller involves specification of both the family of input functions and specification of the switching logic. There is little guidance in the published literature on how the family of input functions should be selected; this is often problem dependent (Wainer).

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S2

S3

S1

Su2

Su3

V2(110)

2 4

4.1 System Analysis The basic circuit of a voltage source inverter (VSI) feeding a Y-connected three-phase load is presented in Fig. 1, where the load has been modelled by phase resistance, inductance and induced voltages. The voltage equation of the Yconnected three-phase load is:

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4. SYSTEM ANALYSIS AND CONTROL

i1 + i2 + i3 = 0 .

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⎧ V0; V(k-1)=V0 ∨ V1 ∨ V3 ∨ V5 ⎪ ⎪ VZ(k) = ⎨ ⎪ V7; V(k-1)=V7 ∨ V2 ∨ V4 ∨ V6 ⎪ ⎪ ⎩

The considered control problem is the tracking of a threephase current reference signal. After the current control error is defined Δis = is − isd , (4) rewritten in error form becomes

Fig. 4. Stator voltage us (Vi ) of three phase inverter.

d (6) Δis + Rs Δis = us (Vi ) − es , dt which contains all the disturbances (exogenous and endogenous) action on the system.

Relation (7) is true for the inverter circuits depicted in Fig. 1. It essentially shows that this particular switching matrix is able to generate three independent control actions denoted as the components S1, S2 and S3 of the control vector

Ls

us (Vi ) = U DC [ S1 , S2 , S3 ] . The components, i.e. switch T

The basic principle of the current control is to manipulate the input voltage vectors us (Vi ) so that the desired current is produced by inverter. This is achieved by choosing an inverter switch combination Si that drives the stator current vector by directly applying the appropriate inverter voltages us (Vi ) to the BLAC machine windings (4). The switch positions of the three-phase inverter are described using the logical variables Vi, dependent if switch Si is ON or OFF. Each variable corresponds to one phase of the inverter (Fig. 1). Three-phase inverter can produce 23 voltage vector combinations; two of them are zero vectors and 6 active vectors, Fig. 4.

position of the inverter are generated by look up table of FPGA controller. 4.2 Discrete Event Control To consider a hysteresis controller as a discrete-event dynamical system, it allows focusing in much more details on the switching actions and will enable a better understanding of the controller design. A discrete-event system reacts only if an event is recognized (Polič at al.). To control the current is , the sector of drive voltage uequ is recognized first, and based on the known sector, the input voltage vector us (Vi ) (the transistor switching pattern) for the current control is selected, respecting the current control error related to Lyapunov stability condition.

The energy flow between the input and output side of the three-phase inverter is controlled by switching matrix. By introducing the binary variables Si that are "1" if particular switch Si is On and "0" if switch Si is OFF (i=1,2,3,…,6) the behaviour of the switching matrix can be described by the three dimensional vector us = U DC L Si , where matrix L and vector S ( S1, S2 , S3 ) are defined as (Sabanovic at al.):

On the basis of Lyapunov stability theory, a positive definite scalar function V , as a candidate for the Lyapunov function, is to be found, such that the total energy of the system is continuously dissipated. In such a case, any nonlinear system must eventually settle down to an equilibrium point. For the Lyapunov function candidate

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(

V = (1/ 2) ΔisT Δis = (1/ 2) is − isd

) (i T

s

)

− isd ,

Moreover, the manifold Δis = 0 is reached in a finite time, and under assumption for fast switching, the trajectories stay in that manifold all the time (Utkin).

(8)

the stability requirement will be fulfilled if control low can be selected as such, that the derivative of the Lyapunov function candidate is negative V = ΔisT Δis ≤ 0 . Derivatives of the current control error (6) may be expressed with the voltage equation

5. SWITCHING STRATEGY The proposed logical event-driven BLAC current control can be realized in the form described in Table 1, where states of current control error are presented by sign(Δis) and currently active voltage sector is presented by sign( uequ).

( d / dt ) ( is − isd ) = (1/ Ls )( −es − Rs is + us (Vi ) ) − ( d / dt ) isd , (9) where isd , is are desired and actual motor current, us (Vi ) is voltage control input, Rs is is resistive voltage drop and es is

Table 1. Look-up table

sign(uequ)

exogenous disturbance.

sign(Δis)

For d ( Δis ) / dt = 0 the equivalent control voltage can be expressed as (Utkin): uequ = es − Rs is − Ls ( d / dt ) isd

Sdi0 Sdi1 Sdi2 Sdi3 Sdi4 Sdi5 Sdi6 Sdi7

(10)

and derivative of Lyapunov function is presented with equivalent control and control input voltage us (Vi ) as: V = ( is − isd ) ( uequ − us (Vi ) ) / Ls < 0

(11)

Considering space vector representation of the stator voltage us (Vi ) , the voltage is represented as vector rotating around the origin. Six active switching vectors of the threephase transistor inverter represent the six active output voltage vectors denoted as V1...V6 ; V0 and V7 are two zero voltage vectors. According to signs of the phase voltages us1 , us 2 and us 3 , the phase plane is divided into six sectors denoted by Su1 … Su6, Fig. 4.

000 100 110 010 011 001 101 111

Su1

Su2

Su3

Su4

Su5

Su6

100

110

010

011

001

101

VZ V1 V2 VZ VZ VZ V6 VZ

VZ VZ VZ VZ VZ V1 VZ VZ VZ V1 V2 V2 VZ VZ VZ V3 V3 V3 VZ VZ VZ V4 V4 V4 VZ VZ VZ V5 V5 V5 VZ VZ VZ V6 V6 VZ VZ VZ VZ VZ ⎧ V0; V(k-1)=V0 ∨ V1 ∨ V3 ∨ V5 VZ(k) = ⎨ ⎩ V7; V(k-1)=V7 ∨ V2 ∨ V4 ∨ V6

To further improve the presentation, active voltage vectors are marked in Table 1 with a blue background. Because the transition between inverter switch states is performed by switching only one inverter leg, switching frequency and current chattering (and consequently torque chattering of BLAC motor) are reduced.

In regards to the situation (Fig. 4), the stator voltage space vector uequ is in sector Su1. In this sector, logical voltage

In the proposed new developed switching voltage space vector modulator voltage vectors Vi can be mapped into the three-phase plane as shown above in Fig. 4. This provides a geometric interpretation of the voltages that can be produced by the inverter. The driving signal of inverter switches can be generated with FPGA circuit. The considered control task is a tracking of three-phase current reference signal. The zero voltage control vector can be consciously used to reduce the transistor's switching frequency (Monmasson at al.).

vectors V0 , V1 , V2 , V6 and V7 are selected for the current control. V0 , V7 are two zero vectors, while V1 , V2 , V6 are three nearest adjacent live output voltage vectors to this sector. With the use of the discrete event system theory, five output voltage vectors V0 , V1 , V2 , V6 and V7 are recognized as discrete states of the system. Events represent allowed transition among the discrete states i.e. allowed switching and are determined via stability issue.

6. IMPLEMENTATION AND PROTECTION ISSUE

A novel, discrete event current controller, evaluates the transistor switching patern, S ( S1, S2 , S3 ) of inverter from the

6.1 Software of Hybrid BLAC Controller

sign of the current error, sign( is − isd ) and the sign of actual voltage sector, sign( uequ ) .

An important advantage of using an FPGA for the controller is that some additional functions like protections, steering, monitoring etc., can be added with no additional resources and almost no drawback in performance showing in Fig. 2. The FPGA concurrency allows executing the logic dedicated to the proposed hybrid BLAC motor control and any additional logic denoted to protections simultaneously. In this way, there is almost no drawback in the controller performance because the control logic is executed as if there were no protections. Even more, the protections are executed

When U dc has enough magnitude that V ≤ 0 , than V → 0 and is → isd . S1, S2 and S3 represent the switching state of the three-phase power converter. Notice that if S1, S2, S3 equal to zero simultaneously, no current is delivered to the BLAC motor.

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continuously, instead of the periodic execution in a DSP. DSP solutions must keep the protections resources at a minimum because the main control is stopped while a protection is verified. In fact, any algorithm can be added to control one while there are available resources.

ISM represents a potential risk for the BLAC motor with inverter if a major driving parameter drifts into it, or if a failure occurs. The circle domain is out of the limits of the safe field of variation of the parameters and system automatically switches off.

The steering function determines the converters operation mode. Three possible modes are concerned: Enabled, Disabled or Fault. They are considered as discrete states of the monitoring function. Initially Disabled mode is active. After the Enable input rises, the transition to Enabled mode occurs. The occurrence of the Fault signal transfers the drive to the Fault mode. Recovery from the Fault mode is possible by applying Reset signal. State graph of proposed function is depicted on Fig. 5.

Fig. 6. State graph of drive protection function.

Fig. 5. State graph drive monitoring function.

The protection function monitors the inverter and AC motor operating conditions and sends Fault signal in the case of a false operating conditions. False operating conditions are detected by comparing input dc link voltage Udc, the machine current is, and the heat sink temperature Temp_PwrStage with the thresholds. Two discrete states are concerned: whether the operating conditions are met or an error appears. The warning signal is used for monitoring purposes. State graph of proposed function is depicted on Fig. 6.

Fig.7. State CAD diagram. 6.2 Hardware of Hybrid Controller The proposed approach is based on fast parallel processing and suitable for a Field Programmable Gate Array (FPGA) implementation. In such implementation, it would be possible to reproduce near ideal switching mode process. However, with FPGA implementation, designer has a difficult task to characterize and describe the hardware architecture that corresponds to the chosen control algorithm. FPGA designers must follow an efficient design methodology in order to benefit from the advantages of the FPGAs and their powerful CAD tools. From a software point of view, HDL modelling

The FPGA based hybrid BLAC motor system is meant to be operated according to certain established rules and principles, which are presented in Fig. 7 as infinite state machine (ISM) in three domain principle. The rectangular ISM describes normal machine operation including start/shut/up/down operation, hybrid controller normal operation. The triangle 457

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Such a scheme for operation occupied 35% of the “number of 4 input LUTs” FPGA circuit and 67% of 18-bits multipliers. VHDL code was created in Xilinx ISE software. Software scheme is divided into individual blocks of better review, Fig. 9. It is composed of a data path and a control unit coded in VHDL. The data path is composed of a fain gain operators such as adders, multipliers, multiplexers and registers. The data transfer between these operators is managed by a control unit, which is synchronized with the clock signal (CLK). The control unit of the developed modules is always activated via Start pulse signal. When the computation time process is over, an End pulse signal indicates that the data outputs of the module are ready.

system is based on using variables that request logic values, too. A low cost Xilinx Spartan 3 FPGA that contains 1.2 M logical gates and includes a 50 MHz oscillator was used as a target component for the implementation of the controller. The architecture of each control algorithm is designed according to an efficient methodology that offers considerable design advantages such as reusability, reduction of development time and optimization of the consumed resources. Each control algorithm is partitioned into elementary modules, which are easier to develop and are more functional. Fig.8 presents the general structure of the different elementary modules. As shown in Fig. 8, Xilinx Spartan XC3S1200E is used to implement the BLAC motor controller. This motor control intellectual property (IP) is divided into three parts. The first part is the driver’s part with ADC and DAC management modules, incremental module for speed and position measurement and RS 232 module for connection of a host PC equipment with Matlab/Simulink program environment. The second part includes PI-speed motor controller. Output of speed controller is measure for the desired torque of BLAC motor and with position signal multiplies the three-phase reference currents of machine. The third part includes look-up table with hysteresis current controller and average output voltage of BLAC motor synchronization – voltage sector selection, Fig. 8.

This generic structure can be easily introduced in an upper hierarchical level architecture and can also be used for future design. Basic step time (clock) for execution of arithmetic operations is 30 ns. Measurement of current is limited, with conversion time of A/D converters, to 2.5 μs. Entire scheme (data lines) is based upon 12-bits signed numbers. The protection of three-phase inverter (dead time) is provided by the transistors drivers. 7. EXPERIMENTAL RESULTS In order to test the proposed Xilinx Spartan 3 FPGA-based controller, experiments were carried out on in-house built FPGA prototyping platform. In all simulation and experimental results, the BLAC motor from Maxon, type EC32 with parameters Rs = 0,56 Ω ,

MATLAB

UDC

Hist Ti , K i , K P

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i

i

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was used.

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tor6 Sec

Switch 1,2,3...

i1 , i2 , i3

Ls = 0.09 mH ,

Ψ PM = 0.0205 Vs , t M = 6.6 ms

Three-phase Inverter

S2

SIN

controller D/A

3 ct or Se V3

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i1 , i2 , i3 clk,conv

INC

Look-up table

Di

iref

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RS232

A/D LTC1400 Clk, conv

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A,B,ri

Fig. 8. FPGA controller of the BLAC motor.

Fig. 9. VHDL scheme of BLAC controller in Xilinx ISE.

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filtering of output voltage

The FPGA controller has the diagnostic features that are necessary for drive installation, test problem detection and elimination. The control algorithm is executed every 2.5 μs, and the switching frequency of three-phase inverter was set with the tolerance band of the hysteresis current control. The control of both, the speed and applied torque, is possible, thus hardware in the loop operation can also be performed.

250 output voltage filtered output voltage

voltage - from state of transistors

200

The experimental results are illustrated in Fig. 10, through, Fig.13. Fig. 10 shows the transient of speed (ω), current of one phase (is1) and voltage vectors (spectrum) Vi(S1,S2,S3) of three-phase inverter during start, which is classical bang-bang regulated and at time t1 is switched to the proposed hybrid discrete event control (DES). Pulse pattern of voltage vector Vi ( Si ) changes transients from conventional bang-bang current control to hybrid DES current control of inverter.

150 100 50 0 -50 -100 -150 -200 -250

0.01

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Fig. 11. Filtering of output voltage for one phase. Step response ( ωref = 0 to 128 rad/s) 500

change of speed from 0 to 100 rad/s

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Fig. 12. Step response of reference speed and currents.

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Fig. 10. Active voltage vectors at progressive change of reference speed.

phase plane of current in a-b reference frame

1000

Fig. 11 shows the transients of output voltage us (Vi ) and average filtered output voltage for one phase of the inverter. This signal is used for voltage vector selection in look-up table. Based on this simplified close-loop, the transfer function is expressed as: Us sB + A ≅ 2 (U s (Vi ) ) s + sB + A

0.06

Ib [m A ]

500

0

-500

(12)

-1000

This transfer function corresponds to a second order system. The observation dynamic is then set using A as frequency and B as damping coefficients of second order low pass filter(12).

-1500 -1500

Step response of speed and currents are shown in Fig. 12. Inverter current response to a step change of mechanical load of motor is illustrated on Fig. 13. The results are recorded using oscilloscope and consequently the measured results are contaminated with noise.

-1000

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Ia [mA]

Fig. 13. Current response to step load change of motor in stator fixed reference frame.

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8. CONCLUSION

REFERENCES

The idea of hybrid-based control of event-driven systems is used for the design of event-driven current (torque) controller and auxiliary steering and protection functions for a BLAC servodrive. Individual control functions have been designed, using the proposed approach, and integrated into the overall functionality description of the inverter. Overall functionality of the inverter and the performance of the proposed event driven current control were checked by simulations and experimentally confirmed. Special attention is paid to the mapping of the proposed design approach into the schematic form for the FPGA implementation.

Boldea, I., and Nasar, S.A. (1999). Electric Drives. CRC Press. London. Bomar, B. (2002). Implementation of a micro programmed control in FPGA. IEEE Trans. Ind. Electronics, vol. 49, no. 2, pp. 415-422. Branicky, M.S., Borkar, V.S., and Mitter, S.K. (1998). A unified framework for hybrid control: model and optimal control theory. IEEE Trans. Autom. Control., vol. 43, no. 1, pp. 31-45. Buisson, J., Richard, P.Y., and Cormerais, H. (2005). On the stabilization of switching electrical power converters. In Hybrid Systems: Computation and Control, pp. 184197. Springer, Berlin / Heidelberg. Buja, G.S., and Kazmierkowski, M.P. (2004). Direct torque control of PWM inverter-fed AC motors – A survey. IEEE Trans. Ind. Electronics, vol. 54, no. 4, pp. 18241842. Cassandras, C.G. and Lafortune, S. (2008). Introduction to Discrete Event Systems. 2nd ed. Springer Science + Business Media. Monmasson, E., and Cirstea, M.N. (2007). FPGA Design Methodology for Industrial Control Systems – A Review. IEEE Trans. Ind. Electronics, vol. 54, no. 4, pp. 1824-1842. Polič, A., and Jezernik, K. (2007). Event-driven current control structure for a three phase inverter. Int. rev. electrical eng. (Testo stamp.), vol. 2, no. 1, pp. 28-35. Sabanovic, A., Jezernik, K., and Sabanovic, N. (2002). Sliding Modes Applications in Power Electronics and Electrical Drives. In Y.U. Xinghuo, X.U. Jian-Xin, (ed.) Lecture notes in control and information sciences, 1st ed., pp. 223–251. Springer Verlag, Berlin. Utkin, V.I. (1992). Sliding Modes in Control and Optimization. Springer Verlag. Berlin. Wainer, G.A (2009). Discrete-Event Modeling and Simulation: A Practitioner's Approach. CRC Press, Taylor and Francis Group. Boca Raton.

In this study, the hybrid control problem of BLAC motor was addressed. The nonlinear behavior of the system, limits the performance of chosen bang-bang controllers, used for this purpose. This paper has successfully demonstrated the design, stability analysis, simulation and tests of a Lyapunov technique approach for the DES control of the BLAC drive. The feedback system is globally asymptotically stable in relationship to the Lyapunov method. The transient state and steady state performances of the logical based controller were improved via Lyapunov stability. Consequently, high performance dynamic, despite the presence of parameter uncertainties or disturbances, is obtained. The simulation and experiments confirmed potential of the presented approach: traditional coding efforts are significantly reduced on one hand, and on the other hand the control algorithm can be verified off-line. Formal mathematical background of the proposed approach and its correspondence to the conventional control system theory opens further possibilities for the design, simulation, and formal analysis of the discrete event systems. The proposed approach offers a promising technique for design of complex and timely critical algorithms. On one hand, future research is oriented towards the optimization of the introduced motion control strategies, and on the other hand towards the formalization of analysis and control design methods.

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