Taos Thesis

CURRENT CONTROL OF PERMANENT MAGNET SYNCHRONOUS MOTORS USING DSP TECHNOLOGIES

by

TAO YANG B.E., Zhejiang University, China, 1998

A THESIS Submitted in partial fulfillment of the

requirements for the degree

MASTER OF SCIENCE

Department of Mechanical and Nuclear Engineering College of Engineering

KANSAS STATE UNIVERSITY Manhattan, Kansas

2004

Approved by:

Major Professor Dr. Dale E. Schinstock

ABSTRACT

This thesis investigates systematic design methods for DSP-based current control

of permanent magnet synchronous motors (PMSM). This includes systematic methods

for finding the gains of the motor current controller and methods for finding the

minimum controller update and pulse width modulation (PWM) frequencies.

In this thesis, two control structures are studied. One is vector current control,

where the controllers are applied in the rotor synchronous, dq0, frame; the other is sine

wave current control, where the controllers are applied in the stationary, abc, frame.

The proportional integral (PI) controller is a standard controller in the current

control of PMSM. Typical tuning methods for PI are reviewed. A system identification

method based on relay feedback is presented for tuning current control loops.

Linear quadratic (LQ) optimization is a time domain controller design method.

The idea of this approach is to select proper controller gains to minimize a performance

index. LQ design is repeated against plant parameters to develop empirical formulae for

specific controllers and plants. This thesis extends previous work in empirical

optimization to apply to a wider range of values in PI control of a 1st order plus time

delay system. It also originally presents a set of empirical optimization formulae for

integral control of a pure time delay system. For these two sets of empirical formulae, the

resulting system bandwidth is formulized originally. The phase margin and gain margin

of the resulting system are also presented.

The effects of digitization and PWM are considered in the systematic design

process. Combining such effects and LQ optimization empirical formulae, systematic

design methods are proposed to determine minimum values for digital update frequency

and PWM frequency according to desired bandwidth and the maximum operation speed.

These methods are the main contribution of this thesis.

i

TABLE OF CONTENTS

LIST OF FIGURES…………………………………………………….……………….iv

LIST OF TABLES………………………...…………………………..………………..vii

ACKNOWLEDGMENTS.……………………………………....……..………….…..viii

CHAPTER 1: INTRODUCTION AND BACKGROUND............................................ 1

1.1. BACKGROUND........................................................................................................... 1

1.2. INTRODUCTION.......................................................................................................... 3

CHAPTER 2: EXPERIMENTAL SYSTEM.................................................................. 6

2.1. OVERVIEW ................................................................................................................ 6

2.2. PMSM BRUSHLESS SERVOMOTOR AND DYNAMOTOR .............................................. 7

2.3. L180 DIGITAL SERVO DRIVE AMPLIFIER .................................................................. 8

2.4. DSP BOARD (MC8-DSP-ISA BOARD) ..................................................................... 9

2.5. POWIRTRAIN .......................................................................................................... 10

2.6. SHAFT POSITION SENSOR ........................................................................................ 11

2.7. ISOLATOR BOARD ................................................................................................... 12

2.8. CURRENT SENSOR BOARD....................................................................................... 12

2.9. POWER SUPPLY ....................................................................................................... 13

CHAPTER 3: PID CONTROL TUNING METHOD REVIEW AND SYSTEM

IDENTIFICATION BY RELAY FEEDBACK............................................................ 14

3.1. INTRODUCTION........................................................................................................ 14

3.2. ZIEGLER-NICHOLS (ZN) METHOD .......................................................................... 14

3.2.1. Open Loop ZN Tuning Method ....................................................................... 15

ii

3.2.2. Closed Loop ZN Tuning Method..................................................................... 17

3.3. RELAY FEEDBACK TUNING ..................................................................................... 19

3.3.1. Astrom-Hagglund Method .............................................................................. 19

3.3.2. Describing Function of a Relay ...................................................................... 20

3.3.3. Describing Function of Relay without Hysteresis .......................................... 21

3.3.4. Describing Function of a Relay with Hysteresis............................................. 22

3.3.5. Describing Function Analysis of a Relay in the Loop .................................... 24

3.3.6. Critical Point Identification............................................................................ 26

3.3.7. Parameter Identification via Relay Test for Specific Systems ........................ 27

CHAPTER 4: OPTIMAL DESIGN FOR PID CONTROLLERS VIA LQ

APPROACH.................................................................................................................... 30

4.1. INTRODUCTION TO LQ DESIGN ............................................................................... 30

4.2. PI/PID CONTROLLER DESIGN BY LQ APPROACH.................................................... 37

4.2.1. PI Controller for FOPTD ............................................................................... 37

4.2.2. PID Controller for the Second Order plus Time Delay System...................... 48

CHAPTER 5: PMSM CURRENT CONTROL ........................................................... 55

5.1. INTRODUCTION........................................................................................................ 55

5.2. VECTOR CURRENT CONTROL .................................................................................. 56

5.3. SINE WAVE CURRENT CONTROL............................................................................. 61

CHAPTER 6: DIGITAL IMPLEMENTATION AND SYSTEMATIC DESIGN OF

PMSM CURRENT CONTROL .................................................................................... 69

6.1. DIGITAL IMPLEMENTATION ARCHITECTURE FOR PMSM CURRENT CONTROL........ 69

iii

6.1.1. Sine Wave Current Control Structure............................................................. 70

6.1.2. Vector Current Control Structure................................................................... 76

6.2. EFFECTS OF DIGITIZATION AND PWM ON THE CURRENT CONTROL SYSTEM DESIGN

....................................................................................................................................... 79

6.3. SIMULATION MODELS ............................................................................................. 87

6.4. SYSTEMATIC DESIGN OF PMSM CURRENT CONTROL SYSTEM ............................... 89

6.4.1. Systematic Design of PMSM Vector Current Control .................................... 89

6.4.2. Systematic Design of PMSM Sine Wave Current Control .............................. 91

6.5. DESIGN EXAMPLES ................................................................................................. 93

6.5.1. Vector Current Control Design Example ....................................................... 93

6.5.2. Sine Wave Current Control Design Example ............................................... 105

6.6. AUTOMATIC TUNING............................................................................................. 114

CHAPTER 7: CONCLUSION..................................................................................... 118

REFERENCES:……......……………………………………………….……………..120

APPENDIX A: Current Control DSP Program Source Code …...….……………..124

APPENDIX B: LQ Design MATLAB Program Code ……….…...….……………..146

iv

LIST OF FIGURES

Figure 1-1: Typical Motion Control Structure.................................................................... 2

Figure 1-2: Three Phase Two-Level Inverter...................................................................... 4

Figure 2-1: Experimental System ....................................................................................... 7

Figure 2-2: The IRPT 1058 PowIRTrain within a Motor Control System....................... 11

Figure 3-1: Open Loop ZN Tuning Method ..................................................................... 16

Figure 3-2: Closed Loop ZN Tuning ................................................................................ 18

Figure 3-3: Relay Tuning Scheme .................................................................................... 19

Figure 3-4: Relay without Hysteresis ............................................................................... 22

Figure 3-5: Relay with Hysteresis..................................................................................... 23

Figure 3-6: Closed Loop System with Tuning Relay ....................................................... 24

Figure 3-7: Stability Analysis of Limit Cycle Operation with Relay ............................... 26

Figure 4-1: Plant with Compensator of Desired Structure................................................ 31

Figure 4-2: Proportional Gain of PI controller vs. NT Delay with ISE............................ 40

Figure 4-3: PI Integral Time Constant vs. NT Delay with ISE ........................................ 40

Figure 4-4: PI Proportional Gain vs. NT Delay with ISTE .............................................. 41

Figure 4-5: PI Integral Time Constant vs. NT Delay with ISTE...................................... 41

Figure 4-6: PI Proportional Gain vs. NT Delay with IST2E ............................................ 42

Figure 4-7: PI Integral Time Constant vs. NT Delay with IST2E.................................... 42

Figure 4-8: Gain Margin of ISE, ISTE, and IST2E Design .............................................. 47

Figure 4-9: Phase Margin of ISE, ISTE, and IST2E Design............................................. 47

Figure 4-10: Crossover Frequency of ISE, ISTE, and IST2E Design............................... 48

Figure 4-11: Integral Gain for ISE.................................................................................... 52

v

Figure 4-12: Integral Gain for ISTE ................................................................................. 52

Figure 4-13: Integral Gain for IST2E................................................................................ 53

Figure 4-14: Gain Margin for ISE, ISTE, and IST2E ....................................................... 53

Figure 4-15: Phase Margin for ISE, ISTE, and IST2E...................................................... 54

Figure 4-16: Crossover Frequency of ISE, ISTE, and IST2E ........................................... 54

Figure 5-1: Phasor Diagram.............................................................................................. 56

Figure 5-2: Simulation Scheme of Vector Current Control.............................................. 59

Figure 5-3: Simulation Result of Vector Current Control for Step Response .................. 60

Figure 5-4: Simulation Result of Vector Current Control for Sinusoidal Response ........ 60

Figure 5-5: Equivalent Phase Circuit ................................................................................ 63

Figure 5-6: Simulation Scheme of Sine Wave Current Control ....................................... 66

Figure 5-7: Simulation Result of Sine Wave Control for Step Response......................... 67

Figure 5-8: Simulation Result of Sine Wave Control for Sinusoidal Response ............... 68

Figure 6-1: System Schematic of Sine Wave Current Control ......................................... 70

Figure 6-2: Program Flow Chart of Sine Wave Current Control...................................... 75

Figure 6-3: Timing Chart of Sine Wave Current Control................................................. 76

Figure 6-4: System Schematic of Vector Current Control................................................ 76

Figure 6-5: Program Flow Chart of Vector Current Control ............................................ 78

Figure 6-6: Time Sequence of Current Sampling, Calculation and PWM Output ........... 81

Figure 6-7: Synchronous Reference Frame Quantities in the Stationary Reference Frame

................................................................................................................................... 82

Figure 6-8: Complex Vector Block Diagram of Vector Current Control ......................... 83

Figure 6-9: System Model of Vector Current Control (1) ................................................ 83

vi

Figure 6-10: System Model of Vector Current Control (2) .............................................. 84

Figure 6-11: Rotation of the Synchronous Reference Frame Corresponding to the Time

Delay ......................................................................................................................... 86

Figure 6-12 Phase Current Reference Generation with the Error Caused by Rotation .... 87

Figure 6-13: One Phase Current Control Loop in Sine Wave Current Control................ 87

Figure 6-14: First Level Vector Current Control Simulation Model................................ 88

Figure 6-15: First Level Sine Wave Current Control Simulation Model ......................... 88

Figure 6-16: Second Level Vector and Sine Wave Current Control Simulation Model .. 89

Figure 6-17: Vector Current Control PI Controller Design Model................................... 90

Figure 6-18: Eigenvalues for Different Speed .................................................................. 95

Figure 6-19: Simulation and Experimental Result at 0 rpm for Step Response ............... 97

Figure 6-20: Simulation and Experimental Result at 1000rpm for Step Response .......... 99

Figure 6-21: Simulation and Experimental Result at 2000 rpm for step response ......... 101

Figure 6-22: Simulation and Experimental Result at 3000rpm ...................................... 103

Figure 6-23: Simulation and Experimental Result at 0 rpm for Sine Wave Command . 105

Figure 6-24: Simulation and Experimental Result at 0 rpm for Step Response ............. 108

Figure 6-25: Simulation and Experimental Result at 1000 rpm for Step Response ....... 110

Figure 6-26: Simulation and Experimental Result at 2000rpm for Step Response ........ 111

Figure 6-27: Simulation and Experimental Result at 0 rpm for 100Hz Sine Wave

Command Response................................................................................................ 113

Figure 6-28: Relay Test Result ....................................................................................... 115

Figure 6-29: Step Responses for Different PI Controller Gain Settings......................... 117

vii

LIST OF TABLES

Table 2-1: Motor Specification........................................................................................... 8

Table 2-2: L180A-310A Drive Specification ..................................................................... 9

Table 3-1: Open Loop ZN Tuning Method....................................................................... 16

Table 3-2: Closed Loop Ziegler-Nichols Tuning.............................................................. 18

Table 4-1: LQ Tracker with Time Independent Performance Index [19]......................... 35

Table 4-2: LQ Tracker with Time Dependent Performance Index [19] ........................... 36

Table 4-3: Empirical Formula Coefficients Table (1) for PI Controllers ......................... 44

Table 4-4: Empirical Formula Coefficients Table (2) for PI Controllers ......................... 45

Table 4-5: Crossover Frequency Formula Coefficients for PI Controllers....................... 46

Table 4-6: Empirical Formula Coefficients Table for Integral Controllers...................... 51

Table 6-1: Inverter State Definitions ................................................................................ 72

Table 6-2: Inverter Switching Patterns ............................................................................. 73

Table 6-3: Auto Tuned PI Controller Gains ................................................................... 116

viii

ACKNOWLEDGMENTS

I am indebted to my advisor, Dr. Dale Schinstock, for the support and guidance

throughout the course of this work. I greatly appreciate his encouragement and patience.

I am also grateful to my committee members, Dr. Warren N. White and Dr. Shelli

K. Starrett, for their participation and advice.

I would like to thank my friends Scott Hammack, Mike Molt, and Michael

Trumble for their help in the past two and a half years.

This material is based upon work supported by the National Science Foundation's

Experimental Program to Stimulate Competitive Research (EPSCoR) program under

Grant No. 0091853. Any opinions, findings and conclusions or recommendations

expressed in this material are those of the author(s) and do not necessarily reflect the

views of the National Science Foundation (NSF).

1

Chapter 1: Introduction and Background

1.1. Background

In recent years, in the field of motion control, permanent magnet ac (PMAC)

machines have been widely used. PMAC machines can be separated into two major

classes according to the back EMF forms. The machines generating trapezoidal back

EMF are referred to as trapezoidal machines [1], and are also known as brushless dc

motors (BDCM) [2]. The machines generating sinusoidal back EMF are referred to as

sinusoidal machines [1], and are also known as the permanent magnet synchronous

motors (PMSM) [2].

In PMAC machines, the field windings are replaced by the permanent magnets,

which make it unnecessary to provide the magnetizing current to the rotor in the power

range of fractional Hp to 30kW, and the motor can operate with higher efficiency than an

induction motor. Slips and brushes are no longer needed which conduct direct current to

the field windings in the traditional synchronous machines, increasing the system

reliability. The use of the permanents magnet can also reduce the motor weight and

result in higher power density (kW/kg).

In motion control applications with PMAC machines, the control structures share

some characteristics. In order to implement the speed or position control, the multi loop

structure is commonly used. Figure 1-1 shows a typical multi loop control system with

PMAC machines.

2

Figure 1-1: Typical Motion Control Structure

The current control loop is used to minimize the machine and inverter

nonlinearity, and move the electrical dynamics to high frequency. It should control the

stator flux position according to the rotor position measured by an angle position sensor

such as a resolver. Generally the electric torque is closely related to the current in PMAC

machines. The current reference to the current loop can directly map the desired torque if

the current control loop is good.

The speed control loop is used to cancel unknown variation in the load torque,

correct the current control loop imperfections and the variation of torque to current ratio.

The position control will correct the position error, and get the desired position trajectory.

In some specific applications, there is only one of the two outer loops outside the

current control loop in the system. For example, in a position control system, the speed

loop can be absent, and if only speed control is desired, the position control will not be

necessary. In some digital implementations of the control system, since the analog speed

may be not available and the derivative calculation is not desirable in the digital control,

only position control loop appears outside the current control loop. Speed control can be

achieved via the position control loop by calculating the position trajectory in real time

Speed Controller

Position Controller

PMAC

Current Controller

ddt

rω

rθ

*rθ rθ∆

+− +

−*rω

rω∆

*I

Angle PositionSensor

3

with desired speed.

1.2. Introduction Currently most PMAC machines use voltage source inverter topologies.

Combined with the current control loop, this kind of configuration yields a current-

regulated voltage source inverter [1]. Physically the current control system for PMSM

and BDCM can share a similar structure. A typical configuration is shown in Figure 1-2.

This configuration is a two-stage configuration with the rectifier in the system and it can

perform ac-dc-ac conversion. In this figure and the experimental system, six transistors,

such as Isolated Gate Bipolar Transistors (IGBT), are used as power switches. Other

possible alternatives include Bipolar Junction Transistors (BJT), MOS Field Effect

Transistors (MOSFET), Gate Turn Off Thyristors (GTO), etc. Since the inverters for

BDCM and PMSM use the same structure, the difference in the switching pattern will be

responsible for the different phase current wave forms. Generally speaking, BDCM

requires six-step switched current waveforms to generate constant torque, while PMSM

requires sinusoidal current waveforms to generate constant torque. For both types, the

maximum torque can only be produced when the phase currents are properly

synchronized with the rotor position. During current control synchronizing phase currents

requires rotor position measurement. As for BDCM, considering the trapezoidal phase

currents, only six discrete points per electrical cycle needed to be detected, and this can

be achieved by using three hall sensors. As for PMSM, the continuous measurement of

the rotor position is necessary to produce the required sinusoidal phase currents. Usually

measurement is achieved using resolvers and encoders. Recently, the sensorless methods

have been attracting research interests toward eliminating the use of the position sensors

4

in the system [3, 4, 5].

Figure 1-2: Three Phase Two-Level Inverter

With the power of digital signal processors, the motor control systems tend to be

implemented with digital techniques recently. The computing power of the digital signal

processor (DSP) makes it possible to employ advanced control algorithms in the control

system. The replacement of hardware by software may reduce the system cost

significantly, and also make the system more flexible.

The importance of the current control loop is obvious in the motor control system.

This thesis will focus on current control for PMSM and its digital implementation.

In the PMSM drive application the current/torque control loop is the inner loop,

and field oriented control is used. Field oriented control in the PMSM drive application is

obtained by simply setting the direct axis current to zero in the rotor synchronous frame

[6].

In this thesis two current control structures are studied. One classical current

control structure is composed of two or three phase current control loops in the stationary,

abc, frame [7, 8, 9, 10, 11, 12]. Since the current references to the controllers are

sinusoidal functions of the rotor position, this structure is referred to as sine wave current

control. Another current control structure includes two current control loops in the rotor

synchronous, dq0, frame [7, 8, 9, 10, 13, 12]. In this thesis, this current control structure

PMACVac

5

is referred to as vector current control.

The further study involves the problems in the digital implementation of the

current control system and system modeling. In the digitally implemented current control

system, the basic plant model is a first order plus time delay system (FOPTD) for each

phase or axis. The systematic design methods of the digital current control system based

on linear quadratic (LQ) optimization are introduced.

Linear quadratic controller design is a time domain method. The basic idea of this

approach is to choose proper controller gains or output feedback gains to minimize the

performance index, which is a linear quadratic function of states, outputs and inputs.

Generally, the LQ design can guarantee good performance in terms of the step response.

To find the empirical formulae for automatic tuning, optimization is repeated for different

values of the normalized plant parameters. In this thesis, the author extends the empirical

optimization formulae for PI control of a FOPTD plant to cover a wider range of the

normalized time delay. The resulting system gain margin and phase margin are presented.

An original development is that the resulting system bandwidth is formulized empirically.

The author also originally presents a set of empirical optimization formulae for an

integral controller for a pure time delay system. Combining these resulting empirical

formulae with the models of the digitally-implemented current control system, the author

proposes systematic design methods to predict required digital update frequency and

PWM frequency in the PMSM current control system design process according to the

desired bandwidth and the maximum operation speed. Furthermore using a relay

feedback tuning process, the current control system can be tuned with these empirical

formulae.

6

Chapter 2: Experimental System

2.1. Overview

The experimental system includes the following hardware components as shown

in Figure 2-1:

• Host Computer

• MC8-DSP-ISA Board

• Interface Board

• Isolation Board

• Integrated Power Stage

• PMSM Brushless Servomotor with Integral Resolver

• Current Sense Board

• Resolver to Digital Converter Board

• Dynamotor Servo Drive

• PMSM Brushless Dynamotor with Integral Resolver

• Power Supply

The software includes the following programs:

• Host PC Interface Program

• L180 Graphical User Interface

• DSP Vector Current Control Program

• DSP Sine Wave Current Control Program

7

Figure 2-1: Experimental System

2.2. PMSM Brushless Servomotor and Dynamotor

In the experimental system, two permanent magnet synchronous motors are used

as the servomotor and dynamotor separately. The servomotor is MOOG’s G412-200, and

the dynamotor is MOOG’s G412-400. Both of them belong to the MOOG G410 Series.

MOOG’s G410 series motors are electronically commutated synchronous AC motors

with permanent magnet field excitation. The motors’ specifications can found in Table 2-

1.

8

Table 2-1: Motor Specification

Metric[English] G412-200 G412-400

Rated voltage, per phase pk pkV − 310 310

Maximum voltage, per phase pk pkV − 360 360

Continuous stall torque Mo [ ]Nm lb in− 0.25 [2.2] 0.50 [4.4]

Continuous stall current lo rmsA 0.65 1.2

Peak torque Mmax [ ]Nm lb in− 0.70 [6.2] 1.5 [13.3]

Peak current lmax rmsA 1.9 3.7

Nominal speed nN rpm 8100 7400

Nominal power PN [ ]W hp 0.15 [0.20] 0.33 [0.44]

Maximum speed nmax rpm 11500 10000

Torque constant kT [ ]rms rmsNm A lb in A− 0.37 [3.3] 0.42 [3.7]

Back EMF constant kb ( )[ ]pk pkV rad s V krpm 0.3[31.42] 0.34[35.60]

Motor terminal resistance Rtt Ω 50.9 20.7

Motor inductance Ltt mH 29.7 15.7

Inertia w/o brake 2 2 -4kg cm [lb - insec ×10 ] 0.09 [0.80] 0.13 [1.2]

Pole Pairs 4 4

2.3. L180 Digital Servo Drive Amplifier

The L180 servo drive amplifier is designed to control MOOG brushless servo

motors. The L180 servo drive is a fully digital drive. It has four operation modes, which

are analog velocity control, analog current/torque control, digital velocity control and

9

digital current/torque control. It can communicate with host computer via the serial port

connection. With “L180 Graphical User Interface”, the servo drive can be configured

according to the motor specification and the desired operation mode. In the experimental

system, the drive model is L180A-310A. The electrical specification is listed in the

following table.

Table 2-2: L180A-310A Drive Specification

3~main line Unit L180A-310A

Continuous current Arms 5

Apeak 7

Maximum current Arms 10

Apeak 14

Continuous power kW 1.9

Maximum power kW 3.9

Single phase operation causes a power reduction to 1/3 of the power rating above

2.4. DSP Board (MC8-DSP-ISA Board)

The MC8-DSP-ISA board is a product of Precision MicroDynamics Inc. It can be

used in data acquisition and motion control.

After the correct installation of both software and hardware on the host PC, the

DSP program can be downloaded directly into the on-chip memory of DSP chip

The board facilities are listed below:

• On-board Analog Devices 21061 (SHARC) DSP, with 128K words of

10

external zero wait-state SRAM. The DSP has full access to all board resources.

All of the DSP’s memory can be read by the host PC.

• 96 bits of digital input, divided into Port A, B and C each of 32 bits. Port A

can generate interrupts.

• 32 bits digital outputs, which can be enabled in 2 banks of 16 bits. Outputs

can be pulled high or low in groups of 8 bits via jumper selection.

• 16 Digital-to-Analog outputs, 14 bits, output range -10V to +10V.

• 8 Analog-to-Digital inputs, 14 bits, input range -10V to +10V.

• 8 Quadrature Decoders, capable of handling up to 5.33 million edges per

second (more if the signals are noise-free).

• Programmable Interval Timer (PIT). Self-loading timer generating an

interrupt and/or an on-board SYNC signal at intervals up to 1030 seconds with

a 240 ns resolution.

• Watchdog Timer: Reloadable countdown timer that zeros the DAC outputs

and generates an external signal when it expires. Up to 125ms delays, in 2us

increments.

• Interrupts: From the PIT signal, SYNC signal (internal and external source)

and from the digital inputs. Interrupts to the on-board DSP and PC are

generated independently.

• 8 PWM Outputs: The minimum PWM update frequency is 7814Hz with

32MHz clock.

2.5. PowIRTrain

The IRPT1058 PowIRTrain from International Rectifier is an integrated power

11

stage for 0.75hp motors. It provides the complete power function for a 0.76 hp variable

frequency, variable voltage AC motor controller. The PowIRTrain combines a power

module IRPT1058A with a Driver-Plus Board IRPT1058D. The power module

IRPT1058A contains a single phase input bridge rectifier, three-phase IGBT inverter,

current sensor shunts and a thermistor. The Drive-Plus Board contains DC link capacitors,

capacitor soft charge function using NTC thermistor, surge suppression MOV, IGBT gate

drivers, DC bus voltage and current feedback signals, protection circuit and local power

supply. PowIRTrain can provide 3-phase PWM power supply with variable frequency

and variable voltage with a user-provided PWM controller. In Figure 2-2, the simplified

block diagram is shown within an AC motor control system.

Figure 2-2: The IRPT 1058 PowIRTrain within a Motor Control System

2.6. Shaft Position Sensor

The motor shaft position is measured by the resolver integrated into the motor and

IRPT1058C

System Power Supply

PWM Generator

Feedback Processing

Keyboard

180-240V Single phase input

PWM variable Frequency output

Feedback (Non-isolated)

AC Motor

12

an external resolver-to-digital-converter board. On the resolver to digital converter board,

the main chips are AD2S99 and AD2S90. AD2S99 is a programmable sinusoidal

oscillator. It provides sine wave excitation for resolvers, and also provides a synchronous

reference output signal (3V p-p square wave) that is phase locked to its SIN and COS

inputs. By setting the corresponding jumpers, AD2S99 can be programmed to generated

2 kHz, 5 kHz, 10 kHz and 20 kHz excitations.

AD2S90 is a complete 12-bit resolution tracking resolver-to-digital converter. A

Type Π servo loop is used to track the input sine and cosine signals and convert them into

digital representation of the input angle. The bandwidth is set internally at 1 kHz. The

maximum tracking rate is 375 RPS at 12-bit resolution. The shaft angle information is

available in two forms, absolute series binary and incremental A quad B. One absolute

binary output has 12 bits of resolution. The encoder emulation output A, B, and NM

continuously produce signals equivalent to a 1024 line encoder.

2.7. Isolator Board

The isolator board is used to provide galvanic isolation between the PowIRTrain

and DSP Board via the Interface Board. The isolated digital signals include 6 PWM drive

signals from the DSP Board to PowIRtrain, the PWMTRIP fault signal from the

PowIRTrain to the DSP Board. The isolated analog signals include the bus voltage

feedback signal from PowIRTrain to the DSP Board.

2.8. Current Sensor Board

Two Current Sensor Boards are stacked. Each sensor board has two current

sensors. Each sensor is connected within the circuit in series and can provide an analog

13

voltage output proportional to measured current. These analog voltages are scaled for the

input to the ADC of the DSP Board. These current sensors can sense the maximum

current of 8 Amps for DC, AC or pulsed currents, and provide suitable output for

sampling.

2.9. Power Supply

In the system, 230V AC power for drives and motors is provided by Line

Isolation-Startup Transformer 298F.

14

Chapter 3: PID Control Tuning Method Review and

System Identification by Relay Feedback

3.1. Introduction

Proportional integral derivative (PID) controllers have been widely used in

industry, and it seems their popularity will persist for the near future. In this chapter the

typical PID tuning methods will be reviewed, and then the relay feedback is introduced in

the application of system identification.

In the past, various forms of the PID controller have been developed for specific

purposes. In the time domain the standard PID controller has the form below.

0

1( ) ( ) ( ) ( )t

p di

du t K e t e t dt T e tT dt

= + +

∫ (3-1)

re y y= − (3-2)

Its standard frequency domain form is

1( ) (1 )c p di

G s K T sT s

= + + (3-3)

3.2. Ziegler-Nichols (ZN) Method

ZN method was first introduced by Ziegler and Nichols in 1940s [14], and has

been widely known as the heuristic method to set the gains of PID controller for good

performance in industrial processes. The ZN tuning method has an open loop version and

a closed loop version.

15

3.2.1. Open Loop ZN Tuning Method

Open loop ZN tuning is applicable to any plant that can be approximated by a first

order plus time delay system (FOPTD). The plant transfer function is

( )1p

K sG s eTs

τ−=+

(3-4)

In this method, the controller gains are set by a simple open loop test. With the

controller in the manual mode, the plant input is changed suddenly. The plant output

follows one of the curves shown in Figure 3-1. In the self regulated system response, a

straight line is drawn tangent to the system response at the inflection point to determine

the plant time constant T , dead time τ and slope S . For an unstable system, the slope S

is determined by drawing a straight line along the ascent of the response as shown Figure

3-1. After the identification of the dead time τ and slope of the rise S , the normalized

slope can be computed by (3-5)

* SSp

= (3-5)

The PID controller gains are given in Table 3-1 by Ziegler-Nichols open loop tuning

method.

16

Figure 3-1: Open Loop ZN Tuning Method

Table 3-1: Open Loop ZN Tuning Method

Controller Type pK iT dT

P *

1Sτ

N/A N/A

PI *

0.9Sτ

3.33τ N/A

PID *

1.2Sτ

2τ 0.5τ

Controller Output

Tτ

p

Time

Self Regulated System Response

Unstable System Response

t

T

T/S T T=

/S T t=

τ

17

The open loop ZN tuning process is simple since only one experiment is

necessary. The system is not required to operate at its stability limit during the

experiment. With the experiment result the controller gains can be calculated very easily

by applying the table above. However, since this experiment is performed under open

loop condition, if a significant load change occurs during the test, no correction is taken

and the test result may be easily distorted. Sometimes if the measurement is noisy, it may

be difficult to determine the slope at the inflection point accurately. This method is also

sensitive to the controller calibration errors.

3.2.2. Closed Loop ZN Tuning Method

The closed loop ZN tuning method is a frequency domain tuning method. In the

beginning of this tuning process, only the proportional controller is applied and all other

controller gains are set to zeros. The proportional gain should be small enough to make

the system stable at the beginning. Then the proportional gain is carefully increased until

a steady oscillation occurs for a small step input. This is often referred to by the term

“continuous cycling,” which means a sustained oscillation with constant amplitude. The

ultimate gain uK is the proportional gain that causes the continuous cycling in the system.

The oscillation period is the ultimate period uP as indicated in Figure 3-2. After finding

the ultimate gain and ultimate period, the PID controller gains can be calculated by Table

3-2 given by the closed loop ZN tuning method.

18

Table 3-2: Closed Loop Ziegler-Nichols Tuning

Controller Type

pK iT dT

P 0.5 uK N/A N/A

PI 0.45 uK 1.2

uP N/A

PID 0.6 uK 2uP

8uP

Limit Cycle

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

1 1.2 1.4 1.6 1.8 2

Time(Sec)

Out

put

Pu

Figure 3-2: Closed Loop ZN Tuning

In practice, the original closed loop ZN method may be difficult to apply in the

working plant. In real world systems, to make the system work at the stability limit may

not be acceptable. Another problem is that this method often ends with poor damping

corresponding to 0.2ζ ≅ [15], which may be too small for many applications. The

resulting closed-loop system can be very sensitive to system parameter variation.

19

3.3. Relay Feedback Tuning

Before the relay was introduced into auto tuning for PID controllers by Astrom

and his co-workers in 1980s [16], it was widely used in the 1950s as an amplifier. It was

also applied in adaptive control in the 1960s. A typical example was the self-oscillating

adaptive controller developed by Minneapolis Honeywell at that time. That controller

could offer an amplitude margin of mA 2= , and was used in several missiles. The details

can be found in [17].

3.3.1. Astrom-Hagglund Method

In 1980s, Astrom and his co-workers introduced the application of the relay into

the PID controller auto tuning procedure [16, 18]. Figure 3-3 corresponds to the Astrom-

Hagglund method. In this tuning procedure, the system is first driven to a convenient

steady state. Then the control switches to the tuning relay. This results in a permanent

oscillation. By measuring the oscillation amplitude and frequency, one point on the

Nyqusit plot of the plant can be approximated by the describing function of the relay.

Figure 3-3: Relay Tuning Scheme

Gp(s)

Tuning Relay

PID

y(t)u(t)y* (t) +

-

20

3.3.2. Describing Function of a Relay

System identification by using a relay is closely related to the describing function

of the relay. The describing function can describe the nonlinear system or component,

and is based on the two basic assumptions:

• If the input to the nonlinear element is sinusoidal, though the output of the

nonlinear element is not sinusoidal, the output of the nonlinear element is

periodic with the same period as the input;

• Only the fundamental harmonic component of the output is significant.

These two assumptions are valid since various nonlinear elements can satisfy

them, such as a relay. Generally, the higher order harmonic components of the output

have much smaller amplitudes than the fundamental harmonic component of the output.

Most of time the linear system is like a low-pass filter and the higher order harmonic

components are attenuated by the linear system.

The describing function is defined as the complex ratio of the fundamental

harmonic component of the output to the input.

11

YN=X

φ∠ (3-6)

Where

N = describing function

X = amplitude of the input sinusoid

1Y = amplitude of the fundamental harmonic component of output

1φ = phase shift of the fundamental harmonic component of output

21

To find the fundamental harmonic component of the nonlinear element output, the

output ( )y t must be expressed in the Fourier series.

01

01

( ) ( cos sin )

( sin( ))

n nn

n nn

y t A A n t B n t

A Y n t

ω ω

ω φ

∞

=

∞

=

= + +

= + +

∑

∑ (3-7)

Where

2

0

1 ( ) cos ( )nA y t n td tπ

ω ωπ

= ∫ (3-8)

2

0

1 ( )sin ( )nB y t n td tπ

ω ωπ

= ∫ (3-9)

2 2n n nY A B= + (3-10)

1tan ( )nn

n

AB

φ −= (3-11)

3.3.3. Describing Function of Relay without Hysteresis

The input and output waveform of a relay without hysteresis is shown in Figure 3-

4. For the sine wave input, the output is a square wave as shown. In the corresponding

Fourier series, 0A is zero since the nonlinearity is symmetric, and the coefficients of the

cosine terms are also zeros since the output waveform is an odd function.

Hence

1( ) sinn

ny t B n tω

∞

==∑ (3-12)

The fundamental harmonic component of ( )y t is

1 1 1( ) sin siny t B t Y tω ω= = (3-13)

22

Where

2

10

2

0

1 ( )sin ( )

1 1 4sin ( ) ( )sin ( )

Y y t td t

MM td t M td t

π

π π

π

ω ωπ

ω ω ω ωπ π π

=

= + − =

∫

∫ ∫ (3-14)

Hence

14( ) sinMy t tωπ

= (3-15)

The describing function is given then by

1Y 4N= 0X

MXπ

°∠ = (3-16)

Figure 3-4: Relay without Hysteresis

3.3.4. Describing Function of a Relay with Hysteresis

For the relay with hysteresis, its output and input wave is shown in Figure 3-5.

input

output

t

t

x

M

M

-M

y(t) y(t)=Y1sin(ωt)

x(t)=Xsin(ωt)

23

Figure 3-5: Relay with Hysteresis

The plot shows that the output of the relay is square wave with a phase lag

11 sin ( )t h Xω −= . The fundamental harmonic component is

1 1 1 1 1( ) cos sin sin( )y t A t B t Y tω ω ω φ= + = + (3-17)

Where

2

1 10

1 4 4( )cos ( ) sinM MhA y t td t tX

π

ω ω ωπ π π

= = − = −∫ (3-18)

2 2

1 1 20

1 4 4( )sin ( ) cos 1M M hB y t td t tX

π

ω ω ωπ π π

= = = −∫ (3-19)

and

11 sin ( )t h Xω −= (3-20)

So

14MYπ

= (3-21)

and

input

output t

t

x

M

M

-M

y(t) y(t)=Y1sin(ωt+Ф)

x(t)=Xsin(ωt)

-h h

t1

h

-h

24

11 sin ( )h

Xφ −= − (3-22)

The describing function can be given then

111

Y 4N= sin ( )X

M hX X

φπ

−∠ = ∠− (3-23)

3.3.5. Describing Function Analysis of a Relay in the Loop

Assuming the plant is a minimum phase system, the closed loop system with a

relay in the control loop is shown in Figure 3-6

Figure 3-6: Closed Loop System with Tuning Relay

If the higher harmonics can be well attenuated, and the relay can be approximated

by its describing function, then the close loop transfer function is:

( ) ( )( )( ) 1 ( )cl

C j NG jG jR j NG j

ω ωωω ω

= =+

(3-24)

The characteristic equation is:

1 ( ) 0NG jω+ = (3-25)

Going further:

1( )G jN

ω = − (3-26)

The 1N

− locus of a relay is a straight line parallel to the real axis in the complex plane

for any given relay as demonstrated by (3-27).

G(s) Tuning Relay c(t)r (t) +

-

25

1 2 21 sin ( )4 4 4

X h hX h jN M X M M

π π π−− = − ∠ = − − − (3-27)

Generally speaking, if the plant’s ( )G jω Nyquist locus goes through the third

quadrant, it may intersect the 1N

− locus, and the system may exhibit a sustained

oscillation or limit cycle at the intersection point.

In Figure 3-7, assume the system originally operates at point A, where the

oscillation amplitude is aX and the frequency is aω . If a slight disturbance is applied to

the system and the input to the relay is increased, the operating point moves from A to B.

Point B corresponds to the critical point or to be -1+j in the complex plane for the linear

system. In this case, since the ( )G jω locus does not enclose point B in Nyquist sense, the

amplitude will decrease and the operating point will move backwards to A. If the slight

disturbance causes the input to the relay to decrease slightly, the operating point moves

from point A to point C. Similar to the case in which the ( )G jω locus encloses the critical

point 1 j− + in linear system analysis, the amplitude of the oscillation will increase, and

the system operating point will move backwards to point A. So the point A possesses

convergent characteristics and the system operation at the point A is stable (i.e. the limit

cycle at this point is stable). By similar analysis, if the point A possesses divergent

characteristics, it corresponds to an unstable limit cycle.

26

Figure 3-7: Stability Analysis of Limit Cycle Operation with Relay

3.3.6. Critical Point Identification

When a relay without hysteresis is used in the system tuning or identification

process, since the describing function of the relay is real, the relay is like a proportional

amplifier and its gain is 4MNXπ

= . If the stable oscillation occurs with this relay, and

considering that the plant is a low pass filter which attenuates the higher harmonics, the

output of the system will be sinusoidal or a sinusoid-like wave having magnitude X.

From (3-25):

( ) 1NG jω = (3-28)

The ultimate gain can be approximated by:

1 4( )u

MKG j Xω π

= = (3-29)

G(jw)

A CB

Re

Im

X →∞ 0X →1N

−

27

The system ultimate gain and ultimate frequency can be easily approximated

using the relay feedback test. After the test, various methods like the ZN method can be

used to calculate the PID or PI controller gains based on the ultimate gain and ultimate

frequency.

Compared with the classical Ziegler-Nichols method, relay feedback is easier to

automate and also avoids tedious changing of the proportional gain.

3.3.7. Parameter Identification via Relay Test for Specific

Systems

For some specific systems, we may know the system model form and only some

parameters are unknown. In this case, the system parameters may be identified by the

relay test. For example, if the plant is a FOPTD system then it is described by (3-4),

which is repeated (3-30)

( )1p

K sG s eTs

τ−=+

(3-30)

Assuming the DC gain K is known, the plant time delay and time constant can be found

by (3-31) and (3-32)

22 1u

u

K KT

ω−

= (3-31)

22arctan( 1)u

u

K Kπτ

ω− −

= (3-32)

If a relay with hysteresis is used in the relay feedback test, the locus of 1N

− is a

straight line parallel to the real axis in the third quadrant, and its offset from the real axis

i.e. the imaginary part of the locus, is determined by h and M . By varying the magnitude

28

of hysteresis h , multiple points on the Nyqusit locus of ( )pG jω can be identified. For

the FOPTD, a two-point test can give enough information to identify the system

parameters.

If a point 1 1 1( , )X ω and a point 2 2 2( , )X ω have been identified, from (3-26), (3-

33) and (3-34) can be found for point 1. (3-35) and (3-36) can be found for point 2. The

system parameter , , and K T τ can be found by solving these equations.

122

141

K XMT

πω

=+

(3-33)

1 1 11 1

1

tan ( ) sin ( )hTX

τω ω π− −+ = − (3-34)

222

241

K XMT

πω

=+

(3-35)

1 1 22 2

2

tan ( ) sin ( )hTX

τω ω π− −+ = − (3-36)

For the second order plus time delay system, the system parameters can also be

identified by a two-point relay test. Using (3-37) to describe this system,

2 2( )2 1pK sG s e

T s Tsτ

ζ−=

+ + (3-37)

After identification of the points 1 and 2, which correspond to 1 1( , )X ω and 2 2( , )X ω ,

using (3-26), equations (3-38) to (3-41) can be solved for the system parameters.

12 22 2 2

11 14(1 ) 4

K XMT T

πω ζ ω

=− +

(3-38)

1 11 1122

11

2tan ( ) sin ( )1

T hXT

ζ ω τω πω

− −+ = −−

(3-39)

29

22 22 2 2

22 24(1 ) 4

K XMT T

πω ζ ω

=− +

(3-40)

1 12 2222

22

2tan ( ) sin ( )1

T hXT

ζ ω τω πω

− −+ = −−

(3-41)

30

Chapter 4: Optimal Design for PID Controllers via LQ

Approach

4.1. Introduction to LQ Design

Linear quadratic controller design is a time domain design method. The basic idea

is to select proper controller gains or output feedback gains to minimize the performance

index, which is a linear quadratic function of states, outputs and inputs.

Most of what follows in this section is a summary and rework of methods

presented by Stevens and Lewis [19] for clarity. The system structure is represented in

Figure 4-1. This approach allows designing a servo control system that has any structure.

The complete structure includes a unity-gain outer loop and the compensator dynamics.

The control gains include two parts. The first part includes the plant output feedback

gains, and the second part includes the compensator output gains. The control gains are

chosen to minimize a quadratic performance index. In this approach, the controller

structure can be determined before the gain design. In some cases this is desired because

there is a wealth of experience and knowledge indicating that some specific controller

structure will yield good control performance for the specific plant. For example, for the

first order plus time delay system, a PI controller can be used in practice to get the good

control performance.

31

Figure 4-1: Plant with Compensator of Desired Structure

The plant in the above figure can be described by

p p p p px A x B u= + (4-1)

p p py C x= (4-2)

p p pz H x= (4-3)

with the state ( )px t , the control input to the plant ( )pu t , the measured output available

for the output feedback ( )py t and the performance output ( )pz t defined in Figure 4-1.

The controller dynamics are described by

w Fw Ge= + (4-4)

v Dw Je= + (4-5)

where ( )w t is the compensator state vector, and ( )v t is the compensator output. The

input to the compensator is the tracking error ( )e t .

( ) ( ) - ( )pe t r t z t= (4-6)

Compensator -L Plant

Hp

Cp -Kpl

Performance Output zp(t)

Measured Output

xp(t) u(t) v(t) e(t) r(t)

yp(t)

32

The matrices , , ,F G D J are known for the given compensator structure. Now the control

input to the plant can be described by

pl pu K y Lv= − − (4-7)

where , plK L should be found by minimizing the performance index.

The complete system with the compensator is described by the following state

equations.

0 0ddt 0

pp p p

p

Ax x Bu r

GH F Gw w = + + −

(4-8)

0 0pp p

p

Cy xr

JH D Jv w

= + − (4-9)

]0 pp

xz H

w

=

(4-10)

The control input to the plant is

] ppl

yu K L

v

= −

(4-11)

The system state equation can be further simplified by redefining the state, output and

matrix variables.

x Ax Bu Gr= + + (4-12)

y Cx Fr= + (4-13)

z Hx= (4-14)

f f fu K y K Cx K Fr= − = − − (4-15)

Here the state ( ) nx t R∈ , control input ( ) mu t R∈ , reference input ( ) qr t R∈ , performance

output ( ) qz t R∈ , and measured output ( ) py t R∈ . The augmented matrices and vectors in

33

(4-12) through (4-15) are defined by comparing to (4-8) through (4-11). The closed loop

system is

( ) ( )

f f

c c

x A BK C x G BK F rA x B r

= − + −

≡ + (4-16)

At this point, it is convenient to define the deviation system because this will

allow the optimal tracking problem to be converted to the regulation form. Denoting the

steady state value by overbars and the deviation from the steady state values by tildes,

and then the state, output and input deviation are given by

( ) ( )x t x t x= − (4-17)

( ) ( )y t y t y Cx= − = (4-18)

( ) ( )z t z t z Hx= − = (4-19)

The tracking error ( ) ( ) ( )e t r t z t= − is given by

( ) ( )e t e t e= + (4-20)

And the error deviation can be given by the follow if the input is a step of size 0r .

0 0( ) ( ) ( ) ( )e t e t e r Hx r Hx Hx z= − = − − − = − = − (4-21)

Ac is nonsingular since in any acceptable design the closed-loop plant will be

asymptotically stable. In steady state,

00 c cA x B r= + (4-22)

10c cx A B r−= − (4-23)

The steady state error is

10 0(1 )c ce r Hx HA B r−= − = + (4-24)

From (4-17),

34

( ) ( )x t x t x= + (4-25)

Plug (4-23) and (4-25) into (4-16), the deviation system can be expressed as

cx A x= (4-26)

y Cx= (4-27)

z Hx e= = − (4-28)

For the tracking problem, the performance index is given in terms of the deviation

system. Considering that the design is based on step-response shaping, and the closed-

loop plant is asymptotically stable, the system state and output will approach their steady

values over time under the step reference command. So the steady values of the state and

output will not be zero, and it is not convenient to incorporate the state and output into

the performance index. On the other hand, it is concerned how well the system state and

output approach their steady values, which can be represented by the dynamics of

deviation system. The time independent performance index is given by (4-29) and the

time dependent index is given by (4-30).

0

1 1( )2 2

T T TJ x Qx u Ru dt e Ve∞

= + +∫ (4-29)

0

1 ( )2

k T T TJ t x Px x Qx u Ru dt∞

= + +∫ (4-30)

For these time independent and time dependent performance indices, local

minimization solutions based on gradient techniques are given in [19]. The details of the

solutions are re-listed in Table 4-1 and Table 4-2. At this point the design of the

controller becomes the selection of the weighting matrices. Generally matrices

, , , P Q R V are symmetric positive semi definite. Engineering experience is important

guidance for the selection of weighting matrices. Stevens and Lewis give some useful

35

suggestions in [19].

Table 4-1: LQ Tracker with Time Independent Performance Index [19]

0

System Model

Control-

Performance Index

1 1( )2 2

Optimal Output Feedback Gain

0=

0=

0= (

f

T T T

T TTmc c f f

Tmc c

TT T T Tmf c

x Ax Bu Gry Cx Frz Hx

u K y

J x Qx u Ru dt e Ve

H A P PA Q C K RK CS

H A S SA XP

H RK CSC B PSC B A PP

∞

−

= + += +=

=

= + +

∂ = + + +∂∂ = + +∂∂ = − +∂

∫

0

0

10

0

10 0

)

with a step magnitude and

where

Optimal Cost 1 1( )2 2

TT T T T Tc

c c

T T TTc c c c

c f

c f

T

H VH xy B A H Vr y

r r

x A B ry Cx Fr

X xx A B r r B A

A A BK CB G BK F

J tr PX e Ve

−

−

− −

+ −

= −= +

= =

= −

= −

= +

36

Table 4-2: LQ Tracker with Time Dependent Performance Index [19]

0

0 0 0

1 1 1 1

-1 1 1 2

System Model

Control-

Performance Index

1 ( )2

Optimal Output Feedback Control

0

0

0

0

f

k T T T

Tc c

Tc c

Tk c k k c k

Tk c k

x Ax Bu Gry Cx Frz Hx

u K y

J t x Px x Qx u Ru dt

g A P P A P

g A P P A P

g A P P A P

g A P

∞

− − −

= + += +=

=

= + +

= ≡ + +

= ≡ + +

= ≡ + +

= ≡

∫

1

1 1

2 2 1

0 0 1

0 0

0

10

0

!

0

0 !

0

0

0= ( )

with a step magnitude and

TTk c k f f

Tc k k c

Tc k k c k

Tc k k c k

Tc c

TT T T T Tmf k k k c k

c c

Tc

P A k P Q C K RK C

A S S A X

A S S A k S

A S S A S

A S S A SH RK CS C B P S P S C B A P xyPr r

x A B ry Cx Fr

X xx A

−

− −

− − −

−

−

−

+ + + +

= + +

= + +

= + +

= + +∂ = − + + +∂

= −= +

= = 10 0

where

Optimal Cost 1 ( )2

T T Tc c c

c f

c f

B r r B A

A A BK C

B G BK F

J tr PX

−

= −

= −

=

37

4.2. PI/PID Controller Design by LQ Approach

In this section, the design methods of PI controllers for FOPTD systems and PID

controllers for the second order plus time delay systems are developed based on LQ

design techniques. Also empirical formulae for the gains of the PI and PID controller are

found for specific performance indices.

4.2.1. PI Controller for FOPTD

The plant is the first order plus time delay system described by (4-31)

( )1p

K sG s eTs

τ−=+

(4-31)

The PI controller has the form

1( ) (1 )ic p p

i

KG s K Ks T s

= + = + . (4-32)

In order to facilitate the procedure of finding the empirical tuning formulae, the

normalized plant and control transfer functions are defined with respect to the plant time

constant T . Also, the normalized plant DC gain is forced to be one by moving the

original DC gain K to the controller proportional gain. Defining the new variable 's Ts= ,

the plant and controller transfer functions can be rewritten as

'' '

'' '

1 1( )1 1p

s sTG s e es s

ττ− −= =

+ + (4-33)

''' '

'

1 1( ) (1 )= (1 )c p pi i

G s KK KT T ssT

= + + . (4-34)

In the normalized plant transfer function, only the normalized time (NT) delay

affects the LQ design result. The Pade approximation is used to approximate the time

38

delay term, and then the normalized plant transfer function is converted to a state

equation. The resulting state equation is shown below,

p p p p px A x B u= + (4-35)

p p p pz y C x= = (4-36)

The PI controller dynamics are described by the state equation

w Fw Ge= + (4-37)

v Dw Je= + (4-38)

where [ ] [ ] 0 10 , 1 , , and

1 0F G D J = = = =

.

Since only the unity-outer loop feedback is used, the complete system dynamics

are

0 00

p pA Bx xd u rw F w Gdt GH

= + + −

(4-39)

[ ] [ ] [ ]xv JH D J r

w

= − +

(4-40)

]0x

z Hw

=

(4-41)

' 'p iu K K v = − (4-42)

The optimal feedback gain vector for the specific performance index is the PI controller

gains. When the optimal feedback gain vector is found, the PI controller gains will be

determined immediately. The design result depends on the performance index selection.

In this thesis, several specific performance indices are studied in detail. The first

performance index studied is the time-independent performance index, where

39

, 0, and 0TQ H H R V= = = . Essentially, this performance index is the well known

integral square error (ISE) performance criterion. The second performance index studied

is the time-dependent performance index where TP H H= , 0Q = , 0R = , 0V = and

2k = . It is the known as the integral squared time weighting error (ISTE) performance

criterion. The third performance index studied is the time-dependent performance index

where , 0, 0, 0, 4.TP H H Q R V and k= = = = = It is known as IST2E. The empirical

formulae for the above three performance indices have been developed by M. Zhuang

and D.P. Atherton in a normalized time delay range of 0.1 to 2 [20]. In this thesis, the

formulae will be developed to cover a wider normalized time delay range for the three

performance indices.

To find the empirical formula for automatic tuning, optimization is repeated for

different values of the normalized delay time Tτ , The time delay term is approximated by

the first order Pade approximation when the normalized time delay is less than 0.1,

otherwise, the 3rd order Pade approximation is used.

From Figure 4-2 to Figure 4-7, the relationships between the optimal proportional

gain, the integral time constant, and the normalized time delay for the performance

indices of ISE, ISTE and IST2 E are shown. In these figures, the original LQ design result

and the empirical formulae are plotted together. It should be noticed that in the

proportional gain figures, the logarithmic coordinates are used.

40

Kp(ISE)

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00

Normalized Time Delay

Kp

LQ Design Emprical Formula

Figure 4-2: Proportional Gain of PI controller vs. NT Delay with ISE

Ti(ISE)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


Ti

LQ Design Empirical Formula

Figure 4-3: PI Integral Time Constant vs. NT Delay with ISE

41

Kp(ISTE)

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Kp

LQ Design Emperical Formula

Figure 4-4: PI Proportional Gain vs. NT Delay with ISTE

Ti(ISTE)

0.00

1.00

2.00

3.00

4.00

5.00

6.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


Ti


Figure 4-5: PI Integral Time Constant vs. NT Delay with ISTE

42

Kp(IST^2E)

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Kp


Figure 4-6: PI Proportional Gain vs. NT Delay with IST2E

Ti(IST^2E)

0.00

0.50

1.00

1.50

2.00

2.50

3.00

3.50

4.00

4.50

5.00

0.00 2.00 4.00 6.00 8.00 10.00 12.00


Ti


Figure 4-7: PI Integral Time Constant vs. NT Delay with IST2E

The empirical formulae are obtained using least square techniques to fit to the LQ

design results. The following general formulae are used. The coefficients of the formulae

43

depend on the performance index and the value of the normalized time delay.

'p

1K ''

bac d

ττ

= ++

(4-43)

2' ' 'iT e f gτ τ= + + (4-44)

The actual PI controller proportional gain and integral time constant can be expressed

then by

p1 1( )bK aK T c d

T

ττ

= + +

(4-45)

2( ) ( )iT e f g TT Tτ τ = + +

(4-46)

The coefficients of the empirical formulae are listed in Table 4-3 and Table 4-4.

44

Table 4-3: Empirical Formula Coefficients Table (1) for PI Controllers

Tτ range 0.01~0.1 0.1~1

Performance

Index

ISE ISTE IST2E ISE ISTE IST2E

a 6.1918E-01

3.0904E-01

6.9730E-02

3.3739E-01

2.1748E-01

-2.4465E+00

b -1.0321E+00

-1.0538E+00

-1.2042E+00

8.6781E-02

2.5543E-01

-1.0251E+00

c 6.8788E+00

3.8624E+00

2.0161E+00

1.3393E+00

1.7348E+00

3.3274E-01

d 3.1912E-01

1.7662E-02

5.2334E-03

4.7205E-04

-2.8402E-03

-2.1071E-03

e 7.0441E+00

3.0128E+00

8.7634E-01

4.3507E-02

-4.8355E-02

1.1961E-01

f -5.3836E-01

6.3030E-02

2.7341E-01

3.8175E-01

4.1762E-01

7.1533E-02

g 1.4664E+00

1.0013E+00

9.9852E-01

1.4373E+00

9.9912E-01

1.0010E+00

45

Table 4-4: Empirical Formula Coefficients Table (2) for PI Controllers

Tτ range 1~5 5~10

Performance

Index

ISE ISTE IST2E ISE ISTE IST2E

a 3.3585E-01

2.4310E-01

3.7763E-01

3.8091E-01

2.9030E-01

3.1022E-01

b 1.1625E-01

1.4896E-01

-1.1864E+00

6.3547E-02

7.4693E-02

-1.2548E+00

c 1.3471E+00

1.8692E+00

-5.7016E-02

1.5063E+00

2.2243E+00

-9.9624E-03

d -1.0742E-02

-5.4684E-02

3.9433E+00

8.3843E-02

1.3721E-01

3.5255E+00

e 1.1244E-02

1.1052E-02

1.2896E-02

1.8999E-03

1.7445E-03

1.5687E-03

f 4.6570E-01

3.5339E-01

2.6528E-01

5.4811E-01

4.3604E-01

3.6336E-01

g 1.3789E+00

1.0022E+00

8.9982E-01

1.1933E+00

8.1472E-01

6.8359E-01

In Figure 4-8 and Figure 4-9, the gain margin and phase margin are shown against

the normalized time delay in the range of 0.01 to 10. The time dependent performance

indices (ISTE and IST2E) give about 60 degrees of phase margin and the gain margin is

in the range of 6dB to 9.5dB. The time independent performance index give a gain

margin of 6dB and the phase margin varies from 45 degrees to 65 degrees. Based on such

empirical formulae, the bandwidth is also predictable. In Figure 4-10, the crossover

frequency vs. the normalized time delay is plotted. Empirical formulae for the bandwidth

as a function of the normal time delay have been found using the least squares fit

46

techniques. These formulae have the form as given below, and their coefficients are

listed in Table 4-2.

( )bc

aT T

τω = (4-47)

The bandwidth frequency bf :

( )2 2

c bb

afT T

ω τπ π

= = (4-48)

Table 4-5: Crossover Frequency Formula Coefficients for PI Controllers

ISE ISTE IST2E

a 0.8127 0.6701 0.5959

b -0.9850 -0.9439 -0.9492

This bandwidth relationship will be useful in the systematic design approach

developed in the chapter 6. They will be used to help choose lower bounds on the update

and PWM frequency of a digital motor controller.

47

Gain Margin

0.00

1.00

2.00

3.00

4.00

5.00

6.00

7.00

8.00

9.00

10.00

0.01 0.10 1.00 10.00


Gai

n M

argi

n (d

B)

ISE ISTE IST^2E

Figure 4-8: Gain Margin of ISE, ISTE, and IST2E Design

Phase Margin

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

80.00

0.01 0.10 1.00 10.00


Phas

e M

argi

n (d

eg)

ISE ISTE IST^2E

Figure 4-9: Phase Margin of ISE, ISTE, and IST2E Design

48

Crossover Frequency (rad/sec)

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Cro

ssov

er F

requ

ency

(rad

/sec

)

ISE ISTE IST^2E

Figure 4-10: Crossover Frequency of ISE, ISTE, and IST2E Design

4.2.2. PID Controller for the Second Order plus Time Delay

System

In the controller design process, if pole-zero cancellation is applied, the system

can be simplified significantly. For example, in the PI controller design for a FOPTD, if

the PI controller zero is designed to cancel the plant pole, the resulting system only

includes an integral controller and a pure time delay. Similarly, in the PID controller

design for a 2nd order plus time delay system, the same simplified system can obtained by

using controller zeros to cancel the plant poles. For this simplified system, the integral

controller gains can be found by the LQ approach. If such design is repeated against the

normalized time delay, empirical formulae can be obtained for the controller gain and

resulting system bandwidth. In this section, the design process is based on a PID

controller for a 2nd order plus time delay system for the purpose of generality, while the

49

result is easily applied on the PI controller design for a FOPTD. These methods are not

used in later sections, although they are presented here. They are presented as an

alternative to the controller design methods presented in the previous section. These

previous methods are used in later sections.

For the second order plus time delay system described by (4-49)

2 2( )2 1pK sG s e

T s Tsτ

ζ−=

+ + (4-49)

The PID controller has the form

2( 1)( ) =

pdi

i i ic p d

KKK s sK K KG s K K ss s

+ += + + (4-50)

Defining 's Ts=

'

2( ')' 2 ' 1

sT

pKG s e

s s

τ

ζ−

=+ +

(4-51)

22

1 1( ' ' 1)( ')= 1 '

pdi

i ic

KKK s sK T K TG s

sT

+ + (4-52)

In order to simplify the PID controller design procedure and find a simple

empirical tuning rule, the zeros of the PID controller can be used to cancel the poles of

the system.

22 '

2

'

1 1( ' ' 1)( ') 1 ' 2 ' 1'

'

pdi

si i T

c p

si T

KKK s sK T K T KG G s e

s ssT

K KT es

τ

τ

ζ−

−

+ +=

+ +

=

(4-53)

After this simplification, the design turns into finding the optimal integral gain iK for the

50

pure time delay system. The Pade approximation is used to approximate the pure delay

term, and another first order term is added to the plant transfer function to make it proper.

The state space equation can be found from the transfer functions like the design

procedure of the PI controller for FOPTD.

The state space equation of the integral controller is

w Fw Ge= + (4-54)

v Dw Je= + (4-55)

where [ ] [ ] [ ] [ ]0 , 1 , 1 , and 0F G D J= = = =

For the different performance indices, the optimal gain settings of the integral

control are found by using the LQ design program. In Figures 4-11, 4-12 and 4-13, the

product of iK K are plotted against the normalized time delay from 0.01 to 10. A uniform

empirical formula has been found to approximate the plots.

( )bi

aKKT T

τ= (4-56)

From (4-53), the proportional gain and the derivative gain can be found by

2p iK K Tζ= (4-57)

2d iK K T= (4-58)

The resultant system crossover frequency and bandwidth can be predicted by the

following formulae for the three performance indices:

( )dc

cT T

τω = (4-59)

( )2

db

cfT T

τπ

= (4-60)

51

Table 4-6: Empirical Formula Coefficients Table for Integral Controllers

Tτ range 0.01~10

Performance Index ISE ISTE IST2E

a 0.7318 0.5886 0.5076

b -1 -1 -1

c 0.7318 0.5886 0.5076

d -1 -1 -1

Phase Margin(deg) 48.1 56.3 60.9

Gain Margin (dB) 6.6 8.5 9.8

The simulation results are plotted in the following figures. Since this design

method offers constant phase margin and gain margin over the range of the normalized

time delay, it can be expected the step response have same shape if we rescale the step

response along the time axis using the normalized time delay.

In this section, the PID LQ design is simplified to an integrator for a pure time

delay plant by pole-zero cancellation. This design approach can be extended to PI design

for FOPTD. Using the PI controller zero to cancel the FOPTD pole will result in a pure

time delay plant and an integrator. The empirical formulae developed in this section can

be used.

52

Ki(ISE)

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Ki


Figure 4-11: Integral Gain for ISE

Ki(ISTE)

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Ki


Figure 4-12: Integral Gain for ISTE

53

Ki(IST^2E)

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Ki


Figure 4-13: Integral Gain for IST2E

Gain Margin

0.00

2.00

4.00

6.00

8.00

10.00

12.00

0.01 0.10 1.00 10.00


Gai

n M

argi

n (d

B)

ISE ISTE IST^2E

Figure 4-14: Gain Margin for ISE, ISTE, and IST2E

54

Phase Margin

0.00

10.00

20.00

30.00

40.00

50.00

60.00

70.00

0.01 0.10 1.00 10.00


Phas

e M

argi

n (d

eg)

ISE ISTE IST^2E

Figure 4-15: Phase Margin for ISE, ISTE, and IST2E

Crossover Frequency(rad/sec)

0.01

0.10

1.00

10.00

100.00

0.01 0.10 1.00 10.00


Cro

ssov

er F

requ

ency

(rad

/sec

)

ISE ISTE IST^2E

Figure 4-16: Crossover Frequency of ISE, ISTE, and IST2E

55

Chapter 5: PMSM Current Control

5.1. Introduction

In the PMSM drive application, the multi-loop structures are commonly used to

implement the current/torque, speed, and position control [1]. Usually, the current/torque

control loop is the inner loop, and its performance is a crucial factor for the whole system.

The current control loop should be able to control the stator flux position and amplitude,

as well as compensate for the nonlinearity of the components in the system. Field

oriented control is commonly used in this loop. Field oriented control was first applied

on the induction motor by Blaaske to get comparable dynamic performance to a DC

motor [21]. Then, this concept was extended to control of other kinds of ac machines

such as permanent magnet synchronous machines. It turns out that field oriented control

on PMSM is easier to implement because of the absence of the slip frequency current

which appears in the induction machine [22].

One classical current control structure is composed of two or three phase current

control loops in the stationary, abc, frame [7, 8, 9, 10, 11, 12], since the current reference

to the controllers are sinusoidal functions of rotor position, this structure is referred to as

sine wave current control. Another current control structure includes two current control

loops in the rotor synchronous, dq0, frame [7, 8, 9, 10, 12, 13] and is referred to as the

vector current control.

In this chapter, these two current control structures are compared under the ideal

condition (analog control), where the ideal controlled voltage source is used and no delay

is considered.

56

5.2. Vector Current Control

In the electric machine analysis, it is convenient to express the angle in the

electrical degrees or electrical radians. In multi pole machine, with P pole pairs, the angle

value measured in electrical units is P times the angle value measured in the actual

spatial unit. Correspondingly the electrical speed and the rotor speed are related by the

following equation

s rPω ω= (5-1)

where sω is the synchronous speed, rω is the rotor speed.

This current control structure is based on the PMSM rotor synchronous ,dq0,

model. The rotor synchronous, dq0, frame is locked to the rotor, and its direct or d axis is

aligned with the rotor flux linkage phasor afλ . Its quadrature or q axis is aligned with the

resulting back EMF phasor. θ is defined as the angle between the q axis and the phase A

axis, and eθ is defined as the angle between the d axis and the stator phase A axis as

demonstrated in Figure 5-1.

Figure 5-1: Phasor Diagram

In the rotor synchronous, dq0, frame, the PMSM can be modeled with following

θ

eθ

id

iq

afλ

57

assumptions [2]:

i. Saturation is neglected although it can be taken into account by

parameter changes;

ii. The induced back EMF is sinusoidal;

iii. Eddy currents and hysterics losses are negligible;

Based on such assumptions, the machine can be modeled in the rotor synchronous,

dq0, frame using following equations.

q q q s ddv Ridtλ ω λ= + + (5-2)

d d d s qdv Ridtλ ω λ= + − (5-3)

q q qL iλ = (5-4)

d d d afL iλ λ= + (5-5)

Here and d qL L are the stator d and q inductance under the condition of alignment with

rotor d and q axes respectively, afλ is the magnet flux linkage amplitude, and R is the

stator resistance.

Variables in the rotor synchronous, dq0, frame and stator a, b, c variables are

related by Park transform. In the following equation, f can be current i or voltage v .

0

2 2cos( ) cos( ) cos( )3 3

2 2 2sin( ) sin( ) sin( )3 3 3

1 1 12 2 2

q a

d b

c

f ff ff f

θ θ π θ π

θ θ π θ π

− + = − +

(5-6)

58

0

cos( ) sin( ) 12 2cos( ) sin( ) 13 32 2cos( ) sin( ) 13 3

a q

b d

c

f ff ff f

θ θ

θ π θ π

θ π θ π

= − −

+ +

(5-7)

The electric torque Te is

e3T [ ( ) ]2 af q d q d qP i L L i iλ= + − (5-8)

The equation of the motor dynamics is

e L r rdT T B Jdt

ω ω= + + (5-9)

where B the damping constant, J is the moment of inertia.

If the PMSM has the uniform magnetic structure like the surface magnet motor,

d qL L= , the electric torque is determined only by the q axis current qi regardless of

whether di is purposely set to zero or not [23]. In field oriented control, good practice is

to force the direct axis current to zero and the make stator flux perpendicular to the rotor

flux. In this case, the electrical torque is proportional to the q axis current.

Rearranging the equation (5-2), (5-3) and (5-10) with (5-4) and (5-5) in the state

space form gives the following.

( ) /q q q s d d s af qd i v Ri L i Ldt

ω ω λ= − − − (5-10)

( ) /d d d s q q dd i v Ri L i Ldt

ω= − + (5-11)

( ) /r e L rd T T B Jdtω ω= − − (5-12)

These equations are nonlinear generally. While in the steady state, the rotor speed is a

59

constant variable or slowly varying variable compared with other state variables and can

be treated as constant. Based on this model, if a PI controller is used in each current

control loop, then zero tracking error can be expected for the constant current reference.

For each loop, the coupling and the back EMF disturbance are constant in steady

state. For the q axis loop, the tracking performance is more important than regulation

performance because qi is set by the outer loop according to the desired torque; while for

the d axis, the regulation performance is much more important because di is usually set

to zero.

A simulation scheme follows with the simulation results for a step input and a

sinusoidal input at 3000rpm. The motor parameters are these given in chapter 2 for the

drive motor and the PI controller gains are Kp=50(V/A) and Ki=85690(V/A-sec).

Figure 5-2: Simulation Scheme of Vector Current Control

60

Vector Current Control

-0.4

0

0.4

0.8

1.2

0.00 0.01 0.01 0.02 0.02 0.03 0.03 0.04 0.04 0.05 0.05

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

Figure 5-3: Simulation Result of Vector Current Control for Step Response

Vector Current Control

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0 0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

Figure 5-4: Simulation Result of Vector Current Control for Sinusoidal Response

61

5.3. Sine Wave Current Control

Sine wave current control is also known as the phase current control. In this

current control structure, two or three controllers are directly applied on the phase current

control loops. In the stationary, abc, frame, the PMSM model can be presented as shown

below.

abc abc abcdv Ridtλ= + (5-13)

In this equation R is the stator resistance, and abcv , abci and abcλ are stator voltage, stator

current and stator flux linkage vectors respectively. These vectors have the form of (5-14)

a

abc b

c

ff f

f

=

(5-14)

The flux linkages of stator phases a, b, c resulting from the currents flowing in the stator

windings are given in [6].

2 2

2 2

2 2

ms msms ls

ms msabcs ms ls abc

ms msms ls

L LL L

L LL L i

L L L L

λ

+ − − = − + − − − +

(5-15)

where and ms lsL L are the stator winding self inductance and leakage inductance. Using

the fact that 0a b ci i i+ + = for the wye connection, it can be found

62

3 0 02

30 02

30 02

3 2

ms ls

abcs ms ls abc

ms ls

ms ls abc

L L

L L i

L L

L L i

λ

+ = + + = +

(5-16)

The flux linkage contributed by the rotor permanent magnet is

cos( ) sin( )2 2cos( ) = sin( )3 32 2cos( ) sin( )3 3

e

abcr af e af

e

θ θ

λ λ θ π λ θ π

θ π θ

= − − + +

(5-17)

where θ is the angle between the rotor q axis and the stator A axis and eθ is the angle

between the rotor direct axis and the stator phase A axis as demonstrated Figure 5-1

sin( )3 2sin( - )2 3

2sin( )3

abc abcs abcr ms ls abc afL L i

θ

λ λ λ λ θ π

θ

= + = + × + +

(5-18)

Substituting the stator flux linkage equation into (5-13) gives,

cos( )( ) ( ) ( )1 2( ) ( ) ( ) cos( )

3( ) ( ) ( ) 2cos( )

3

as as asaf s

bs bs bss s s

cs cs cs

i t i t v td Ri t i t v tdt L L L

i t i t v t

θλ ω

θ π

θ π

= − + − −

+

(5-19)

where 32s ms lsL L L= + is defined as the synchronous inductance [24]. The phase equivalent

63

circuit of the PMSM is shown in Figure 5-5, where cos( )af saf

se

Lλ ω

θ= .

Figure 5-5: Equivalent Phase Circuit

In the sine wave current control structure, the two or three PI controllers are used

to control the phase currents. The advantages and disadvantages of two or three loops

have been discussed [7, 11]. The phase current references are generated according to the

rotor position and the equivalent current amplitudes in the rotor synchronous, dq0, frame,

which is directly associated with the electrical torque. The equation for the phase current

reference generation for the stator flux perpendicular to the rotor is listed in (5-20) and

(5-21).

0


aref qref

bref dref

cref ref

i ii ii i

θ θ

θ π θ π

θ π θ π

= − −

+ +

(5-20)

Since generally the direct axis current is required to be zero so that the stator flux can be

perpendicular to the rotor, the phase current reference can be simplified as

cos( )2cos( )32cos( )3

aref

bref qref

cref

ii ii

θ

θ π

θ π

= −

+

(5-21)

RsL

afe

+

−av

+

−

ai

64

The wave current reference is a sinusoidal function of θ and its amplitude is the q axis

current.

The back EMF on each phase is a sinusoidal disturbance whose amplitude is

proportional to the rotor speed and whose frequency is associated with the rotor speed.

The phase current controller should be able to reject this kind disturbance effectively if

the rotor speed is small enough.

If the torque or q axis current control reference is sinusoidal with an frequency of

qf , the reference current to the phase current controller is

cos(2 )cos( )

1 [cos(2 ) cos(2 )]2

qref q s

qref q s q s

I f t t

I f t t f t t

π ω α

π ω α π ω α

−

= − + + + − (5-22)

where α is 2 20, and 3 3π π− for the phase a, b, and c. The higher frequency component

of the reference current is at 2

sqf

ωπ

+ . The phase current controllers must have much

higher bandwidth than 2

sqf

ωπ

+ to satisfy the application requirements. Since the phase

current reference is sinusoidal, it is not possible to get zero tracking error under

conditions of constant speed and constant q axis current reference. The tracking error will

increase when the rotor spins faster.

In steady state, the electrical angular velocity is sω , and stθ ω= , the closed loop

amplitude ratio and the phase lag are 1A and 1α at sω . So the phase current can be

presented as

65

1

1 1

1

cos( )2cos( ) 32cos( )3

a

b ref

c

ii A Ii

θ α

θ π α

θ π α

−

= − −

+ −

(5-23)

Transforming the phase current into the rotor synchronous, dq0, frame by Park Transform,

such phase lag and amplitude attenuation degrade the motor performance as shown in (5-

24)

1

1 1

0

cos( )sin( )

0

q

d ref

ii A Ii

αα

=

(5-24)

The transfer function for each phase back EMF can be presented as

( )( )

1 ( ) ( )p

dc p

G sG s

G s G s= −

+ (5-25)

If the plant pole is cancelled by the PI controller zero, ( )dG s can simplified to (5-26)

2( )( )d

s p i

sG sL s R K s K

= −+ + +

(5-26)

The back EMF on phase a, b, c can be presented as

cos( )( )2( ) cos( )3

( ) 2cos( )3

ads

bds af s

cds

v tv tv t

θ

λ ω θ π

θ π

= −

+

(5-27)

The currents corresponding to such disturbances are

66

2

2 2

2

cos( )( )2( ) cos( )3

( ) 2cos( )3

sads

bds af s s

cds

s

ti ti t A ti t

t

ω α

λ ω ω π α

ω π α

−

= − −

+ −

(5-28)

The amplitude damping ratio 2A and the phase lag 2α can be determined from the

transfer function with the corresponding frequency sω . 2α is larger than 90 .

Transforming such currents into the rotor synchronous, dq0, frame,

2

2 2

0

( ) cos( )( ) sin( )( ) 0

qds

dds af s

ds

i ti t Ai t

αλ ω α

=

(5-29)

Finally, the currents represented in the rotor synchronous, dq0, frame are

1 2

1 1 2 2

0

( ) cos( ) cos( )( ) sin( ) sin( )( ) 0 0

qs

ds ref af r

s

i ti t A I Ai t

α αα λ ω α

= +

(5-30)

In simulation and experimental results, it is obvious that such disturbance cause

the current offsets on the q and d axes from the desired values. The simulation scheme in

MATLAB/SIMULINK is shown in Figure 5-6.

Figure 5-6: Simulation Scheme of Sine Wave Current Control

67

In Figure 5-7 and Figure 5-8, the simulated system responses for a sinusoidal

input and step input are shown for simulation at 3000RPM. The quadrature and direct

axis currents obviously deviate from the references. The motor parameters are these

given in chapter 2 for the drive motor and the PI controller gains are Kp=50(V/A) and

Ki=85690(V/A-sec).

Sine Wave Current Control

-0.8

-0.4

0

0.4

0.8

1.2

0 0.01 0.02 0.03 0.04 0.05

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

Figure 5-7: Simulation Result of Sine Wave Control for Step Response

68

Sine Wave Current Control

-2

-1.5

-1

-0.5

0

0.5

1

1.5

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

Figure 5-8: Simulation Result of Sine Wave Control for Sinusoidal Response

69

Chapter 6: Digital Implementation and Systematic

Design of PMSM Current Control

In Chapter 5, the continuous time domain models of PMSM and corresponding

current control methods are presented, and the two current control structures are

compared theoretically.

Recently, the developments in DSP techniques and power electronic components

have made it possible to implement high performance current control for PMSM via

digital implementation. This chapter will focus on the digital implementations for these

two current control structures. The digital implementations of these current control

structures have at least two differences from the analog implementation: the PWM

technology application and the effects caused by digitization.

The experimental system also includes a second motor driven by an off-the-shelf

amplifier, the L-180. For the experimental data in this thesis the second motor is used to

drive the system at a constant velocity.

6.1. Digital Implementation Architecture for PMSM Current

Control

The two current control structures, sine wave current control and vector current

control, share the same hardware configuration in our experimental system. The

experimental system is composed of a host computer, a motion control DSP card (MC8),

an integrated power stage, a permanent magnet synchronous motor and necessary sensors.

The differences between the two current control structures lie in the control algorithm of

70

the DSP programs.

6.1.1. Sine Wave Current Control Structure

The sine wave current control structure for the PMSM is described in Figure 6-1

[7, 8, 9, 10, 11, 12]. In the digital implementation of this sine wave current control system

all control algorithms are implemented in the DSP program. Compared with analog

implementation or pure hardware implementation, this implementation method is the

most flexible. The components implemented in DSP program in Figure 6-1 are the phase

reference generator, the three or two PI controllers and the duty cycle computation for the

PWM.

Figure 6-1: System Schematic of Sine Wave Current Control

• Phase Current Reference Generator

The phase current reference generator generates the phase current references for

Resolver to Digital Converter

A/D Fire Circuits

Phase current Reference Generator

Controller

Duty computation

-

+

Iaref

Ibref

Icref

…

…

ia

Current Amplitude Reference

βa βb βc

ib ic

VDC

+

-

71

the each current control loop. The inputs to the reference generator are the rotor

position rθ and desired current amplitude. The rotor position comes from the

resolver/quadrature encoder and can be easily translated into the electric angle θ by the

equation rPθ θ= . The desired current amplitude is proportional to with the desired

torque. In most applications it would come from an outer speed or position control loop.

The equation that the phase current reference generator uses is

cos( )2cos( )32cos( )3

aref

bref qref

cref

ii ii

θ

θ π

θ π

= −

+

(6-1)

• PI Controller

The PI controller is designed in the continuous time domain and implemented by

the approximation equation:

10

( ) ( ) ( )k

k p k i s nn

u t K e t K T e t+=

= + ∑ (6-2)

Where 1( )ku t + is the controller output, ( )ke t is the error to the controller, sT is the

sampling period, and pK and iK are the proportional and integral gains respectively. For

the good approximation, the sampling frequency should be sufficiently fast.

• Duty Cycle Computation

Duty cycle computation uses the outputs of the PI current controllers to generate

the PWM drive signals with the calculated duty cycle. In the experimental system, a

center-aligned PWM modulation method is used. The behavior of the duty cycle

computation is examined below.

72

At any moment, one switch of each leg in the inverter is on and the other switch is

off. The three legs can be driven to 8 unique states. For each leg, one is used to represent

the state that the top switch is on and bottom switch is off. Zero is used to represent the

state that the top switch is off and the bottom switch is on. The inverter states are encoded

by three digits and listed in Table 6-1.

Table 6-1: Inverter State Definitions

U000 U100 U110 U010 U011 U001 U101 U111

abc 000 100 110 010 011 001 101 111

The states of U000 and U111 are zero states, which correspond to the situation

that all three motor terminals are connected to the lower bus or upper bus of the DC

supply.

The immediate outputs of the PI controllers are compared with each other, and the

minimum one will be used to offset the phase voltage references. As a result, there is

always one phase voltage reference at zero with respect to the ground line of the DC

supply. In this PWM period, the leg associated with this phase has no switching action

and the phase stay at the ground. Less switch action means less energy loss caused by

inverter switching.

The duty cycle calculation equation is

'ii

Bus

vV

β = . (6-3)

where , ,i a b c= for each phase.

Aligning the ON state of each leg to the center of the PWM period, the switch

73

patterns are listed in Table 6-2. Every PWM period starts from U000 and ends in U000,

and at each state transition, only one switching action occurs.

Table 6-2: Inverter Switching Patterns

a b cv v v> >

a c bv v v> >

b a cv v v> >

b c av v v> >

c a bv v v> >

c b av v v> >

'av a cv v− a bv v− a cv v− 0 a bv v− 0

'bv b cv v− 0 b cv v− b av v− 0 b av v−

'cv 0 c bv v− 0 c av v− c bv v− c av v−

Switching

Patterns

000100110100000

UUUUU

⇒⇒⇒⇒

000100101100000

UUUUU

⇒⇒⇒⇒

000010110010000

UUUUU

⇒⇒⇒⇒

000010011010000

UUUUU

⇒⇒⇒⇒

000100110100000

UUUUU

⇒⇒⇒⇒

000001011001000

UUUUU

⇒⇒⇒⇒

In the practical system implementation, the dead time of the power stages need to

be considered. Dead time is the time between turning one switch in a leg off and the other

one on. Correction terms are added to the duty cycle calculation to minimize the

distortions caused by the dead time. Detailed discussion of this can be found in [25]

Using the minimum value of the three phase voltage references to offset the

reference voltage will not change the voltage vector in the rotor synchronous, dq0, frame.

This can be verified by (6-4), where only the zero-sequence variable is changed by the

offset of the phase variables.

74

0

2 2cos( ) cos( ) cos( )3 3

2 2 2sin( ) sin( ) sin( )3 3 3

1 1 12 2 2

2 2cos( ) cos( ) cos( )3 3

2 2 2sin( ) sin( ) sin( )3 3 3

1 1 12 2 2

q a offset

d b offset

c offset

v v vv v vv v v

θ θ π θ π

θ θ π θ π

θ θ π θ π

θ θ π θ π

− + − = − + − −

− +

= − +

02 03

32

a

b

coffset

vvv v

−

(6-4)

• DSP Program Implementation

The DSP control programs were written in C. The development environment is

VisualDSP++ for the Sharc processor family. The flow chart for the sine wave current

control program is presented below.

75

Figure 6-2: Program Flow Chart of Sine Wave Current Control

The control algorithm is implemented in an interrupt service subroutine driven by

a clock running at the sampling frequency. From the program flow chart, the timing

scheme of the server subroutine is presented below.

Start

Initialize pointes, global variables, interrupt

Handle message exchange with the host PC

Read the sampling measurement.Current/Position

Phase Current Reference Generation

PI Control

Generate Command

Generate Synchronous Signals to start sampling and PWM output

Calculate PWM Duty Cycle

Return to Main Program

Store Data if Required

Interrupt Entrance

76

Figure 6-3: Timing Chart of Sine Wave Current Control

From Figure 6-3, the delay between the sampling and the corresponding output is

one sampling period if we neglect the sampling time and output delay caused by PWM.

6.1.2. Vector Current Control Structure

Figure 6-4: System Schematic of Vector Current Control

The digital implementation structure is described in Figure 6-4 [7, 8, 9, 10, 12,

Resolver to Digital Converter

A/D Fire Circuits

Duty computation

Iqref

βa βb βc

Ia Ib Ic -> Iq Id Electrical Position

Controller-

+

Controller-

+

Uqd0 -> Ua Ub Uc

Idref=0

iq

id

tTk Tk+1 Tk+2

Computation

Sampling and PWM Update

Computation


Computation


77

13]. The vector current control shares the hardware structure with sine wave current

control. Some components of the control algorithm are also the same, such as duty cycle

computation, and the PI controller algorithm. In the vector current control structure,

controlled variables are not directly the stator current; instead, the controlled variables are

the currents and d qi i in the rotor synchronous, dq0, frame. The components that do not

appear in the sine wave current control structure are listed below.

• Conversion from the stationary, abc, frame to the rotor synchronous, dq0, frame

This conversion is used to transform the stator current from the stationary, abc,

frame into the rotor synchronous, dq0, frame. The inputs to this conversion are the rotor

position and the current sensor outputs. The transformation equation is Park transform (6-

5).

0

2 2cos( ) cos( ) cos( )3 3

2 2 2sin( ) sin( ) sin( )3 3 3

1 1 12 2 2

q a

d b

c

f ff ff f

θ θ π θ π

θ θ π θ π

− + = − +

(6-5)

• Conversion from the rotor synchronous, dq0, frame to the stationary, abc, frame

This conversion is used to transform the variables from the rotor synchronous,

dq0, frame into the stationary, abc, frame. The inputs to this conversion are the PI

controller outputs, which are the voltages in the rotor synchronous, dq0, frame. The

transformation equation is the inverse Park transform (6-6)

78

0


a q

b d

c

f ff ff f

θ θ

θ π θ π

θ π θ π

= − −

+ +

(6-6)

• DSP Program Implementation

The DSP program for vector current control shares the program structure

with sine wave current control. The program flow chart is presented below.

Figure 6-5: Program Flow Chart of Vector Current Control

Start

Initialize pointes, global variables, interrupt

Handle message exchange With the host PC

Read the sampled measurement. Current/Position

Current Conversion (abc -> dq0)

PI Control

Generate Command

Generate Synchronous Signal to start sampling and PWM output

Calculate PWM Duty Cycle

Return to main program

Store Data if Required

Voltage Conversion (dq0 -> abc)

Interrupt Entrance

79

The timing diagram of the vector control program is basically same with the sine

wave current structure as shown in Figure 6-3.

6.2. Effects of Digitization and PWM on the Current Control

System Design

In digital implementation of the current control for permanent magnet

synchronous motors time delay is inevitable. On the motion control board in our system

the sensor circuits are synchronized by a signal, which will trigger A/D conversion, latch

DAC output and quadrature decoder counts (the decoders will provide the rotor position

in our system). From the timing diagram shown in Figure 6-3, the time delay is one

sampling period. Further delay is caused by PWM. This time delay is complex to analyze,

but one-half of the PWM update period is a good approximation [26, 27].

Generally for the permanent magnet synchronous motor current control design,

the delay effects in the digital implementation can be neglected in the case that the time

delay is small enough compared with the electric time constant of the motor, and the

motor maximum speed is low. In this case, the system can be designed using continuous

time models. Sometimes, the digital update period and PWM update period are not small

enough to be neglected and the delay effects should be taken into account in the system

design and tuning process to guarantee good system performance. Usually, faster PWM

update and digital update can result in better system performance, but considering that the

inverter loss will increase when the switching frequency increases, if the system

performance can be satisfied, lower switching frequency may be preferred. Another

limitation for the small digital update period is the DSP computation power; a small

digital update period requires a more powerful digital signal processor and may raise the

80

system cost.

Another effect of the digital implementation of the current control is the ripples

caused by the discrete time control. Actually it is the nature of the discrete time control

since the output of the controller is not smooth, instead, the output of the controller is

stair step pattern. Since the plant is usually a low pass filter it will filter out the higher

frequency components and output smooth movement.

With the popularity of the digital implementation of motor control system, the

practical problems in such implementations have attracted a lot of interest. T.A. Sakharuk

and his coworkers presented the analytic result to quantify the effects of low-switching

frequency in ac drives supplied by pulse-width modulation power electronic inverters in

[28]. Their studies focus on the effects of the PWM. Different from the conventional

modeling procedure where the PWM switching frequency is assumed high enough and

the resulting waveform can be replaced by its first order harmonics in the control-

oriented model, their model provides a new PWM gain which accounts for the effects of

low switching frequency. Also a formula for the magnitude of the current ripple is given

using a piecewise linear approximation for the current. The investigation was also made

by Vladimir Blasko and his coworkers on the sampling methods of voltage and current

signals in electric drives. They analyzed the limitation of the bandwidth of a current

controller according to the time delay in the sampling and PWM in [26]. Further analysis

is presented by Bon-Ho Bae in [27].

Using PWMT to indicate the PWM update period and sT to indicate the digital

update period, sT must be greater than PWMT since it is meaningless if the PWM update

period is larger than the digital update period. Usually the digital update period is set as a

81

multiple of the PWM update period, i.e. ( 1, 2,...)s PWMT nT n= = . The total time delay

caused by the digital implementation and PWM is the summation of one digital update

period and one half the PWM update period. Using dT to indicate the total time delay,

2PWM

d sTT T= + [26, 27].

In Bae’s work, the analysis starts from complex vector analysis and the similar

time sequence to that used in our experimental system, which is represented below.

Figure 6-6: Time Sequence of Current Sampling, Calculation and PWM Output

The relationship of vector presentations in the stationary dsqs0s frame and the

rotor synchronous, dq0, frame is demonstrated in Figure 6-7. The superscripts s and e

indicate the stationary and synchronous reference variables respectively, and superscript

* indicates the reference value. The angle between the stationary reference frame and the

synchronous reference frame is eθ , the transformation of the vector present in the two

reference frames is

* *ejs e

dqs dqsv v e θ= (6-7)

* *eje s

dqs dqsi i e θ−= (6-8)

t t+Ts t+2Ts

Current Sampling

Calculation PWM Output

PWM Update

θe θe+Tsωs θe+2Tsωs

82

Figure 6-7: Synchronous Reference Frame Quantities in the Stationary Reference Frame

Under the assumption that the synchronous speed sω is constant during the time

delay, the stator refereed voltage reference for the PWM generation *

_sdqs digitalv by

averaging is [27]

* * *2

( ) (1.5 )_

1 ( , )s

s e s s e

s

Tj j Ts e e

dqs digital dqs s s dqss T

v v e d K T e vT

ω τ θ ω θτ ω+ += =∫ (6-9)

where

2( , ) sin( )2s s

s ss s

TK TT

ωωω

= (6-10)

The error caused by the frame rotation is

*

*

( 1.5 ) ( )

(1.5 ) ( , ) ( , )( , )

e s s D

s s e

j e j T jdqs

error j T es s s ss s dqs

e v e efK T K TK T e v

θ ω θ

ω θ ω ωω

− −

+= = = (6-11)

In Bae’s work, the time delay is replaced by a first order term, whose time

constant is the delay. It is approximated by one sampling period and one half of a PWM

update period. In the design procedure, the delay term will remain. The block diagram

ds

d

qsq

θe

*edqsv

*edsv

*eqsv

83

follows.

Figure 6-8: Complex Vector Block Diagram of Vector Current Control

The frame rotation causes errors on the magnitude and vector phase of the voltage

vector. The magnitude of the voltage is multiplied by the scalar value 1/ ( , )e sK Tω , and

the phase of the voltage vector is delayed by Dθ , which is the rotating angle of the

reference frame during one and a half sampling periods.

The error in the voltage phase causes additional coupling between the d axis and q

axis. It is more obvious if the system is presented as below.

Figure 6-9: System Model of Vector Current Control (1)

p iK s Ks+

( 1.5 )

( , )

s sj T

s s

eK T

ω

ω

−

dT se−1

s s sL s R j Lω+ +

edqsi

*edqsi

edqse

edqsi

Plant Controller

Error Caused By Digital

Implementation

Processing And PWM Delay

p iK s Ks+ 1

( , )s sK Tω dT se−1

sL s R+

s sLω

*eqsi

sqse

-

++

-eqsi

Plant Controller

cos( )Dθ

sin( )Dθ

p iK s Ks+ 1

( , )s sK Tω dT se−1

sL s R+*e

dsisdse

+

++

-edsicos( )Dθ

sin( )Dθs sLω

-

-

+

+

+

-

84

Since the complete system is a multi-input multi-output system, a convenient way

to describe the system is using the state space equation. Using a first order term to replace

the output time delay term in the system, 11

dT s

d

eT s

− ≈+

, and defining the state variables

as indicated in the figure,

Figure 6-10: System Model of Vector Current Control (2)

The state space equation can be defined as below.

x Ax Bu= + (6-12)

y Cx= (6-13)

*

*qs

ds

iu

i

=

, qs

ds

iy

i

=

(6-14)

e sLω

*eqsi

sqse

eqsi

Plant Controller

*edsi

sdse

edsi

e sLω

−

1( , )e sK Tω cos( )Dθ

sin( )Dθ

1s iK

pK

cos( )Dθ

sin( )Dθ

1( , )e sK Tω

1s iK

pK

1s

1

dT−

1

dT1s

s

RL

−

1

sL

1s

s

RL

−

1

sL1s

1

dT−

1

dT

1x

+ +

+ + +

+

+

+

+

−

− −

−

+ +

+

+

+

+

+

+

+

+

+

2x

3x

4x

5x

6x

−

85

1 0 0 0

10 0 0

cos( ) sin( ) cos( ) sin( )1 0( , ) ( , ) ( , ) ( , )

sin( ) cos( ) sin( ) cos( )10( , ) ( , ) ( , ) ( , )1 0 0 0 0 0

ss s

ss s

D p D p D i D i

d s s d s s d d s s d s s

D p D p D i D i

d s s d s s d d s s d s s

RL L

RL L

k k k kA

T K T T K T T T K T T K Tk k k k

T K T T K T T T K T T K T

ω

ω

θ θ θ θω ω ω ωθ θ θ θω ω ω ω

− −

−

− − −=

− − −

−0 1 0 0 0 0

−

,

0 00 0

cos( ) sin( )( , ) ( , )

sin( ) cos( )( , ) ( , )1 00 1

D p D p

d s s d s s

D p D p

d s s d s s

k kT K T T K T

Bk k

T K T T K T

θ θω ωθ θω ω

− =

, 1 0 0 0 0 00 1 0 0 0 0

C

=

(6-15)

In the sine wave current control structure, the digitization and rotor reference

frame rotation cause error on the phase current reference generation. A similar process to

that used in [27] can be used to find such error. Considering the phase current reference

generation process, its purpose is essentially to transform the desired current vector from

the rotor synchronous, dq0, frame to the stationary, abc, frame and results in the phase

current references for each phase. Usually the d axis component of the reference current

vector is zero as discussed previously.

86

Figure 6-11: Rotation of the Synchronous Reference Frame Corresponding to the Time Delay

With the time sequence shown in Figure 6-3, the synchronous reference frame

rotation and the trace of the reference current vector move as shown in Figure 6-11.

Similar to the process to develop (6-11), for the given time t, the corresponding reference

current vector output is from ( st T+ ) to ( 2 st T+ ), under the assumption that the

synchronous frequency sω is constant during the time delay and the current reference

vector for the control loop is derived by averaging the current vector reference from

( st T+ ) to ( 2 st T+ ).

* * *2

( ) (1.5 )_

1 ( , )s

s e s s e

s

Tj j Ts e e

dqs digital dqs s s dqss T

i i e d K T e iT

ω τ θ ω θτ ω+ += =∫ (6-16)

where

2( , ) sin( )2s s

s ss s

TK TT

ωωω

= (6-17)

The error caused by the conventional transformation which is same with (6-12) can be

ds

d(t)

qs

q(t)

θe

*edqsi

d(t+Ts)

d(t+2Ts)

q(t+Ts)

q(t+2Ts)

87

calculated as,

*

*

( 1.5 ) ( )

(1.5 ) ( , ) ( , )( , )

s s D

s s e

j e j T jdqs

error j T es s s ss s dqs

e i e efK T K TK T e i

θ ω θ

ω θ ω ωω

− −

+= = =

Differing from the vector current control structure, the error caused by the synchronous

reference frame rotation is not in the phase current control loops; instead, it is in the

phase current reference generation process as shown in Figure 6-11. One phase current

control loop is shown in Figure 6-12 including the time delay term separately.

Figure 6-12 Phase Current Reference Generation with the Error Caused by Rotation

Figure 6-13: One Phase Current Control Loop in Sine Wave Current Control

6.3. Simulation Models

In the development of current control methods and system modeling, two levels

of simulation models have been developed. The first level simulation model for vector

current control uses the scheme in Figure 6-9, and the first level simulation model for

sine wave current control used the scheme in Figure 6-12 and Figure 6-13. At this level,

all components are continuous. The effects of rotor rotation and digitization are emulated

( )

( , )

Dj

s s

eK T

θ

ω

− Inverse Park’s

Transform

iq*

id*

ia_digital*

ib_digital *

ic_digital *

p iK s Ks+

dT se−1

sL s R+*

_a digitali

afe

-

++

-

ai

Plant Controller

88

by a corresponding continuous gain and time delay. The second level simulation models

use the PMSM model and universal bridge from the SimPowerSystems library in

MATLAB SIMLINK. The controllers are written in C and emulate the control program

used in the experimental system. In Figure 6-14 and 6-15, the first level simulation

models for vector current control and sine wave current control are shown. In Figure 6-16,

the second level simulation model for both of the current control structures is shown.

Figure 6-14: First Level Vector Current Control Simulation Model

Figure 6-15: First Level Sine Wave Current Control Simulation Model

89

Figure 6-16: Second Level Vector and Sine Wave Current Control Simulation Model

6.4. Systematic Design of PMSM Current Control System

For a given current control structure, the design procedure will be to find the

proper digital update frequency and PWM update frequency, and determine the PI

controller gains to satisfy the system requirements like the bandwidth and the maximum

motor speed. In the coming sections, the design procedure will be presented for both

current control structures.

6.4.1. Systematic Design of PMSM Vector Current Control

The complete model of digital implementation of vector current control is

presented in the previous section. In this model, the system is a time varying system. The

coupling between the two axis loops depends on the motor speed and the sampling period.

In the case of low speed, the coupling caused by digital implementation between

90

the d and q axis loops is weak and the electric coupling in the motor is also weak. While

in the high speed condition, the couplings between the two loops become stronger and

even affect the system stability.

Neglecting the coupling between the two loops, each loop can be simplified as

shown in Figure 6-17.

Figure 6-17: Vector Current Control PI Controller Design Model

The term cos( )( , )

D

e sK Tθ

ω is approximately one and can be neglected in the design procedure.

Suppose the desired bandwidth is bf , and the maximum speed is sω . Define the

plant transfer function as 1( ) dT sp

s

G s eL s R

−=+

. The following design is based on the LQ

optimization equations since LQ optimization can guarantee system stability and good

performance. The steps are listed below.

• According to the plant transfer function ( )pG s , the time constant sLTR

= .

Plug it into the equation ( )2

bb

afT T

τπ

= (4-48), the time delay can be found

by 12( )s b s b

dL f LTR aR

πτ= = , where values of a and b depend on the selected LQ

performance index. The resulting time delay is the maximum time delay for

the desired bandwidth. With this resulting time delay and correctly set PI

controller gains, the system is optimized with the desired bandwidth in the

p iK s Ks+ cos( )

( , )D

s sK Tθ

ω dT se−1

sL s R+*

i + -

i

91

sense of the selected performance index.

• From 2

PWMd s

TT T= + , the digital update period and PWM update period can

be determined. Usually s PWMT kT= for 1,2...k = . Considering the issue of the

ripples caused by PWM, it is a good practice to have the PWM update period

less than one-fifth of the motor electrical time constant.

• Determine the PI controller proportional gains pK and integral time constant

iT by (4-45) and (4-46) and corresponding iK .

• Check the system stability by using the complete system state space equation

at maximum speed by (6-12). If the system is unstable at desired high speed,

use a higher bandwidth to repeat the design process.

From this procedure, the digital update frequency and PWM update frequency

will be determined for good performance at low speed. If the system becomes unstable at

the desired high speed, the digital update frequency or PWM update frequency should be

increased. The gain margin and phase margin of the resultant system varies depending on

the LQ design performance index. For the given desired bandwidth, among the three

performance indices, ISE, ISTE and IST2E, ISE usually result in an aggressive design.

This means the resultant system’s digital update frequency and PWM update frequency

are lowest and its stability margin is smallest. Conversely, IST2E usually result in a

conservative design, meaning the resultant system’s digital update frequency and PWM

update frequency are highest and its stability margin is largest.

6.4.2. Systematic Design of PMSM Sine Wave Current Control

The current control system design based on sine wave current control structure

92

has a similar design procedure as the vector current control design procedure.

Since the electric torque is determined by the q axis current in the rotor

synchronous, dq0, frame, it is reasonable to give the system design requirements in the

term of q axis current bandwidth. In the sine wave current control structure, as discussed

previously, the current reference is a sinusoidal function of the rotor position and the

frequency is the electrical frequency of the motor. If the reference current in the rotor

synchronous reference is also sinusoidal, and its frequency is the desired current loop

bandwidth bf , the higher frequency component of the reference current is at 2

sbf

ωπ

+ .

Therefore the current loop bandwidth should be larger than2

sbf

ωπ

+ . The design

procedure is listed below.

• Determine the desired phase current loop bandwidth ' ( )2

sb bf k f ω

π= + . Since

the phase lag will cause the current shift on the d and q axis, and eventually

degrade the torque response, the phase current bandwidth is set to k multiples

of +2

sbf

ωπ

with k=5…10 for the smaller phase lag.

• Plug the phase time constant sLTR

= and the desired phase current loop

bandwidth 'bf into the equation ( )

2b

bafT T

τπ

= (4-48) to determine the total

time delay 12 '( )s b s b

dL f LTR aR

πτ= = . The resulting time delay is the maximum

time delay for the desired bandwidth. With this resulting time delay and

correctly set PI controller gains, the system is optimized with the desired

93

bandwidth in the sense of the selected performance index.

• Use 2

PWMd s

TT T= + to determine the digital update period and PWM update

period. Usually s PWMT kT= for 1,2...k = . Considering the issue of the ripples

caused by PWM, it is a good practice to have the PWM update period be less

than one-fifth of the load time constant, i.e. the motor electrical time constant

in this application.

• Since the error in the phase current reference generation process (see Fig. 6-

15) caused by rotor rotation is not in the closed loop, check the value of

cos( )( , )

D

s sK Tθ

ω, which represent the DC gain on the q axis. To limit the resultant

error to be less than 1%, cos( ) 1 0.01( , )

D

s sK Tθ

ω− < .

• Determine the PI controller proportional gains pK and integral time constant

iT by (4-45) and (4-46), and the corresponding iK

6.5. Design Examples

In this section, the design examples will be presented for the two current control

structures. Both simulated and experimental results will also be presented.

6.5.1. Vector Current Control Design Example

Current Control loop specification:

Bandwidth: 200bf Hz=

Maximum Speed: 2000rpm

94

(1) Maximum Time delay calculation

Use ISTE optimal criterion, 0.6701, -0.9439a b= = for (4-48).

12( ) 5.3040e-004secs b s bd

L f LTR aR

π= =

(2) In this example, the motor electrical time constant is 0.6 msec. The digital update

frequency and PWM update frequency can be found by2

PWMd s

TT T= + , Let

4s PWMT T= , which results in 4.7147e-004sec, 1.1787e-004secs PWMT T= = .

(3) Determine the PI controller gains by the LQ optimal equations. In this example,

the ISTE is used.

22(V/A), 27611(V/A-sec).p iK K= =

In the following figures, the current step responses at different speeds are

presented with simulation data and experimental data. The step response overshoot tend

to increase with increasing speed. This can be explained by observing the roots of the

closed loop system. One pair of complex roots move to right side and will pass the

imaginary axis when the motor operating speed is high enough. This is demonstrated in

Figure 6-18. The simulation and the experimental results show that the model is accurate

in predicting the system stability.

95

Eigenvalues at Different Speeds

-2500.00

-2000.00

-1500.00

-1000.00

-500.00

0.00

500.00

1000.00

1500.00

2000.00

2500.00

-2500.00 -2000.00 -1500.00 -1000.00 -500.00 0.00 500.00

Real

1000 rpm 2000 rpm 3000 rpmImag

Figure 6-18: Eigenvalues for Different Speed

96

Simulation (Level-1, 0rpm)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

(a) First Level Simulation


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command

(b) Second Level Simulation

97

Experimental Result (0rpm)

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad

(c) Experimental Result

Figure 6-19: Simulation and Experimental Result at 0 rpm for Step Response

98


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


99


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-20: Simulation and Experimental Result at 1000rpm for Step Response

100


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


101


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02 0.025 0.03

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-21: Simulation and Experimental Result at 2000 rpm for step response

102


-30

-20

-10

0

10

20

30

0 0.01 0.02 0.03 0.04 0.05

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-8

-6

-4

-2

0

2

4

6

0 0.01 0.02 0.03 0.04 0.05

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


103


-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

0 0.01 0.02 0.03 0.04 0.05

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-22: Simulation and Experimental Result at 3000rpm

104

Simulation (Level-1, 200Hz, 0rpm)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.01 0.02 0.03 0.04

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.01 0.02 0.03 0.04

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


105

Experimental Result (200Hz, 0rpm)

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

0 0.01 0.02 0.03 0.04

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-23: Simulation and Experimental Result at 0 rpm for Sine Wave Command

The simulation and experimental results show that the first and second level

models can demonstrate the main features of the system, and the second level model

tends to be more accurate in the details.

6.5.2. Sine Wave Current Control Design Example

Current Control loop specification:

Bandwidth: 100bf Hz=

Maximum Speed: 1000rpm ( 419 / secs radω = )

(1) Redefine the phase current loop bandwidth

' 5( ) 8332

sb bf f Hzω

π= + =

106

(2) Maximum total time delay calculation

Use ISTE optimal criterion, 0.6701, -0.9439a b= = for (4-48)

12 '( ) 1.1694e-4secs b s bd

L f LTR aR

π= = .

(3) Using a digital update frequency and a PWM update frequency that are equal. The

total time delay will be 1.5 times the digital update period.

/1.5 7.7963e-5secs dT T= = .

(4) At the desired maximum speed, 0.0490radDθ = , cos( ) 1 0.0012 0.01( , )

D

s sK Tθ

ω− = < .

(5) Determine the PI controller gains by the LQ optimal equations.

77(V/A), 122840(V/A-sec)p iK K= =

107


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


108


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad



109


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


110


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad




-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


111


-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time(Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-26: Simulation and Experimental Result at 2000rpm for Step Response

112


-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command



-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Command


113

Experimental Result (100Hz, 0rpm)

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad


Figure 6-27: Simulation and Experimental Result at 0 rpm for 100Hz Sine Wave Command Response

The simulation and experimental results show that the first and second level

simulation models can demonstrate the main system characteristics. The second level

simulation tends to be more accurate in details. In order to control currents effectively at

high speed, the phase current control loops should have higher bandwidth, which requires

a fast digital update frequency and a fast PWM update frequency. Unfortunately a higher

digital update frequency and a higher PWM update frequency will make the system

impractical or raise the cost. The previous results also demonstrate a systematic approach

to the selection of minimum update and PWM frequencies given desired bandwidth and

motor speed.

114

6.6. Automatic Tuning

In the previous sections, the two current control structures have been discussed

and the systematic design procedures are also presented based on LQ design results. In

this section, the tuning process is to be presented based on relay feedback and the LQ

design result. The example system uses the vector current control structure. The digital

update frequency and PWM update frequency are 10kHz . In this example, the relay

without hysteresis is used, and it is implemented in the DSP program. It is applied on the

q axis current control loop.

For the q axis current control loop, the plant can be model as a first order plus

time delay system as discussed previously.

( )1p

K sG s eTs

τ−=+

The DC gain is first tested using open loop test. The applied DC voltage is 20volts,

and the q axis current is 0.6 amps when the motor rotor is held stationary, so the DC gain

can be calculated as

0.6 0.03(A/V)20

K = = .

Figure 6-28 is the experimental test result using the relay with an output

amplitude of 40volts (M=40). The ultimate period can be identified as 0.6ms (Tu=0.6ms),

and the ultimate gain by (3-29)

4 134(V/A)uMkXπ

= =

The time constant and the time delay are approximated below by (3-31) and (3-32).

115

22

22

15.11 4sec

arctan( 1)1.68 4sec

u

u

u

u

K kT e

K ke

ω

πτ

ω

−= = −

− −= = −

The identified time delay approximately 1.5 times the digital update period. This

shows that the modeling of the digital implementation is valid. The identified time

constant also approximates the actual motor time constant.


-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0 0.005 0.01

Time (Sec)

Cur

rent

(Am

p)

Iq Id

Figure 6-28: Relay Test Result

Using the relay test result, the PI controller gains with various tuning methods

discussed in the previous sections are listed in Table 6-3. In Figure 6-29, the step

responses are shown for the different gain settings.

116

Table 6-3: Auto Tuned PI Controller Gains

Kp(V/A) Ti(sec) Ki(V/A-sec)

ZN 60 5e-4 120000

ISE 86 8e-4 107160

ISTE 64 5.8e-4 111050

Experimental Result (ZN, 0rpm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad

(a) ZN

117

Experimental Result (ISE, 0rpm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad

(b) ISE

Experimental Result (ISTE, 0rpm)

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0 0.005 0.01 0.015 0.02

Time (Sec)

Cur

rent

(Am

p)

Iq Id Iq Commnad

(c) ISTE

Figure 6-29: Step Responses for Different PI Controller Gain Settings

118

Chapter 7: Conclusion

The objective of this thesis is to investigate the systematic design methods for

DSP-based PMSM current control. Chapter 2 gives a detailed description of the

experimental system used in the thesis. In Chapter 3, the ZN PID control tuning was

reviewed and the relay test was introduced to identify the plant and facilitate the

automatic tuning process. For the first order plus time delay and second order plus time

delay plants, equations have been developed to identify the system parameters. In

Chapter 4, the optimal design of PI/PID control is proposed based on LQ optimization.

For the performance indices of ISE, ISTE and IST2E, empirical formulae for setting the

PI controller gains of a FOPTD plant are developed in a wider range of the normalized

time delay compared with known empirical formulae for the same purpose. The resulting

systems gain margin and phase margin are also presented. Furthermore, the resulting

bandwidth is formulized originally in this thesis as another empirical formula. Like the

case of PI controller design for a FOPTD plant, the set of empirical formulae for an

integral controller design for a pure time delay system are presented originally for the

gain setting and resulting system bandwidth. This set of formulae is useful when pole-

zero cancellation is applied in the PID/PI controller design process. The resulting system

has constant gain margin and phase margin in the studied range of the normalized time

delay, and the values only depend on the selected performance index. In Chapter 5,

vector current control structure and sine wave current structure were examined under

ideal conditions.

In Chapter 6 more accurate system models were presented for the digitally

implemented current control system for both control methods while considering the

119

digital update frequency and PWM frequency. The systematic design processes were

developed for vector current control and sine wave control to determine minimums for

the digital update frequency and PWM frequency, and PI gains according to the desired

bandwidth and maximum motor speed. The validity of the proposed design processes was

tested by simulation and experiments. For a given system, the automatic relay tuning

process can be used to determine the PI controller gains for good performance after it is

implemented. The research shows that the vector current control structure is better than

the sine wave current structure especially in the high speed application. The automatic

tuning example demonstrates the auto tuning process for a vector current control system.

This auto tuning process can be useful for final tuning of the system to account for the

errors in the design model.

In this thesis, the PMSM current loop bandwidth, digital update frequency and

PWM update frequency, and PI controller gains are set by empirical formulae from LQ

design. Actually, the same process can be applied on other known PI controller automatic

tuning rules such as ZN tuning method.

In this thesis, the proposed systematic design methods focus on the system

specifications of bandwidth, gain margin and phase margin. The resulting system is

reasonable in terms of control performance. However, in future research other system

performance measurements such as current ripple might be investigated. The

relationships between current ripple and the digital update frequency and PWM

frequency may be determined to improve the proposed design methods.

120

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124

Appendix A: Current Control DSP Program Source Code

• File Name: dmcmsg.h

/************************************************************************/

/************** HOST-DSP COMMUNICATION CONSTANTS **********************/

#define GET_STATUS 0x30 // dsp updates the status structure with its values

#define GET_GAINS 0x40 // dsp updates the gain structure with its values

#define GET_POS 0x50 // dsp updates the postion structure with its values

#define CHANGE_GAINS 0x60 // dsp reads gains from gain structure

#define RESET 0x70 // reset all encoders and commands

#define CHANGE_PARAMS 0x80 // dsp reads the mag used for steps and periodic waveforms

#define ENABLE_LIMIT 0xA0 // set error limit mode

#define DISABLE_LIMIT 0xB0 // disable error limit mode

#define START_STORE 0xC0 // start storing data

#define STOP_STORE 0xD0 // stop storing data

/************************************************************************/

/*********** Constants for Drive motor ***************************/

#define DRV_STEP_DN_I 0x1100 // decrement the current command

#define DRV_STEP_UP_I 0x1110 // increment the current command

#define DRV_STEP_DN_P 0x1200 // decrement the pos command

#define DRV_STEP_UP_P 0x1210 // increment the pos command

#define DRV_STEP_DN_V 0x1300 // decrement the vel command

#define DRV_STEP_UP_V 0x1310 // increment the vel command

#define DRV_NOWAVE 0x1500

#define DRV_START_SINE 0x1510 // start a sine wave

#define DRV_START_SQUARE 0x1520 // start a square wave

#define DRV_CURRENT_CONTROL 0x1600 //ste the drive motor into the current control loop

#define DRV_POS_CONTROL 0x1610 //set the drive motor into the position control loop

#define DRV_VEL_CONTROL 0x1620 //set the drive motor into the velocity control loop

#define ENABLE_DRV_CNTRL 0x1710 //enable the Drive motor Amps

#define DISABLE_DRV_CNTRL 0x1700 //disable the Drive motor Amps

125

/****************************************************************************/

/************** STRUCTURES SHARED BY HOST AND DSP ********************/

typedef struct

float Drv_Kp,Drv_Ki,Drv_Kd,Drv_NH,Drv_PosErrLimit; // Parameters for position control for the

//drive motor

float Drv_Kpi,Drv_Kii,Drv_Kt,Drv_IErrLimit, Drv_Feedback_Kp; // Parameters for current/torque

//control for the drive motor

float Dyn_Kp, Dyn_Ki, Dyn_Kd, Dyn_NH, Dyn_PosErrLimit; //Parameters for the

//position control loop of the dyno motor

float IaOffset, IaScale;

float IbOffset, IbScale;

float IcOffset, IcScale;

float IBusOffset, IBusScale;

float VBusOffset, VBusScale;

float TorqOffset, TorqScale;

GainStruct;

typedef struct

float Drv_Iq, Drv_IqCom, Drv_Th, Drv_ThCom, Drv_Vel, Drv_VelCom;

float Dyn_IqCom, Dyn_Th, Dyn_ThCom, Dyn_Vel, Dyn_VelCom;

float Drv_Ibus, Drv_Vbus,Torque;

unsigned int Diagnose, ComCount, ErrLimitFlag, DataReady;

PosStruct;

typedef struct

float CurMag, ThMag, VelMag, WaveFreq, SampleFreq;

unsigned int ErrLimitMode, StoreMode;

unsigned int Drv_CntrlMode, Drv_WaveMode, EnableDrvCntrl;

unsigned int Dyn_CntrlMode, Dyn_WaveMode, EnableDynCntrl;

unsigned int DataPtr, TestAxis;

StatusStruct;

typedef struct

float StrokeTime; // Time for the Stroke

ProfileStruct;

126

/************************************************************************/

/************** EXTERNAL MEMORY MANAGEMENT **************************/

// There are 131072 (128K) words of external memory on the MC8 board

// The following allocates some of this

// The dsp program probably allocates 10,000 words for a cos lookup at the end

// of the external memory

#define EXMEM_BASE 0x400000 //start of external memory

#define HANDSHAKE EXMEM_BASE //handshake integer

#define GAINLEN 27 //length of gain structure

#define POSLEN 18 //length of position structure

#define STATUSLEN 15 //length of status structure

#define PROFILELEN 1 //length of profile structure

#define GAINSTART (EXMEM_BASE + 1)

#define POSSTART (GAINSTART + GAINLEN)

#define STATUSSTART (POSSTART + POSLEN) //start of status structure

#define PROFILESTART (STATUSSTART + STATUSLEN) //start of profile structure

#define DATA1START (PROFILESTART + PROFILELEN) //start of data buffer1

#define DAT_NUM 28 // number of data elements stored each sample

#define SAMPLE_NUM 2048 // number samples stored in circular buffer changed from 2000 to

//2048/yt

#define DATA2START (DATA1START + DAT_NUM*SAMPLE_NUM) //start of data buffer2

127

• File Name: dmc.c

#include <21060.h>

#include <signal.h>

#include <stdlib.h>

#include <math.h>

#include "dmcmsg.h"

/************************************************************************/

/************** internal function prototypes **************************/

void servo_loop(int sig);

void get_velocities(void);

void get_all_adcs(float adcs[8]);

void output_dacs07(float dacs[8]);

void init_pointers(void);

void init_interrupts(void);

void handle_msg(unsigned int msg_type);

void gen_commands(void);

void store_data(void);

void gen_cos_table(void);

void update_status(void);

void update_gains(void);

void update_pos(void);

void init_globals(void);

void output_3ph_pwms(float Va, float Vb, float Vc, float Vbus);

int read_resolver_absolute(void);

void reset_gate_driver(void);

void vector_cntrl(void);

void B_PI(void); //biliear PI controller

void sine_cntrl();

void Sinewave_PI(void)

void cmd_reset();

/************************************************************************/

/************** constants *********************************************/

#define BASE 0x480000 // base register address for MC8 registers

#define DT 0.00047 //0.0001 //0.00011//0.0001 // inverse of servo freq (delta T)

#define QMODEVAL 0x1310 // rollover bit always zero, don't allow rollover,

// Tmin is 64 us, reset the encoder

#define PWMPERIOD 0xDBF // Tpwm=0.11msec

128

#define PWMDEAD 0x020 // 0x20 is a 1 usec dead time with a 32 MHz clock

#define VBUS 330.0 // Nominal bus voltage of inverter

#define PHASE_MAX_CURRENT 1.9 //peak stall current of the drive motor

/************************************************************************/

/*** pointers to registers and structures, should be constants ********/

int *DIGOUTD,*DAC,*ADC,*SOFTSYNC,*OEDIG,*DIGINA,*QDCOUNT,*QDMODE, *VEL;

int *PWMtop,*PWMbot,*PWMCnt,*PWMEn,*PWMLE;

float *COSTABLE, *DATA1, *DATA2;

unsigned int *HNDSHK;

GainStruct *GAIN;

PosStruct *POS;

StatusStruct *STATUS;

ProfileStruct *PROF;

/************************************************************************/

/************** general global variables **************************************/

float CurMag, ThMag, VelMag, WaveFreq, SampleFreq;

unsigned int ErrLimitMode=1, StoreMode=0, ErrLimitFlag=0;

unsigned int DataPtr=0, StoreDelay=20;

unsigned int Diagnose = 0, ComCount = 0;

unsigned int DataReady = 0, DataReadyPtr;

volatile unsigned int IntrptCount=0;

float DataTime=0;

float Th1Scale, Th2Scale,TorqScale,TorqOffset,VbusScale,VbusOffset;

float IaScale,IaOffset,IbScale,IbOffset,IcScale,IcOffset,IbusScale,IbusOffset;

float Torque;

/******Global variables for the drive motor control*********/

float Drv_Kp,Drv_Ki,Drv_Kd,Drv_NH,Drv_PosErrLimit; // Parameters for position control for the

//drive motor

float Drv_Kpi,Drv_Kii,Drv_Kt,Drv_IErrLimit, Drv_Feedback_Kp; // Parameters for current/torque

//control for the drive motor

float Drv_Id, Drv_Iq, Drv_IqCom, Drv_Th, Drv_ThCom, Drv_Vel, Drv_VelCom; //Position ststus of

//the drive motor

unsigned int Drv_CntrlMode, Drv_WaveMode, EnableDrvCntrl;

float Drv_Umax, Drv_Imax; //for the device safety

float Drv_UOffset; //To compensate the d/a offset

float Drv_IComOffset,Drv_ThComOffset,Drv_VelComOffset; //for the wave mode of the drive motor

//int ThElec;

129

float Drv_Ibus,Drv_Ia,Drv_Ib,Drv_Ic,Drv_Vbus,Drv_Va,Drv_Vb,Drv_Vc,Drv_Vd,Drv_Vq;

float Drv_Orig_Va, Drv_Orig_Vb, Drv_Orig_Vc; //the exact values just after PI controller

float Drv_IaCom, Drv_IbCom, Drv_IcCom; // for the comparation with sinewave control

/************************************************************************/

/************************************************************************/

void main (void)

unsigned int i, msg_type;

int flag = 0;

float PhaseVolts[3];

init_pointers();

init_globals();

gen_cos_table();

init_interrupts();

*PWMCnt=(PWMPERIOD-1); *PWMEn=0xff00; //set PWM freq and disable PWM outputs

*OEDIG=0x0003; // enable digital output port

*DIGOUTD=0x0000; //yt According to the circuit logic, clear PD3 to disable L180

for (i=0;i<4;i++)

QDCOUNT[i]=0; QDMODE[i]=QMODEVAL;

// reset all encoders

while (!IntrptCount);

i=read_resolver_absolute();

QDCOUNT[0]=i; QDMODE[0]=QMODEVAL;

QDCOUNT[1]=i; QDMODE[1]=QMODEVAL; // reset both encoders

Drv_IqCom=0; Drv_ThCom=Th1Scale*i;

cmd_reset();

*HNDSHK = 0;

while(1) // loop looking for message from host

if ((*HNDSHK) & 0x80000000) // if msg from host

msg_type=(*HNDSHK)&0x0ffffff0;

handle_msg(msg_type); // handle the msg

*HNDSHK = 0; // write a handshake

ComCount++; // DIAGNOSE CAN BE PUT ANYWHERE AND SET TO

//ANYTHING

// IT WILL BE PRINTED TO THE SCREEN EVERY 0.25 Sec

130

/************************************************************************/

/************************************************************************/

void handle_msg(unsigned int msg_type)

unsigned int i;

if(msg_type == ENABLE_DRV_CNTRL)

Drv_ThCom=Drv_Th; Drv_IqCom=0.0; Drv_VelCom=0.0;

reset_gate_driver(); EnableDrvCntrl=1;

else if(msg_type == DISABLE_DRV_CNTRL)

EnableDrvCntrl = 0;

else if(msg_type == GET_STATUS)

update_status();

else if(msg_type == GET_GAINS)

update_gains();

else if(msg_type == GET_POS)

update_pos();

else if(msg_type == CHANGE_GAINS)

EnableDrvCntrl = 0;

Drv_Kp=GAIN->Drv_Kp; Drv_Ki=GAIN->Drv_Ki; Drv_Kd=GAIN->Drv_Kd;

Drv_NH=GAIN->Drv_NH;

Drv_PosErrLimit=GAIN->Drv_PosErrLimit;

Drv_Kpi=GAIN->Drv_Kpi; Drv_Kii=GAIN->Drv_Kii; Drv_Kt=GAIN->Drv_Kt;

Drv_IErrLimit=GAIN->Drv_IErrLimit;

Drv_Feedback_Kp = GAIN->Drv_Feedback_Kp;

IaOffset = GAIN->IaOffset; IaScale = GAIN->IaScale;

IbOffset = GAIN->IbOffset; IbScale = GAIN->IbScale;

IcOffset = GAIN->IcOffset; IcScale = GAIN->IcScale;

IbusOffset = GAIN->IBusOffset; IbusScale = GAIN->IBusScale;

VbusOffset = GAIN->VBusOffset; VbusScale = GAIN->VBusScale;

TorqOffset = GAIN->TorqOffset; TorqScale = GAIN->TorqScale;

else if(msg_type == RESET)

cmd_reset();

131

else if(msg_type == CHANGE_PARAMS)

EnableDrvCntrl = 0;

CurMag=STATUS->CurMag; ThMag=STATUS->ThMag; VelMag=STATUS->VelMag;

if (STATUS->WaveFreq > 1.0/(10.0*DT))

WaveFreq = 1.0/(10.0*DT);

else if (STATUS->WaveFreq < 0.01)

WaveFreq = 0.01;

else

WaveFreq = STATUS->WaveFreq;

if ((STATUS->SampleFreq > 1.0/DT) || (STATUS->SampleFreq < 10.0))

StoreDelay = 1;

else

StoreDelay = 1/DT/STATUS->SampleFreq;

else if(msg_type == ENABLE_LIMIT)

ErrLimitMode = 1;

else if(msg_type == DISABLE_LIMIT)

ErrLimitFlag=ErrLimitMode = 0;

else if(msg_type == DRV_STEP_DN_I)

Drv_IqCom -= CurMag;

else if(msg_type == DRV_STEP_UP_I)

Drv_IqCom += CurMag;

else if(msg_type == DRV_NOWAVE)

Drv_WaveMode=DRV_NOWAVE;

else if(msg_type == DRV_START_SINE)

Drv_IComOffset=Drv_IqCom; Drv_ThComOffset=Drv_ThCom;

Drv_VelComOffset=Drv_VelCom;

Drv_WaveTime = 0; Drv_WaveMode=DRV_START_SINE;

else if(msg_type == DRV_START_SQUARE)

Drv_IComOffset=Drv_IqCom; Drv_ThComOffset=Drv_ThCom;

132

Drv_VelComOffset=Drv_VelCom;

Drv_WaveTime = 0; Drv_WaveMode=DRV_START_SQUARE;

else if(msg_type == DRV_CURRENT_CONTROL)

Drv_CntrlMode = DRV_CURRENT_CONTROL;

else if(msg_type == START_STORE)

DataPtr=0; DataTime=0; StoreMode=1;

else if(msg_type == STOP_STORE) StoreMode=0;

/************************************************************************/

/***** update the status structure read and written by the host *********/

void update_status(void)

STATUS->CurMag=CurMag; STATUS->ThMag=ThMag; STATUS->VelMag=VelMag;

STATUS->WaveFreq=WaveFreq; STATUS->SampleFreq=1.0/DT/StoreDelay;

STATUS->Drv_WaveMode=Drv_WaveMode; STATUS->Drv_CntrlMode=Drv_CntrlMode;

STATUS->Dyn_WaveMode=Dyn_WaveMode; STATUS->Dyn_CntrlMode=Dyn_CntrlMode;

STATUS->ErrLimitMode=ErrLimitMode;

STATUS->EnableDrvCntrl = EnableDrvCntrl;

STATUS->EnableDynCntrl = EnableDynCntrl;

STATUS->StoreMode=StoreMode; STATUS->DataPtr=DataPtr;

/************************************************************************/

/**** update the message structure read and written by the host *********/

void update_gains(void)

GAIN->Drv_Kp=Drv_Kp; GAIN->Drv_Ki=Drv_Ki;

GAIN->Drv_Kd=Drv_Kd; GAIN->Drv_NH=Drv_NH;

GAIN->Drv_PosErrLimit=Drv_PosErrLimit; GAIN->Drv_Kpi=Drv_Kpi;

GAIN->Drv_Kii=Drv_Kii; GAIN->Drv_Kt=Drv_Kt;

GAIN->Drv_IErrLimit=Drv_IErrLimit;

GAIN->Drv_Feedback_Kp = Drv_Feedback_Kp;

GAIN->IaOffset = IaOffset; GAIN->IaScale = IaScale;

GAIN->IbOffset = IbOffset; GAIN->IbScale = IbScale;

GAIN->IcOffset = IcOffset; GAIN->IcScale = IcScale;

GAIN->IBusOffset = IbusOffset; GAIN->IBusScale = IbusScale;

GAIN->VBusOffset = VbusOffset; GAIN->VBusScale = VbusScale;

133

GAIN->TorqOffset = TorqOffset; GAIN->TorqScale = TorqScale;

/************************************************************************/

/**** update the message structure read and written by the host *********/

void update_pos(void)

POS->Drv_Iq=Drv_Iq; POS->Drv_IqCom=Drv_IqCom;

POS->Drv_Th=Drv_Th; POS->Drv_ThCom=Drv_ThCom; POS->Drv_Vel=Drv_Vel;

POS->Drv_VelCom=Drv_VelCom;

POS->Drv_Ibus=Drv_Ibus; POS->Drv_Vbus=Drv_Vbus; POS->Torque=Torque;

POS->Diagnose=Diagnose; POS->ComCount=ComCount;

POS->ErrLimitFlag=ErrLimitFlag, POS->DataReady=DataReady;

/************************************************************************/

/** with pwm outputs high for >12 usec pull reset pin lo for >2 usec ****/

void reset_gate_driver(void)

EnableDrvCntrl=0; *PWMEn=0xff00; //disable PWM outputs

IntrptCount=0; while(IntrptCount<2); // Wait for at least one full servo cylcle

*DIGOUTD=*DIGOUTD&0xFFFB; // Clear RESET


*DIGOUTD=*DIGOUTD|0x0004; // Set RESET

/************************************************************************/

/************************************************************************/

void servo_loop(int sig)

float dacs[8], adcs[8];

int i;

static float buf_Ia[5]=0, 0, 0, 0, 0, buf_Ib[5]=0, 0, 0, 0, 0, buf_Ic[5]=0, 0, 0, 0, 0;

static int buf_pos=0;

float ave_DrvIa, ave_DrvIb, ave_DrvIc;

IntrptCount++;

*DIGOUTD = *DIGOUTD|0x0010; // Set PD4 for scope watch

*DIGOUTD = *DIGOUTD^0x0020; // Toggle PD5 for scope watch

get_all_adcs(adcs);

Drv_Ia=IaScale*(adcs[0]+IaOffset); Drv_Ib=IbScale*(adcs[1]+IbOffset);

Drv_Ic=IcScale*(adcs[2]+IcOffset); Drv_Ibus=IbusScale*(adcs[3]+IbusOffset);

Drv_Vbus=VbusScale*(adcs[4]+VbusOffset); Torque=TorqScale*(adcs[5]+TorqOffset);

134

Drv_Th=Th1Scale*QDCOUNT[0]; // read encoders

ThElec=QDCOUNT[0]; ThElec=ThElec-(ThElec/1024)*1024; //ThElec=ThElec-(ThElec/1024)*1024-

//256; inverse qd axis used in blocked rotor test for d axis

if (ThElec<0) ThElec=1024+ThElec;

get_velocities();

gen_commands();

//Depend on the used control structure, the unused used control structure should be commented

//vect_cntrl() for vector current control

//sine_cntrll for sine wave current control

vector_cntrl(); //vector current control

//sine_cntrl(); //sine wave current control

if (StoreMode) store_data(); //store data in external memory

output_dacs07(dacs);

*SOFTSYNC=0x0001; // writing any value generates a sync signal

if (ErrLimitFlag) EnableDrvCntrl=0; // do after store_data

if (EnableDrvCntrl)

*PWMEn=0xFFFF; // enable PWM outputs

else

*PWMEn=0xFF00; // disable PWM outputs

*DIGOUTD = *DIGOUTD & 0xFFEF; // clear bit PD4 for scope watch

/************************************************************************/

/************************************************************************/

void vector_cntrl(void)

int Th2pi3, Th_2pi3, Th_90, Th2pi3_90, Th_2pi3_90;

float P[2][3];

Th2pi3=ThElec+341; if(Th2pi3 > 1023) Th2pi3 -= 1024;

Th_2pi3=ThElec-341; if(Th_2pi3 < 0) Th_2pi3 += 1024;

Th_90=ThElec-256; if(Th_90 < 0) Th_90 += 1024;

Th2pi3_90=ThElec+85; if(Th2pi3_90 > 1023) Th2pi3_90 -= 1024;

Th_2pi3_90=ThElec-597; if(Th_2pi3_90 < 0) Th_2pi3_90 += 1024;

P[0][0]=COSTABLE[ThElec]; P[0][1]=COSTABLE[Th_2pi3]; P[0][2]=COSTABLE[Th2pi3];

P[1][0]=COSTABLE[Th_90]; P[1][1]=COSTABLE[Th_2pi3_90]; P[1][2]=COSTABLE[Th2pi3_90];

//According the Parkson transform, a factor 2/3 needed from the phase coordinte to the dq coordinate

Drv_Id = 0.666666667*(P[1][0]*Drv_Ia+P[1][1]*(-Drv_Ia-Drv_Ic)+P[1][2]*Drv_Ic);

Drv_Iq = 0.666666667*(P[0][0]*Drv_Ia+P[0][1]*(-Drv_Ia-Drv_Ic)+P[0][2]*Drv_Ic);

B_PI(); //ErrLimitFlag set in here

135

Drv_Va = (P[0][0]*Drv_Vq+P[1][0]*Drv_Vd);// /VBUS;

Drv_Vb = (P[0][1]*Drv_Vq+P[1][1]*Drv_Vd);// /VBUS;

Drv_Vc = (P[0][2]*Drv_Vq+P[1][2]*Drv_Vd);// /VBUS;

if ((Drv_Vc <= Drv_Va)&&(Drv_Vc <= Drv_Vb))

Drv_Va=Drv_Va-Drv_Vc; Drv_Vb=Drv_Vb-Drv_Vc; Drv_Vc=0;

if ((Drv_Va >= Drv_Vb)&&(Drv_Va > VBUS))

Drv_Vb=Drv_Vb/Drv_Va*VBUS; Drv_Va=VBUS;

else if (Drv_Vb > VBUS)

Drv_Va=Drv_Va/Drv_Vb*VBUS; Drv_Vb=VBUS;

else if ((Drv_Vb <= Drv_Va)&&(Drv_Vb <= Drv_Vc))

Drv_Va=Drv_Va-Drv_Vb; Drv_Vc=Drv_Vc-Drv_Vb; Drv_Vb=0;

if ((Drv_Va >= Drv_Vc)&&(Drv_Va > VBUS))

Drv_Vc=Drv_Vc/Drv_Va*VBUS; Drv_Va=VBUS;

else if (Drv_Vc > VBUS)

Drv_Va=Drv_Va/Drv_Vc*VBUS; Drv_Vc=VBUS;

else

Drv_Vb=Drv_Vb-Drv_Va; Drv_Vc=Drv_Vc-Drv_Va; Drv_Va=0;

if ((Drv_Vb >= Drv_Vc)&&(Drv_Vb > VBUS))

Drv_Vc=Drv_Vc/Drv_Vb*VBUS; Drv_Vb=VBUS;

else if (Drv_Vc > VBUS)

Drv_Vb=Drv_Vb/Drv_Vc*VBUS; Drv_Vc=VBUS;

Diagnose=ThElec;

output_3ph_pwms(Drv_Va,Drv_Vb,Drv_Vc,VBUS);

/************************************************************************/

136

/************************************************************************/

void output_3ph_pwms(float Va, float Vb, float Vc, float Vbus)

float duty, Voltage[3];

int HalfPeriod, TopCount, BotCount, i;

Voltage[0]=Va; Voltage[1]=Vb; Voltage[2]=Vc;

HalfPeriod = (int)(PWMPERIOD/2);

for(i=0;i<3;i++)

duty = Voltage[i]/Vbus;

if (duty > 1.0) duty = 1.0;

else if (duty < 0.0) duty = 0.0;

BotCount = (int)(duty*HalfPeriod); TopCount =(int) (BotCount-PWMDEAD);

if (TopCount < 0) TopCount = 0;

PWMtop[i]=0x80000000 + ((HalfPeriod+TopCount)<<16) + (HalfPeriod-TopCount);

PWMbot[i]=((HalfPeriod+BotCount)<<16) + (HalfPeriod-BotCount);

*PWMLE = 0x00;

/************************************************************************/

/************************************************************************/

int read_resolver_absolute(void)

// PD0 = SCLK, PD1=NotCS, PA0 = Data

int ResolverValue=0, i, PlaceValue=2048;

*DIGOUTD=*DIGOUTD&0xFFFE; // Clear SCLK

*DIGOUTD=*DIGOUTD|0x0002; // Set Chip Select


*DIGOUTD=*DIGOUTD&0xFFFD; // Clear Chip Select


for(i=0;i<12;i++)

*DIGOUTD=*DIGOUTD|0x0001; // Set SCLK


*DIGOUTD=*DIGOUTD&0xFFFE; // Clear SCLK


ResolverValue += PlaceValue*(*DIGINA & 0x0001); //Read a bit of data

PlaceValue = PlaceValue>>1; // Divide place holder value by 2

*DIGOUTD=*DIGOUTD|0x0002; // Set Chip Select

return (ResolverValue);

137

/************************************************************************/

/************************************************************************/

void get_velocities(void)

float velocity[2];

int DxDt, Dx, Dt, i;

for(i=0;i<2;i++)

DxDt = VEL[i];

Dt = DxDt & 0xffff;

if (Dt == 0xffff)

velocity[i] = 0.0;

else

Dx = (DxDt>>16) & 0x3ff;

if (Dx & 0x200) Dx = Dx | 0xfffffc00;

velocity[i] = Dx*32.0E6/Dt;

Drv_Vel = Th1Scale*velocity[0]/6.0;

/************************************************************************/

/************************************************************************/

void store_data(void)

static int count=0;

unsigned int mem_ptr, i;

count++;

if (count >= StoreDelay)

count = 0; DataPtr++;

if (DataPtr > (SAMPLE_NUM-1)) DataPtr=0;

DataTime = DataPtr*DT*StoreDelay;

mem_ptr = DataPtr*DAT_NUM;

//Time,Th1,Th2,ThCom,Vel1,Vel2,VelCom,Toruqe,IqCom,Iq,Id,Ibus,Ia,Ib,Ic,Vbus,Va,Vb,Vc,

//Uamp

//Drv_IaCom Drv_IbCom Drv_IcCom Drv_Orig_Va Drv_Orig_Vb Drv_Orig_Vc Drv_Vq

//Drv_Vd

DATA1[mem_ptr]=DataTime; DATA1[mem_ptr+1]=Drv_Th;

DATA1[mem_ptr+3]=Drv_ThCom; DATA1[mem_ptr+4]=Drv_Vel;

138

DATA1[mem_ptr+6]=Drv_VelCom; DATA1[mem_ptr+7]=Torque;

DATA1[mem_ptr+8]=Drv_IqCom; DATA1[mem_ptr+9]=Drv_Iq;

DATA1[mem_ptr+10]=Drv_Id; DATA1[mem_ptr+11]=Drv_Ibus;

DATA1[mem_ptr+12]=Drv_Ia; DATA1[mem_ptr+13]=Drv_Ib;

DATA1[mem_ptr+14]=Drv_Ic; DATA1[mem_ptr+15]=Drv_Vbus;

DATA1[mem_ptr+16]=Drv_Va; DATA1[mem_ptr+17]=Drv_Vb;

DATA1[mem_ptr+18]=Drv_Vc; DATA1[mem_ptr+22]=Drv_IcCom;

DATA1[mem_ptr+23]=Drv_Orig_Va; DATA1[mem_ptr+24]=Drv_Orig_Vb;

DATA1[mem_ptr+25]=Drv_Orig_Vc; DATA1[mem_ptr+26]=Drv_Vq;

DATA1[mem_ptr+27]=Drv_Vd;

/************************************************************************/

/************************************************************************/

void gen_commands(void)

int Drv_SinIndex;

Drv_WaveTime += DT;

if (Drv_WaveTime >= 1/WaveFreq) Drv_WaveTime -= 1/WaveFreq;

Drv_SinIndex = 1024.0*(Drv_WaveTime*WaveFreq) - 256;

if (Drv_SinIndex < 0) Drv_SinIndex += 1024;

/*********Drive motor command generator*************/

if(Drv_CntrlMode==DRV_CURRENT_CONTROL)

if(Drv_WaveMode==DRV_NOWAVE)

Drv_IqCom=Drv_IqCom;

else if(Drv_WaveMode==DRV_START_SINE)

Drv_IqCom = Drv_IComOffset + CurMag*COSTABLE[Drv_SinIndex];

else if(Drv_WaveMode==DRV_START_SQUARE)

if (Drv_WaveTime < 0.5/WaveFreq)

Drv_IqCom = Drv_IComOffset + CurMag;

else

Drv_IqCom = Drv_IComOffset - CurMag;

139

/************************************************************************/

/************************************************************************/

void B_PI(void) // PI Current Controller

float Errd, Errq, VId, VIq,Vsat;

static float VId_1, VIq_1, Errd_1=0.0, Errq_1=0.0;

Errd = -Drv_Id*Drv_Feedback_Kp; Errq = Drv_IqCom-Drv_Iq*Drv_Feedback_Kp;

VId = VId_1 + Drv_Kii*DT*(Errd+Errd_1)/2; VIq = VIq_1 + Drv_Kii*DT*(Errq+Errq_1)/2;

Errd_1 = Errd; Errq_1 = Errq;

Vsat=VBUS/2.0; //this limit will hold down the integrator output, and then degrade the compensatation

//capacity for high speed operation

if (VId > Vsat) VId = Vsat;

else if (VId < -Vsat) VId = -Vsat;

if (VIq > Vsat) VIq = Vsat;

else if (VIq < -Vsat) VIq = -Vsat;

if (!EnableDrvCntrl) VId = 0.0; VIq = 0.0;

Drv_Vd = VId + Drv_Kpi*Errd; Drv_Vq = VIq + Drv_Kpi*Errq;

Vsat=VBUS;

if (Drv_Vd > Vsat) Drv_Vd = Vsat; // with saturation limits

else if (Drv_Vd < -Vsat) Drv_Vd = -Vsat;

if (Drv_Vq > Vsat) Drv_Vq = Vsat; // with saturation limits

else if (Drv_Vq < -Vsat) Drv_Vq = -Vsat;

if (ErrLimitMode)

if ((fabsf(Errd)>Drv_IErrLimit)||(fabsf(Errq)>Drv_IErrLimit))

ErrLimitFlag = ErrLimitFlag | 0x01;

VId_1 = VId; VIq_1 = VIq;

/************************************************************************/

/************************************************************************/

void get_all_adcs(float adcs[8])

int hi_lo, hi_word, lo_word, i;

for (i=0;i<4;i++)

hi_lo=ADC[i];

if (hi_lo & 0x2000) lo_word = hi_lo | 0xffffc000; // bit 13 is sign bit for

else lo_word = hi_lo & 0x00001fff; // the lo word

140

if (hi_lo & 0x20000000) hi_word = (hi_lo>>16) | 0xffffc000; // bit 29 is sign bit for

else hi_word = hi_lo>>16; // the hi word

adcs[2*i]=lo_word*10/8192.0; adcs[2*i+1]=hi_word*10/8192.0;

/************************************************************************/

/************************************************************************/

void output_dacs07(float dacs[8])

unsigned int hi_lo, hi_word, lo_word, i;

for (i=0;i<4;i++)

if (dacs[2*i]<-10.0) lo_word = 0;

else if (dacs[2*i]>9.998) lo_word = 0x3fff;

else lo_word = (unsigned int)(dacs[2*i]*8192.0/10.0+8192.0);

if (dacs[2*i+1]<-10.0) hi_word = 0;

else if (dacs[2*i+1]>9.998) hi_word = 0x3fff;

else hi_word = (unsigned int)(dacs[2*i+1]*8192.0/10.0+8192.0);

hi_lo = lo_word | (hi_word<<16);

DAC[i] = hi_lo;

/************************************************************************/

/************************************************************************/

void gen_cos_table(void)

int i;

float theta;

for (i=0;i<1024;i++)

theta = 2.0*3.1415927*i/1024.0;

COSTABLE[i] = cos(theta);

/************************************************************************/

/************************************************************************/

void init_pointers(void)

DAC=(int *)BASE+0x0004; // address for DAC 0

ADC=(int *)BASE+0x000c; // address for ADC 0

QDCOUNT=(int *)BASE+0x0020; // address for Quad Decoder 0 Count Reg

QDMODE=(int *)BASE+0x0028; // address for Quad Decoder 0 Mode Reg

141

VEL=(int *)BASE+0x0030; // address for DX/DT 0 Reg

DIGOUTD=(int *)BASE+0x0013; // address for digital output port

DIGINA=(int *)BASE+0x0010; // address for digital input port A

OEDIG=(int *)BASE+0x0019; // address for digital output enable

SOFTSYNC=(int *)BASE+0x001f; // address for SOFTSYNC signal

PWMtop=(int *)BASE+0x0024; // address for first four PWM channels

PWMbot=(int *)BASE+0x0034; // address for second four PWM channels

PWMCnt=(int *)BASE+0x002C; // address for PWMCnt Reg

PWMEn=(int *)BASE+0x002D; // address for PWMEn Reg

PWMLE=(int *)BASE+0x002F; // address for PWMLE Reg

COSTABLE = (float *)(EXMEM_BASE + 121072);

GAIN = (GainStruct *)GAINSTART;

POS = (PosStruct *)POSSTART;

STATUS = (StatusStruct *)STATUSSTART;

PROF = (ProfileStruct *)PROFILESTART;

DATA1 = (float *)DATA1START;

DATA2 = (float *)DATA2START;

HNDSHK = (unsigned int *)HANDSHAKE;

/************************************************************************/

/************************************************************************/

void init_globals(void)

int index;

int SegLen = DAT_NUM*SAMPLE_NUM;

Drv_Kp=1.0; Drv_Ki=0.0; Drv_Kd=0.0; Drv_NH=2.0*360.0/4096.0; Drv_PosErrLimit =90.0;

Drv_Kpi=90.3053; Drv_Kii=160221.2; Drv_Kt=3.27; Drv_IErrLimit=0.25;

Drv_Feedback_Kp = 1.0;

Drv_Imax=1.5; Drv_Umax=220;

Drv_UOffset =0.0;

Drv_IComOffset=0.0; Drv_ThComOffset=0.0;

Th1Scale=360.0/4096.0; Th2Scale=360.0/4096.0;

TorqScale=1.0631/*1.017262*/; TorqOffset=0.0; VbusScale=119.05; VbusOffset=0.0;

IaScale=0.502589;IaOffset=-2.024005-1.46E-03;

IbScale=0.5018039;IbOffset=-2.010417-2.35E-02;

IcScale=0.4985377;IcOffset=-2.02; IbusScale=0.5042214;IbusOffset=-2.05;

CurMag=0.05; ThMag=60.0; VelMag=5.0; WaveFreq = 1.0;

142

//initialize the data store segment

for(index=0; index<SegLen; index++)

DATA1[index] = 0.0;

/************************************************************************/

/************************************************************************/

void init_interrupts(void)

unsigned int timer_period;

// contants or globals DT

timer_period = 40000000.0*DT-1; // set up timer 0 to be a clock at the

timer_set(timer_period,10); // desired servo update frequency

interruptf(SIG_TMZ0,servo_loop); // set up servo_loop() as the interrupt

timer_on(); // handler for timer 0 interrupt

/************************************************************************/

/************************************************************************/

void cmd_reset()

int i;

EnableDrvCntrl=0;

ErrLimitMode=1; ErrLimitFlag=0;

Drv_WaveMode=DRV_NOWAVE; Drv_CntrlMode=DRV_CURRENT_CONTROL;

//To make the relative angle with respect to the phase A

i=read_resolver_absolute()-256;

if (i > 1024) i -= 1024;

QDCOUNT[0]=i; QDMODE[0]=QMODEVAL;

QDCOUNT[1]=i; QDMODE[1]=QMODEVAL; // reset both encoders

Drv_IqCom=0.0; Drv_ThCom=Th1Scale*i; Drv_VelCom=0.0;

Drv_IComOffset=0.0; Drv_ThComOffset=0.0;

/************************************************************************/

/************************************************************************/

//In the function, a sine wave controller was implemented. The command is the current sinewave amplitute,

//which can be got from the Iq command on the controller interface

void sine_cntrl(void)

143

int Th2pi3, Th_2pi3, Th_90, Th2pi3_90, Th_2pi3_90;

float P[2][3];

Th2pi3=ThElec+341; if(Th2pi3 > 1023) Th2pi3 -= 1024;

Th_2pi3=ThElec-341; if(Th_2pi3 < 0) Th_2pi3 += 1024;

Th_90=ThElec-256; if(Th_90 < 0) Th_90 += 1024;

Th2pi3_90=ThElec+85; if(Th2pi3_90 > 1023) Th2pi3_90 -= 1024;

Th_2pi3_90=ThElec-597; if(Th_2pi3_90 < 0) Th_2pi3_90 += 1024;

P[0][0]=COSTABLE[ThElec]; P[0][1]=COSTABLE[Th_2pi3]; P[0][2]=COSTABLE[Th2pi3];

P[1][0]=COSTABLE[Th_90]; P[1][1]=COSTABLE[Th_2pi3_90]; P[1][2]=COSTABLE[Th2pi3_90];

Drv_Id = 0.66666667*(P[1][0]*Drv_Ia+P[1][1]*(-Drv_Ia-Drv_Ic)+P[1][2]*Drv_Ic);

Drv_Iq = 0.66666667*(P[0][0]*Drv_Ia+P[0][1]*(-Drv_Ia-Drv_Ic)+P[0][2]*Drv_Ic);

Drv_IaCom = (P[1][0]*0.0+P[0][0]*Drv_IqCom);

Drv_IbCom = (P[1][1]*0.0+P[0][1]*Drv_IqCom);

Drv_IcCom = (P[1][2]*0.0+P[0][2]*Drv_IqCom);

Sinewave_PI();

Drv_Orig_Va = Drv_Va*VBUS; Drv_Orig_Vb = Drv_Vb*VBUS; Drv_Orig_Vc = Drv_Vc*VBUS;

Drv_Vd = 0.66666667*(P[1][0]*Drv_Orig_Va+P[1][1]*Drv_Orig_Vb+P[1][2]*Drv_Orig_Vc);

Drv_Vq = 0.66666667*(P[0][0]*Drv_Orig_Va+P[0][1]*Drv_Orig_Vb+P[0][2]*Drv_Orig_Vc);

if ((Drv_Vc <= Drv_Va)&&(Drv_Vc <= Drv_Vb))

Drv_Va=Drv_Va-Drv_Vc; Drv_Vb=Drv_Vb-Drv_Vc; Drv_Vc=0;

if ((Drv_Va >= Drv_Vb)&&(Drv_Va > 1))

Drv_Vb=Drv_Vb/Drv_Va; Drv_Va=1;

else if (Drv_Vb > 1)

Drv_Va=Drv_Va/Drv_Vb; Drv_Vb=1;

else if ((Drv_Vb <= Drv_Va)&&(Drv_Vb <= Drv_Vc))

Drv_Va=Drv_Va-Drv_Vb; Drv_Vc=Drv_Vc-Drv_Vb; Drv_Vb=0;

if ((Drv_Va >= Drv_Vc)&&(Drv_Va > 1))

Drv_Vc=Drv_Vc/Drv_Va; Drv_Va=1;

else if (Drv_Vc > 1)

Drv_Va=Drv_Va/Drv_Vc; Drv_Vc=1;

else

144

Drv_Vb=Drv_Vb-Drv_Va; Drv_Vc=Drv_Vc-Drv_Va; Drv_Va=0;

if ((Drv_Vb >= Drv_Vc)&&(Drv_Vb > 1))

Drv_Vc=Drv_Vc/Drv_Vb; Drv_Vb=1;

else if (Drv_Vc > 1)

Drv_Vb=Drv_Vb/Drv_Vc; Drv_Vc=1;

Drv_Va *= VBUS; Drv_Vb *= VBUS; Drv_Vc *= VBUS; Diagnose=ThElec;

output_3ph_pwms(Drv_Va,Drv_Vb,Drv_Vc,VBUS);

/************************************************************************/

/************************************************************************/

void Sinewave_PI(void)

float ErrIa, ErrIb, ErrIc, VIa=0, VIb=0, VIc=0, Vsat;

static float VIa_1, VIb_1, VIc_1;

ErrIa = Drv_IaCom-Drv_Ia*Drv_Feedback_Kp; ErrIb = Drv_IbCom-Drv_Ib*Drv_Feedback_Kp;

ErrIc = Drv_IcCom-Drv_Ic*Drv_Feedback_Kp;

VIa = VIa_1+Drv_Kii*DT*ErrIa; VIb = VIb_1+Drv_Kii*DT*ErrIb;

Vsat=VBUS/5.0;

if (VIa > Vsat) VIa = Vsat;

else if (VIa < (-Vsat)) VIa = -Vsat;

if (VIb > Vsat) VIb = Vsat;

else if (VIb < (-Vsat)) VIb = -Vsat;

if (!EnableDrvCntrl) VIa = 0.0; VIb = 0.0; VIc =0.0;

Drv_Va = VIa + Drv_Kpi*ErrIa; Drv_Vb = VIb + Drv_Kpi*ErrIb;

Drv_Vc = -(Drv_Va+Drv_Vb); //calculate the Drv_Vc by Drv_Va and Drv_Vb instead of a PI

//controller

Vsat=VBUS;

if (Drv_Va > Vsat) Drv_Va = Vsat;

else if (Drv_Va < -Vsat) Drv_Va = -Vsat;

if (Drv_Vb > Vsat) Drv_Vb = Vsat;

else if (Drv_Vb < -Vsat) Drv_Vb = -Vsat;

if (Drv_Vc > Vsat) Drv_Vc = Vsat;

else if (Drv_Vc < -Vsat) Drv_Vc = -Vsat;

if (ErrLimitMode)

145

if ((fabsf(ErrIa)>Drv_IErrLimit)||(fabsf(ErrIb)>Drv_IErrLimit)||(fabsf(ErrIc)>Drv_IErrLimit))

ErrLimitFlag = ErrLimitFlag | 0x01;

VIa_1 = VIa; VIb_1 = VIb; //VIc_1 = VIc;

Drv_Va = Drv_Va/VBUS; Drv_Vb = Drv_Vb/VBUS; Drv_Vc = Drv_Vc/VBUS;

146

Appendix B: LQ Design MATLAB Program Code • File Name: lyapunov.m

//Written by Dr. Dale Schinstock

function P=lyapunov(A,Q)

[N,N]=size(Q); M=N*(N+1)/2;

L=zeros(N,N); V=zeros(M,M);

Rv=zeros(M,1);

k=1;

for i=1:N

for j=i:N

L(i,j)=k; L(j,i)=k;

Rv(k)=Q(i,j);

k=k+1;

end

end

for i=1:N

for j=1:N

for k=1:N

V(L(i,k),L(j,k))=A(j,i)+V(L(i,k),L(j,k));

end

end

end

for i=1:N

for j=1:M

V(L(i,i),j)=2*V(L(i,i),j);

end

end

Pv = -V\Rv;

P=zeros(N,N);

k=1;

for i=1:N

for j=i:N

P(i,j)=Pv(k); P(j,i)=Pv(k);

k=k+1;

end

end

147

• File Name: LineSearch.m

%Written by Dr. Dale Schinstock

function [J,K,check] = LineSearch(Kstart,Jstart,Gradstart,NewtonStep,FunHandle)

Alpha=1e-4; Lam = 1;

Slope = Gradstart'*NewtonStep; DoInterps = true;

FirstInterp=true; check = false;

while (DoInterps)

K = Kstart + Lam*NewtonStep;

J = feval(FunHandle,K);

if (J < 0) % Use negative J to indicate instability

Lam = Lam/2; % Cut Lambda and start all over

FirstInterp=true;

else if (J <= (Jstart+Alpha*Lam*Slope))

DoInterps=false; % Sufficient PI decrease

LamNew = Lam;

else if (FirstInterp)

% Minimum lambda for a quadratic fit

LamNew = -Slope/(2*(J-Jstart-Slope));

FirstInterp=false;

else

% Model as cubic polynomial

A=[1/Lam^2 -1/Lam2^2; -Lam2/Lam^2 Lam/Lam2^2];

B=[J - Slope*Lam - Jstart; J2 - Slope*Lam2 - Jstart];

% Cubic Coefficients

C = A*B/(Lam - Lam2); a =C(1); b=C(2);

if (a == 0)

LamNew = -Slope/(2*b);

else

disc = b^2-3*a*Slope;

if (disc < 0)

LamNew = Lam/2;

elseif (b <= 0) LamNew = (-b+sqrt(disc))/(3*a);

else LamNew = -Slope/(b+sqrt(disc));

end

end

end

if (LamNew > Lam/2) LamNew = Lam/2; end % At least cut step in half

148

end

Lam2 = Lam;

J2 = J;

Lam = max(LamNew,0.1*Lam); % Don't cut step too quickly

end

end % Repeat interpolations

149

File Name: DFPmin.m

//Written by Dr. Dale Schinstock

function [Kopt,Jopt,NumIters,Termination,Grad] = DFPmin(FunHandle,initK,MaxIter,Gradtol)

if (nargin < 3) MaxIter = 1000; end

if (nargin < 4) Gradtol = 1e-7; end

EPS = 3e-8; [m1,p1]=size(initK);

N=m1*p1; Hess=eye(N);

K=zeros(m1*p1,1); K(:)=initK; % Form K vector

[J,Grad]=feval(FunHandle,K); NewtonStep = -Hess*Grad;

DoIters=true; NumIters=0; k=0; while (DoIters)

k=k+1; NumIters=NumIters+1;

[Jnew,Knew] = LineSearch(K, J, Grad, NewtonStep,FunHandle);

[Jnew,Gnew]=feval(FunHandle,Knew); % Jnew should same as previous line

dK=Knew-K; dG=Gnew-Grad; dJ=Jnew-J;

HdG=Hess*dG;

J = Jnew; Grad = Gnew; K = Knew;

den1=dK'*dG; den2=dG'*HdG;

if (den1 > sqrt(EPS*(dG'*dG)*(dK'*dK))) % skip Hess update if not positive enough

Hess = Hess + dK*dK'/den1 - HdG*HdG'/den2;

end

if k==10*N

k=0; Hess=eye(N);

end

NewtonStep = -Hess*Grad;

if (NumIters >= MaxIter) DoIters=false; Termination=1; end

if (norm(Grad)*max(norm(NewtonStep),1)/max(J,1)<Gradtol); DoIters=false; Termination=2; end

Kprint=K'

J=J

Gnew=Gnew;

end

Kopt=K;

Jopt=J;

150

• Name File: T0XCostGrad.m

//The original file was written by Dr. Schinstock, this is modified version by Tao Yang

function [J,Grad,P]=T0XCostGrad(Kvector)

global A B C G F H r0 R Q P V m p n

%%%%% Time-independed Cost Function %%%%%%%%%%%%%%%%%%%%

K=zeros(m,p); K(:)=Kvector; % Form K matrix from vector

Ac = A-B*K*C; Aci=inv(Ac);

Bc = G-B*K*F;

xbar = -Aci*(Bc*r0);

ybar = C*xbar + F*r0;


ebar = r0-H*xbar;

X = xbar*xbar';

Qlyap=C'*K'*R*K*C+Q;

P=lyapunov(Ac,Qlyap);

J=0.5*(trace(P*X)+ebar'*V*ebar) % Cost function

[Pcholesky,Pcheck]=chol(P);

if (Pcheck)

J=-1;

end

if nargout > 1 % Called with 2 output arguments

S=lyapunov(Ac',X);

dHdK = R*K*C*S*C'-B'*P*S*C'+B'*Aci'*P*xbar*ybar'; % Gradient of Hamiltonian

dHdK = dHdK + B'*Aci'*H'*V*(H*xbar*ybar'-r0*ybar'); % Gradient of Hamiltonian

Grad=zeros(m*p,1); Grad(:)=dHdK; % Form vector gradient from matrix

End

151

File Name: T2XCostGrad.m

//The original file was written by Dr. Schinstock, this is modified version by Tao Yang



%%%%%% Time-depended Cost Function %%%%%%%%%%%%%%%%%%%%



Bc = G-B*K*F;




ebar = r0-H*xbar;

X = xbar*xbar';

P0=lyapunov(Ac,P);

P1=lyapunov(Ac,P0);

Qlyap=C'*K'*R*K*C+Q + 2*P1;

P2=lyapunov(Ac,Qlyap);

J=0.5*(trace(P2*X)+ebar'*V*ebar) % Cost function

[Pcholesky,Pcheck]=chol(P2);

if (Pcheck)

J=-1;

end


S2=lyapunov(Ac',X);

S1=lyapunov(Ac',2*S2);

S0=lyapunov(Ac',S1);

% Gradient of Hamiltonian

dHdK = R*K*C*S2*C'-B'*(P0*S0+P1*S1+P2*S2)*C'+B'*Aci'*P2*xbar*ybar';



End

152

• File Name: T4XCostGrad.m

%The original file was written by Dr. Schinstock, this is modified version by Tao Yang



%%%%%% Time-depended Cost Function %%%%%%%%%%%%%%%%%%%%



Bc = G-B*K*F;




ebar = r0-H*xbar;

X = xbar*xbar';

P0=lyapunov(Ac,P);

P1=lyapunov(Ac,P0);

P2=lyapunov(Ac,P1);

P3=lyapunov(Ac,P2);

Qlyap=C'*K'*R*K*C+Q + 4*3*2*P3;

P4=lyapunov(Ac,Qlyap);

J=0.5*(trace(P4*X)+ebar'*V*ebar) % Cost function

[Pcholesky,Pcheck]=chol(P4);

if (Pcheck)

J=-1;

end


S4=lyapunov(Ac',X);

S3=lyapunov(Ac',4*3*2*S4);




dHdK = R*K*C*S4*C'-B'*(P0*S0+P1*S1+P2*S2+P3*S3+P4*S4)*C'+B'*Aci'*P4*xbar*ybar';

% Gradient of Hamiltonian



end

153

• File Name: LQ4PIFOPTD.m

% PI controller

function [ tolerance, Kopt, Kp_opt, Ki_opt, Jopt, Termination]= LQ4PIFOPTD (tau_ratio, Kinit)


T = 1; %the plant system time constant

k = 1; %the plant dc gain

tau=T*tau_ratio; %time delay

% Define original dynamic systems

pade_ord = 3; %the order of pade approximation for the time delay term

Tdelay = tf(1,1,'inputdelay', tau);

PTdelay = pade(Tdelay,pade_ord);

[dtnum dtden] = tfdata(PTdelay,'v');

Tplant = zpk(tf(k,[T 1])*PTdelay);

[pnum pden] = tfdata(Tplant, 'v');

[Ap Bp Cp Dp] = tf2ss(pnum, pden); %the plant state equation

Hp = Cp;

% Define PI controller dynamics

Fc = [0]; Gc = [1];

Dc = [0;1]; Jc = [1;0];

% Define the complete system dynamics with the controller dynamics

A = [Ap zeros(pade_ord+1,1);

-Gc*Hp Fc ];

B = [Bp; 0];

G = [zeros(pade_ord+1,1); Gc];

C = [-Jc*Hp Dc];

F = [Jc];

H = [Hp 0];

% LQT OPTIMIZATION

m=1; p=2; n=pade_ord+2;

P = H'*H;

% P = eye(n);

% P = zeros(n,n);

% Q = H'*H;

% Q=eye(n);

Q = zeros(n,n);

rho=0.0;

%rho=0.163*tau^1.7; %for the 10% overshoot ISE

154

R=rho*eye(m);

V=[0];

initK = Kinit;

r0=[1];

[Kopt,Jopt,NumIters,Termination,Grad] = DFPmin(@T4XCostGrad,initK,3000,1e-3); %IST^2E

varioustolerance = Jopt/100000;

tolerance = varioustolerance;

cnt = 0;

while(Termination==1 && cnt<20)

cnt = cnt+1;

initK = Kopt';

[Kopt,Jopt,NumIters,Termination,Grad] = DFPmin(@NavionLatCostGrad,initK,200,varioustolerance);

end;

Kp_opt = Kopt(1);

Ki_opt = Kopt(2);

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