Director
MARCUS BERNER
Masters’ Degree Project
Stockholm, Sweden June 2009
XR-EE-RT 2009:012
Directors designed for observation and fire control in naval environments consist of a mechanical pedestal moved by two electrical motors. To meet the high demands on director precision, a servo solution based on feedback control is used. The digital servo controller has to be tuned to meet demands on performance and stability. This report presents methods for automatic tuning, intended to replace today’s manual tuning procedures. System iden- tification based on relay feedback and recursive least-squares approximations are combined with the Ziegler-Nichols and AMIGO tuning procedures for PI controllers are evaluated.
Evaluations are performed in simulations, for which a SIMULINK model is constructed. Results indicate that the automatic tuning may perform well compared to the manual tuning used today, and that it could bring consid- erable reduction in the time required for tuning.
Keywords– Automatic tuning, Relay feedback, Recursive Least Squares approximation, Ziegler-Nichols, AMIGO, Anti-windup
Acknowledgements
I would like to thank both my supervisor Fredrik S¨odersr¨om at Saab Systems and my examinator Mikael Johansson at KTH for all the support and good feedback during the project.
1 Introduction 7
1.1 Auto-tuning . . . 7
1.2 Problem formulation . . . 8
1.3 System Overview . . . 9
1.3.1 Director structure . . . 9
1.3.2 Servo sensors . . . 10
1.3.3 Servo system architecture . . . 11
1.4 Thesis outline . . . 11
2 Modeling 13 2.1 Controller . . . 13
2.2 Motors . . . 14
2.3 Commutation routine . . . 16
2.4 Motor control electronics . . . 18
2.5 One-phase motor equivalent . . . 20
2.6 Mechanics . . . 21
2.7 Decoupled SISO model . . . 22
2.8 Resonances due to non-stiff mechanics . . . 23
2.9 Disturbances . . . 24
2.9.1 Ship movements . . . 24
2.9.2 Wind . . . 24
2.9.3 Friction torque . . . 26
2.10 Measurement Noise . . . 26
2.10.1 Gyro signal noise . . . 26
2.10.2 Resolver signal noise . . . 32
3 Model Validation 34 3.1 Triangular wave input . . . 34
3.2 Relay feedback test . . . 37
3.3 Step response . . . 40
2
4 System identification 45 4.1 Identification of frequency response with a frequency response
analyzer . . . 45
4.2 Relay feedback identification . . . 49
4.2.1 Additional time delay due to sampling . . . 49
4.2.2 Recursive frequency and amplitude estimation . . . . 50
4.2.3 Oscillation estimation convergence . . . 52
4.2.4 Results from simulation . . . 52
4.2.5 Robustness to resonances . . . 55
4.2.6 Sensitivity to signal noise . . . 56
4.3 Parameter fitting with Recursive Least-squares (RLS) algorithm 59 4.3.1 Model structure . . . 61
4.3.2 Control signal . . . 62
4.3.3 Identification results . . . 63
5 Control Design 66 5.1 The PID-controller . . . 66
5.2 Ziegler-Nichols Tuning based on relay feedback . . . 67
5.3 AMIGO Tuning . . . 67
5.4 Gyro Loop properties . . . 69
5.5 Anti-Windup . . . 71
6 Simulation Results 72 6.1 Reference following . . . 72
6.2 Disturbance suppression . . . 72
7 Conclusions 77 7.1 Proposals for system modifications . . . 77
7.2 Future work . . . 78
A Extract from motor data sheets 80
B Optimal weighting of AWGN signals 81
C Transfer of torque over a non-stiff axle 84
1.1 Electro-optical director EOS-500. . . 9
1.2 Optronic tracking system main components. . . 10
1.3 Pedestal motion angles. . . 10
1.4 Servo system architecture. . . 11
2.1 Control algorithm cycle. . . 13
2.2 Model of digital controller with AD/DA converters. . . 14
2.3 Torque (τ ) generated by the stator field (Bstator) and the rotor field (Bstator) seperated by an angle of (δ). . . 15
2.4 Approximation of motor electronics. . . 16
2.5 Direction of the magnetic field from the phases in the stator (¯eR, ¯eS, ¯eT) and the total magnetic field (¯eB). . . 17
2.6 Commutation routine functionality. . . 18
2.7 Voltages in the PWM circuit. . . 18
2.8 Linear approximation of the PWM circuit. . . 19
2.9 Feedback loop TR-Driv. . . 19
2.10 One-phase approximation of the commutation routine, TR- Driv and motor. . . 20
2.11 Model of rotating mass with applied torque. . . 21
2.12 Model of simplified system. . . 22
2.13 Ship movements. . . 24
2.14 System model with disturbances from ship movements (R), wind (W ), friction torque (F ) and signal noise (NG, NR). . . 25
2.15 Gyro signal channel model. . . 27
2.16 Noise in azimuth gyro signals. . . 28
2.17 Correlation of the azimuth gyro noise. . . 29
2.18 Periodogram of the azimuth gyro noise. . . 30
2.19 Noise in elevation gyro signals. . . 32
2.20 Auto-correlation estimate of resolver signal noise. . . 33
3.1 Gyro signal from low frequency triangular wave input exper- iment. . . 35
4
3.2 Gyro signal from experiment and simulation with the control signal in Figure 3.1 and 21 Nm Coulomb friction torque. . . . 36 3.3 Approximated angular acceleration at different control signal
levels with a rectangular wave input in simulation (upper) and real experiment (lower). . . 38 3.4 Block diagram of a process with relay feedback. . . 38 3.5 Friction torque model used in simulation compared to stan-
dard Coulomb friction torque. . . 40 3.6 Resulting gain and oscillation frequency of azimuth gyro from
relay feedback. . . 41 3.7 Resulting gain and oscillation frequency of elevation gyro
from relay feedback. . . 42 3.8 Azimuth gyro step response for different step amplitudes. . . 43 3.9 Azimuth gyro step response for 7.5V and 10V steps compared
to respective integrator approximation. . . 44 4.1 Block diagram of system during measurements of the fre-
quency response with the frequency response analyzer. . . 45 4.2 Frequency response of the open loop with controller given
by eq. 4.9 estimated by the frequency response analyzer with different disturbance levels compared to the linear model. . . 47 4.3 Frequency response of the system estimated by the frequency
response analyzer with different disturbance levels compared to the linear model. . . 48 4.4 Frequency response estimate generated by frequency response
analyzer measurements on the simulation model. . . 49 4.5 Control signal (u) in oscillating system with output (y) under
relay feedback with a sampled relay. . . 50 4.6 Gyro signal at start of a relay feedback test. . . 52 4.7 Estimation convergence of oscillation amplitude and frequency
in azimuth gyro relay feedback. . . 53 4.8 Oscillation amplitude and frequency for simulation of azimuth
gyro relay feedback with different relay amplitudes. . . 54 4.9 Oscillation amplitude and frequency for simulation of eleva-
tion gyro relay feedback with different relay amplitudes. . . . 54 4.10 Control signal amplitude required to saturate the PWM with
a current feedback gain of 1.6. . . 55 4.11 Oscillation estimation results from simulation of azimuth gyro
relay feedback with a resonance frequency of fr Hz. . . 56 4.12 Oscillation estimation results from simulation of elevation
gyro relay feedback with a resonance frequency of fr Hz. . . . 57 4.13 Oscillation estimation results from simulation of azimuth gyro
relay feedback with different levels of white signal noise. . . . 57
4.14 Oscillation estimation results from simulation of elevation gyro relay feedback with different levels of white signal noise. 58 4.15 Oscillation estimation results from simulation of azimuth gyro
relay feedback with different levels of 50Hz signal noise. . . . 58 4.16 Oscillation estimation results from simulation of elevation
gyro relay feedback with different levels of 50Hz signal noise. 59 4.17 Model parameter convergence from simulations in azimuth
with RLS for model (4.33). . . 63 4.18 Model parameter convergence from simulations in elevation
with RLS for model (4.33) . . . 64 4.19 Model parameter convergence from simulations in elevation
with RLS for model (4.33) and a control signal limit of 1.5V. 64 4.20 Model parameter convergence from triangular wave experi-
ment data in azimuth with RLS for Model 2 (4.33). . . 65 5.1 Bode diagram of nominal open azimuth gyro loop with a time
delay of 3/512s and PI controller parameters from Table 5.4. 70 5.2 Bode diagram of nominal closed azimuth gyro loop with PI
controller parameters from Table 5.4. . . 70 6.1 Step response (1 rad/s) of closed azimuth gyro loop for dif-
ferent controllers. . . 73 6.2 Step response (1 rad/s) of closed elevation gyro loop for dif-
ferent controllers. . . 73 6.3 Step response (0.1 rad/s) of closed azimuth gyro loop for dif-
ferent controllers. . . 74 6.4 Step response (0.1 rad/s) of closed elevation gyro loop for
different controllers. . . 74 6.5 Azimuth gyro loop responce to a 20 Nm torque step disturbance. 75 6.6 Elevation gyro loop responce to a 5 Nm torque step disturbance. 75 6.7 Azimuth gyro loop suppression of a sinusoidal angle distur-
bance with amplitude 0.5 rad and frequency 1 rad/s. . . 75 6.8 Elevation gyro loop suppression of a sinusoidal angle distur-
bance with amplitude 0.5 rad and frequency 1 rad/s. . . 76 C.1 Model of a non-stiff axel. . . 84 C.2 Bond graph of the non-stiff axle. . . 85
Introduction
A director is a combination of sensors used for observation or target tra- jectory tracking. The sensors can for example be radars, cameras or lasers.
Often a combination of sensors is used to get more information of the target position.
Common for the sensors used is that they have a narrow field of view and thus have to be directed toward the target. To be able to direct the sensors they are mounted on a pedestal. The pedestal has two motors allowing it to be directed in the demanded direction. With a narrow field of view and distant targets a small deviation in angle can result in lost targets. This puts high demands on precision requiring a servo system based on feedback.
The directors can have different configurations of sensors and hence dif- ferent dynamic properties. The directors may also differ in components like motors and bearings. Thus every configuration must be tuned individually to achieve acceptable performance and to guarantee stability.
Today the tuning is done manually by measuring the frequency re- sponses. From the results a suitable controller is designed. This method is time consuming and requires experience. A desirable feature would be if the tuning could be performed automatically in the controller software.
1.1 Auto-tuning
Automatic tuning or auto-tuning is basically a built in function for tuning the controller parameters. The tuning can be initiated by the operator but it performes all tests and designs automatically. Thus a minimal experience in servo controller design is required to perform the tuning.
The advantages are fast tuning with minimal experience. The drawback may be that the identification and control design leaves no room for the flexibility of manual tuning.
Auto-tuning is normally performed in three steps. The procedure is similar to how manual black or gray box identification and tuning is done
7
manually.
1. A disturbance is generated. The most common disturbance is a tran- sient, e.g., a step or pulse, on the input.
2. The response of the system is evaluated. A system model can then be fitted to the data. Other information, needed for the control design, can also be extracted.
3. Controller parameters are computed based on the information from the evaluation.
The experiments can be performed in open- or closed loop, depending on the nature of the system and operative conditions.
1.2 Problem formulation
The goal of this thesis is to investigate the possibility of applying auto- tuning on the electro-optical director EOS-500 developed by Saab Systems.
The tuning algorithm should be applicable on the current hardware config- uration. This poses the following constraints on the algorithm:
• Measurable states are restricted to the gyro and resolver signals de- scribed in section 1.3.2.
• The algorithm must be executed at exactly 512 Hz.
• The algorithm must be recursive due to the systems inability to store large batches of data.
The aim is to answer the questions:
• Are the advantages sufficient to motivate automatic tuning over the manual tuning used today?
• Can the auto-tuning algorithm be made robust to variations in the system?
• What modifications could be made to make auto-tuning and control in general perform better?
Testing and evaluation is carried out in simulations. The system is mod- eled and a simulation model is constructed in SIMULINK . The controllerR is represented with an embedded MATLAB block executing at 512 HzR with an additional time delay.
Two different approaches of system identification are tested.
The first approach is based on relay feedback [2]. It is a feedback based method for detecting the amplitude margin and frequency where the phase- shift is −180 degrees. The pre-tuning is minimal but the results are limited in terms of received system knowledge.
The second approach is based on model parameter fitting with the Recur- sive least-squares estimation (RLS) algorithm [3]. This is merely a recursive variation of linear regression often used in adaptive control. The method can potentially give more system information compared to relay feedback but requires a suitable model structure.
The performance of some different linear control design methods will be compared in simulations. The evaluation will comprise reference following, disturbance suppression and robustness. The advantage of anti-windup will also be invesigated.
1.3 System Overview
The director is mounted on the deck of a naval ship and is a part of a combat system consisting of radars, weapons, command stations etc. EOS- 500, seen in Figure 1.1, is designed to be compact and light weight. The main structure of the pedestal is similar to the larger radar directors and has inherited much of their servo system architecture.
Figure 1.1: Electro-optical director EOS-500.
1.3.1 Director structure
The sensors used for target tracking are mounted on a pedestal. The pedestal can turn both in azimuth and elevation. The two axes of the pedestal are mounted with ball bearings and directly connected to electrical motors.
By turning the pedestal the optronic sensors can be directed freely in azimuth but the elevation has mechanical limitations of +85◦ to −35◦.
Rate gyro
Azimuth resolver Azimuth motor
Elevation motor
Elevation resolver Optronic sensors
Figure 1.2: Optronic tracking system main components.
There are two resolvers, one for each axis, measuring the angular posi- tions. There is also a gyro measuring the angular velocity, mounted together with the optronic sensors.
1.3.2 Servo sensors
The direction of the system is managed in a spherical coordinate system.
That way each axis of the director only moves in one of the coordinates, like illustrated in Figure 1.3.
el
az
Figure 1.3: Pedestal motion angles.
By using these coordinates in feedback the motion in azimuth and eleva- tion can be seen as two separate SISO (Single input, Single output) systems.
There is still some coupling between the two axes due to, e.g., the gyroscopic effects of rotating bodies. These effects can however be neglected due to the slow velocities used in operation.
The resolvers measure the angles relative to the ship deck where the
director is mounted. When the director is mounted on a steady platform the angle measured by the resolver is the integral of the velocity measured by the gyro. When in operation this is normally not the case because the ship movements are picked up by the gyro but not by the resolvers.
1.3.3 Servo system architecture
The purpose of the servo system is to control the director position or velocity with the use of feedback from the gyro and resolvers. Desired values for the angles or angular velocities are sent to the controller from the Combat Management System (CMS). The local controller will then use feedback from the gyro and resolvers to follow the demanded values. The controller is run on a standard PC with Windows XP. From the controller, current references are sent through a D/A-converter to an analog motor control card named TR-Driv. The control card supplies the motor with voltage to achieve the demanded current.
Angles (Resolver) Angular rates (Rate gyro) Current
refernce
Director Controller PC TR-Driv
Voltage State reference from
Combat Management System
Figure 1.4: Servo system architecture.
1.4 Thesis outline
The thesis is divided into the chapters described below.
In Chapter 2 the system is broken down into its components for mathe- matical modeling. The result will be used to construct the simulation model and the design of the identification algorithm.
In Chapter 3 simulation results are compared to measurements from the real system for validation.
Chapter 4 describes the identification methods used and why some meth- ods are not suitable for the system.
Chapter 5 reviews the different control design methods used based on the identification results.
In Chapter 6 the performance of the controllers are evaluated by simu- lation results.
In Chapter 7 the conclusions from the results are presented together with proposals for future work.
Modeling
To produce the simulation model the system will be broken down into its components for mathematical modeling. Some simplifications will be made.
Some of them are due to lack of process knowledge and some are made when it can be argued that the effects on the model behavior are minimal compared to the simplicity gained.
2.1 Controller
The algorithm in the controller PC is executed at a rate of 512 Hz. Every cycle is executed in the four steps illustrated in Figure 2.1.
3. Apply the comuted control
4. Wait 1. Read one sample
from the gyro & resolvers 2. Compute the new control signal
512 Hz
Figure 2.1: Control algorithm cycle.
13
1. The values from the gyro and resolvers are sampled and quantized.
2. The values from step one and old sample values are used to compute a new control signal.
3. When the new control signal is computed it is applied through a D/A- converter to the motor control card. The output is constant until the next value is computed in the following cycle.
4. In the remaining time of the cycle no computations are made. The system waits for the next sample.
The digital controller can be modeled as in Figure 2.2. The sampling and quantizers represents the A/D-converters and the (Pulse Amplitude Modu- lator) PAM the D/A-converter. The delay is due to the time of computation.
The time delay can not exceed one sampling period since that would imply that the computations take more than one sampling period, which is not possible. The delay can be assumed to be constant if the same algorithm is executed in every sample.
Digital
filtering e
-sT Gyro
Resolver PAM
Sampling
Sampling
Quantizer
Quantizer
Control effort
Digital servo controller
Reference from CMS (Digital)
Figure 2.2: Model of digital controller with AD/DA converters.
The reference from the CMS typically has a lower sampling rate (∽ 50 Hz) than the local controller.
2.2 Motors
The motors used to control the pedestal are three-phase brushless DC (BLDC) motors. A three-phase BLDC motor consists of a stator and a rotor. The stator is the static part that is fixed to the mounting and the rotor is the rotating part.
The stator consists of a number of windings. When current runs through the windings an electromagnetic field is induced. Each of the three electrical phases has a set of windings distributed around the stator in such a way that
the magnetic fields generated by each phase are separated by 2π/3 radians.
By applying different currents in the different windings the strength and direction of the resulting magnetic field can be controlled, due to its super positioning property.
The rotor consists of permanent magnets. Thus no current runs from the stator to the rotor. When a magnetic field is induced in the stator, torque, trying to align the magnetic fields in the stator and rotor, will arise. The torque (τ ) depends on the strength of the stator and rotor fields and the difference in angle (δ) as described in (2.1) and illustrated in Figure 2.3.
τ ∝ |Bstator| · |Brotor| · sin (δ) (2.1) Since the rotor has permanent magnets, the magnetic field has constant magnitude and the direction depends only on the alignment of the motor.
The controllable parameters are therefore reduced to Bstator and δ. Thus (2.1) can be reduced to (2.2).
τ ∝ |Bstator| · sin (δ) (2.2) Brotor
Bstator
!
Figure 2.3: Torque (τ ) generated by the stator field (Bstator) and the rotor field (Bstator) seperated by an angle of (δ).
To generate the current, voltage is applied to the three phases (R, S and T ). The electronics can be approximated with the symmetrical circuit in Figure 2.4.
The resistors represent the resistance in the wires and the inductor comes from the magnetic fields generated in the windings. The circuit has no ground or zero. Thus the sum of the currents must be zero according to Kirchoff’s first law.
iR+ iS+ iT = 0 (2.3)
UR
US UT
iR
iS iT
R L L
L
R R
Figure 2.4: Approximation of motor electronics.
When the motor is turning, voltage is induced in the stator windings.
This voltage is called Back EMF (Electromotive force) and works against the motor velocity, acting as a brake. The EMF is proportional to the angular velocity.
2.3 Commutation routine
The magnetic field generated in each phase in the stator is proportional to the current running through the winding. The voltages applied to the three phases are used to control the strength and direction off the total stator field.
Because the phases are evenly distributed the direction of the field from each phase will be separated by 2π/3 radians as illustrated in Figure 2.5.
The total field will be the sum of the three fields.
Btote¯B= BR¯eR+ BSe¯S+ BTe¯T (2.4) With a perfectly symmetric motor the relation between the current and the magnetic field strength will be the same for all phases. This implies that to fulfill (2.3), the sum of the field strengths must be zero.
BR+ BS+ BT = 0 (2.5)
The preferred way to choose the fields would be:
BR= B · sin (ϑ) BS= B · sin (ϑ +23π)
BT = B · sin (ϑ +43π) = −BR− BS
(2.6)
eS
eT
eR 3
2
! eB
3 2
Figure 2.5: Direction of the magnetic field from the phases in the stator (¯eR, ¯eS, ¯eT) and the total magnetic field (¯eB).
If the contribution to the magnetic field from each phase could be chosen like in (2.6) the resulting field would be:
| ¯Bstator| = 32B
φ = π2 − ϑ (2.7)
There are now two parameters (B and ϑ) used to adjust the fields in the stator phases. The only reason for applying current in the motor is to generate torque. Thus it is desirable to keep the torque to current ratio as high as possible. According to (2.2) that is achieved when:
sin (δ) = 1 → δ = π
2 (2.8)
If the rotor field angle (ϕ) is known, the value of ϑ that gives the highest torque to current ration can be derived by (2.9).
ϑ = −ϕ (2.9)
Now the only parameter is B, which will be proportional to the torque.
Because the magnetic field is proportional to the current, torque can be controlled by applying the phase currents as in (2.10).
iR= I · sin (−ϕ) iS = I · sin (23π − ϕ) iT = −iR− iS
⇒ τ ∝ I (2.10)
The currents are not directly adjustable. They are controlled by the motor control card TR-Driv described in section 2.4. TR-Driv takes scaled current references for two of the phases as inputs (iRref, iSref).
The output from the controller algorithm (u) is a scaled torque reference for each motor. By using the measurement of the motor angel from the resolver (ϕ), iRref and iSref can be computed according to (2.10). This is done by the commutation routine that is integrated in the controller software illustrated in Figure 2.6.
Controller algorithm u Commutation iRref Sref
i
Figure 2.6: Commutation routine functionality.
2.4 Motor control electronics
The motor control uses PWM (Pulse Width Modulation). This is a very commonly used technique in power electronics. The principle is to only use two levels of voltage instead of various levels and switch between the levels very fast.
A triangular wave with a frequency of 19.2 kHz and magnitude of 10V is generated in the hardware. The triangular wave is compared to a reference voltage (uref). If the reference voltage gets below the voltage of the triangu- lar wave, the voltage of the power supply (Umax) is applied. Otherwise the used voltage is the same as the ground of the power supply. The voltages in the PWM circuit are illustrated in Figure 2.7.
t t Umax
+10V
-10V
Output
uref
0V V
Figure 2.7: Voltages in the PWM circuit.
The electrical circuit in the motor has low-pass characteristics. The switching frequency of the PWM can be considered to be so high that only the average of the output voltage has any effect on the current. By neglecting
the high frequency content of the PWM output, the relation between uref
and U can be approximated by (2.11).
Umean=
Umax
20 · uref+Umax2 , |uref| ≤ 10
0 , uref < −10
Umax , uref > 10
(2.11)
The motor has no connection to ground. Because the voltages between the phases are relative to each other an equal potential offset to all the phases has no effect. Thus the offset in (2.11) can be removed without affecting the behavior of the model. The relation between uref and U can now be approximated by a gain and saturation as in (2.12) and Figure 2.8.
U =
Umax
20 · uref , |uref| ≤ 10
Umax
2 · sgn(uref) , |uref| > 10
(2.12)
20 Umax
10
ref PWM
u U uref U
Figure 2.8: Linear approximation of the PWM circuit.
The outputs from the controller software are desired values of the cur- rents. To be able to control the current and thereby the torque in the motor the hardware has built in current feedback. This will increase the bandwidth compared to direct voltage control.
The input is a scaled current reference (iref) and the actual current is measured, forming a closed loops like in Figure 2.9.
Kj
Ka !10
!5
20 Umax
iref uref
i
U
Figure 2.9: Feedback loop TR-Driv.
Approximate values of Ka and Kj are:
Ka≈ 60
Kj ≈ 1.6 (2.13)
Each of the three phases has a PWM circuit. For phase one and two the reference comes from the commutation routine. The commutated phase references are converted with a DA-converter and used as input to TR-Driv.
The third reference is generated in the control card so that the sum of the currents are zero.
2.5 One-phase motor equivalent
If the commutation routine works perfectly as described in section 2.3, the simplified model of TR-Driv in section 2.4 is considered accurate and the motor is symmetrical, a one-phase approximation like in Figure 2.10 can be made. The simplifications may not capture all dynamics. It is however sufficient considering the limited knowledge and detail in modeling of TR- Driv. With a more detailed modeling of TR-Driv a three-phase model may be needed to capture eventual non expected behavior.
Commutation
Routine TR-Driv Motor
u ref
iR
Sref
i
US
UR
UT
sL R !
1
i
K
j"
K
a #10 U20maxK
t5
#
u
UE
"
Figure 2.10: One-phase approximation of the commutation routine, TR-Driv and motor.
!
r
r
V J
Figure 2.11: Model of rotating mass with applied torque.
2.6 Mechanics
When torque is generated by the motor the axis starts to accelerate. The energy consumed in the motor is transformed into kinetic energy in the mass of the moving parts. The amount of kinetic energy depends on the angular velocity and moment of inertia (J). The relationship between torque (τ ) and angular acceleration ( ¨ϕ) is given by (2.14).
¨ ϕ = 1
Jτ (2.14)
If the center of mass of the elevating part is not in the center of rotation it will act as a disturbance. Torque will be generated by gravity and control has to be applied to keep it steady. Counterweights are therefore used to move the center of gravity to the axis of rotation. The balancing will be assumed to be perfect and the disturbance due to unbalanced mass is zero.
The counterweights will however increase the total mass and thereby the moment of inertia.
The moment of inertia of a point mass (m) with the distance r to the center of rotation is computed as in (2.15).
Jm= mr2 (2.15)
The moment of inertia for an entire rotating body can thus be computed like in (2.16), where ¯r is the point in space, r is the perpendicular distance to the rotation axis and ρ(¯r) is the density at ¯r.
J = Z Z Z
V
r2ρ(¯r)dV (2.16)
To compute the exact value of J for the director would require very detailed information about every component. It is not necessary for the purpose at hand. The values used in the simulation model will instead be approximations based on the total weight and size. The following values are used in the model:
Jaz≈ 3 kg m2
Jel≈ 2 kg m2 (2.17)
Another assumption is that the pedestal is stiff. This is however not valid for the larger radar directors. Because they are much heavier, the torque will make the axis between the motor and moving mass distort if the acceleration is sufficient. This can cause problems at higher frequencies where the system can get oscillatory under feedback. No such behavior has been detected on EOS-500 at relevant frequencies.
2.7 Decoupled SISO model
By combining the component models, the pedestal motion in azimuth and elevation can be modeled as decoupled SISO-systems as in Figure 2.12.
sL R
1 Resolver
! ! !
"
i
Kb
# Kj
#
Ka Kt
J 1 20
Umax
10
$ 5
$ u
Motor control Electronics
Mechanics
Back EMF
Gyro
10
$
Figure 2.12: Model of simplified system.
If none of the signals are saturated the model is linear and can be ex- pressed as a transfer function from the control signal to the angular velocity by (2.18).
Go(s) =
KaUmax
20
Kb+ s
J R
Kt +KaK20KjJ Umax
t
+ s2 JLK
t
(2.18)
With the parameters from the data sheets in appendix A, the approxima- tion made in (2.17) and the used power supply voltage of 80V , the functions in (2.19) and (2.20) can be derived.
Gaz(s) ≈ 240
0.0064s2+ 175s + 7.6 (2.19) Gel(s) ≈ 240
0.01s2+ 59s + 13 (2.20)
Both systems are second order and very stiff. The poles of respective systems are placed as follows.
Poles of azimuth transfer function:
p1 ≈ −0.043
p2 ≈ −2.7 · 104 (2.21)
Poles of elevation transfer function:
p1 ≈ −0.22
p2 ≈ −6.0 · 103 (2.22)
The systems can therefore be approximated by (2.23) and (2.24) while in the linear region.
Gaz(s) ≈ 1.37
s + 0.043 (2.23)
Gel(s) ≈ 4.06
s + 0.22 (2.24)
2.8 Resonances due to non-stiff mechanics
There may be many sources of model errors due to the lack of system knowl- edge and simplifications made. The larger radar directors have a resonance at 30 − 60 Hz. This comes from lack of stiffness in the pedestal and mount- ings. One way of modeling the non-stiff pedestal is derived in Appendix C.
The additional dynamics caused by the pedestal not being completely stiff may potentially include complex conjugated poles. These poles will yield mechanical resonances and possibly an oscillating closed loop system if performance is pushed without applying specially designed compensation filters. Because EOS-500 is both lighter and more compact than the radar directors the problem with resonances will not be as severe, but they still needs to be considered.
The influence of these resonances on identification and control will be investigated both for robustness reasons and as a step in transferring the auto-tuning to the radar directors.
2.9 Disturbances
A large part of the control effort will be used to compensate for disturbances acting on the system. Some of them will be modelled to enable investigations of different controllers suppression properties through simulations.
2.9.1 Ship movements
When the ship is out at sea it will not be steady. The different ship motions are illustrated in Figure 2.13.
Yaw Pitch Roll
Heave
Figure 2.13: Ship movements.
Because the director is mounted on the ship deck it will follow the ship’s movements. The resolvers will however not pick up these movements due to the fact that the sensors are mounted on the ship and measures the angles in the ship coordinate system. The motion will though be picked up by the rate gyro. If the information gained from the servo sensors on the director is not complemented with information about the ship position, only the gyro signal can be used in feedback to compensate.
The movements, denoted R in Figure 2.14, have low frequency. A normal test signal is the low frequency sinusoidal in (2.25).
R = 0.5 sinπ 3t
rad (2.25)
2.9.2 Wind
When winds hits the director, torque will be generated, just like the force from the sails on a sailing boat. Wind is a bigger problem with the larger radar directors. The radar dish has a size and shape that can capture a lot of wind compared to the smoother shaped of EOS-500. The disturbance from wind must however be considered. The wind is modeled as a torque disturbance, denoted W in Figure 2.14. Wind can have very different char- acteristics e.g., constant, steps or pulse, see [12].
sLR
1 Resolver !
"#i bK$
jK$ aKtK J1 20maxU 10%
5% u
Gy ro
dt
d R$
WRNGN 10% )(&$fF
""
Figure 2.14: System model with disturbances from ship movements (R), wind (W ), friction torque (F ) and signal noise (NG, NR).
2.9.3 Friction torque
The director must be able to withstand heavy mechanical strain. The ball bearings must therefore be tight to prevent vibrations from causing too much wear. The tight bearings unfortunately have a lot of friction torque. This can be a serious problem when using the electrical motors. With the high friction torque and limited torque, lot of the control effort will be spent on overcoming the friction torque. The friction torque can, like the wind, be seen as a torque disturbance.
One distinction between friction torque and the other disturbances dis- cussed is the strong dependence of the internal states of the system. There are many different friction torque models. A good summary of the most common models can be found in [5].
The starting point of the friction torque model used in simulations will be Coulomb friction torque, see (2.26). The friction torque model will be based on experiment data. Some modifications will therefore be made to the Coulomb friction torque model in chapter 3 to better fit the real system behaviour.
F =
fc , ˙ϕ < 0 0 , ˙ϕ = 0
−fc , ˙ϕ > 0
(2.26) Due to how the bearings are constructed the friction torque is not even around the orbit. In some positions it is higher than other. The friction torque is also dependent on temperature. When the temperature changes the material in the bearing components will expand or contract. This will have effects on friction torque because the bearing gets tighter or looser.
Temperature can also change the properties of the lubrication, thus affecting the friction torque.
2.10 Measurement Noise
The measured signals from the resolvers and gyro are quantized and noisy.
This may have effects both on identification and control performance. Thus it will be modeled based on measurements to enable simulation of these influences.
2.10.1 Gyro signal noise
There are two gyro signals available for each axis of the pedestal. The two signals comes from the same gyro but are scaled differently before the DA converter. This is done to get better resolution at low speeds. The two signals can also be combined to reduce noise. An approximation of the two gyro signal channels is depicted in Figure 2.15
Gyro
N1
N2
6 . 2
!
6 . 2
!
Software
4 41 a
b
b a "
1 ˆ
2
~
1
~
Figure 2.15: Gyro signal channel model.
The saturations come from the DA converters. The fact that the DA converters also quantize the signals will here be neglected. This is because the noise levels are much higher than the quantizer steps. The gyro signals may also contain an offset due to gyro drift. There are algorithms to com- pensate for this drift. These algorithms will here be considered perfect. To model the noise, measurement data will be used. The measurements are made indoors with the director mounted on a steady platform.
By keeping the control signal at zero the system is at rest and the mea- sured signal is noise only. The offset in the signal is compensated by sub- tracting the mean value from each sample.
Azimuth gyro noise
The 2000 sample azimuth noise signals in Figure 2.16 will be used to model the noise. The straight forward approach would be to approximate the variance by the mean square as in (2.27) and (2.28), then combine the two signals as described in Appendix B. The combined noise N3, is then given by (2.29).
s21= 1 2000
2000
X
l=1
N12(l) ≈ 2.13 · 10−6 (2.27)
s22= 1 2000
2000
X
l=1
N22(l) ≈ 1.82 · 10−6 (2.28) By combining the two signals the estimated variance (eq. 2.30) is re- duced, but not as much as expected (eq. 2.31). This indicates some correla- tion between N1 and N2.
N3 = s22N1+ s21N2
s21+ s22 (2.29)
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−5 0 5
x 10−3 N
1
sample
rad/s
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−5 0 5
x 10−3 N2
sample
rad/s
Figure 2.16: Noise in azimuth gyro signals.
1 2000
2000
X
l=1
N32(l) ≈ 1.17 · 10−6 (2.30) s21s22
(s21+ s22) ≈ 0.98 · 10−6 (2.31) The autocorrelation of each signal and the cross-correlation between them are depicted in Figure 2.17. The autocorrelation indicates that both N1 and N2 consist of an uncorrelated part and a part with high correlation, like a single frequency sinusoidal. The cross-correlation indicates that the white part of N1 and N2 are uncorrelated with each other while the corre- lated parts seems almost identical. This indicates that the correlated part originate from the same source.
The spectra of the noise signals in Figure 2.18 reveals two sinusoidal waves at approximately 1.5 Hz and 49 Hz. The noise signal can be approx- imated by (2.32). The 49 Hz disturbance most likely comes from the power grid. This part of the disturbances may vary with different sources of powers and how the wiring is done.
−2000 −1000 0 1000 2000
−5 0 5 10 15 20
x 10−7
lag (k) rN1(k)
−2000 −1000 0 1000 2000
−5 0 5 10 15 20
x 10−7
lag (k) rN2(k)
−2000 −1000 0 1000 2000
−5 0 5 10 15 20
x 10−7
lag (k) rN1 N2(k)
Figure 2.17: Correlation of the azimuth gyro noise.
N1 ≈ A11sin (3πt) + A12sin (98πt) + W1 W1∼ N(0, σ12) N2 ≈ A21sin (3πt + φ1) + A22sin (98πt + φ2) + W2 W2∼ N(0, σ22) N3 ≈ A31sin (3πt + Ψ1) + A32sin (98πt + Ψ2) + W3 W3∼ N(0, σ32)
(2.32) The relation between the periodogram amplitude (P ) and the sinusoidal amplitude (A) is given by: A = 2√
P . The values of Anm received from the periodograms in Figure 2.18 can be found in Table 2.1.
Parameter Approximated value (rad/s)
A11 5.38 · 10−4
A12 7.32 · 10−4
A21 4.20 · 10−4
A22 6.57 · 10−4
A31 4.74 · 10−4
A32 6.90 · 10−4
Table 2.1: Approximated amplitudes of sinusoidals in the azimuth gyro signal noise
If N1 and N2 are combined like in (2.29) the theoretical sinusoidal am- plitudes in N3 are given by (2.33).
A3m= 1 s22+ s21
q
s42A21m+ s41A22m+ 2s22s21A1mA2mcos (φm) (2.33) An estimate of the phase-shift between the channels is given by (2.34).
φm= cos−1 (s22+ s21)2A23m− s42A21m− s41A22m 2s22s21A1mA2m
(2.34)
0 50 100 150 200 250 300 0
0.5 1 1.5
x 10−7
frequency (Hz) PN1(ej2π f)
0 50 100 150 200 250 300
0 0.5 1 1.5
x 10−7
frequency (Hz) PN2(ej2π f)
0 50 100 150 200 250 300
0 0.5 1 1.5
x 10−7
frequency (Hz) PN3(ej2π f)
Figure 2.18: Periodogram of the azimuth gyro noise.
Using the values in Table 2.1, (2.27) and (2.28), gives the phase-shift estimations as shown in (2.35).
φˆ1 ≈ 0.076 rad
φˆ2 ≈ 0.13 rad (2.35)
Such small phase differences can be neglected and the weighted averaging between the sinusoidal parts can be approximated by weighted averaging of the amplitudes. Thus the optimal weighting of the sinusoidal signals is to only use the one with least amplitude.
Approximations of the white noise variance in the noise signal are given in (2.36).
˜
σ21 = s21− 12A211−12A212≈ 1.71 · 10−6
˜
σ22 = s22− 12A221−12A222≈ 1.51 · 10−6
˜
σ23 = s23− 12A231−12A232≈ 8.24 · 10−7
(2.36) The reduction of white noise is very close to what could be expected, see (2.37).
s42σ˜21+ s41σ˜22
(s21+ s22)2 = 8.03 · 10−7 ≈ ˜σ32 (2.37) The complete model of the azimuth gyro noise can be found in (2.38).
NGaz(n) ≈ 5 · 10−4sin
3π fsn
+ 7 · 10−4sin
98π fs n
+ Waz(n)
E[Waz2(n)] ≈ 8 · 10−7
E[Waz(n)Waz(n + k)] = 0, k 6= 0
(2.38)
Elevation gyro noise
The elevation gyro noise in Figure 2.19 can be modeled in a similar way as for the azimuth signal. The same sequence of measured data yields the noise model in (2.39).
NGel(n) ≈ 0.0013 sin
98π fs n
+ Wel(n)
E[Wel2(n)] ≈ 8 · 10−7
E[Wel(n)Wel(n + k)] = 0, k 6= 0
(2.39)
The white parts of the azimuth and elevation noise have similar power.
The source of the low frequency disturbance detected in azimuth is unknown.
The lack of that part in elevation may depend on that the measurements
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−5 0 5
x 10−3 N1
sample
rad/s
0 200 400 600 800 1000 1200 1400 1600 1800 2000
−5 0 5
x 10−3 N
2
sample
rad/s
Figure 2.19: Noise in elevation gyro signals.
were not performed simultaneously and the disturbance was generated from an external source only present during the azimuth measurement.
There is some difference in power of the 49 Hz disturbance between azimuth and elevation. If the disturbance is generated by the power grid this difference may depend on the power consumption of the motors in the different measurements.
These explanations are only speculations but will not be investigated further here.
2.10.2 Resolver signal noise
Compared to the quantization resolution the noise level of the gyro signal is much higher than the resolver signal. The effects of the quantizer can thus not be neglected. The overall noise level is however relatively low. The variance estimated from a 2000 sample measurement with the system steady can be seen in (2.40) below.
˜
σr2= 1 2000
2000
X
n=1
NR2(n) ≈ 2.9 · 10−9 (2.40) The noise has low correlation between the samples. This is illustrated by the auto-correlation estimate from the same measurement in Figure 2.20.
−1000 −800 −600 −400 −200 0 200 400 600 800 1000
−1.5
−1
−0.5 0 0.5 1 1.5 2 2.5
3x 10−9
k (samples) r N R(k)
Figure 2.20: Auto-correlation estimate of resolver signal noise.
Model Validation
To ensure that the main dynamics and limiting factors of the system are captured by the simulation model, some experiment data will be compared to simulation results. The data from real measurements are limited, espe- cially in elevation where open loop experiments are not suitable, due to the mechanical limitations.
3.1 Triangular wave input
A low frequency triangular wave is used as input in azimuth. Because the triangular wave has most of its energy at low frequencies the current can be assumed to follow the reference when the PWM is not saturated. The generated torque is then given by (3.1).
τ ≈ Kt
Kj
u (3.1)
From the experiment results depicted in Figure 3.1 an approximate value of the Coulomb friction torque can be determined.
With a gradually increasing torque and the system at rest, the Coulomb friction torque is obtained as the level of torque which makes the system move. From the measurements that is approximatlly when the control sig- nal reaches 5V. The Coulomb friction torque approximation can then be obtained by (3.2).
fc ≈ Kt
Kju ≈ 6.6
1.6· 5 ≈ 21 Nm (3.2)
Simulation results with the Coulomb friction torque level in (3.2) and the same control signal as in Figure 3.1 are depicted in Figure 3.2. The level of control signal needed to get the system moving seems to be the same.
That is expected since the level of friction torque was derived from the same
34
0 2 4 6 8 10 12 14 16 18 20 -3
-2 -1 0 1 2
time (s)
angular velocity (rad/s)
0 2 4 6 8 10 12 14 16 18 20
-10 -5 0 5 10
time (s) control signal-Fc
Fc
Figure 3.1: Gyro signal from low frequency triangular wave input experi- ment.
experiment. The gain of the simulation model seems to be higher than the actual system. There can be many reasons for that.
The moment of inertia was approximated based on the total size and weight of the system, thus it is not very exact. A higher moment of inertia would not affect the level of torque required to get the system moving, but it would affect the gain. A model change is however not motivated due to the uncertainties in the friction torque model and lack of measurement data.
Even if the error lies in the friction torque model the data available is not enough to motivate a more complex friction torque model. The simu- lation results can thus be considered acceptable with respect to the above measurement.
0 2 4 6 8 10 12 14 16 18 20
−3
−2
−1 0 1 2 3
time (s)
angular velocity (rad/s)
Simulation Experiment
Figure 3.2: Gyro signal from experiment and simulation with the control signal in Figure 3.1 and 21 Nm Coulomb friction torque.
If none of the internal states are saturated the system is expected to act like a first order system according to section 2.7. The angular acceleration is then given by (3.3).
˙ω ≈ 1.37u − 0.043ω (3.3)
At reachable velocities the approximation given by (3.4) can be made.
1.37u >> 0.043ω ⇒ ˙ω ≈ 1.37u (3.4) With the 5V of the control signal used to overcome the friction torque, three different modes (3.5a, 3.5b, 3.5c) can be expected.
ω = 0
|u| < 5
⇒ ˙ω = 0 (3.5a)
ω > 0 ⇒ ˙ω = 1.37(u − 5) (3.5b) ω < 0 ⇒ ˙ω = 1.37(u + 5) (3.5c) How the angular acceleration depends on the control signal is illustrated in Figure 3.3, both for simulation and experiment data. The angular accel- eration can be computed by differentiating the gyro signal. Unfortunately this amplifies the noise a lot. The noise can be removed from the simulation model but not from the real measurements. To reduce the noise, the gyro signal is therefore low-pass filtered before differentiating.
Not unexpected, the approximation is almost a perfect match for the simulation results. The exceptions are the loops at the endpoint where the control signal is close to 10V or −10V. This depends on that the PWM circuit gets saturated when the back EMF and the control signal gets too high. In the real experiment data, the same thing can be observed but only at control signals close to −10V and not to the same extent as in the simulations.
By looking at the angular velocity from the experiments in Figure 3.1 it can be seen that it is higher in the negative direction. The difference most likely depends on non symmetrical friction torque. At low frequancies the suspected saturation in the PWM circuit occurs when the condition in (3.6) is fulfilled.
Umax
2 · sgn(u) − ωKb< R
Kj · u (3.6)
From Figure 3.3 it seems like the PWM is saturated when the control signal drops below −9V. With the constants used in the simulation model the required angular velocity to saturate the PWM is given by (3.7).
ω <
R
Kj · u − Umax2 · sgn(u)
Kb ≈ 1.38 rad/s (3.7)
The critical level (3.7) is reached in the negative but not in the positive direction during the measurement. When the power supply voltage is not sufficient, the increased control signal will not result in a higher acceleration.
This results in the loops seen at the end points in Figure 3.3..
3.2 Relay feedback test
Relay feedback experiments were performed both in azimuth and elevation.
The relay feedback set up is illustrated in Figure 3.4.
−10 −5 0 5 10
−5 0 5
Simulation Results
control signal (u) angular acceleration (rad/s2 )
−10 −5 0 5 10
−5 0 5
Experiment Data
control signal (u) angular acceleration (rad/s2 )
Figure 3.3: Approximated angular acceleration at different control signal levels with a rectangular wave input in simulation (upper) and real experi- ment (lower).
d d
!
1
e u y
r
Figure 3.4: Block diagram of a process with relay feedback.
The experiment scheme can be described by (3.8).
u = d · sgn(e) e = r − y r = 0
⇒ u = −d · sgn(y) (3.8)
If the system has a phase shift of −π radians at any frequency (ωπ) it will get into sustained oscillations at that frequency. The input will then be a rectangular wave with the same frequency and the same amplitude as the relay (d).
By expressing a rectangular wave with amplitude d as a Fourier series it can be seen that it is a sinusoidal with the same frequency and amplitude 4d/π and higher frequency transients with lower amplitude. Because the system has low-pass properties the transients will be damped much more and the output will be close to a single frequency sinus.
Since the control signal has opposite sign of the output the phase shift will be −π radians. With oscillation amplitude A the gain of the system at ωπ can be approximated by (3.9).
Kπ ≈ Aπ
4d (3.9)
The output in the relay experiments are the gyro signals. In the linear region the system is expected to behave like a first order systems with a delay, see (3.10).
G(s) =ˆ α
s + βe−sL (3.10)
The system pole is very slow. Thus the expected oscillation frequency can be given by (3.11).
arctan ωπ
β
+ ωπL = π ⇒ {ωπ >> β} ⇒ ωπ ≈ π
2L (3.11)
The delay depends on the computation time in the controller. There may also be some additional delay caused by the DA/AD converters. To get similar results in simulation as in experiments, the delay is set to three period times of the controller. It is important to notice that the delay may also be caused by some un-modeled dynamics. To be sure of this, further investigation would be required.
L ≈ˆ 3
512 ≈ 0.006 s (3.12)
At the low velocity oscillations generated by the relay feedback the fric- tion torque seems to have almost no effect. This behavior may be better described with a displacement friction torque model [5]. The measurement
