2. Table of Contents
1. Introduction 3
1.1 The introduction of the hardware of CE150 3
1.2 The introduction of the software of CE150 3
2. Inputs and outputs 4
3. Setting equation 5
3.1 Dynamic elevation 6
3.2 Dynamic azimuth 6
4. Dynamics of the complete system 7
4.1.non-linear model 7
4.2. Linear Model 8
5. Hardware of the helicopter and software of control 8
5.1 MF 624--multifunction I/O card 9
5.2 AD 622—data acquisition card 9
5.3 Modelling 9
6. The software of CE150 10
6.1 Demonstration Program Description 10
6.2 Demonstration Program Algorithms 11
6.3 A/D Converter 12
6.4 AD 622 13
7. Controller Design 14
7.1 Pole placement design of state feedback controller 14
8. Conclusion 14
3. 1. Introduction
1.1 The introduction of the hardware of CE150
The system studied is a helicopter model (CE
150) produced by the company Humusoft. This
model is connected to a computer through a
PCI card, allowing achieving real-time
simulations in Matlab / Simulink environment.
The helicopter has two rotors, a main rotor and
a tail rotor, trained by two DC motors. In
addition, a servo can manipulate its center of
gravity. The model has two degrees of freedom:
in the vertical elevation plane α and β in
azimuth in the horizontal plane. These two
angles are measured by sensors.
1.2 The introduction of the software of CE150
There are five types of software environment available to the user:
Demo program - directly executable program written in C language. Friendly user
interface facilitates first experiments with PID control.
Matlab demo program - more complex demonstration program with graphical user
interface running under Matlab. This program allows to export recorded data directly
into the Matlab environment which facilitates tasks like simulation, data analysis or
model tuning.
Matlab is a high performance software package for scientific and numeric
computation, signal processing and graphics in an environment where problems and
solutions are expressed just as they are written mathematically - without traditional
programming. The power of Matlab environment is further extended by Simulink - a
block oriented environment for simulation of dynamic systems and numerous
toolboxes. Some of them are highly recommended for the experimentation with the
CE150 model: Control System Toolbox, System Identification, Toolbox, Optimization
4. Toolbox, Real-Time Windows Target and Virtual Reality Toolbox. An absolute
minimum to conduct experiments from this manual is the Control Systems Toolbox
and Simulink.
The Real Time Toolbox for Matlab - supplied with the CE150 Helicopter Model
enables the user to communicate directly with the system from the Matlab
environment.
The Real-Time Windows Target - this environment offers best control performance.
Real-Time Windows Target is not included, but offered separately by The MathWorks.
Simulink controller examples are included.
User developed environment - this option might be useful for solving special tasks not
supported by Matlab and Real Time Toolbox, e.g. from the area of nonlinear control.
2. Inputs and outputs
To allow the user to control the system in real time via the Matlab interface, all inputs
/ outputs are in the range -1.1 * +. These signals do not have a physical dimension.
Their unity is denoted MU (Machine Unit).
Rotations of the model 50 ° in elevation
160 ° in azimuth
main Engine DC motor with permanent magnets
Maximum voltage: 12 V, 0-6 A
Maximum speed: 9000 RPM
secondary engine DC motor with permanent magnets
Maximum voltage: 6 V, 0-4 A
Maximum Speed: 12,000 RPM
The system studied is then a MIMO (Multi Input Multi Outputs) with three inputs and
two outputs:
5. Definition inputs
Symbol Name Unit
u1 Voltage applied to the main
engine
MU
U2
voltage applied to the
secondary motor
MU
U3 position servo (weight) MU
Definition outputs
Symbol Name Unit
α Elevation angle MU
β
Azimuth angle MU
To simplify the model, it is assumed that the weight does not move during the
simulations model. So we have u3 = 0.
3. Setting equation
The mathematical model described below is a simplified but adequate model in our
study.
6. 3.1 Dynamic elevation
By a balance of forces acting on the helicopter in the vertical plane is obtained:
With: α elevation angle * rad +
I1 the moment of inertia of the helicopter about the horizontal axis kg.m ² + *
τ1 the main rotor torque [N · m]
τc centrifugal torque [N · m]
τG the gyroscopic torque [Nm] τw the gravitational torque [Nm] τf1 friction
torque [Nm]
Resolving the forces in the vertical plane acting on the helicopter body
3.2 Dynamic azimuth
By a balance of forces acting on the helicopter in the horizontal plane is obtained:
With: the azimuth angle β * rad +
I2 the moment of inertia about the vertical axis kg.m ² + *
τ2 torque tail rotor [N · m]
τr the torque caused by the main rotor [Nm]
τf2 friction torque
7. 4. Dynamics of the complete system
4.1. Non-linear model
After establishing the dynamics of the various components of the helicopter, we can
build
Block diagram of the complete system in Simulink
The values of various parameters of the system are given by the manufacturer and
have the following values
Symbol Value Unit
T1 0,3 s
a1 0,105 N.m/MU
b1 0,00936 N.m/MU²
I1 4,37e-3 kg.m²
B1 1,84e-3 kg.m²/s
Tg 3,83e-2 N.m
T2 0,25 s
8. a2 0,033 N.m/MU
b2 0,0294 N.m/MU²
Tor 2,7 s
Tpr 0,75 s
Kr 0,00162 N.m/MU
I2 4,14e-3 kg.m²
B2 8,69e-3 kg.m²/s
KGyro -0,015 kg.m/s
kα π/1024 rad
kβ π/1024 rad
yα0 275 π/1024 rad
4.2. Linear Model
Using the linmod Matlab function, we can linearize our model.
5. Hardware of the helicopter and software
of control
9. 5.1 MF 624--multifunction I/O card
The MF 624 multifunction I/O card is designed for the
need of connecting PC compatible computers to real
world signals. The MF 624 contains 8 channel fast 14 bit
A/D converter with simultaneous sample/hold circuit, 8
independent 14 bit D/A converters, 8 bit digital input port
and 8 bit digital output port, 4 quadrature encoder inputs
with single-ended or differential interface and 5
timers/counters.
The card is designed for standard data acquisition and
control applications and optimized for use with Real Time
Toolbox for Simulink. MF 624 features fully 32 bit
architecture for fast throughput.
5.2 AD 622—data acquisition card
The AD 622 data acquisition card is designed for the need of connecting PC
compatible computers to real world signals. The AD 622 contains 8 channel fast 14
bit A/D converter with simultaneous sample/hold circuit, 8 independent 14 bit D/A
converters, 8 bit digital input port and 8 bit digital output port. The card is designed
for standard data acquisition and control applications and optimized for use with Real
Time Toolbox for Simulink. AD 622 features fully 32 bit architecture for fast
throughput.
5.3 Modelling
An attempt to model the system dynamics in detail leads to extremely complicated,
not readable and not useful model. The engineer should decide what is the model used
for and under what conditions it will work. In our case the model will be used for
investigating the system dynamics with respect to control tasks. The system will
10. operate in some working conditions and not all of the dynamical properties will be
invoked. This leads to the assumptions which will simplify the derivation of the
model.
We propose two ways how to get the model. The first one is a systematic modeling
method based on variation approach, i.e. Lagrange's equations. The second approach
is a direct derivation of the model by computing the force balances. The second
approach is described in the following text. Both methods lead to the model in the
form of nonlinear differential equations.
6. The software of CE150
6.1 Demonstration Program Description
The menu screens are mostly selfexplaining. To choose one of menu options
(trajectory etc...) type number corresponding to your choice and press Enter. To accept
default values (in angle brackets), just press Enter. To modify parameter value, type
row number of the parameter and press Enter. When the prompt appears, enter data
followed by Enter. Note that you can use arithmetic operators, MATLAB variables
and a predefined variable named default holding old parameter value. Vectors may but
need not be enclosed in angle brackets. After the experiment is finished, all
parameters and experimental data stay in the workspace. They are used as defaults for
the next experiment. If you want to start with defaults, simply clear all variables.
There are two versions of the demo program on your diskette. The basic one,
HE2DEMO, controls the helicopter in both degrees of freedom. The HE1DEMO
controls only the elevation, azimuth must be fixed. It may be simpler to experiment
with the constants of this controller, at least for the beginning.
11. 6.2 Demonstration Program Algorithms
The following describes some details of the demonstration program.Don't bother to
read this unless you actually want; rather skip to Chapter 2 and return here when you
will be more familiar with the model.
Coefficients of discrete polynomial controller can be entered either as coefficient of
continuous PID controller or directly as polynomials in z operator.
PID controller is defined either by its coefficients Kp , Ki , Kd and Kdd
or by total gain G and time constants Ti , Td and Tdd ,
where
Ts is sampling period
y(z) is system output
e(z) is output error w(z) - y(z)
u(z) is system input
F(z) is first order discrete filter with time constant Tf .
Polynomial controller may be entered either in the general form
or in the r-s-t-form with common denominator
or as open-loop compensator.
12. The input filter is implemented as discrete polynomial filter in z operator with
sampling period Tsf . The sampling period may be higher than the controller sampling
period Ts .
where
yf(z) is filtered system output
y(z) is physical system output.
6.3 A/D Converter
Features List
The MF 624 offers following features:
14 bit A/D converter with simultaneous sample & hold circuit
13. Conversion time 1.6 μs for single channel or 3.7 μs for 8 channels
8 channel single ended fault protected input multiplexer
Input range ±10V
Internal clock & voltage reference
8 D/A converters with 14 bit resolution and ±10V output range
4 quadrature encoder inputs with single-ended or differential interface
Software selectable digital input noise filter (0.3 μs)
The MF 624 multifunction I/O card is designed for the need of connecting PC
compatible computers to real world signals. The MF 624 contains 8 channel fast 14
bit A/D converter with simultaneous sample/hold circuit, 8 independent 14 bit D/A
converters, 8 bit digital input port and 8 bit digital output port, 4 quadrature encoder
inputs with single-ended or differential interface and 5 timers/counters. The card is
designed for standard data acquisition and control applications and optimized for use
with Real Time Toolbox for Simulink. MF 624 features fully 32 bit architecture for
fast throughput.
6.4 AD 622
The AD 622 offers following features:
32-bit architecture
14 bit A/D converter with simultaneous sample & hold circuit
Conversion time 1.6 μs for single channel or 3.7 μs for 8 channels
8 channel single ended fault protected input multiplexer
Input range ±10V
Internal clock & voltage reference
8 D/A converters with 14 bit resolution and ±10V output range
8 bit TTL compatible digital input port
8 bit TTL compatible digital output port
Interrupt
The AD 622 data acquisition card is designed for the need of connecting PC
compatible computers to real world signals. The AD 622 contains 8 channel fast 14
bit A/D converter with simultaneous sample/hold circuit, 8 independent 14 bit D/A
converters, 8 bit digital input port and 8 bit digital output port. The card is designed or
standard data acquisition and control applications and optimized for use with real
Time Toolbox for Simulink®. AD 622 features fully 32 bit architecture for act
throughput.
14. 7. Controller Design
7.1 Pole placement design of state feedback controller
The main idea of this design method is that the location of the closed loop poles of a
linear system determines the closed loop system dynamics. In this problem the poles
of a closed loop system are placed according to the position specified by the designer.
This may be done only by the state feedback. The block structure of the closed loop
system enabling us to place the poles arbitrarily is shown, where A, B, and C are the
matrices of the linearized model identified in the Problem 3.9 and K is a matrix of
feedback gains to be designed. The discrete time model is considered, but the same
procedure is applied when continuous time model of the process is used.
8. Conclusion
This paper is devoted to mathematical modeling. General model structure valid under
some simplifications is derived in terms of nonlinear state space description and the
corresponding block diagram. Identification of system parameters is done in item 3.
Number of experiments for direct and indirect measurement of system parameters is
described. Typical numerical values are derived and used for initial tuning of the
system model. The model is linearized around the different set points. Transfer
function matrix as an Input/Output model and linear state space model are derived.
The following chapters are devoted to controllers design. Item 4 gives guidelines for
selection of the sampling period for digital control systems. Item 5 shows the design
of the conventional PID control and State Feedback control by pole placement.
Figures from measurements can be found in Item 6.
