Fault Tolerant Control

Applied to a Helicopter CE 150

ESTACA CAED Program

30 May 2013

Hongyi HUANG

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

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

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:

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.

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

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

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

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

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.

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.

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

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.

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.