Predictive functional control (PFC) for use in autopilot design.
The
University
Of
Sheffield.
Automatic
Control &
Systems
Engineering.
Predictive Functional Control (PFC)
For Use in Autopilot Design
By: Hyreil Anuar Kasdirin
MSc in Control Systems: August 2006
Supervisor: Dr J A Rossiter
A dissertation submitted in partial fulfilment of the requirements for the degree of
Master of Science in Control Systems
ABSTRACT
This paper discusses the design and implementation of PFC as a controller for an
autopilot missile. Two linear continuous time missile models which are derived from
nonlinear model produced by Horton [13] and another from the basic Ballistic Missile
[10] are used for the prediction models. The PFC algorithm is developed based on the
models. The PFC algorithm developed is seems intuitive and computationally simple as
the missile need not to be very complicated as it will explode as it reaches the target.
Furthermore, the analysis and issues of the implementation relating linear discrete-time
stable and unstable process are being discussed. In addition, PFC tuning parameters play
an important part of the autopilot controller. Thus, the result indicated that the PFC
control law is built better when choosing the dynamic pole of the missile mode to be the
desired time constant, 'P and small coincidence horizon n1 as performing in single
coincidence point. The implementation of PFC on the missiles-scenario is also developed
for Model Missile I and 2. As a result, some positive results is illustrated and discussed
as the both missile followed its reference trajectory during simulation using MATLAB
7.0.
Keywords: Predictive Functional Control (PFC), autopilot design, discrete-time state-
space models
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
EXECUTIVE SUMMARY
Introduction/Background
This paper will concentrate on the basic handling of PFC as a controller for autopilot
missile. The formulation of PFC will be developed as well as how PFC handles with
stable and unstable process. A particular type of missile and onboard guidance system has
not been specified in the reference missile model. However, the paper briefly explained
some missile models and its missile guidance control. Thus, the result and
implementation of the PFC algorithm as a controller of autopilot missile will be further
discussed later on.
Aims and Objectives
The main aim of this project is to understand the design the PFC as a controller for an
autopilot missile. The principle objectives of this project are to understand and develop
the basic of PFC methodology. It also intended to analyze issues relating stable and
unstable process on PFC algorithm. Lastly, the objectives of this paper are to analyze the
results from the design and tests using PFC algorithm on missile models using MATLAB
7. 0 environment.
Achievements
At the end of this project the implementation of PFC is discussed. This paper will
concern on linear continuous-time stable and unstable missile model that only concern on
the aerodynamic control of the missiles. Furthermore, this paper shows that PFC is
developed successfully with one coincidence point, n 1• The tuning parameters of the PFC
algorithm; which are the desired time constant, 'I' and the control horizon, n are being
manipulated in order to give the best performance of the controller. As a result, it is
clearly illustrates that it is best to chose the dynamic pole of the missile mode to be the
desired time constant, 'I' and small coincidence horizon n 1 as performing in single
coincidence point. This section also has successfully implementing PFC on unstable
system with a single unstable pole. However, the system need to perform pre-stabilise
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
2
before implementing Closed-Loop Prediction (CLP) on the modified PFC control law to
give stable and good response and performance. The implementation of PFC on the
missile-scenario is also illustrated by showing the result of discrete linear missile model 1
and 2. The section 4.5 clearly showing that PFC perform well as a missile autopilot
controller as it gave good trajectory as the reference trajectory moved and tum in
different direction. As a result, some problems faced and suggestions are also being
discussed.
Conclusion I Recommendations
At the end of this project it was recommended to improve the missile model as the
prediction model and develop the scenario of missile-target to see whether the PFC could
be used as a controller for the autopilot missile. In all, based on the achievement/result
the implementation of PFC algorithm seems intuitive and computationally simple. This
truly important as the missile controller need not to be very complicated as it will explode
as it reaches the target.
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
3
ACKNOWLEDGEMENT
I thank God Al-Mighty for giving me the opportunity to achieve my goal.
Here, I would like to express my gratitude and acknowledgement to Dr J. A. Rossiter for
his guidance and continuous encouragement as well as motivation throughout the project.
I also would like to express my gratitude to my lovely wife and daughter for their
understanding, patience and moral support.
I would like to thank to the Kolej Universiti Teknikal Kebangsaan Malaysia (KUTKM),
for their financial support of this course. Last but not least, many thank to my family,
friends and all of the lecturers of The Department of Automatic Control and Systems, for
continuous support and encouragement to my study at the University of Sheffield, United
Kingdom.
HYREIL ANUAR KASDIRIN
August 2006
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
4
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
1
Executive Summary
... .. .. .. ........... .. ... ... . .. .. .......................... . ..... ........ .. ....... .. ..
2
List of Figures
.. . .. ... .... .... ..... .... .. .. .... ..... . .... . .... .... ... ... ... .. . ..... ......... ........
8
List of Tables
.. ... .. . .... .............. ... ..... ... ................ .. . .... ... . ........... .. .........
9
Abstract
SECTION 1: BACKGROUND PROJECT
1.1 An Autopilot Missile
... .... . .... ... . .. ... .. . .. ... . ... .. . ... ......... ... ...... .. . ... . ... .. ... .. .. 10
1.2 Predictive Functional Control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Thesis Scenario: Missile Guidance Control
1.4 Aim and Objectives
1.5 Chapter Outline
10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
.. . . . . . .. .. . . .. . . . . . . . . . . . ... . .. . . .. . . . . . . .. . . .. .. . . .. . . . . .. . . . . .. . . . .. . . ... 12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
SECTION 2: PREDICTIVE FUNCTIONAL CONTROL (PFC)
14
2.1 Predictive Control and MPC Algorithm ...... ..... .... ... .... .. .. . .. . ... . ..... ... ... ...... .. . . ... 14
2.2 Optimal Control of MPC
.. .. .. .. . .. . . .. ... ...... ... .... ..... ... .. . ... . ... .. .... ... . . ....... 14
2.3 Formulation of PFC Algorithm
. .. ... .. . .... .... .. ..... ... ... . ..... ... ... . .... .. .. ... . .. .. . ... .. .. 15
2.3.1 Models
.... .. ... .. . .... .. .... . . .... .. .. . ... ... ... ... .... .. .. . . . . ... . .... .. .. ... .... 15
2.3.2 State-space Model
.. .. .. ... .. ... . ..... .. .. ... .... ... .. ... . ... . . ... ... . .... .. .. . .... .. 15
2.3.3 Reference trajectory formulation
.... .. ... . .. . .. ... ... . .... . .... .. .. .. .. ... . .. . ... 17
. . . .. . . . .. . . . . . . .. . . .. . . . . . . . .. . . .. . . . . . . . . . . . . . . . .. . . . . .. 17
2.3.4 The coincidence points
2.3.5 Paramerisation of the d.o.f/future control trajectory
. . .. . . .. . . . . . .. . . .. . . . . .. . . . . .. 18
2.3.6 Computational of control law
.. . . . . . .. . . . . .. . .. . .. . . . . . . .. .. . . .. . . . . .. . . . . . . .. . .. . . . . .. 18
2.3.7 Tuning Parameters of PFC
. . ... . ... .. . ... . .... . ... ... . .. ... . .. ... .. . ... ... . .. . ..... . 19
2.4 PFC for unstable Process
2.4.1 Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. .. ... ... .. ... ..... .. .. . .. ... . ......... .... ... ......... .... .... .. ... .... .... 19
2.4.2 Unstable Open-Loop Problems
2.4.3 Predictive Stabilisation
. . .. . . . . . . .. . . . . . . .. .. .. . . . . . . . . .. . . . . . . . .. . .. . .. 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.4 Closed-Loop Paradigm (CLP) Concepts
2.4.5 CLP Predictions
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . .. . . . . . . . ... . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . ... . . . . . . . . . . . .. 22
5
2.5 Summary
·· ··· ··· ·· ·· ··········· ··· · ·· ········ · · ·· ·· ··· ····· ·· ·· ·· ·· ··· ······················ · 24
25
SECTION 3: AIR MISSILE & AUTOPILOT CONTROL
......... ... ......... . . .... ... .. .... ......... . ............................... 26
3 .1 Missile Airframe
26
3.1 .1 The Basics of Flight Dynamics
3.1.2 Equations of Motion
.. .... ... .. ... .... .................... ... .. ........ ..... .. .... ... ... 27
3.1.3 Assumptions Made
.. .. ... ............................... ............... . .. ............ 28
3 .2 Missile Modelling Design 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
. . . . .. . .. . . . . . . .. . . .. . . . . .. . . . . .. . . .. .. .. . . . .. .. . . . . . . . .. .. 28
3 .2.1 The Dynamic Equation
3.2.2 The Linearised Continuous-Time Models
3.3 Missile Modelling Design 2: Unstable Case
.... ..................................... 29
..... ....... ............... ... ......... . .......... 31
3.3.1 The Dynamic Equation
···· ··· ··· ···· · · ················ ··· ···· ············· ······· 32
.. ... ... .... ... .. . .... ... .. .... ... .. ..... 33
3.3.2 The Linearised Continuous-Time Model
.. . ... .. .. .. .. . ... . ... ......... .......... 34
3 .4 Missile Control System and Autopilot Design
3.5 Summary
.......... .. .. . .... .. .. ... . . .. ... .. .... .. .. .... . ... .. .. .. .. . .. ........................ 35
SECTION 4: IMPLEMENTATION OF PFC
4.1 Implementation of PFC algorithm
4.1.1 Prediction Model
36
................ ........ .. .. .... . .. ..... ... .... .. ........ .. 36
.. ... . .......... ......... . ....... .. .. ........... ... ... .............. 36
4.1 .2 Model 1: The Horton Missile Model
....... . ....... . ..... ... ... ....................... 36
4.1.3 Model 2: The Ballistic Missile Model .......... ... ..................................... 37
4.1.4 Set-up PFC Algorithm for Discrete Linear Model Missile 1
4.2 The Tuning Parameters and Its relation with Model Dynamic
4.2.1 Comparison of different desired time constant
....................... 38
. . .... .... ...... .... .... .... .... 40
................................ 40
4.2.1.1 Control and Output Response/or 'P = 0.3
................................ 41
4.2.1.2 Control and Output Response for IJf = 0.6
............... .. .. .. ....... .. .. 42
4.2.1.3 Control and Output Response for IJf = 0.9
............ ..... . .............. 43
4.2.1.4 Discussion
....... ... . . ... .. .. .. .. ....... ..... ... .................. ... . ........ 43
4.2.2 Relation \f with poles of the discrete linear model, G(z)
4.2.3 Discussion
....................... 44
........ .... .. .. .. ... .. .... ...... .... ...... . ................. ................ 46
4.3 The Implementation of PFC for unstable process
4.3.1 The Open Loop Response of Model 2
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
....... . ... . ... .... . .. ...... ...... ....... 47
....... . .. ... ... ... .. ... .. ............ ... 47
6
4.3 .2 The Observability of the Example 2
..... . . .... . .... .. ... .... .. . ..... .... ... .. ..... ... 48
4.3.3 Solving the Prediction Mismatch by Pre-stabilised Prediction
4.3.4 Discussion
.... .. ........ 49
.. .. .. .. .. . .... .. ....... .. ........ . ... .. ... ...... .. .. . ... .. . ................. 50
4.4 Implementation PFC as Missile Autopilot Control
.... ... ... .. .. . .. .... .. ... . ... . ... .. . .. .. 51
4.4. l Implementation PFC on Missile Model 1
..... .................. . ... .... ... .. ... .. 51
4. 4.1.1 PFC as an Autopilot Control of the Missile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
4.4.1.2 Missile Constraint
... .... ... ... .. .... .. .................... .. .. ... .. ..... .. .. 53
4.4.2 Discussion
··· ····· ················ ·· ········ ··· ·· ···· · ······················· · ········· 54
4.5 Implementation PFC on Missile Model 2: An Unstable Process Example
. . . . . . . . . . . ... 54
4.5. 1 Model Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
4.5.1.1 PFC as an Autopilot Control ofthe Missile 2
4.5.2 Discussion
4.6 Summary
.. .. .. .... .... .. .. ..... 55
.......... . ........ .... . ...... .. . .. ... . ... . .. ....... .. .. .. .. ....... .. .. .. . ... 57
. .. ....... .. . .... . ..... ... . ... . .... .. .. ...... .. .. .. .. . ... . .. ... ... .. ........... .. ..... 59
SECTION 5: CONCLUSION AND RECOMMENDATIONS
59
REFERENCES
61
APPENDIXES
63
APPENDIX A: International Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
APPENDIX B: Missile Model 1 and 2
... .... . .... ......... ..... . ... ..... ......... .. .... 64
APPENDIX C: PFC Algorithm for nl = 2
....... . .... .. . ..... .... .. . . ............... ....... 67
APPENDIX D: PFC Algorithm for nl = 3
.. . .. ... ........................... . .... ... ....... 68
APPENDIX E: PFC Algorithm for nl = 6
...... .... ..... .. ... .. .. ...... .. . ... ... .. .... ... .. 69
APPENDIX F: PFC Algorithm for nl = 10 .. .. .... . ..... .. .. ......... ... ..... . ... ..... ... ... 70
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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LIST OF FIGURES
Figure 1.1 : Pitch, Yaw and Roll of a Missile
Figure 2.1 Basic structure of MPC
.............. . ... . ... ............... .. .. .. ... .... 10
.. ... ........ .. ....... .. .. .. .. .. . ... .. ...... ..... ... .. . .. .. .. ..... .. 14
Figure 3 .1: The Airframe of Basic Missile
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 26
Figure 3.2: Ballistic missile axis system.
. ... .............................. ...... .. .. .... ........... 31
Figure 3.3: Sketch of an aerodynamic missile and axis system
Figure 3.4: Basic Layer of Automatic Missile Control
.... ... .. .. .... . .... . ... ... ..... 34
.. .. ..................... ......... .. ..... 34
Figure 4.1: The Control and Output Response for Example 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 4.2: The Closed-loop Step Response for 'I'= 0.3
.. ............ ........... . ............... 41
Figure 4.3: The Closed-loop Step Response for 'I'= 0.6
..... ... ..... .. ............... . .......... 42
Figure 4.4: The Closed-loop Step Response for 'I'= 0.9
........... .. ... ...... .. ... .............. 43
Figure 4.5: The Controller Response of Example 1 for 'I' = 0.94
.... . ..... ......... .. ... .. ...... 46
Figure 4.6: The Root Locus and Bode Diagram for Missile Model 2
...... ... .. ... 47
Figure 4.7: The Controller Response of Example 1for'I'=0.94
... ..... .. .. .... ... ... .......... 50
Figure 4.8: Sketch of an aerodynamic missile of Missile Model 1
... ... .. .. .. .. . ... ... .. .... ..... 52
Figure 4.9: The Missile Controller Response of Missile Model 1
.. ... ............ .. ......... .... 53
Figure 4.10: Sketch of an aerodynamic missile of Missile Model 2
.... . .... .. ... ... . ..... 55
Figure 4.11: The Missile Controller Response of Missile Model 2
....................... 56
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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LIST OF TABLES
Table 3 .1: The Motion Variables of the Basic Missile Model
................................ 27
Table 3.2: The Characteristics and the coefficients of Horton Missile Model 1
.. ..... .... ... 29
Table 3.3: Characteristics and coefficient of basic ballistic Missile Model 2
....... .. ..... 32
Table 4.1: Parameters of Example 1 .. ............... ....... .. .... ..... ... .............................. 38
Table 4.2: Parameters Used for Tuning Parameters of PFC Algorithm of Example 1
40
Table 4.3: Parameters Used for Developed of PFC control law for Missile Model 1
52
Table 4.4: Parameters Used for Developed of PFC control law for Missile Model 2
56
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
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SECTION 1: BACKGROUND PROJECT
1.1 An Autopilot Missile
A missile is a projectile, often self propelled, which delivers a payload to a target.
Missiles have various launch platforms, ranges, targets, payloads and are typically guided
either remotely or automatically. Basically, there are 6 degrees of freedom which need to
be controlled. These angles are conventionally called yaw, pitch and roll (refer Figure
1.1) which all of these measures by the on-board sensors. In this paper, there are two
models that have been selected and they are only two degree of freedom.
Figure 1.1: Pitch, Yaw and Roll of a Missile
In this paper, the main point of research is the autopilot system of the missile. An
autopilot is a mechanical, electrical or hydraulic system used to guide a vehicle without
assistance from human being. Missile autopilot system is one of the examples of an
autopilot system. The main purpose of the autopilot missile is to enable the missile to
accomplish their mission autonomously, without any (or with minimal) input from the
missile operator. It includes the missile automated take-off or target hit, depends on its
mission.
Nowadays, modern autopilot missiles use computer software to control the missile. It
uses the missile state information provided by the on-board sensors to drive the control
surface actuators (servos) and gives feedback to the missile control surface. The autopilot
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
10
missile is also designed to hold certain parameters constant, for example its direction,
speed, altitude etc.
The control and guidance system is the brain of the autopilot missile. The on-board
control circuit need not to be too complicated or big as its mission is only target-hit
approach. Numerous missile designers and researcher have attempted to build effective
control of an autopilot missile such as using H-infinity controller [14] and Adaptive
controller [15]. In addition, this paper is continuing the work of Ben [16] and Nick [17] for
implementing Model Predictive Control as a controller for autopilot missile. They have
successfully developed missile model and simulation of target-hit of the model. However,
they only concentrate briefly on the control section using Model Predictive Control (MPC).
Hence, this paper will only concern of the missile autopilot controller using Predictive
Functional Control (PFC).
1.2 Predictive Functional Control (PFC)
Predictive Functional Control (PFC), which developed by Richalet [1] is one of Model
Predictive Control (MPC) techniques that have been developed as a powerful algorithm
for controlling process plants. In this paper, the focus is on the implementation of the
predictive functional control (PFC) on the missile dynamic models. PFC is based on the
same approach with all MPC strategies i.e., prediction of the future outputs, and
calculation of the manipulated variables for an optimal control. Therefore, PFC is also
based on the same principles which are using an internal model, specification of a
reference trajectory and determination of the control law.
1.3 Thesis Scenario: PFC as a controller of Autopilot Missile
This paper will concentrate on the basic handling of PFC as a controller for autopilot
missile. The formulation of PFC will be developed as well as how PFC handles with
stable and unstable process. A particular type of missile and onboard guidance system has
not been specified in the reference missile model. However, the paper briefly explained
some missile models and its missile guidance control. Thus, the result and
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
11
implementation of the PFC algorithm as a controller of autopilot missile will be further
discussed later on.
1.4 Aims and Objectives
Therefore, regarding the thesis scenario of this project, the aim of this project is as
follows:
•
Understand the design the PFC as a controller for an autopilot missile.
In addition, the objectives of this project are to:
•
Understand and develop the basic of PFC methodology.
•
Analyze issues relating stable and unstable process on PFC algorithm.
•
Analyze the results from the design and tests using PFC algorithm on missile
models using MATLAB 7.0 environment.
1.5 Chapter Outline
Based on the aim and objectives given above, this project is investigating the design an
autopilot control system of a guided missile using PFC controller.
The second section will be looked on some basic theory of PFC as a controller so that it
can be formulated and implemented in the following sections. However, at first, this
section will describe the basic of MPC algorithm and its optimal control that has become
an efficient control strategy for a large number of processes [2). After that, it followed by
the introduction of PFC algorithm and the formulation of its control law. This section
also will be discussing the way PFC handle the unstable process by pre-stabilise the
unstable plants to implement the stabilizing linear PFC control formulation.
After that, section three will be introducing some basic models of missile and its autopilot
control. The main content of the section is to show two basic missile models that will be
used for PFC implementation as its controller in the following section. The autopilot
control for the first model is the control of the deflection of control surfaces, whereas the
second model; ballistic missiles are controlled by deflecting the thrust vector [10].
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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Lastly, section four will illustrate the implementation of PFC. At first, this section will be
fully explained how PFC algorithm could work in given stable and unstable model using
the models that have been introduced in the section three. The second sub-section will
then further the implementation of PFC whether PFC could work as a controller on fast
process such as autopilot missile.
The last section of this project tries to conclude the project as it developed from previous
section. The summary of the project will be discussed and some recommendation will be
noted for further analysis and research.
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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SECTION 2: PREDICTIVE FUNCTIONAL CONTROL (PFC)
This section is basically will be discussing the theoretical part of PFC as a controller so that it
can be formulated and implemented in the following sections. However, at first, this section will
describe the basic of MPC algorithm and its optimal control that has become an efficient control
strategy for a large number of processes [2). After that, it followed by the introduction of PFC
algorithm and the formulation of its control law.
2.1 Predictive Control and MPC Algorithm
Predictive Control or so called Model Predictive Control (MPC) has being developed for
more than 20 years, both in industry and academic community. The principles of MPC
are universal, and can be found in many textbooks [3], [4], [5). A wide range of MPC
algorithm was developed, where it developed to suit given types of industrial application.
Some of the most popular MPC algorithms as follow;
a.
Generalised Predictive Control (GPC), [2]
b.
Dynamic Matrix Control (DMC) from Culter and Ramaker,
c.
Model Algorithmic Control (MAC) from Richalet,
d.
Predictive Functional Control (PFC), developed by Richalet and
ADERSA. [1], [3]
2.2 Optimal Control of MPC
Reference
trajectory
Past inputs
and outputs
.
Future
inputs
Predicted
outputs
Model
+
-
Optimiser
Cost
function
'"
Future errors
Constraints
Figure 2.1 Basic structure of MPC
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
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The basic structure to implement MPC strategy shows in figure 2.1. There are two
important blocks as illustrates above, which are model and optimiser. Hence, it is true
that the essence of MPC as well as PFC is to optimise, over the manipulability inputs,
predicting of the process behaviour given [6]. A model is used to predict the future plant
outputs, based on past and current values and on the proposed optimal future control
actions. These actions are calculated by the optimizer taking into account the cost
function as well as the constraints.
2.3 Formulation of PFC Algorithm
2.3.1 Models
The model is the essential element of an MPC controller [6] and hence, also for
PFC controller. PFC can use many forms of model i.e.; internal model (IM),
including state space, transfer function, Finite Impulse Response (FIR), fuzzy
rules, and etc. The use of IM is important in PFC to capture the process dynamics
of the system and also continue to calculate the PFC control law later on.
2.3.2 State-space Model
Hence, for this section, the model will be developed in state-space form. The
discussion of PFC and other MPC algorithms in state-space form has several
advantages, including easy generalisation to multivariable systems, ease of
analysis of closed-loop properties, and online computation [6].
Given the general state space model, of the form:
! k+1
=A!k + B?!.k
ｾ ｫ＠ =C!k +D?!.k
(2.1)
Prediction with a strictly proper system (D = [O])
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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:!k+2
= A:!k+i + ｂｾｫＫＱ＠
(2.2)
Y
- k+2 -Cx
- -k+2
Substituting Equation 2.1 into 2.2;
:!k+3
= A 2[A:!* + ｂｾｊＫ＠
ｾｫＫＳ＠
=C:!k+3
ａｂｾｫＫｩ＠
+ ｂｾｫＫＲ＠
This process is simply an iteration of a one step ahead prediction, repeated
substitutions result in the prediction matrices, P and H.
:!k+n
ｾｫＫｮ＠
= A n:!k + An-IBｾｫ＠ + An-2BｾｫＫｉ＠
= c[An :!k + An-1 ｂｾＪ＠ + An-1 ｂｾｫＫＱ＠
+ •• ·+ B ｾｫＫｮ
+ ... + ｂｾｫＫｮＭＱ｝＠
Ｍ Ｑ＠
(2.3)
State Prediction Equation
!
k+l
!
!
k+2
!
k+n
k+J
B
0
0
AB
2
A B
B
AB
0
B
f..xx
... 0
... 0
... 0
... 0
... B
Hxx
Y. k
Y. k+l
Y. k+2
Y.
k+n-1
ｾ＠
k-1
(2.4)
Output Prediction Equation
.l::'. k+l
.l::'. k+2
.l::'. k+J
}:'. k+n
=
0
CB
CAB
=
H
0
0
CB
0
0
0
0
Y. k
Y. k +l
Y. k +2
CB
Y. k+n-1
ｾ＠
k-1
(2.5)
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
16
From Equation 2.4 and 2.5 above, it shows the model used is a linear one that
represent by;
ｾｫ＠
= Pxx!k + HxlJ.k-1
ｾｫ＠
=P!_k +HlJ.k-1
where !le is the state model, ｾ＠
(2.6)
is the input model, »is the measured output model,
P= Hxx, P and Hare respectively, matrices or vectors of the right dimension.
Below, the algorithm for PFC is outlined as found in [5]. There is an element of
derivation here, however its inclusion is necessary as it helps to explain the main
concepts behind PFC. This intuitive approach is one of the key selling points.
2.3.3 Reference trajectory formulation
PFC formulates the reference trajectory by placing the desired closed-loop
dynamic into the reference trajectory. Given the actual set point is r, the loop set
point w is a first order lag [3].
(2.7)
where Yk is the most recent measured output and '¥ ( 0 < '¥ < 1 ) is scalar and a
tuning parameter setting the desired closed-loop pole.
The predictive essence of control strategy is completely included in Equation 2. 7
above. Indeed, the aim is to track the set point trajectory following the reference
desired closed-loop behaviour.
2.3.4 The coincidence points
The control law is determined by using the d.o.f to enforce equality of the
predictions and the reference trajectory at a number of points, that is, by solving
the control moves such that:
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
17
Y
k+n
=W
k+n, n
(2.8)
= n1,n2, ... .
These equalities are called coincidence points. Typically, there only have one or
two coincidence points. However, in this paper will only concentrate with one
coincidence points only.
2.3.5 Paramerisation of the d.o.f/future control trajectory
The PFC takes the trajectory as the sum of a step change, a ramp, a parabola, etc.
The precise components to be included are selected to match the expected
characteristics in the set point.
2.3.6 Computational of the control law:
At a single coincidence points, and using equation 2. 7 and 2.8, the control law is
determined by;
(2.9)
Hence, substituting Equation 2.6 with 2.9;
(2.10)
By assuming g
k + ;
= g "' the control law can be computed by rewriting the
Equation 2.10 and obtain;
I
M. k
=-H [P !
M. k
=-K !
where; K
fJ
k
k
.
+ ( rk -
(rk -
+ fJ Tk
=-H
I
y,J IP 1 )
]
(2.11)
.
( P - IP 1 Yk)
= -HI (I - IP;)
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
18
Thus, it can easily express as a fixed linear feedback law in the form of prediction
algorithm. Hence conventional a posterior stability and sensitivity analysis can be
applied in straightforward manner.
2.3. 7 Tuning Parameters of PFC
The tuning parameters of PFC are generally the coincidence horizon, e.g. n1
= 1,
2, . . . and the desired time constant, \f. The typical procedure with one
coincidence point [3] would be as follows:
1. Choose the desired \f.
2. Do a search for n 1 = 1, ... large and find the associated control law for each
n1.
3. Select the n 1 which gives closed-loop dynamics closest to the chosen \f.
4. Simulate the proposed law. Otherwise, reselect \f and go to step 2.
Hence, the tuning reduces to a global search, but this requires only relatively
trivial computations and hence would be quite quick. With two coincidence
points, the global search would be more involved but should still be quick.
2.4 PFC for unstable Process
2.4.1 Introduction
PFC algorithm is defined in the previous sub-section is basically open-loop
process control applications. In the contrary, in industry applications, open-loop
unstable processes do also occur. Yet, these systems are difficult to control.
Hence, systematic control design tools are needed to handle complex instability
without a high on-line computational load. ADERSA have successfully applied
PFC on many unstable systems [3]. This section will discuss the theoretical tools
to pre-stabilise the unstable plants to implement the stabilizing linear PFC control
formulation.
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
19
2.4.2 Unstable Open-Loop Problems
PFC, as well as other MPC algorithms is weak with both non-minimum phase
problems and some unstable process [3], or called prediction mismatch. If the
process open-loop is unstable, PFC and also MPC are ill-posed because prediction
cannot match desired behaviour of the process, i.e. diverging. Therefore,
divergence open-loop prediction is the main cause of the prediction mismatch.
Hence, there is a must to stabilise the prediction. There are two ways of prestabilise the predictions which are inserting a stabilising loop and another by
shaping the future inputs, algebraically so that the outputs are stable. However,
this section only focused on solving algebraically the unstable process as
discussed by [7] and [8].
2.4.3 Predictive Stabilisation
Removal of the prediction mismatch is essential for PFC to work. Hence, the
model needs to have prediction stabilisation. One method is by cancelling the
unstable modes and starts working with stable predictions. This means that PFC
control law process must be modified. Therefore, in solving this problems PFC
will lead to good closed-loop performance if the predictions used are a good
match to the consequent closed-loop behaviour.
The illustration below shows the state space method of predictions to cancel the
unstable modes [3]. Let a state-space matrix have some unstable eigenvalues.
Decompose the system into stable and unstable modes using eigenvalue
decomposition;
A
[ w. W.J diag[A. A.J
｛ｾ＠
(2.12)
where subscript s is used for stable and u for unstable. Clearly if a state lies solely
in the stable manifold of A, then it must satisfy:
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
20
(2.13)
Given this, the predicted state evolution would follow
ｾ＠
=A
k+ ilk
i
i
ｾ＠
k
= Ws As Vs
T
(2.14)
ｾ＠ k
then the mode 2 predictions are given by
(2.15)
!:!. k+ nc+ ;=O
By assuming there is no disturbances or measurement noise, then continue with
predict by iterating the model (Equation 2.1 ). So we get
ｾ＠
］ａｾ＠
k + 111c
k
ｾ＠
k + 211c = A
ｾ＠
k
ｾ＠
;
k + ilk = A ＠ｾ
k
+ B!:!. k!k
+ 111c + B!:!. k + 111c
+ A ;.1 B
!:!. klk
= A
2
ｾ＠
k +AB !:!. k!k
+ ..... + B
!:!. k + i -1/k
which can be summarized as
,i; "
v.
ｾ＠
+ B!:!. k+ 11k
A; ,i; , + [ A;.JB + A;.iB ... B ) [
!:!.!:!.k k!k
+llk
!:!. k +
l
i -1/ k
M
(2.16)
or in common form
(2.17)
To ､ｲｩｶ･ｾＫ＠
k
+ ilk = 0, from equation 2.13, we know that
VuT X k+ ilk=
0,
(2.18)
Hence, by substituting equation 2.18 with 2.17 and 2.12, we get;
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
21
T
.
Vu [A'-! k + M !! k
.
]
= 0,
T
So as clearly for Wu Au' Vu = 0. Therefore;
T
i
i
T
Vu [A -! k + Ws As Vs !! k +
i]
=0
(2.19)
The control law becomes,
!! k + ;
.J! k
= - [ Ws A/ V/ r 1 V/ Ai -! k +HP
=-K
-! k
+ HP
where K = [ Ws A/ Vs T
r 1 Vu
(2.20)
T Ai
and
p could be choose freely.
2.4.4 Closed-Loop Paradigm (CLP) Concepts
As mentioned above, PFC will lead to good closed-loop performance if the
predictions used are a good match to the consequent closed-loop behaviour.
Hence, the closed-loop paradigm (CLP) is introduced here. CLP, which was
originally proposed as part of an algorithm stable generalised predictive control
(SGPC, [9]) will be implemented in the modified PFC after the pre-stabilised
prediction.
2.4.5 CLP Predictions
Based on the prediction on the PFC algorithm, the equations within the prestabilised loop during prediction are;
-! k+ilk =A-! k+i-llk + B!! k +;;
!! k+i = Ｍｾ＠
k+ilk
+ c k+i;
(2.21)
By removing the dependent variable y_ k + ; one gets;
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
22
:! k+ilk = /A - BK);! k+i-Ilk + B H,k +1,·
H. k+i ］Ｍｾ＠
(2.22)
c k+i;
k+ilk +
Hence, simulating these forward in time with 2
B
0
0
2B
B
!
!
!
k+J
!
k+n
"
:! k+n
f. cl
k+J
k+2
J
!k
+
University
Of
Sheffield.
Automatic
Control &
Systems
Engineering.
Predictive Functional Control (PFC)
For Use in Autopilot Design
By: Hyreil Anuar Kasdirin
MSc in Control Systems: August 2006
Supervisor: Dr J A Rossiter
A dissertation submitted in partial fulfilment of the requirements for the degree of
Master of Science in Control Systems
ABSTRACT
This paper discusses the design and implementation of PFC as a controller for an
autopilot missile. Two linear continuous time missile models which are derived from
nonlinear model produced by Horton [13] and another from the basic Ballistic Missile
[10] are used for the prediction models. The PFC algorithm is developed based on the
models. The PFC algorithm developed is seems intuitive and computationally simple as
the missile need not to be very complicated as it will explode as it reaches the target.
Furthermore, the analysis and issues of the implementation relating linear discrete-time
stable and unstable process are being discussed. In addition, PFC tuning parameters play
an important part of the autopilot controller. Thus, the result indicated that the PFC
control law is built better when choosing the dynamic pole of the missile mode to be the
desired time constant, 'P and small coincidence horizon n1 as performing in single
coincidence point. The implementation of PFC on the missiles-scenario is also developed
for Model Missile I and 2. As a result, some positive results is illustrated and discussed
as the both missile followed its reference trajectory during simulation using MATLAB
7.0.
Keywords: Predictive Functional Control (PFC), autopilot design, discrete-time state-
space models
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
EXECUTIVE SUMMARY
Introduction/Background
This paper will concentrate on the basic handling of PFC as a controller for autopilot
missile. The formulation of PFC will be developed as well as how PFC handles with
stable and unstable process. A particular type of missile and onboard guidance system has
not been specified in the reference missile model. However, the paper briefly explained
some missile models and its missile guidance control. Thus, the result and
implementation of the PFC algorithm as a controller of autopilot missile will be further
discussed later on.
Aims and Objectives
The main aim of this project is to understand the design the PFC as a controller for an
autopilot missile. The principle objectives of this project are to understand and develop
the basic of PFC methodology. It also intended to analyze issues relating stable and
unstable process on PFC algorithm. Lastly, the objectives of this paper are to analyze the
results from the design and tests using PFC algorithm on missile models using MATLAB
7. 0 environment.
Achievements
At the end of this project the implementation of PFC is discussed. This paper will
concern on linear continuous-time stable and unstable missile model that only concern on
the aerodynamic control of the missiles. Furthermore, this paper shows that PFC is
developed successfully with one coincidence point, n 1• The tuning parameters of the PFC
algorithm; which are the desired time constant, 'I' and the control horizon, n are being
manipulated in order to give the best performance of the controller. As a result, it is
clearly illustrates that it is best to chose the dynamic pole of the missile mode to be the
desired time constant, 'I' and small coincidence horizon n 1 as performing in single
coincidence point. This section also has successfully implementing PFC on unstable
system with a single unstable pole. However, the system need to perform pre-stabilise
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
2
before implementing Closed-Loop Prediction (CLP) on the modified PFC control law to
give stable and good response and performance. The implementation of PFC on the
missile-scenario is also illustrated by showing the result of discrete linear missile model 1
and 2. The section 4.5 clearly showing that PFC perform well as a missile autopilot
controller as it gave good trajectory as the reference trajectory moved and tum in
different direction. As a result, some problems faced and suggestions are also being
discussed.
Conclusion I Recommendations
At the end of this project it was recommended to improve the missile model as the
prediction model and develop the scenario of missile-target to see whether the PFC could
be used as a controller for the autopilot missile. In all, based on the achievement/result
the implementation of PFC algorithm seems intuitive and computationally simple. This
truly important as the missile controller need not to be very complicated as it will explode
as it reaches the target.
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
3
ACKNOWLEDGEMENT
I thank God Al-Mighty for giving me the opportunity to achieve my goal.
Here, I would like to express my gratitude and acknowledgement to Dr J. A. Rossiter for
his guidance and continuous encouragement as well as motivation throughout the project.
I also would like to express my gratitude to my lovely wife and daughter for their
understanding, patience and moral support.
I would like to thank to the Kolej Universiti Teknikal Kebangsaan Malaysia (KUTKM),
for their financial support of this course. Last but not least, many thank to my family,
friends and all of the lecturers of The Department of Automatic Control and Systems, for
continuous support and encouragement to my study at the University of Sheffield, United
Kingdom.
HYREIL ANUAR KASDIRIN
August 2006
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
4
CONTENTS
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ..
1
Executive Summary
... .. .. .. ........... .. ... ... . .. .. .......................... . ..... ........ .. ....... .. ..
2
List of Figures
.. . .. ... .... .... ..... .... .. .. .... ..... . .... . .... .... ... ... ... .. . ..... ......... ........
8
List of Tables
.. ... .. . .... .............. ... ..... ... ................ .. . .... ... . ........... .. .........
9
Abstract
SECTION 1: BACKGROUND PROJECT
1.1 An Autopilot Missile
... .... . .... ... . .. ... .. . .. ... . ... .. . ... ......... ... ...... .. . ... . ... .. ... .. .. 10
1.2 Predictive Functional Control
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.3 Thesis Scenario: Missile Guidance Control
1.4 Aim and Objectives
1.5 Chapter Outline
10
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
.. . . . . . .. .. . . .. . . . . . . . . . . . ... . .. . . .. . . . . . . .. . . .. .. . . .. . . . . .. . . . . .. . . . .. . . ... 12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
SECTION 2: PREDICTIVE FUNCTIONAL CONTROL (PFC)
14
2.1 Predictive Control and MPC Algorithm ...... ..... .... ... .... .. .. . .. . ... . ..... ... ... ...... .. . . ... 14
2.2 Optimal Control of MPC
.. .. .. .. . .. . . .. ... ...... ... .... ..... ... .. . ... . ... .. .... ... . . ....... 14
2.3 Formulation of PFC Algorithm
. .. ... .. . .... .... .. ..... ... ... . ..... ... ... . .... .. .. ... . .. .. . ... .. .. 15
2.3.1 Models
.... .. ... .. . .... .. .... . . .... .. .. . ... ... ... ... .... .. .. . . . . ... . .... .. .. ... .... 15
2.3.2 State-space Model
.. .. .. ... .. ... . ..... .. .. ... .... ... .. ... . ... . . ... ... . .... .. .. . .... .. 15
2.3.3 Reference trajectory formulation
.... .. ... . .. . .. ... ... . .... . .... .. .. .. .. ... . .. . ... 17
. . . .. . . . .. . . . . . . .. . . .. . . . . . . . .. . . .. . . . . . . . . . . . . . . . .. . . . . .. 17
2.3.4 The coincidence points
2.3.5 Paramerisation of the d.o.f/future control trajectory
. . .. . . .. . . . . . .. . . .. . . . . .. . . . . .. 18
2.3.6 Computational of control law
.. . . . . . .. . . . . .. . .. . .. . . . . . . .. .. . . .. . . . . .. . . . . . . .. . .. . . . . .. 18
2.3.7 Tuning Parameters of PFC
. . ... . ... .. . ... . .... . ... ... . .. ... . .. ... .. . ... ... . .. . ..... . 19
2.4 PFC for unstable Process
2.4.1 Introduction
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
. .. ... ... .. ... ..... .. .. . .. ... . ......... .... ... ......... .... .... .. ... .... .... 19
2.4.2 Unstable Open-Loop Problems
2.4.3 Predictive Stabilisation
. . .. . . . . . . .. . . . . . . .. .. .. . . . . . . . . .. . . . . . . . .. . .. . .. 20
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.4 Closed-Loop Paradigm (CLP) Concepts
2.4.5 CLP Predictions
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
. . . . . . . .. . . . . . . . ... . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . ... . . . . . . . . . . . .. 22
5
2.5 Summary
·· ··· ··· ·· ·· ··········· ··· · ·· ········ · · ·· ·· ··· ····· ·· ·· ·· ·· ··· ······················ · 24
25
SECTION 3: AIR MISSILE & AUTOPILOT CONTROL
......... ... ......... . . .... ... .. .... ......... . ............................... 26
3 .1 Missile Airframe
26
3.1 .1 The Basics of Flight Dynamics
3.1.2 Equations of Motion
.. .... ... .. ... .... .................... ... .. ........ ..... .. .... ... ... 27
3.1.3 Assumptions Made
.. .. ... ............................... ............... . .. ............ 28
3 .2 Missile Modelling Design 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 28
. . . . .. . .. . . . . . . .. . . .. . . . . .. . . . . .. . . .. .. .. . . . .. .. . . . . . . . .. .. 28
3 .2.1 The Dynamic Equation
3.2.2 The Linearised Continuous-Time Models
3.3 Missile Modelling Design 2: Unstable Case
.... ..................................... 29
..... ....... ............... ... ......... . .......... 31
3.3.1 The Dynamic Equation
···· ··· ··· ···· · · ················ ··· ···· ············· ······· 32
.. ... ... .... ... .. . .... ... .. .... ... .. ..... 33
3.3.2 The Linearised Continuous-Time Model
.. . ... .. .. .. .. . ... . ... ......... .......... 34
3 .4 Missile Control System and Autopilot Design
3.5 Summary
.......... .. .. . .... .. .. ... . . .. ... .. .... .. .. .... . ... .. .. .. .. . .. ........................ 35
SECTION 4: IMPLEMENTATION OF PFC
4.1 Implementation of PFC algorithm
4.1.1 Prediction Model
36
................ ........ .. .. .... . .. ..... ... .... .. ........ .. 36
.. ... . .......... ......... . ....... .. .. ........... ... ... .............. 36
4.1 .2 Model 1: The Horton Missile Model
....... . ....... . ..... ... ... ....................... 36
4.1.3 Model 2: The Ballistic Missile Model .......... ... ..................................... 37
4.1.4 Set-up PFC Algorithm for Discrete Linear Model Missile 1
4.2 The Tuning Parameters and Its relation with Model Dynamic
4.2.1 Comparison of different desired time constant
....................... 38
. . .... .... ...... .... .... .... .... 40
................................ 40
4.2.1.1 Control and Output Response/or 'P = 0.3
................................ 41
4.2.1.2 Control and Output Response for IJf = 0.6
............... .. .. .. ....... .. .. 42
4.2.1.3 Control and Output Response for IJf = 0.9
............ ..... . .............. 43
4.2.1.4 Discussion
....... ... . . ... .. .. .. .. ....... ..... ... .................. ... . ........ 43
4.2.2 Relation \f with poles of the discrete linear model, G(z)
4.2.3 Discussion
....................... 44
........ .... .. .. .. ... .. .... ...... .... ...... . ................. ................ 46
4.3 The Implementation of PFC for unstable process
4.3.1 The Open Loop Response of Model 2
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
....... . ... . ... .... . .. ...... ...... ....... 47
....... . .. ... ... ... .. ... .. ............ ... 47
6
4.3 .2 The Observability of the Example 2
..... . . .... . .... .. ... .... .. . ..... .... ... .. ..... ... 48
4.3.3 Solving the Prediction Mismatch by Pre-stabilised Prediction
4.3.4 Discussion
.... .. ........ 49
.. .. .. .. .. . .... .. ....... .. ........ . ... .. ... ...... .. .. . ... .. . ................. 50
4.4 Implementation PFC as Missile Autopilot Control
.... ... ... .. .. . .. .... .. ... . ... . ... .. . .. .. 51
4.4. l Implementation PFC on Missile Model 1
..... .................. . ... .... ... .. ... .. 51
4. 4.1.1 PFC as an Autopilot Control of the Missile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 52
4.4.1.2 Missile Constraint
... .... ... ... .. .... .. .................... .. .. ... .. ..... .. .. 53
4.4.2 Discussion
··· ····· ················ ·· ········ ··· ·· ···· · ······················· · ········· 54
4.5 Implementation PFC on Missile Model 2: An Unstable Process Example
. . . . . . . . . . . ... 54
4.5. 1 Model Scenario . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 55
4.5.1.1 PFC as an Autopilot Control ofthe Missile 2
4.5.2 Discussion
4.6 Summary
.. .. .. .... .... .. .. ..... 55
.......... . ........ .... . ...... .. . .. ... . ... . .. ....... .. .. .. .. ....... .. .. .. . ... 57
. .. ....... .. . .... . ..... ... . ... . .... .. .. ...... .. .. .. .. . ... . .. ... ... .. ........... .. ..... 59
SECTION 5: CONCLUSION AND RECOMMENDATIONS
59
REFERENCES
61
APPENDIXES
63
APPENDIX A: International Standard Atmosphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 63
APPENDIX B: Missile Model 1 and 2
... .... . .... ......... ..... . ... ..... ......... .. .... 64
APPENDIX C: PFC Algorithm for nl = 2
....... . .... .. . ..... .... .. . . ............... ....... 67
APPENDIX D: PFC Algorithm for nl = 3
.. . .. ... ........................... . .... ... ....... 68
APPENDIX E: PFC Algorithm for nl = 6
...... .... ..... .. ... .. .. ...... .. . ... ... .. .... ... .. 69
APPENDIX F: PFC Algorithm for nl = 10 .. .. .... . ..... .. .. ......... ... ..... . ... ..... ... ... 70
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
7
LIST OF FIGURES
Figure 1.1 : Pitch, Yaw and Roll of a Missile
Figure 2.1 Basic structure of MPC
.............. . ... . ... ............... .. .. .. ... .... 10
.. ... ........ .. ....... .. .. .. .. .. . ... .. ...... ..... ... .. . .. .. .. ..... .. 14
Figure 3 .1: The Airframe of Basic Missile
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 26
Figure 3.2: Ballistic missile axis system.
. ... .............................. ...... .. .. .... ........... 31
Figure 3.3: Sketch of an aerodynamic missile and axis system
Figure 3.4: Basic Layer of Automatic Missile Control
.... ... .. .. .... . .... . ... ... ..... 34
.. .. ..................... ......... .. ..... 34
Figure 4.1: The Control and Output Response for Example 1
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
Figure 4.2: The Closed-loop Step Response for 'I'= 0.3
.. ............ ........... . ............... 41
Figure 4.3: The Closed-loop Step Response for 'I'= 0.6
..... ... ..... .. ............... . .......... 42
Figure 4.4: The Closed-loop Step Response for 'I'= 0.9
........... .. ... ...... .. ... .............. 43
Figure 4.5: The Controller Response of Example 1 for 'I' = 0.94
.... . ..... ......... .. ... .. ...... 46
Figure 4.6: The Root Locus and Bode Diagram for Missile Model 2
...... ... .. ... 47
Figure 4.7: The Controller Response of Example 1for'I'=0.94
... ..... .. .. .... ... ... .......... 50
Figure 4.8: Sketch of an aerodynamic missile of Missile Model 1
... ... .. .. .. .. . ... ... .. .... ..... 52
Figure 4.9: The Missile Controller Response of Missile Model 1
.. ... ............ .. ......... .... 53
Figure 4.10: Sketch of an aerodynamic missile of Missile Model 2
.... . .... .. ... ... . ..... 55
Figure 4.11: The Missile Controller Response of Missile Model 2
....................... 56
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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LIST OF TABLES
Table 3 .1: The Motion Variables of the Basic Missile Model
................................ 27
Table 3.2: The Characteristics and the coefficients of Horton Missile Model 1
.. ..... .... ... 29
Table 3.3: Characteristics and coefficient of basic ballistic Missile Model 2
....... .. ..... 32
Table 4.1: Parameters of Example 1 .. ............... ....... .. .... ..... ... .............................. 38
Table 4.2: Parameters Used for Tuning Parameters of PFC Algorithm of Example 1
40
Table 4.3: Parameters Used for Developed of PFC control law for Missile Model 1
52
Table 4.4: Parameters Used for Developed of PFC control law for Missile Model 2
56
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
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SECTION 1: BACKGROUND PROJECT
1.1 An Autopilot Missile
A missile is a projectile, often self propelled, which delivers a payload to a target.
Missiles have various launch platforms, ranges, targets, payloads and are typically guided
either remotely or automatically. Basically, there are 6 degrees of freedom which need to
be controlled. These angles are conventionally called yaw, pitch and roll (refer Figure
1.1) which all of these measures by the on-board sensors. In this paper, there are two
models that have been selected and they are only two degree of freedom.
Figure 1.1: Pitch, Yaw and Roll of a Missile
In this paper, the main point of research is the autopilot system of the missile. An
autopilot is a mechanical, electrical or hydraulic system used to guide a vehicle without
assistance from human being. Missile autopilot system is one of the examples of an
autopilot system. The main purpose of the autopilot missile is to enable the missile to
accomplish their mission autonomously, without any (or with minimal) input from the
missile operator. It includes the missile automated take-off or target hit, depends on its
mission.
Nowadays, modern autopilot missiles use computer software to control the missile. It
uses the missile state information provided by the on-board sensors to drive the control
surface actuators (servos) and gives feedback to the missile control surface. The autopilot
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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missile is also designed to hold certain parameters constant, for example its direction,
speed, altitude etc.
The control and guidance system is the brain of the autopilot missile. The on-board
control circuit need not to be too complicated or big as its mission is only target-hit
approach. Numerous missile designers and researcher have attempted to build effective
control of an autopilot missile such as using H-infinity controller [14] and Adaptive
controller [15]. In addition, this paper is continuing the work of Ben [16] and Nick [17] for
implementing Model Predictive Control as a controller for autopilot missile. They have
successfully developed missile model and simulation of target-hit of the model. However,
they only concentrate briefly on the control section using Model Predictive Control (MPC).
Hence, this paper will only concern of the missile autopilot controller using Predictive
Functional Control (PFC).
1.2 Predictive Functional Control (PFC)
Predictive Functional Control (PFC), which developed by Richalet [1] is one of Model
Predictive Control (MPC) techniques that have been developed as a powerful algorithm
for controlling process plants. In this paper, the focus is on the implementation of the
predictive functional control (PFC) on the missile dynamic models. PFC is based on the
same approach with all MPC strategies i.e., prediction of the future outputs, and
calculation of the manipulated variables for an optimal control. Therefore, PFC is also
based on the same principles which are using an internal model, specification of a
reference trajectory and determination of the control law.
1.3 Thesis Scenario: PFC as a controller of Autopilot Missile
This paper will concentrate on the basic handling of PFC as a controller for autopilot
missile. The formulation of PFC will be developed as well as how PFC handles with
stable and unstable process. A particular type of missile and onboard guidance system has
not been specified in the reference missile model. However, the paper briefly explained
some missile models and its missile guidance control. Thus, the result and
ACS6200: Predictive Functional Control (PFC)
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implementation of the PFC algorithm as a controller of autopilot missile will be further
discussed later on.
1.4 Aims and Objectives
Therefore, regarding the thesis scenario of this project, the aim of this project is as
follows:
•
Understand the design the PFC as a controller for an autopilot missile.
In addition, the objectives of this project are to:
•
Understand and develop the basic of PFC methodology.
•
Analyze issues relating stable and unstable process on PFC algorithm.
•
Analyze the results from the design and tests using PFC algorithm on missile
models using MATLAB 7.0 environment.
1.5 Chapter Outline
Based on the aim and objectives given above, this project is investigating the design an
autopilot control system of a guided missile using PFC controller.
The second section will be looked on some basic theory of PFC as a controller so that it
can be formulated and implemented in the following sections. However, at first, this
section will describe the basic of MPC algorithm and its optimal control that has become
an efficient control strategy for a large number of processes [2). After that, it followed by
the introduction of PFC algorithm and the formulation of its control law. This section
also will be discussing the way PFC handle the unstable process by pre-stabilise the
unstable plants to implement the stabilizing linear PFC control formulation.
After that, section three will be introducing some basic models of missile and its autopilot
control. The main content of the section is to show two basic missile models that will be
used for PFC implementation as its controller in the following section. The autopilot
control for the first model is the control of the deflection of control surfaces, whereas the
second model; ballistic missiles are controlled by deflecting the thrust vector [10].
ACS6200: Predictive Functional Control (PFC)
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Lastly, section four will illustrate the implementation of PFC. At first, this section will be
fully explained how PFC algorithm could work in given stable and unstable model using
the models that have been introduced in the section three. The second sub-section will
then further the implementation of PFC whether PFC could work as a controller on fast
process such as autopilot missile.
The last section of this project tries to conclude the project as it developed from previous
section. The summary of the project will be discussed and some recommendation will be
noted for further analysis and research.
ACS6200: Predictive Functional Control (PFC)
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SECTION 2: PREDICTIVE FUNCTIONAL CONTROL (PFC)
This section is basically will be discussing the theoretical part of PFC as a controller so that it
can be formulated and implemented in the following sections. However, at first, this section will
describe the basic of MPC algorithm and its optimal control that has become an efficient control
strategy for a large number of processes [2). After that, it followed by the introduction of PFC
algorithm and the formulation of its control law.
2.1 Predictive Control and MPC Algorithm
Predictive Control or so called Model Predictive Control (MPC) has being developed for
more than 20 years, both in industry and academic community. The principles of MPC
are universal, and can be found in many textbooks [3], [4], [5). A wide range of MPC
algorithm was developed, where it developed to suit given types of industrial application.
Some of the most popular MPC algorithms as follow;
a.
Generalised Predictive Control (GPC), [2]
b.
Dynamic Matrix Control (DMC) from Culter and Ramaker,
c.
Model Algorithmic Control (MAC) from Richalet,
d.
Predictive Functional Control (PFC), developed by Richalet and
ADERSA. [1], [3]
2.2 Optimal Control of MPC
Reference
trajectory
Past inputs
and outputs
.
Future
inputs
Predicted
outputs
Model
+
-
Optimiser
Cost
function
'"
Future errors
Constraints
Figure 2.1 Basic structure of MPC
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The basic structure to implement MPC strategy shows in figure 2.1. There are two
important blocks as illustrates above, which are model and optimiser. Hence, it is true
that the essence of MPC as well as PFC is to optimise, over the manipulability inputs,
predicting of the process behaviour given [6]. A model is used to predict the future plant
outputs, based on past and current values and on the proposed optimal future control
actions. These actions are calculated by the optimizer taking into account the cost
function as well as the constraints.
2.3 Formulation of PFC Algorithm
2.3.1 Models
The model is the essential element of an MPC controller [6] and hence, also for
PFC controller. PFC can use many forms of model i.e.; internal model (IM),
including state space, transfer function, Finite Impulse Response (FIR), fuzzy
rules, and etc. The use of IM is important in PFC to capture the process dynamics
of the system and also continue to calculate the PFC control law later on.
2.3.2 State-space Model
Hence, for this section, the model will be developed in state-space form. The
discussion of PFC and other MPC algorithms in state-space form has several
advantages, including easy generalisation to multivariable systems, ease of
analysis of closed-loop properties, and online computation [6].
Given the general state space model, of the form:
! k+1
=A!k + B?!.k
ｾ ｫ＠ =C!k +D?!.k
(2.1)
Prediction with a strictly proper system (D = [O])
ACS6200: Predictive Functional Control (PFC)
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:!k+2
= A:!k+i + ｂｾｫＫＱ＠
(2.2)
Y
- k+2 -Cx
- -k+2
Substituting Equation 2.1 into 2.2;
:!k+3
= A 2[A:!* + ｂｾｊＫ＠
ｾｫＫＳ＠
=C:!k+3
ａｂｾｫＫｩ＠
+ ｂｾｫＫＲ＠
This process is simply an iteration of a one step ahead prediction, repeated
substitutions result in the prediction matrices, P and H.
:!k+n
ｾｫＫｮ＠
= A n:!k + An-IBｾｫ＠ + An-2BｾｫＫｉ＠
= c[An :!k + An-1 ｂｾＪ＠ + An-1 ｂｾｫＫＱ＠
+ •• ·+ B ｾｫＫｮ
+ ... + ｂｾｫＫｮＭＱ｝＠
Ｍ Ｑ＠
(2.3)
State Prediction Equation
!
k+l
!
!
k+2
!
k+n
k+J
B
0
0
AB
2
A B
B
AB
0
B
f..xx
... 0
... 0
... 0
... 0
... B
Hxx
Y. k
Y. k+l
Y. k+2
Y.
k+n-1
ｾ＠
k-1
(2.4)
Output Prediction Equation
.l::'. k+l
.l::'. k+2
.l::'. k+J
}:'. k+n
=
0
CB
CAB
=
H
0
0
CB
0
0
0
0
Y. k
Y. k +l
Y. k +2
CB
Y. k+n-1
ｾ＠
k-1
(2.5)
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From Equation 2.4 and 2.5 above, it shows the model used is a linear one that
represent by;
ｾｫ＠
= Pxx!k + HxlJ.k-1
ｾｫ＠
=P!_k +HlJ.k-1
where !le is the state model, ｾ＠
(2.6)
is the input model, »is the measured output model,
P= Hxx, P and Hare respectively, matrices or vectors of the right dimension.
Below, the algorithm for PFC is outlined as found in [5]. There is an element of
derivation here, however its inclusion is necessary as it helps to explain the main
concepts behind PFC. This intuitive approach is one of the key selling points.
2.3.3 Reference trajectory formulation
PFC formulates the reference trajectory by placing the desired closed-loop
dynamic into the reference trajectory. Given the actual set point is r, the loop set
point w is a first order lag [3].
(2.7)
where Yk is the most recent measured output and '¥ ( 0 < '¥ < 1 ) is scalar and a
tuning parameter setting the desired closed-loop pole.
The predictive essence of control strategy is completely included in Equation 2. 7
above. Indeed, the aim is to track the set point trajectory following the reference
desired closed-loop behaviour.
2.3.4 The coincidence points
The control law is determined by using the d.o.f to enforce equality of the
predictions and the reference trajectory at a number of points, that is, by solving
the control moves such that:
ACS6200: Predictive Functional Control (PFC)
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Y
k+n
=W
k+n, n
(2.8)
= n1,n2, ... .
These equalities are called coincidence points. Typically, there only have one or
two coincidence points. However, in this paper will only concentrate with one
coincidence points only.
2.3.5 Paramerisation of the d.o.f/future control trajectory
The PFC takes the trajectory as the sum of a step change, a ramp, a parabola, etc.
The precise components to be included are selected to match the expected
characteristics in the set point.
2.3.6 Computational of the control law:
At a single coincidence points, and using equation 2. 7 and 2.8, the control law is
determined by;
(2.9)
Hence, substituting Equation 2.6 with 2.9;
(2.10)
By assuming g
k + ;
= g "' the control law can be computed by rewriting the
Equation 2.10 and obtain;
I
M. k
=-H [P !
M. k
=-K !
where; K
fJ
k
k
.
+ ( rk -
(rk -
+ fJ Tk
=-H
I
y,J IP 1 )
]
(2.11)
.
( P - IP 1 Yk)
= -HI (I - IP;)
ACS6200: Predictive Functional Control (PFC)
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Thus, it can easily express as a fixed linear feedback law in the form of prediction
algorithm. Hence conventional a posterior stability and sensitivity analysis can be
applied in straightforward manner.
2.3. 7 Tuning Parameters of PFC
The tuning parameters of PFC are generally the coincidence horizon, e.g. n1
= 1,
2, . . . and the desired time constant, \f. The typical procedure with one
coincidence point [3] would be as follows:
1. Choose the desired \f.
2. Do a search for n 1 = 1, ... large and find the associated control law for each
n1.
3. Select the n 1 which gives closed-loop dynamics closest to the chosen \f.
4. Simulate the proposed law. Otherwise, reselect \f and go to step 2.
Hence, the tuning reduces to a global search, but this requires only relatively
trivial computations and hence would be quite quick. With two coincidence
points, the global search would be more involved but should still be quick.
2.4 PFC for unstable Process
2.4.1 Introduction
PFC algorithm is defined in the previous sub-section is basically open-loop
process control applications. In the contrary, in industry applications, open-loop
unstable processes do also occur. Yet, these systems are difficult to control.
Hence, systematic control design tools are needed to handle complex instability
without a high on-line computational load. ADERSA have successfully applied
PFC on many unstable systems [3]. This section will discuss the theoretical tools
to pre-stabilise the unstable plants to implement the stabilizing linear PFC control
formulation.
ACS6200: Predictive Functional Control (PFC)
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2.4.2 Unstable Open-Loop Problems
PFC, as well as other MPC algorithms is weak with both non-minimum phase
problems and some unstable process [3], or called prediction mismatch. If the
process open-loop is unstable, PFC and also MPC are ill-posed because prediction
cannot match desired behaviour of the process, i.e. diverging. Therefore,
divergence open-loop prediction is the main cause of the prediction mismatch.
Hence, there is a must to stabilise the prediction. There are two ways of prestabilise the predictions which are inserting a stabilising loop and another by
shaping the future inputs, algebraically so that the outputs are stable. However,
this section only focused on solving algebraically the unstable process as
discussed by [7] and [8].
2.4.3 Predictive Stabilisation
Removal of the prediction mismatch is essential for PFC to work. Hence, the
model needs to have prediction stabilisation. One method is by cancelling the
unstable modes and starts working with stable predictions. This means that PFC
control law process must be modified. Therefore, in solving this problems PFC
will lead to good closed-loop performance if the predictions used are a good
match to the consequent closed-loop behaviour.
The illustration below shows the state space method of predictions to cancel the
unstable modes [3]. Let a state-space matrix have some unstable eigenvalues.
Decompose the system into stable and unstable modes using eigenvalue
decomposition;
A
[ w. W.J diag[A. A.J
｛ｾ＠
(2.12)
where subscript s is used for stable and u for unstable. Clearly if a state lies solely
in the stable manifold of A, then it must satisfy:
ACS6200: Predictive Functional Control (PFC)
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(2.13)
Given this, the predicted state evolution would follow
ｾ＠
=A
k+ ilk
i
i
ｾ＠
k
= Ws As Vs
T
(2.14)
ｾ＠ k
then the mode 2 predictions are given by
(2.15)
!:!. k+ nc+ ;=O
By assuming there is no disturbances or measurement noise, then continue with
predict by iterating the model (Equation 2.1 ). So we get
ｾ＠
］ａｾ＠
k + 111c
k
ｾ＠
k + 211c = A
ｾ＠
k
ｾ＠
;
k + ilk = A ＠ｾ
k
+ B!:!. k!k
+ 111c + B!:!. k + 111c
+ A ;.1 B
!:!. klk
= A
2
ｾ＠
k +AB !:!. k!k
+ ..... + B
!:!. k + i -1/k
which can be summarized as
,i; "
v.
ｾ＠
+ B!:!. k+ 11k
A; ,i; , + [ A;.JB + A;.iB ... B ) [
!:!.!:!.k k!k
+llk
!:!. k +
l
i -1/ k
M
(2.16)
or in common form
(2.17)
To ､ｲｩｶ･ｾＫ＠
k
+ ilk = 0, from equation 2.13, we know that
VuT X k+ ilk=
0,
(2.18)
Hence, by substituting equation 2.18 with 2.17 and 2.12, we get;
ACS6200: Predictive Functional Control (PFC)
For Use in Autopilot Design
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T
.
Vu [A'-! k + M !! k
.
]
= 0,
T
So as clearly for Wu Au' Vu = 0. Therefore;
T
i
i
T
Vu [A -! k + Ws As Vs !! k +
i]
=0
(2.19)
The control law becomes,
!! k + ;
.J! k
= - [ Ws A/ V/ r 1 V/ Ai -! k +HP
=-K
-! k
+ HP
where K = [ Ws A/ Vs T
r 1 Vu
(2.20)
T Ai
and
p could be choose freely.
2.4.4 Closed-Loop Paradigm (CLP) Concepts
As mentioned above, PFC will lead to good closed-loop performance if the
predictions used are a good match to the consequent closed-loop behaviour.
Hence, the closed-loop paradigm (CLP) is introduced here. CLP, which was
originally proposed as part of an algorithm stable generalised predictive control
(SGPC, [9]) will be implemented in the modified PFC after the pre-stabilised
prediction.
2.4.5 CLP Predictions
Based on the prediction on the PFC algorithm, the equations within the prestabilised loop during prediction are;
-! k+ilk =A-! k+i-llk + B!! k +;;
!! k+i = Ｍｾ＠
k+ilk
+ c k+i;
(2.21)
By removing the dependent variable y_ k + ; one gets;
ACS6200: Predictive Functional Control (PFq
For Use in Autopilot Design
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:! k+ilk = /A - BK);! k+i-Ilk + B H,k +1,·
H. k+i ］Ｍｾ＠
(2.22)
c k+i;
k+ilk +
Hence, simulating these forward in time with 2
B
0
0
2B
B
!
!
!
k+J
!
k+n
"
:! k+n
f. cl
k+J
k+2
J
!k
+