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Automatic Flight Control System

Classical approach and modern control perspective

Said D. Jenie and Agus BudiyonoDepartment of Aeronautics and Astronautics, ITB

Jl. Ganesha 10Bandung 40132

IndonesiaPhone: +62-22-250-4529

Fax: +62-22-253-4164Email: [email protected]

Copyright c Bandung Institute of Technology

January 12, 2006


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2

Abstract

The document is used as a lecture note for the graduate course on flightcontrol system at the Malaysian Institute of Aviation Technology (MIAT),Kuala Lumpur, Malaysia. Some parts of the document represent an en-hanced version of the material given as an elective undergraduate courseand a graduate course in optimal control engineering in the Departmentof Aeronautics and Astronautics at ITB, Bandung, Indonesia. SinggihS Wibowo helped prepare the typesetting of the formulas and check theMATLAB programs. The constructive input and feedbacks from studentsare also gratefully acknowledged.


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Contents

I Classical Approach 9

1 Introduction 111.1 Types of Automatic Control System . . . . . . . . . . . . . . . . 11

1.1.1 AFCS as the trimmed flight holding system . . . . . . . . 111.1.2 AFCS as the stability augmentation system of the aircraft 131.1.3 AFCS as the command augmentation system of the aircraft 141.1.4 AFCS as the stability provider and command optimizer . 16

1.2 Elements of Automatic Flight Control System . . . . . . . . . . . 171.2.1 Front-end interface offlight control system . . . . . . . . 171.2.2 Back-end interface offlight control system . . . . . . . . . 191.2.3 Information processing system . . . . . . . . . . . . . . . 201.2.4 Control Mechanism System . . . . . . . . . . . . . . . . . 21

2 Autopilot System 272.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272.2 Working Principle of Autopilot System . . . . . . . . . . . . . . . 272.3 Longitudinal Autopilot . . . . . . . . . . . . . . . . . . . . . . . . 34

2.3.1 Pitch Attitude Hold System . . . . . . . . . . . . . . . . . 342.3.2 Speed Hold System . . . . . . . . . . . . . . . . . . . . . . 442.3.3 Altitude Hold System . . . . . . . . . . . . . . . . . . . . 53

2.4 Lateral-Directional Autopilot . . . . . . . . . . . . . . . . . . . . 692.4.1 Bank Angle Hold (Wing Leveler System) . . . . . . . . . 692.4.2 Heading Hold System . . . . . . . . . . . . . . . . . . . . 772.4.3 VOR-Hold System . . . . . . . . . . . . . . . . . . . . . . 84

3 Stability Augmentation System 93

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 933.2 Working Principle of Stability Augmentation System . . . . . . . 933.3 Longitudinal Stability Augmentation System . . . . . . . . . . . 94

3.3.1 Pitch Damper System . . . . . . . . . . . . . . . . . . . . 953.3.2 Phugoid Damp er . . . . . . . . . . . . . . . . . . . . . . . 99

3.4 Lateral-Directional Stability Augmentation System . . . . . . . . 1023.4.1 The Dutch-roll stability augmentation: Yaw Damper . . . 102

3


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4 CONTENTS

II Modern Approach 115

4 Introduction to optimal control 117

4.1 Some Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . 1174.2 Linear Systems. . . . . . . . . . . . . . . . . . . . . . . . . . . 119

4.2.1 Controllability . . . . . . . . . . . . . . . . . . . . . . . . 1204.2.2 Conventions for Derivatives . . . . . . . . . . . . . . . . . 1214.2.3 Function Minimization . . . . . . . . . . . . . . . . . . . . 123

4.3 Constrained Problems . . . . . . . . . . . . . . . . . . . . . . 1264.3.1 Elimination . . . . . . . . . . . . . . . . . . . . . . . . . 1274.3.2 Method of Lagrange . . . . . . . . . . . . . . . . . . . . . 127

4.4 Inequality Constraints . . . . . . . . . . . . . . . . . . . . . . . . 1294.5 Sensitivity of Cost to Constraint Variations . . . . . . . . . . . . 131

4.6 Dynamic Programming . . . . . . . . . . . . . . . . . . . . . . . . 132

5 Discrete time Optimal Control 137

5.1 Higher Dimension Control Problems . . . . . . . . . . . . . . . . 137

5.2 Discrete time optimal control problem . . . . . . . . . . . . . . . 142


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List of Figures

1.1 Autopilot Control System Diagram: Example of Pitch Channel . 121.2 Stability Augmentation System Diagram. Example of Pitch Chan-

nel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

1.3 Command Augmentation System Diagram. Example of PitchChannel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

1.4 The Stability Provider and Control Power Optimizer Control Sys-tem Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

1.5 The automatic and manual flight control loop . . . . . . . . . . . 181.6 The aicraft cockpit as the interface between controller and con-

trolled systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.7 Mechanical Control System: example of longitudinal channel . . 221.8 Hydraulically power assisted control system: example of longitu-

dinal channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231.9 Hydromechanical Control System . . . . . . . . . . . . . . . . . . 241.10 Electrohydromechanical Control System . . . . . . . . . . . . . . 25

1.11 Electrohydraulic Control System . . . . . . . . . . . . . . . . . . 26

2.1 SAS as an inner loop of aircraft autopilot system . . . . . . . . . 282.2 CN235-100 Autopilot control system APS-65 . . . . . . . . . . . 292.3 CN235-100 Autopilot control system APS-65 . . . . . . . . . . . 312.4 The location of auto pilot system components in the standard

control diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342.5 Functional diagram of pitch attitude hold system . . . . . . . . . 352.6 Pitch attitude hold system . . . . . . . . . . . . . . . . . . . . . . 362.7 N250-100 aircraft prototype 2 Krincing Wesimanufactured by

the Indonesian Aerospace Inc. . . . . . . . . . . . . . . . . . . . . 382.8 Root locus of Pitch attitude hold system for N250-100 PA-2 aircraft 412.9 Root locus of Pitch attitude hold system for N250-100 PA-2

aircraft enlarged to show the phugoid mode . . . . . . . . . . . 422.10 Time response of (t) due to step ref = 5

o with and withoutpitch attitude hold system. . . . . . . . . . . . . . . . . . . . . . 43

2.11 Time response for u(t) and (t) for Kct = 8.9695 . . . . . . . . 442.12 Speed hold system functional diagram . . . . . . . . . . . . . . . 452.13 Mathematical diagram of speed hold system . . . . . . . . . . . . 462.14 Root locus diagram for the speed hold system of N250-100 PA-2 50

5


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6 LIST OF FIGURES

2.15 Root locus diagram for the speed hold system of N250-100 PA-2zoomed around the pitch oscillation and phugoid modes . . . . 51

2.16 Time response ofu(t) to maintain uref = 1 with and without thespeed hold system, K = 17.3838 . . . . . . . . . . . . . . . . . . 53

2.17 Time response of(t) and (t) to maintain uref = 1, with thespeed hold gain K = 17.3838 . . . . . . . . . . . . . . . . . . . . 54

2.18 Tail air brake (Fokker F-100/70) . . . . . . . . . . . . . . . . . . 552.19 Outer-wing air brake (Airbus A-320/319/321) . . . . . . . . . . . 562.20 Functional diagram of altitude hold system . . . . . . . . . . . . 572.21 Kinematic diagram of aircraft rate of climb . . . . . . . . . . . . 582.22 Mathematical diagram of the altitude hold system . . . . . . . . 592.23 Root locus for the altitude hold system for the N250-100 with

h e feedback with gain Kct < 0 . . . . . . . . . . . . . . . . 602.24 Root locus for the altitude hold system for the N250-100 with

h e feedback with gain Kct < 0 zoomed to show thephugoid mode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2.25 Altitude hold system with an attitude hold system as the innerloop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

2.26 Altitude hold system with inner loop having gain Kct . . . . . . 632.27 Root locus diagram of the outer loop of the altitude hold system

for N250-100 PA-2 aircraft . . . . . . . . . . . . . . . . . . . . . . 642.28 Root locus diagram of the outer loop of the altitude hold system

for N250-100 PA-2 aircraftzommed to show the phugoid mode . 652.29 Time response of h(t) for an input of href = 1 from a system

with and without altitude hold for N250-100 . . . . . . . . . . . . 662.30 Altitude hold system with forward acceleration ax as the inner

loop feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 672.31 Altitude hold system with inner loop using forward accelerationfeedback ax . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2.32 Root locus diagram of inner control loop u e for N250-100altitude hold system . . . . . . . . . . . . . . . . . . . . . . . . . 69

2.33 Root locus diagram of inner control loop u e for N250-100altitude hold systemzoomed around the phugoid mode . . . . . 70

2.34 Root locus diagram of outer control loop u e for N250-100altitude hold system . . . . . . . . . . . . . . . . . . . . . . . . . 71

2.35 Time response ofh(t) for N250-100 altitude hold system with theinner loop of ax e feedback . . . . . . . . . . . . . . . . . . . 72

2.36 Functional diagram of Bank angle hold system . . . . . . . . . . 732.37 Mathematical diagram of bank hold system . . . . . . . . . . . . 74

2.38 Root locus diagram of the bank hold system of N250-100 aircraftfor a number ofs values . . . . . . . . . . . . . . . . . . . . . . 76

2.39 Time response of(t) for a = 1/s = 10, 5, 2 due to an impulsefunction input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

2.40 Time response of (t) for a = 1/s = 10, 5, 2 due to a stepfunction input . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

2.41 Forces equilibrium during the turn maneuver . . . . . . . . . . . 79


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LIST OF FIGURES 7

2.42 Functional diagram of heading hold system . . . . . . . . . . . . 802.43 Mathematical diagram of heading hold system . . . . . . . . . . 812.44 Heading hold system with bank angle hold as an inner loop . . . 822.45 Root locus diagram of the heading hold system of N250-100 air-

craft for a number of a = 1/s values . . . . . . . . . . . . . . . . 832.46 Time response of (t) and (t) of the heading hold system of

N250-100 aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 842.47 Effect of wind to the aircraft flight path . . . . . . . . . . . . . . 852.48 VOR guidance path geometry . . . . . . . . . . . . . . . . . . . . 862.49 The functional diagram of VOR-hold guidance-control system . . 872.50 The mathematical diagram of the VOR-hold guidance-control

system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 882.51 Navigation and guidance system: VOR hold . . . . . . . . . . . . 892.52 Root locus diagram for the VOR offset (s) with respect to the

bearing ref of the N250-100 aircraft . . . . . . . . . . . . . . . 902.53 Time response of(t) of the VOR Hold system of the N250-100

aircraft with gains of: k = 1.1152 (guidance loop), k

i =4.5943 (outer loop control) and ki = 8.9344 (inner loop control) . 91

3.1 The functional diagram of the pitch damper system . . . . . . . 943.2 The mathematical diagram of the pitch damper system . . . . . 953.3 Root locus of the inner control loop: pitch damper system of

N250-100 PA-2 at cruise condition with V = 250 KIAS and h =150000 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

3.4 Root locus of the inner control loop: pitch damper system ofN250-100 PA-2 at cruise condition with V = 250 KIAS and h =

15000

0enlarged around the phugoid mode . . . . . . . . . . . . 983.5 Time response of(t) and q(t) with and without pitch damper of

N250-100 aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . . 993.6 The mathematical diagram of the pitch attitude hold with the

pitch damper as the inner loop . . . . . . . . . . . . . . . . . . . 1003.7 Root locus of outer control loop: phugoid damper of N250-100

PA2 at cruise condition . . . . . . . . . . . . . . . . . . . . . . . 1013.8 Root locus of outer control loop: phugoid damper of N250-100

PA2 at cruise conditionenlarged around phugoid mode . . . . . 1023.9 Time response of(t) of the phugoid damper with K = 3.7979

and Kq = 0.2432 . . . . . . . . . . . . . . . . . . . . . . . . . . 1043.10 The functional diagram of the yaw damper system with r r

feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

3.11 The mathematical diagram of the yaw damper system with r r feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

3.12 Root locus of the yaw damper system with r r feedback ofN250-100 aircraft . . . . . . . . . . . . . . . . . . . . . . . . . . 108

3.13 Root locus of the yaw damper system with r r feedback ofN250-100 aircraftenlarged around the dutch-roll mode . . . . . . 109

3.14 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110


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8 LIST OF FIGURES

3.15 Time response of(t),(t) and (t) of the N250-100 PA2 equippedwith the yaw damper . . . . . . . . . . . . . . . . . . . . . . . . . 113

3.16 Phase portrait of (t) vs p(t) for three cases: no yaw damper,with yaw damper and yaw damper+wash-out . . . . . . . . . . . 114

4.1 Minimum of a cost function J(x) at a stationary point . . . . . . 1234.2 Minimum ofJ(x) at the boundary . . . . . . . . . . . . . . . . . 1244.3 Minimum ofJ(x) at a corner . . . . . . . . . . . . . . . . . . . . 1244.4 Constrained minimum vs unconstrained minimum . . . . . . . . 1284.5 Inequality constraints . . . . . . . . . . . . . . . . . . . . . . . . 1304.6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1314.7 Principle of Optimality . . . . . . . . . . . . . . . . . . . . . . . . 1324.8 Multistage decision example . . . . . . . . . . . . . . . . . . . . . 1334.9 Flight Planning application . . . . . . . . . . . . . . . . . . . . . 134

4.10 One dimensional scalar state problem . . . . . . . . . . . . . . . 135

5.1 Discrete grid ofx and t . . . . . . . . . . . . . . . . . . . . . . . 1385.2 Two-dimensional array of states . . . . . . . . . . . . . . . . . . . 1405.3 Double interpolation of Cost function . . . . . . . . . . . . . . . 1415.4 Interpolation in the grid of u0s . . . . . . . . . . . . . . . . . . . 143


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Part I

Classical Approach

9


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Chapter 1

Introduction

The content of the book is centered on the discussion of automatic controlwithout the pilot in the control loop. The role of the automatic flight controlsystem is to support the pilot in performing his job as the steerer and the missionexecutor so as to reduce the pilots load. As a steerer, the pilot has two maintasks which are controlling and guiding the aircraft. The control is performed inorder to maintain the aircraft at the desired equilibrium flight attitude, whereasthe guidance is the task to bring the aircraft from a certain equilibrium state toanother equilibrium state. As a mission executor, the pilots task is dependenton the type of the aircrafts mission. For instance, for a fighter aircraft, thepilot has the tasks to find and to investigate the target and then to aim andshoot it after the process of search and investigation has been determined.

The Automatic Flight Control System (AFCS) is designed to ease the pi-lot in performing the above tasks in such a way that his physical as well aspsychological load can be reduced.

1.1 Types of Automatic Control System

Based on the tasks level of difficulty, the Automatic Flight Control System canbe categorized into four different types:

1. AFCS as the trimmed flight holding system

2. AFCS as the stability augmentation system

3. AFCS as the command augmentation system4. AFCS as the stability maker and command optimization

1.1.1 AFCS as the trimmed flight holding system

This type of automatic control system is commonly known as the auto-pilotwhich is the abreviation of the automatic-pilot (AP). The AP system has the

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1.1. TYPES OF AUTOMATIC CONTROL SYSTEM 13

1.1.2 AFCS as the stability augmentation system of the

aircraftThe type of automatic flight control system that adds stability to the aircraftis usually called the Stability Augmentation System or SAS. This type of auto-matic flight control system improves the stability of an aircraft at certain flightconfigurations and conditions within the flight envelope. For conventional air-crafts, the stability augmentation will be needed during the flight at low speedand low altitude for instance during landing or approach. The control opti-mization of typical aircrafts are conducted only at a certain flight configurationsuch as cruise configuration. This makes the aircraft stability at other flightconfigurations namely approach, landing or other special configurations tend todeteriorate. The stability augmentation is therefore necessary for those configu-rations. The stability augmentation can be achieved by increasing the damping

ratio of the existing aerodynamic damping ratio through the application of feed-back control system.

Sensor

SAS

Basic Control System

Motion sensor Aircraft Motion

SASLoop

Figure 1.2: Stability Augmentation System Diagram. Example of Pitch Channel

The types of the SAS, for example, are:

Damping ratio augmentation system such as pitch damper, yaw damperand roll damper

Dynamic compensation supplier system such as wing leveler and turn co-


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14 CHAPTER 1. INTRODUCTION

ordinator

Fig.1.2 shows the example of pitch damper SAS implemented for the air-craft pitch longitudinal channel. Note that the SAS signal comes out of FCC(Flight Control Computer) which processes the logic of stability augmentation.This signal directly enters the ECU and is combined with the command signalfrom the pilot to move the elevator. The SAS signal coupled with the aircraftdynamics will improve the pitch damping ratio such that the aircraft dynamicsis more stable.

It can be observed that the SAS is different from AP in some ways. In the APsystem, the output from the AP computer is used to move the control stick inlieu of the pilot input. In the SAS system, the output from the SAS computer isentered into the ECU which forms a closed loop in order to increase the stabilityof the aircraft. Thus, the SAS system will keep working even though there is an

input command from the pilot. Whereas in the AP system, the AP loop will beautomatically off once the pilot moves the controller stick. The SAS thereforehas higher level of authority compared to AP system. The SAS system is calledthe flight control system with partial authority. To deactivate the SAS, the pilotcan turn the SAS switch.

1.1.3 AFCS as the command augmentation system of theaircraft

The automatic flight control system with this type of task is commonly calledCommand Augmentation Systems or CAS. This system adds the power of inputcommand of the pilot by processing the input command and the generatedaircraft motion to optimize the input command to the aerodynamic controlsurface. The working principle of this system can be likened to that of thepower steering of the ground vehicle.

The types of this CAS system, for example, are:

Pitch oriented flight control system

Roll oriented flight control system

Yaw oriented flight control system

Fig. 1.3 shows the example of CAS system for the pitch oriented columnsteering. From the diagram, it is evident that input command from the con-troller stick is processed to follow the pilots desired pitch angle. The commandsignal is then corrected by the actual pitch angle and is processed and sentthrough the ECU to the ECHP (electronically controlled hydraulically pow-ered) actuator. It can also be inferred from the diagram that the pilots desiredpitch angle can effectively be achieved by an appropriate control stick input.

In summary, the differentiating features of the CAS, SAS and AP are:

CAS reacts due to control stick input and results in the desired orienta-tion. If the pilot does not move the control stick, CAS is not operating


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1.1. TYPES OF AUTOMATIC CONTROL SYSTEM 15

Sensor

CAS



CASLoop

Comparator

comm

Figure 1.3: Command Augmentation System Diagram. Example of Pitch Chan-nel

SAS reacts continuously regardless the motion of the controller stick.When the SAS is operating, the stability of the aircraft is increased.

AP is operating in the condition that the control stick is not moved.When AP is working, the aicraft will maintain its trimmed condition asdesired by the pilot.

From the perspective of control circuit, the following feature distinguishesCAS, SAS and AP:

CAS the circuit is closed through the Flight Control Computer at thejunction point of controller stick and output from the aircraft motionsensor

SAS the circuit is closed through the Flight Control Computer directlyto the actuator

AP the circuit is closed by the motion of the AP electromotor at thecontroller stick

From the above comparison, it is clear that the CAS system has a higherauthority than the SAS does because it always reacts to follow the desiredattitude set by the pilot.


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Sensor

ACT



Actuator

Computer

FCCSensor

Stick Motion

sensor

Figure 1.4: The Stability Provider and Control Power Optimizer Control SystemDiagram

1.1.4 AFCS as the stability provider and command opti-mizer

This kind of automatic flight control is commonly called Super AugmentationFlight Control System. This control system is typically used to create an arti-ficial stability for the class of aircrafts which are statically unstable. The samesystem is simultaneously used to optimize the control power through the appli-cation of control laws provided by the Flight Control Computer. The domainof this type of control system is electronic and hydraulic. The super augmentedcontrol system is often called electro (opto) hydraulic flight control system orFly by Wire (Light) flight control system which is abbreviated as FbW or FbL.Fig. 1.4 shows the example of the FbW control system diagram which is usedby the F-16 fighters of the Indonesian Air Forces.

From the diagram, it can be observed that the function of the FCC consists

of combination of three activities, namely:

Superaugmentation: providing an artificial stability and optimizing thecontrol power of the aircraft. This subsystem works continuously and cannot be overridden by the pilot.

Autopilot: taking over some parts of pilots routine tasks. If this systemis in operation, the pilot does not need to hold the control stick. This


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1.2. ELEMENTS OF AUTOMATIC FLIGHT CONTROL SYSTEM 17

subsystem can be overruled by the pilot by moving the controller stick

Control Law: governing the optimization of the aircraft motion outputfollowing the desired mission. Using the control law, the aircraft motionis optimized in such a way that it will not always be the same as themotion due solely to the input command from the pilot. The control lawis also used for protection or limit of the state variables of the aircraft atthe a certain flight configuration.

The artificial stability provided by the superaugmentation system is thelongitudinal and/or lateral directional static stability. This static stability iscreated through the continuos feedback process in such a way that the trimmedcondition of the aircraft is maintained.

It can be observed that this type offl

ight control system is a control systemwith a full authority. Without the availability of this type of system, the aircraftsthat are statically unstable will not be able to fly. Thus, the characteristic ofthis control system is flight critical.

1.2 Elements of Automatic Flight Control Sys-tem

The basic elements in the control information loop are plant (the controlledsystem) and the controller. For an aircraft, the controlled system consists ofcontrol apparatus, control surface and the aircraft. Whereas the controller part

consists of three subsystems namely aicraft motion sensor, aircraft motion infor-mation processor and control command generator. Fig.1.5 shows the functionaldiagram of the manual and automatic control system for an aircraft. The di-agram shows that the primary interface between the controlled and controllersystems can be divided into two parts: front-end interface which is the aircraftsensory system and back-end interface which is the control command generator.

1.2.1 Front-end interface offlight control system

The front-end interface of the flight control system is the part where the motionof the aircraft is observed, recorded and displayed in the presentation map or

transmitted in the form of information signal to the aicraft motion informationprocessor system. In the study of control engineering, the ability to observe theaircraft motion and to reconstruct it as motion information signal is called ob-servability. In the aircraft flight control system, the front-end interface elementis located inside the cockpit (flight deck) consisting of front window, side win-dow, instrumentation display dashboard, pilot vision or aircraft motion sensorssuch as pitot-static system, vane and inertial platform to name a few.


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FCC

Information

Processor

RLG,Accel,PitotActuator

Command generatorFluids dynamics/

Inertial sensors

Control apparatus Control surfaces Aircraft

AUTOMATIC CONTROLLER

Pilots brainPilots hands, feetvoices

Cockpit display,window view, pilotseyes and ears

Command generator Information processor Motion sensor

HUMAN CONTROLLER

Aircraft

motion

Controller

motion

Figure 1.5: The automatic and manual flight control loop


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Front-end

interface

Back-endinterface

Figure 1.6: The aicraft cockpit as the interface between controller and controlledsystems

1.2.2 Back-end interface offl

ight control system

In the back-end interface, the control command generator is the part of thesystem through which pilot command is inputted namely the controller manip-ulator (stick, steering wheel or pedal) and propulsion controller manipulator(power lever and condition lever). In the study of control engineering, the abil-ity to move the control manipulator is called the controllability. In the aircraftflight control system, the back interface element is located inside the cockpit(flight deck) and is composed of among others the controller manipulator andautopilot actuator. See Fig.1.6.

It can be concluded that the cockpit or the flight deck is the flight front-end

and is the most important part of the aircraft since it is in this part that the twomain elements of the control system namely controlled and controller parts areconnected. Particularly for unmanned aerial vehicles such as drones, missiles orsatellites, the two interfaces are combined into the ground control station sys-tem. The ground control station is typically composed of display system whichcomparable to conventional aircraft cockpit and control and navigation interfacewhere ground pilot can enter control commands or navigation waypoints.


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1.2.3 Information processing system

Another very critical element of the controller is the information processor sys-tem. In the manual controller (human controller), this system is representedby the pilots brain supported by the basic information processing computersto speed up the decision making process. In the automatic controller, the infor-mation processing element is represented by a Flight Control Computer (FCC).The FCC works continuously in real time depending on the authority level ofthe implemented automatic control system. The software inside the FCC thatmanipulates the input of the FCC to be converted to the desired control signalby the control system is called the control law. The control law can be in theform of simple instructions which typically used by the autopilot. Some of the

examples of control law are:

Constant Gain. The FCC repre-sents a multiplier or an amplifieronly. KK

FCC

K - constant

outin

out iny Ku=

Variable Gain (Gain scheduling).The FCC works as a modulatedtransformer. The gain can be reg-ulated as a function of one or moreparameters.

FCC

outin

( )out i iny f u=

( )iK f =

K

Robust Gain. The FCC gives the

value of the gain K in the admis-sible control region. The robustproperty means that control lawwill still work when there existssome level of uncertainty or param-eter changes in the plant.

FCC

out

iny

Ry


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Optimal Gain. The FCC calcu-lates the optimal gain based ona certain predetermined optimiza-tion criterion such as minimumcontrol power, minimum time andminimum fuel.

FCC

outu x

yoptimal

gain

Reconstruction

Adaptive Gain. The FCC deter-mines the varying gain that adjuststo the most suitable model of a cer-tain configuration.

Other than the above control laws, there are many other approaches that aregetting more applications in the automatic flight control design, namely: neuralnetworks, fuzzy logic, H2 and H control, and passitivity-based control.

1.2.4 Control Mechanism System

The control mechanism is the element of control system as a whole that is alsoimportant. This system allows the transmission of the control command to theaircraft control surfaces. Based on the physical domain of the control commandtransmission, the control mechanism is categorized into the following types:

1. Mechanical Control System

2. Hydro-mechanical Assisted Control System

3. Hydro-mechanical Powered Control System

4. Electro hydro-mechanical Control System

5. Electrohydraulic Control System (Fly-By-Wire)

6. Opto Hydraulic Control System (Fly-By-Light)

The following figures show the illustration of the physical diagram of eachcontrol mechanism type.

Mechanical Control SystemA mechanical control system is composed of physical object components withtranslation and rotation mechanical domain. Fig.1.7 shows an example of amechanical control system in the longitudinal channel. The general componentsof this type of control system are among others:

push-pull rod


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mounting controllerstick

push-pull rod

campulley

spring/damper

spring/damper

cable

pulley

pulley

cam

mounting

tensionregulator

elevator

Figure 1.7: Mechanical Control System: example of longitudinal channel

pulleys for rotational transmission

cable tension regulator, springs

pulley roller cables

component mountings

mechanical damper

The dynamic of mechanical control system is influenced by their physicaldynamic properties such as mass, inertia, damping, friction and stiffness. Tosome extend, these physical dynamic properties reduce the performance of thecontrol system as a control command transmission from the pilot to the aircraft.On the other hand, the greatest advantage of the mechanical control system isthe fact that the pilot through the mechanical control linkage can retain thefeel of a direct connection with the aircraft that he is controlling. Another im-portant property of the mechanical control system is the ability to imitate themotion transmission from the controller stick to the control surface by the con-trol surface motion back to the controller stick. Hence, the mechanical controlsystem is referred to as a reversible control system.

Hydro-mechanical Assisted Control System

A hydromechanically assisted control system is a control system with mechan-ical domain where some parts of the subsystem are strengthened by hydraulicactuators. See Fig.1.8. The typical components of this control system include:

mechanical components (as in mechanical control system)


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mounting

controllerstick

push-pull rod

campulley

spring/damper

spring/damper

cable

pulley

pulley

cam

mounting

tension

regulator

elevator

hydraulicactuator

hydraulicactuator

hydraulicactuator

Figure 1.8: Hydraulically power assisted control system: example of longitudinalchannel

hydraulic actuators mounted on a number of locations to strenghten me-chanical parts such as push-pull rod, cam and elevator

Note that the hydraulic actuators are employed to assist the performance ofthe existing mechanical components. These actuators do not directly connectthe pilot and the aircraft. Thus if these actuators fail to work or are out of or-der, the pilot can still control the aircraft through the mechanical linkage whichmaintains a direct connection between the pilot and the aircraft. This type ofcontrol system is commonly called Power Assisted Flight Control System. Simi-lar to a fully mechanical control system, it is also reversible meaning the motiontransmission from the controller stick to the control surface can be reversed.

Hydro-mechanical Powered Control System

The hydromechanically powered control system consists of components withthe domain of mechanical and hydraulic. In this control system, the hydraulic

component directly connects the pilot and the aircraft. Consequently, the pilotdoes not have the feel of directly controling the aircraft. The detachment of thepilot from the aircraft increases the risk of aircraft operation. To overcome thisproblem, an artificial feel system is introduced to provide the feel to pilot thathe is directly controlling the aircraft. As a result, the integration of the pilotand the aircraft can be maintained.

The primary components of this type of control system are:


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controllerstick

cam

mounting

tensionregulator

elevator

primary

hydraulic

actuator

hydraulicactuator

artificialfeel system

hydraulicpower supply

Figure 1.9: Hydromechanical Control System

mechanical components

hydraulic actuators

artificial feel system

See Fig.1.9. From direct observation, it is clear that the system is irre-versible. The motion from the controller stick to the control surface can not bereturned by the elevator motion to the controller stick.

Electro hydro-mechanical Control System

The electrohydromechanical control system consists of components with thedomain of electrical, mechanical and hydraulic. The automatic control systemsas discussed earlier are of this class of control system if the basic control systemdomain is hydromechanical. A specific feature of this control system is theexistence of the closed-loop between the control system and the aircraft throughthe aircraft motion sensor as shown in Fig.1.10. The electronic circuit closes

the control loop through the aircraft motion sensor, control stick motion sensorand information processing computer for control strategy. Due to the hydraulicactuator that directly links the controller stick to the control surface, an artificialfeel system is introduced to provide the feel to pilot that he is directly controllingthe aircraft. As a result, the system is irreversible. In the longitudinal channel,for instance, the motion from the controller stick to the control surface can notbe returned by the elevator motion to the controller stick.


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controllerstick

cam

mounting

tensionregulator

elevator

primaryhydraulicactuator

hydraulicactuator

artificialfeel system

Sensor


FCC

electrical

commandsignal

electricalsignal

control sensor

Figure 1.10: Electrohydromechanical Control System

Electrohydraulic Control System (Fly-By-Wire)

The electrohydraulic control system consists of components with the domainof electrical and hydraulic. In this control system, all mechanical componentsare eliminated. The basic philosophy behind this type of control system is thereduction of weight and space while simplifying the installation mechanism.Fig.1.11 shows the schematic diagram of the electrohydraulic control system,popularly known as Fly by Wire control system. A distinct feature of thiscontrol system is related to the closed-loop through the aircraft motion sensor,controller stick sensor and information processing computer. The Fly by Wirecontrol system is irreversible.

In this electrohydraulic control system, the primary components consist of

processor or intrument blocks namely computer block, sensor block, hydraulicactuator block and power supply block. Each of the block is connected by elec-trical transmission cables. The nature of the system consisting the blocks isthus modular. This modular feature allows for the flexible and simple installa-tion and repair which makes maintenance job easier and less time consuming.The feature also enables the design of control system with minimum weight andspace requirement.


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controller

stick

elevator

primaryhydraulicactuator

artificial

feel system


FCC

control sensor

informationprocessor

Figure 1.11: Electrohydraulic Control System

Opto Hydro-Mechanical (Fly-By-Light)

The optohydraulic control system consists of components with the domain ofoptronic (opto-electronic) and hydraulic. In this control system, all mechanical

components are eliminated. The working principle and basic philosophy behindthis type of control system is similar to that of Fly by Wire control system.The distinction is on the data transmission which is done through the opticaldomain.


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Chapter 2

Autopilot System

2.1 Introduction

An autopilot is flight condition/attitude holding system of an aircraft. In anumber of textbooks, this flight condition/attitude holding system is referredto as displacement autopilot due to its task to restore the state variable thatit maintains to the original desired value. The autopilot will work well forthe aircraft with good stability characteristic. For the aircraft with marginalstability, the autopilot can have a better performance if a stability augmentationsystem is installed onboard the aircraft as an inner loop of the autopilot systemas illustrated in Fig.2.1. Some types of autopilot system that is commonlyused for conventional transport aircraft are

1. Longitudinal mode

(a) Pitch attitude hold

(b) Speed/Mach number hold

(c) Altitude hold

(d) Glide-slope hold

2. Lateral Directional mode

(a) Heading hold

(b) Bank angle hold or wing leveler

(c) VOR-hold

(d) Turn coordinator

2.2 Working Principle of Autopilot System

Fig.2.2 shows an example of the AP system used for the Indonesian Aerospacetransport aircraft CN-235. The system is of type APS-65 produced by an avionic

27


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28 CHAPTER 2. AUTOPILOT SYSTEM

AP sensor

Aircraft motion

AP loop

SAS sensor

SAS loop

AFCScomputerAP-SAS

Control linkages Aircraft flightattitude/condition

Ref.attitude

Figure 2.1: SAS as an inner loop of aircraft autopilot system

manufacturer Collins. The figure specifically shows the panels inside the cockpitin conjunction with the operation of an AP system. The associated panels,indicators and control manipulator are listed in the following figures. Fig.2.3further shows the components of an AP system mounted on a number of parts ofthe aircraft. Refer to Fig.2.2 and 2.3 for the explanation of the working principleof autopilot system. Fig.2.4 shows the location of the autopilot components inthe standard control diagram.


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2.2. WORKING PRINCIPLE OF AUTOPILOT SYSTEM 29

(B)

(K)

(E) The status of autopilot andautomatic control system

Control Manipulator

Circuit breaker

(G) Attitude DirectionIndicator

(D) AP switch panel

Figure 2.2: CN235-100 Autopilot control system APS-65


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(B)

(E) The status of autopilot andautomatic control system

Circuit breaker

(D) AP switch panel

(K) Control Manipulator

(G) Attitude Direction


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(6) Aileronservo

(9) Elevatortrim servo

(7) Elevatorservo

(8) Rudder servo

(3) Normalaccelerometer

(2) Slide slipsensor

(4) Avionic rack

(1) Air datasensor (5) Autopilot

computer

Figure 2.3: CN235-100 Autopilot control system APS-65


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Basically, an AP system is easy to operate. In this case, a pilot can selectthe state variable to be maintained as the reference by bringing the aircraftto the trimmed condition at the value of the the selected reference variable.

The pilot executes this process of trimming by the use the control manipulator(K), wheels, control stick or pedal in consideration of the information from theattitude direction indicator instrument (G). If the trim condition is achieved,the pilot can then press the AP knob (D) so that the AP system is activated andworking to maintain the trim condition. The trim condition is maintained by theAP system through the AP actuators (68) and AP data processing system (5)which continuously keeping the aircraft trim condition at the value of the state


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variable selected as the reference. On condition that the AP system is working,the pilot can release his hand and feet from the control manipulators, wheels,and controller stick or p edal. All he needs to do is occasionaly checking upon the aircraft trim condition as displayed by the Attitude-Direction-Indicator(ADI). To disengage the AP system, the pilot would just need to grasp controllerstick/wheel/pedal and to displace it a little bit. In this case, the AP controlloop will be automatically disengaged and the pilot regains the full control ofthe aircraft. With this characteristic, the autopilot is often referred to as a lowauthority system.

6,7,8,9

1,2,3

CN2354,5+

-

Figure 2.4: The location of auto pilot system components in the standard controldiagram

2.3 Longitudinal Autopilot

The following subsections will be focused on the longitudinal autopilot designfor a transport aircraft. The discussion covers the design for pitch attitudehold, speed hold and altitude hold systems. The elaboration of the altitudehold system include a number of inner loop feedback designs including the pitchattitude hold, forward acceleration and compensator integration.

2.3.1 Pitch Attitude Hold System

The block diagram of a pitch altitude hold system is shown in Fig.2.5. Note

that the aircraft is modeled by block (1) with the output of motion states inthe longitudinal mode x = {u,, , q} where the pitch angle is sensed by thevertical gyro, represented by block (2). The output of the vertical gyro is the

signal m(t) which is then entered to and processed by the autopilot computer,block(3). The computer receives an input from the pilot in the form of the

desired value of pitch angle as the reference angle to be maintained, ref(t).

In the autopilot computer (APC) the signal ref(t) is compared to m(t) by


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2.3. LONGITUDINAL AUTOPILOT 35

using a comparator circuit and the result is amplified by an amplifier circuitwhich yields the output signal

. The signal

is in turn sent to the autopilot

servo motor that moves the steering stick and afterward conveys the signal tothe elevator through the control mechanism, block(4). The elevator deflectione changes the aircraft attitude and the new pitch angle (t) is sensed by thevertical gyro. The whole process in the autopilot loop is repeated until thedesired value = 0 is reached. This condition means that m(t) = ref(t) orthe aircraft pitch angle (t) is the same as the reference pitch angle ref desiredby the pilot.

( )u t

( )t

( )t

( )q t

( )t ( )m t

ref +

T

e

(2) Vertical gyro

(1) aircraft

(4) AP servo to control stickelevator

AP computer

Amplifier

(3) AP comparator

Figure 2.5: Functional diagram of pitch attitude hold system

Since the process from block(2) until block(4) is performed in the electricaldomain, it can be considered very fast. Thus the result is the maintenance ofthe aircraft pitch angle (t) at the value of the reference pitch angle ref desiredby the pilot.

The above functional diagram can be described in the following mathemat-ical diagram shown in Fig.2.6:

In the above model, the transfer functions of the pitch attitude hold closedloop system consists of:

1. Aircraft transfer function matrix, for T = 0,

GA/C(s)T=0

=Nlong(s)|r=0long(s)

=x(s)

e(s)(2.1)


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( )u s

( )s

( )s

( )q s

+

Vertical gyro

aircraft

( )ctG s

( )vgG s

( )T s

( )e s

[ ( )]

( )

long

long

N s

s

Servo AP

( )t ( )m t

ref

Figure 2.6: Pitch attitude hold system

2. Vertical gyro transfer function. The time response of the gyro in sensingthe pitch angle (t) is considered very fast, thus:

Gvg(s) = Svg constant (2.2)

3. Autopilot computer transfer function:

b(s) = brefbm(s) (2.3)4. Autopilot servo and control mechanism transfer function. The block is

modeled as a system with time response ofs representing the first orderlag factor in the transmission of signal to the aircraft elevator.

Gct (s) =Kct

s + 1/s(2.4)

The characteristic polynomial of the pitch attitude hold closed loop systemis given by:

cl(s) = long(s) + keNe(s)

s + 1/s(2.5a)


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where

ke SvgKct (2.5b)

Therefore, the characteristic equation of the closed loop system can be writ-ten as

1 + keNe(s)

long(s) [s + 1/s]= 0 (2.6)

The above equation can be written in a simpler form as

1 + keGol (s) = 0 (2.7)

where

Gol (s) =Ne(s)

long(s) [ss + 1](2.8a)

ke = SvgKcts (2.8b)

In the case of a conventional aircraft, GOL(s) will have two real zeros and fivepoles consisting of two pairs of complex poles of the aircraft and one real poleassociated with the autopilot control mechanism servo. Observing the closed

loop characteristic polynomial, Eq.2.7, note that Kct (servo gain) is the onlygain that can still be changed or varied. The vertical gyro sensitivity Svg andservo lag factor 1

sare typically of a constant value.

As a case study, the pitch attitude hold system of the N250-100 aircraftprototype 2 Krincing Wesi manufactured by the Indonesian Aerospace Inc. ispresented. See Fig.2.7.

The longitudinal mode transfer function of the N250-100 aircraft duringcruise at the altitude of h = 15000 ft and the speed of Vc = 250 kts is given asthe following (for the elevator control channel):

Nue(s) = Sue s

37.6823 1

s

4.2634+ 1

(2.9a)

Ne(s) = Se s

91.1168+ 1 s

0.009 +j0.0831+ 1 s

0.009j0.0831+ 1

Ne(s) = Se

s1.2860

+ 1 s

0.0222+ 1

Nqe(s) = sN

e

(s)

with the static sensitivity coefficient of


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Figure 2.7: N250-100 aircraft prototype 2 Krincing Wesi manufactured bythe Indonesian Aerospace Inc.

Sue = 3598.817 (2.9b)Se = 1.992

Se = 8.1443

and the characteristic polynomial of

long(s) = s4 + 3.3115s3 + 8.1448s2 + 0.1604s + 0.0554 (2.10a)

having the characteristic roots:

p1,2 = 1.6472j2.317 (2.10b)

p3,4 = 0.0085j0.0824

The N250-100 utilizes ring laser gyroscope (RLG) as its Inertial NavigationSystem (INS), therefore the time response can be considered very fast. Hence,the gain of the vertical gyro, Svg, in sensing the changes of the pitch angle (t),can be taken as:


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Svg = 1 (2.11)

Also, since the N250-100 employs Fly by Wire control system in its longi-tudinal and lateral/directional channels, the domain of the autopilot servo istherefore electromechanical with a relatively fast time response. As a result,the time response of the AP servo is taken as

s = 0.1 sec (2.12)

Subtituting Eqs.2.9b until 2.12, to Eqs.2.7 and 2.8a, the following closed loopcharacteristic polynomial of the pitch altitude hold system can be obtained:

1 + keGol (s) = 0

where

ke = SvgKctsSe (2.13a)

= 0.81443Kct

and

Gol (s) = s

1.286 + 1 s

0.0222 + 1 sp1 + 1 sp2 + 1 sp3 + 1 sp4 + 1 (ss + 1) (2.13b)The control algorithm is implemented in MATLAB codes. The programs

for generating root locus diagram and time response for the design of the pitchattitude hold is presented along with the commentary below.

% Pitch Attitude Hold -- Root locus drawing

close all

% Static sensitivity coefficient:

Sude = 3598.817;

Sade = -1.9920;

Stde = -8.1443;

% coefficients of characteristic polynomial:D_long = [1 3.3115 8.1448 0.1604 0.0554];

% finding the roots:

p = roots(D_long) ;

% time response of Auto Pilot (AP) servo

ts = 0.1;

% open loop zeros


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zero = [-1.286 -0.0222]

% open loop poles

pole = [p(1) p(2) p(3) p(4) -1/ts]

% open loop gain

Pipole = pole(1)*pole(2)*pole(3)*pole(4)*pole(5);

kol = real(Stde*Pipole/(zero(1)*zero(2)));

% Open loop transfer function

[N_ol,D_ol] = zp2tf(zero,pole,kol);

tf(N_ol,D_ol)

% Vertical gyro gain

Svg = 1;

%Drawing the root locus

sys_ol = zpk(zero,pole,kol);

figure(1);

set(1,Name,Open Loop Root Locus);rlocus(sys_ol);

grid on

For a positive value of gain Kct, the feedback gain Ke will be negative.Fig.2.8 shows the root locus diagram of the pitch attitude hold system of theN250-100 aircraft.

From the root locus, it is evident that for a negative gain the natural fre-quency of the pitch oscillation mode tends to increase and the damping ratio Dalso decreases. Nevertheless, it is still acceptable to the extent that the dampingratio is still greater than 0.35. It is interesting to note that, the phugoid modetends to be more damped and for an even higher value of negative Ke this

mode breaks up into two phugoid subsidence modes. This condition is advan-tageous for maintaining the pitch angle b to be at its reference value bref sincethe aircraft speed is also maintained due to the overdamped phugoid dampingratio.

As an example, the pitch oscillation damping ratio PO is taken to be 0.35.The corresponding closed loop characteristics roots are given as follows

= 0.35 Kct = 8.9695 (2.14)

the roots of the polynomial are:

psv = 11.2777 (2.15)

ppo1 = 0.5390 +j4.1205

ppo2 = 0.5390j4.1205

pph1 = 0.9305

pph2 = 0.0254


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-12 -10 -8 -6 -4 -2 0 2-8

-6

-4

-2

0

2

4

6

0.160.340.50.640.760.86

0.94

0.985

0.160.340.50.640.760.86

0.94

0.985

24681012

Root Locus

Real Axis

Ima

ginaryAxis

phugoidpitch oscillation

AP servo

Figure 2.8: Root locus of Pitch attitude hold system for N250-100 PA-2 aircraft

The following figure shows the bref step response for two kind of cases:Case 1 Without a feedback

Case 2 With a feedback of b e with Ke = 8.9695% Pitch attitude hold -- Closed loop time response analysis

Kct = 8.9695; Ktde = ts*Svg*Kct;

% Closed loop

N_cl = Ktde*N_ol; D_cl = D_ol + Ktde*N_ol;

[zero_cl,pole_cl,Kcl] = tf2zp(N_cl,D_cl); pole_cl; tf(N_cl,D_cl);sys_cl = zpk(zero_cl,pole_cl,Kcl);

% Time response for step input

t= 0:0.2:700; u = t*0; u(find(t>1)) = 5/57.3;

% Open loop time response

y_ol = lsim(sys_ol,u,t);

% Closed loop time response


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-0.35 -0.3 -0.25 -0.2 -0.15 -0.1 -0.05 0 0.05

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.160.340.50.640.760.86

0.94

0.985

0.160.340.50.640.760.86

0.94

0.985

0.050.10.150.20.250.35

Root Locus

Real Axis

ImaginaryAxis

phugoid

Figure 2.9: Root locus of Pitch attitude hold system for N250-100 PA-2 aircraftenlarged to show the phugoid mode

y_cl = lsim(sys_cl,u,t);

figure(Name,Time Respons Theta); subplot(211);

plot(t,u*57.3,t,y_ol*57.3,-m); ylabel(open loop teta); grid on

subplot(212); plot(t,u*57.3,t,y_cl*57.3,-r);axis([0 35 0 7]);

ylabel(closed loop teta);grid on

% u response

zero_u = [37.6823 -4.2634];

kcl_u = Kcl*Sude/Stde*zero(1)*zero(2)/(zero_u(1)*zero_u(2));

[Nu_cl,Du_cl] = zp2tf(zero_u,pole_cl,kcl_u);

[zero_ucl,pole_ucl,Kucl] = tf2zp(Nu_cl,Du_cl) ;

pole_ucl; tf(Nu_cl,Du_cl);sysu_cl = tf(Nu_cl,Du_cl);yu_cl = lsim(sysu_cl,u,t);

% alpha response

zero_a = [ -91.1168 -0.009+0.0831j -0.009-0.0831j]

kcl_a = -Kcl*Sade/Stde*zero(1)*zero(2)/real(zero_a(1)*zero_a(2)*zero_a(3))

[Na_cl,Da_cl] = zp2tf(zero_a,pole_cl,kcl_a);tf(Na_cl,Da_cl);

sysa_cl = tf(Na_cl,Da_cl);ya_cl = lsim(sysa_cl,u,t);


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figure(Name,Close Loop Time Respons)

subplot(211);plot(t,yu_cl,-g);ylabel(closed loop u [m/s]);

axis([0 250 -35 0]);grid on

subplot(212);plot(t,ya_cl*57.3,-b);grid on

axis([0 25 -1 4]);ylabel(closed loop \alpha [deg]);grid on

It is apparent that for the case with the pitch attitude hold, the value of can be quickly maintained in less than 15 time units, while when the pitchattitude hold is not present, Ke = 0, the time it takes to reach the steady level

of bref = 0 is more than 500 time unit or 35 times as along as that of the systemwith the pitch attitude hold. Also note that when the pitch attitude hold is off,there exists an offset angle of about 15o whereas with the pitch attitude holdthe offset angle is only approximately 0.25o. Fig.2.11 shows the time responseof velocity, u(t), and angle of attack, (t). The bleed-off speed to hold the pitch

angle = 5o is about 35 kts while the angle of attack increases by 0.5o.

0 100 200 300 400 500 600 700-100

0

100

200

openloopteta

0 5 10 15 20 25 30 350

2

4

6

closedloopteta

Figure 2.10: Time response of(t) due to step ref = 5o with and without pitchattitude hold system.

This pitch attitude hold system can further be improved by pitch oscillationstability augmentation system (SAS) which will be discussed in more detail in


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0 50 100 150 200 250

-30

-20

-10

0

closedloopu[m/s]

0 5 10 15 20 25-1

0

1

2

3

4

closedloo

p[

deg]

Figure 2.11: Time response for u(t) and (t) for Kct = 8.9695

the next chapter.

2.3.2 Speed Hold System

In the aircraft application, the speed hold system can be categorized into twotypes:

Speed Hold System: for a low speed aircraft (low subsonic)

Mach Hold System: for a high speed aircraft (high subsonic to transonic)

Only the speed hold system will be covered in further detail in this course.Refer to Fig.2.12. The figure illustrates the speed hold system that is typicallyused for an aircraft. The aircrafts airspeed is sensed by the pitot static (2)and the result is sent to the autopilot computer (3) to be compared with the

reference flight speed (the speed which will be maintained by this speed holdsystem). The speed difference bu will be sent by the AP computer to the enginecontrol system (power lever to propulsion control mechanism) (4). The result isa throttle deflection, th applied to the aircraft engine (5). The aircraft enginein turn changes the thrust of the aircraft by T. The aircraft (1) will react to thethrust input T and its velocity u(t) will change accordingly. The process thencontinues as before and finally stops when the aircraft velocity u(t) has reached


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the value of the reference speed

buref(t). In this condition the signal of speed

difference bu from the computer will be zero. When the autopilot is working,the pilot can release the power lever and let the engine control manipulatormoves by itself following the closed loop process of the speed hold autopilotsystem. Hence this type offlight speed holding system is commonly called anAuto Throttle system.

( )u t

( )t

( )t

( )q t

( )u t ( )mu t

refu u+

th T

e

(2) Pitot static: velocity sensor

(1) aircraft(5) engine andpropeller

(4) engine controlsystem

(3) AP computer

Figure 2.12: Speed hold system functional diagram

The following figure describes the mathematical diagram of the aircraft speedhold system. This speed hold system has the following transfer functions

1. Aircraft transfer function matrix, for e = 0.

GA/C(s)e=0

=Nlong(s)|r=0long(s)

=x(s)

T(s)(2.16)

2. Pitot static transfer function. In measuring the flight speed, the pitot ismodeled by first order lag system which is called pitot-lag

Gps(s) =1/ps

s + 1/ps(2.17)

3. Autopilot computer transfer function. The comparator computer is mod-eled by: bu(s) = burefbum(s) (2.18)


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( )u t( )t

( )t

( )q t

( )u t ( )mu t

refu u+

( )th

s

pitot static

aircraftenginepropulsion control

( )PC

G s ( )ENG

G s

( )PS

G s

( )T s

( )e

s

[ ( )]

( )

long

long

N s

s

Figure 2.13: Mathematical diagram of speed hold system

4. Propulsion control transfer function. For the current existing advancedpropulsion technology, the transfer function can be modeled by first orderlag as given by

Gpc (s) = Kpc1/pc

s + 1/pc

(2.19)

5. Engine (and propeller) transfer function. For the present technology, com-pared to the aircraft dynamics, the corresponding transfer function canalso be modeled as a first order lag :

Geng (s) = Ke1/e

s + 1/e(2.20)

The closed loop characteristic polynomial of the speed hold system can beexpressed as the following:

cl(s) = long(s) + kuTNuT(s)

[s + 1/ps] [s + 1/pc] [s + 1/e] (2.21a)

where

kuT KpcKe/ [pspce] (2.21b)


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Thus, the closed loop characteristic equation is given by:

1 + KuTGol (s) = 0 (2.22)

where

KuT = KpcKe (2.23a)

Gol (s) =NuT(s)

[pss + 1] [pcs + 1] [es + 1]long(s)(2.23b)

The speed hold system is usually used during the approach and landing inorder to reduce the work load of the pilot who has been primarily occupied bythe aircraft guidance task.

As a case study, the speed hold system of the N250-100 aircraft manufactured

by the Indonesian Aerospace during the landing confi

guration will be analyzed.In this configuration, the aircraft velocity is 125 kias at the altitude of h = 0from sea level. The transfer functions associated with this configuration aregiven by:

NuT(s) = SuT

s0.3519

+ 1 s

0.8248 +j0.809+ 1

s

0.8248j0.809+ 1

NT(s) = ST

s

0.0342 +j0.1992+ 1

s

0.0342j0.1992+ 1

NT(s) = ST s

0.9093+ 1 s

0.0706 1 (2.24a)

NqT(s) = sNT

(s)

with the static sensitivity coefficients as follows,

SuT = 25.9337 (2.24b)

ST = 0.1276

ST = 0.1912

and characteristic polynomial:

long(s) = s

4

0.0588 + s

3

0.02876 + s

2

0.03296 + s0.6164 + 1 (2.25a)

with the characteristic roots:

p1,2 = 1.0147 j0.8304 (2.25b)

p3,4 = 0.0076 j0.1848


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The aircraft velocity is sensed by the the pitot static system, with the transferfunction modeled by a first order lag having a time response of:

ps = 0.2 sec (2.26)

The N250-100 propulsion control system uses the Full Authority DigitalEngine Control (FADEC) technology and thus the time response is fast. Thevalue of the time response is assumed to be

pc = 0.1333 (2.27)

To generate thrust, the engine and propeller system of the N250-100 aircraftemploys advanced six bladed propeller yielding a relatively fast time response.The value of the time response is taken to be

e = 1 (2.28)

Substituting Eqs.2.24b2.28 to Eqs.2.222.23a, the following equation is ob-tained

1 + KuTGol (s) = 0 (2.29)

where

KuT = KpcKeSuT (2.30a)

Gol (s) =

s0.3519 + 1

s0.8348+j0.809 + 1

s0.8348j0.809 + 1

s5 + 1

s7.5 + 1

(s + 1)

sp1

+ 1

sp2

+ 1

sp3

+ 1

sp4

+ 1

(2.30b)

The following figure shows the root locus diagram of the speed hold system ofthe N250-100 aircraft for positive KuT gain. The root locus shown in the figureis varied as a function of the gain K = KPCKe. Notice that the pitch oscillationmode does not change at all since it is constrained by the two complex zero ofNuT(s) = 0. Nonetheless, the damping ratio of the phugoid mode increases upto its maximum value before moving to the unstable region.

The MATLAB code implementation for the root locus drawing of the speedhold system is presented below:

% Static sensitivity coefficient:

Sudt = 25.9337;

Sadt = -0.1276;

Stdt = 0.1912;

% coefficients of characteristic polynomial:


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D_long = [1/0.0588 1/0.02876 1/0.03296 1/0.6164 1];

% roots:

p = roots(D_long) ;

% time constants

tps = 0.2; tpc = 0.13333; te = 1;

% open loop zeros

zero = [-0.3519 -0.8348-0.809i -0.8348+0.809i]

% open loop poles

pole = [p(1) p(2) p(3) p(4) -1/tps -1/tpc -1/te]

% open loop gain

Pipole = pole(1)*pole(2)*pole(3)*pole(4)*pole(5)*pole(6)*pole(7);

kol = real(Pipole/(zero(1)*zero(2)*zero(3)));

% Open loop transfer function

[N_ol,D_ol] = zp2tf(zero,pole,kol);tf(N_ol,D_ol)

% Drawing the root locussys_ol = zpk(zero,pole,kol);

figure(1)

set(1,Name,Open Loop Root Locus);

rlocus(sys_ol);zoom(3); grid on

To achieve the desired most favorable performance, a point in the root locusassociated with the highest phugoid damping ratio is chosen. Refer to the abovefigure, this point is given by

max = 0.0898 K = 17.3838

and its corresponding roots of the polynomial characteristics are

psv1,2 = 6.5126 j0.2513 (2.31)

ppo1,2 = 0.8657 j0.9481

pph1,2 = 0.1729 j0.9474

psv3 = 0.4422

Using the above values of poles and gain K, the closed loop characteristicpolynomial of this speed hold system can be written as

cl (s, K) = s7 + 15.5445s6 + 79.3848s5 + 163.9051s4 (2.32)

+222.9647s3 + 185.1533s2 + 114.5985s + 28.7110

The MATLAB code implementation for the gain selection process and the as-sociated time response analysis is given below:

% Selecting the gain

[Ku_dt,pcl] = rlocfind(sys_ol);D_cl = poly(pcl);

D_ps = [1 5];N_cl = Ku_dt/5*conv(N_ol,D_ps);

% Preserve the dim of N_cl

N_cl = N_cl(2:length(N_cl));D_cl = D_ol + Ku_dt*N_ol;


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-10 -8 -6 -4 -2 0

-6

-4

-2

0

2

4

6 0.160.340.50.640.760.86

0.94

0.985

0.160.340.50.640.760.86

0.94

0.985

246810

Root Locus

Real Axis

ImaginaryAxis

prop control system

pitot static

engine and

propeller

pitch oscillation

phugoid

Figure 2.14: Root locus diagram for the speed hold system of N250-100 PA-2

[zero_cl,pcl,Kcl] = tf2zp(N_cl,D_cl) ;pcl

tf(N_cl,D_cl)

sys_cl = zpk(zero_cl,pcl,Kcl);

% Time response

t = 0:0.2:700;u = t*0;u(find(t>1)) = 1;

% open loop time response

y_ol = lsim(sys_ol,u,t);

% closed loop time response

V = 250*1.8*1000/(60*60);y_cl = lsim(sys_cl,u,t);

figure(Name,Time Respons)

subplot(211);plot(t,u,t,y_ol,-m);ylabel(open loop u);grid on

subplot(212);plot(t,u,t,y_cl,-r);axis([0 50 0 2]);ylabel(closed loop u);grid on

% pitch angle response

zero_theta = [-0.9093 0.0706 -5];

Pipcl = pcl(1)*pcl(2)*pcl(3)*pcl(4)*pcl(5)*pcl(6)*pcl(7);

kcl_theta = real(Stdt*Pipcl/(zero_theta(1)*zero_theta(2)*zero_theta(3)));

[Ntheta_cl,Dtheta_cl] = zp2tf(zero_theta,pcl,kcl_theta);


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-1.4 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

10.160.340.50.640.76

0.86

0.94

0.985

0.160.340.50.640.76

0.86

0.94

0.985

0.20.40.60.811.21.4

Root Locus

Real Axis

Ima

ginaryAxis

engine and

propeller

pitch oscillation phugoid

Figure 2.15: Root locus diagram for the speed hold system of N250-100 PA-2zoomed around the pitch oscillation and phugoid modes

tf(Ntheta_cl,Dtheta_cl)

systheta_cl = zpk(zero_theta,pcl,kcl_theta);

ytheta_cl = lsim(systheta_cl,u,t);

% angle of attack response

zero_a = [ -0.0342-0.1992j -0.0342+0.1992j -5];

kcl_a = real(Sadt*Pipcl/(zero_a(1)*zero_a(2)*zero_a(3)));

[Na_cl,Da_cl] = zp2tf(zero_a,pcl,kcl_a);

tf(Na_cl,Da_cl)

% Closed loop time responsesysa_cl = zpk(zero_a,pcl,kcl_a);

ya_cl = lsim(sysa_cl,u,t);

V = 250*1.8*1000/(60*60);

figure(Name,Close Loop Time Respons)

subplot(211);plot(t,ytheta_cl*57.3,-g);axis([0 50 -60 60]);

ylabel(closed loop \theta [deg]);grid on


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subplot(212);plot(t,ya_cl*57.3,-b);grid on

axis([0 50 -40 40]);ylabel(closed loop \alpha [deg]);grid on

The time response due to a step input, u(t), can be obtained through theinverse Laplace transformation of the closed loop transfer function as the fol-lowing

u (t) = 1 [Gcl(s)uref(s)] (2.33a)

where

Gcl(s) =K (s + 1/ps) N

T

(s)

cl(s, K)(2.33b)

The following figures show the comparison of the time response u(t) due toa unit step uref = 1, between the aircraft without speed hold system and onewith speed hold system operated with gain K. The effectivity of the speed holdsystem can be observed from the comparison figures. The speed hold systemallows the achievement of the steady state value of u(t) at 35 time units. Whilewithout the speed hold system, the steady state condition has not been reacheduntil 350 time units. An offset from the steady state value uref = 1 can becompensated through the use of a gain adjuster which is governmed by theAuto Pilot computer of the speed hold system.

The corresponding equations for (t) and (t) can be derived as follows:

(t) =1 K

(s + 1/pl) NT(s)

cl(s, K) uref (2.34) (t) = 1

"K (s + 1/pl) NT(s)

cl(s, K)uref

#The time responses of (t) and (t) are shown by the following figures. Inthe above analysis, the gain used for the variant in determining the root locusis K = KPCKe. Typically, for a throttle lever system, the gain KPC can bealtered. The only factor that can be varied therefore is the gain associated withengine and propeller, Ke.

It is evident from the figure that the faster the time response of the engineand propeller, the higher the phugoid damping ratio and the faster the timeresponse of the speed hold system.

For a number of jet transport aircrafts such as, among others, Fokker-100,Airbus A-320/330/340 and Lockheed Tristar L-1011, an in-flight air brake isprovided in the aircraft to govern the speed during the approach and landingphases. The following figure describes the speed brake system as the flight speedregulator of the Fokker F-100 and Airbus A 319/321.

Through the use of the airspeed brake, the aircraft flight speed can be con-trolled by modulating the brake relative to its open-close position.


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0 100 200 300 400 500 600 7000

0.5

1

1.5

2

openloopu

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

closedloopu

Figure 2.16: Time response of u(t) to maintain uref = 1 with and without thespeed hold system, K = 17.3838

2.3.3 Altitude Hold System

The altitude hold system is a standard system for medium and long range trans-port aircrafts. This system maintains a cruise altitude that has been selectedby the pilot. The system clearly reduces the pilots work load significantly.

The basic principle of the altitude hold system is the use of a signal pro-portional to the measured aircraft altitude as a feedback to the elevator in sucha way that the elevator motion enable the aircraft to maintain its prescribed

altitude. Fig.2.20 shows the functional diagram of the system. The flight alti-tude is measured by a pitot static system and the elevator is moved by the basiccontrol mechanism through a servo motor. At glance this system looks similarto the flight attitude holding system. The difference is on the fact that flightaltitude variable is not part of the aircraft state variables.

Since the flight altitude h(t) is not part of the motion state variable x, inthe mathematical analysis, firstly it has to be modeled. The variable h(t) is


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0 5 10 15 20 25 30 35 40 45 50-60

-40

-20

0

20

4060

closedloop

[deg]

0 5 10 15 20 25 30 35 40 45 50-40

-20

0

20

40

closedloop[de

g]

Figure 2.17: Time response of (t) and (t) to maintain uref = 1, with the

speed hold gain K

= 17.3838

categorized as the output variable y(t) of the aircraft.

From the flight performance analysis, the model of rate of climb can be givenas the following. See Fig.2.21.

dH

dt= Vss sin (2.35)

where, H the aircraft altitude with respect to sea levelVss the aircraft steady state velocityddt(.) rate of change with respect to time

If this steady state condition is perturbed by a small disturbance, then thefollowing relations apply:

H = Ho + h (2.36)

= o +


55/145


Air brake on the tail

Figure 2.18: Tail air brake (Fokker F-100/70)

The equation of the rate of change of flight altitude can then be obtainedas:

dh

dt= Vss (2.37)

In the non-dimensional form, it can be rewritten as:

1

Vss

dh

dt h = (2.38)

From the kinematic diagram given in Fig.2.21, it can be deduced that:

= (2.39)

therefore,

dh

dt= h =

Zt( ) d (2.40)


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Air brake on the outer wing

Figure 2.19: Outer-wing air brake (Airbus A-320/319/321)

In the Laplace domain, the flight altitude can then be obtained as:

h (s) =1

s[ (s) (s)] (2.41)

This equation for h(s) is called the output equation of the altitude holdsystem.

Using the available mathematical model of h(s), the mathematical diagramof the altitude hold system can then be illustrated as the following figure.

Based on Eqs.2.4 and 2.17, the transfer function of the control mecha-nism/servo and the pitot static can be given respectively as follows:

Gct (s) =Kct

s + 1/s(2.42)

Gps (s) =1/s

s + 1/ps

In this model, s and PS have been specified using the instrument data,while the gain Kct can be varied to satisfy the control criterion of the altitudehold system. Referring to Eq.2.41, the transfer function of the flight altitudeh(s) with respect to the elevator deflection e(s) can be obtained as follows:


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58/145


sx

sy

ssV

dV

dt

dH

dt

Figure 2.21: Kinematic diagram of aircraft rate of climb

Based on Fig.2.22, the closed loop transfer function h(s)href(s) can be written as

follows:

h (s)

href= Gcl (s) (2.46)

where

Gcl (s) =Ncl (s)

cl (s)(2.47a)

Ncl (s) = Gct (s) Ghe (s) = KctNhe (s)

s (s + 1/s)long (s)(2.47b)

cl (s) = 1 + (Kct/ps)Nhe (s)

s (s + 1/s) (s + 1/ps)

long (s)

(2.47c)

To better understand the work principle of the altitude hold system, a casestudy will be taken for the Indonesian Aerospace N250-100 PA2 Krincing Wesiat the cruise flight configuration with the speed of V = 250 kias and at thealtitude of h = 15000 ft. From Eqs.2.9b2.10a, the data of the altitude holdsystem of the N250-100 PA2 can be obtained as the following. For s = 0.1 andPS = 0.2, then the numerator polynomial for h due to e can be given as


59/145


( )u s( )s

( )s

( )q s

h+

pitot static

aircraft

( )CtG s

( )PSG s

( )T s

( )e s

[ ( )]

( )

long

long

N s

s

refh

mh

control mechanismand servo

1

s

( )h s h+

Figure 2.22: Mathematical diagram of the altitude hold system

Nhe (s) = She

s10.8454

1 s

10.8389+ 1

s0.0167

+ 1

(2.48a)

and the denumerator polynomial is:

h (s) = s(s +1

s)(s +

1

PS)long(s) (2.48b)

h (s) = s

s6 + 18.3115s5 + 107.8173s4

+287.9076s3 + 409.7023s2 + 8.8517s + 2.7708

(2.48c)

where

She = 0.1230 (2.48d)

The following figure illustrates the root locus of this altitude hold system. Itcan be observed that for a negative gain, Kct < 0, the system quickly leads toinstability through its phugoid mode. The damping ratio for the pitch oscillationmode is not substantially increased either. For the positive gain, Kct > 0, the

condition is worse since the integrator.

h directly moves to the right half plane ofthe root locus. Thus it can be concluded that the altitude hold system using theh e feedback fails to give a stable solution in maintaining the flight altitude.


60/145


Therefore a different strategy needs to be devised so that the instability of thephugoid mode can be delayed and the damping ratio of the pitch oscillation canbe increased.

-40 -30 -20 -10 0 10 20 30-30

-20

-10

0

10

20

30Root Locus

Real Axis

ImaginaryA

xis

pitch oscillation

pitot static

servo

Figure 2.23: Root locus for the altitude hold system for the N250-100 withh e feedback with gain Kct < 0

To achieve the above objective a number of methods by adding inner feed-back loop are in order, namely:

1. pitch attitude feedback,

2. forward acceleration feedback, ax

3. compensator addition, Gc(s)

Altitude hold system using an attitude hold as its inner loop

Since the first method has been discussed in Sec.2.3.1, the design of this pitchattitude hold system is readily implementable for the inner loop of the altitudehold system of the aircraft. See the mathematical diagram of the correspondingin Fig.2.25.


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-0.1 -0.05 0 0.05 0.1

-0.1

-0.05

0

0.05

0.1

Root Locus

Real Axis

ImaginaryAxis

phugoid

integrator

Figure 2.24: Root locus for the altitude hold system for the N250-100 withh e feedback with gain Kct < 0 zoomed to show the phugoid mode

Note that Sh is the conversion gain from signal bh to signal ref which isa constant. From the elaboration given in Sec.2.3.1, a good pitch attitude holdcontrol loop has been obtained with Kct = 8.9695. The corresponding rootsof the closed loop polynomial have been given in Eq.2.15. Thus the inner closedloop characteristics polynomial can be expressed as:

cli (s, K

ct) = s5 + 13.3115s4 + 41.2598s3 + 223.1724s2 + 186.8562s + 4.6024

(2.49)

By using this inner loop, the mathematical diagram of the altitude hold sys-tem can be simplified as illustrated in Fig.2.26. In the diagram, the numeratorpolynomial Nhe can be calculated as follows:

Nhe = K

ct

Ne N

e

(2.50)

Thus, the result is:

Nhe (s) = 1.5535s3 0.0159s2 + 182.6204s + 3.0580 (2.51)


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( )u s( )s

( )s

( )q s

h

+

( )Ct

G s

( )PS

G s

( )T s

( )e

s

[ ( )]

( )

long

long

N s

s

ref

h

mh 1

s

( )h s h+

vgS

hS

ref

m

Figure 2.25: Altitude hold system with an attitude hold system as the innerloop

From the above diagram, the equation for h(s) can be expressed as:

h (s) = Gclo (s) href (2.52a)

where

Gclo (s) Khps (s + 1/ps) N

he

(s)

pss (s + 1/ps)cli (s) + KhNhe

(s)(2.52b)

=Nhecloclo

In this case, the root locus equation for the Gclo(s) is clo(s) = 0 or

1 +

Khps

Nhe (s)

s (s + 1/ps)cli (s) = 0 (2.53)

Fig.2.27 shows the root locus of the altitude hold system using the attitudehold inner loop. From the root locus diagram, it is manifest that the dampingratio of the pitch oscillation mode increases for the low value of gain Kh whereasthe phugoid mode moves from the subsidence to a long period oscillation beforetending to unstable oscillatory condition.


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( )u s

( )s

( )s

( )q s

h+

( )PSG s

*

*

[ ( )]( , )

i

ct long

cl ct

K N s

s K

refh

mh 1

s

( )h s ( )h s+

hK

( )s( )s

Outer gain

Inner loop with gain *ctK

Figure 2.26: Altitude hold system with inner loop having gain Kct

If the gain is chosen such that:

Khps

= 1.7372 Kh = 0.17372 (2.54)

the following characteristic roots are obtained:

p1 = 11.2729 (2.55)

p2 = 5.0514

p3 = 0.0163

p4,5 = 0.5727 j4.1047

p6,7 = 0.4102 j0.4051

In this case, the outer closed loop gives the damping ratio of the pitchoscillation and phugoid modes as follows:

Pitch oscillation po = 0.098Phugoid ph = 0.356

If this damping can be accepted, then h(t) can be calculated by using

Eq.2.52a.Fig.2.29 illustrates the time response of h(t) for href = 1, with and without

the altitude hold system.

Case 3 For the case without altitude hold system, the step input href = 1 cannot be maintained. As a result, the aircraft will keep losing the altitude unlessthe pilot interferes with corrective actions


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-10 -5 0 5 10-10

-8

-6

-4

-2

0

2

4

6

8

10

Root Locus

Real Axis

ImaginaryAxis

pitch oscillation

phugoid

pitot

static

Figure 2.27: Root locus diagram of the outer loop of the altitude hold systemfor N250-100 PA-2 aircraft

Case 4 For the case with altitude hold system, where a pitch attitude hold sys-tem is employed as an inner loop, the value of href = 1 can be quickly main-tained in not more than 10 time unit. This system works perfectly and yieldsno offset at all. The altitude hold system employing an attitude hold system asthe innner loop demonstrates reliable performance. A better performance of thistype of system can be further achieved if a pitch stability augmentation systemis employed.

Altitude hold system using a forward acceleration feedback as its

inner loop

A second method that can be used for the inner loop of the altitude hold systemis the feedback of forward acceleration dudt . For illustration of the inner controlloop with ax feedback, see the following diagram given in Fig.2.30.

From the above diagram, the following expression for u(s) can be derived:


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-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-1.5

-1

-0.5

0

0.5

1

1.5

0.20.380.560.7

0.81

0.89

0.95

0.988

0.20.380.560.70.81

0.89

0.95

0.988

0.511.52

Root Locus

Real Axis

ImaginaryAxis

phugoid

integrator

Figure 2.28: Root locus diagram of the outer loop of the altitude hold systemfor N250-100 PA-2 aircraftzommed to show the phugoid mode

u (s) =

"Nue(s)Gct (s)

long(s) + sGct (s) SaccNue(s)

#aref (2.56)

Using the model of control mechanism and elevator servo as given in Eq.2.4with the value of s = 0.1 sec., the polynomial of the inner control loop usingthe forward acceleration feedback can be expressed as follows:

1 + KisNue(s)

(s + 1/s)long(s)= 0 (2.57)

where,

Ki = Sacc/s

Fig.?? shows the root locus of the inner control loop. It is clear that the phugoidmode gets more stable as the gain Ki goes up, due to the increasing damping


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0 20 40 60 80 100 120 140 1600

10

20

30

40

openlooph

0 5 10 15 20 25 30 350

0.5

1

1.5

closedloophdenganp

itchattitudehold

Figure 2.29: Time response of h(t) for an input of href = 1 from a system withand without altitude hold for N250-100

ratio. However, the pitch oscillation mode becomes oscill

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