Design and Flight Control of Miniature Aerial Vehicle

Design and Flight Control of Miniature Aerial Vehicle

A THESIS

SUBMITTED FOR THE DEGREE OF

Master of Engineering

IN THE FACULTY OF ENGINEERING

By

Ch.Sunil Kumar

Department of Aerospace Engineering

Indian Institute of Science

Bangalore-560 012

July 2012

iii

Acknowledgements

I would like to sincerely thank my research advisor Prof. M SeetharamaBhat for introducing me to this area of research.His constant guidance and dedication towards research helped me in understanding the practical difficulties faced by the people working in this area. He taught me the way of interpreting the results which helped me in strengthening my basic knowledge.

I am very thankful to all my course instructors who helped me in understanding the subject in a better way. They have enriched my knowledge on various researchtopics that are currently being used.

I would like to thankTitas,Harikumar, and Madhumita for being a great support in my research work. They helped me in analyzing the results and understanding the purpose of the experiment.

I would like to thank NspVarma, Amarender, Reginald, Venkatesh, Akhilesh and Ramjee for their support in academic and personal life. Without them my stay in IISc would have not been possible. Their constant support helped in understanding the way to lead my life in a better way.

I would also like to thank Dr. Prashant and Mr.Sam for coming to the institute to fly the vehicle whenever needed.

I am grateful to my parents and my brotherfor bringing me up to what I am now. In spite of the family background, they encouraged and supported me in pursuing higher education. They were always supportive in all the decisions in my life.

Finally I would like to thank all the people staying in IISc for making my stay in IISc a memorable one.

v

Abstract

The term ‘Miniature air vehicle’ (MAV) is used for unmanned aerial vehicles which are man-portable. A typical MAV has a weight of around 2kg to 5 kg, wingspan of 1m to 2m. These MAV’s are used in military and civilian applications.

The design and methodology of constructing a miniature airvehicle and its flight control is discussed in this thesis. The vehicle should have a maximum take-off weight of 1.5 kg and an endurance of 45-60 minutes. Its range should be 1-2 km with an operating height of 100-200 meters and a cruise velocity of 15m/s and shouldbe electrically propelled. It should have a low stall speed which makes it possible to be hand launched. The vehicle should be installed with Kestrel autopilot system.Itis capable of autonomous and semi-autonomous flights after installation and tuning of feedback loops.

The vehicle operates at Reynolds number between 90,000 and 215,000 at flight velocity of 8m/s to 18m/s. The wing and tail configurations of the vehicle are designed. Due to the availability of a similar kind of vehiclematching the requirements, instead of constructing a new vehicle, the existing vehicle-Fpv Raptor is procured from the market.This vehicle is selected as it has a stable operation in radio controlled mode. It has good glide properties and hence it will land gently in case of power/battery failure. It is readily available at low cost since it is manufactured in large numbers and sold to hobby flyers. The vehicle is mathematically modeled. It has a low phugoid and dutch roll damping. This low damping makes it difficult for the pilot to fly the vehicle.

Artificial stabilization is required to improve the flying qualities of the vehicle. The

feedback for stabilization is implemented on the kestrel autopilot hardware. It allows only PID based feedback structures to be implemented, hence gives no choice to the designer to implement higher order control. The digital integrator and differentiator implementations are non-ideal which further reduces the effectiveness of the control. The limitations of the vortex-lattice simulation based sofwares used for modeling increases the uncertainty in the plant model. The controller also needs to reject wind gust disturbances during flight. All the above requirements can be best addressed in robust control design. The classical robust control design suffers either

Abstract vi

from over conservativeness or computational intractability. The statistical learning theory overcomes both the above problems but suffers with confidence and accuracy limitations. To combine the effectiveness of robust control design and statistical learning theory, a combined Monte Carlo and modified Iterative Linear Matrix Inequality (ILMI) algorithm is used to design the controller.

In the longitudinal dynamics, pitch rate and pitch angle feedback are used to improve the

damping of the phugoid mode. The PID control required for stabilizing the longitudinal plant model is designed by using the combined algorithm. The constraints on pole placement, gain and phase margin, damping of closed loop poles, and performance are simultaneously applied in the algorithm. The designed controller is tested and the simulation results showed improvement in the performance of closed loop plant.

vii

Contents

Acknowledgements iii

Abstract v

Nomenclature and Abbreviations xi

List of Tables xv

List of Figures xvii

1. Introduction 1 1.1 Motivation andObjective…………………………………………………………..……..1 1.2 Literature Survey…………………………………………………………………….........2

1.2.1 Miniature Air Vehicles……………………………………………………............2 1.2.2 Design Process of vehicle…………………...…………………………….............3 1.2.3 Mathematical Modeling…………………………………………………………...4 1.2.4 Robust Control Techniques………………………………………………………..4 1.2.5 Linear Matrix Inequality…………………………………………………………..5 1.2.6 Statistical Learning Theory and Randomized Algorithms………………………...7 1.2.7 Path Planning and Following Algorithms…………………………………………8 1.2.8 Conclusion……………………………………………………………………….11

1.3 Scope of the Thesis……...……………………………………………………………….11

1.3.1 Contribution……………………………………………………………………...11 1.3.2 Organization of the Thesis……………………………………………………….12

2. MAV Design Process 15 2.1 Introduction…………………………………………………………………………........15 2.2 MAV Design……………………………………………………………………………..17

2.2.1 Design Requirements…………………………………………………………….17 2.2.2 Technology Availability…………………………………………………………18 2.2.3 Initial Sizing……………………………………………………………...............22 2.2.4 Aerodynamic Design…………………………………………………………….22

2.3 Fpv Raptor……………………………………………………………………….............30

Contents viii

2.4 Conclusion………………………………………………………………………………34

3. Mathematical Modeling of Fpv Raptor 35

3.1 Introduction……………………………………………………………............................35 3.2 Mathematical Modeling…………………………………………………….....................35

3.2.1 Linearized decoupled state space model…………………………………………39 3.2.2 Estimation of Moments of Inertia……………………………………………......39 3.2.3 Stability and Control Derivatives…………………………………………...........40 3.2.4 Throttle Modeling………………………………………………………………..40 3.2.5 Actuator modeling……………………………………………………………….42

3.3 Open Loop responses for Longitudinal and lateral Model ……………………………...43

3.3.1 Response to step input………….………………………………………………...44 3.3.2 Response to pulse input………………………………………………..................46 3.3.3 Response to 3-2-1-1 input………………………………………………………..48

3.4 Conclusion……………………………………………………………………………….49 3.5 Appendix…………………………………………………………………………………50

3.5.1 Generating Control and Stability Derivatives……………………………………50

4. Iterative Linear Matrix Inequality with Statistical Learning Theory 53

4.1 Introduction………………………………………………………………………………53 4.2 Linear Matrix Inequalities………………………………………………………………..54

4.2.1 LMI for pole placement on the left side of σ = α line in complex s-plane………........................................................................................................54

4.3 Static Output Feedback for Non-Ideal PID Implementation………………………….…57 4.4 Generalized Plant for H2/H∞Control design…………………………………………….61

4.5 Modified ILMI Algorithm……………………………………………………………….62 4.5.1 The steps of modified ILMI algorithm……………………………………………62

4.6 Sampling Based ILMI Control Design…………………………………………………..65 4.7 Monte Carlo based modified ILMI Algorithm…………………………………………..66 4.8 Conclusion……………………………………………………………………………….67

Contents ix

5. Controller Design 69

5.1 Introduction………………………………………………………………………………69 5.2 Longitudinal Control Design…………………………………………………………….69

5.2.1 Classical feedback Analysis……………………………………………………...70 5.2.2 Feedback design using Monte Carlo based modifiedILMIalgorithm ………….73 5.2.3 Attitude Hold Control System…………………………………………………...80

5.3 Lateral Control Design…………………………………………………………………...83 5.4 Open loop and Closed loop simulation…………………………………………………..84 5.5 Conclusion……………………………………………………………………………….86

6. Conclusions and Future Work……………………………………………………………..87 References 89

xi

Nomenclature and Abbreviations

AR - Wing aspect ratio b - Wing span c - Mean aerodynamic chord

- 2-D Drag coefficient - Normalized aerodynamic force in the X direction

- Normalized aerodynamic force in the Y direction

- Normalized aerodynamic force in the Z direction

- 2-D Drag Coefficient - 3-D Drag Coefficient - 3-D Lift Coefficient - Normalized aerodynamic rolling moment

- Variation of rolling moment with roll rate

- Variation of rolling moment with yaw rate

- Variation of rolling moment with sideslip

- Maximum lift coefficient

- Pitching moment coefficient - Normalized aerodynamic pitching moment

- Variation of pitching moment with forward velocity

- Variation of pitching moment with pitch rate

- Variation of pitching moment with angle of attack

- Variation of pitching moment with angle of attack

- Normalized aerodynamic yawing moment

- Variation of yawing moment with roll rate

- Variation of yawing moment with yaw rate

- Variation of yawing moment with sideslip

- Coefficient of thrust - Variation of X force with pitch rate

Nomenclature and Abbreviations xii

- Variation of X force with forward velocity - Variation of X force with angle of attack - Variation of X force with rate of change of α

- Variation of Y force with roll rate - Variation of Y force with yaw rate - Variation of Y force with sideslip - Variation of Y force with rollangle - Variation of Y force with heading - Variation of Z force with pitch rate

- Variation of Z force with forward velocity

- Variation of Z force with angle of attack

- Variation of Z force with rate of change of α

- Error in pitch angle - Moment of inertia around body fixed X, Y and Z axis

- Cross moment of inertia between body fixed x and z axis - Derivative gain - Derivative gain for pitch rate feedback

- Derivativegain for pitch angle feedback

- Integral gain for pitch angle feedback

- Integral gain - Proportional gain for pitch rate feedback

- Proportional gain for pitch angle feedback L - Lift force of airplane L/D - Lift-to-drag ratio also called glide ratio M - Pitching moment of airplane p - Roll rate q - Pitch rate

- Dynamic pressure = r - Yaw rate Re - Reynolds number S - Surface area of the wing

Nomenclature and Abbreviations xiii

u - Forward velocity

- Elevator commanded α - Angle of attack β - Angle of sideslip γ - Flight path angle

- Aileron deflection - Elevator deflection

θ - Pitch angle ξ - Damping of phugoid, dutch roll od short period modes ω - Phugoid, dutch, short period frequency ρ - Density of air in Kg/ Φ - Roll angle ψ - Yaw angle μ - Dynamic viscosity of air AVL - Athena Vortex Lattice software AXI - Brushless motor (nomenclature used by model motors) BMI - Bilinear matrix inequality CFD - Computational fluid dynamics EPP - Extended polypropylene (Type of foam used for model airplanes) Fpv Raptor - MAV used in this thesis GM - Gain margin ILMI - Iterative Linear Matrix Inequality LiPo - Lithium Polymer LMI - Linear matrix inequality LQR - Linear Quadratic Regulator MAV - Miniature air vehicle PID - Proportional Integral Derivative controller PM - Phase margin PWM - Pulse width modulation QMI - Quadratic Linear matrix inequality UIUC - University of Illinois at Urbana-Champaign GM-15 - Glenn Martin-15 airfoil XFLR5 - Xfoil software used for mathematical modeling similar to AVL

xv

List of Tables

1.1 List of autonomous electrically powered miniature air vehicles……………………………...3

2.1 MAV Design Requirements………………………………………………………………….18

2.2 Motor Survey………………………………………………………………………………...19

2.3 Battery survey………………………………………………………………………………..21

2.4 Initial Weight Estimate………………………………………………………………………22

2.5 Airfoils having 10-12% thickness shortlisted from database………………………………..24

2.6 Airfoils having5-7% thickness………………………………………………………………26

2.7Characteristics of the vehicle designed using S1223 and Clark Y airfoils…………………..29

2.8 Comparison of airfoil properties……………………………………………………………..32

2.9 Comparison of vehicle properties……………………………………………………………32

2.10 Properties of Fpv Raptor……………………………………………………………………34

3.1 Theoretical and experimental moment of inertia values……………………………………..40

3.2 Characteristics of the plant model……………………………………………………………43

5.1 Eigen Vector analysis for longitudinal motion………………………………………………70

5.2 Longitudinal Control design requirements…………………………………………………..70

5.3 Uncertainty considered for the stability derivatives…………………………………………73

5.4 Longitudinal PID gains………………………………………………………………………76

5.5 Eigenvalue-Eigenvector of closed loop plant………………………………………………..76

5.6 Open loop and Closed loop characteristics…………………………………………………..77

5.7 Lateral Control design requirements…………………………………………………………83

xvii

List of Figures

1.1 Description of RRT algorithm………………………………………………………………9

1.2 Reactive Obstacle avoidance algorithm……………………………………………………...10

1.3 Construction of waypoint path in Reactive Obstacle Avoidance Algorithm………………...10

2.1 Flow chart for design and development of MAV…………………………………………...16

2.2 Power available and Power required curves versus velocity for 11×7 propeller…………….21

2.3 Airfoil of Fpv Raptor………………………………………………………………………...31

2.4 Fpv Raptor modeled in XFLR……………………………………………………………….31

2.5 Comparison of characteristics of Fpv Raptor and vehicle using Clark Y airfoil…………….33

3.1 Coefficient of Thrust versus Advance Ratio for 8×4 inch propeller………………………41

3.2 Power available and Power required curves versus velocity………………………………...41

3.3 Response of servo actuator to step input……………………………………………………..42

3.4 Response for unit step input to elevator……………………………………………………...44

3.5Response for unit step input to aileron………………………………………………………45

3.6Response for unit step input to rudder……………………………………………………….45

3.7Response to unit pulse elevator………………………………………………………………46

3.8 Response to unit pulse aileron……………………………………………………………….47

3.9 Response to unit pulse rudder………………………………………………………………..47

3.10 Response to 3-2-1-1 input to elevator………………………………………………………48

3.11 Response to 3-2-1-1 input to aileron………………………………………………………..48

3.12 Response to 3-2-1-1 input to rudder………………………………………………………..49

4.1 General feedback plant………………………………………………………………………55

List of Figures xviii

4.2 Closed loop system with PID-PD feedback………………………………………………….57

4.3 SOF closed loop system……………………………………………………………………...60

4.4 Generalized Plant…………………………………………………………………………….61

5.1 Longitudinal Nominal Plant………………………………………………………………….71

5.2 Root locus for pitch rate feedback…………………………………………………………...71

5.3 Step response of the closed loop system……………………………………………………..72

5.4 Longitudinal Generalized plant………………………………………………………………74

5.5 Nominal Lateral Plant………………………………………………………………………..77

5.6 Bode plot of the system obtained by breaking at the input…………………………………..78

5.7 Step response of the closed loop system for plant model at 8m/s…………………………...79

5.8 Bode plot of the system obtained by breaking at the input (plant model at 8m/s)…………79

5.9 Step response of the closed loop system for plant model at 18m/s………………………….79

5.10 Bode plot of the system obtained by breaking at the input (plant model at 18m/s)………..80

5.11 Altitude Hold Control System……………………………………………………………...81

5.12 Nominal Lateral Plant………………………………………………………………………82

5.13 Open loop response for a commanded elevator (step input)………………………………..83

5.14 Closed loop response for a commanded elevator (step input)……………………………...84

5.15 Step Response of the closed loop system…………………………………………………..86

1

Chapter 1

Introduction

1.1 Motivation and Objective The term ‘Miniature Air Vehicle’ (MAV) is used for Unmanned Air Vehicles which are

man-portable. A typical MAV has a weight of around 2kg to 5 kg, wingspan of 1m to 2m. Vehicles of smaller size and weight (micro air vehicles) are being developed around the world. These MAV’s can carry a larger payload (hence more useful work) when compared to the smaller vehicles. They can operate in wind gusts which are intolerable for the micro air vehicles. These vehicles can be used in situations of natural disasters like earthquakes, floods or cyclones where human being gets trapped. There are helicopters used to survey the area and gather information but they are expensive and cannot be launched in large numbers at short notices. The MAV’s equipped with a reasonable high resolution cameras can directly send video to the ground station giving an overview of the situation.

The general payload carried by these vehicles is a high resolution camera which can directly transfer video to the ground station. They can be used inautomobile traffic monitoring, wild land monitoring, pipelines and power line monitoring and for surveillance in areas having risk of illegal mining. They can be pre-programmed to visit a desired location and can be rescheduled during flight. In military use, the soldiers can carry these portable vehicles with them during any operation. They can be flown over the enemy territory in order to know the position of the enemy ground force before proceeding any further. They also can be used for border patrol.These vehicles can be equipped with explosives or biochemical weapons. These can be used to directly execute the enemy artillery in order to reduce the loss of our human life.

With the evolution of present sensor technology, there are high quality, light weight sensors available for collecting weather information, detecting explosives and land resources. The MAV research is also interesting for a researcher because of the opportunity to build a real airplane and learn flying it. Flying the vehicle under wind disturbances and practically testing the

Chapter 1. Introduction 2

control system designed by them is very interesting and challenging. It involves knowledge from multiple disciples which improves the awareness of the researcher in various fields.

The objective of this thesis is to develop a miniature air vehicle which is man portable, can be hand launched and easily flyable. The total weight of the vehicle including the payload should not exceed 1.5 kg. It should be electrically propelled in order to reduce the engine noise. For surveillance purposes, the vehicle should fly a fairly stabilized flight at a constant altitude so that the information from the camera installed onboard is useful. The handling qualities of these vehicles are poor due to their low damped dynamics. Hence artificial flight stabilization is required in order to obtain a stabilized flight. Therefore, development of flight control and testing its performance during flight is the highest priority. Flight stabilization improves the quality of the flight and reduces the pilot effort required. This allows higher navigation loops to be added for autonomous operation of the vehicle.

1.2 Literature Survey

1.2.1 Miniature Air Vehicles

There are different types of MAV’s such as Fixed Wing, Tailless aircraft, Rotary Wing. The conventional fixed wing aircraft have good gliding ability and good handling qualities. There are various MAV’s already designed and being used in military applications given in [1, 2] and some of them are tabulated in table 1.1 along with theirspecifications. The military specifications that these vehicles satisfy are given in [3].

The MAV’s given in the table 1.1 are used in military applications. These vehicles satisfy the military specifications [3]and are being widely used for surveillance applications. The typical weight of these vehicles is around 2 Kg. They are carried by the soldiers in their backpack. The wing span is typically below 1.5 m which makes them less detectable during flight and also easily portable. The typical characteristic for these military vehicles is that they can be assembled and hand launched within a few minutes. They have an endurance of 1 hour and are operated at a height of 300-600 meters above ground level. The ceiling of the aircraft is from 3 km to 4.5 km. Their operating range should be more than 5 km as the ground force should not be too close to the enemy territory. They use electric propulsion producing minimum noise and heat signature which makes them undetectable to the enemies while flying over enemy territory. The


typical payload carried by these vehicles is a high resolution camera or infrared camera which can directly send the video to the ground station.

Table1.1List of autonomous electrically powered miniature air vehicles

MAV name

Maximum Weight

Endurance Operating Altitude (meters)

Wing span

(meters)

Range (km)

Payload

Aladin 3Kg 45 min

150-250 1.46 5km Electro-optical camera

Bird Eye 100

1.3Kg 60min ---------- 0.85 5km --------

Bush master

2Kg 90min 300-600 ---- 8km Color TV-camera

(fuselage mounted) Swift Eye

1.8Kg 40 min 300-600 1.42 ---- Color TV-camera

Dragon Eye

2.7Kg 45-60 min 100-300 1.1 10km Motion camera or

electro-optical camera (nose assembly)

RQ-11A (Raven)

1.9Kg 80 min 1000 1.3 10km Video and Infrared

camera

1.2.2Design Process of vehicle

The Design process includes three phases and is disused in detail in [16]. The level of the detail of the design increases as we go through conceptual, preliminary and detail design. The brief overview of the design process is given below:

i. Conceptual Design: In this phase, the design requirements are used to guide and evaluate the development of the overall aircraft configuration arrangement. This design includes the size and shape of the wing, the required tail geometry and its configuration, location of the payload, battery, motor.

ii. Preliminary Design: In this phase only minor changes are made to the configuration of the vehicle (wing or tail configuration). Extensive wind tunnel testing and CFD


calculations are done in this phase. At the end of this phase the wing, tail configuration, fuselage configuration, placement of componentsare finalized and detailed drawings of the airplane are made.

iii. Detail Design: In this phase, the detailed design of each and every part (including the nuts and bolts) to be fabricated is done.The size, number and location of each component to be fabricated are calculated and detailed drawings with actual fabrication geometries and dimensions are made. At the end of this phase, the aircraft is ready to be fabricated.

1.2.3Mathematical Modeling

After design and fabrication of the vehicle, mathematical modeling of it is done using vortex latticebased simulation softwares like XFLR5 and AVL [4, 5]. These vortex based simulation softwares are not professional softwares and thus do not offer any guarantees of robustness and accuracy. For the aircraft the mathematical model isto describe both a physical system (i.e. aircraft and its surroundings) and a process(i.e. control of the aircraft along some desired path). The equations of the modeldefine relationship between system variables. The aircraft is a nonlinear system and this system is linearized into longitudinal and lateral modes which illustrate the longitudinal and lateral motion respectively. These are thoroughly discussed in [6]. The mathematical equations and values are used to convert motion sensor inputs intomeasurements of aircraft state, compute guidance parameters, derive servo and throttlecommands, and perform co-ordinate frame transformations. The main goal of mathematical model of aircraft is 1. Predicting the aircraft state using motion inputs. 2. Estimating the command inputs needed to guide the aircraftalong a selected path following some predefined waypoints. Using the above model we can design navigation and guidance systems capable of autonomously guiding an aircraft through predefined waypoints.

1.2.4 Robust Control Techniques

Robust control techniques are widely used in aerospace applications because of the uncertainties present in their dynamics and also in the environment in which they fly. The uncertainty is also introduced intentionally when the non-linear aircraft model is linearized for


control design. The uncertainty is also introduced due to modeling errors. The modeling errors are due to the inability of the wind tunnel to simulate the environment in which the vehicles actually fly or due to the approximations done by flow simulation software used to model the aircraft. The environment being an open space is difficult to predict and this introduces uncertainty in the vehicle behavior during flight. This uncertainty is more like a disturbance the vehicle experiences during flight. Therefore, in the flight control design we have to incorporate certain robustness to these uncertainties.

The uncertainty in the system can be handled by specifying bounds on the uncertainties and designing the control which gives satisfactory performance for uncertainties within these bounds. This is a robust control way of handling the uncertainties. The uncertainties can also be handled using a stochastic control technique in which the uncertainty is modeled as probabilistic variables and designing the control after introducing them in the dynamics.

The modeling errors and the wind gust disturbances are the major source of uncertainties in the aircraft application. In [7], the Eigen structure assignment is used in which the modeling uncertainties are not considered. In [8] LQR design is used in which all the states should be available for feedback. In the aircraft application all the states are not available for feedback which requires a Kalman filter to estimate the unknown states which is a LQG design discussed in the same paper. A trajectory tracking system for UAV application is studied in [9]. Robust control theory is used to design the flight control system. The navigation control commands are produced by the deviation of real position data with the preset flight path.

In case of an aircraft, if we know the bounds on each of the parameters then the uncertainties can be modeled using Linear Fractional Transformation (LFT). In [10] μ-synthesis is done on the LFT model and the resulting controller performance is shown to be better than the H-infinity based controller. In [11], the classical PID-controller, Integral separated PID controller and Adaptive proportion PID controller are compared for a UAV application.The adaptive proportion PID controller is good in both stability and dynamic performance of pitch attitude control. In [12], all the robust control methods are briefly discussed.

1.2.5 Linear Matrix Inequality

The modern control design problem is to solve for a good solution for a problem with conflicting requirements which is similar to solving an optimization problem or feasibility


problem. These optimization problems are convex or quasi convex problems and they can be expressed as Linear or Bilinear Matrix inequalities (LMI and BMI). A linear matrix inequality is an expression of the form [13]

where, (1.1)

• is a vector of real numbers which are decision variables of the inequality

• are real symmetric matrices • is negative definite.

All eigenvalues of F(x) are negative

The linear matrix inequality of the form Eq. 1.1 is always convex because the condition for convexity (Eq. 1.2) is always satisfied by F(x). These inequalities are always convex and therefore can be solved by convex optimization software.

(1.2)

A bilinear matrix inequality is of the form

(1.3)

where and are vectors of real numbers which are decision variables of the inequality.

In case of aircraft, satellite or missile, the closed loop plant is required to satisfy worst case stability criteria in the face of uncertainties and disturbances which works out to be a BMI problem. The BMI problem is not a convex problem and is difficult compared to LMI which is always convex. In case of convex LMI problems, interior point methods are used which cannot be used for BMI due to the non-convexity. A number of algorithms to solve BMI are present


since 1990s. There is also research going on converting the BMI to solvable forms or approximating it and finding the global solution iteratively.

Though analytical solutions are used by most of the control design techniques, finding the solution is often difficult and many times requires searching. If these are reformulated in linear matrix inequality feasibility or optimization problems then their solutions can easily be found by numerical methods. Multiple LMI constraints can be converted to a single LMI constraint by augmenting them into a single LMI [14]. After the formulation of the LMI, the efficient algorithms are used to solve the LMI [16].There are many control design problems which have equivalent LMI formulations and these are discussed as follows.

The design of state output feedback controller that satisfies additional constraints on the closed loop pole location is discussed in [15]. In this paper, sufficient conditions for feasibility are derived for a general class of convex regions of the complex plane. These conditions are expressed in terms of LMI’s and subsequently solved. For multi input and multi output (MIMO) systems which cannot be well approximated by the first or second order systems, the tuning of the PID controllers is very difficult. Therefore, these design problems are transformed into static output feedback controller design problems which can be done solved using LMI’s. In [14, 17] the design of multivariable PID controllers is done using LMI approach. H2 and H-infinity suboptimal control is also done with PID controllers.

1.2.6 Statistical Learning Theory and Randomized Algorithms

The classical robust control techniques discussed in the sections 1.2.4, 1.2.5 are either over conservative or computationally intractable [18]. Probabilistic and Randomized techniques have been used in various applications to tackle computationally difficult problems that are too hard to be treated using exact deterministic methods [18]. If an efficient or poly-time algorithm exists for a particular problem, then the problem is soft or tractable otherwise the problem is hard or intractable. The starting point of the probabilistic approach is to assume that the uncertainty affecting the control system has a stochastic nature and to provide probabilistic assessments on the system characteristics i.e. a characteristic ( norm, norm, gain and phase margin) of the system is robustly satisfied in a probabilistic sense if it is guaranteed against most of the possible uncertainty outcomes.


In the aircraft application, the aircraft model is obtained from wind tunnel data or

modeling software. The mathematical model is different for different trim conditions. The model considered for the controller design is for a particular trim condition (cruise velocity of the vehicle). Due to the wind disturbances and other environmental factors, the trim condition cannot be maintained during the whole flight duration. The uncertainties and disturbances are introduced as parametric uncertainties in the stability derivatives in the mathematical model of the plant. The probabilistic method gives the controller which satisfies the plant with the presence of these uncertainties. The price to be paid is that it works in most of the cases but fails in some. The introduction to Statistical Learning Theory and Monte Carlo simulation is given in [18, 19]. The use of Uniform Distribution in robustness analysis is discussed in [20].

1.2.7 Path Planning and Following Algorithms:

The controller designed for an MAV improves its handling qualities which result in a stabilized flight. The vehicles having stable flight with good damping characteristics can be operated autonomously with the help of path planning and obstacle avoidance algorithms. Some of these algorithms are discussed in [2, 36]. The following three algorithms are widely used in this application and are discussed briefly.

i. Rapidly-Exploring Random Tree (RRT) algorithm ii. Vector Field Path Following

iii. Reactive Obstacle and Terrain Avoidance

RRT Algorithm This algorithm builds a tree that uniformly explores the search space. The input to the

algorithm is a start ( ) and an end ( ) point. The algorithm uniformly searches the space between start and end point by randomly sampling from a uniform probability distribution. The algorithm randomly selects a sample p from the space and then selects a waypoint ( ) at a fixed distance (D) from the start point, on the line joining the start point and the point p. It checks whether the waypoint collides with an obstacle. If the waypoint collides then it is discarded and a new point p is selected or it adds the waypoint to the random tree. In this way the algorithm finds a complete path through the obstacle area. It is a probabilistic approach. It considers the dynamic constraints like turn radius limitations and airspeed. The disadvantage is that it cannot be applied for detecting moving obstacles.


Figure 1.1 Description of RRT algorithm

Vector Field Path Following This algorithm produces a field of desired course commands that drives the MAV to the

current path segment. At any point, the desired course commands are calculated. This desired course is used to command heading and roll control loops to guide the MAV onto the desired path. This method is effective in tracking paths with wind speeds up to 50% of the airspeed of MAV. This algorithm considers dynamic limitations, imprecise sensors and controls, wind disturbances.


Reactive obstacle avoidance

This algorithm uses a laser ranger to detect and avoid obstacles.Consider a situation as shown in the figure 1.2. Let the vehicle has a minimum turn radius R. When the aircraft detects an obstacle, it constructs an internal map obstacle, a cylinder with radius R as the actual size of the obstacle is unknown. It has two paths around the cylinder and it chooses randomly if both are similar (either free or obstructed).The endpoints of the waypoint paths are selected so that the new way point paths are tangent to the obstacles in the internal map. The new waypoints are located at a distance from the original waypoint, where d is the turn away distance from obstacle as shown in the figure 1.3. As it tries to overcome the map obstacle, it detects the actual obstacle again and creates a new map obstacle as shown in figure 1.2. This process is continued till it overcomes the actual obstacle.

Figure 1.2 Reactive Obstacle Avoidance Algorithm

Figure 1.3 Construction of waypoint path in Reactive Obstacle Avoidance Algorithm The waypoint path is constructed so that it is perpendicular to the map obstacle. The radius R ensures collision free passage around the map obstacle.


1.2.8 Conclusion

The miniature air vehicle designed in this thesis is similar to the vehicles already available as shown in the Table 1.1. The objective of this thesis is to develop a vehicle of lesser weight and with better endurance than the vehicles already available. These two improvements make the vehicle useful for long endurance surveillance operations. The miniature air vehicle has a low damped dynamics and therefore requires artificial stabilization. The modified ILMI is used in designing the controller for UAV applications. This robust control method gives an over conservative design. In the case of unavailability of wind tunnel data, the accuracy of the model decreases and the above method cannot give good results. In these applications, the modified ILMI algorithm is applied in a probabilistic framework which combines the advantages of both the methods. This combined algorithm is used to design the controller in this thesis.

1.3 Scope of the Thesis In this thesis, the preliminary design of wing and tail geometries of a miniature air

vehicle is done. The preliminary design satisfies the design requirements of the vehicle. In the design process, we realized that the detailed design of the vehicle requires extensive wind tunnel data, structural calculations before fabricating the vehicle. After fabricating the vehicle, the vehicle should be tested and again redesigned if any of the design requirements are not satisfied.This whole process cannot be done within the specified time limit. Therefore, we thought of procuring a vehicle, which satisfies the requirements, from the market. The vehicle –Fpv Raptor is procured from the market which is widely used by hobbyists and has good gliding capabilities. The vehicle is modeled using Xfoil softwares to obtain the mathematical model. The vehicle has a low damped dynamics and requires artificial stabilization to improve its flying qualities. Longitudinal controller is designed for stabilizing the plant using Monte Carlo approach and modified ILMI algorithm. The closed loop plant is tested in Matlab for its improvement in performance.

1.3.1 Contribution

The contribution of this thesis can be summarized as follows

• The preliminary design of the MAV involves the initial size estimates of the wing and tail geometry. The existing airfoils available in literature are analyzed and the characteristics


of each airfoil are tabulated and compared to choose an airfoil suitable for the design requirements. The wing and tail geometry calculations are done based on the weight and stability calculations following the procedure available in literature.

• During the design process, it is understood that the detailed design requires wind tunnel data and structural calculations before fabrication of the vehicle. After fabricating the vehicle, the vehicle should be tested and again redesigned if any of the design requirements are not satisfied.This whole process cannot be done within the specified time limit. Therefore the vehicle-Fpv Raptor which satisfies the design requirements is procured from the market. The airfoil of the Fpv Raptor is analyzed. The vehicle is mathematically modeled after experimental calculation of the Moment of Inertia values.

• A systematic method of designing control for the vehicle is done using Matlab tools. The modified ILMI algorithm is combined with Monte Carlo based simulation to design the controller. The additional dynamics introduced by non-ideal implementation of differentiator and integrator is taken into account in designing the controller.

• The above algorithm is used to design a longitudinal controller. The controller designed is tested for the improvement in the performance of the closed loop system. The open loop response for a step input is compared with that of the closed loop response. The controller showed improvement in the handling qualities (improved the damping) of the longitudinal mode.

1.3.2 Organization of the Thesis

The remainder of the thesis is organized in four chapters followed by conclusion and a list of bibliography. The contents of each chapter are given below.

• Chapter 2:The preliminary design of the MAV involves the initial size estimates of the wing and tail geometry. The existing airfoils available in literature are analyzed and the characteristics of each airfoil are tabulated and compared to choose an airfoil suitable for the design requirements. The wing and tail geometry calculations are done based on the weight and stability calculations following the procedure available in literature. During the design process, it is understood that the detailed design requires wind tunnel data and structural calculations before fabrication of the vehicle. After fabricating the vehicle, the vehicle should be tested and again redesigned if any of the design requirements are not satisfied.This whole process cannot be done within the specified time limit. Therefore the


vehicle-Fpv Raptor which satisfies the design requirements is procured from the market. The vehicle is reconstructed using modeling softwares and the lift, drag and pitching moment characteristics of the vehicle are compared with those of the vehicle constructed in the design process. Chapter 3: The mathematical model of the vehicle is obtained using XFLR5 and AVL softwares. The open loop responses for step and pulse inputs are analyzed to understand the performance of the plant. The vehicle is tested in flight by the pilot by giving pulse and step inputs to the elevator. The model is verified with the data obtained from the test flights. Chapter4:The modified ILMI algorithm is used to develop a multi-loop robust pole placement PID design. The LMI’s used in the algorithm are for pole placement, and

constraints. Thisrobust design is done in a probabilistic framework using Monte Carlo simulation in order to reduce the over conservativeness of the classical robust control technique. The combined algorithm implementing modified ILMI algorithm in probabilistic framework is finally presented.

Chapter 5: In this chapter the longitudinal controller is designed for the Fpv Raptor vehicle using the modified ILMI algorithm in probabilistic framework discussed in the previous chapter. The designed controller is tested for its closed loop performance using Matlab simulation. The results of the closed loop responseshow an improvement in the damping characteristics (phugoid damping is improved) of the vehicle.

15

Chapter 2

MAV Design Process

2.1 Introduction The miniature air vehicle is designed to be fixed wing, portable, easily launched and

recovered. The different MAV’s compared in table 1.1 are portable and can be carried in the backpack by a soldier. This is possible due to the characteristics like smaller wing span and lower weight. They can also be easily dismantled for storage and assembled when required. These characteristics must be met while designing the MAV. The design process is a sequential approach from initial sketch to final fabrication and testing of the model. The detailed design process is discussed in [21] which consist of three main phases, conceptual, preliminary and detailed design followed by fabrication of the model. This process is an iterative process and these three phases are interlinked with each other. If in some phase the design goes out of the mission specifications, then the design is reiterated to meet all the goals and this process is continued until the detailed designed matches the design specifications. The vehicle is then fabricated and tested for its performance.

In this chapter, design of a miniature air vehicle is discussed. The design is challenging due to low stability of these kinds of vehicles. The airfoil selection is done based on analysis of existing airfoils. The wing and tail configurations are selected based on the available information about these kinds of vehicles.The initial wing and tail geometries are estimated using calculations based on initial sizing. The stability and control analysis of the vehicle is performed. The dihedral and sweep characteristics of the wing, the area of the control surfaces are tested and their initial estimatesare obtained.During the design process, it is understood that the detailed design requires wind tunnel data and structural calculations before fabrication of the vehicle. After fabricating the vehicle, the vehicle should be tested and again redesigned if any of the design requirements are not satisfied.Designing and building airframes is expensive and highly

Chapter 2. MAV Design Process 16

Design Requirements

Technology Availability

Market survey for components

Concept Sketch Initial Sizing

Aerodynamic Design

Stability and Control Analysis

Review Design Requirements and Reiterate the Design

Fabrication

Test Flight

Review performance and Reiterate the Design till requirements are satisfied

Figure 2.1 Flow chart for design and development of MAV


time consuming and the quality is also sacrificed during manufacturing process (as small quantity is manufactured). Therefore the vehicle-Fpv Raptor which satisfies the design requirements is procured from the market. This vehicle is selected as it has a stable operation in radio controlled mode. It has good glide properties and hence it will land gently in case of power/battery failure. It is readily available at low cost since it is manufactured in large numbers and sold to hobby flyers.

2.2 MAV design The design process is an iterative process and is done keeping into account the

specifications and available hardware components. The overall design can be split into four major fields. These are Aerodynamics, Propulsion, Structure, Stability and Control. These fields are considered as the subsystems in the design process. The payload comprising of the avionics is also a subsystem. Each of these subsystems consists of various factors to be decided while designing. These are discussed in detail in this section. These subsystems are interrelated to each other and cannot be dealt individually. The flowchart in figure 2.1 shows the whole design process. The flowchart is taken from [22] and modified with the help of overall design process in [21]. We will follow this flowchart for designing the MAV.

2.2.1 Design Requirements

The design of any aircraft begins with specific set of design requirements established by the customer. These requirements are based on the final application (military or civilian) of the aircraft. They include parameters such as aircraft range and payload, takeoff and landing distances, maneuverability and speed requirements. They also depend on set of military and civil design specifications such as stall speed, structural design limits. It is these requirements that the final design should satisfy in order to satisfy the customer. Therefore, it is necessary to check the design requirements after each step of the design process. These requirements are generally obtained from existing aircraft designs and modified for the present application.

The MAV’s being used in military applications are listed in table 1.1. Their characteristics like weight, wing span, operating altitude, endurance, propulsion system and payload are suitable to their application as discussed in section 1.2.1. The MAV being designed in this thesis is used for a similar application. The design requirements of the MAV are given in table 2.1.


The maximum takeoff weight of the MAV should not exceed 1.5 kg including the

payload. The wing span should be less than 1.5 m for easy portability. The endurance of the vehicle should be at least 45 minutes which is required for its application of surveillance. The vehicle is being designed for surveillance in civilian areas like illegal mining areas unlike the military application as discussed in section 1.2.1. The range can be limited to 1-2 km based on availability of the communication equipment. The operating altitude should be around 100 to 200 meters with a cruise speed of 12 to 15 m/s. It should use electric propulsion to reduce the noise in civilian areas. The vehicle should be hand launched from any terrain. The payload is a high resolution camera which can take photos and videos and directly send them to the surveillance officer at the ground station.

Table 2.1 MAV Design Requirements

Maximum Take Off Weight 1.5Kg

Endurance 45-60 min

Range 1-2Km

Operating Altitude 100-200 m

Cruise speed 12-14 m/s

Maximum speed 18 m/s

Propulsion Electric Motor

Launch and Recovery Hand Launch and Skid Recovery

Payload Camera

2.2.2 Technology Availability

This step is very crucial in the design process. If the vehicle should have long endurance then it should be light weight with a high battery capacity. If the vehicle should be highly maneuverable and flying at high speeds then the structure should be rigid and the propulsion


system should produce the extra thrust required. The availability of the technology should be checked before proceeding with design.

Propulsion: The electric motor is a clean, low maintenance and less noisy source of propulsion. The manufacturer gives the details like the maximum current drawn, thrust available in no load condition for various motor-propeller combinations. The maximum current drawn by the motor, servo actuators and the avionics can be calculated and the battery capacity required can be estimated.

Table 2.2 Motor Survey

Motor Weight No. of cells (Voltage)

Propeller

(inches) Maximum thrust

No load current

AXI 2814/20 106 g 3 (11.1V) 13×8 1700g (weight of model)

0.7A 11×7 1500g(weight of model)

EMAX GT2218

76 g 3 (11.1V) 11×3.8 1380g(weight of model) 0.6 A

Table 2.2 shows the motors suitable to the requirements. The AXI 2814/20 motor gives the required thrust of 1500 to 1700 gm. The EMAX GT2218 motor gives a thrust of 1400g with a weight of just 76g.

Thrust Modeling: The AXI motor can be used with two 13×8 inch and 11×7 inch propellers. The wind tunnel measurements for the 11×7 inch propeller are taken from UIUC Propeller Database [23]. Empirical thrust coefficient ( ) versus advance ratio (J) curves for different propellers are given in [23]. The advance ratio J is given by the expression

(2.1)

where V is the forward flight velocity, n is the rpm of the motor, D is the diameter of the propeller.

The thrust coefficient is given by the expression


(2.2)

where is the thrust in newtons, ρ is the air density.

The curve for the propeller is available for different rpm with only slight difference. For a particular velocity, advance ratio is calculated. Using the curve the thrust coefficient is determined. The thrust available is calculated from the Eq 2.2. The power available is calculated from the relation where is the power available in watts, is thrust available in newtons, V is the forward velocity in m/s.

The thrust required is calculated using the following equilibrium steady flight equations.

(2.3)

(2.4)

where L is the lift force generated, D is the drag force generated, W is the weight of the vehicle, is the thrust required.

The following equations are used in calculating the drag force experienced by the vehicle.

(2.5)

(2.6)

where S is the wing surface area, are the 3D lift and drag coefficients of the vehicle.

The thrust required is calculated using Eq 2.4. The power required is calculated from the relation where is the power required in watts, is thrust required in newtons, V is the

forward velocity in m/s. The power required and power available curves for 11×7 propeller are shown in the figure 2.2 for 5000 rpm of the motor.


Figure 2.2 Power available and Power required curves versus velocity for 11×7 propeller

From the figure, it is understood that the flight regime is from 7 to 18 m/s where power available is greater than power required. This is a good flight regime of operation and therefore, this propeller motor combination is chosen. The EMAX motor is considered as they are available in the local market and can be procured easily. This AXI motor is widely used by hobbyists around the world and also gives the required thrust.The manufacturer provides information like discharge rating, weight and capacity of the battery.

Table 2.3 Battery survey

Battery Weight No. of cells Capacity (mAh) Discharge rating

Zippy Flightmax 8000 3S1P 30C 644 g 3 8000 30C

Turnigy nano tech 8400mAH 641 g 3 8400 40C

Table 2.3 shows the battery cells suitable to the requirements. The battery weight is around 650 g which is half of the weight of the aircraft. The batteries compared are almost identical with Turnigy which is having a high discharge rate. The discharge rate limits the


maximum current that can be drawn from the battery pack. The 30C discharge of Zippy Flightmax battery gives a current of 24 A which is equal to the current capacity of the AXI motor. It is also widely used and has good performance compared with the Turnigy battery packs. Therefore, the Flightmax battery is chosen.

2.2.3 Initial Sizing of the vehicle

The wing and tail geometries, propulsion system location and placement of the components are approximately finalized in the initial sizing. The initial weight estimate required for aerodynamic calculations is obtained here. The initial weight estimate of the MAV is given in table 2.4.

Table 2.4 Initial Weight Estimate

Subsystem Quantity Weight

Motor(AXI 2814/20) with propeller 1 116g

Zippy Flightmax 8000 1 644g

Kestrel autopilot with hardware

+ Autopilot Battery

1 80g+ 25g

JETI speed controller(30A) 1 25g

Servo actuators 4 20g

Structure 590g

Total 1500g

2.2.4 Aerodynamic Design

The airfoil, the planform shape, wing and tail configurations are the various aspects in aerodynamic design. The airfoil effects the cruise speed, stall speed and handling qualities and overall aerodynamic efficiency during all phases of flight.The wind tunnel testing of airfoils is not done due to the time limitations. The vortex lattice simulation based software,XFLR5 (an XFOIL software) [24] is used for analyzing the airfoils.


XFLR5:XFLR5 is an analysis tool for airfoils, wings and planes operating at low Reynolds Numbers. It includes:

• XFoil's Direct and Inverse analysis capabilities • Wing design and analysis capabilities based on the Lifting Line Theory, on the Vortex

Lattice Method, and on a 3D Panel Method

Validation:This software is validated for the results before being used further in order to find the accuracy of the predicted data. The XFLR5 data of some existing airfoils are comparedwith the wind tunnel data available in the literature. There is slight deviation of the predicted data from the wind tunnel data which is acceptable.

The aerodynamic design is combined with the stability and control analysis.The stability analysis is used to estimate the size of the tail and control surfaces, wing dihedral, wing sweep.The procedure followed is outlined here. This procedure is sequentially followed in this section.

• Choosing an airfoil • Calculation of Wing loading • Choosing the AR (aspect ratio) and calculating the Wing span • Wing Configuration and geometry • Tail configuration • Tail Sizing

Thesekinds of vehiclesaregenerally designed with airfoils of thickness around 10%. The airfoils are chosen by shortlisting from the UIUC airfoil database and AID Airfoil Investigation Database [25, 26]. Table 2.5shows the list of the airfoils along with the XFLR data. The simulation in XFLR is done for a particular Reynolds number which is calculated using the formula given in Eq. 2.1.

(2.7)

where ρ is density of air=1.1 kg/m3(density in Bangalore)

v is the velocity of air ( cruise velocity of the MAV) =15m/s

L is the Mean Aerodynamic chord (considered as 20cm)


μ is the dynamic viscosity of the fluid =

Re obtained is 1,85,000 and this number is used for the simulation in XFLR. The cruise velocity and the mean aerodynamic chord are considered from the data collected about similar kinds of vehicles tabled in 1.1 in chapter 1.

Table 2.5 Airfoils having 10-12% thickness shortlisted from database

Airfoil Thickness Clmax Cl/Cd max Cm @Cl/Cd max

E374 10.91% 1.02 @ α=110 72 @ α=60 -0.045

E224 10.17% 1.1 @ α=100 67 @ α=50 -0.05

E168 12.44% 1.02 @α=110 52 @ α=7.50 0.002

DF102 11% 1.3 @ α=120 63 @ α=3.50 -0.052

Dae51 9.31% 1.35 @α=120 85 @ α=60 -0.1

Clark k 11.70% 1.2 @ α =100 63@ α=60 -0.08

S1223 12.43% 2.2 @ α=120 71 @ α=3.50 -0.273

Clark y 11.71 % 1.39 @ α=12.50 72 @ α=4.50 -0.075

The following properties are considered in choosing the airfoils:

i. The structural weight of the wing and body should be as low as possible. This benefits us in the way that a higher capacity battery pack can be used which increases the endurance of the vehicle.The weight of the wing depends on the thickness of the airfoil. The thickness should be small enough to reduce the weight and large enough to give overall strength to the wing.


The thickness of the airfoil is generally chosen by the amount of volume required for storing fuel. This is an electrically powered vehicle and this way of choosing the thickness cannotbe done.

ii. The Lift coefficient should be high. Maximum Lift Coefficient (Clmax) of the airfoil should be greater than 1.5.

iii. The moment coefficient should be low which reduces the tail volume required for trimming.Pitching moments of the order 0.08 are acceptable.

iv. The ratio (glide ratio) should be high. v. The operating angle of attack (α), where the glide ratio is high, should be far from the

stall angle of attack ( ).This margin gives a wide range of operating α for the pilot.

vi. The material to be used for construction of the vehicle depends on the required structural rigidity and overall weight of the vehicle.As the structural weight of the vehicle should be small,the vehicle should be made ofcomplete EPP (Extend poly propylene) foam (density 20 kg/m3) or foam reinforced with balsa (density 120-200 kg/m3) to get enough strength for the wing.

The airfoils chosen have good pitching moment coefficients but the coefficient of lift is low which increases the required wing area. The thickness of the airfoil is greater than 10% and if we use foam reinforced with balsa for construction, the overall weight of the wing increases. Therefore, airfoils of lesser thickness are chosen and similar analysis is done. The results are shown in the table 2.6.

The airfoils having good coefficient of lift has very high pitching moment coefficient and vice versa. So there should be some tradeoff between the required values and this depends on

i. The value should be such that it does not give high wing loading for the required

stall speed. The wing span required should be less than 1.5m with an aspect ratio of 10-12.

ii. The value should be such that the tail volume required for trimming the vehicle should be practically feasible. The tail volumes can be found out using existing formulae in the literature [21].


Table 2.6Airfoils having5-7% thickness

Airfoil Thickness Clmax (Cl/Cd)max Cm @ (Cl/Cd)max

a18sm 7.28% 1.3@ α=90 85@ α=4.50 -0.115

bw-3 5.02% 1.44@ α=90 70 @ α=40 -0.120

E61 5.67% 1.45@ α=7.50 96@ α=3.50 but very sharpcurve

-0.22

Gm15sm 6.74% 1.36@ α=100 89.5 @ α=3.50 -0.13

S6062 7.94% 0.9 @ α=90 59 @ α=40 -0.035

Lrn1007 7.27% 1.1@ α=50 91 @ α=50but very sharp curve

-0.12

NACA M6 7.81% 0.9@ α=90 64 @ α=6.50 -0.035

The wing structure can either be made from foam reinforced with balsa or using only foam which is available in the laboratory. In order to verify the structural stiffness,we made a rough model of the vehicle using foam and balsa. The airfoil “GM15sm” (5% thickness) is used for constructing the wing. It has good glide ratio and good . The wing is constructed using

foam and reinforced with carbon fiber and balsa is used for reinforcing at the bottom part of the wing. The wing is then covered by glass fiber tape.It is observed that the wing is very weak and that it flutters when the vehicle flies. It has to be reinforced again which increases the weight beyond the required value. Instead of using strong reinforcements for a thin wing and spoiling its airfoil shape, a thick wing would be better.

The airfoils are chosen from the shortlisted airfoils of 10% thickness. S1223 and Clark Y are considered and analyzed thoroughly and the results are shown according to the guidelines shown before. S1223 airfoil: This Selig airfoil has a good coefficient of lift (Clmax=2.2) but the pitching moment coefficient (Cm=-0.273) is too large. The stall angle is around α=120 which is far away from the operating angle of attack α=3.50.


Wing Loading: In most airplane designs, wing loading is determined by considerations of

and landing distance. As landing distance is not considered in the design, the wing loading is determined purely by . The stall velocity is the minimum velocity with which the aircraft can fly without stalling. The stall velocity for these kinds of vehicles is considered as 6 to 7 m/s.

Aspect Ratio(AR):It is one of the design considerations that should be taken into account. The high-aspect ratio-wings do not experience much loss of lift and increase in drag due to tip effects compared to a low-aspect ratio-wing of equal area. The maximum subsonic L/D (lift to drag ratio) of an aircraft increases approximately by the square root of an increase in aspect ratio but the wing weight also increases with aspect ratio. Due to the reduced effective angle of attack at the tips, a low aspect ratio wing stalls at higher angle of attack.Thisvehicle which should have a good gliding capability and so the aspect ratio should be that used for the sail planes. An aspect ratio of 10-12 can be used.

From the formula

b, the wing span can be calculated.

Wing Configuration

The wing configuration has certain aspects like wing sweep, taper ratio, wing twist and dihedral. The wing configuration is generally finalized from the data available in literature about similar kind of existing vehicles. The wing sweep is used for reducing the transonic and supersonic drag. It reduces the coefficient of lift at low speeds. The vehicle travels at very low speeds, so the sweep is not necessary.

The wing taper ratio is the ratio between the tip chord and the root chord. By the Prandtl Lifting Line theory, induced drag reduces when the lift is distributed in an elliptic fashion. This demands that the wing shape be an ellipse which is difficult to construct. The tapering of the wing makes the Lift distribution elliptical. Most wings of low sweep have a taper ratio of 0.4-0.5 whereas most swept wings have taper ratio of 0.2-0.3.The taper ratio is taken as 0.45 which is generally used for many designs.


The wing twist is typically used for unswept and untapered wings. It is used to prevent tip

stall and to give an elliptic lift distribution. This vehicle does notrequire any wing twist as it makes its construction difficult.The dihedral is used for lateral stability. Positive dihedral tends to roll the aircraft level and thus increases the lateral stability of the vehicle. The dihedral is chosen from historical data and this depends on the high, low or mid type wing configuration.

The mid wing configuration is suitable for this vehicle as it is easy to manufacture and does not add additional drag. A dihedral of to is generally used for a mid unswept wing.The wing is constructed in XFLR and simulation is done. A dihedral of 30 proved effective from the simulation.

Fuselage Length: For the initial estimate of the Fuselage length, the formula given in the [21] is used. For a sail plane Fuselage Length is given by L= where a=0.71 and C=0.48 and is the weight of the vehicle.

Tail Configuration There are different tail arrangements that can be possibly used.

The T-tail configuration has the disadvantage that at high angles of attack (during climb), the horizontal tail comes under downwash and if the wing stalls by any chance, the tail which is in the downwash also gets stalled.Thus the vehicle cannot be recovered from the stall region. This makes the vehicle unstable at higher angles of attack.

Without further analyses, the conventional tailis considered as it is easy to manufacture and also structurally strong.

Calculating the planform area of the horizontal and vertical tail:

where VHT and VVT are the horizontal and vertical tail volume coefficients.

=Horizontal distance between the C.G of the airplane and the aerodynamic center of the horizontal tail.

= Planform area of the horizontal tail


=Horizontal distance between the C.G of the airplane and the aerodynamic center of the

vertical tail.

= Planform area of the vertical tail

= Mean Aerodynamic chord of the wing

Tail Sizing The geometry of the tail can be calculated similar to the wing assuming it as rectangular tapered wing.Tail aspect ratio and taper ratio for sail plane are taken from the historical data given in [21].The above discussed characteristics are calculated for both the airfoils considered, S1223 and Clark Y and are listed in table 2.7.

Table 2.7 Characteristics of the vehicle designed using S1223 and Clark Y airfoils

The table shows the wing and tail geometries of both the vehicles. The wing span of the vehicle should be within 1.5 cm according to the design requirements. A high aspect ratio wing has good

Characteristics S1223 airfoil vehicle Clark Y airfoil vehicle

Wing Loading 59.722 kg/m2 49.284 kg/m2

Wing surface area 0.246 m2 0.3 m2

Wing span 1.6m for AR=10 1.73m for AR=10

1.55 for AR=8

Rectangular tapered wing

(root and tip chord) 21.5cm , 10cm 24cm, 11cm

Horizontal tail

Span 38 cm 44cm

Root and tip 7.5cm , 4cm 10cm, 5cm

Vertical tail Span 14 cm 16cm

Root and tip 10cm , 5 cm 12cm, 6.5 cm


gliding capabilities which are required for the surveillance applications. The aspect ratio of 10 used in sail planes is used in this design. The wing span of 1.6 m for the S1223 vehicle is acceptable. The wing span of the Clark Y airfoil is greater than 1.7 m which gives makes the vehicle difficult to carry. The tail surface area is calculated from existing formulae from literature. The wing and tail geometries are considered to be rectangular and the root and tip chords are mentioned in the table. The tail surface is mainly used for stability of the vehicle.

The vehicles are constructed in XFLR5 and stability analysis is performed to calculate the areas of the control surfaces. The analysis shows that the tail surface of the S1223 vehicle is not sufficient for stabilizing the vehicle longitudinally. This is due to the high pitching moment coefficient of the airfoil as given in table 2.5. The Clark Y airfoil vehicle is longitudinally and laterally stable and the initial estimate of the control surface areas are obtained from the analysis. The Clark Y airfoil with a reduced AR of 9 is finalized due to its good stability characteristics. The final estimates of the weights, wing and tail geometries are not calculated as a vehicle-Fpv Raptor is procured from the market. This vehicle satisfies the design requirements and is used instead of constructing a new vehicle. In the next section, these characteristics of the vehicle are compared with the Fpv Raptor vehicle.

2.3 Fpv Raptor During the design process, similar kind of high aspect ratio vehicles are searched in the

market. The vehicle - Fpv Raptor is chosen as it satisfies the design requirements.The weight of the UAV is 1.3 kg including battery and autopilot which is well within the range(1.5 kg). The wing span of the vehicle is 1.6m while the wing span for the vehicle designed with Clark Y airfoil is 1.7m. The Fpv vehicle has a pusher configuration (propulsion system). This makes the vehicle easy to handle as the propeller does not damage during skid recovery.The UAV is procured from the market and is modeled in XFLR in the following way.

Modeling the airfoil and wing: The airfoil of the wing is obtained approximately from tracing it on a graph sheet. The coordinates of the airfoil are entered in text file and stored in “.dat” format. The XFLR takes this file as input and can simulate the airfoil coefficients for required Reynolds numbers. The tail is constructed using NACA 0006 (National Advisory Committee for Aeronautics) airfoil. This airfoil is almost similar to a flat plate and is generally used for construction of tail. The body of the vehicle is constructed as shown in the figure 2.2.The wing body interactions are considered to some extent in this software. The predicted data in the


presence of wing-body interaction is compared with the data without body. The results show a decrease in lift coefficient, glide ratio and increase in drag coefficient.

Figure 2.3 Airfoil of Fpv Raptor

The wing and tail dimensions are calculated and reconstructed using XFLR5.The model of the FPV Raptor is given in figure 2.6.

Figure 2.4Fpv Raptor modeled in XFLR


The Fpv airfoil is analyzed in XFLR5 and the resulting data is compared with that of the Clark Y airfoil data in the table2.7.

Table 2.8 Comparison of airfoil properties

Clark Y Fpv Raptor

1.39 @ α= 12.50 1.05 @ α= 100

72 @ α=4.50 62 @ α=30

-0.075 -0.08

The glide ratio, and coefficient of lift and pitching moment is better for Clark Y airfoil when compared to the Fpv Raptor in simulation. The 3-D wing properties are shown in the table 2.8. Figure 2.5 shows the coefficient of lift, coefficient of drag, coefficient of pitching moment and glide ratio plotted with α (angle of attack) for both Clark Y and Fpv vehicles.

Table 2.9 Comparison of vehicle properties

Clark Y Fpv Raptor

24 @ α=4.50 16 @ α=10

operating 0.42 @ α=4.50 0.5 @ α= 10

The coefficient of lift at operating alpha is greater in case of Fpv Raptor but the glide ratio is smaller than that for Clark-Y vehicle. The Fpv vehicle is trimmed with zero elevator angleand has good handling qualities.

The coefficient of lift of the Fpv vehicle is slightly less than that of the Clark Y vehicle but increases with the same slope. The coefficient of drag of Fpv vehicle increases steeply for angle of attack greater than . This is not a good characteristic as the overall glide ratio comes down at higher angles of attack. The pitching moments of both the vehicles are good. The vehicle can be made stable in the longitudinal motion with a slight elevator angle without saturating the servo. The glide ratio of the Fpv vehicle comes down steeply which is not a good flying characteristic. The Fpv is flown by many hobbyists and it is understood that the flying


characteristics of this vehicle are good. The anomaly between this and the simulation results might be due to the inability of the XFLR software in estimating the properties of the new Fpv airfoil. It might also be due to the human error in exactly reproducing the airfoil shape. The properties of the vehicle are given in the table 2.5.

Figure 2.5 Comparison of characteristics of Fpv Raptor and vehicle using Clark Y airfoil


Table 2.10 Properties of the Fpv Raptor

Name Fpv Raptor

Weight (Flying weight) 1.360kg

Wing span 1.6m

Wing Area 0.327 m2

Root chord and M.A.C 19.8 cm ,18.56 cm

C.G position from nose x= 35cm y=0cm z=8.5 cm

Wing incidence 4.60

2.4 Conclusion The design of an MAV according to the design requirements is done in this chapter. The

Stability and Control analysis is done to finalize tail geometry and area of control surfaces. The thrust calculations, current calculations, final evaluation of aerodynamic design are not done as the vehicle is not designed for fabrication. The Fpv vehicle is modeled and its performance is compared with the simulation results of the vehicle usingClark Y airfoil. The results show a degraded performance of Fpv but it is due to the inability of the software in accurately predicting its properties.

35

Chapter 3

Mathematical Modelingof Fpv Raptor

3.1 Introduction The aircraft dynamics is nonlinear and coupled system of equations. For the purpose of

designing flight control understanding of the system, these equations are linearized and decoupled into longitudinal and lateral model. These decoupled equations are analyzed to understand the motion of the aircraft in both longitudinal and lateral mode. The mathematical model of the aircraft, a state space realization is used for the flight control system and simulation of aircraft under external disturbances.

In this chapter, the XFLR5 and AVL softwares are used to estimate the mathematical model of the Fpv Raptor vehicle. The compound pendulum method is used to estimate the inertia values of the vehicle. Various inputs (step input and pulse input) are applied to the open loop plant in both longitudinal and lateral modes and their responses are analyzed. 3.2 Mathematical Modeling

For a fixed wing aircrafts, six degrees of freedom equations of motion are used to define motion in flight. The aircraft motion can be categorized as longitudinal motion, lateral motion or a combination of both. The purely longitudinal motion does not include lateral motion and purely lateral motion has very little effect on longitudinal motion. Therefore, to reduce complexity, the longitudinal and lateral motion are decoupled and defined as two separate motions with three degrees of freedom each. It is further considered that the aircraft experiencesvery small perturbationsfrom its equilibrium condition during flight.

3.2.1 Linearized Decoupled State Space Model

The decoupled perturbed linear equations of motion are taken from [27]. The equations are repeated here for the sake of completeness.

Chapter 3. Mathematical Modeling of Fpv Raptor 36

(3.1)

(3.2)

(3.3)

(3.4)

(3.5)

(3.6)

The equations 3.1-3.3 define the longitudinal motion of the aircraft and 3.4-3.6 define the lateral motion. The lateral (roll) and directional motion (yaw) are highly coupled motions and cannot be decoupled.

Linearized Longitudinal model The longitudinal motion is the motion of the airplane in the X-Z plane. This motion can be defined by a minimum of four states. They are identified as the forward velocity, velocity in the z-direction, pitch, and pitch rate. The longitudinal motion is controlled by a deflection of elevator and change in throttle. The linear model is given here.

U is the velocity of the vehicle at equilibrium state of the aircraft. The system matrix entries are a function of stability derivatives. These depend on the vehicle to be modeled and its desired equilibrium condition.


where A= , B= , C=

Linearized Lateral model The lateral motion is the motion of the aircraft in the X-Y and Y-Z planes. Its motion can be defined by a minimum of four states. These are the angle of side slip, roll, roll rate and yaw rate. The lateral motion is controlled by the differential deflection of ailerons and rudder.


where A= , B= , C=

The procedure to obtain the mathematical model from simulation softwares is discussed here. The mathematical model of the vehicle is the state space representation of the linearized longitudinal and lateral modes. The state matrix consists of stability derivatives and the input matrix consists of control derivatives. The vehicle is reconstructed in XFLR5 software as shown in Figure 2.2. The wing, tail and the control surface geometries are calculated from the vehicle and are used in the wing and plane design in XFLR5 software. The trim condition is calculated from the stability analysis provided by the software. The geometry of the vehicle is exported to AVL software. The AVL software does not have an editor and is very difficult to build a model directly. It takes the geometries of wing, tail and control surfaces in the form of a text file. This


text file is obtained from the XFLR5 software which has a user friendly editor. The AVL also takes a mass distribution file as input which requires the center of gravity (C.G) location and moment of inertia values of the vehicle. The estimation of the inertia values is discussed in the next section. The AVL takes the geometric values and inertia values of the vehicle as input.The model of the vehicle is calculatedby entering the required trim velocity or angle of attack α. The output file is a text file which contains both lateral and longitudinal model of the vehicle at the specified trim condition.

The mathematical model for different trim velocities can be obtained in a similar manner. The XFLR software could not predict the characteristics of the 2-D airfoil at higher α. This is due to the inefficiency of reproducing the actual airfoil shape. These result in unstable plant models at high α. In order to overcome this difficulty, the stability derivatives for lower α are used to predict the plant models for higher α. This method produced stable plant models for all the trim conditions. This process is discussed in detail in section 3.2.3.

3.2.2 Estimation of Moments of Inertia

The moment of inertia values are computed using compound pendulum method. The method is widely used to estimate the inertia values of rigid bodies and is discussed in [28-30].The theoretical calculations of inertia can be done using second moment equations. Thename is because of the squared moment arm that multiplies each infinitesimal volume during integration. The constant mass and density is assumed throughout the analysis, we can use the simplified estimation equations


where xi, yi are the positions of the individual masses which make the UAV. The masses are estimated approximately and each weight is taken as a point weight acting at xi, yi. Both Ixy and Iyz are zero because of symmetry and the C.G location is taken as xC.G = 35 cm and yC.G = 8.5cm.The inertia values estimated using above method are compared with the theoretical values in table 3.1 Table 3.1 Theoretical and experimental moment of inertia values

Moment of Inertia Theoretical value in kgm2 Experimental value in kgm2

Ixx 0.0548 0.130254

Iyy 0.080712 0.145874

Izz 0.1088 0.233184

Ixz 0.004098 -----------

3.2.3 Stability and control derivatives

The stability and control derivatives are obtained from XFLR and AVL data and modified using Matlab. The procedure is discussed in detail in Appendix given at the end of this chapter. The stability and control derivatives for a particular trim velocity are given by the numerical equations given in the Appendix.

3.2.4 Throttle Modeling

Fpv Raptor is throttled by an electric motor and 8×4 inch propeller. The wind tunnel data of propellers are given in [23]. The advance ratio (J) versus coefficient of thrust ( ) curve for the 8×4 inch propeller is taken from the data given in [24] and is shown in Figure 3.1. The power required and power available curves versus forward velocity are obtained using Eqs 2.1-2.6 in Section 2.2.2.


Figure 3.1 Coefficient of Thrust versus Advance Ratio for 8×4 inch propeller

Figure 3.2 Power available and Power required curves versus velocity


Figure 3.2 shows the flight regime in which Fpv Raptor can fly. The minimum velocity is limited by the stall velocity. The maximum velocity is limited because of the unavailability of required power from the motor.

3.2.5 Actuator Modeling

The stability derivatives of the vehicle uniquely define its dynamics whereas the control of the aircraft also depends on the actuator used for control surface deflection. The actuators in the Fpv vehicle have limited performance due to its small size and low weight. The actuator response is slower compared to the vehicle response which reduces the pilot authority to correct the motion. Therefore, it becomes necessary to accurately model these actuators and use it while designing the control.

The servo is assumed as a first order transfer function and its time constant is calculated from a simple experiment using a servo driving kit [31]. The servo is driven with a PWM signal to move the horn form one position to another. The horn is connected to a spring load and a potentiometer. The output response is a step response and the time constant of the system is calculated from it. The transfer function of the actuator is

(3.7)

The gain of the servo is the ratio of the amount of deflection obtained to the amount of deflection needed. The gain of the servo in no load condition is 1 and this is assumed to be true in loaded condition. The figure 3.1 shows the response of the servo to a step input.

Figure 3.3 Response of servo actuator to step input


3.3 Open Loop responses of Longitudinal and lateral Model The mathematical model for a trim velocity of 12 m/s is given below. The longitudinal model has (forward velocity, velocity in z direction, pitch rate and pitch angle) states. It takes the elevator deflection as input. The lateral model has (velocity in y direction, roll rate, yaw rate and roll angle) states. It takes the aileron and rudder deflection as inputs.

are the longitudinal state and input matrices respectively. are the lateral state and input matrices respectively. The characteristics of the plant are given in table 2.2.

Table 3.2 Characteristics of the plant model

Aircraft Mode Damping (ξ) Frequency ( ) in rad/sec

Phugoid mode 0.03 0.982

Short period mode 0.62 7.72

Dutch Roll Mode 0.0282 1.65

The military specifications for these kinds of vehicles are given in [3]. The plant

characteristics are compared with the given military specifications. The phugoid mode frequency is within the specifications but its damping is very low and has to be improved. The short period frequency and damping are within the specifications. The dutch roll damping is very low and


also has to be improved. The open loop responses of longitudinal and lateral plant modelfor unit step, unit pulse are discussed in the sections3.3.1-3.3.3.

3.3.1 Response to Step input

The open loop plant is driven with unit step input to the elevator, aileron and Rudder and the responses are shown in the figures 3.2-3.4.

Figure 3.4Response for unit step input to elevator

The figure shows that if the elevator is deflected by 1 degree, the variation in the longitudinal variables are within bounds and also practically feasible. The phugoid damping of the model is low causing the low damping observed in the plot. The positive step is causing negative θ which can be understood by the θ/δ transfer function. The positive deflection of elevator is downwards. The downward deflection causes the aircraft to pitch down which results in negative θ. The aircraft’s velocity increases when the aircraft is pitching down and this result


in a positive deflection of forward velocity. The vertical velocity changes as there is a component of forward velocity acting downwards as the aircraft is pitching down.

Figure 3.5Response for unit step input to aileron

Figure 3.6Response for unit step input to rudder


The response for the aileron and rudder inputs suggests that the damping of the dutch roll

mode is poor and has to be improved. The spiral mode is unstable due to the right hand side pole. These problems must be overcomed in the closed loop system to be designed. The positive step input to the aileron rolls the aircraft to its right whereas a positive step input to rudder yaws the aircraft towards negativey-direction.

Similar analysis holds for the pulse input givenin the next section and it can be seen that the response to the elevator input is within bounds but that of the lateral model is not. This is due to the unstable spiral mode. 3.3.2 Response to Pulse input

A pulse of 1 degree deflection is given for a very short period of time (around 0.1sec) for the control surfaces and the responses are shown in the figures.

Figure 3.7Response to unit pulse elevator


Figure 3.8Response to unit pulse aileron

Figure 3.9Response to unit pulse rudder


3.3.3Response to 3-2-1-1 input

The 3-2-1-1 input is chosen such that the all the modes are excited. This is ensured by calculating the width of the pulses using the formula available in Appendix B of [34].

For the short period mode to be excited the value of Δt is 0.15 and for the dutch roll mode, the value is 0.9. The output responses are shown in the figure 3.11-3.13.

Figure 3.10 Response to 3-2-1-1 input to elevator

Figure 3.11 Response to 3-2-1-1 input to aileron


Figure 3.12 Response to 3-2-1-1 input to rudder

3.4 Conclusion

The Moment of Inertia values of the vehicle are calculated using Compound Pendulum method. The compound pendulum method is widely used for calculating the moment of inertia of the unmanned aerial vehicles. This method is proven to give better results compared to other methods. The servo actuator used in the vehicle is modeled using an experimental setup. It is assumed as a first order system and its transfer function is determined. Although this process is not so accurate, it works well in our application. The open loop step and pulse input responses are plotted. They indicate the behavior of the aircraft for disturbances during flight. The responses indicate poor damping characteristics and the need for a controller to improve the handling qualities of the aircraft.


3.5 Appendix

3.5.1 Generating Control and Stability Derivatives

The xfoil softwares could not calculate all the coefficients for the required angles of attack. This caused instabilities in calculating the stability derivatives for some angles of attack. These instabilities cause an unstable plant model. In order to get a stable plant model for all velocities, the XFLR and the AVL data is exportedto Matlab and curve fitting is done. The variation of stability and Control derivatives with alpha are numerically obtained. The variation of alpha with velocity is also obtained. So for a given a velocity, the angle of attack, stability and control derivatives can be obtained using the numerical equations shown below.

The angle of attack (α) for a given velocity (V) is given by the relation:

The longitudinal and lateral stability and control derivatives for a given angle of attack α are given below:

53

Chapter 4

Iterative Linear Matrix Inequality Algorithm in

Probabilistic Framework

4.1 Introduction The mathematical model of the Fpv Raptor Vehicle is obtained using Xfoil softwares.

There is no wind tunnel data to validate the results or to check the accuracy of the results. The test flights conducted on the vehicle provides a way to test the performance of the model but with certain limitations. The magnitude of the wind gusts or the wind disturbances during the test flights is unknown which puts a limit on the accuracy of the flight data. Therefore, the uncertainty in the model parameters is high. The non-linear equations are linearized in order to reduce the complexity of the controller design and this again limits the performance of the controller. The low cost, low performance MEMS based sensors used on these vehicles have their own error bound, which is not negligible. In order to get robust performance to overcome the above uncertainties and disturbances, modern control synthesis technique has to be used. The objective of the control is to improve the flying qualities of these vehicles by placing the poles of their dynamics at desired regions in s-plane. The hardware on which the control is implemented often has limited scope for implementing a generic feedback structure. The onboard implementable control structure on Kestrel [32] Autopilot hardware is PID based. This makes the control design problem a fixed order multi-objective control design exercise.

The iterative linear matrix inequality (ILMI)[15, 17] is a multiple objective automatic tuning algorithm. The solution is iteratively found by loosening the performance criteria which needs to be satisfied while optimally placing the poles to minimize the real part of the largest eigenvalue. The solution is also constrained to satisfy the performance criteria. Optimal pole placement is done to maximize stability, gain and phase margin while achieving the desired performance. The robust stability criterion is also satisfied by the obtained solution.

Chapter 4. Iterative Linear Matrix Inequality Algorithm in Probabilistic Framework 54

Several problems in robustness analysis and synthesis of control systems are NP-

complete or NP-hard. The solutions to these problems are computationally intensive. An approach gaining popularity for solving these problems is the use of randomized algorithms.In the classical robust control techniques, the objective function to be minimized is the worst case performance index which results in over-conservative designs. In statistical learning theory, the objective function is to minimize the average performance index of the controller.This approach of using randomized algorithms helps to solve the intractable problems. The penalty is however, these algorithms are probabilistically complete. The probability of failure ofrandomized algorithms can be arbitrarily made close to zero but can never be exactly equal to zero. In [33] the probabilistic robust design with linear quadratic regulators is discussed.

4.2 Linear Matrix Inequalities Linear Matrix Inequalities (LMIs) and LMI techniques have emerged as powerful design tools in areas ranging from control engineering to system identification and structural design. The LMI formulations, for checking if a particular control solution satisfies the various , or pole placement based criteria, leads to a convex optimization problem. But the problem of finding a control solution satisfying these criteria is not computationally easy. They are often bilinear in nature and have to be modified into solvable convex quadratic or linear matrix inequalities. The modification of one such bilinearmatrix inequality(BMI) for pole placement to minimize the largest eigenvalue of the system is presented in this section.

4.2.1 LMI for pole placement on the left hand side of line in complex s-

plane

Consider an open loop plant which has the system matrix [Ref Eq. 4.1] and its closed loop system [Ref Eq.4.2] such that (Fig 4.1). The feedback gain is K.

(4.1)

(4.2)


Fig 4.1 General feedback plant

Lyapunov Conditions for Pole Clustering If D is a sub region of the complex left half plane then a dynamical system as given above is called D-stable if all its poles lie in D. If D is considered as an entire left half plane, then we look for a symmetric matrix,X satisfying

(4.3) such that is stable, [14] If the pole placement is on the left side of line then corresponding LMI’sare

(4.4)

Similar LMI formulations are also proposed for cone, disk and other shaped region in the complex plane. The closed loop matrix can be expressed in terms of open loop system matrices if the controller K is static or is a real value matrix as given below.

Substituting this equation in (Eq. 4.2)

(4.5)

G(s)

K

y u

Open loop plant

Closed loop plant


Here the unknowns are X, K and α. Therefore we can solve a feasibility problem for X and K for a given α or we can solve an optimization problem to minimize α and find solution for X and K. In both the cases the problem becomes bilinear which is computationally difficult to solve. The available Matlab tools [16] can solve only quadratic matrix inequalities (QMIs) and LMIs. Therefore above problem is converted into QMI in [17]. Substituting X by P in Eq. 4.3 and not considering the inequality P > 0 for time being, we can write

(4.6)

Adding and subtracting gives

Now a term is added on the left hand side. This term is always positive. Assume that inequality holds even after adding this term. The justification of the assumption is done later in the derivation.

It can be written as

(4.7) The proof for the above result is in [17]. The Eq. 4.5 is convex quadratic in nature except the negative term, ( ). This term prevents the equation from transforming into a solvable LMI. To make the inequality quadratic, we introduce another variable X such that,

(4.8)

for any X and P of the same dimension. Solving the above equation gives

where equality hold when X=P Taking transpose of the above equation and considering the property that X and P are symmetric results in

Substituting the above equation in Eq. 4.5 results in

(4.9)


Theα stability synthesis matrix inequality can now be written in QMI form as

(4.10)

(4.11) where ∑ = , The result of adding a positive definite term is that even for a system with all its poles in the left half complex plane, the α solution can be positive. Therefore, the term added makes the solution slightly over conservative.

4.3 Static Output Feedback for Non Ideal PID

Implementation The ILMI algorithm and its disadvantages are discussed in [22]. The implementation of static output feedback for ideal PID implementation is also discussed in [22]. In this thesis, the modified ILMI is used and its advantages over the ILMI algorithm are discussed in the next section. Figure 4.2 Closed loop system with PID-PD feedback

PD

X G(s)

+ PID

-

+

+


In the Kestrel Autopilot, the differentiator and integrator are of standard form as given in Eqs. 4.11, 4.12.

(4.12)

(4.13)

The values of these constants cannot be changed by the designer and are given in [35] The PID and the PD blocks have the following transfer functions

(4.14)

(4.15) The are the gains of the PD block corresponding to and are the gains of the PID block corresponding to output . Let three blocks of G, PD, PID be respectively and the corresponding state matrices be

, ( ) and ( ) and the state space representation is as shown in the equations below.

(4.16)

(4.17)

(4.18) Substituting , in the above equations and solving for the state output feedback form, we get,

(4.19)


(4.20)

The above equation can be represented as

can be written as and for r and u inputs respectively.

Similarly

Substituting , , in equations 4.15-4.17, then the closed loop system with r as an input can be defined as

(4.21) The above equation can be written as

where can be expressed in terms of as in Eq. 4.21

(4.22)

where K=[- ] is the unknown gain matrix.The are the unknown matrices of the PD, PID controllers and needs to be determined. The LMIs. 4.9, 4.10 are solved using efficient algorithms to obtain the K matrix.


If K=[ ] then , , , . The SOF closed loop system is shown in the figure 4.3.

Figure 4.3 SOF closed loop system

The K matrix is the feedback gain matrix but we need to calculate the PID gains from this K matrix. Let where the elements of the matrix are the PD,PID gains which are unknown.

(4.23)

where (4.24)

(4.25)

The matrix M can be singular only if any one or more of are zero. But they cannot be zero because:

1. If are zero then the integrator and differentiator transfer functions are zero respectively.

2. If are zero then the integrator and differentiator transfer function have their pole and zero on the origin respectively, which makes the corresponding transfer function a constant gain system.

P(s)

K

u

r y


4.4 Generalized Plant for H2/H∞Control Design

In order to do a robust control design, the SOF plant is augmented with frequency weighting functions. The weighting functions are to minimize the error and control in the low frequency region and the weighting function is to improve the performance. The Generalised plant after augmenting the weighting functions is shown in figure 4.4. Let be the sates of the closed loop plant and r is the plant input and u is the control input.

G(s)

Figure 4.4 Generalized Plant

(4.26)

(4.27)

(4.28)

(4.29)

The generalized plant is of the form:

(4.30)

K

r

u

y


(4.31) The generalised feedback plant after feedback has a H2performance index γ and all its poles are at the left hand side of σ=α/2 in the complex plane. If there exists a SOF gain matrix K and P such that

(4.32) ∑= (4.33)

(4.34) (4.35)

The above formulation is derived by combining the LMI formulation for pole placement and the LMI formulation for performance. Using the above formulation, for a given performance index γ we can solve an optimization problem for KandX with an objective to minimize α while constraining the solution to satisfy the above inequalities.

4.5 Modified ILMI Algorithm The ILMI algorithm given in [22] is modified for the non-ideal implementation of the PID

controllers as is required in Kestrel Autopilot. The modifications done on the ILMI algorithm are listed below:

1. The dynamics introduced by the non-ideal implementation of PID controllers are incorporated and absorbed in the SOF formulation.

2. Constraints are put in terms of gain and phase margin on the closed loop system which are the best indicator of system stability in the classical sense. Constraints are also put on the damping of the closed loop poles. If the constraints are not met, then the optimal performance index is increased and the result is computed based on the new performance index.


3. The ILMI algorithm keeps on minimizing the trace(X) when no negative solution for α is

obtained. The negative solution for α is not obtained due to two factors. Here the significance of will be clear.

• The solution obtained for K, laces the poles in the left half plane such that α is a negative or zero solution but the positive definite term which was added makes the solution more conservative.

• If in the first iteration, for best performance index, the lower order feedback structure cannot achieve stabilizing control.

In both the above cases, the solution is to increase the optimal performance index.

4.5.1 The steps of modified ILMI Algorithm

The modified ILMI algorithm used to design the controller in this thesis is given below.

1. Find out the SOF representation of the system as in Eqs. 4.18, 4.19. 2. Augment the SOF output to obtain the generalized plant . Define (gain and

phase margin constraints) and any damping constraints on the control design. 3. Compute the optimal guaranteed performance index, , using function of

MATLAB. This serves as the starting point. 4. Initialize and also initialize the incremental step in γ as dγ. 5. For , obtain the initial X from the riccati equation below.

(4.36) where The Q is taken as Identity matrix. Initialize j=1

6. Using X from previous step, solve for P, , to minimize amd satisfy the LMIs below,

(4.37) ∑= (4.38)

(4.39) (4.40)

7. If then put


j=j+1 Goto step 5

8. Else if j=1 or if n>0 and n<10 Solve for and P , to minimize trace(P). The value of is obtained from the previous step.

(4.41) ∑= (4.42)

(4.43) (4.44)

i. If P has a feasible solution then Put X=P , n=n+1 , j=j+1 Goto step 6

ii. Else Put Goto step 6

9. Else if j and Put j=j+1, Goto step 6

10. Else if j and Form the closed loop system with feedback gain .

Find using normhinf function in Matlab. Find the Gain Margin (GM) and Phase Margin (PM) and the damping (Z) of the closed loop system.

i. If and and and then Goto step 11. ii. Else

j=j+1, Goto step 5 11. Final Gain 12. Find 13. END


4.6Sampling based ILMI Controller Design Designing a robust controller in the presence of structured uncertainty is computationally intractable or it is an NP-hard problem. Even when the uncertainty is defined in probabilistic framework, the exact computation of the probabilistic density function is computationally intractable. In order to get rid of these computationally intractable problems, the design of the robust controller is done in the probabilistic sense. This guarantees robust satisfaction of a certain characteristic or performance level of the system, if the controller works well for most (but not all) of the systems.

In the aircraft application, the uncertainty is mainly due to two reasons:

i. The model of the vehicle is obtained from XFLR5 and AVL which have their limitations in predicting the performance of the vehicle. There is no wind tunnel data on the vehicle which can be used to test the accuracy of the software used.

ii. The controller is finally designed for a certain trim condition. But during flight it is not possible to fly the vehicle in or nearby the trim condition. When the vehicle is flying at a different trim condition, then the model is different from that used to design the controller.

In order to overcome these difficulties, we introduce parametric uncertainty in the model and then design a controller which satisfies the performance index in a probabilistic sense. The stability derivatives in the mathematical model are perturbed by adding a percentage of uncertainty on each of them. The plant is constructed by choosing a perturbed plant and the controller is designed such that it stabilizes most of the plants within the perturbed region.

Monte Carlo Approach The parametric uncertainties in the stability derivatives of the model generate a plant parameter space. The plant parameter space is the product space formed by varying each parameter by its uncertainty.Using uniform sampling, we generate plant models, which we took as an overall representative of entire plant parameter space. The number of samples generated using Monte Carlo estimates, as

(4.45)

where is the confidence and ε is the accuracy parameter.


For δ=0.01 and ε=0.05, it is enough to generate 1060 plant models.

4.7 Monte Carlo based modified ILMI algorithm For each controller, the norm is calculated for all the plants used and the controller which gives the minimum value is chosen as the final controller. Once m number of plants are selected, a modified ILMI approach is used to yield a robust controller for each of the plant.

4.8.1 Steps of the combined Monte Carlo and modified ILMI algorithm

1. Sample number of plants mfrom plant parameter space

where is the confidence and ε is the accuracy parameter 2. For each plant

Apply modified ILMI algorithm for a plant from the plant space P. i. If a controller ( ) is feasible for then store both .

i=i+1, count=count+1 ii. Else i=i+1

3. For each controller Compute the closed loop system for each and find the norm.

4. Find the average norm for each controller and find the controller which gives the minimum value.

5. END

4.8 Conclusion Iterative linear matrix inequality algorithm is combined with Monte Carlo approach to obtain better results. The modified ILMI algorithm finds the optimal solution by simultaneously putting constraints on performance, norm, pole placement and gain and phase margin of the closed loop system. The damping of the closed loop poles is also taken into account. These constraints result in a controller which is robust. With respect to It also takes into account the non-ideal implementation of the PID logic.

69

Chapter 5

Controller Design

5.1 Introduction The dynamics of these kind vehicles is often lowly damped with a high natural frequency of operation. This implies that the handling qualities of these vehicles must be improved for a less-skilled pilot to fly the vehicle. This vehicle is designed for surveillance applications. It has to be flown by different pilots and also remain in the air for a longer time. These requirements will be satisfied only when the handling qualities of the vehicle are improved. Feedback control is useful in improving the characteristics of the aircraft. The guidance loop commands the attitude of the aircraft which is followed by the feedback control loop. From takeoff to landing, the vehicle operates at different operating points. A linear controller can either be designed at each of these operating points or a single controller can be designed to give satisfactory performance throughout the flight. The operating point in the middle of the flight velocity regime is selected and a controller is designed for that operating point. In a classical design the above method is followed and it is ensured that the controller works well for the other operating conditions with stable but degraded performance. Using a robust design in a probabilistic framework gives us the confidence that even with the presence of certain uncertainties in modeling the vehicle and in the environment, the controller designed will stabilize the vehicle and also gives better performance. The choice of the feedback control structure is limited as the designed controller is implemented using the Kestrel Autopilot.

5.2 Longitudinal Control Design The longitudinal motion is dominated by the phugoid and short period mode. The eigen values and corresponding Eigenvectors of the longitudinal mode for a trim velocity of 12m/s is provided in the table 5.1.

Chapter 5. Controller Design 70

Table 5.1 Eigen Vector analysis for longitudinal motion

Modes States

Phugoid Short period

u -0.9851 -0.0489 0.0505i w 0.0968 0.0122i -0.8541 q -0.0980 0.0167i -0.1417 0.4911i θ 0.0206 0.0992i -0.0385 0.0539i

It is clear from the table 5.1 that the pitch rate feedback (q) and velocity (w) will improve the short period frequency. Likewise, the feedback of attitude (θ) and forward speed (u) will enhance the phugoid mode characteristics. However, large gain for θ and u can saturate the actuator which is not desired. The phugoid and short period damping and frequencies compared with the required values are given in the table below. The minimum required values are based on MIL-F-8785C specifications.

Table 5.2 Longitudinal Control design requirements

Actual Required Phugoid damping 0.03 >0.3

Phugoid frequency 0.982 rad/sec 0.7-1.2 rad/sec Short period damping 0.62 0.35-0.8 Short period frequency 7.72 rad/sec 2-20 rad/sec

From the table we can understand that the phugoid damping should be improved. The Kestrel autopilot hardware allows programming of pitch rate feedback and an error in pitch angle to the elevator. We perform a classical design for a pitch error and pitch rate feedback.

5.2.1 Classical Feedback Analysis

The Nominal longitudinal plant is shown in the figure 5.1.The open loop plant transfer function (q/u) is given in Eq. 5.1.


(5.1) Figure 5.1 Longitudinal Nominal Plant

Pitch rate feedback The classical root locus design is done to improve the phugoid damping to a value of 0.4

while the other parameters remaining constant. The root locus has to be changed in order to improve the damping. The root locus must satisfy the magnitude and angle condition at a point if it has to pass through it. These conditions are used to get a stabilizing controller for the plant. Figure 5.2 Root locus for pitch rate feedback

PD

G(s) +

PID

+

+


Pitch Control

The vehicle is designed to undertake surveillance missions. During the mission it is often required to maintain constant attitude. In an actual flight the vehicle is subjected to disturbances, therefore maintaining attitude is cumbersome for the pilot. Pitch tracking control can reduce the pilot effort by maintaining the pitch. One more advantage of the pitch feedback is the improvement in phugoid damping. The final closed loop system is

(5.2) and its step response is

Figure 5.3 Step response of the closed loop system (classical design)

From the step response, it is understood that the rise time is less than 4 seconds and the settling time is 20 seconds. The low rise time indicates that in tracking , the aircraft responds quickly but the high settling time makes the aircraft to achieve the required angle in around 20 seconds. The low rise time is due to the phugoid pole which has a damping 0.3. The high settling time is due to the pole at -0.1961 which is closer to the origin than the phugoid pole. This dominant pole has a damping of 1 which makes the system as over damped resulting in the sluggish response. If the dominant pole is moved towards the left in complex s-plane, then the phugoid pole dominates the response. The response shows an overshoot and decrease in settling


time. This indicates that the aircraft overshoots the required pitch angle and then settles quicker than the previous case. If the dominant pole is moved towards the right, then it becomes the dominant pole. The response shows a sluggish response increasing both rise time and settling time. The phugoid mode does not appear in the response which results in high rise time.

5.2.2 Feedback design using Monte Carlo based modified ILMI algorithm

The plant space for the longitudinal plant is formed by perturbing the stability derivatives and the control derivatives. The state matrix for the longitudinal case is

(5.3) The table 5.3 summarizes the uncertainty considered in the stability derivatives. The probability density function is considered as uniform as mentioned before.

Table 5.3 Uncertainty considered for the stability derivatives

The control derivatives are given a perturbation of and the stability derivatives with respect to w are given a perturbation of . The stability derivatives with respect to u, q are given a perturbation of .

Stability Derivatives

Uncertainty in percentage

Stability Derivatives

Uncertainty in percentage

40 20 40 20 40 40

20 40

20 20

20


The operating point is 13m/s and so the optimal plant which is perturbed is the model obtained at the operating point. The plant space is formed by adding the above specified uncertainties to the optimal plant. The pitch and pitch rate feedback are done simultaneously.

Generalized Plant The plant (longitudinal mode with actuator dynamics) in the equivalent SOF form is

augmented with the weighting functions to get a generalized plant.The control system components in the closed loop system should minimize the weighted outputs . The

are performance measures whereas gives the robust performance measure. The generalized plant is shown in the figure 5.4

Figure 5.4 Longitudinal Generalized plant The selected weights are

(5.4)

(5.5)

PD

G(s) + PID

+ +

-

+


(5.6) The control effort should be minimized at the low frequency range where the vehicle

generally operates. The weighting function ( ) is a low pass filter which has high gain in low frequency region and the gain starts decreasing for frequency greater than 0.21 rad/sec. The higher gain in the low frequency region ensures that the control effort is less. As the gain decreases, the required control effort increases but this occurs at higher frequency beyond our operating range. The weight ( ) minimizes the error dynamics in the low frequency range. The weight ( ) determines the performance. This weighing function is complement to the

weighting function and is thus a high pass filter as given in Eq 5.6. The criterion for choosing weights is given in [36].

and are the minimum gain and phase margin to be satisfied by the closed loop plant. and

The algorithm can now be used to design the P,I,D gains for the pitch loop and pitch rate loop. The gains to be put in the Virtual Cockpit window of Kestrel autopilot are given in the table 5.4.

Controller obtained using the Algorithm

Pitch rate loop (PD):

(5.7) Pitch loop (PID):

(5.8) The integrator and differentiator transfer functions are fixed and they are given in Eqs. 4.11 and 4.12 which are repeated here

(5.9)

(5.10) The values of the constants are fixed and their values are given below


Table 5.4 Longitudinal PID gains

Eigenvalue and Eigen-vector analysis The Eigenvalue-Eigenvector analysis gives us the information about the effect of

eigenvalues on each state. It is given in table 5.5. The individual modes are also indicated in the table. The , are the states of the PID controller. The is the state of the PD controller.

Table 5.5 Eigenvalue-Eigenvector of closed loop plant

The closed loop and open loop phugoid and short period characteristics are compared in the table 5.6.

Kp Ki Kd

Pitch rate loop -1.0611 0 0.0210

Pitch loop 1.5481 1.1673 -0.1797

Eigen values -5.22 10.16 i -6.0574 -3.1083 -0.3971 + 0.7498i -0.1577 -2.913

States Eigen Vectors u 0.0080 0.021i 0.0785 0.2622 0.9712 -0.9591 0

w 0.178 0.54i -0.9935 -0.6742 -0.1901 0.1004i 0.2788 0

q 0.8049 0.0530 -0.1252 0.0474 0.0348i -0.0029 0

θ -0.017 0.05i -0.0088 0.0403 0.0102 0.0686i 0.0185 0

-0.0165 0.1492i -0.0597 -0.0386 -0.0087 0.0078i 0.0152 0

-0.0002 0.0002i 0.0005 -0.0664 0.0252 0.0183i 0.0426 0.3182

0.0031 0.0017i -0.0028 0.2063 0.0037 0.0261i -0.0067 -0.9269

-0.0085 0.0537i -0.0169 0.6411 0.0211 0.0076i -0.011 0.1992

Modes Short period Phugoid


Table 5.6 Open loop and Closed loop characteristics

Mode Open loop Closed loop Phugoid damping 0.03 0.4 Phugoid frequency 0.982 rad/sec 0.848 Short period damping 0.62 0.5 Short period frequency 7.72 rad/sec 11.42 rad/sec

From the table we can observe that the phugoid damping in increased which gives a

better response. The short period damping decreased from 0.62 to 0.5 but is well within the limit of the specifications. The short period frequency increased from 7.72 to 11.42 but is well within the specified range. The step response for the closed loop system is given in figure 5.5.

Figure 5.5 Step response of the closed loop system (Robust analysis)

The step response indicates a low rise time but a large settling time. This is due to the pole which is close to the origin. This dominant pole has a damping of 1 and this gives the sluggish response leading to high settling time. The low rise time is due to the phugoid damping of 0.4. If the dominant pole is moved to its left in the complex s-plane then the phugoid mode becomes dominant and the system response becomes under damped. This results in overshoot and reduced rise time and settling time. If the dominant pole is moved towards its right in the


complex s-plane, then it becomes more dominant and the response becomes more sluggish. This results in sluggish response with high rise time and settling time.

The pitch angle and pitch rate feedback increased the damping of the phugoid mode and

thus improved the handling qualities of the vehicle. This closed loop system is driven by a reference pitch angle ( ). The gain margin and phase margin of the system obtained by breaking at the input is shown in the figure 5.6.

Figure 5.6 Bode plot of the system obtained by breaking at the input

The Bode plot shows negative gain and phase margin. Negative gain margin tells us that the return gain is less than 1 i.e. if there is any disturbance in the system then the system can tolerate the disturbance without amplifying it. The magnitude of the disturbance that gets attenuated is given by the gain of this system. The controller obtained for the plant at 12m/s is tested for 8m/s and 18 m/s and the resultant responses are shown in figures 5.7-5.10.


Figure 5.7 Step response of the closed loop system for plant model at 8m/s

Figure 5.8 Bode plot of the system obtained by breaking at the input (plant model at 8m/s)

Figure 5.9 Step response of the closed loop system for plant model at 18m/s


Figure 5.10 Bode plot of the system obtained by breaking at the input (plant model at 18m/s) The figures show that the controller performs well in the higher flight velocities and underperform at lower flight velocities. At 8m/s, the response is sluggish with high rise time compared to the response at 12m/s.In the autonomous mode, this is given by an outer loop which can be altitude hold loop or velocity hold loop. The altitude hold control system is described in the next section.

5.2.3 Altitude hold control system

The altitude hold control system helps the aircraft in holding a particular altitude [27]. The input to this system is given by a path planning algorithm implemented in the autopilot for autonomous flying. In this algorithm, the aircraft is made to follow a pre-specified path given in terms of waypoints. If the aircraft has to go from one way point to another at a constant height or at a steady climb rate, then the aircraft has to maintain its altitude at each point of time given by this algorithm. This is achieved by the altitude hold control system. The outer loop of this system is to track the altitude. The altitude is fed back using a PID controller. The longitudinal nominal plant ( ) given in Fig 5.1 forms the inner loop of the system. The pitch rate and pitch angle feedback helps to hold the required pitch angle ( ) given by the outer loop which is an altitude hold loop. Thus the is tracked by the output and the aircraft attains the altitude as required by the path planning algorithm


Figure 5.5 shows the altitude hold control system. If the altitude of the aircraft decreases

due to a sudden gust, then the aircraft has to come back to its original altitude.This requires a positive pitch angle which is driven to the inner loop. This angle is tracked by the inner loop. The

becomes zero when the altitude is achieved.

Instead of using altitude hold control as the outer loop, velocity control can be used. In this system, the forward velocity is tracked instead of altitude. This system is generally used in cases in which a particular trim velocity has to be maintained. If there is any change in the forward velocity, then the control system tries to correct it.

Figure 5.11 Altitude Hold Control System

The figure shows the altitude hold control system with the inner loop having pitch rate and pitch angle feedback. The input is given from a path planning algorithm.The output h tries to follow the reference input. The input is given by the PID controller and the inner loop tries to track this input as shown in the figure 5.5. The transfer function is computed from the relation shown in Eq 5.11.

which can be solved as (5.11)

PID +

+ +

PD

G(s) PID -

+ -


(5.12)

The transfer functions of , can be computed from the plant model and can be obtained. The can be obtained from Eq. 5.13

(5.13)

The closed loop response for the altitude hold control system is shown in the figure 5.6. The inner loop is the closed loop system from the previous section designed using robust synthesis.

Figure 5.12 Closed loop response of altitude hold control system

The step response indicates a low rise time but a large settling time. This is due to the pole which is close to the origin. This dominant pole has a damping of 1 and this gives the sluggish response leading to high settling time. The low rise time is due to the phugoid damping of 0.4. The aircraft flying autonomously with a path planning algorithm will track the reference altitude with a sluggish response as shown in the figure 5.6.


5.3 Lateral Control Design The Lateral motion comprises of thedutch roll mode, the spiral mode and the roll subsidence mode. The Kestrel autopilot allows feedback of roll rate and yaw rate to the aileron and rudder respectively. Roll angle feedback to aileron is also possible for attitude tracking. The table 5.6 shows the actual plant characteristics and the minimum required characteristics based on MIL-F-8785C specifications.

Table 5.7Lateral Control design requirements Actual Required Dutch Roll damping 0.19-0.36 0.0282

Dutch Roll frequency 1.2 – 4 rad/sec 1.65 rad/sec Roll subsidence < -6.8 -10.4 Spiral < 0 0.0836

The Dutch roll damping has to be increased a lot in order to meet the requirements. The spiral mode is unstable and it must be made stable.The nominal lateral plant model is shown in the figure 5.7.

Figure 5.13 Nominal Lateral Plant

PD

G(s) + PID

-

+

+

u

PD


Both classical and robust controller design should be performed on the lateral plant similar to the longitudinal plant. 5.4 Open loop and Closed loop simulation Matlab simulation is done to test the controller performance. Simulation is done to compare theopen loop and closed loop response for a commanded elevator. The open loop simulation is without any feedback. Pilot command is the only input to the elevator. The open loop response to a step elevator input is shown in the figure 5.6.

Figure 5.14 Open loop response for a commanded elevator (step input) From the response we can observe that the damping is very low which is causing the large settling time. The low damping of the response is due to the poor damping of the phugoid mode (0.03) as given in the table 5.2. The pitch rate goes to zero whereas the forward and vertical velocity and pitch angle does not go to zero. This is true because a step input is applied to the elevator which changes the elevator position. The elevator remains in its new position which gives new values of pitch angle and forward and vertical velocities. The closed loop response is given in the figure 5.7.


Figure 5.15 Closed loop response for a commanded elevator (step input) The closed loop response shows the improvement in the settling time. This is due to the increase in the phugoid damping to 0.4.The short period damping decreased from 0.62 to 0.5 but is well within the limit of the specifications. The short period frequency increased from 7.72 to 11.42 but is well within the specified range. The open and closed loop characteristics are compared in table 5.6 which explains the improvement in the response. The pitch and pitch rate is in rad/sec while the forward and vertical velocity is in m/sec. It is observed that the forward velocity changes in greater magnitude than the vertical velocity which is expected. The pitch rate goes to zero at a faster rate compared to the open loop. This signifies that the vehicle’s response to the wind gusts will be quicker which helps the plane to become stable within less time. If a unit step input is applied to a closed loop system, the response tells us the tracking nature of the system, i.e. how well the output tracks the input. It indicates the performance of the feedback configuration. In our application the PD, PID controllers constitute the feedback. Therefore, a step input to the closed loop system indicates the role of the designed controllers in improving the performance of the plant. In order to study this, a step input is applied to the closed loop system at the input ( ). The response is shown in the figure 5.8.


Figure 5.16Step Response of the closed loop system The input ( ) is driven with a step input. The output (θ) is shown following the input with a nonzero settling time. The output is taking a while to settle to its steady state value ( ) which is clear from the figure. The responses of the forward velocity, vertical velocity and the pitch rate are also observed. The pitch rate is settling to zero and the forward and vertical velocities are settling to a new value which is due to change in the pitch angle. The performance of the closed loop system is good which implies that the controllers designed using the combined Monte Carlo and modified ILMI algorithm made the open loop plant stable and also improved the performance. 5.5 Conclusion The combined Monte Carlo and modified ILMI algorithm developed in the previous chapter is used to design the pitch rate and pitch angle feedback controllers for the vehicle’s longitudinal plant. The closed loop system characteristics are compared with the open loop characteristics. The short period damping and frequency worsened from the open loop plant but well within the limits of the specifications mentioned. The improvement in the phugoid damping reduced the settling time of the responses. The open loop and closed loop responses are compared for a step elevator input which clearly indicates the improvement in the overall system. The tracking nature of the closed loop system is studied by driving the input with a step input and studying the response of the system. The controller designed using the algorithm definitely improved the performance of the system.

87

Chapter 6

Conclusions and Future Work

The design and flight control of miniature air vehicle is addressed in this thesis. The vehicle - Fpv Raptor is choseninstead of constructing a new vehicle.This vehicle is selected as it has a stable operation in radio controlled mode. It has good glide properties and hence it will land gently in case of power/battery failure. It is readily available at low cost since it is manufactured in large numbers and sold to hobby flyers.The vehicle is modeled using vortex lattice simulation based softwares. The softwares are limited in their prediction of the properties of the wing and wing-body combination. The wind tunnel data is not available which can be used to check the accuracy of the model. The longitudinal model indicates a short period frequency of 7.72 rad/sec with damping of 0.62 and phugoid damping of 0.982rad/sec with poor damping of 0.0359. The lateral model indicates a dutch roll mode of frequency 1.61 rad/sec with damping of 0.0125, stable roll subsidence mode and an unstable spiral mode. The characteristics show a poor phugoid and dutch roll damping far lower than that specified in the military specifications. The handling qualities of the vehicle must be improved by artificial stabilization.

The uncertainties in the model and the wind gust disturbances during flight insist on a robust control design. The classical robust design is either over conservative or computationally intractable. Therefore, the robust design is done in probabilistic framework in order to overcome both the above problems. The statistical learning theory which is based on the probabilistic framework suffers from the limitation that it works for most of the plants but not all. The uncertainties discussed above are represented as parametric uncertainties in the stability derivatives. The controller designed stabilizes the plant in the presence of these parametric uncertainties.Monte Carlo simulation is combined with modified ILMI algorithm to design the controller which satisfies the constraints on H2 and H∞performance, and also on gain and phase

margins.

Chapter 6. Conclusions and Future Work 88

The pitch rate feedback PD controller and pitch angle feedback PID controller is designed for an optimal plant model at 12m/s after introducing the parametric uncertainties. The closed loop system has improved phugoid damping of 0.4. The short period damping reduced to 0.5 which is within the range of military specifications. The short period frequency increased to 11 rad/sec but is within the specifications. A step input is applied at the input of the closed loop system and is found that the input tracking performance is improved. The controller is checked for its performance for different plant models in the flight regime and encouraging results are obtained. The response of the altitude hold control system showed the improvement in performance due to controller. Future Work

Due to the limitation of time, controller for the lateral plant is not designed. In the extension of the above work, the SOF plant model for the lateral model should be constructed and a controller should be designed using the steps followed for the design of the longitudinal plant controller. The performance of these controllers on nonlinear plant models can be verified before the actual flight. The gains are entered in the kestrel autopilot before flying the vehicle. The attitude tracking can be employed in order to verify the performance of the vehicle. The recorded flight data can be compared with the simulation data and the accuracy of the design can be verified. Pilot feedback is also useful in determining the improvement in the handling qualities of the vehicle.The kestrel autopilot is not open source which makes it difficult to change its path planning algorithm. If a different autopilot is used, then path planning and obstacle avoidance algorithms can be implemented and verified in-flight.

89

References

1. http://www.defence-update.com

2. http://www.globalsecurity.org

3. MIL-F-8785C Flying qualities for piloted airplanes November5, 1980.

4. AVL software http://web.mit.edu/drela/Public/web/avl/

5. XFLR5 http://www.xflr5.com/xflr5.htm

6. Bernard Etkin, Lloyd Duff Reid , Dynamics of Flight- Stability and Control, Wiley India Pvt. Ltd.

7. Hassan Noura Francois Bateman Control ofanUnamanned Aerial Vehicle Proceeding of the 7th International Symposium on Mechatronics and its Applications (ISMA10), Sharjah, UAE, April 20-22, 2010.

8. Prof. Dr. Szabolcsi, RóbertUAV Controller Synthesis using LQ-based Design Methods International Conference of Scientific Paper AFASES 2011 Brasov, 26-28 May 2011.

9. Yusong Jiao, Juan Du, Xinmin Wang, and RongXie, H-infinity State Feedback Control for UAV Maneuver Trajectory TrackingInternational Conference on Intelligent Control and Information Processing August 13-15, 2010 - Dalian, China.

10. Li Meng, Liu Li, S.M. VeresEmpirical Aerodynamic Modeling for Robust Control Design of An Oceanographic Uninhabited Aerial Vehicle, International Conference on Electronics and Information Engineering (ICEIE 2010).

11. Chenggong Huang Qiongling Shao, Pengfei Jin, Zhen Zhu, Bihui Zhang, Pitch Attitude Controller Design and Simulation for a Small Unmanned Aerial Vehicle 2009 International Conference on Intelligent Human-Machine Systems and Cybernetics.

12. Dr. M. SeetharamaBhat Modern Control for Aerospace Vehicles Department of Aerospace Engineering Indian Institute of Science Bangalore 560012, INDIA.

13. Carsten Scherer and SiepWeiland DISC Course on Linear Matrix Inequalities in Control - 2004/2005 http:/www.cs.ele.tue.nl/sweiland,lmi.html, November 24, 2004.

References 90

14. Mahmoud Chilali and Pascal Guidance H-infinity Design with Pole Placement

Constraints: An LMI Approach IEEE Transactions on Automatic Control, Vol.41, NO.3 March 1996

15. FengZheng, Qing-Guo Wang, Tong Heng Lee, On the design of multivariable PID controllers via LMI Approach,Automatica 38 (2002) 517-526

16. Pascal Gahinet, ArkadiNemirovski, Alan J.Laub, Mahmoud Chilali, LMI control Toolbox For use with Matlab, The Mathworks user’s guide version 1.

17. Yong-Yan Cao, James Lam and You-Xiam Sun, Static Output Feedback Stabilization : An ILMI Approach, Automatica, Vol. 34, No. 12, pp. 1641-1645, 1998.

18. M.Vidyasagar, Statistical Learning Theory and Randomized Algorithms for Control, IEEE Control Systems, December 1998.

19. M.Vidyasagar Randomized algorithms for Robust Controller Synthesis using Statistical Learning Theory, Automatica 37(2001) 1515-1528.

20. B.R Barmish, C.M LagoaThe Uniform Distribution: A Rigorous Justification for its use in Robustness Analysis, Proceedings of the 13th conference on Decision and Control Kobe, Japan December 1996.

21. Daniel P.Raymer, Aircraft Design: A Conceptual Approach American Institute of Aeronautics and Astronautics, Inc.

22. NehaSatak,Design, Development and Flight Control of Sapthami –A Fixed Wing Micro Air vehicle M.Sc ThesisDepartment of Aerospace Engineering, Indian Institute of Science, Bangalore December 2008.

23. UIUC Propeller Database http://www.ae.illinois.edu/m-selig/props/propDB.html

24. XFLR5http://www.xflr5.com/xflr5.htm.

25. UIUC Airfoil Database http://www.ae.illinois.edu/m-selig/ads/coord_database.html

26. Airfoil Investigation Database http://www.worldofkrauss.com/foils/search

27. John H. Blakelock, Automatic Control of Aircraft and Missiles, John Wiley & Sons, Inc.

28. M.P Miller An Accurate Method of Measuring the Moments of Inertia of Airplanes, Langley Memorial Aeronautical Laboratory, Washington October,1930.

References 91

29. M.W Green, Measurement of Moment of Inertia of Full Scale Airplanes, Langley

Memorial Aeronautical Laboratory, Washington September, 1927.

30. FlavioLuiz de Silva Bussamra, Carlos Miguel MontestruqueVilchez, Julio Cesar Santos Experimental Determination of Unmanned Aircraft Inertial Properties, ITA – InstitutoTecnológico de Aeronáutica, Divisão de EngenhariaAeronáuticaPraçaMarechal Eduardo Gomes, 50 - CEP 12228-900 – São José dos Campos – SP – Brasil.

31. http://www.pololu.com

32. Kestrel Autopilot, Procerus Technologies, http://www.procerusuav.com.

33. B.T. Polyak, R.Tempo, Probabilistic robust design with linear quadratic regulators, Systems and Control Letters 43(2001) 343-353.

34. J.R.Raol, G.Girija, J.Singh, Modeling and Parameter Estimation of Dynamic Systems, IET Control Engineering Series 65.

35. Kestrel User Guide, Kestrel autopilot system Procerus Technologies.

36. R.W. Beaven, M.T Wright and D.R Seaward, Weighting function selection in H∞ design process, Control Engineering practice, Vol 4, No. 5, pp 625-633, 1996.

Design and Flight Control of Miniature Aerial Vehicle

Documents

Transcript of Design and Flight Control of Miniature Aerial Vehicle