ae483 metu automatic control project

8/10/2019 ae483 Metu automatic control project

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MIDDLE EAST TECHNICAL UNIVERSITY

DEPARTMENT OF AEROSPACE ENGINEERING

AE483AUTOMATIC CONTROL SYSTEMS II

PROJECT

Submitted by:

Mustafa GRLER 1746916

Hasan ENER 1747302

Ali YILDIRIM 1747146

Submission Date:26/01/2014


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Table of Contents

1. INTRODUCTION ............................................................................... 3

1.1 BASIC UNDERSTANDINGS OF A HELICOPTER .............................. 3

1.2 HELICOPTER FLIGHT CONTROLS .................................................. 4

1.2.1 COLLECTIVE PITCH CONTROL ................................................ 4

1.2.2 THROTTLE CONTROL ............................................................. 5

1.2.3 CYCLIC PITCH CONTROL ........................................................ 5

1.2.4 ANTI-TORQUE PEDALS .......................................................... 5

2. RESULTS & DISCUSSION ................................................................... 63. CONCLUSION ................................................................................. 44

4.REFERENCES .................................................................................. 45

5.APPENDIX ...................................................................................... 46


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1. INTRODUCTION

1.1 BASIC UNDERSTANDINGS OF A HELICOPTER

A helicopter is an aircraft that is lifted and propelled by one or more horizontal rotors which

are consisting of two or more rotor blades. Helicopters classified as rotorcraft or rotary-wing

aircraft because they generate lift from the rotor blades rotating around a mast.

The main advantage of the helicopters is providing lift without requiring the aircraft to move

forward due to rotor blades revolving through air. This eliminates the need of large runway

areas and provides vertical take-off and landing. For this reason, in congested places where

fixed-wing aircrafts could not take-off or land, helicopters are generally used. Moreover,

provided lift by rotors allows the helicopter to hover in one area and to do so more efficiently

than other forms of vertical takeoff and landing aircraft, allowing it to accomplish tasks that

fixed wing aircraft are unable to perform.

Figure 1: Search and rescue helicopter landing in a confined area

Piloting a helicopter requires a great deal of training and skill, as well as continuous attention

to the machine. The pilot must think in three dimensions and must use both arms and both

legs constantly to keep the helicopter in the air. Coordination, control touch, and timing are

all used simultaneously when flying a helicopter. Although most previous designs used more

than one main rotor, it was the single main rotor with an anti-torque tail rotor configuration

design that would come to be recognized worldwide as the helicopter.


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Uses: Due to its ability to take off and land vertically, to hover for extended periods of time,

and the aircrafts handling properties under low airspeed conditions, helicopter has been

chosen to conduct tasks that were previously not possible with other aircrafts. Today,

helicopters are used for transportation, construction, firefighting, search and rescue, and avariety of other jobs that require its special capabilities.

Rotor System: The helicopter rotor system is the rotating part of a helicopter that generates

lift. A rotor system may be mounted horizontally, as main rotors are, providing lift vertically;

it may be mounted vertically, such as a tail rotor, to provide lift horizontally as thrust to

counteract torque effect. In the case of tilt rotors, the rotor is mounted on a nacelle that

rotates at the edge of the wing to transition the rotor from a horizontal mounted position,

providing lift horizontally as thrust, to a vertical mounted position providing lift exactly as a

helicopter. Tandem rotor helicopters have two large horizontal rotor assemblies; instead of

one main assembly and a smaller tail rotor. Single rotor helicopters need a tail rotor to

neutralize the twisting momentum produced by the single large rotor. However, tandem rotor

helicopters, use counter-rotating rotors, with each canceling out the others torque. Counter-

rotating rotor blades wont collide with and destroy each other if they flex into the other

rotors pathway. This configuration also has the advantage of being able to hold more weight

with shorter blades, since there are two sets. Also, all of the power from the engines can be

used for lift, whereas a single rotor helicopter uses power to counter the torque. Because of

this, tandem helicopters are among some of the most powerful and fastest.

1.2 HELICOPTER FLIGHT CONTROLS

In this section, it is assumed that the helicopter has a counterclockwise main rotor blade

rotation. In the other case, there will be a need of reversing left and right references in the

areas of rotor blade pitch angle, anti-torque pedal movement, and tail rotor trust.

There are four basic controls used during flight. They are the collective pitch control, the

throttle, the cyclic pitch control, and the anti-torque pedals.

1.2.1 COLLECTIVE PITCH CONTROL

As the name implies, the collective pitch control changes the pitch angle of all main rotor

blades simultaneously. As the collective pitch control is changed, there is a simultaneous and

equal change in pitch angle. Using pitch control changes the angle of attack of each blade and


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thus leads to change in drag which affects the rpm. of the main rotor. In order to keep a

constant rotor rpm, the throttle control is used.

1.2.2 THROTTLE CONTROL

The throttle is responsible for regulating engine rpm. If the systems such as correlator or governor

system are not installed, the throttle has to be moved manually in order to maintain rpm.

1.2.3 CYCLIC PITCH CONTROL

The cyclic pitch control tilts the main rotor disc by changing the pitch angle of the rotor blades

in their cycle of rotation. When the main rotor disc is tilted, the horizontal component of lift

moves the helicopter in the direction of tilt. For example, if the cyclic is moved forward, the

angle of attack decreases as the rotor blade passes the right side of the helicopter andincreases on the left side. This results in maximum downward deflection of the rotor blade in

front of the helicopter and maximum upward deflection behind it, causing the rotor disc to tilt

forward.

1.2.4 ANTI-TORQUE PEDALS

The main purpose of the tail rotor is to counteract the torque effect of the main rotor. Since

torque varies with changes in power, the tail rotor thrust must also be varied. The pedals are

connected to the pitch change mechanism on the tail rotor gearbox and allow the pitch angle

on the tail rotor blades to be increased or decreased.

Besides counteracting torque of the main rotor, the tail rotor is also used to control the

heading of the helicopter while hovering or when making hovering turns. Hovering turns are

commonly referred to as pedal turns.


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2. RESULTS & DISCUSSION

1. Trimming helicopter at hover and 60knots forward flight at 1000ft altitude using Heli-

Dyn.

Finding the position of trim point xeis to let net forces and moments summation be equal to

zero (including aerodynamic, gravity and inertial forces) under control ue. The mathematical

manipulation is to solve xeand uefrom the equation F(xe, ue)=0.

After trimmed the helicopter at hover, trim results at 1000ft altitude are:

Collective control (deg) 13.9785 Main rotor inflow(ft/s) 28.2247

Pedal control (deg) 8.77541 Tail rotor inflow (ft/s) 40.4225

Longitudinal control (deg) 0.144992 Forward velocity(knots) 0

Lateral control (deg) -3.27302 Altitude(ft) 1000

Phi(deg) 0.0384227 Psi(deg) 0

Theta(deg) 5.28265

Same procedure can be applied for the 60knots forward flight and trim results at given

altitude are:

Collective control (deg) 12.9083 Main rotor inflow(ft/s) 7.62113

Pedal control (deg) 4.26039 Tail rotor inflow (ft/s) 13.7441

Longitudinal control (deg) 0.327579 Forward velocity(knots) 60

Lateral control (deg) -2.63501 Altitude(ft) 1000Phi(deg) 0.155926 Psi(deg) 0

Theta(deg) -2.1929

2. Linearization of the helicopter dynamics at trim values:

The six degree of freedom rigid body motion of helicopter can be described as following:


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where U, V, W, and P, Q, R are linear and angular velocities, respectively, and all referred to

the fuselage (body-fixed) axes system; Ixx, Ixz,, etc, are the moments of inertia of the

helicopter; ms is the helicopter's mass. Forces (Fx,Fy,Fz) and moments (L,M,N) include the

effects coming from aerodynamics, gravity, and propulsion.

After constructing the helicopter nonlinear model, it is noticed that in designing control laws,

linearized model is more often used. Therefore, obtaining the linear model becomes more

important for controller design. To linearize the system, there are three procedures; flight

conditions, trim condition calculation and linearization. Trimming is important since if working

with random point rather than trim point, there will be residiual forces and moments which

ruin the exact nonlinear character.

The linearized helicopter dynamics can be written in the state space form as:

= + where X is the state, U is the control vector and A is the system matrix, and B is the control

matrix.The state vector X=[u w q v p r]Tand the control vector is U=[long coll lat pedal]T.


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After linearizing the system in Heli-Dyn, A&B matrices for hover and 60 knots forward flight

condition can be found as:

At hover;

=0.0128 0.0362 2.5520 32.0042 0.0021 1.0028 0 00.0192 0.4069 0.5149 2.9592 0.0214 0 0.0215 00.0028 0.0016 0.4151 0 0.0003 0.1057 0 00.000028 0.000016 0.9958 0 0 0.0011 0 0.00070.0021 0.0230 1.0028 0.0020 0.0445 2.6031 32.0042 0.73750.000027 0.0003 0.5184 0 0.0086 1.3419 0 0.12970.000002 0.000023 0.0052 0 0.0001 0.9864 0 0.09340.0020 0.0214 0.0710 0 0.0135 0.1554 0 0.3508

=

32.9444 39.1248 0.3694 00 423.1459 0 03.4738 0.000002 0.0389 00.0347 0.0001 0.0004 0.00010.3694 24.8403 32.9444 16.56310.1910 6.1388 17.0309 3.83670.0019 0.0513 0.1725 0.03100.0262 10.9209 2.3339 7.9409

At 60knots forward flight;

=

0.0768 0.0676 5.8709 32.1172 0.0031 0.7843 0 0.01050.0242 1.0252 107.6114 1.2298 0.0445 2.8832 0.0874 00.0015 0.0200 0.5187 0 0.0001 0.0980 0 00.000015 0.0002 0.9948 0 0.000002 0.0010 0 0.00270.0039 0.0449 0.6473 0.0033 0.0957 6.7758 32.1170 99.04620.0035 0.0177 0.3763 0 0.0171 1.4166 0 0.34870.000034 0.0002 0.0038 0 0.0002 0.9858 0 0.03440.0027 0.0073 0.1247 0 0.0366 0.0329 0 1.0201

=

[

25.7672 36.2087 0.2891 095.0907 518.4319 1.0563 03.2190 1.1076 0.0361 00.0322 0.0112 0.0004 0.00034.0184 23.8968 32.4654 19.11451.5394 9.5066 16.7773 4.8541

0.0157 0.0965 0.1669 0.05210.7331 3.6934 2.2886 9.1923]

After linearized the system, it should be decided whether the system is stable or not. In order

to do this, eigenvalues are calculated as

=

[

1.4479 + 0.00000.3359 + 0.00000.4023 + 0.36110.4023 0.36110.0318 + 0.0000

0.3688 + 0.00000.1592 + 0.38430.1592 0.3843

]


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Figure 2: Eigenvalues of Hover Flight

Since there are two roots on the right hand side, the system is not stable in hover position at

1000ft altitude.

=

1.6361+0.0000i0.4663+1.8591i0.46631.8591i

0.7327+1.4417i0.73271.4417i0.0398+0.1628i0.03980.1628i0.0392+0.0000i

Figure 3: Eigenvalues of 60knots Forward Flight


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As seen in eigenvalues of the system matrix A in 60 knots forward flight, all roots are in left hand

side and the system is stable in 60knots forward flight at 1000 ft altitude.

3. Obtaining reduced-order linear dynamics where the body angular velocities (p,q,r) are the

states and the longitudinal cyclic, lateral cyclic and tail rotor (pedal) control are the controls.

From linearized matrices A&B, for reduced order linear dynamics, A&B matrices can be found as

following where given X and U matrices.

=0.0128 0.0362 2.5520 32.0042 0.0021 1.0028 0 00.0192 0.4069 0.5149 2.9592 0.0214 0 0.0215 00.0028 0.0016 0.4151 0 0.0003 0.1057 0 00.000028 0.000016 0.9958 0 0 0.0011 0 0.00070.0021 0.0230 1.0028 0.0020 0.0445 2.6031 32.0042 0.73750.000027 0.0003 0.5184 0 0.0086 1.3419 0 0.12970.000002 0.000023 0.0052 0 0.0001 0.9864 0 0.09340.0020 0.0214 0.0710 0 0.0135 0.1554 0 0.3508

=

[

32.9444 39.1248 0.3694 00 423.1459 0 03.4738 0.000002 0.0389 00.0347 0.0001 0.0004 0.00010.3694 24.8403 32.9444 16.56310.1910 6.1388 17.0309 3.83670.0019 0.0513 0.1725 0.03100.0262 10.9209 2.3339 7.9409]

, = 0.4151 0.1057 00.5184 1.3419 0.12970.0710 0.1554 0.3508where =

, = 3.4738 0.0389 00.1910 17.0309 3.83670.0262 2.3339 7.9409where =

=0.0768 0.0676 5.8709 32.1172 0.0031 0.7843 0 0.01050.0242 1.0252 107.6114 1.2298 0.0445 2.8832 0.0874 00.0015 0.0200 0.5187 0 0.0001 0.0980 0 00.000015 0.0002 0.9948 0 0.000002 0.0010 0 0.00270.0039 0.0449 0.6473 0.0033 0.0957 6.7758 32.1170 99.04620.0035 0.0177 0.3763 0 0.0171 1.4166 0 0.34870.000034 0.0002 0.0038 0 0.0002 0.9858 0 0.03440.0027 0.0073 0.1247 0 0.0366 0.0329 0 1.0201


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=25.7672 36.2087 0.2891 095.0907 518.4319 1.0563 03.2190 1.1076 0.0361 00.0322 0.0112 0.0004 0.00034.0184 23.8968 32.4654 19.11451.5394 9.5066 16.7773 4.85410.0157 0.0965 0.1669 0.05210.7331 3.6934 2.2886 9.1923

, = 0.5187 0.0980 00.3763 1.4166 0.34870.1247 0.0329 1.0201where =

, = 3.2190 0.0361 01.5394 16.7773 4.85410.7331 2.2886 9.1923

where =

4. Checking controllability of systems for reduced order linear dynamics.

A linear system is controllable at t0if it is possible to find an input function u(t), defined over the

time of interest, that will transfer the initial state x(t 0) to the origin in finite time. If this is true

regardless of the initial time and initial condition, the system is said to be completely controllable.

Controllability will depend on the A and B matrices and defined as:

If the controllability matrix C is in full rank, it can be concluded that the system is controllable.

Then,

= , , , , , = , , , , ,

() =3 () = 3Since controllability matrices Choverand C60knotsare in full rank, they are both controllable.


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5. Constructing non-linear model with step inputs to each control and plotting the outputs.

Using Heli-Dyn, outputs will be observed for unit inputs to each control. With the help of non-

linear solver in the Heli-Dyn, responses to unit inputs of longitudinal cyclic, collective cyclic,

lateral cyclic and tail rotor control are observed.

Linear and angular velocity responses to 1 unit input are observed. At hover;

Figure 4: Angular velocities versus time for nonlinear open loop system of hover flight

Figure 5: Linear velocities versus time for nonlinear open loop system of hover flight


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Figure 6: Angular velocities versus time for nonlinear open loop system of 60knots forward flight

Figure 7: Linear velocities versus time for nonlinear open loop system of 60knots forward flight

When 1rad unit input is given to all inputs, it is seen that there is a discontinuity after

approximately 4.6 seconds. Especially at hover, outputs are very large values and this result can

be evaluated as that the helicopter may tumble and cannot respond the given input value.

Therefore, better way is to give 0.5rad step input to all inputs. Then, the outputs at hover and

60knots forward flight can be obtained as follows:


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Figure 8.1: Nonlinear open loop system responses to 0.5 unit input to each control of 60knots

forward flight


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Figure 8.2: Nonlinear open loop system responses to 0.5 unit input to each control of 60knots

forward flight


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At hover;

Figure 9.1: Nonlinear open loop system responses to 0.5 unit input to each control of hoverflight


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Figure 9.2: Nonlinear open loop system responses to 0.5 unit input to each control of hover

flight


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These responses for both flight; hover and forward flight, can be evaluated as unstable. Since the

helicopter is not stable at hover trim position, unit input makes the system more unstable and

the responses diverge. At forward flight, the helicopter is stable at trim position. However, the

system does not have any controller, with a step input, system responses diverge. This is theexpected conclusion of open loop systems.

6. Constructing linear model with step inputs to each control and plotting the outputs for linear

and reduced-order linear systems.

By Simulink, state space representation of the linear system is created. Here, A&B matrices are

8x8 and 8x4 linearized matrices at trim position, C is identity matrix and D is zero matrixes.

Figure 10: Simulink model of linear system

In Simulink, after giving 0.5 step input to each input, result are obtained as follows:


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At 60knots forward flight with linear model dynamics:

Figure 11.1: Open loop linear system responses to 0.5 unit input to each control of 60knots forward

flight


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Figure 11.2: Open loop linear system responses to 0.5 unit input to each control of 60knots forward

flight

At hover with linear model dynamics:

Figure 12.1: Open loop linear system responses to 0.5 unit input to each control of hover flight


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Figure 12.2: Open loop linear system responses to 0.5 unit input to each control of hover flight

By Simulink, state space representation for reduced order linear dynamics is also created. Here,

A&B matrices are reduced matrices at trim position, C is identity matrix and D is zero matrixes.

Figure 13: Simulink model of reduced order linear system


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After giving 0.5 step input to each input, result are obtained as follows:

At 60knots forward flight with reduced order linear dynamics:

Figure 14: Open loop reduced order linear system responses (p, q, r) to 0.5 step input of 60knots

forward flight

At hover with reduced order linear dynamics:

Figure 15: Open loop reduced order linear system responses (p, q, r) to 0.5 step input of hover

flight


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While in nonlinear system, responses oscillate much, by linearizing the system, oscillations

decrease. However, because it is still open loop, the system responses continue to diverge for

linear model.

By constructing reduced order linear model, it is seen that pitch, roll and yaw rate responses

converge to certain numbers for both hover and forward flight cases. Therefore, it is concluded

that the helicopter is stable in reduced order linear model where the responses are pitch, roll

and yaw rate.

7. Design a Stability Augmentation System (SAS), by using the angular velocities (p, q, r) as

feedback and the three controls, longitudinal cyclic, lateral cyclic and tail rotor (Pedal) control.

For this part, desired poles should be selected. Choosing one pole arbitrarily, other poles will be

obtained from;

, = where = 1 For 60knots forward flight:

For s1=-10, =0.95 and n=10, s2,3=-9.5 3.1225i;

Figure 16: Forward flight desired Poles for s1=-10, =0.95 and n=10

For this desired poles, Kdesired,1is found as:

= 261.7766 93.0243 0.0063

3.1384 0.5982 0.26380.0498 0.1507 1.1376


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For s1=-3.5, =0.85 and n=2, s2,3=-1.70 1.0536i;

Figure 17: Forward flight desired Poles for s1=-3.5, =0.85 and n=2


= 37.0231 33.4040 0.00210.5037 0.3591 0.09090.0169 0.0243 0.3699For s1=-3.5, =0.9 and n=1.2, s2,3=-1,80 0,7454i;

Figure 18: Forward flight desired Poles for s1=-3.5, =0.9 and n=1.2


= 39.9047 24.5251 0.00210.5190 0.2540 0.09090.0119 0.0258 0.3699


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After for different poles, desired controller gains are determined. By Simulink, system responses

are observed for three cases. Here, the equilibrium for a linear model is u=0, x=0, since everything

in the linear model is a perturbation. Thus trim values are all zero.

Figure 19: Simulink Model for different SAS controller gains

Then, p-q-r responses are plotted for different desired gain matrices.

At forward flight

Figure 20: Forward flight pitch, roll and yaw rate versus time for K1


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Figure 21: Forward flight pitch, roll and yaw rate versus time for K2and K3

As seen in Figure 19 and Figure 20, K1converges to responses faster than K2and K3. In addition, difference

between input and responses (error) is the lowest when system is controlled by K1.

At hover flight;

The same formulation was used to predict desired A matrices. The poles of the system stay same

but the desired matrices and feedback gains changed.

= 2.7585 0.0354 0.00280.0614 0.4447 0.25060.0000 0.1504 1.0785

Figure 22: Hover flight pitch, roll and yaw rate versus time for K1


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= 0.3688 0.3338 0.00100.0926 0.0118 0.09090.0171 0.0242 0.3699


= 0.3978 0.2452 0.00100.0760 0.0183 0.09090.0121 0.0258 0.3699



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Again, as seen in Figure 22-23 and 24, system is best controlled by K1.

We designed this controller gain K1for the linear system. We should test our controller with

nonlinear model. 0.07rad=4.01degree unit input is given to all inputs.

Figure 25: Simulink model for nonlinear system with controller


Figure 26: Forward flight u, v, w, phi, theta and psi versus time for non linear model with

controller gain K1


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Figure 27: Forward flight X, Y, Z and roll rate versus time for non linear model with controller

gain K1

Figure 28: Forward flight pitch and yaw rate versus time for non linear model with controller

gain K1

At hover flight;

Figure 29: Hover flight phi, theta, psi, X, Y and Z versus time for non linear model with

controller gain K1


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Figure 30: Hover flight p, q and r versus time for non linear model with controller gain K1

After tested K1in the nonlinear model, we can conclude that it could not control the system

very fast, but after approximately 10 seconds, the controller converges to pitch, roll and yaw

rates.

8. Designing the same controller of part 7 using LQR.

Linear Quadratic Regulator

Advantage of the linear quadratic regulator (LQR) over the pole placement method is to

provide a systematic way of computing state feedback control gains matrix.

Consider a linear system equation

= + determines the matrix K of the optimal control vector.


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() =()to minimize performance index

= ( + ) where the Q is positive definite (or positive-semi definite) Hermitian(H) or real symmetric

matrix and R is a positive definite Hermitian or real symmetric matrix. Q and R matrices have

relative importance of the error and expenditure energy of this system. If second term is

greater than the first which are lie on the right hand side of the performance index, the cost

function is dominated by control effort . So the controller minimizes the control actionitself.If the first term is greater, the cost function is dominated by the output errors y, andthere is no penalty for using large .

Figure 31: Optimal Regulator System

Feedback gain must to be determined to minimize performance index, then () =()is optimal for any initial state (0).Optimal K appears in LQR as follows

= The matrix P in above equation must satisfy the following reduced equation which is called

the reduced-matrix Ricatti Equation.

+ + = 0Optimal K can be found after solving Ricatti Equation for P and putting in optimal feedback

gain equation. If P is positive-definite matrix and it exists then system is stable.

Forward Flight;

In forward flight for reduced system matrices it is tried to minimize performance index by

minimizing equation


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= ( + ) It has been looked for to find optimal feedback gain which lessens the sum of control effort

and error contribution terms. Simple matlab code was written the optimize performance

index by integrating the time and doing the simulation for all new R matrix while Q matrix is

kept as unity matrix. In Q matrix it was put the same importance for all states. Therefore it

was chosen as unit matrix. However as it can be shown in graphs pitch rate goes steady state

more slowly than the roll and yaw rate. It can be solved by changing and increasing the value

of the first row first column of Q matrix. However it is not a big deal in our circumstance.

For 100 different R matrixes, 100 simulations was held and each time lqr problem was solved.

Then the following optimum feedback gain was obtained.

= 0.8462 0.0672 0.05030.0446 0.8886 0.24760.0659 0.2392 0.8626 = 1 0 00 1 0

0 0 1

=0.6274Following graphs were obtained for forward flight with LQR.

Figure 32: Optimization of Performance Index for Forward Flight


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Figure 33: q, p, r vs. time for Forward FlightHover Flight;

Same procedure is done for hover flight also.

= 0.8876 0.0101 0.00050.0114 0.8986 0.22560.0046 0.2249 0.9312

Figure 34: Optimization of Performance Index for Hover


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Figure 35: q, p, r vs. time for Hover

At 60 knots forward flight;

Figure 36: Forward flight p, q and r versus time for nonlinear model with LQR


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At hover flight;

Figure 37: Hover flight p, q and r versus time for nonlinear model with LQR

9. Adding integral controllers to the channels

Again, we should test our controllers for the nonlinear system. Similarly, 0.07rad=4.01degree

unit input is given to all inputs.


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Figure 38: Simulink model for nonlinear system with integral gain controller


Figure 39: Forward flight u, v, w, phi, theta and psi versus time for non linear model with

integral gain controller


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Figure 40: Forward flight X, Y, Z and roll rate versus time for non linear model with integral

gain controller

Figure 41: Forward flight pitch and yaw versus time for non linear model with integral gain

controller

After tested our controller gain found with LQR in the nonlinear model for 60 knots forward

flight, we can say that it could not control the system very fast, but after approximately 10

seconds, the controller almost converges to pitch, roll and yaw rates.

10. Investigation of the real stability requirements of ADS-33

The summary of Aeronautical Design Standard 33;

Aeronautical Design Standard 33 is a performance specification for the handling qualities of

rotorcraft which its missions ranging from scout and attack to utility and cargo. The handling

qualities criteria and metrics of ADS-33 depend primarily on the mission the helicopter has

to execute rather than its role or size. ADS-33 indicates the specification of aircraft response


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characteristics depends on numerical quantitative criteria in both the frequency and time

domains, and qualitative criteria based on pilot ratings. Handling qualities of helicopter is

achieved by comprehensive assessment of well defined tasks. During these particular tasks

three pilots rate the helicopter response according to Cooper Harper scale which shows thetask performance of helicopter and workload of pilot.

Figure 42: Definition of Handling Qualities Levels

For flight within the operational flight envelope, Level 1 handling qualities are required. Level

2 is acceptable in extreme situations i.e. the case of failed and emergency situations. However

Level 3 is completely unacceptable. The specifications of the Mission Task Element (MTE), the

Usable Cue Environment (UCE) and the response type must be included to satisfy Level 1

handling which ADS-33 requires.


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In this part the first level requirement of ADS-33 was used to design stability augmentation

system. Rather than using bandwidth and phase delay data limits on pitch (roll) oscillations

for hover and low speed flight which are the cases for our helicopter.

Table 1: Response limits for pitch, roll and yaw rate according to ADS-33

Firstly damping ratio and natural frequency was

estimated.

= 0.82 =0.609

=0.5

Secondly first pole is estimated and respectively

second and third poles were found. By applying pole

placement technique the desired feedback matrix

was found.

, =0.49940.3486 Figure 43: Pitch (Roll) Oscillations Criteria


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11. Design a SAS using the requirements of ADS-33

Nonlinear Model, at 60knots forward flight;

Figure 44: Forward Flight p, q and r versus time for nonlinear model with SAS

The values in figure 44 states that p, q and r diverge with a step input even though it has a

controller. Lets look at it by using the linear model obtained. The integral controller also

were connected to the channels and analyzed.


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Linear Model

At hover;

According to above poles by using pole placement technique gain for hover as appears as

follows and the stability augmentation

system feedback gain;

= 0.0237 0.1302 0.00010.0499 0.0520 0.01110.0056 0.0047 0.0155

Figure 45: q, p, r vs. time for ADS-33 Hover

System does not go to given response value, actually it is very far from that value. So the

integral controller is added to system as below.

A_tilda matrix eigenvalues;

=0.2098 2.7456 =0.1657 0.3977 =0.3740 0.6950 Integral gain;

= 0.0500 0.1000 0.10000.1000 0.5000 0.10000.1000 0.5000 0.1500

Figure 46: q, p, r vs. time for ADS-33 Hover with Integral Controller


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42

At 60knot forward flight;

Stability augmentation system feedback gain;

= 0.0065 0.1382 0.00000.0444 0.0431 0.00410.0020 0.0039 0.0576

Figure 47: q, p, r vs. time for ADS-33 Forward Flight

At below the integral controller was added to system to be stable at given input.

A_tilda matrix eigenvalues;

=0.2053 2.5831 =0.2276 0.3133 = 0.3165 0.9078 Integral gain is the same as in hover flight

= 0.0500 0.1000 0.10000.1000 0.5000 0.10000.1000 0.5000 0.1500


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Figure 48: q, p, r vs. time for ADS-33 Forward Flight with Integral Controller

For both hover and 60knots forward flight, the angular velocities, p, q and r converges to the

unit step input value of 0.5. There are some oscillations seen in the graph due to the fact that

the eigenvalues of integral controlled system remains at the right of the feedback system

obtained by real stability requirements of ADS-33.


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3. CONCLUSION

In this project, we are asked to design a controller system for a Uh-1h helicopter using

requirements from the ADS-33 Rotorcraft Handling Qualities specifications using the Heli-Dyn.

Firstly, the helicopter was trimmed for hover and 60knots forward flight using Heli-Dyn

software. After trimming the system, it was linearized around its equilibrium points.

Next, reduced order linear dynamics matrices which shows the pitch, roll and yaw rates were

found. The controls were taken as, longitudinal cyclic, lateral cyclic and tail rotor (Pedal)

control are the controls. The linear dynamic model was set up for this reduced order system.

Then, the controllability of this system was checked and it came out as controllable for both

flight. After, using Heli-Dyn, nonlinear model was solved in Simulink for step inputs. Outputswere plotted for reduced and nonlinear system and it was seen that both were unstable for

step inputs since they were in open loop system.

Moreover, in the project, Stability Augmentation System was investigated and according to

this, a controller gain was designed in reduced order linear system by tuning desired poles.

However, it was used for nonlinear system. Then, by keeping the collective control on its trim

position and giving the others step inputs, the responses were analyzed. The same controller

was designed with LQR and same procedures were applied.

In order to decrease steady state error in the system, the integral controller were added all

the channels. Allthough steady state error became zero for linear system, it could not be done

for nonlinear system. The reason for this may be is the fact that the controller gained from the

reduced order linear system (p, q and r) does not work for all the states when it was put in

nonlinear system. It is also possible that other responses apart from p, q and r affect in the

negative manner.

Furthermore, ADS-33 requirements were researched for real stability characteristics for the

helicopter. Its limits and levels were observed; then, desired poles were adjusted according to

these requirements. With these desired poles, a controller gain was designed for again

reduced order linear dynamics. However, for the same reason explained above, it did not work

for nonlinear system dynamics.

It was all honour to become a part of this magnificant project which contributed to us in terms

of consciousness of controller design and real analysis.


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REFERENCES

1. Abbott W. C., Engineering Evaluation of Aeronautical Design Standard (ADS)-33C,

Handling Qualities Requirements for Military Rotorcraft, Utilizing an AH-64A

Apache Helicopter,1991.

2. ADS-33 Handling Qualitieshttp://arc.aiaa.org/doi/abs/10.2514/6.2009-6059

3. Modern Control Engineering Fifth Edition, Katsuhiko Ogata

4. http://www.fai.org/rotorcraft

5. http://www.heli-dyn.com/Helidyn/
http://arc.aiaa.org/doi/abs/10.2514/6.2009-6059http://arc.aiaa.org/doi/abs/10.2514/6.2009-6059http://arc.aiaa.org/doi/abs/10.2514/6.2009-6059http://www.fai.org/rotorcrafthttp://www.fai.org/rotorcrafthttp://www.heli-dyn.com/Helidyn/http://www.heli-dyn.com/Helidyn/http://www.heli-dyn.com/Helidyn/http://www.fai.org/rotorcrafthttp://arc.aiaa.org/doi/abs/10.2514/6.2009-6059http://arc.aiaa.org/doi/abs/10.2514/6.2009-6059


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APPENDIX

APPENDIX A

SAS controller design

A_red_60knots=[ -0.5187 0.0980 0;% -0.3763 -1.4166 0.3487;% -0.1247 -0.0329 -1.0201];B_red_60knots=[-3.2190 -0.0361 0;1.5394 16.7773 4.8541;-0.7331 2.2886 -9.1923];A_red_hover=[ -0.4151 0.1057 0;

-0.5184 -1.3419 0.1297;-0.0710 -0.1554 -0.3508];

B_red_hover=[-3.4738 -0.0389 0;-0.1910 17.0309 3.8367;

-0.0262 2.3339 -7.9409];ksi_1= 0.95;wn_1 = 10;wd_1 = wn_1*sqrt(1-ksi_1^2);p1_1 = -10;p2_1 = -ksi_1*wn_1+wd_1*1i;p3_1 = -ksi_1*wn_1-wd_1*1i;

Poles_sas_1 = [p1_1;p2_1;p3_1];K1_sas = place(A_red_hover,B_red_hover,Poles_sas_1);

ksi_2= 0.85;

wn_2 = 2;wd_2 = wn_2*sqrt(1-ksi_2^2);

p1_2 = -3.5;p2_2 = -ksi_2*wn_2+wd_2*1i;p3_2 = -ksi_2*wn_2-wd_2*1i;


ksi_3= 0.9;wn_3 = 1.2;wd_3 = wn_3*sqrt(1-ksi_2^3);

p1_3 = -3.5;p2_3 = -ksi_3*wn_2+wd_3*1i;p3_3 = -ksi_3*wn_2-wd_3*1i;


K_lqr_forward=[-0.8462 0.0672 -0.0503; 0.0446 0.8886 0.2476; 0.0659 0.2392-0.8626];K_lqr_hover=[-0.8876 -0.0101 -0.0005;-0.0114 0.8986 0.2256;0.0046 0.2249 -0.9313];


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APPENDIX B

LQR Optimization of Performance Index

clear all;clc;

%%%Forward Flight Reduced Order%%%% A=[ -0.5187 0.0980 0;% -0.3763 -1.4166 0.3487;% -0.1247 -0.0329 -1.0201];% B=[-3.2190 -0.0361 0;% 1.5394 16.7773 4.8541;% -0.7331 2.2886 -9.1923];%%%Hover%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% A=[ -0.4151 0.1057 0;

-0.5184 -1.3419 0.1297;-0.0710 -0.1554 -0.3508];

B=[-3.4738 -0.0389 0;-0.1910 17.0309 3.8367;

-0.0262 2.3339 -7.9409];%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% C=eye(3);D=zeros(3);Q=[1 0 0 ;0 1 0 ;0 0 1]; %selected state cost matrixH=[1:100];%%%Optimization of Cost Function%%%fori=1:100R=eye(3)*i;[K,S,E]=lqr(A,B,Q,R);j1=0.;j2=0.;sim('hehe.mdl')

int1=0.;int2=0.;forj=1:100

delta_t=t(i+1)-t(i);int1=x(j,:)*Q*x(j,:)'*delta_t;j1=j1+int1;U=-K*x(j,:)';int2=U'*R*U*delta_t;j2=j2-1.*int2;

endj1_last(i)=j1;j2_last(i)=j2;j_sum(i)=0.5*(j1_last(i)+j2_last(i));end%%%PLOT%%%figure(1)plot(j2_last,j1_last,'-r','linewidth',2)xlabel('J2')ylabel('J1')title('Optimization of Cost Function for LQR - Hover')grid on%%%Optimum R Matrix%%%[min I]=min(j_sum);fprintf('min(j_sum)=')disp(min)%%%Optimum Feedback Gain%%%R_best=eye(3)*I;[K_best,S,E]=lqr(A,B,Q,R_best);fprintf('Optimum K=')


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ae483 metu automatic control project

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