ae483 metu automatic control project
TRANSCRIPT
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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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At 60knots forward flight;
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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At 60knots forward flight;
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
For this desired poles, Kdesired,2is found as:
= 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
For this desired poles, Kdesired,3is found as:
= 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
Figure 23: Hover flight pitch, roll and yaw rate versus time for K2
= 0.3978 0.2452 0.00100.0760 0.0183 0.09090.0121 0.0258 0.3699
Figure 24: Hover flight pitch, roll and yaw rate versus time for K3
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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
At 60knots forward flight;
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
At 60knots forward flight;
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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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;
Poles_sas_2 = [p1_2;p2_2;p3_2];K2_sas = place(A_red_hover,B_red_hover,Poles_sas_2);
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;
Poles_sas_3 = [p1_3;p2_3;p3_3];K3_sas = place(A_red_hover,B_red_hover,Poles_sas_3);
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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