three dimensional trajectory linearization control for ... · 1110 g.d. zhu and z.j. shen /...

Procedia Engineering 99 ( 2015 ) 1108 – 1119

1877-7058 © 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA)doi: 10.1016/j.proeng.2014.12.646

ScienceDirectAvailable online at www.sciencedirect.com

“APISAT2014”, 2014 Asia-Pacific International Symposium on Aerospace Technology, APISAT2014

Three Dimensional Trajectory Linearization Control for Flight of Air-breathing Hypersonic Vehicle

G. D. Zhu*, Z. J. Shen School of Aeronautic Science and Engineering, Beijing University of Aeronautics and Astronautics, Beijing 100191,China

Abstract

An integrated guidance and control design approach for air-breathing hypersonic flight vehicle (ABHV) with strong aero-propulsion couplings and aerodynamic constraints is proposed in this paper. By exploiting the inherent nature of multi-time scale of hypersonic flight vehicle dynamics, the guidance and control system is designed based on time scales of subsystem dynamics, with the fast dynamics serving as the pseudo-control for the slow dynamics. Feedback gains are computed online as symbolic functions of the state variable along the reference trajectory, and no explicit gain scheduling or mode-switching is needed. Comparison with linear quadratic regulator control is conducted and performance of the proposed approach is demonstrated via high fidelity simulations with considerable aerodynamic uncertainty and environmental perturbation. © 2014 The Authors. Published by Elsevier Ltd. Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).

Keywords: trajectory linearization control; air-breathing hypersonic vehicle; time scale separation; nonlinear control; trajectory tracking

1. Introduction

Air-breathing hypersonic vehicle has attracted a lot of attention for its military and civil potential since the 1980s. The dynamics of ABHV demonstrate unique characteristics compared with conventional airplanes. The strong aero-angle constraints, highly coupled aero-propulsive-structural characteristics and fast time-varying dynamics caused by wide flight envelop and mass consumption make control system design a very challenging work [1]. A detailed introduction to the control law used in the Hyper-X program is presented in [2]. This control law is based on rigid body dynamics and designed by using classical control theory. Ref. [3] investigates the application of linear

* Corresponding author. Tel.: +86 18811434182

E-mail address: [email protected]

© 2015 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA)

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1109 G.D. Zhu and Z.J. Shen / Procedia Engineering 99 ( 2015 ) 1108 – 1119

Nomenclature

, ,r radial distance from the center of earth to the vehicle, longitude, latitude , ,V earth-relative velocity, flight path angle, heading angle

Vx,Vy,Vz scalar components of velocity in body axes , ,x y z scalar components of angular velocity in body axes

Jx,Jy,Jz moments of inertia about (x,y,z) axes , , angle of attack, side-slip angle, bank angle , , pitch angle, yaw angle, roll angle (Euler angles)

L,C,D,T aerodynamic lift force, side force, drag and thrust force ,, eg gravitation, atmosphere density, earth-rotation rate

Dx,Dy,Dz aileron, rudder,elevator deflection (control surface deflection) and fuel equivalent ratio ( thrust control) ,EB BE angular velocity with respect to earth fixed axes, transform matrix from earth fixed axes to body axes

quadratic method on the longitudinal control design of ABHV, and reduce the difficulty of selection weight matrix by using implicit model following design. In Ref. [4], random robust control is employed in control system design. Genetic algorithm based Monte Caro simulation is performed to search the optimal design parameter in the whole design space. Model predictive control is described in Ref. [5]. The formulation explicitly accommodates nonlinear constraints involving both state and control variables. Ref. [6] points out that model adaptive control has great potential in ABHV control but needs to address the problem of parameter tuning. Ref. [7] researches the application of nonlinear robust adaptive controller. Intelligent control methods have also been applied to ABHV control, like Genetic algorithm [8] and fuzzy control [9][10].

Trajectory linearization control (TLC) is a nonlinear control method which is based on differential algebra theory. TLC consists two essential parts: a dynamic inverse of the plant to compute the nominal control for any given nominal output, this provides agile tracking responses and facilitates the linearization of nonlinear time-varying error dynamics; a tracking error stabilizing control law to account for modeling uncertainties, disturbances and initial conditions. TLC can be viewed as ideal gain-scheduling control and feedback gains can be computed symbolically, therefore no slow time-varying constraints are imposed. In Ref.[11], TLC is used in the attitude control of X33. The controller could pass high fidelity simulation test with large dispersion and perturbation. TLC is also combined with time-varying notch filter which could detect the elastic frequency online and attenuate the oscillation [12][13]. In Ref. [14], TLC is used to design a four sub-loop controller for a fixed wing general airplane.

The main contribution of this paper is extending the application of TLC on ABHV control from longitudinal channel to BTT strategy based full 3DOF flight, deriving the expressions of four sub-loops feedback gains in the body axes and comparing the performance of TLC based integrated G&C controller with LQR method.

2. Entry Dynamics and Aircraft Model

2.1. 3DOF dynamic equations

For guidance law design purpose, attitude changes are considered as fast dynamics and can be ignored. Furthermore, rotation-Earth effects are not important and can be easily compensated by the feedback nature of guidance law. Therefore, the 3DOF point-mass dynamics of ABHV over non-rotating spherical earth are described by the following equations [15]:

2

2

= sin( cos ) / sin

[ sin )cos / ( )cos ] /

[( sin )sin / ( cos ) cos sin tan

/

/ ] /

r VV T D m g

T L m V g V

T L m r V

r

V

(1)

The aero-propulsive coupling has its effect mainly on the longitudinal motion, embodied by the term tnC . When


taking into consideration of this coupling effect, lift force is calculated by 2 /= ( ) 2y tn refV c c SL .

2.2. 6DOF dynamic equations

As a result of the difficulty in directly measuring aero-angles and the high requirement of data precision, Obtaining ABHV aero-angles is largely depend on the on-board inertial measure unit. The force equations used in this paper are projected into body axis(2), which will benefit the numerical simulation of Strapdown Inertial Navigation System. It needs to point out that the original of geographic frame (G) is projection of vehicle CM on the horizontal geodetic datum. OYG coincides with the normal line of oblate earth. OXG coincides with the reference spheroid pointing to North Pole. OZG ,OXG and OYG form a right-handed coordinate system. Vehicle body-fixed system origins at vehicle CM and axes aligned with vehicle reference direction with OYB pointing upward.

21 31

31 11

11 21

( ) 2/ / ( )

/ / ( )

/ / (

( ) 2

( ) 2 )

x x z yEB zEB e z BE y BE

zEB xEB e BE BE

xEB yEB e BE

y

y y x z x z

z z y x y x BE

dV F V VdV F V VdV F V V

dt m V Vdt m V Vdt m V V

(2)

The symmetrical plane of ABHV is OXBYB, therefore the product inertial Jxz=Jyz=0. Rotation dynamics and Euler differential equations for this system of coordinates are:

2 2

/ ( / )

/ ( / )

( )

( )

( )/ ( )

x x y z y z xy y x z x

y y z x x z xy x y z y

z x y x y xz y zxy

J dt J dt MJ d

d J J dd J d

d J

t J J dt M

J dt J J M

cos sin

1 ( sin cos )cos

tan ( sin cos )

y z

y z

x y z

(3)

It worth noting that when heading angle 90 , the second equation of Euler differential equations become singular. However, due to the limited maneuverability of ABHV, through adjusting the heading baseline, singular case can always be avoided.

2.3. Aero-propulsion data processing and subsystem modeling

The aero-propulsion data is procured through CFD computation. The original format of aero-propulsion data is a high dimensional look-up table. To acquire data at arbitrary values of independent variables in simulation and controller design, linear interpolation algorithm is employed. The aero-propulsion data at trim condition (no aileron or rudder deflection and pitch moment equals zero) is used in the guidance loop design.

A second order system (4) is used to model the time delay between accepting control command and establishing flow field in the combustion chamber in the scramjet engine:

2

2 22c E nE nE

nE

s s (4)

The response of actual control surface deflection to control order is also modeled by a second order delay(4) to simulate the limited servo power.

3. Trajectory linearization control theory

The strategy of integrated TLC controller is based on singular perturbation and time-scale separation principles [17][19]. The full 6DOF nonlinear rigid body dynamics of aircraft can be divided into two loops: guidance and attitude loop and could be further partitioned according to time scale into four sub-loops: altitude (state variable has very slow dynamic), velocity (slow dynamic), attitude (fast dynamic) and attitude angular velocity (very fast dynamic) sub-loop. The aim of separating dynamics into four subsystems is to address the problem that there are fewer inputs than states to be controlled.


4. Guidance loop design

Altitude LTV PIcontroller

Velocity LTVPI controller

Altitude pseudo-inversion

Velocity pseudo-inversion

Cr

Vr

V

r

C

Angular velocityLTV

PI controller

Angular velocitypseudo-inversion

y

x

z

y

x

z

MM

M

y

x

z

Attitude LTVPI controller

Attitude pseudo-inversion

Sub-loop1

Sub-loop2 Sub-loop3x

y

zx

z

y

Sub-loop4Guidance LoopAttitude Loop

Con

trol

Allo

cati

on

Fig. 1. Structure of TLC Controller

4.1. Attitude Sub-loop (Sub-loop 1) design

In the altitude sub-loop, path angle is fast dynamic compared with altitude, and therefore used as pseudo-control

variable to control the altitude. The state variable and its nominal value are 1 ,[ ]Trdt rx 1 ,T

r rx. Define

the tracking error 1 1 1xx x and linearize the derivative of tracking error at the original of error dynamics

coordinate:

cos

0 1 00 0

rdt rdtVrr

(5)

The expected closed loop error dynamics of sub-loop 1 is:

1111 112

0 1CA (6)

After transforming (5) into canonical form and assigning the PD spectrum, feedback control can be calculated as

111 1121 1( )= - ,( ) , -

cos cosT

r r tV V

tK K (7)

4.2. Velocity Sub-loop (Sub-loop 2) design

The control goal of velocity sub-loop is to use angle of attack, bank angle and fuel equivalence ratio to track path angle, heading angle and velocity. For the conventional aircraft, the velocity control can be decoupled from the path and heading angle control. But for the aero-propulsive coupled ABHV, it is necessary to take consideration of the three control variables together.

For the trajectory tracking guidance, control commands that satisfy the constraint of dynamic equations(1) have been stored along with state variables in the trajectory, and can be directly used as nominal control. The tracking error in sub-loop2 is 2 [ , , , , , ]Tdt dt Vdt Vx , feedback control variable 2 , , T

u . Linearize the

derivative of 2x and get:


2 2 2 2 2( ( ))t tx xA B u (8) The expression of 2B needs to be listed below:

2 1 3 210 211 212 1 3 230 231 232 1 3 250 251 252( ) ( ) ( ) ( ) ( )( ) ; ( ); ; ( ) ( ) (; );t t t t t t t t tt b b b b b b b b b0 0B 0 (9) Where

210 sin cos cos /) (( )T Lb VT m 211 ( sin )sin / ( )b T L mV 212 ( sin cos) / ( )b mT L V

230sin cos si

os( ) n

cT

bT L

mV 231

( sin + )coscos

T LbmV

232( sin sin

co)

sT L

Vb

m 250 ( cos s /n )ib T T D m

251 0b 252 ( cos ) /Db mT

The expected closed loop error dynamics of sub-loop 2 is:

211 212

2221 222

231 232

0 1 0 0 0 0( ) ( ) 0 0 0 0

0 0 0 1 0 00 0 ( ) (

( )

) 0 00 0 0 0 0 10 0 0 0 ( )

C

t t

t t

t t

A (10)

The expression of feedback control of sub-loop 2: *2 2 2 2 2 2( -)- =tu x T KK x . 2T is the inverse of the matrix

combined by the second, fourth and sixth row in 2B .

* *211 212 216

* * * *2 222 221 224 226

*231 236

- 0 0 0

= 0 - 0

0 0 0cos

k k

k k k

g k

K (11)

*216 2 2 2

( sin ) cos ( sin )cos ( ) o1 s( ) cMa Ma VT L M V T L grmV

k tVm V

2*212 312

1 (( ) )sinVk gV r

t

2

*222 2

( sin )sin sin + sin sin tancos

( )k tm

T L VrV

*224 222 cos cos tan( ) V

Rk t

*226 2

( sin sin ( sin )sin cos sin tan+cos cos

)( ) Ma M VT L T L

mV rmVM

k t

*236 232

( co )( ) -

s VMa Ma Mk

Tt

Dm

The aero-angle command will be passed to the attitude loop as nominal signal; whereas fuel equivalent ratio command will be directly send to the 6DOF nonlinear aircraft model to drive the simulation.

4.3. Brent algorithm based guidance equation trim method

For cases that no reference control commands are stored in the trajectory, nominal controls need to be calculated on-board which demands the employment of a nonlinear equations solving method. This method should on one hand guarantee convergence, on the hand should be highly efficient to meet the real-time requirement. Brent algorithm is an efficient nonlinear equation root finding method that combines the superlinear convergence with the sureness of bisection [20]. A new method is proposed based on Brent algorithm to generate the nominal control. This new method exploits the monotonic feature of guidance equations with respect to control variables to get iterative solution. Solution interval can also be specified by the designer and convergence can be guaranteed. When no root exists in the specified interval, the nearest boundary value of the interval to the root is provided as solution.

When the required accelerations of three body axes are known, the guidance equations can be written as:


2ngi

2

cos sin

sin cos / )cos

( sin )sin / cos cos sin

( ) (

+ tan /L

a

La

o

ti

mN

r g mN

T m

T D mg

T

r m

L

N

m V

L V

(12)

Na NLongi and NLati are axial, longitudinal and latitudinal acceleration command respectively. The left hands of the three equations are monotonic with fuel equivalent ratio, angle of attack and bank angle. The trimming of guidance loop is transformed into solving three singe variable equations iteratively by the following algorithm:

Fix the thrust command in the first equation, and substitute this value into the following two equations, which then only depends on and .

Fix in the second equation and substitute into the third equation. Then the third equation becomes a single variable equation and monotonic with regard to bank angle , which can be easily solved using Brent algorithm. Denote the solution of the third equation as .

Substitute back into second equation, which now only depends on and also monotonic with . Us Brent algorithm to solve the second equation and use the solution to update . Similarly, substitute and back into the first equation and get the solution of the first equation * .

If *| | tol , iteration stops. Otherwise, return to 1), use * to update , and then repeat the whole iteration.

To improve the efficiency, when trimming the guidance equations continuously, the last trimming solution is used as the initial value of new iteration. Exploiting the property of the continuity of solution, the length of intervals can also be shortened based on the last trim solution. When the velocity does not change significantly, as in the case of tactical cruise, the first equation can be ignored and the number of equations to be solved reduces to two. The computation time required would further decreases.

5. Attitude loop design

5.1. Attitude Sub-loop (Sub-loop 3) design

The control goal of attitude sub-loop is to track angle of attack and bank angle command. In the attitude controller design, earth rotation effect is ignored. The derivatives of aero-angles have the following relationship with angular velocity in body axis.

cos sin ( cos cos sin ) / ( )

cos tan sin tan ( sin ) / ( cos )

sin ( sin ) tan / ( ) ( cos sin ) / cos

z x

y x z

x z

mg L mV

C T mV

T

C T mV

(13)

Inverse (13) yields the expression of nominal angular velocity:

sin + cos cos - cos cos sin ( sin cos sin cos sin ) / ( )

sin sin sin ( cos cos sin ) / ( )

cos cos sin cos sin sin [cos ( sin ) sin ( sin )] / (

-

)z

x

y

L C mg mV

C T mV

L mg T C mV

(14)

In(14), , ,T

is obtained through pseudo-differentiator (Section 6.2). Since 0 ,the expression of (14) can

actually be further simplified. The error state variable of sub-loop 3 is 3 [ , , , , , ]Tdt dt dtx , feedback

control 3 , ,T

x y zu . The expected closed loop error dynamics of sub-loop 3 has the same expression as (10).

3 3 3(- )tu xK (15)


312 314 331 332

3 322 321 324 331 332

311 332 334

0 cos cossin s

0= 0

0i

0n

0

k kk kk k

K (16)

312 sin sec ) / ( ) t( [( an) ]xCk t C mV

314 ( ) [ sin / ( )( ) ]zk t C T mV 2

322 322sec tan / ( )) ) n( ta( xmVk t C C

324 322 (( )cos / )) (k t TC mV 332 312sin cos( ) ( ) / ( )k T Vt L T m

334 ( ) s ( )in( ) /xk t L T mV

5.2. Attitude Angular Velocity Sub-loop (Sub-loop 4) design

The control goal of sub-loop 4 is to track nominal angular velocity. The nominal moment of force can be obtained via inversing the rotation dynamic equation(3).

2 2

( ) + ( )

( ) ( )

( ) ( )

x x x y z y z xy x z y

y y y z x x z xy y z x

z z z x y x y xy y x

J J J

J J J

M J

M

M J

J

J J J

(17)

The tracking error and feed back control of sub-loop 4 is 4 [ , , , , , ]Tx x y y z zdt dt dtx and

4 , ,T

x y zM M Mu. The expected closed-loop dynamics of sub-loop4 is the same as (10). In the preliminary design, aerodynamic

damping derivatives and aerodynamic cross derivatives can be ignored, feedback control is 4 4 4(- )tu xK

411 412 412 414 416

4 411 422 421 424 426

432 434 431 432

0

0

0 0

x xy

xy y

z z

k J k

K J k k

k J

k

J k

k J

J

(18)

412 412( ) xy z xk t J J 414 422( ) ( )y zzxyk t J J J 416 )( () xy x yy zk JJt J

422 412( ) ( )z x zxyk t J J J 424 422( ) xy z yk t J J

426 ( ) ( )x z x xy yk t J J J

432 ( 2 )) (xy x x y yJ J Jk t 434 ( ) 2 ( )xy y x y xk t J J J

5.3. Attitude control allocation

Attitude control allocation is used to transform the commanded moments generated by the attitude loop to the control surface deflections. Moment of force got by present fuel equivalent ratio command C and last surface deflection commands * * *, ,[ ]Tx y zD DD is denoted asM

. Define CM MM .The linearized equation is:

*{ { , , }, }

=

y z

y z

y

x

x

Com

zx

DD Dx x xx x

DD Dy

DD D z

y y y y

zx yz z Dz z

M M M M

M M MDD

DM M M

M

M

(19)

The square matrix is called the control effectiveness matrix. Inverse (19) yields [ , , ]Tx y zD D D ,which is the control surface deflection difference that needs to be added to previous value in order to generate the demanded torque. Final surface deflection *D D D . The initial values of *

xD and *yD are set to be zeros, and *

zD equals to the deflection at trim condition.


6. Controller parameter tuning

6.1. Closed-loop PD spectrum design

( )ijl t , 1,2,3,4, 1,2,3, 1,2i j l in matrix (10) is obtained through second order closed loop PD Spectrum:

2

1

2

( ) ( )

( ) 2 ( ) ( ) / ( )ij nij

ij ij nij nij nij t

t

t t

t

t (20)

ij is constant damp ratio and nij is time-varying band-width. When 0t and 0,( ) ( ) 0ij nijt t , the system is closed loop stable. Since TLC design is based on singular perturbation theory, therefore the band width of inner loop should sufficiently larger than the outer loop ( 1 2 3 4n j n j n j n j ). The physical meaning of ij and nij is analogous to LTI second order system: the damp ratio determines the overshoot and natural frequency determines the response time. The tuning of ij can start from the “optimal damp ratio” 0.707 and increase gradually according

to time domain response. The selection of nij can consult the value used in X33 design in reference [11] and do some adjustments to meet the response speed requirement.

Table 1 Four Loops TLC Controller Parameters

Sub-loop1 Sub-loop2 Sub-loop3 Sub-loop4

11

11

2.40.03n

21

21

2.40.1n

22

22

2.40.08n

23

23

2.40.2n

31

31

3.40.3n

32

32

3.40.3n

3

33

3

3.40.3n

41

41

3.41.2n

4

42

2

3.41.2n

4

43

3

3.41.5n

6.2. pseudo-differentiator design

In the attitude loop, in order to get nominal angular velocity and nominal moment of force, it is necessary to derivate the aero-angle commands and angular velocity commands. These derivatives are computed using a first order pseudo-differentiator represented by the following transfer function:

,

,( ) n diff

diffn diff

ss

sG (21)

,n diff is the bandwidth of the low-pass filter. The selection of ,n diff should concern the problem of reducing noise as well as retaining useful information. The bandwidth of sub-loop 3 is at first set to be zero and then gradually increase, meanwhile adjusting the bandwidth of sub-loop 4 to be 3~5 times greater than sub-loop, until clear oscillation can be seen. These are the largest possible values the two bandwidths can reach. Then fine tune the parameters below the largest possible values. The final values selected are 3, =0.3n diff , 4, 1.5n diff .

7. Tests and Simulation results

Linear quadratic regulator method is a linear optimal control design algorithm that has been successfully applied to MIMO system control. In this paper, LQR method is used as baseline control method to compare with the performance of TLC based G&C integrated Controller. In the guidance loop, angle of attack, bank angle and fuel equivalent ratio are used to track altitude, path angle, heading angle and velocity. The linearization result of guidance equations is(22). Weight matrixes are selected as Q=diag[0.1,5.28e6,1.28e6,1.] R=diag[1.31e4 3.28e3 1.e4].


Guid GuidA B

r r

VV

(22)

In the longitudinal attitude channel, elevator is used to track angle of attack command. In the lateral-directional attitude channel, aileron and rudder are employed to track bank angle command and keep side-slip angle near zero. The linearized longitudinal and lateral-directional equations are listed below:

Longi Longi z Latx

i Latiyzz x

yy

x

A bD

AD

BD (23)

Longitudinal and lateral weighting matrixes are selected as Q=diag[2.28e7 0.],R=diag[800.], Q=diag[2.28e5 1.28e7 3.28e3 3.28e4],R=diag[1.e3 3.e3] respectively.

The test item selected to compare the two controllers requires ABHV to track an S curve and in the same time keep flight at constant height and velocity. This test item entails a fast establishment of large bank angle to meet the demand of lateral acceleration at the start. The angle of attack and side-slip angle will also experience a violent transitory process and will hence influence the thrust control. The aim of this test item is to investigate the tracking accuracy of the controller when performing large three dimensional maneuver. Its application background is to avoid no-fly zone via latitudinal maneuver. The uncertainty intervals of atmospheric density, lift and drag coefficient are set to be 10%, 10% and 15%.

-78 -77.8 -77.6 -77.4 -77.2 -77 -76.8 -76.629.27

29.28

29.29

29.3

29.31

29.32

29.33

Longi(deg)

Lati(

deg)

Latitudinal Maneuvering 5KM

FinalHeading

OriginalHeading

180 190 200 210 220 230 24019.95

20

20.05

timeInFlight(sec)

alt(

KM

)

Ref

TLC

LQR

180 190 200 210 220 230 2401395

1400

1405

timeInFlight(sec)

Vel

ocity

(m/s

)

Ref

TLC

LQR

Fig. 2. Diagram of Latitudinal S Maneuver Fig. 3. Altitude and Velocity Tracking History


180 190 200 210 220 230 24084

85

86

87

88

89

90

91

timeInFlight(sec)

psi(D

eg)

TLC

LQR

180 190 200 210 220 230 2402

3

4

5

6

timeInFlight(sec)

alph

a(D

eg)

TLC

TLCGuid

180 190 200 210 220 230 2402

3

4

5

6

timeInFlight(sec)

alph

a(D

eg)

LQR

LQRGuid

Fig. 4. Heading Angle Tracking History Fig. 5. Angle of Attack Tracking History

180 190 200 210 220 230 240-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

timeInFlight(sec)

beta

(Deg

)

TLC

LQR

180 190 200 210 220 230 240-50

0

50

100

timeInFlight(sec)

sigm

a(D

eg)

TLC

TLCGuid

180 190 200 210 220 230 240-50

0

50

100

timeInFlight(sec)

sigm

a(D

eg)

LQR

LQRGuid

182 184 186 18840

45

50

55

182 184 186 18840

50

60

Fig. 6. Sideslip Angle Tracking History Fig. 7. Bank Angle Tracking History

180 190 200 210 220 230 240-1.2

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

timeInFlight(sec)

Dx(

Deg

)

TLC

LQR

180 190 200 210 220 230 240-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

timeInFlight(sec)

Dy(

Deg

)

TLC

LQR

Fig. 8. Aileron Deflections Fig. 9. Rudder Deflections


180 190 200 210 220 230 240-0.5

0

0.5

1

1.5

2

2.5

timeInFlight(sec)

Dz(

Deg

)

TLC

LQR

180 190 200 210 220 230 2400.8

0.85

0.9

0.95

1

1.05

1.1

1.15

1.2

1.25

timeInFlight(sec)

eta

TLC

LQR

Fig. 10. Elevator Deflections Fig. 11.Fuel Equivalence Ratio Command History

Remarks: It can be seen from Fig3 that both LQR and TLC could stabilize the ABHV at specified altitude and velocity (it

also indicates that LQR is a rather good controller). However, with nearly the same amount of control surface usage (Fig 8~11), TLC has a better performance in tracking angle of attack, sideslip angle and bank angle commands(Fig 5 ~ 7).

8. Conclusion

1) Since BTT guidance law generates angle of attack, bank angle and zero side-slip angle, which facilitates the attitude loop to track aero-angles directly, BTT strategy can better satisfy strict aero-angle constraint.

2) TLC controller could achieve accurate tracking under considerable aerodynamic uncertainties. Compared with Linear Quadratic Method, TLC is a fully onboard nonlinear control method which has fewer parameters to tune, a more visualized way to select parameters.

3) TLC enables the unification and design integration of the control structure in the guidance and attitude tracking law, which facilitates parameter-matching and real-time adjusting of guidance and attitude loops.

References

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