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Procedia Engineering 00 (2014) 000–000
www.elsevier.com/locate/procedia
“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 aeropropulsion 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 aeroangle 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: judenchd@hotmail.com
1877-7058 © 2014 The Authors. Published by Elsevier Ltd.
Peer-review under responsibility of Chinese Society of Aeronautics and Astronautics (CSAA).
2
G.D.Zhu Z.J.Shen / Procedia Engineering 00 (2014) 000–000
Nomenclature
radial distance from the center of earth to the vehicle, longitude, latitude
r , , 
earth-relative velocity, flight path angle, heading angle
V , ,
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
g ,  , e 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]:
r =V sin 
V  (T cos   D) / m  g sin 
(1)
  [T sin   L) cos  / m  (V 2 / r  g ) cos  ] / V
  [(T sin   L)sin  / (m cos  )  V 2 cos  sin tan  / r ] / V
The aero-propulsive coupling has its effect mainly on the longitudinal motion, embodied by the term
Ctn
. When
G.D.Zhu Z.J.Shen/ Procedia Engineering 00 (2014) 000–000
3
taking into consideration of this coupling effect, lift force is calculated by L =V 2 (cy  ctn ) Sref / 2 .
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.
dVx / dt  Fx / m  (Vz yEB  VyzEB )  2e (Vz  BE 21  Vy BE 31 )
dVy / dt  Fy / m  (VxzEB  VzxEB )  2e (Vx BE 31  Vz BE11 )
(2)
dVz / dt  Fz / m  (VyxEB  Vx yEB )  2e (Vy BE11  Vx BE 21 )
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:
   y cos   z sin 
J x d x / dt  ( J y  J z ) yz  J xy (d  y / dt  xz )  M x
1
(3)
J y d  y / dt  ( J z  J x )xz  J xy (d x / dt   yz )  M y

( y sin   z cos  )
cos 
J z d z / dt  ( J x  J y )x y  J xy (x2   y2 )  M z
  x  tan  ( y sin   z cos  )



90
It worth noting that when heading angle
, 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
nE

 2
(4)
2
c s  2 E nE s  nE
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.
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G.D.Zhu Z.J.Shen / Procedia Engineering 00 (2014) 000–000
4. Guidance loop design
Altitude pseudoinversion
rC
r
Altitude LTV PI
controller
r

V 
 
 
 
 

Velocity pseudoinversion
Sub-loop2
Velocity LTV
PI controller
V 
Guidance Loop 
 
 
 
 
 
 
 
 
 
 x 
 
 y 
 
Attitude pseudo-  z 
inversion
 x 
 
Sub-loop3
 y 
 
Attitude LTV  z 
Angular velocity
pseudo-inversion
Angular velocity
LTV
PI controller
PI controller
 
 
 
 
Attitude Loop
 x 
 
 y 
 
 z
Mx 
 
My 
 
Mz 
C
Control
Allocation
Sub-loop1
x
y
z
Sub-loop4
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
T
x1    r , r 
x1  [  rdt , r ]T
variable to control the altitude. The state variable and its nominal value are
，
. Define
the tracking error
x1  x1  x1 and linearize the derivative of tracking error at the original of error dynamics
coordinate:
 rdt  0 1   rdt   0 
 
   


0
0
 r  
 V cos  
  r
The expected closed loop error dynamics of sub-loop 1 is:
1 
 0
A1C  

 111 112 
(5)
(6)
After transforming (5) into canonical form and assigning the PD spectrum, feedback control can be calculated as
T

 

  K1 (t )   r r  , K1 (t )= - 111 , - 112 
(7)
 V cos  V cos  
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 x2  [  dt ,  ,  dt , ,  Vdt ,V ]T , feedback control variable u2   ,  ,  . Linearize the
T
derivative of x2 and get:
G.D.Zhu Z.J.Shen/ Procedia Engineering 00 (2014) 000–000
5
(8)
x2  A2 (t ) x2  B2 (t )u2
B
2
The expression of
needs to be listed below:
(9)
B2 (t )  013 ; b210 (t ) b211 (t ) b212 (t ) ; 013 ; b230 (t ) b231 (t ) b232 (t ); 013 ; b250 (t ) b251 (t ) b252 (t )
Where
b210  (T sin   T cos   L ) cos  / (mV ) b211  (T sin   L)sin  / (mV ) b212  (T sin   L ) cos  / (mV )
(T sin   T cos   L ) sin 
mV cos 
 (T cos   T sin   D ) / m
(T sin  +L) cos 
mV cos 
b251  0
b230 
b250
b231 
b232 
(T sin   L ) sin 
mV cos 
b252  (T cos   D ) / m
The expected closed loop error dynamics of sub-loop 2 is:
A2C
1
0
0
0
0 
 0
  (t )  (t )
0
0
0
0 
212
 211
 0
0
0
1
0
0 


0
 221 (t )  222 (t )
0
0 
 0
 0
0
0
0
0
1 


0
0
0
 231 (t )  232 (t ) 
 0
(10)
The expression of feedback control of sub-loop 2: u2  - K 2 (t ) x2 =-T2 K2* x2 . T2 is the inverse of the matrix
combined by the second, fourth and sixth row in B2 .
- 211

*
K2 =  0

 0
*
k216
(t )  
(TMa sin   LMa )MV V cos 
mV 2
(T sin   L)sin  sin 

*
k212
0
0
0
*
k222
- 221
*
k224
0
g cos 
0
0
 231
(T sin   L)cos 
mV 2
1 g
(  2 ) cos 
r V
* 
k216

*
k226

* 
k236 
*
k212
(t )   312 
(11)
1 V2
(
 g )sin 
V r
2
V
V
*
k224
(t )   222  cos  cos tan 
sin  sin tan 
R
r
mV cos 
(T sin   LM )MV sin  (T sin   L)sin  cos  sin tan  *
(T cos   DMa ) M V
*
k236 (t )  - 232  Ma
k226
(t )   Ma
+

2
m
mV cos 
r
mV cos 
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.
*
k222
(t )  
2
+
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:
6
G.D.Zhu Z.J.Shen / Procedia Engineering 00 (2014) 000–000
T cos   D  mg sin   mN a
(T sin   L) cos   m(V 2 / r  g ) cos   mN Longi
(12)
(T sin  +L)sin  / cos   mV cos  sin tan  / r  mN Lati
2
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.
   z cos    x sin   (mg cos   L  T cos  sin  ) / (mV )
   y   x cos  tan    z sin  tan   (C  T sin  ) / (mV cos  )
(13)
   sin   (C  T sin  ) tan  / (mV )  ( x cos    z sin  ) / cos 
Inverse (13) yields the expression of nominal angular velocity:
x  - sin  + cos  cos  - cos  cos  sin   ( L sin   C cos  sin   mg cos  sin  ) / ( mV )
 y     sin   sin  sin   (C cos   T cos  sin  ) / ( mV )
(14)
z   cos    cos  sin   cos  sin  sin   [cos  ( L  mg sin  )  sin  (T  C sin  )] / ( mV )
T
In(14),  ,  ,   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 x3  [   dt ,  ,   dt ,  ,   dt ,  ]T , feedback
T
control u3   x ,  y ,  z  . The expected closed loop error dynamics of sub-loop 3 has the same expression as (10).
(15)
u3  - K3 (t ) x3
G.D.Zhu Z.J.Shen/ Procedia Engineering 00 (2014) 000–000
k312
 0
K 3 =  0
k322
 311 k332
0
k314
321 k324
0
k334
331 cos 
331 sin 
0
332 cos  
 332 sin  

0
k312 (t )  [(C sin   C sec  ) / (mV )  x tan  ]
k314 (t )  [(C  T ) sin  / (mV )  z ]
k322 (t )  (C  C sec ) tan  / (mV )  x tan 2   322
k324 (t )  322  (C  T )cos  / (mV )
k332 (t )  ( L  T sin   T cos  ) / (mV )   312
k334 (t )  x  (L  T sin  ) / (mV )
7
(16)
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).
M x  J xx  ( J y  J z ) yz +J xy (xz   y )
M y  J y y  ( J z  J x ) x z  J xy ( y z   x )
M z  J z z  ( J x  J y )x y 
J xy ( y2
(17)
 x2 )
The tracking error and feed back control of sub-loop 4 is x4  [   x dt ,  x ,   y dt ,  y ,   z dt ,  z ]T and
T
u4   M x , M y , M z 
.
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 u4  - K 4 (t ) x4
  411 J x

K 4    411J xy
 0

k412
 412 J xy
k422
 421 J y
k424
0
k432
0
k434
 431 J z
k414
0
k416 

k426 
 432 J z 
k412 (t )  J xyz  412 J x
k414 (t )  422 J xy  ( J y  J z )z
k416 (t )  J xyx  ( J y  J z )y
k422 (t )  412 J xy  ( J z  J x )z
k424 (t )  J xyz  422 J y
k426 (t )  J xyy  ( J z  J x )x
k432 (t )  2 J xyx  ( J x  J y )y
k434 (t )  2J xyy  ( J x  J y )x
(18)
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 [ Dx* , D*y , Dz* ]T is denoted as M . Define M  MC  M .The linearized equation is:
Dy
D
M xDz 
 M x  M x x M x
 Dx 




 
Dy
Dx
D
z
My
My 
(19)
M y  = M y
 Dy 

 M   Dx


Dy
 z   M z
 Dz 
Mz
M zDz  *
{ D { x , y , z},Com }
The square matrix is called the control effectiveness matrix. Inverse (19) yields [ Dx , Dy , Dz ]T ,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 Dx* and D *y are set to be zeros, and Dz* equals to
the deflection at trim condition.
8
G.D.Zhu Z.J.Shen / Procedia Engineering 00 (2014) 000–000
6. Controller parameter tuning
6.1. Closed-loop PD spectrum design
ijl (t ) , i  1, 2,3, 4, j  1, 2,3, l  1, 2 in matrix (10) is obtained through second order closed loop PD Spectrum:
2
ij1 (t )  nij
(t )
ij 2 (t )  2ijnij (t )  nij (t ) / nij (t )
(20)
ij is constant damp ratio and nij is time-varying band-width. When t  0 and ij (t )  0, nij (t )  0 , 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 ( n1 j  n2 j  n3 j  n4 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
11  2.4
n11  0.03
Sub-loop2
Sub-loop3
Sub-loop4
 21  2.4
n 21  0.1
31  3.4
n31  0.3
 41  3.4
n 41  1.2
 22  2.4
n 22  0.08
32  3.4
n32  0.3
 42  3.4
n 42  1.2
 23  2.4
n 23  0.2
33  3.4
n33  0.3
 43  3.4
n 43  1.5
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 s
(21)
Gdiff ( s) 
n,diff  s
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 n3,diff =0.3 , n4,diff  1.5 .
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].
G.D.Zhu Z.J.Shen/ Procedia Engineering 00 (2014) 000–000
9
r 
r 
 
 
 
 
 


(22)
   AGuid    BGuid  

 
 
 
V 
V 
 
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:
 
 
 
 
 
 
   A    B  Dx 

A

b
D
(23)
 

Longi 
Lati
Lati 
 Longi z
 
 x 
Dy 
 z 
x
 z 

 
 
 y 
 y 
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%.
29.33
20.05
alt(KM)
Original
Heading
29.32
Ref
TLC
LQR
20
19.95
180
190
200
29.3
Final
Heading
29.29
29.28
29.27
-78
Latitudinal Maneuvering 5KM
-77.8
-77.6
-77.4
-77.2
Longi(deg)
-77
-76.8
Fig. 2. Diagram of Latitudinal S Maneuver
-76.6
210
timeInFlight(sec)
220
230
1405
Velocity(m/s)
Lati(deg)
29.31
240
Ref
TLC
LQR
1400
1395
180
190
200
210
timeInFlight(sec)
220
230
Fig. 3. Altitude and Velocity Tracking History
240
10
G.D.Zhu Z.J.Shen / Procedia Engineering 00 (2014) 000–000
91
6
alpha(Deg)
TLC
LQR
90
4
3
88
2
180
87
6
alpha(Deg)
psi(Deg)
89
86
85
84
180
190
200
210
timeInFlight(sec)
220
230
210
timeInFlight(sec)
220
230
240
LQR
LQRGuid
3
190
200
210
timeInFlight(sec)
220
230
240
Fig. 5. Angle of Attack Tracking History
100
TLC
LQR
0.5
55
TLC
TLCGuid
50
sigma(Deg)
0.6
200
4
Fig. 4. Heading Angle Tracking History
0.7
190
5
2
180
240
TLC
TLCGuid
5
50
45
40
182 184
0
186
188
beta(Deg)
0.4
-50
180
0.3
0.2
190
200
sigma(Deg)
-0.1
-0.2
180
190
200
210
timeInFlight(sec)
220
230
40
182
184
186
188
0
190
200
210
timeInFlight(sec)
220
230
240
Fig. 7. Bank Angle Tracking History
0.3
TLC
LQR
0.6
240
LQR
LQRGuid
50
Fig. 6. Sideslip Angle Tracking History
0.8
230
50
-50
180
240
220
60
100
0.1
0
210
timeInFlight(sec)
TLC
LQR
0.2
0.4
0.1
0.2
Dy(Deg)
Dx(Deg)
0
-0.2
-0.4
-0.6
0
-0.1
-0.2
-0.8
-0.3
-1
-1.2
180
190
200
210
timeInFlight(sec)
Fig. 8. Aileron Deflections
220
230
240
-0.4
180
190
200
210
timeInFlight(sec)
Fig. 9. Rudder Deflections
220
230
240
G.D.Zhu Z.J.Shen/ Procedia Engineering 00 (2014) 000–000
11
1.25
2.5
TLC
LQR
TLC
LQR
1.2
2
1.15
1.1
1.05
eta
Dz(Deg)
1.5
1
1
0.95
0.5
0.9
0
0.85
-0.5
180
190
200
210
timeInFlight(sec)
220
Fig. 10. Elevator Deflections
230
240
0.8
180
190
200
210
timeInFlight(sec)
220
230
240
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.
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