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Nonlinear Missile Autopilot Design with − Dθ Technique
Ming Xin*, S.N. Balakrishnan**
University of Missouri-Rolla, Rolla, MO 65409
and
Donald T. Stansbery+ Ernest J. Ohlmeyer++
Boeing Company, St. Louis, MO 63166 Naval Surface Weapon Center, Dahlgren, VA 22448-5000
Abstract
In this paper, a new nonlinear control method is used to design a full-envelope, hybrid bank-to-
turn (BTT)/skid-to-turn (STT) autopilot for an air-breathing air-to-air missile. Through this new
approach, called the Dθ − method, we find approximate solutions to the Hamilton-Jacobi-
Bellman (HJB) equation. As a result, the resulting nonlinear feedback law can be expressed in a
closed form. In this paper, a Dθ − outer-loop and inner-loop controller structure is used in an
autopilot design. A hybrid BTT/STT autopilot command logic is used to convert the commanded
accelerations from the guidance laws to reference angle commands for the autopilot. The outer-
loop Dθ − controller converts the angle-of-attack, sideslip, and bank angle commands to body
rate commands for the inner loop. An inner-loop Dθ − controller converts the body rate
commands to fin commands. This design is evaluated using a detailed six-degrees-of-freedom
simulation. Numerical results show that the new controllers achieve excellent tracking
performance and exhibit insensitivity to parameter variations over a wide flight envelope.
Keywords: Nonlinear Systems, Optimal Control, Missile Autopilot, Bank-to-turn/Skid- to-Turn Missile
* Postdoctoral Research Fellow, Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, MO 65401. [email protected]. Member AIAA.
** Professor, Department of Mechanical and Aerospace Engineering and Engineering Mechanics, University of Missouri-Rolla, MO 65401. [email protected]. Associate Fellow AIAA.
+ Research Scientist, Phantom Works, Boeing Company at Saint Louis, P.O. Box 516, MO 63166. [email protected]. Senior Member AIAA. ++: Scientist, Naval Surface Weapon Center, Attn: G23, 17320 Dahlgren Road, Dahlgren, VA 22448-5000 Associate Fellow AIAA
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Nomenclature
α Angle of attack
β Side-slip angle
µ Bank angle about the velocity vector
γ Flight path angle
p,q,r Body-frame roll, pitch and yaw rate
V Missile speed
M Mach number
m Missile mass
q Dynamic pressure
S Missile cross-sectional area
d Missile Diameter
g Gravity
h Missile flight altitude
, ,x y zI I I Moments of inertia about body-frame
, ,p q rδ δ δ Aileron, elevator, and rudder fin deflections
AC Axial force coefficient
YC Side force coefficient
( )Z NC C− Normal force coefficient
0 0,N YC C Normal and side force coefficients with zero fin
deflections
, ,p q rN N NC C C
δ δ δ Normal force coefficients with respect to aileron, elevator
and rudder fin respectively
, ,p q rY Y YC C C
δ δ δ Side force coefficients with respect to aileron, elevator and
rudder fin respectively
lC Roll moment coefficient
mC Pitch moment coefficient
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nC Yaw moment coefficient
0lC Roll moment coefficient with zero fin deflections
plC Roll moment coefficient with respect to roll rate
, ,p q rl l lC C C
δ δ δ Roll moment coefficient with respect to aileron, elevator
and rudder fin respectively
0mC Pitch moment coefficient with zero fin deflections
qmC Pitch moment coefficient with respect to pitch rate
, ,p q rm m mC C C
δ δ δ Pitch moment coefficient with respect to aileron, elevator
and rudder fin respectively
0nC Yaw moment coefficient with zero fin deflections
rnC Yaw moment coefficient with respect to yaw rate
, ,p q rr r rC C C
δ δ δ Yaw moment coefficient with respect to aileron, elevator
and rudder fin, respectively
Iya , I
za : Acceleration along the inertial y and z axis
T Thrust
1. Introduction:
Missile and aircraft autopilot designs have been mostly dominated by classical control
techniques. They require a great deal of tuning and ad hoc modifications are often unavoidable.
Cloutier et al. [1] surveyed on the classical designs as well as a variety of gain-scheduled modern
control designs such as LQG/LTR and eigenstructure assignment in the context of bank-to-turn
missile autopilot designs. Gain-scheduling has produced many highly reliable and effective
control systems. The drawback of this method is that information about the actual nonlinear
behavior is usually discarded. Williams et al. [2] proposed a gain-scheduled LQG controller to
account for nonlinear kinematic coupling terms by scheduling the linear pitch/yaw channel gains
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as a function of both dynamic pressure and body-axis roll rate. This approach achieved good
performance over a range of operating conditions although unsuitable for high angle of attack
maneuvering since it does not consider the aerodynamic nonlinearities. Krause and Stein [3]
presented an adaptive longitudinal autopilot scheduled on the basis of an estimated pitch control
derivative. Kamen et al. [4] synthesized an adaptive autopilot for a discrete-time linear-time-
varying (LTV) missile model. Tan et al. [5] applied linear parameter-varying (LPV) control
theory to the design of a gain-scheduled missile autopilot.
A number of nonlinear control methods have also been proposed for the missile autopilot design.
Dynamic Inversion (DI) was used for the inner-loop design in the inner/outer loop control
structure first proposed by Adams [6,7]. In the inner loop the plant dynamics are equalized
across the flight envelope using DI. µ -synthesis was employed to design the outer loop to
achieve performance and robustness requirements. McFarland and D’Souza [8] also combined
dynamic inversion control and µ -synthesis in the missile autopilot design. Since DI replaces the
set of existing dynamics with a designer selected set of dynamics, Georgie and Valasek [9]
attempted to quantify the particular form of desired dynamics which produce the best closed-
loop performance and robustness in a DI flight controller. Schumacher and Khargonekar [10]
compared gain-scheduled H∞ control with dynamic inversion using linearized analysis and
found that the latter lacked in robustness. Wise and Sedwick [11] applied nonlinear H∞ theory to
a missile with coupled aerodynamic and thrust-vectoring control. However, they showed that the
nonlinear design did not significantly improve the performance of a well-designed gain-
scheduled linear autopilot. There are also examples of missile autopilot designs using sliding
mode control [12-15].
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Another recently emerging technique that systematically solves the nonlinear regulator problem
is the State Dependent Riccati Equation (SDRE) method (Cloutier et al., 1996) [16]. By turning
the equations of motion into a linear-like structure, this approach permits the designer to employ
linear optimal control methods such as the LQR methodology and the H∞ design technique for
the synthesis of nonlinear control systems. It can be used for a broad class of nonlinear regulator
problems. It has been employed to design advanced guidance algorithms in [17] and used in [18]
for integrated missile guidance and control design. Wise and Sedwick [19] also reformulated
their earlier nonlinear H∞ missile autopilot design [11] using the SDRE framework, which led to
a simplified feedback control form for this nonlinear control application. The SDRE method
however, needs online computation of the algebraic Riccati equation at each sample time. The
Dθ − method developed in this study though similar to the SDRE technique has a closed form
solution.
In this paper, the Dθ − approach is formulated as a way to find an approximate solution to the
Hamilton-Jacobi-Bellman (HJB) equation. By introducing an intermediate variable θ , the co-
state λ can be expanded as a power series in terms of θ . The HJB equation is then reduced to a
set of recursive algebraic equations. By adding perturbations to the cost function and
manipulating these terms appropriately semi-global asymptotic stability can be achieved [20]. By
adjusting the parameters in the perturbation terms, we are also able to modulate the transient
performance of the system.
There are two basic modes of controlling the attitude of a missile to achieve the acceleration
commanded by the guidance law: skid-to-turn (STT) and bank-to-turn (BTT). In the STT mode,
the roll angle may be held constant or uncontrolled. The major advantage of STT is its faster
response. A BTT missile allows only positive angles of attack while maintaining small sideslip
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angles to prevent missile maneuvers from shading the inlet in order to increase engine efficiency
and thereby maximize range. In this paper, we apply the Dθ − technique to design a hybrid
BTT/STT autopilot for an air-to-air missile to use their respective advantages.
It is an accepted result in flight mechanics that the flight control problem is structured in two
layers. The motion of the center of gravity is addressed in the outer loop while the angular
motion around the center of gravity is taken care of by the inner loop. Likewise, the control
structure can also be structured into an outer-loop control which provide the servo tracking of
prescribed flight angle command and the inner-loop control providing servo tracking of angular
rates. Tournes and Johnson [21] developed an inner-loop controller for a combat aircraft based
on Subspace Stabilization Control Theory to track the attitude rates. The outer-loop controller
was designed with the Linear Adaptive technique to follow the flight path and ground track
angles. Good results were obtained in both tracking performance and robustness. The SDRE
technique is used in [22] to design the inner/outer loop, full-envelope autopilot for Bank-to-
Turn/Skid-to-Turn missiles from an optimal control point of view
This paper is organized as follows: The θ -D approximation method is formulated in Section 2.
In Section 3, the missile dynamics are described. The Dθ − inner/outer loop autopilot design is
presented in Section 4. In Section 5, the BTT/STT command logic is presented. Simulation
results and analysis are given in Section 6. Conclusions are made in Section 7.
2. Dθ − Suboptimal Control Method
In this paper we consider optimal control of systems of the form
( ) ( )x f x B x u= + (1)
with a cost function given by
0
1( ( ) )
2T TJ x Q x x u Ru dt
∞= +∫ (2)
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where , , , , ,n n n m m n n m mx R f R B R u R Q R R R× × ×∈ ∈ ∈ ∈ ∈ ∈ ; Assume that x ∈Ω and Ω is a compact
set in nR ; ( )Q x is semi-positive definite and R is a positive definite constant matrix; It is
assumed that f(0)=0;
To ensure that the control problem is well posed we assume that a solution to the optimal control
problem (1), (2) exists. We also assume that ( )f x is of class 1C in x on a compact set Ω and
zero state observable through Q.
The optimal solution of the infinite-horizon nonlinear regulator problem can be obtained by
solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation [23]:
11 1( ) ( ) ( ) ( ) 0
2 2
T TT TV V V
f x B x R B x x Q x xx x x
−∂ ∂ ∂− + =∂ ∂ ∂
(3)
where V(x) is the optimal cost , i.e. 0
1( ) m in ( ( ) )
2T T
uV x x Q x x u R u dt∞
= +∫ (4)
We assume that V(x) is continuously differentiable and V(x)>0 with V(0)=0.
Optimal control is given by 1 ( )T Vu R B x
x− ∂= −
∂ (5)
The HJB equation is extremely difficult to solve in general, rendering optimal control techniques
of limited use for nonlinear systems.
Now consider perturbations added to the cost function:
0
1
1[ ( ( ) ) ]
2T i T
ii
J x Q x D x u Ru dtθ∞∞
=
= + +∑∫ (6)
where θ and iD are chosen such that 1
( ) ii
i
Q x Dθ∞
=+∑ is semi-positive definite.
Write the original state equation as:
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0 0
( ) ( )( ) ( ) ( ) ( )
A x g xx f x B x u A x g uθ θ
θ θ = + = + + +
(7)
where A0 and 0g are constant matries such that 0 0( , )A g is a stabilizable pair and
0 0[( ( )),( ( ))]A A x g g x+ + is pointwise controllable.
Also write the perturbed cost function as
00
1
( )1[ ]
2T i Tx
ii
Q xJ x Q D x u Ru dtθ θ
θ
∞∞
=
= + + + ∑∫ (8)
such that 0( ) ( )xQ x Q Q x= + and 0Q is a constant matrix.
Define V
xλ ∂=
∂ (9)
By using (8) and (9) in HJB equation (3) we have the perturbed HJB equation:
10
1
( )1 1( ) ( ) ( ) 0
2 2T T T T ix
ii
Q xf x B x R B x x Q D xλ λ λ θ θ
θ
∞−
=
− + + + =
∑ (10)
Assume a power series expansion of λ in terms of θ
0
ii
i
VT x
xλ θ
∞
=
∂= =∂ ∑ (11)
where iT are assumed to be symmetric and to be determined.
Substitute (11) into the HJB equation (10) and equate the coefficients of powers of θ to zero to
get the following equations:
10 0 0 0 0 0 0 0 0 0T TT A A T T g R g T Q−+ − + = (12)
1 1 1 10 01 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1
( ) ( ) ( )( ) ( )
T TT T T T xT A x A x T Q xg g
T A g R g T A T g R g T T g R T T R g T Dθ θ θ θ θ
− − − −− + − = − − + + − − (13)
1 1 1 1 11 12 0 0 0 0 0 0 0 0 2 0 0 1 0 0 1 0 0
( ) ( )( ) ( )
T T TT T T TT A x A x T g g g g
T A g R g T A T g R g T T g R T T R g T T R Tθ θ θ θ θ θ
− − − − −− + − = − − + + +
1 1 11 0 0 1 1 0 0 1 0 0 2
TT Tg g
T g R g T T g R T T R g T Dθ θ
− − −+ + + − (14)
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11 1 1 11 1
0 0 0 0 0 0 0 0 0 0 10
( ) ( )( ) ( ) ( )
T TnT T T Tn n
n n n j n jj
T A x A x T g gT A g R g T A T g R g T D T g R R g T
θ θ θ θ
−− − − −− −
− −=
− + − = − − − + +∑
2 1
1 12 0 0
0 1
n nT T
j n j j n jj j
T gR g T T g R g T− −
− −− − −
= =
+ +∑ ∑ (15)
Since the right hand side of equations (12)-(15) involve x and θ , iT would be a function of x
and θ . Thus we denote it as ( , )iT x θ . The expression for control can be obtained in terms of the
power series:
1 1
0
( ) ( ) ( , )T T ii
i
Vu R B x R B x T x x
xθ θ
∞− −
=
∂= − = −∂ ∑ (16)
It is easy to see that the equation (12) is an algebraic Riccati equation. The rest of equations are
Lyapunov equations that are linear in terms of Ti ( 1i n= ).
We construct the following expression for , 1,iD i n= :
1 1 10 01 1 0 0 0 0 0 0
( ) ( ) ( )[ ]
T Tl t T xT A x A x T Q xg g
D k e T g R T T R g Tθ θ θ θ θ
− − −= − − + + − (17)
2 1 1 1 1 1 11 12 2 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1 0 0 1
( ) ( )[ ]
T T T Tl t T T TT A x A x T g g g g g g
D k e T g R T T R g T T R T Tg R T T R g T Tg R g Tθ θ θ θ θ θ θ θ
− − − − − − −= − − + + + + + + (18)
1 2 11 1 1 11 1
0 0 1 2 0 00 0 1
( ) ( )[ ( ) ]n
T T Tn n nl t T Tn n
n n j n j j n j j n jj j j
T A x A x T g g g gD k e T g R R g T T R T T g R g T
θ θ θ θ θ θ
− − −− − − − −− −
− − − − −= = =
= − − + + + +∑ ∑ ∑ (19)
where ik and 0, 1,il i n> = are adjustable design parameters.
The idea in constructing iD in this manner is based on the observation that large initial states
may give rise to large initial control due to the state dependent term ( )A x on the right-hand side
of the equations (13)-(15). It happens when there are some terms in A(x) which could grow to a
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high magnitude as x is large. To see this, for example, when ( )A x includes a cubic term, its
magnitude could be large if x is large. This large value will be reflected into the solution for iT ,
i.e. the left-hand side of equations (13)-(15). Since iT will be used in the next equation to solve
for 1iT + , this large value will be propagated and amplified and consequently cause higher control
or even instability. So if we choose iD such that
1 2 11 1 1 11 1
0 0 1 2 0 00 0 1
( ) ( )( )
T T Ti i iT Ti i
j i j j i j j i j ij j j
T A x A x T g g g gT g R R g T T R T T g R g T D
θ θ θ θ θ θ
− − −− − − −− −
− − − − −= = =
− − + + + + −∑ ∑ ∑
1 2 11 1 1 11 1
0 0 1 2 0 00 0 1
( ) ( )( )[ ( ) ]
T T Ti i iT Ti i
i j i j j i j j i jj j j
T A x A x T g g g gt T g R R g T T R T T g R g Tε
θ θ θ θ θ θ
− − −− − − −− −
− − − − −= = =
= − − + + + +∑ ∑ ∑ (20)
where ( ) 1 il ti it k eε −= − (21)
is a small number, iε can be used to suppress this large value from propagating in the equations
(13) through (15). ( )i tε is chosen to satisfy some conditions required in the proof of
convergence and stability of the above algorithm [20]. On the other hand, the exponential term
il te− with 0il > is used to let the perturbation terms in the cost function and HJB equation
diminish as time evolves.
Remark 2.1
Solving equations (12)-(15) is carried out offline successively from top to bottom, i.e. nT can
be solved from 1nT − . Equation (12) is a standard algebraic Riccati equation. Once 0 0 0, ,A g Q and
R are determined, 0T can be solved to be a constant matrix. The rest of the equations (13)-(15)
are linear equations in terms of 2, , nT T with constant coefficients 10 0 0 0( )TA g R g T−− and
10 0 0 0( )T TA T g R g−− once we obtain the solution for 0T from Eq. (12). With some linear algebra,
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the equations (13)-(15) can be rearranged in the form of 0ˆ ( , , )i iA T Q x tθ= . As a result, iT can be
written in a closed form, 10
ˆ ( , , )i iT A Q x tθ−= where 0A is a constant matrix. So we can get closed
form solutions to 2, , nT T with just one matrix inverse operation. The expression of ( , , )iQ x tθ
on the right hand side of the equations is already known and only needs simple matrix
multiplications and additions.
Remark 2.2:
The construction of iD in (17)-(20) serves three functions. The first is to suppress the large
control if it happens. The second is to provide an appropriate iε to guarantee the convergence of
power series expansion 0
( , ) ii
i
T x θ θ∞
=∑ and stability of the closed loop system [20]. The third
function is to allow flexibility to modulate the system performance by tuning the parameters of
ik and il in the iD . This will be shown in the autopilot design.
Remark 2.3:
θ is just an intermediate variable and its value can be kept as unity.
3. Missile Dynamics
Equations of motion of a generic air-to-air missile dynamics are given below [22] in terms of
missile’s speed V, angle of attack α , sideslip β , bank angle µ , roll rate p, pitch rate q, and yaw
rate r:
0 0
cos coscos cos sin sin cos sinA Y N
qS qS qSV C C C g T
m m m m
α βα β β α β γ= − + − − +
[( sin sin cos ) ( sin sin cos ) ( sin sin cos ) ]p p q q r rY N Y N Y N
qSC C p C C q C C r
m δ δ δ δ δ δβ α β δ β α β δ β α β δ+ − − − − − (22)
0tan cos tan sin cos cos cos sin
cos cos cosN A
g qS qSq p r C C
V mV mVα β α β α γ µ α α
β β β= − − + − +
12
sin cos( )
cos cos p q rN N N
qST C p C q C r
mV mV δ δ δ
α α δ δ δβ β
− − − − (23)
0 0
cos sinsin cos cos sin sin sin cos sin cosN A Y
g qS qS qSp r C C C T
V mV mV mV mV
α ββ α α γ µ α β α β β= − + + + + −
[( cos sin sin ) ( cos sin sin ) ( cos sin sin ) ]p p q q r rY N Y N Y N
qSC C p C C q C C r
mV δ δ δ δ δ δβ α β δ β α β δ β α β δ+ + − − − − (24)
cos sin[sin sin tan cos sin cos tan sin tan ]
cos cos
Tp r
mV
α αµ α µ γ α β µ γ α ββ β
= + + − +
0 0[ sin (sin tan tan ) cos cos sin tan sin cos sin tan cos cos tan ]A A N Y
qSC C C C
mVα µ γ β α µ β γ α µ β γ µ β γ− + + + +
cos cos tan [ ][cos (sin tan tan ) sin cos sin tan ]p q rN N N
g qSC p C q C r
V mV δ δ δγ µ β δ δ δ α µ γ β α µ β γ− + − − + +
[ ]cos cos tanp q rY Y Y
qSC p C q C r
mV δ δ δδ δ δ µ β γ+ − − (25)
0 0
2
( cos sin ) ( cos sin )2 p r
y zl n l n
x x x
I I qSd qSdp qr C p C r C C
I I V Iα α α α
−= + − + −
[( cos sin ) ( cos sin ) ( cos sin ) ]p p q q r rl n l n l n
x
qSdC C p C C q C C r
I δ δ δ δ δ δα α δ α α δ α α δ+ − − − − − (26)
0
2
( )2 q p q r
z xm m m m m
y y y
I I qSd qSdq rp C q C C p C q C r
I I V I δ δ δδ δ δ−
= + + + − − (27)
0 0
2
( sin cos ) ( sin cos ) [( sin cos )2 p r p p
x yl n l n l n
z z z z
I I qSd qSd qSdr pq C p C r C C C C p
I I V I I δ δα α α α α α δ
−= + + + + + +
( sin cos ) ( sin cos ) ]q q r rl n l nC C q C C r
δ δ δ δα α δ α α δ− + − + (28)
4. Dθ − Autopilot Design
4.1 Outer-loop and Inner-loop Tracking Structure
The autopilot design is performed in a two-loop structure shown in Figure 1. Acceleration
commands from the missile guidance law are converted to ,c cα β and cµ which are angle-of-
attack, sideslip, and bank angle commands respectively by using the BTT/STT logic which will
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be presented in Section 5. The outer-loop converts ,c cα β and cµ to roll-rate, pitch-rate, and
yaw-rate commands ,c cp q and cr respectively for the inner loop. The inner loop then converts
,c cp q and cr to fin commands ,c cp qδ δ and
crδ for the actuators.
Figure 1: Dθ − Outer-loop and Inner-loop Autopilot Structure
In order to design the autopilot to perform command following, the Dθ − controller is
implemented as an integral servo-mechanism. This is accomplished as follows. First, the state x
is decomposed as
[ ]T
T Nx x x= (29)
where Tx consists of the tracked states which should track a reference command cr . Nx consists
of the nontracked states. The state vector x is then augmented with Ix , the integral states of Tx :
[ ]T
I T Nx x x x= (30)
The augmented system is given by
( ) ( )x f x B x u= + (31)
where 0
( )( )
B xB x
=
(32)
So the problem addressed here is to find an optimal controller to minimize the cost functional
0
1( ( ) )
2T TJ x Q x x u Ru dt
∞= +∫ (33)
BTT/STT Logic
Dθ − outer-loop servo
Dθ − inner-loop servo
Actuator
Missile Dynamics
ca c
c
c
αβµ
+ -
+
-
pc
qc
r c
δδδ
p
q
r
δδδ
p,q,r , ,α β µ
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The Dθ − integral controller is given by
1
0
( ) ( ) ( , )
I c
T nn T c
n
N
x r dt
u R x B x T x x r
x
θ θ∞
−
=
− = − −
∫∑ (34)
where cr is the command that it is desired to track. In the Dθ − controller design, use of the first
three terms in 0
( , ) nn
n
T x θ θ∞
=∑ has been found to be accurate enough for this problem (as well as
some others that have been solved).
In the Dθ − formulation, we choose the factorization of nonlinear equation (7) in this way:
0 00 0
( ) ( ) ( ) ( )( ) ( )
A x A x B x B xx A x x B x uθ θ
θ θ − −= + + +
(35)
The advantage of choosing this factorization is that in the Dθ − formulation 0T is solved from
0A and 0g in Eq. (7) and Eq. (12). If we select 0 0( )A A x= and 0 0( )g B x= , we would have a
good starting point for 0T because 0( )A x and 0( )B x keep much more system information than
an arbitrary choice of 0A and 0g .
4.2 Dθ − Outer-loop Design For the outer loop, the state space is chosen to be:
[ ]TI I Ix V α α β β µ µ= (36)
with control vector u given by: [ , , ]Tu p q r= (37)
The tracking command is: [ ]Tc c c cr α β µ= (38)
This command is obtained from the guidance law by using the BTT/STT Logic which will be
presented in Section 5.
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The outer-loop state-dependent coefficient factorization ( )OLA x is chosen as:
11 13 15
31 33
51 53 55 57
71 73 75 77
0 0 0 0
0 0 1 0 0 0 0
0 0 0 0 0
( ) 0 0 0 0 1 0 0
0 0 0
0 0 0 0 0 0 1
0 0 0
OL
a a a
a a
A x
a a a a
a a a a
=
(39)
where
11 cos cos sin cos cos ( sin sin cos )p pA Y N
qS g T qSa C C C p
mV V mV mV δ δα β γ α β β α β δ= − − + + −
( sin sin cos ) ( sin sin cos )q q r rY N Y N
qS qSC C q C C r
mV mVδ δ δ δβ α β δ β α β δ− − − − (40)
013
sincosN
qSa C
m
α βα
= − (41)
015
sinY
qSa C
m
ββ
= (42)
031 2 2 2cos cos cos cos ( )
cos cos cos p q rN N N N
g qS qSa C C p C q C r
V mV mV δ δ δγ µ α α δ δ δ
β β β= − − − − (43)
33
sin sin
cos cos A
T qSa C
mV mV
α αβ α β α
= − + (44)
051 2cos ( cos sin sin ) ( cos sin sin )
p p q qY Y N Y N
qSa C C C p C C q
mV δ δ δ δβ β α β δ β α β δ= + + − +
( cos sin sin )r rY NC C r
δ δβ α β δ− + (45)
053
1 sinsin
2 N
qSa C
mV
α βα
= (46)
055
1 sin sinsin ( ) cos
2 N A
qS qS Ta C C
mV mV mV
β βα αβ β
= + − (47)
16
57
sincos
ga
V
µγµ
= (48)
071 2( ) cos cos tan
p q rY Y Y Y
qSa C C p C q C r
mV δ δ δδ δ δ µ β γ= + − − (49)
73
1 sin 1 sin 1 sinsin tan tan sin tan
2 2 2 A
T T qSa C
mV mV mV
α α αµ γ β µ γα α α
= + −
0
1 sin 1 sintan ( ) cos sin tan
2 2 p q rA N N N N
qS qSC C C p C q C r
mV mV δ δ δ
α αβ δ δ δ µ β γα α
− + + − − (50)
75
sin 1 sin 1 sincos cos tan cos sin sin
2 2 cos A
T T qSa C
mV mV mV
β β βα µ γ β α αβ β β β
= − + −
0
sin sin( )cos cos cos tan
cos p q rN N N N A
qS qSC C p C q C r C
mV mVδ δ δ
β βδ δ δ α α µ γβ β β
+ + − − +
0
1 sin sin( )sin cos tan cos cos
2 cosp q rN N N N
qS gC C p C q C r
mV Vδ δ δ
β βδ δ δ α µ γ γ µβ β β
+ + − − − (51)
77
1 sin 1 sinsin tan sin tan
2 2 A
T qSa C
mV mV
µ µα γ α γµ µ
= −
0
sin( )cos tan
p q rN N N N
qSC C p C q C r
mV δ δ δ
µδ δ δ α γµ
+ + − − (52)
Remark 4.1:
There are numerous way of factorizing ( )f x as ( )A x x . The principle of choosing ( )A x peculiar
to this missile problem is to find the terms with sinusoid function such as sinα , sin β or sin µ
and write these terms as sin(*)
* in which * represents , orα β µ . In this way we can pull out
the state variables from ( )f x and guarantee sin(*)
* is regular when * goes to zero.
The ( )OLB x becomes:
17
0 0 0
0 0 0
tan cos 1 tan sin
0 0 0( )
sin 0 cos
0 0 0
cos sin0
cos cos
OLB x
β α β α
α α
α αβ β
− − = −
(53)
Since p, q and r would be the control in the outer-loop, the current values of the fin deflections
pδ , qδ , and rδ are used in this loop during the simulation which is based on Eqs (22)-(25).
4.3 Dθ − Inner-loop Design
The objective in the inner-loop controller is to follow the commands roll rate cp , pitch rate cq
and yaw rate cr , respectively, which are produced in the outer-loop controller.
In the inner-loop design, we would not adopt the integral servo since our major aim is to follow
the angular commands , ,α β and µ , respectively, which is realized in the outer-loop controller.
Tracking of the body rate commands p, q and r in the inner loop is used to improve the transient
response of the missile.
Consider Eqs. (26)-(28); We notice that there are state-independent terms
0 0( cos sin )l n
x
qSd C C
I
α α−, 0m
y
qSdC
I and 0 0
( sin cos )l n
z
qSd C C
I
α α+ that won’t go to zero when
the inner-loop states approach zero. To see this, note that V and α , the state variables in the
outer-loop, are considered as constants in the inner-loop. As a result, q is not a function of any
state variable of the inner-loop state space. In order to impose the f(0)=0 assumption, these terms
can be manipulated by multiplying and dividing these terms by a state that will never go to zero.
18
Since p, q, r and pδ , qδ , rδ can all go to zero, an additional state s with stable dynamics is
added to the state space in order to absorb the biases.
ss sλ= − (54)
At each pass through the inner loop, s is set to its initial value (assumed small).
In the inner-loop, a hard bound of 30 degree is imposed on the fin commands. This is achieved
by replacing pδ , qδ , and rδ in Eqs. (26)-(28) with saturation sine functions. That is:
sin(30, )
sin(30, )
sin(30, )
sat pp
q sat q
r sat r
δδδ δδ δ
=
(55)
where the actuator dynamics for pδ , qδ , and rδ are modeled as first-order linear systems:
.
( )p pp p uδ λ δ= − − (56)
.
( )q qq q uδ λ δ= − − (57)
.
( )r rr r uδ λ δ= − − (58)
and the saturation sine function is defined as
*sin( )2
sin( , )2
2
m z for z
sat m z m for z
m for z
π
π
π
<= ≥− ≤ −
(59)
Thus, for the inner loop, the state space is given by
[ ]Tx p q r s p q rδ δ δ= [ ]Tp q ru u u u= (60)
The inner-loop controller becomes
19
1
0
( ) ( ) ( , ) T cT IL nIL IL n
n N
x ru R x B x T x
xθ θ
∞−
=
− = −
∑ (61)
where [ ]TTx p q r= , [ ]T
Nx s p q rδ δ δ= (62)
We follow the same factorization form as Eq. (35). The corresponding inner-loop state-
dependent coefficient factorization is given by [22]:
11 12 13 14 15 16 17
21 22 23 24 25 26 27
31 32 33 34 35 36 37
( ) 0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0 0 0
IL s
p
q
r
a a a a a a a
a a a a a a a
a a a a a a a
A x λλ
λλ
= − −
− −
(63)
where
2
11 cos2 lp
x
qSda C
I Vα= ; 12
1
2y z
x
I Ia r
I
−= ;
2
13
1sin
2 2 r
y zn
x x
I I qSda q C
I I Vα
−= −
0 014 ( cos sin )l nx
qSda C C
sIα α= − ; 15
sin(30, )( cos sin )
p pl nx
qSd sat pa C C
I pδ δ
δα αδ
= −
16
sin(30, )( cos sin )
q ql nx
qSd sat qa C C
I qδ δ
δα αδ
= − −
17
sin(30, )( cos sin )
r rl nx
qSd sat ra C C
I rδ δ
δα αδ
= − −
21
1
2z x
y
I Ia r
I
−= ; 2
22 2 qmy
qSda C
I V= ; 23
1
2z x
y
I Ia
I
−=
024 my
qSda C
sI= ; 25
sin(30, )pm
y
qSd sat pa C
I pδ
δδ
= ;
26
sin(30, )qm
y
qSd sat qa C
I qδ
δδ
= − ; 27
sin(30, )rm
y
qSd sat ra C
I rδ
δδ
=
20
2
31
1sin
2 2 p
x yl
z x
I I qSda q C
I I Vα
−= + ; 32
1
2x y
z
I Ia p
I
−= ;
2
33 cos2 rn
x
qSda C
I Vα=
0 034 ( sin cos )l nz
qSda C C
sIα α= + ; 35
sin(30, )( sin cos )
p pl nz
qSd sat pa C C
I pδ δ
δα αδ
= +
36
sin(30, )( sin cos )
q ql nz
qSd sat qa C C
I qδ δ
δα αδ
= − +
37
sin(30, )( sin cos )
r rl nz
qSd sat ra C C
I rδ δ
δα αδ
= − +
The linear factorization is taken except that sin(30, )sat ∆
∆ replaces ( ∆ ) wherever it appears. ∆
represents pδ , qδ , or rδ .
Also
0 0 0
0 0 0
0 0 0
( ) 0 0 0
1 0 0
0 1 0
0 0 1
ILB x
=
(64)
5. Bank-to-Turn/Skid-to-Turn Command Logic
A hybrid BTT/STT autopilot command logic is used to convert the commanded accelerations
from the guidance laws to reference angle commands for the autopilot. In the midcourse and
terminal phases of a missile flight, the BTT control is employed to prevent any engine flameout.
As the missile approaches the endgame phase and passes a preset time-to-go threshold, BTT
commands are executed simultaneously with STT commands to improve transient responses.
During this interval, the missile is flying as a hybrid BTT/STT.
21
The BTT mode is broken into three commanded-acceleration magnitude regions in the inertial
frame (I) as defined below [22]:
If 0.1Ica g< then STT control: cµ µ= (65)
If 0.1 1Icg a g≤ < then reduced BTT control: 1tan
1c
c
I Iy c
c Iz
a a
a gµ µ µ−
−= + −
(66)
If 1Ica g≥ then full BTT control: 1tan
Iyc
c Izc
a
aµ − −
=
(67)
where [ ]c c
I I I Tc y za a a= is the commanded acceleration in the inertial frame.
The idea of choosing command logic this way is to attempt to reduce the effect of noise affecting
the controller. When the commanded acceleration is small, it is desired to make the error
between commanded angles and actual angles small such that the controller would not respond to
the noise if the noise magnitude is comparable to the commanded accelerations. As the
commanded accelerations go up, the weight on errors would be increased accordingly. If above
1g, the true desired angle commands are used.
In the full BTT mode: ( / ) ( )c Z Z
c
Z
a m qS C C
Cα
α
αα
− −= , 0cβ = (68)
where [ ]c c
Tc y za a a= is the commanded acceleration in the body frame.
In the reduced BTT mode, ( / ) ( )
cos( )c Z Z
c full c
Z
a m qS C C
Cα
α
αα µ µ
− −= − (69)
where absolute value is used to ensure that angle of attack command for BTT mode is positive.
( / ) ( )
sin( )c Y Y
c full cY
a m qS C C
Cβ
β
ββ µ µ
− −= − (70)
22
where 1tan c
c
Iy
full Iz
a
aµ −
−=
(71)
Note that STT is used for small acceleration commands to prevent the missile from performing
0180 rolls to achieve insignificant accelerations. STT control is also used in the endgame for
quicker response.
In the STT mode:
( / ) ( )
cz Z Zc
Z
a m qS C C
Cα
α
αα
− −= ,
( / ) ( )cy Y Y
cY
a m qS C C
Cβ
β
ββ
− −= , cµ µ= (72)
As the missile approaches the endgame phase and passes a preset time-to-go threshold, the STT
commands are switched into the BTT commands over a preselected time interval to attenuate
transient responses,
(1 )c c c
c c c
c c cSTT BTT
α α αβ ρ β ρ βµ µ µ
= + −
(73)
where ρ is a parameter that varies linearly from zero to one over the specified time interval.
6. Simulation and Analysis of the Results
6.1 Simulation setup
Numerical results are obtained using a six degree-of-freedom BTT/STT missile model with a
Simulink setup developed in [22]. To ensure that the autopilot only command deflections that are
achievable, a 0200 / sec hard limit is imposed upon commanded roll rate p , pitch rate q and yaw
rate r respectively. The fin deflection limit is set to 030 . The actuator dynamics is modeled as a
first order system with a time constant of 0.01.
23
The goal of this study is to design an autopilot that maintains a good response throughout a wide
flight envelope. The state weighting on Iα ,α , Iβ and β are varied at low, medium and high
altitudes until comparable performance is obtained over a range of altitudes. The state weights
are chosen according to [22]. They are curve fit to a quadratic function of dynamic pressure, q ,
which in turn is a function of missile velocity, one of the states in the outer-loop.
22 33 44 55 66( ) 0, ( ), ( ), ( ), ( ), ( ),120OLQ x diag q x q x q x q x q x= , ( ) 0.5,1, 0.5OLR x diag= (74)
where *diag represents the diagonal matrix;
11 2 522 ( ) 11.6742 10 ( ) 2.0182 10 ( ) 0.6137q x q x q x− −= × + × +
10 2 433( ) 11.4251 10 ( ) 2.1170 10 ( ) 105.7985q x q x q x− −= × − × +
44 22( ) ( )q x q x= , 55 33( ) ( )q x q x= , 266 ( ) 1.2*min100,[( )180 / ] 0.001cq x µ µ π= − +
The inner-loop state and control weightings are chosen in a similar manner to those in the outer-
loop design:
22 33( ) 16, , ,0,0,0,0ILQ x diag q q= ( ) 0.04,0.3,0.06ILR x diag= (75)
where 10 2 422 1.4609 10 1.6816 10 48.7078q q q− −= × − × +
10 2 433 1.3880 10 1.6573 10 99.5426q q q− −= × − × +
0Q and ( )xQ x required in the Dθ − method (Eq. (10)) are chosen in the same manner as in the
Eq. (35), i.e. 0 0( )Q Q x= and 0( ) ( ) ( )xQ x Q x Q x= −
The time constants used the Eqs. (54), (56)-(58) are given by
1p q r sλ λ λ λ= = = = (76)
In order to show the effectiveness of the Dθ − autopilot design for a wide flight envelope, the
simulation is evaluated at three different flight conditions: The first condition is the initial design
24
condition: flight Mach number M=2.7 and flight altitude h=20,000ft. The second and third
conditions are at M=2.7, h=100ft (low altitude flight) and M=2.7, h=40,000ft (high altitude
flight), respectively. At the start of simulation, a commanded inertial acceleration of
10 , 10c cy za g a g= = − is fed into the system. These values are rotated into the body frame to be
used as inputs to the BTT/STT command logic. Since these commanded accelerations are larger
than 1g, the missile responded in the BTT mode.
6.2 iD design with the Dθ − controller:
In this simulation, first three terms, i.e. 0 1,T T and 2T , in the control equation (16) are used to
compute the necessary control. The simulation results will show that they are sufficient to
achieve satisfactory performance. The Dθ − outer/inner controller was designed based on the
first flight condition. It was found that the controller worked very well for the second and third
conditions using the same design parameters. The iD matrices for the outer-loop controller were
chosen as:
10 1 10 01 0 0 0 0 0 0
( ) ( ) ( )( ) ( )0.9
T Tt TOL OL xT A x A x T Q xg x g x
D e T g R T T R g Tθ θ θ θ θ
− − − = − − + + −
(77)
10 11 12 0 0 1
( ) ( ) ( )0.9
T Tt OL OLT A x A x T g x
D e T g R Tθ θ θ
− −= − − +
1 1 1
0 0 1 0 0 1 0 0
( ) ( ) ( ) ( )T TTg x g x g x g x
T R g T T R T T g R Tθ θ θ θ
− − −+ + +
1 11 0 0 1 0 0 1
( ) T Tg xT R g T T g R g T
θ− − + +
(78)
Note that in the outer-loop design, ( )B x is state dependent. Therefore there would be a ( )g x
term in 1D and 2D . Also recall in Remark 2.3 that θ is an intermediate variable that does not
affect the results.
The inner-loop controller is designed with
25
0.01 0 01
( ) ( ) ( )0,0,0,0,0.7 ,0,0.99
Tt t IL IL x ILT A x A x T Q x
D diag e eθ θ θ
− − = − − −
(79)
5 0.01 11 12 1 0 0 1
( ) ( )0.9 ,0,0.9 ,0,0.9 ,0,0.99
Tt t t t TIL ILT A x A x T
D diag e e e e Tg R g Tθ θ
− − − − − = − − +
(80)
Note that in the inner-loop design, ( )B x is a constant matrix. So there would be no ( )g x term in
1D and 2D .
As seen in Section 2, iD matrices play an important role in the Dθ − method design. ik and il
in the iD equations (17)-(19) are the design parameters. As pointed out in Remark 2.2, they can
be used to adjust the system performance as well as ensure the stability of the closed-loop
system. The selection of ( , )i ik l is based on the following method.
This approach is based on the observation that the Dθ − approach gives an approximate closed-
form solution to the State Dependent Riccati Equation (SDRE)
( )T -1 TF x P(x)+ P(x)F(x) - P(x)B(x)R B (x)P(x)+ Q = 0 (81)
where 0F(x)= A + A(x) .
To see this, assume that V
P(x)x∂ =∂x
(82)
and P(x) is symmetric. Substituting Eq. (82) into the HJB equation (3) and writing nonlinear
( )f x in a linear like structure 0( ) ( ) [ ( )]f x F x x A A x x= = + leads to the state dependent Riccati
equation (81). This is in fact the idea of the recently emerging State Dependent Riccati Equation
(SDRE) technique [16]. Compared with the SDRE approach, the Dθ − method solves a
perturbed HJB equation (10) and assumes that
0
T ( , ) ii
i
Vx xθ θ
∞
=
∂ =∂ ∑x
(83)
As can be seen, the Dθ − method is similar to the SDRE approach in the sense that both bring
the nonlinear equation into a linear-like structure ( ) ( )f x F x x= and solve the optimal control
problem. The former gives an approximate closed-form solution while the latter solves the
26
algebraic Riccati equation at each state. It can be predicted that the Dθ − solution is close to the
SDRE solution, i.e.
P( ) P( )D SDREθ − ≈
where 0
P( ) T ( , ) ii
i
D xθ θ θ∞
=− =∑ is obtained from solving Eqs. (12)-(15) offline and P( )SDRE is
obtained by solving Eq. (81) online.
An efficient procedure for finding the ( , )i ik l was determined as follows: The SDRE controller is
used to generate a state trajectory and then the maximum singular value of P( )SDRE , i.e.
max (P( ))SDREσ , is computed at each state point. Similarly, the ( , )i ik l parameters determine
P( )Dθ − and its associated max (P( ))Dσ θ − . Curve fits are then applied to max (P( ))SDREσ and
max (P( ))Dσ θ − , and the ( , )i ik l are selected to minimize the difference between these singular
value histories in a least squares sense . The ( , )i ik l can all be determined in one least squares run
off-line.
6.3. Numerical results and analysis
Figure 2 shows the performance of the outer/inner loop autopilot at the first flight condition. The
three figures on the left column of Figure 2 demonstrate the outer-loop tracking, i.e ,c cα β and
cµ . The right column shows the inner-loop tracking, i.e. ,c cp q and cr . Likewise, the results for
the second and third flight conditions are presented in Figure 3 and Figure 4, respectively. As can
be seen, both the outer-loop and inner-loop controllers provide excellent tracking response over a
large region of the operating envelope.
Figures 5-7 show the required fin deflections for these three cases. It can be seen that the fin
deflections are all well-behaved. Since the missile is flying in the BTT mode, the sideslip should
be as small as possible. From the results of these three altitude simulations it can be observed
that the sideslip angle is kept at less than 1 degree throughout. It can also be seen that the settling
27
time of the tracking responses at 40,000ft is longer than the other two. This is due to the variation
of the air density at high altitudes. Lower density at higher altitude leads to reduce aerodynamic
forces and moments. The same commanded accelerations need bigger aerodynamic angles to
achieve the necessary forces and moments. Therefore, the control response at 40,000ft shows
larger fin deflections compared to lower altitude cases. At 100ft altitude, the missile goes
through lowest angle of attack, sideslip angle and control effort. Figure 8 presents the autopilot
tracking of the acceleration commands coming from the guidance law. As can be seen, they are
tracked very well in all three altitudes.
To evaluate the sensitivity of the autopilot design to speed, the autopilot designed at M=2.7,
h=20,000ft is also evaluated at two other different Mach numbers, M=2.0 and M=3.2. Figures 9-
11 show the tracking responses and achieved fin deflections at these two Mach numbers,
respectively.
It can be seen that the autopilot still performs very well. Note that at the higher Mach number the
required angle of attack and control level are smaller than those at the lower Mach number. The
reason is that aerodynamic coefficients associated with the aerodynamic angles and fin
deflections are larger when the Mach number is higher. Also, at a given altitude, the dynamic
pressure is higher if the Mach number is higher. Thus for the same force commands, they need
less aerodynamic angles α and β and fin deflections. Since the missile is flying at the BTT
mode, the bank angle should be kept constant. Figure 12 demonstrates the comparison of bank
angle response at all the five different flight conditions. It can be seen that the autopilot performs
very well in all these cases.
From the simulations, several advantages of the Dθ − method can be observed. First, the Dθ −
outer/inner loop controllers show excellent tracking characteristics and are insensitive in
28
operating flight conditions. Second, the Dθ − based autopilot design is based on optimal
control of a cost function. It allows the designer to consider the tracking performance and control
energy in one integrated design. The third important advantage of the Dθ − method is that it is
available in closed-form expression if we take finite terms in Eq. (16). Compared with the
SDRE technique [22], the Dθ − method does not need excessive on-line computations. This
implementation advantage is more significant in the missile autopilot design problem since it
involves 7 7× matrices in both outer and inner loop designs. Solving 7 7× Riccati equations on-
line would be very time-consuming. The final remark is that the tuning of design parameters in
iD matrices is not an involved process. Usually a diagonal matrix is picked and multiplied by the
state dependent terms which are given in Eqs. (17)-(19). The numerical experiment also shows
that system performance is not sensitive to the variations around the values of ik and il in iD .
7. Conclusions
In this paper, a new nonlinear suboptimal control synthesis technique, Dθ − method, has been
used to design a hybrid Bank-to-Turn/Skid-to-Turn autopilot for a generic air-to-air missile. We
formulate this autopilot design as an optimal control problem. An outer loop and an inner loop
control structures are used. The outer-loop converts commanded angle-of-attack, sideslip, and
bank angle to body rate commands for the inner loop. The inner loop converts the body rate
commands to fin deflection commands. The simulation results demonstrate excellent tracking
performance and insensitivity to flight conditions over a wide flight envelope. The Dθ − design
gives a closed-form solution to this nonlinear optimal control problem and is easy to implement.
Acknowledgement
Grant from Anteon Corporation in support of this study for Naval Surface Warfare Center is
gratefully acknowledged.
29
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Figure 2: (M=2.7, h=20,000ft) Tracking responses of , , , , ,p q rα β µ
32
Figure 3: (M=2.7, h=100ft) Tracking responses of , , , , ,p q rα β µ
33
Figure 4: (M=2.7, h=40,000ft) Tracking responses of , , , , ,p q rα β µ
34
Figure 5: (M=2.7, h=20,000ft) Achieved
fin deflections
Figure 6: (M=2.7, h=100ft) Achieved fin
deflections
Figure 7: (M=2.7, h=40,000ft) Achieved
fin deflections
Figure 8: Total body YZ acceleration
tracking at different altitudes
35
Figure 9: (M=2.0, h=20,000ft) tracking response of , , , , ,p q rα β µ
36
Figure 10: (M=3.2, h=20,000ft) Tracking response of , , , , ,p q rα β µ
37
Figure 11: (h=20,000ft) Fin deflection comparison at different Mach numbers
38
Figure 12: Bank angle comparison at different flight conditions