1

Missile Longitudinal Autopilot Design Using A New

Suboptimal Nonlinear Control Method

Ming Xin* and S.N. Balakrishnan**

xin@umr.edu , bala@umr.edu

Department of Mechanical and Aerospace Engineering and Engineering Mechanics

University of Missouri-Rolla, Rolla, MO 65401, USA

Abstract

In this paper, a missile longitudinal autopilot is designed using a new nonlinear control

synthesis technique (called theθ -D approximation). The particular Dθ − methodology used

in this paper is referred to as the Dθ − 2H design. The Dθ − technique can achieve

suboptimal solutions to a class of nonlinear optimal control problems in the sense that it

solves the Hamilton-Jacobi-Bellman (HJB) equation approximately by adding perturbations

to the cost function. By manipulating the perturbation terms both semi-globally asymptotic

stability and suboptimality properties can be obtained. An interesting feature of this method is

that the expansion terms in the expression for suboptimal control are nothing but solutions to

the state dependent Riccati equations associated with this class of problems. The Dθ − 2H

design has the same structure as that of the linear 2H formulation except that the two Riccati

equations are state dependent. In using the Dθ − technique, we do not require online

solutions of the Riccati equation like the recently popular State Dependent Riccati Equation

(SDRE) technique. Development of the Dθ − technique is presented and an acceleration

tracking autopilot for a missile is designed. Numerical simulations are presented that

* Ph.D student ** Professor and contact person

2

demonstrate the potential of this technique for use in an autopilot design. Furthermore, these

results are evaluated in comparison with the SDRE 2H method.

Keyword: Nonlinear Systems, Optimal Control, Perturbation Methods, Missile Autopilot

1. Introduction

Many numerous techniques exist for the synthesis of control laws for nonlinear systems. One

popular method of formulation has been the optimal control of nonlinear dynamics with

respect to a mathematical index of performance [1-7]. A major difficulty in this line of

approach is finding solutions to the resulting Hamilton-Jacobi-Bellman (HJB) equation. The

HJB equation is extremely difficult to solve in general rendering optimal control techniques

of limited use for nonlinear systems. Consequently, a number of papers investigated methods

to find suboptimal solutions to nonlinear control problems.

One such technique is the power series expansion based method [1-7]. Garrard et.al.[1]

expanded the optimal cost function as a power series in terms of an artificial variableε and

utilize a similar technique to that for the linear systems. But this technique can only be

applied to a certain class of nonlinear systems in which the nonlinearity can be considered as

small perturbations. Nishkawa et.al.[2] proposed a method to determine the coefficients of a

power series. But the convergence of the series is not guaranteed particularly when the

nonlinearity is large. In addition, higher-order approximations do not necessarily give better

results. Wernli and Cook [3] developed an approach by bringing the original system into an

apparent linearization form. Their suboptimal control involves finding the Taylor expansion

of the solution to a state dependent Riccati equation. But the convergence of this series is not

guaranteed and the resulting control law leads to a large control effort when the initial states

are large. Garrard [4][5] also formulated another approach that expanded both the optimal

3

cost and the nonlinear dynamics as a power series of the states and used the same idea as

before. This idea is applicable to more general nonlinear systems. However, this method has

to assume the structure of the optimal cost as a scalar polynomial with undetermined

coefficients which contains all possible combinations of products of the elements of the state

vector. As the system order increases, the complexity of determining these coefficients

increases dramatically. Zhang et.al. [6] and Krikelis et.al.[7] formulated a method for

calculating the coefficients of a series solution. But it can only guarantee stability around the

origin. The common problem with these methods is that they do not offer a way to ensure that

the system is asymptotically stable in the large.

Saridis et.al.[8] developed a technique from an inverse point of view. Given an arbitrarily

selected admissible feedback control, a recursive algorithm solving the Generalized

Hamilton-Jacobi-Bellman (GHJB) equation was proposed for sequential improvement of the

control law that converges to the optimal. In [9], Saridis and Wang also extended this theory

to stochastic nonlinear systems and proposed design procedures using upper and lower

bounds of the cost function. But finding an appropriate value function to the GHJB equation

is still a very difficult task. Beard et.al. [10] adopted the Galerkin approximation to solve the

GHJB equation. Since the control laws are given as a series of basis functions, they are

inherent complex though. In addition, to find an admissible control to satisfy all the ten

conditions proposed in that paper is not an easy task.

Another recently emerging technique that systematically solves the nonlinear regulator

problem is the State Dependent Riccati Equation (SDRE) method (Cloutier et al., 1996)[11].

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

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class of nonlinear regulator problems. It has been employed to design advanced guidance

algorithms in [12] and used in [13] for a nonlinear benchmark problem. The SDRE technique

was also briefly discussed in [14]. The SDRE method however, needs online computation of

the algebraic Riccati equation at each sample time. The method developed in this study

however, has a closed form solution.

In this paper, a new suboptimal nonlinear controller synthesis (θ -D approximation)

technique based on approximate solution to the Hamilton-Jacobi-Bellman (HJB) equation is

proposed. By introducing an artificial variableθ , the co-stateλ 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 we are able to achieve semi-globally asymptotic stability. This has been proved

by using the Lyapunov stability theory. In addition, this technique can overcome the problem

of large-control-for-large-initial-states encountered by using the control law in [3]. By

adjusting the parameters in the perturbation terms, we are also able to modulate the transient

performance of the system.

Modern aircrafts or missiles often operate in flight regimes where nonlinearities

significantly affect dynamic response. For example, a high-performance missile must be

quickly responsive to and follow accurately any guidance commands, so that it can intercept

fast moving and agile targets. Fighter aircraft may operate at high angles of attack where the

lift coefficient cannot be accurately represented as a linear function of angle of attack or at

high roll rates where nonlinear, inertial cross-coupling may result in instabilities. Traditional

gain-scheduling design of autopilots was performed separately on each axis of the missile.

The drawback of this method is that we discard information about the actual nonlinear

behavior. Many nonlinear control methods have been proposed for the missile autopilot

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design. In [4], a power series expansion based method was used in the design of nonlinear

automatic flight control systems operating with high angles of attack. This method was also

applied to the control of highly maneuverable aircraft in [5] and is compared with the

performance of a conventional PI gain scheduled controller. Tan et.al. [15] applied linear

parameter-varying (LPV) control theory to the design of a gain-scheduled missile autopilot.

A nonlinear, coupled, 3-axis generic missile model is used and the LPV method yielded

excellent results. There are also examples of pitch-axis missile autopilot designs using sliding

mode control [16], H∞ synthesis [17] and µ synthesis [18]. These methods presented the

robust missile autopilot design under the nonlinear uncertain models.

In this paper, we extend the standard linear 2H optimal control method to the nonlinear

problems using the Dθ − technique. The linear 2H control problem has been studied and

implemented since 1960’s [22]. It is to find a proper controller that stabilizes the system

internally and minimize the 2H norm of the transfer function from the exogenous input to the

performance output. It is defined in the frequency domain. In the state-space solution, it can

be referred to the standard LQR or LQG design that has been used widely. With output

feedback, the 2H design ends up with having to solve two Riccati equations. There are some

studies that examined the use of 2H optimal controller design in nonlinear systems. In [19],

the SDRE 2H method was used to design a full-envelope pitch autopilot. However, solving

two Riccati equations on-line is very time-consuming. In this paper, Dθ − 2H design is

proposed addressing the same problem as that in [19]. However, the Dθ − 2H design does

not require on-line computation of Riccati equation and gives an approximate closed form

solution to the two state dependent Riccati equations. Although the linear 2H (LQG)

solutions provide no global guaranteed robustness properties possessed by LQR solutions, we

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have found that the extension to the nonlinear 2H design with the Dθ − technique have the

robustness to some extent which will be shown in this paper. In a broader sense, since the

Dθ − method is an effective tool to solve the state dependent Riccati equation off-line, it can

be easily extended to the nonlinear H∞ problem. The linear H∞ problem still involves two

Riccati equations. Cloutier et.al.[11] have extended it to the nonlinear H∞ problem by

solving two state dependent Riccati equations. In this sense, the Dθ − technique is more

applicable in this area.

The formulation of theθ -D approximation method is presented in section 2. In section 3,

the Dθ − 2H method is used to design the missile longitudinal autopilot and is compared

with the SDRE 2H method. Numerical results and analysis are given in section 4.

Conclusions are given in section 5. Proof of convergence of the cost function expansion and

bounds on the error in cost are presented in the appendix.

2. Suboptimal Control of a Class of Nonlinear Systems

In this paper we restrict ourselves to the state feedback control problem for the class of

nonlinear time-invariant systems described by

( ) ( )x f x B x u= +& (1)

with the cost function:

0

1 ( )2

T TJ x Qx u Ru dt∞

= +∫ (2)

where , : , , : , ,n n n m m n n m mx R f R B R u R Q R R R× × ×∈Ω⊂ Ω→ ∈ Ω→ ∈ ∈ ; Q is semi-definite

matrix and R is positive definite matrix; f(0)=0;

To ensure that the control problem is well posed we assume that a solution to the optimal

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control problem (1), (2) exists. We also assume that ( )f x is locally Lipschitz in x on a

compact set Ω and zero state observable through Q.

The optimal solution to the infinite-horizon nonlinear regulator problem can be obtained by

solving the Hamilton-Jacobi-Bellman (HJB) partial differential equation [20]:

11 1( ) ( ) ( ) 02 2

T TT TV V Vf x B x R B x x Qx

x x x−∂ ∂ ∂

− + =∂ ∂ ∂

(3)

where (0) 0V = .

The optimal control is given by 1 ( )T Vu R B xx

− ∂= −

∂ (4)

and V(x) is the optimal cost , i.e. dtRuuQxxxV TTu )(

21min)(

0

+= ∫∞

(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 [ ( ) ]2

T i Ti

iJ x Q D x u Ru dtθ

∞∞

=

= + +∑∫ (6)

where θ and iD are chosen such that 1 2

ii

iD θ

∞

=∑ is small compared to

2Q .

For later use, we rewrite the original state equation as:

0 0

( ) ( )( ) ( ) [ ( )] [ ( )]A x g xx f x B x u A x g uθ θθ θ

= + = + + +& (7)

where A0 is a constant matrix such that 0 0( , )A g is a stabilizable pair.

Define xV∂∂

=λ (8)

8

By using (8) in (3), we have

1 21 2

1 1( ) ( ) ( ) ( ) 02 2

T T T T nnf x B x R B x x Q D D D xλ λ λ θ θ θ−− + + + + + =L (9)

Assume a power series expansion of λ as 0

ii

i

T xλ θ∞

=

= ∑ (10)

where iT are to be determined and assumed to be symmetric.

By substituting (10) into the HJB equation (3) and equating the coefficients of powers of θ to

zero to get the following equations:

10 0 0 0 0 0 0 0 0T TT A A T T g R g T Q−+ − + = (11)

1 1 1 10 01 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1

( ) ( )( ) ( )T T

T T T TT A x A x T g gT A g R g T A T g R g T T g R T T R g T Dθ θ θ θ

− − − −− + − = − − + + − (12)

1 1 1 1 1 11 12 0 0 0 0 0 0 0 0 2 0 0 1 0 0 1 0 0 1 0 0 1

( ) ( )( ) ( )T T T

T T T T TT A x A x T g g g gT A g R g T A T g R g T T g R T T R g T T R T T g R g Tθ θ θ θ θ θ

− − − − − −− + − =− − + + + +

1 11 0 0 1 0 0 2

TTg gT g R T T R g T D

θ θ− −+ + − (13)

11 1 1 11 1

0 0 0 0 0 0 0 0 0 0 10

( ) ( )( ) ( ) ( )T Tn

T T T Tn nn n j n j

j

T A x A x T g gT A g R g T A T g R g T T g R R g Tθ θ θ θ

−− − − −− −

− −=

− + − =− − + +∑

2 1

1 12 0 0

0 1

n nT T

j n j j n j nj j

T gR g T T g R g T D− −

− −− − −

= =

+ + −∑ ∑ (14)

Since the right hand side of equations (11)-(14) involve x andθ , iT would be the function

of x andθ . 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

u R B x R B x T x xλ θ θ∞

− −

=

= − = − ∑ (15)

Note that equation (11) is an algebraic Riccati equation. The rest of equations are

Lyapunov equations that are linear in terms of Ti(x). In the rest of this paper we will call this

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method θ -D approximation technique. The algorithm without iD term is called the θ

approximation. The algorithm in [3] would result in the θ approximation although with a

different set of arguments. One of the problems with theθ approximation, however, is that

large initial conditions may give rise to large control or even instability. We construct the

following expression for Di:

1 0 01 1

( ) ( )[ ]T

l t T A x A x TD k eθ θ

−= − − (16)

2 1 12 2

( ) ( )[ ]T

l t T A x A x TD k eθ θ

−= − − (17)

M

1 1( ) ( )[ ]n

Tl t n n

n nT A x A x TD k e

θ θ− − −= − − (18)

where ik and 0, 1,il i n> = L are constants.

The idea in constructing D in this manner is that the aforementioned large control

results from the state dependent term ( )A x on the right hand side of the equations (11)-(14). It

happens when there are some terms in A(x) which could grow to a high magnitude as x is

large. For example, when ( )A x includes a cubic term, the higher initial state would result in

higher initial iT and consequently higher initial control. So if we choose Di such that

1 1 1 1( ) ( ) ( ) ( )( )[ ]T T

i i i ii i

T A x A x T T A x A x TD tεθ θ θ θ

− − − −− − − = − − (19)

where ( ) 1 il ti it k eε −= − and ( )i tε is a small number, iε can be used to suppress this large

value to propagate in the equations (12) through (14). ( )i tε is chosen to satisfy some

conditions required in the proof of convergence and stability of the above algorithm [21]. On

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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 the time evolves.

Remark 2.1

Solving equations (11)-(14) is carried out offline from top to bottom. Equation (11) is a

standard algebraic Riccati equation. The rest of the equations (12)-(14) are linear equations in

terms of 2, , nT TL with constant coefficients 10 0( )TA gR g T−− and 1

0 0( )T TA T gR g−− . Just

with some linear algebra, the equations can be rearranged as: 0ˆ ( , , )i iA T Q x tθ= and

10

ˆ ( , , )i iT A Q x tθ−= where 0A is a constant matrix. So we can get the closed form solutions

for 2, , nT TL with just one matrix inverse operation. The expression of ( , , )iQ x tθ on the right

hand side of the equations is already known and need just some simple matrix multiplications

and additions.

Remark 2.2:

The construction of iD in (16)-(19) serves two functions. One of them is to provide an

appropriate iε to guarantee the convergence of power series expansion 0

( , ) ii

iT x θ θ

∞

=∑ and

stability of the closed loop system [21]. The other purpose is to modulate the system transient

performance by tuning the parameters of ik and il in the iD . Among other things it can be

used to reduce large control in transient response if it occurs.

Remark 2.3

θ is just an intermediate variable. The introduction of θ is just for the convenience of power

series expansion. The value of θ does not matter since it will be cancelled by the choice of

iD matrices (see equations (16)-(18)). Because iθ appears linearly in equations (11)-(14), it

get canceled when iT multiplies iθ in the control.

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Theoretical work has been done to prove the convergence of series expansion of

0

( , ) ii

i

T x θ θ∞

=∑ , semi-globally asymptotic stability of the Dθ − method and estimation of the

error in the cost by choosing appropriate Di matrices. These proofs can be found in [21].

3. Missile Longitudinal Autopilot Design

3.1 Formulation of the Dθ − 2H problem

Consider the general nonlinear system

( ) ( ) ( )w ux f x B x w B x u= + +& (20)

( ) ( )z zuz c x D x u= + (21)

( ) ( )y ywy c x D x w= + (22)

where w is the exogenous input including tracking command and noises injected into the

system; u is the control, z is the performance output and y is the measurement output.

Now the nonlinear dynamics is rewritten to have a linear-like structure as

( ) ( ) ( )w ux A x x B x w B x u= + +& (23)

( ) ( )z zuz C x x D x u= + (24)

( ) ( )y ywy C x x D x w= + (25)

Then the following formulation is similar to the standard linear 2H problem except that the

coefficent matrices of x, u and w are state-dependent. This has the same formulation as

SDRE 2H at this point [19].

The linear 2H problem leads to solving two Riccati equations given below in terms of their

Hamiltonians

12

1 1

2 12 21 1

1 12 2 12 2 12( )

T Tu u u

T T Tu

A B R R B R BR R R R A B R R

− −

− −

− − − + − −

(26)

1 1

12 2 21 1

1 12 2 12 12 2

( )( )

T Ty y y

Ty

A V V C C V CV V V V A V V C

− −

− −

− − − + − −

(27)

where

1 12 2T T T

w w w yw yw ywV B B V B D V D D= = = 1 12 2T T Tz z z zu zu zuR C C R C D R D D= = = (28)

If we write the nonlinear system (20)-(22) as linear-like systems (23)-(25), the nonlinear 2H

problem needs to solve the state dependent Riccati equation (26) and (27). Here the argument

x has been omitted for brevity.

Construct the nonlinear feedback controller via

c cdx A x B ydt

= + (29)

cu C x= (30) where , ,c c cA B and C are:

c u c c yA A B C B C= + − (31)

112 2[ ]T

c yB QC V V −= + (32)

12 12[ ]T T

c uC R B P R−= − + (33)

It is interesting to note that solving the state dependent Riccati equation (26) is equivalent to

solving the following nonlinear optimal control problem:

Finding u to minimize cost function: 1 11 12 2 12 20

[( ) ]T T TJ x R R R R x u R u dt∞ − −= − +∫ (34)

subject to the nonlinear differential constraint:

12 12[ ( ) ( ) ] ( ) ( ) ( )T

u u ux A x B x R R x B x u f x B x u−= − + = +& (35)

This class of nonlinear optimal control problem can be solved by using the Dθ − technique.

The same is true for the second state dependent Riccati equation (27). In the next section we

will use this Dθ − 2H approach to design a missile longitudinal autopilot.

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3.2 Missile Longitudinal Dynamics

The missile model used in this paper is taken from [19]. This model assumes constant

mass, no roll rate, zero roll angle, no sideslip, and no yaw rate. The nonlinear equations of

motion for a rigid airframe reduce to two force equations, one moment equation, and one

kinematic equation.

XBFU QW

m+ = ∑& (36)

ZBFW QU

m− = ∑& (37)

Y

Y

MQ

I= ∑& (38)

Qθ =& (39)

The force and moments about the center of gravity are

sin cos sinXBF L D mgα α θ= − −∑ (40)

cos sin cosZBF L D mgα α θ= − − +∑ (41)

YM M=∑ (42)

where α is angle of attack; θ is pitch angle;

212 LL V SCρ= , 21

2 DD V SCρ= , 212 mM V SdCρ= (43)

The normal force coefficient is used to calculate the lift and drag coefficients:

cosL NC C α= − , 0

sinD D NC C C α= − (44)

The nondimensional aerodynamic coefficients for the missile at 6096m altitude are:

3 (2 )3N n n n nMC a b c dα α α α δ= + + − + (45)

3 8( 7 )3m m m m m mMC a b c d e Qα α α α δ= + + − + + + (46)

14

The numerical values for the coefficients are:

019.373, 31.023, 9.717, 1.948, 0.300

40.440, 64.015, 2.922, 11.803, 1.719n n n n D

m m m m m

a b c d C

a b c d e

= = − = − = − =

= = − = = − = − (47)

In this paper, we adopt Mach number M, angle of attack α , flight path angle γ , and pitch

rate Q as the longitudinal variables since they appear in the aerodynamic coefficients. The

relationships required are:

2 2 2tan , , ,W VV U W MU a

α γ θ α= = + = = − (48)

and ,V UU WWM Va V

+= =& & &

& & (49)

Expanding these equations in terms of the aerodynamic coefficients yields

0

200.7 [ ( sin ] sinD NP S gM M C C

ma aα γ−

= − −& (50)

00.7 cos cosNP S gMC Q

ma aMα α γ= + +& (51)

00.7 cos cosNP S gMC

ma aMγ α γ= − −& (52)

200.7m

Y

P SdQ M CI

=& (53)

and we get the following equations by substituting aerodynamic data:

2 3 2 2

2 2

0.4008 sin 0.6419 sin 0.2010 (2 ) sin3

0.0062 0.0403 sin 0.0311sin

MM M M M

M M

α α α α α α α

αδ γ

= − − −

− − −

& (54)

30.4008 cos 0.6419 cos 0.2010 (2 ) cos3

cos0.0403 cos 0.0311

MM M M

M QM

α α α α α α α α

γαδ

= − − −

− − +

&

(55)

15

30.4008 cos 0.6419 cos 0.2010 (2 ) cos3

cos0.0403 cos 0.0311

MM M M

MM

γ α α α α α α α

γαδ

= − + + −

+ +

&

(56)

2 3 2 2

2 2

849.82 78.86 3.60 ( 7 )3

14.54 2.12

MQ M M M

M M Q

α α α α

δ

= − + − +

− −

& (57)

Actuator dynamics are included in the design and analysis. The model used is

2 2

0 1 02 c

a a a

δδδ

ω ζω ωδδ

= + − −

&

&&& (58)

where 0.7ς = , aω =50

The measurement used in this study is the normal acceleration (in g’s) which is described by

the equation:

200.7cos cos( )ZBz N

F P Sn M Cmg mg

θ γ α= + = + +∑ (59)

In terms of the flight conditions at 6096m (see Eqs. (45) and (47))

2 3 2 2 212.901 20.659 6.471 (2 ) 1.297 cos( )3zMn M M M Mα α α α δ γ α= − − − − + + (60)

3.3 Dθ − Controller Design

The objective in this paper is to design an optimal controller which is able to drive the

system to track the commanded normal acceleration (in g). The tracking block diagram is

shown in Figure 1.

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The control weight is cρ . The plant and output disturbance weights are wρ and ρ∆

The performance weighting function for tracking error r my y− is chosen to be [19]:

1( )0.001tW s

s=

+ (61)

or in state space form: 0.001 ( )ct t z zx x n n= − + −& (62)

The augmented state space x is given as:

[ , , , , , , ]Ttx M Q xα γ δ δ= & (63)

The control variable is the fin deflection:

cu δ= (64)

The state dependent coefficient matrix A(x) in (23) augmented by tx is chosen as [19]:

1K + +

( )uB x

cρ

cz

∫ C(x

A(x

2K

tW tz

my r zcy n= wρ

w

+ + +

+ -

ρ∆

∆

Figure 1: 2H Tracking Block Diagram

17

2 2 2

22

2

2

2

2

0.4008 sin 0.6419 sinsin0.0062 0.0311 0 0.0403 sin 0 0

0.2010 (2 )sin3

0.4008 cos 0.6419 coscos0.0311 0 1 0.0403 cos 0 0

0.2010 (2 )cos3

0.4008 cos 0.6419 coscos( ) 0.0311

0

M MM MMM

M MMMM M

M MA x

M

α α α αγ α

γα

α α α αγ α

α

α α α αγ

−− − −

− −

−−

− −

−=

−

2 2 2 2 2 2

2 2 2 2 2

0 0 0.0403 cos 0 0.2010 (2 )cos

380 49.82 78.86 3.6 ( 7 ) 0 2.12 14.54 0 0

30 0 0 0 0 1 00 0 0 0 2500 70 2500

cos( ) 12.901 20.659 6.471 (2 ) 0 0 1.297 0 03

MMM

MM M M M M

MM M M MM

αα

α α

γ α α α

− −

− + − + − − − − +− − + + −

(65)

2[0 0 0 0 0 0]Tu aB ω= (66)

The measurement is [ ]Tm zy n M Q= (67)

The acceleration command cr zy n= (68)

So the measured output is: [ ]c

Tz zy n n M Q= (69)

The corresponding ( )yC x is:

2 2 2 2 2

2 2 2

22

2 2 2 2 2 2

0 0 0 0 0 0 0cos( ) 12.901 20.659 6.471 (2 ) 0 0 1.297 0 0

30.4008 sin 0.6419 sin( ) sin0.0062 0.0311 0 0.0403 sin 0 0

0.2010 (2 )sin3

80 49.82 78.86 3.6 ( 7 ) 0 2.12 14.54 03

y

MM M M MM

M MC xM MMM

MM M M M M

γ α α α

α α α αγ α

γα

α α

+− − −

−=− − −

− −

− + − + − − 0

(70)

The exogenous input is: [ ]c z

Tz plant n M Qw n= ∆ ∆ ∆ ∆ (71)

The plant noise weights are chosen to be:

[0.2 0.01 0.01 0.2 0.01 0.01]Twρ = (72)

The measurement noise weights on zn , M and Q respectively are:

18

0.01 0 00 0.001 00 0 0.01

ρ∆

=

(73)

The control weight cρ is used as the design parameter and chosen to be 0.5, 2.

In order to avoid overflow problem in the numerical simulation, sinγγ

is set to 1 when γ

is less than 410− radian.

In the Dθ − formulation, we choose the partition of nonlinear equation (7) in this way:

0 00 0

( ) ( ) ( ) ( )( ) ( )A x A x B x B xx A x x B x uθ θθ θ

− − = + + + & (74)

with A(x) defined by (65). The advantage of choosing this partition is that in the Dθ −

formulation 0T is solved from 0A and 0g in (7) and (11). 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 that arbitrarily choosing 0A and 0g would have.

4. Numerical Results and Analysis

The simulation scenario is to initially command a zero g normal acceleration. At one

second we use a square wave normal accerleration command of magnitude 10g’s returning to

zero at 3 seconds.The initial states chosen for the simulation are: Mach number 2.5 with the

rest of variables being 0. The simulation is run at 100 samples per second. In solving the two

state dependent Riccati equations (26) and (27) with the Dθ − method, we pick 0T , 1T and 2T

three terms in theλ expansion (10).Three terms have been found to be sufficient for a good

approximation in this problem (as well as some others that we have solved). More terms

could be added if needed. The Dθ − design parameters are chosen as:

19

40 0 01 7

( ) ( )0.9999 [ ]T

t T A x A x TD e Iθ θ

−= − − , 40 1 12 7

( ) ( )0.9999 [ ]T

t T A x A x TD e Iθ θ

−= − − (75)

The above 1D and 2D are what we think the optimal parameters after some tuning

process. The numerical experiment with these parameters shows that the system performance

is not sensitive to the variations around these selected values. For comparison, we also use

the SDRE 2H method to design this missile autopilot based on the same model and

parameters. The parameterizations of f(x) and ( )yc x are the same as (65) and (70), i.e. same

A(x) and ( )yC x .

The results are presented in Figure 2-17. Figure 2 shows the commanded and achieved

normal acceleration and the control usage when the weight cρ of control is 0.5. Both the

SDRE method and the Dθ − method track very well and have reasonable transient

responses. We can see that during the second jump both controller show the maximum effort.

It is because changing from –10g to +10g is a significant jump for the missile to track and

need much more control energy. The SDRE controller needs more effort at this jump seen

from the control plot. Figure 3 shows that the state histories are similar for both methods.

Figure 4-5 represent the effects of increase in the weight on control to 2. While the control

usage is reduced, the normal accelereation tracking shows more lag. Figure 6-9 demonstrate

the effect of increasing the duration of the wave in the commanded normal acceleration from

1 second to 1.5 second with different cρ . We can observe that the SDRE controller shows an

obvious spike in the second jump. However, the Dθ − controller does not exhibit this

phenomenon.

Figure 10-11 show the responses in tracking a 20g normal acceleration with a wave of 1

second duration and 2cρ = . As can be seen, the normal acceleration, control usage and pitch

20

rate Q show some significant oscilations using SDRE 2H design. However, the Dθ − 2H

results do not have these spikes and go smoothly. It is obvious that SDRE control deflection

and pitch rate have exceeded the maximum allowable deflection. Figure 12-13 show the same

phenomenon when we consider a 1.5 second wave.

To test the robustness of both controllers, we investigate the performance robustness to

parameter variations. For this test the control weight of two, 20g normal acceleration tracking

and 1 second wave time are chosen as the design parameters. All aerodynamic coefficients

are then changed by 10%± in the missile model while keeping elevator coefficients

unaltered. Figure 14-15 show the effects on the Dθ − design while Figure 16-17 show the

effects on the SDRE design. As can be seen, the performance and control usage for both

methods do not change significantly with these parameter variations.

As far as the implementation issue is concerned though, the θ -D algorithm needs a matrix

inverse operation only one time offline when solving the linear Lyapunov equations (12)-(14)

and solution to the first algebraic Riccati equation (11) only one time, offline. That is to say,

when solving (12)-(14), we only need to rearrange the left hand side of the equations such

that they form a linear matrix equation: 0ˆ ( , , )i iA T Q x tθ= and then 1

0ˆ ( , , )i iT A Q x tθ−=

where 0A is a constant matrix and ( , , )iQ x tθ is the right hand side of (12)-(14). When

implemented online, this method involves only two 7 7× matrix multiplications and three

7 7× matrix additions if we take three terms. However, in comparison, SDRE needs

computation of the 7 7× algebraic Riccati equation at each sample time

21

5. Conclusions

In this paper, a new suboptimal nonlinear control synthesis technique was applied to the

missile longitudinal autopilot design. The new nonlinear Dθ − 2H design extends the

applicability of the linear 2H design. Approximate closed-form solutions to the two state

dependent Riccati equations in the nonlinear 2H formulation can be obtained by the Dθ −

method. Compared with the SDRE 2H design, this approach does not need the intensive on-

line solutions of the Riccati equation. In addition, simulation results demonstrated that the

Dθ − 2H design performed better than SDRE 2H design in both tracking performance and

in terms of the control effort they needed.

Acknowledgement

Grant from Anteon Corporation in support of this study for Naval Surface Warfare Center

is gratefully acknowledged.

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24

Figure 2: 10g (1 sec) Normal Acceleration Tracking and Control Usage when 0.5cρ =

Figure 3: Trajectories of M, Q, α and γ with 10g (1 sec) command tracking when 0.5cρ =

25

Figure 4: 10g (1 sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 5: Trajectories of M, Q, α and γ with 10g (1 sec) command tracking when 2cρ =

26

Figure 6: 10g (1.5sec) Normal Acceleration Tracking and Control Usage when 0.5cρ =

Figure 7: Trajectories of M, Q, α and γ with 10g (1.5sec) command tracking when 0.5cρ =

27

Figure 8: 10g (1.5sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 9: Trajectories of M, Q, α and γ with 10g (1.5sec) command tracking when 2cρ =

28

Figure 10: 20g (1 sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 11: Trajectories of M, Q, α and γ with 20g (1 sec) command tracking 2cρ =

29

Figure 12: 20g (1.5sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 13: Trajectories of M, Q, α and γ with 20g (1.5 sec) command tracking 2cρ =

30

Figure 14: Dθ − 20g (1 sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 15: Dθ − Trajectories of M, Q, α and γ with 20g (1 sec) command tracking 2cρ =

31

Figure 16: SDRE 20g (1 sec) Normal Acceleration Tracking and Control Usage when 2cρ =

Figure 17: SDRE Trajectories of M, Q, α and γ with 20g (1 sec) command tracking 2cρ =