We report on the design of a threeaxis missile autopilot using multiobjective control synthesis via linear matrix inequality techniques. This autopilot design guarantees
H
_{2}
/
H
_{∞}
performance criteria for a set of finite linear models. These models are linearized at different aerodynamic roll angle conditions over the flight envelope to capture uncertainties that occur in the highangleofattack regime. Simulation results are presented for different aerodynamic roll angle variations and show that the performance of the controller is very satisfactory.
1. Introduction
The increased demand for high maneuverability of modern missiles requires excellent autopilot performance over a large flight envelope. The main difficulty in missile autopilot design is the influence of high angleofattack aerodynamic phenomena on stability characteristics (Hemsch, 1992). As a missile executes a high angleofattack maneuver, the forebody vortices can become asymmetric and give rise to significant lateralforces and yawing moments. Classically, missile autopilots are designed using gain scheduling approaches in roll, pitch, and yaw channels (Blakelock, 1991; Shamma and Athans, 1992; Shamma and Cloutier, 1993; White et al., 2007). However, the performance of singleaxis autopilot design approaches are limited within some flight boundaries because aerodynamic coupling effects and their parameter dependencies in large angleofattack aerodynamics could not be considered. In this context, the design of a threeaxis missile autopilot has been studied using linear and nonlinear control approaches (Choi et al., 2008; Devaud et al., 2001; Kim et al., 2008). These methods are based on gain scheduling techniques by local linearization, and demonstrate some improvements from a practical point of view.
In this paper, a threeaxis missile autopilot design with a multiobjective outputfeedback control theory is presented. Because high angleofattack aerodynamics is highly nonlinear and estimates of the aerodynamic coefficients are very imprecise, a mix of
H
_{∞}
and
H
_{2}
criteria is employed to guarantee the performance of the threeaxis missile autopilot. This control methodology, which uses linear matrix inequality (LMI) techniques, was motivated by the work in (Schere et al., 1997).
This paper is organized as follows. In Section 2, we give an overview of multiobjective outputfeedback control theory. Section 3 describes our formulation of the missile control problem. Section 4 describes strategies and numerical simulation results for the designed autopilot. Our conclusions are provided in Section 5.
2. MultiObjective Control
This section gives a brief overview of multiobjective outputfeedback synthesis via the LMI approach described in (Chilali and Gahinet, 1996; Gahinet and Apkarian, 1994). For the missile autopilot design problem, the control objectives include
H
_{∞}
and
H
_{2}
performance.
H
_{∞}
performance is a convenient way to enforce robustness and model uncertainty with frequency domain specifications.
H
_{2}
performance is a useful way to address the output power (or error) of a generalized system due to unit intensity white noise input.
Consider a generalized plant with
where A∈R
_{nxn}
, D
_{12}
∈R
^{p1xm2}
, and D
_{21}
∈R
^{p2xm2}
. In addition,
x, u, z, y
, and
w
are the state, control input, controlled output, measurement output, and exogenous input, respectively. The multiobjective control problem is to find an LTI control law
u
=
K(s)y
that minimizes an upper bound for the
H
_{2}
gain subject to an
H
_{∞}
gain constraint.
 2.1 MultiObjective LMI Formulation
The LTI controller
K(s)
can be represented in statespace form by
For the LMI approach to multiobjective synthesis, nonlinear terms added in the output feedback case should be eliminated by an appropriate change of controller variables. This change of controller variables is implicitly defined in terms of the Lyapunov matrix
P
as follows:
The new controller variables can be written as
The mixed
H
_{2}
/
H
_{∞}
synthesis would then involve finding
X, Y
, and new controller variables such that Eqs. (59) hold while minimizing γ and
v
:
with the shorthand notation
 2.2 Controller Computations
Note that the multiobjective outputfeedback synthesis is an LMI problem of the form
After solving the synthesis LMIs, the controller computation proceeds as follows:

Find a nonsingular matrixM, Nto satisfyMNT=IXYvia SVD.

Solve the linear equation (Eq. 4) for the controller K(s) by computing (Eq. 13).
This LMI optimization problem can be efficiently solved with the LMI Control Toolbox (Gahinet et al., 1995). The solution of the LMI problem gives us an upper estimate of the suboptimal
H
_{∞}
and
H
_{2}
performance.
3. Missile Control Problem Formulation
 3.1 Missile Model and Autopilot Requirements
We considered a threeaxis nonlinear missile model of a skidtoturn cruciform missile with a high angleofattack. Sensors and actuator modeling were also taken into account. We assumed that the missile plant had 5 degrees of freedom (DOFs) with the fixed altitude and longitudinal velocity. Also, rigid body dynamics was considered. Under these assumptions, the 5DOF rigid body equations of motion were expressed using differential equations describing the translational motion and rotational motion as follows:
We considered a threeaxis nonlinear missile model of a skidtoturn cruciform missile with a high angleofattack. Sensors and actuator modeling were also taken into account. We assumed that the missile plant had 5 degrees of freedom (DOFs) with the fixed altitude and longitudinal velocity. Also, rigid body dynamics was considered. Under these assumptions, the 5DOF rigid body equations of motion were expressed using differential equations describing the translational motion and rotational motion as follows:
where δ, δ
^{cmd}
, Ω
^{act}
, and ξ
^{act}
are the actual fin deflection, commanded fin deflection, natural frequency, and damping coefficient, respectively. The statespace form of the missile model is written as
where
In this expression, α , β , and φ are the angle of attack, sideslip angle, and bank angle, respectively; δ
_{z}
, δ
_{y}
, and δ
_{r}
are the actual tail deflections in the pitch, yaw, and roll axes, respectively; and η
_{z}
and η
_{y}
are pitch and yaw acceleration commands, respectively. The inputs to the missile plant are commanded tail deflections δ
_{zc}
, δ
_{yc}
, and δ
_{rc}
. Therefore, the accelerations, bank angle, and angular rates are available for feedback. The total incidence angle α ' = acos(cosα cosβ) was used for scheduling, but the aerodynamic roll angle φ ' = atan(tanβ sinα ) was not measurable in this missile problem.
The aim of the proposed autopilot design is to steer the missile to track the acceleration guidance commands generated by an outer loop and stabilize the missile airframe at a given bank angle. The stabilization of the bank angle is a critical requirement for controlling a highly maneuverable missile. However, the aerodynamic roll angle is not measurable in a practical context. To deal with this problem, we found a dynamic controller
K(s)
to satisfy the given multiobjective for the multimodel using LMI techniques.
 3.2 Interconnected System Model
The performance specifications for the closedloop system in terms of multiobjective control are as follows:

Guarantee an upper bound on the L2induced gain γ of the operator mapping z to Ω.
Minimize an upper bound on the variance of z2 due to the disturbance Ω.
Control structure and interconnection for missile plant.
Frequency responses of the weighting function.
These performance requirements of the closedloop system were transformed into the multiobjective control framework with weighting functions. Weight functions were selected to account for the relative magnitudes of signals, frequency dependence, and relative importance. This method allows the controller designer to apply classical loopshaping concepts to obtain good performance while optimizing the response near the system bandwidth to achieve robust stabilization. The continuous time synthesis plant
P(s)
was readily obtained from the connections shown in
Fig. 1
, and incorporated in the missile model
G(s)
and weighting functions
W
_{ref}
,
W
_{err}
, and
W
_{u}
. These frequencydependent weights were tuned by performing a few synthesis and simulation trialanderror it erations on s for the nominal plant.
Figure 2
shows the frequency responses of the weighting functions.
4. Simulation Results
The angleofattack α and the sideslip angle β were not available by measurement only. In this missile control problem, α' and φ' can capture the nonlinearities of the missile plant during a high angleofattack maneuver. However, φ' is not available for scheduling while α' can be estimated. Modern missiles suffer from variations in aerodynamic roll angle because of their extended flight regimes. Thus, an autopilot should be robust against aerodynamic roll angle variations. In this section, we present our simulation results for the threeaxis missile autopilot design using LMI techniques.
Linear timeinvariant (LTI) models of a nonlinear missile plant were derived at three fixed operating points.
Table 1
shows the trim conditions of each operating point. V t is the total velocity of the plant. The multimodel includes Modela, Modelb, and Modelc. Since LMI constraints can incorporate multiple models with the design criteria, we can formulate the proposed autopilot design strategy systematically.
Trim conditions for local linearization
Trim conditions for local linearization
Pitch acceleration step responses.
Yaw acceleration step responses.
Bank angle stabilization.
Figures 3

5
show the tracking performance of the proposed scheme. Notice that the autopilot guarantees
H
_{2}
/
H
_{∞}
performance criteria for multiple linear models linearized at the different aerodynamic roll angle conditions. From the simulations results, we note that the performance goals were satisfied for each linearized model. However, the proposed controller cannot guarantee multiobjective performance for the entire flight trajectory in spite of its robustness. To overcome this conservativeness problem, studies of aerodynamic model uncertainties in the missile plant and additional linearized models for a wide range of aerodynamic roll angles are needed.
5. Conclusions
We proposed a threeaxis missile autopilot design with multiobjective control synthesis. Our simulation results for the missile plant demonstrate the effectiveness and performance of the proposed control strategy in spite of its limitations. The results show that the designed autopilot provided satisfactory performance for nonlinearities over high angleofattack flight boundaries and uncertainties in the aerodynamic roll angle estimation.
Acknowledgements
This research was sponsored in part by the Agency for Defense Development in Korea under grant number ADD 09010303.
Blakelock J. H
1991
Automatic Control of Aircraft and Missiles
2nd ed
Wiley
New York
Chilali M
,
Gahinet P
1996
H(infinity) design with pole placement constraints: an LMI approach
IEEE Transactions on Automatic Control
41
358 
367
DOI : 10.1109/9.486637
Choi B. H
,
Kang S. H
,
Kim H. J
,
Won D. Y
,
Kim Y. H
,
Jun B. E
,
Lee J. I
2008
Rollpitchyaw integrated H(infinity) controller synthesis for high angleofattack missiles
KSAS International Journal
9
66 
75
DOI : 10.5139/IJASS.2008.9.1.066
Devaud E
,
Harcaut J. P
,
Siguerdidjane H
2001
Threeaxes missile autopilot design: From linear to nonlinear control strategies
Journal of Guidance Control and Dynamics
24
64 
71
DOI : 10.2514/2.4676
Gahinet P
,
Apkarian P
1994
Linear matrix inequality approach to H(infinity) control
International Journal of Robust and Nonlinear Control
4
421 
448
Gahinet P
,
Nemirovski A
,
Laub A. J
,
Chilali M
1995
The LMI Control Toolbox User's Guide
The MathWorks Inc
Hemsch M. J
1992
Tactical Missile Aerodynamics: General Topics
American Institute of Aeronautics and Astronautics
Washington DC
Kim Y. H
,
Won D. Y
,
Kim T. H
,
Tahk M. J
,
Jun B. E
,
Lee J. I
,
An J. Y
2008
Integrated rollpitchyaw autopilot design for missiles
KSAS International Journal
9
129 
136
DOI : 10.5139/IJASS.2008.9.1.129
Schere C
,
Gahinet P
,
Chilali M
1997
Multiobjective outputfeedback control via LMI optimization
IEEE Transactions on Automatic Control
42
896 
911
DOI : 10.1109/9.599969
Shamma J. S
,
Athans M
1992
Gain scheduling: potential hazards and possible remedies
IEEE Control Systems Magazine
12
101 
107
DOI : 10.1109/37.165527
Shamma J. S
,
Cloutier J. R
1993
Gainscheduled missile autopilot design using linear parameter varying transformations
Journal of Guidance Control and Dynamics
16
256 
263
DOI : 10.2514/3.20997
White B. A
,
Bruyere L
,
Tsourdos A
2007
Missile autopilot design using quasiLPV polynomial eigenstructure assignment
IEEE Transactions on Aerospace and Electronic Systems
43
1470 
1483
DOI : 10.1109/TAES.2007.4441752