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Anti-Windup Command Filtered Adaptive Backstepping Autopilot Design for a TailControlled Air-Defense Missile
Conference Paper · August 2013
DOI: 10.2514/6.2013-5013
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Anti-Windup Command Filtered Adaptive Backstepping
Autopilot Design for a Tail-Controlled Air-Defense Missile
Florian Peter1, Fabian Hellmundt2, Florian Holzapfel3, Farhana Chew4
Institute of Flight System Dynamics, Munich, Germany, 85748
Two missile autopilots based on Command Filtered Adaptive Backstepping (CFAB) are
developed for a tail-controlled missile, which is steered in Skid-To-Turn (STT) mode. First, a
standard CFAB approach is tailored to the missile application. The second control law
constitutes a modification of standard CFAB design by incorporating an anti-windup strategy.
This novel Anti-Windup CFAB (AWCFAB) control law accounts explicitly for the limitations
of the actuator dynamics in order to prevent anti-windup effects. Since the missile dynamics
is subject to a large spectrum of uncertainties, an adaptive control layer is needed, which
compensates for the main disturbances, uncertainties and modeling errors. Therefore, the
desired, nominal performance and tracking quality can be preserved even in cases of a
degraded plant model.
Two evaluation series are accomplished with a high-fidelity 6DOF missile model, in which
a realistic spectrum of uncertainty combinations and actuator limitations are considered. In
the first evaluation series the performance of standard CFAB and the developed AWCFAB
autopilot is compared under nominal conditions. For this purpose a demanding Skid-To-Turn
scenario is chosen, which requires the full actuator capability. The benefits of the adaptive
autopilot are demonstrated via Monte-Carlo simulations.
Nomenclature
(𝒂𝐴𝑖 )𝐵
(𝐶𝑖𝑗 )𝐵
𝑇
𝑖
𝑖
𝑖
𝑎𝑦,𝐴
𝑎𝑧,𝐴
= [𝑎𝑥,𝐴
]𝐵 acceleration due to aerodynamic forces at the reference position 𝑖 denoted in the
𝐵-frame
= aerodynamic coefficient
(𝐶𝑖𝑗,𝑥 )𝐵
𝒇𝑖
(𝑭𝐴𝐺 )𝐵
(𝒈𝐺 )𝐸
𝑮𝑖
=
=
=
=
=
(𝑰𝐺 )𝐵𝐵
=
𝑀𝑎
𝑴𝐸𝐵
(𝑴𝐴𝐺 )𝐵
𝑞̅
(𝒒)𝐵
(𝒓𝐺 )𝐸
𝒖
𝒖𝑐𝑚𝑑
(𝑽𝐺𝐾 )𝐸𝐵
=
=
=
=
=
=
=
=
=
derivative of the aerodynamic coefficient wrt the state 𝑥
nominal nonlinear functions of the respective dynamics
aerodynamic forces considered at the c.g.
gravity vector given in the 𝐸-frame
nominal input effectiveness of the respective dynamics
𝐺
𝐼𝑥𝑥
0
0
𝐺
0 ] moment of inertia given in the 𝐵-frame
[ 0 𝐼𝑦𝑦
𝐺
0
0 𝐼𝑧𝑧
𝐵𝐵
Mach number
transformation matrix from 𝐵-system to 𝐸-system
aerodynamic moments
dynamic pressure
quaternion vector
[𝑥 𝐺 𝑦 𝐺 𝑧 𝐺 ]𝑇𝐸 position vector of the missile’s c.g. given in the 𝐸-frame
aerodynamic equivalent controls
control output of the FCS
[𝑢𝐾𝐺 𝑣𝐾𝐺 𝑤𝐾𝐺 ]𝐸,𝑇
𝐵 velocity vector given in the 𝐵-frame
1
Ph.D. Candidate, Student Member AIAA; florian.peter@tum.de
Master Student, fabian.hellmundt@tum.de
3
Professor, Senior Member AIAA; florian.holzapfel@tum.de
4
Ph.D. Candidate, Student Member AIAA; farhana.chew@tum.de
2
1
American Institute of Aeronautics and Astronautics
𝒘𝑖
𝒙
𝒚
𝒚𝑃
𝒚𝑃𝑓𝑖𝑙
𝒛𝑖
𝛼𝐾
𝜶𝝎
𝛽𝐾
̂𝑖
𝜣𝑖 , 𝜣
̃
𝜣𝑖
(𝝎𝐸𝐵
𝐾 )𝐵
=
=
=
=
=
=
=
=
=
=
=
=
unachieved portion in case of AWCFAB
generalized state vector
controlled variable
controlled variable considered at virtual point 𝑃
filtered output at virtual point 𝑃
unachieved portion for CFAB
angle of attack
intermediate control
angle of side-slip
unknown plant parameter, estimated plant parameter
parameter error
[𝑝𝐾 𝑞𝐾 𝑟𝐾 ]𝐸𝐵,𝑇
angular rates given in the 𝐵-frame
𝐵
I. Introduction
D
esigning a flight control system (FCS) for a highly agile air-defense missile demands to the fulfillment of two
top-level requirements over the entire flight envelope: first, the full physical capabilities of the missile system
should be exploited; and second, the closed-loop behavior should compensate for unavoidable deviations from the
assumed model. The autopilot is designed to fulfill demanding STT-maneuvers, which consists of lateral and
longitudinal acceleration commands by maintaining a roll angle of zero.
The extended flight envelope, which consists of a large velocity and angle of attack range demands extensive
efforts in the modeling process. Due to the extended flight envelope, the identification of the aerodynamic data set is
the main source of modeling errors. The need for a highly accurate missile model stems from the requirement of
certain model-based control design methodologies like classical Backstepping [1] or Nonlinear Dynamic Inversion
(NDI) [2], which require a precise knowledge of the system. In order to reduce the cost and effort in the model
identification process, an adaptive autopilot is developed by taking only the predominant effects of the missile’s
dynamics into account.
Backstepping [1] is a recursive control design procedure for nonlinear systems similar to NDI or Feedback
Linearization [3]. A significant advantage is the derivation of the control law, which goes hand in hand with Lyapunov
stability proof for the closed-loop system. Due to the Lyapunov-driven control design, it is possible to leave parts of
the plant dynamics unchanged, which contributes in a positive way to the closed-loop stability. Missiles in general
can exhibit varying time-scale separation properties over the flight envelope. Since Backstepping approaches do not
require time-scale separation, they are more suitable compared to cascaded control concepts. In its classical form,
Backstepping requires analytical derivatives of the intermediate pseudo control laws [1], [4], [5], [6]. This fact makes
the Backstepping approach impractical for systems with complex dynamics [7], [8]. In recent years a command filtered
(CFB) version of the Backstepping approach has been developed [7], which provides a theoretical framework for
generating the pseudo control derivatives via filter algorithms. The paper [8] presents an adaptive version of the CFB
approach, which can compensate matched and unmatched uncertainties due to its recursive nature. Besides the remedy
of calculating the analytical derivative, CFB and CFAB approaches can be applied to the broader class of nonlinear
systems and are not limited - like classical Backstepping - to systems in strict-feedback form [7], [5].
Due to the non-strict-feedback form of the missile dynamics [5], the need for compensation of matched and
unmatched uncertainties [9] and the complex analytical expression of the pseudo control laws [4], CFAB is perfectly
suitable for missile autopilot design. Extensive investigations of missile autopilot design using Backstepping and
Adaptive Backstepping have been conducted in [10] and [4] respectively. Both approaches use the classical
Backstepping design without command filters leading to quite complex control laws. The approaches presented in
[11], [12] use a Dynamic Inversion-based baseline autopilot augmented by an adaptive layer compensating only
matched uncertainties. [13] follows a similar approach with a linear baseline controller.
Nevertheless, all Backstepping-based approaches have one drawback in common: the considered system has to
exhibit a minimum phase input/output characteristic [1], [7]. Considering the longitudinal and lateral accelerations as
outputs and the tail-fins as inputs, the tail-controlled missile exhibits (depending on the IMU position) a non-minimum
phase behavior. In order to render the missile minimum phase, the longitudinal and lateral acceleration measurements
are transformed to a virtual point [11].
Within this paper, two control approaches (CFAB and AWCFAB) are applied to a generic but realistic air-defense
missile model. In order to achieve the missile’s maximum performance and to exploit the entire missile’s flight
envelope, the preliminary control design goal is to fully exploit the missile’s actuator dynamics. Tracking highly agile
2
American Institute of Aeronautics and Astronautics
targets can lead to guidance commands that demand stressful maneuvering of the missile. Therefore, the autopilot
shall guarantee a safe continuation of the mission, even in cases where the actuators are saturated. Preserving closedloop stability in case of input saturations is achieved by incorporating an anti-windup strategy within the autopilot
design. Since the standard CFAB approach does not explicitly account for input saturation effects, a novel AWCFAB
approach is derived, which accounts for the actuator dynamics within the Lyapunov-based control design procedure
by maintaining the main advantages of CFAB.
The autopilot within this paper is designed for a high-fidelity, generic model of a surface-to-air missile [9], [12].
A nonlinear 6DOF simulation model including second order actuator dynamics with saturation effects, high-fidelity
aerodynamics in tabular form and a dynamic sensor model is used for evaluation purposes. To demonstrate the
robustness with respect to modeling uncertainties and actuator constraints, a realistic parameter and measurement
uncertainty spectrum is chosen.
This paper is organized as follows. Section II describes the 6DOF simulation model of the generic tail-controlled
missile. The control approach is presented in Section III. Simulation results for demanding STT-maneuvers exploiting
the full physical capabilities of the missile are presented in Section IV. Finally, Section V summarizes concluding
remarks.
II. Simulation Model of the Missile
A. Rigid body equations of motion
The generic tail-controlled air defense missile considered
within this paper is depicted in Figure 1. The following section
presents a short description of the missile dynamics. A more
detailed representation of the missile model can be found in [9].
For the purpose of evaluating the designed control algorithm,
the most challenging part of the missile flight phase is
considered: the terminal phase. Within this flight phase, high
demands in terms of agility, performance, and tracking accuracy
are posed on the missile system - especially the flight control
system (FCS). The booster is assumed to be burned out by the
beginning of this flight phase [14]. Therefore, no propulsion
force acts on the missile. Since an air defense missile covers only
a relatively short distance within the endgame, it is legitimate to Figure 1. Generic tail-controlled missile with
body fixed 𝐵-frame [9]
consider the earth as non-rotating and flat. With this assumption,
two main coordinate frames are sufficient to describe the
missile’s rigid body dynamics: a body-fixed frame (𝐵-frame) and an Earth Centered Earth Fixed frame (𝐸-frame).
The body-fixed frame is attached to the missile’s center of gravity (c.g.) with the x-axis pointing to the missile’s cone
(see Figure 1). The 𝐸-frame, which is used to describe the missile’s position dynamics, has its center located at sealevel with the 𝑥- and 𝑦-axis pointing in north- and east-directions, respectively, while the 𝑧-axis points downward,
perpendicular to the earth’s surface.
By using the feasible assumptions of a flat and non-rotating earth, the missile’s velocity (𝑽𝐺𝐾 )𝐸𝐵 =
𝐺
[𝑢𝐾 𝑣𝐾𝐺 𝑤𝐾𝐺 ]𝐸,𝑇
𝐵 results from the translational dynamics
𝐸𝐵
(𝑽̇𝐺𝐾 )𝐵 = 𝑴𝐵𝐸 ⋅ (𝒈𝐺 )𝐸 +
1
𝐺 𝐸
⋅ (𝑭𝐴𝐺 )𝐵 − (𝝎𝐸𝐵
𝐾 )𝐵 × (𝑽𝐾 )𝐵
𝑚
(1)
with (𝒈𝐺 )𝐸 and (𝑭𝐴𝐺 )𝐵 describing the gravitational acceleration and the aerodynamic forces, respectively. Besides the
velocity, another valid representation of the translational dynamics is obtained by the absolute velocity 𝑉𝐾𝐺 , the angle
of attack 𝛼𝐾 , and the angle of sideslip 𝛽𝐾 . Equation (2) depicts the analytical relationship between the velocities and
the alternative translational states.
3
American Institute of Aeronautics and Astronautics
𝑉𝐾𝐺 = √((𝑢𝐾𝐺 )𝐸𝐵 )2 + ((𝑣𝐾𝐺 )𝐸𝐵 )2 + ((𝑤𝐾𝐺 )𝐸𝐵 )2 ,
𝐸
𝛼𝐾 = arctan
𝛽𝐾 = arctan
𝐺
(𝑤𝐾
)
𝐵
𝐸
𝐺)
(𝑢𝐾
𝐵
𝐺
(𝑣𝐾
)
,
(2)
𝐸
𝐵
𝐸 2
𝐸 2
√((𝑢𝐺 ) ) +((𝑤𝐺 ) )
𝐾 𝐵
𝐾 𝐵
.
By integrating the velocity described in the 𝐸-frame
(𝒓̇ 𝐺 )𝐸𝐸 = 𝑴𝐸𝐵 ⋅ (𝑽𝐺𝐾 )𝐸𝐵 ,
(3)
the missile’s position (𝒓𝐺 )𝐸 = [𝑥 𝐺 𝑦 𝐺 𝑧 𝐺 ]𝑇𝐸 results.
Considering the aerodynamic moments (𝑴𝐴𝐺 )𝐵 and the missile’s moment of inertia (𝑰𝐺 )𝐵𝐵 , the dynamics of the
𝐸𝐵
𝑇
𝑞𝐸𝐵
𝑟𝐸𝐵
body-rates (𝝎𝐸𝐵
𝐾 ]𝐵 are given by
𝐾 )𝐵 = [𝑝𝐾
𝐾
𝐺
𝐵
𝐺 −1
𝐸𝐵
𝐺
𝐸𝐵
(𝝎̇𝐸𝐵
𝐾 )𝐵 = (𝑰 )𝐵𝐵 [(𝑴𝐴 )𝐵 − (𝝎𝐾 )𝐵 × (𝑰 )𝐵𝐵 (𝝎𝐾 )𝐵 ].
(4)
With the quaternion definition presented in [15], a singularity-free description of the missile’s attitude is obtained.
The quaternions (𝒒)𝐵 = [𝑞0 𝑞1 𝑞2 𝑞3 ]𝑇𝐵 result from integration of
1
(𝒒̇ )𝐵 = (𝒒)𝐵 ∘ (𝝎𝐸𝐵
𝐾 )𝐵 ,
(5)
2
where ∘ denotes the quaternion product [15].
The aerodynamic forces and moments used in the dynamical equations (1) and (4) are calculated based on the
angle of attack 𝛼𝐾 , angle of sideslip 𝛽𝐾 and Mach number 𝑀𝑎 by applying the following application rule for the force
(𝑭𝐴𝐺 )𝐵
𝐶𝑥0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝑥𝜉 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎, 𝜉)
= 𝑞̅ ⋅ 𝑆𝑟 ⋅ [𝐶𝑦0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝑦𝜁 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎, 𝜁)]
𝐶𝑧0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝑧𝜂 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎, 𝜂)
(6)
𝐵
and moment coefficients
𝐶𝐿
(𝑴𝐴𝐺 )𝐵 = 𝑞̅ ⋅ 𝑆𝑟 ⋅ 𝑙𝑟 ⋅ [𝐶𝑀 ]
𝐶𝑁 𝐵
𝑙𝑟𝑒𝑓
⋅ (𝑝𝐾𝐸𝐵 )𝐵 + 𝐶𝐿𝜉 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅ 𝜉
2𝑉𝐾𝐺
𝑙𝑟𝑒𝑓
= 𝑞̅ ⋅ 𝑆𝑟 ⋅ 𝑙𝑟 ⋅ 𝐶𝑀0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝑀𝑞 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅ 𝐺 ⋅ (𝑞𝐾𝐸𝐵 )𝐵 + 𝐶𝑀𝜂 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅ 𝜂 .
2𝑉𝐾
𝑙𝑟𝑒𝑓
𝐶𝑁0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝑁𝑟 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅ 𝐺 ⋅ (𝑟𝐾𝐸𝐵 )𝐵 + 𝐶𝑁𝜁 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅ 𝜁
2𝑉𝐾
[
]𝐵
𝐶𝐿0 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) + 𝐶𝐿𝑝 (𝛼𝐾 , 𝛽𝐾 , 𝑀𝑎) ⋅
(7)
B. Modeling of the actuator, sensor subsystem, and estimation unit
The air-defense missile considered within this paper is controlled via four aerodynamic fins attached to the
missile’s rear section (Figure 1). Each of the four fins denoted with 𝛿𝑖 , 𝑖 = 1, … ,4 can be deflected independently and
include a sensor system, which measures the angular deflection. The fins are modeled as a dynamical system of the
̇
̈
order two, including limits for deflection 𝛿𝑖,𝑙𝑖𝑚𝑖𝑡 , deflection rate 𝛿𝑖,𝑙𝑖𝑚𝑖𝑡
, and deflection acceleration 𝛿𝑖,𝑙𝑖𝑚𝑖𝑡
. Since the
4
American Institute of Aeronautics and Astronautics
aerodynamic data set (6), (7) is expressed in the aerodynamic equivalent controls 𝒖 = [𝜉 𝜂 𝜁]𝑇 , these entities serve
as inputs for control design purposes. The transformation between the aerodynamic equivalent, pseudo-controls 𝒖 and
the real physical control inputs 𝜹 = [𝛿1 𝛿2 𝛿3 𝛿4 ]𝑇 is obtained via the linear mapping rule in [15].
The measurement system includes an Inertial Measurement Unit (IMU), which is assumed to output the
aerodynamic acceleration
(𝒂𝐴𝐺 )𝐵 =
1
⋅ (𝑭𝐴𝐺 )𝐵 .
𝑚
(8)
at the c.g. and the body-rates (𝝎𝐸𝐵
𝐾 )𝐵 .
Besides the outputs measured via the IMU, additional states are necessary for the purpose of control design. Since
these states are non-measureable in missile application (e.g. angle of attack 𝛼𝐾 , dynamic pressure 𝑞̅), they have to be
estimated online. Due to the estimation process, which takes measurements and model information into account, the
estimated states are subject to uncertainties. These uncertainties are modeled as constant biases within a certain range.
In addition to the estimation uncertainties, the plant parameters – especially the aerodynamic data set – is subject to
modeling errors. The range of the estimation and modeling errors are summarized in Table 1.
Table 1. Overview of the considered uncertainties (from [1]).
Variable
𝐼𝑥𝑥 , 𝐼𝑦𝑦 , 𝐼𝑧𝑧
𝑚
𝑥𝑐𝑔
(𝐶𝑥0 )𝐵 , (𝐶𝑦0 ) ,
𝐵
(𝐶𝑧0 )𝐵
(𝐶𝐿0 )𝐵 , (𝐶𝑀0 )𝐵 ,
(𝐶𝑁0 )𝐵
(𝐶𝑥𝜉 ) , (𝐶𝑦𝜁 ) ,
Uncertainty Range
±5%
±1%
±50
Unit
[−]
[−]
[𝑚𝑚]
±10%
[−]
±20%
[−]
(𝐶𝑧𝜂 ) , (𝐶𝐿𝜉 ) ,
±20%
[−]
±20%
[−]
±2.5
±5%
±10%
[𝑑𝑒𝑔]
[−]
[−]
𝐵
𝐵
𝐵
𝐵
(𝐶𝑀𝜂 ) , (𝐶𝑁𝜁 )
𝐵
𝐵
(𝐶𝐿𝑝 )𝐵 , (𝐶𝑀𝑞 )𝐵 ,
(𝐶𝑁𝑟 )𝐵
𝛼𝐾 , 𝛽𝐾
𝑞̅
𝑀𝑎
III. Command Filtered Adaptive Backstepping Design
In this section, the CFAB control law is derived. First, the control task, the control variables and the inputs are
defined. The physically motivated transformation of the accelerations to a virtual point, in order to render the missile
system minimum phase is briefly explained. Based on this redefined output, the model used for control design purposes
is presented. A nonlinear reference model described in [12] transforms the commanded signal 𝒚𝑐𝑚𝑑 to a feasible
physical reference input 𝒚𝑟𝑒𝑓 and its corresponding time derivative 𝒚̇ 𝑟𝑒𝑓 .
Within this paper, a STT-autopilot for the tail controlled missile described in Section II is developed. For the
purpose of controlling the three body-axes of the missile, the roll angle around the velocity vector
𝑡
𝜙𝐾 = ∫𝑡 [cos 𝛼𝐾 ∙ cos 𝛽𝐾 ∙ (𝑝𝐾𝐸𝐵 )𝐵 + sin 𝛽𝐾 ∙ (𝑞𝐾𝐸𝐵 )𝐵 + sin 𝛼𝐾 ∙ cos 𝛽𝐾 ∙ (𝑟𝐾𝐸𝐵 )𝐵 ] 𝑑𝑡,
0
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American Institute of Aeronautics and Astronautics
(9)
𝐼𝑀𝑈
𝐼𝑀𝑈
the longitudinal acceleration (𝑎𝐴,𝑧
)𝐵 and lateral acceleration (𝑎𝐴,𝑦
) measured at the IMU specify the control
𝐵
variables. The vector
𝒚=
𝜙𝑣
𝐼𝑀𝑈
[(𝑎𝐴,𝑧 )𝐵 ]
(10)
𝐼𝑀𝑈
(𝑎𝐴,𝑦
)
𝐵
summarizes these three control variables. Due to the input-affine structure of the aerodynamic data set, the equivalent
controls 𝒖𝑇 = [𝜉 𝜂 𝜁] can be used as pseudo controls for control design purposes. The output of the FCS is the
transformation of 𝒖 to the respective fin deflections 𝜹𝑇 = [𝛿1 𝛿2 𝛿3 𝛿4 ] via the above mentioned mapping rule
[15]. Those commanded fin deflections describe the output of the FCS.
Figure 2 depicts the structure of the overall CFAB autopilot. The different subsystems are described in detail in the
text that follows.
Figure 2. Structure of the AWCFAB/CFAB autopilot.
A. Rendering the missile system minimum phase
𝐼𝑀𝑈
𝐼𝑀𝑈
The longitudinal (𝑎𝐴,𝑧
)𝐵 and lateral accelerations (𝑎𝐴,𝑦
) measured at the IMU position, which are controlled
𝐵
via the axis-wise, equivalent controls 𝜂 and 𝜁, render the missile input/output characteristics non-minimum phase.
Since Backstepping approaches require the plant to exhibit a minimum phase input/output characteristic (see [1], [4]),
a physically motivated transformation of the acceleration is applied according to [12]. A transformation is proposed,
which shifts the IMU acceleration to a virtual point 𝑃 ahead of the missile’s center of percussion. The distance between
𝐼𝑀𝑈
𝐼𝑀𝑈
the IMU and the virtual point 𝑃 is labeled with 𝑥𝐼𝑀𝑈,𝑃 . The longitudinal (𝑎𝐴,𝑧
)𝐵 and lateral accelerations (𝑎𝐴,𝑦
) at
𝐵
𝐼𝑀𝑈
𝑃 result through the addition of acceleration increments (Δ𝑎𝐴,𝑧 )𝐵 and (Δ𝑎𝐴,𝑦 )𝐵 to the measured accelerations (𝑎𝐴,𝑧
)𝐵
𝐼𝑀𝑈
and (𝑎𝐴,𝑦
) :
𝐵
[
𝐼𝑀𝑈
(𝑎𝐴,𝑧
)
(𝑝𝐾𝐸𝐵 )𝐵 ∙ (𝑟𝐾𝐸𝐵 )𝐵 ⋅ 𝑥𝐼𝑀𝑈,𝑃
−(𝑞̇ 𝐾𝐸𝐵 )𝐵𝐵 ⋅ 𝑥𝐼𝑀𝑈,𝑃
]
=
[
]
+
[
]
+
[
]
𝐸𝐵
𝐵
𝐼𝑀𝑈
𝑃
(𝑝𝐾𝐸𝐵 )𝐵 ∙ (𝑞𝐾𝐸𝐵 )𝐵 ⋅ 𝑥𝐼𝑀𝑈,𝑃
(𝑎𝐴,𝑦
)
(𝑎𝐴,𝑦
)
⏟(𝑟̇𝐾 )𝐵 ⋅ 𝑥𝐼𝑀𝑈,𝑃
𝐵
𝐵
𝑃
(𝑎𝐴,𝑧
)
(11)
(Δ𝑎𝐴,𝑧 )
=[
]
(Δ𝑎𝐴,𝑦 )
𝐵
The acceleration increments describe the relative accelerations of the virtual point 𝑃 with respect to the IMU location
𝐵
due to the rotation of the missile body. Due to the fact that the angular accelerations (𝝎̇𝐸𝐵
𝐾 )𝐵 vanish in the case of
trimmed flight condition, the acceleration increments are zero. Therefore, the measured and virtual accelerations are
equal. Besides this compliance in steady-state flight, the difference in transient behavior between both signals remains
within certain bounds for an appropriate choice of 𝑥𝐼𝑀𝑈,𝑃 . Since the acceleration increments depend on the model
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information and measurements, the capability of approximating the measured acceleration by (11) decreases with
increased deviation from the assumed, nominal plant. Therefore, tracking the virtual acceleration (11) in cases of plant
𝐼𝑀𝑈
𝐼𝑀𝑈
disturbances would lead to an unsatisfactory tracking behavior of the actual control variables (𝑎𝐴,𝑧
)𝐵 and (𝑎𝐴,𝑦
) .
𝐵
To overcome this drawback without violating the gained minimum phase characteristic of (11), the low-pass filtered
versions of the increments are subtracted from (11):
[
𝑃
(𝑎𝐴,𝑓𝑖𝑙,𝑧
)
𝐼𝑀𝑈
(𝑎𝐴,𝑧
)
(Δ𝑎𝐴,𝑧 )
(Δ𝑎𝐴,𝑧 )
]
=
[
] +[
] − 𝑮𝐿𝑃 (𝑠) [
]
𝐼𝑀𝑈
𝑃
(𝑎𝐴,𝑦 )
(Δ𝑎𝐴,𝑦 ) 𝐵
(Δ𝑎𝐴,𝑦 ) 𝐵
(𝑎𝐴,𝑓𝑖𝑙,𝑦 )
𝐵
(12)
𝐵
By an appropriate choice of the low-pass filter 𝑮𝐿𝑃 (𝑠) and the distance 𝑥𝐼𝑀𝑈,𝑃 , a minimum phase output is generated,
which approximates the IMU acceleration even in cases of deviations from the nominal model.
For the sake of a compact notation, the acceleration definitions (11) and (12) are merged in the vectors
𝜙𝑣
𝑃
𝒚𝑃 = [ (𝑎𝐴,𝑧 )𝐵 ]
𝑃
(𝑎𝐴,𝑦
)
(13)
𝐵
and
𝒚𝑃𝑓𝑖𝑙
=
𝜙𝑣
𝑃
[ (𝑎𝐴,𝑓𝑖𝑙,𝑧 ) ],
𝑃
(𝑎𝐴,𝑓𝑖𝑙,𝑦
)
(14)
respectively. In summary, the output (13), which contains the unfiltered acceleration, constitutes the output of the
model used for control design. The filtered version (14) states the redefined, minimum-phase control variable. By a
feasible choice of the low-pass filters 𝑮𝐿𝑃 (𝑠) and the distance 𝑥𝐼𝑀𝑈,𝑃 [12], 𝒚𝑃𝑓𝑖𝑙 approximates 𝒚 sufficiently precisely.
Therefore, if the chosen minimum-phase output 𝒚𝑃𝑓𝑖𝑙 tracks the desired command 𝒚𝑐𝑚𝑑 (subordinate control goal), the
superior control goal 𝒚 = 𝒚𝑟𝑒𝑓 is fulfilled.
B. Model used for control design purposes
Before starting with the derivation of the control law, an appropriate model of the missile dynamics has to be
defined. For the purpose of control design, only the predominant dynamical effects missile dynamics (Section II.A)
are taken into account. It is feasible to consider only the minimum phase output
𝜙̇𝑣
𝐸𝐵
𝑃
𝒚̇ 𝑃 = [ (𝑎̇ 𝐴,𝑧 )𝐵 ] = 𝒇𝒚 + 𝑮𝒚 ⋅ (𝝎 𝐾 ) + 𝜣𝒚 ⋅ 𝝋𝒚
𝐵
𝑃
(𝑎̇ 𝐴,𝑦
)
(15)
𝐵
(𝝎̇𝐸𝐵
𝐾 )𝐵 = 𝒇𝝎 + 𝑮𝝎 ⋅ 𝒖 + 𝜣𝝎 ⋅ 𝝋𝝎 .
(16)
𝐵
and the rotational dynamics
Due to non-measureable states, modeling errors and negligible dynamical effects, a nominal model with feasible
complexity is used for control design purposes.
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sin(𝛽𝐾 ) (𝑞𝐾𝐸𝐵 )𝐵 + sin(𝛼𝐾 ) cos(𝛽𝐾 )(𝑟𝐾𝐸𝐵 )𝐵
𝑆𝑟 𝑞̅
𝑆𝑟 𝑙𝑟 𝑞̅
⋅ [𝐶𝑧,𝛼 𝑍𝛼 + 𝐶𝑧,𝛽 𝛽̇ ] − 𝑥𝐼𝑀𝑈,𝑃
[(𝐶𝑀0,𝛼 + 𝐶𝑀𝜂,𝛼 ) ⋅ 𝑍𝛼 + (𝐶𝑀0,𝛽 + 𝐶𝑀𝜂,𝛽 ) ⋅ 𝛽̇ ]
𝐼𝑦𝑦
𝒇𝒚 = 𝑚
,
𝑆𝑟 𝑞̅
𝑆𝑟 𝑙𝑟 𝑞̅
⋅ [𝐶𝑦,𝛼 𝛼̇ + 𝐶𝑦,𝛽 𝑌𝛽 ] + 𝑥𝐼𝑀𝑈,𝑃
[(𝐶𝑁0,𝛼 + 𝐶𝑁𝜁,𝛼 ) ⋅ 𝛼̇ + (𝐶𝑁0,𝛽 + 𝐶𝑁𝜁,𝛽 ) ⋅ 𝑌𝛽 ]
[ 𝑚
]
𝐼𝑧𝑧
(17)
𝑙𝑟𝑒𝑓
⋅ (𝑝𝐾𝐸𝐵 )𝐵
2𝑉𝐾𝐺
𝑙𝑟𝑒𝑓
𝐸𝐵
𝐺
𝐸𝐵
⋅ (𝑞𝐾𝐸𝐵 )𝐵 + (𝑰𝐺 )−1
𝒇𝝎 = 𝑞̅ 𝑆𝑟 𝑙𝑟 (𝑰𝐺 )−1
𝐵𝐵 ⋅ 𝐶𝑀0 + 𝐶𝑀𝑞 ⋅
𝐵𝐵 ⋅ [−(𝝎𝐾 )𝐵 × (𝑰 )𝐵𝐵 (𝝎𝐾 )𝐵 ]
2𝑉𝐾𝐺
𝑙𝑟𝑒𝑓
𝐶𝑁0 + 𝐶𝑁𝑟 ⋅ 𝐺 ⋅ (𝑟𝐾𝐸𝐵 )𝐵
2𝑉𝐾
[
]
(18)
and
𝐶𝐿0 + 𝐶𝐿𝑝 ⋅
with the abbreviations
𝛼̇ 𝐾 = 𝑍𝛼 + (𝑞𝐾𝐸𝐵 )𝐵
𝛽𝐾̇ = 𝑌𝛽 − cos(𝛼𝐾 ) (𝑟𝐾𝐸𝐵 )𝐵
(19)
describe the nominal, nonlinear functions of the respective dynamics.
The nominal input effectiveness of (15) and (16) is depicted by the diagonal matrices,
cos 𝛼𝐾 ∙ cos 𝛽𝐾
0
𝑮𝒚 = 𝑞̅𝑆𝑟 𝑙𝑟 (𝑰𝐺 )−1
𝐵𝐵 ⋅
[
0
𝑆𝑟 𝑞̅
𝑆𝑟 𝑙𝑟 𝑞̅𝑥𝐼𝑀𝑈,𝑃
𝐶 −
[𝐶𝑀0,𝛼 + 𝐶𝑀𝜂,𝛼 ]
𝑚 𝑧,𝛼
𝐼𝑦𝑦
0
0
0
(20)
𝑆𝑟 𝑞̅
𝑆𝑟 𝑙𝑟 𝑞̅𝑥𝐼𝑀𝑈,𝑃
𝐶 −
[𝐶𝑁0,𝛽 + 𝐶𝑁𝜁,𝛽 ]
𝑚 𝑦,𝛽
𝐼𝑦𝑦
]
0
and
𝑮𝝎 =
𝑞̅ 𝑆𝑟 𝑙𝑟 (𝑰𝐺 )−1
𝐵𝐵
𝐶𝐿𝜉
⋅[ 0
0
0
𝐶𝑀𝜂
0
0
0 ],
𝐶𝑁𝜁
(21)
respectively. To account for the structure of CFAB [8], deviations from the assumed, nominal dynamics are modeled
by the product of ideal constant parameter matrices 𝜣𝒚 ∈ ℝ3×𝑔1 , 𝜣𝝎 ∈ ℝ3×𝑔2 and nonlinear regressor vectors 𝝋𝒚 ∈
ℝ𝑔1 ×1 , 𝝋𝝎 ∈ ℝ 𝑔2×1 .
C. Command Filtered Backstepping Autopilot
Backstepping approaches are recursive procedures, which start at the output dynamics and derive subsystem-wise
intermediate control laws. Those pseudo controls 𝑥𝑖+1,𝑑 = 𝛼𝑖 , 𝑖 = 1, … , 𝑛 − 1 are derived based on Lyapunov’s
second method of stability by considering the state 𝑥𝑖+1 of the proximate ODE as control input to the considered
subsystem [1]. In the next step the considered subsystem is extended by the proximate ODE. The pseudo control
𝑥𝑖+1,𝑑 = 𝛼𝑖 serves as a reference input to the extended subsystem with the control variable 𝑥𝑖+1 . Again, the proximate
state 𝑥𝑖+2 is the pseudo control for the considered, extended subsystem. Based on a control Lyapunov function, the
pseudo control 𝑥𝑖+2,𝑑 = 𝛼𝑖+1 is designed to asymptotically stabilize the tracking error 𝑒𝑖+1 = 𝑥𝑖+1 − 𝛼𝑖 . This
procedure is repeated until the innermost subsystem is reached and the final control law 𝑢 = 𝛼𝑛 is derived. Each
pseudo control law 𝛼𝑖 requires the time derivative 𝛼̇ 𝑖−1 of the previous one. Since the pseudo control laws contain
model information, Backstepping is prohibitive for complex dynamics (e.g. aerodynamics) and systems possessing an
order larger than two [7]. In contrast to classical Backstepping concepts, CFAB and CFB design procedures use
(command) filters to generate the necessary time derivative 𝛼̇ 𝑖−1 of the intermediate control laws [5], [7], [8] for the
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proximate subsystem. This control technique accounts for the reaction deficit caused by the command filters.
Additionally, the actuator dynamics were incorporated in the layout by treating it as a command filter.
First, the standard CFAB autopilot design is presented within this section. The second subsection includes the
novel AWCFAB control design, which extends the CFAB theory to systems with input saturations.
1. Standard CFAB
𝐸𝐵
In the first step of deriving the CFAB autopilot law, the desired pseudo control (𝝎 𝐾 )
𝐵,𝑑𝑒𝑠
= 𝜶𝝎 for subsystem
(15) is calculated. Before starting with the Lyapunov-based design, it is necessary to introduce the tracking errors.
The tracking error 𝒆𝒚 and its integral portion 𝒆𝒚,𝐼 of the control variable 𝒚𝑃𝑓𝑖𝑙 are summarized in the vector
𝒚𝑃𝑓𝑖𝑙 − 𝒚𝑟𝑒𝑓
𝒆𝒚
𝑬𝒚 = [𝒆 ] = [
].
𝒚,𝐼
∫(𝒚𝑃 − 𝒚 )𝑑𝑡
𝑓𝑖𝑙
(22)
𝑟𝑒𝑓
For filtering the intermediate control 𝜶𝝎 a linear second order filter (23), defined by frequency 𝝎𝑓𝑖𝑙 and damping
𝜻𝑓𝑖𝑙 , is used.
[
𝜶̇ 𝝎,𝑓𝑖𝑙
𝟎
] = [−𝝎2
𝜶̈ 𝝎,𝑓𝑖𝑙
𝑓𝑖𝑙
𝜶𝝎,𝑓𝑖𝑙
𝑰
𝟎
2 ] 𝜶𝝎
−2𝜻𝑓𝑖𝑙 𝝎𝑓𝑖𝑙 ] [𝜶̇ 𝝎,𝑓𝑖𝑙 ] + [𝝎𝑓𝑖𝑙
(23)
The filtered pseudo control 𝜶𝝎,𝑓𝑖𝑙 serves as a reference input for the 𝝎-subsystem (16). Therefore, the deviation
𝐸𝐵
𝒆𝝎 = (𝝎 𝐾 ) − 𝜶𝝎,𝑓𝑖𝑙
(24)
𝐵
𝐸𝐵
between the filtered intermediate control 𝜶𝝎,𝑓𝑖𝑙 and the body-rates (𝝎 𝐾 ) completes the tracking error definition
𝐵
for the simplified missile model (15) and (16). By filtering the desired pseudo control via the command filter, the
filtered signal 𝜶𝑖,𝑓𝑖𝑙 is delayed compared to the desired signal 𝜶𝑖 . This reaction deficit caused by the filter leads to a
deficit in the transient of the respective controlled variable. The transient error
𝒛𝑖 = 𝒙𝑖 (𝜶𝑖,𝑓𝑖𝑙 ) − 𝒙𝑖 (𝜶𝑖 )
(25)
due to the reaction deficit of the filter is called the unachieved portion and is labeled with 𝒛𝑖 . Since the actuator
subsystem (Section II.) delays the final control law 𝒖𝑐𝑚𝑑 analogous to the command filters, this unachieved portion
caused by the actuator dynamics is incorporated in the CFAB design and constitutes a deviation from the standard
procedure presented in [8]. Removing this unachieved portion from the respective tracking errors (22) and (23), leads
to the definition of the compensated tracking errors:
𝒆 𝒚 − 𝒛𝒚
𝒚𝑃𝑓𝑖𝑙 − 𝒚𝑟𝑒𝑓 − 𝒛𝒚
𝒆̃𝒚
̃
𝑬𝒚 = [ ] = [
]=[
],
𝒆̃𝒚,𝐼
∫(𝒆𝒚 − 𝒛𝒚 )𝑑𝑡
∫(𝒚𝑃𝑓𝑖𝑙 − 𝒚𝑟𝑒𝑓 − 𝒛𝒚 )𝑑𝑡
𝐸𝐵
𝒆̃𝝎 = 𝒆𝝎 − 𝒛𝝎 = (𝝎 𝐾 ) − 𝜶𝝎,𝑓𝑖𝑙 − 𝒛𝝎 .
𝐵
(26)
(27)
̂ 𝑖 and the nominal parameter 𝜣𝑖 , 𝑖 = 𝒚, 𝝎 is denoted with
The difference between the estimated parameter 𝜣
̃ 𝑖 = 𝜣𝑖 − 𝜣
̂ 𝑖.
𝜣
(28)
̂ 𝒚 for the 𝒚-dynamics (15) is derived based
In the first step, the stabilizing function 𝜶𝝎 and the parameter update law 𝜣
on the Lyapunov candidate
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American Institute of Aeronautics and Astronautics
1 𝑇
1
̃ 𝒚 𝜞−𝟏
̃𝑇
̃ 𝑷 𝑬
̃ + 𝑇𝑟(𝜣
𝑉1 = 𝑬
𝒚 𝜣𝒚 ),
2 𝒚 𝒚 𝒚 2
(29)
which weights the compensated tracking error by 𝑷𝒚 ∈ ℝ6×6 and the parameter error by the adaptive learning rate 𝜞𝑖 ,
𝑖 = 𝒚, 𝝎. 𝑷𝒚 depicts the weighting matrix of the compensated tracking error 𝒆̃𝒚 and its integral 𝒆̃𝒚,𝐼 (22), respectively.
The intermediate control law 𝜶𝝎 is designed to render the time derivative of (29)
1 ̇𝑇
1 𝑇
̃ 𝒚 𝜞−𝟏
̃̇ 𝑇
̃ 𝒚 𝑷𝒚 𝑬
̃𝒚 + 𝑬
̃ 𝑷 𝑬
̃̇ + 𝑇𝑟 (𝜣
𝑉1̇ = 𝑬
𝒚 𝜣𝒚 )
2
2 𝒚 𝒚 𝒚
(30)
𝐸𝐵
negative semi-definite. Substituting the body-rates from equation (27) with (𝝎 𝐾 ) = 𝒆̃𝝎 + 𝜶𝝎,𝑓𝑖𝑙 + 𝒛𝝎 , the error
𝐵
dynamics of the compensated tracking error results in
̂
̃
̃̇ 𝒚 = [𝒇𝒚 + 𝑮𝒚 ⋅ (𝒆̃𝝎 + 𝜶𝝎,𝑓𝑖𝑙 + 𝒛𝝎 + 𝜶𝝎 − 𝜶𝝎 ) + 𝜣𝒚 ⋅ 𝝋𝒚 + 𝜣𝒚 ⋅ 𝝋𝒚 − 𝒚̇ 𝑟𝑒𝑓 ] − 𝒁̇𝒚 .
𝑬
𝒆𝒚
(31)
Since the model parameters 𝜣𝑖 are assumed to be constant, the parameter error dynamics is equal to the dynamics of
̂ 𝑖:
the estimated parameter 𝜣
̃̇ 𝑖 = −𝜣
̂̇ 𝑖
𝜣
(32)
𝐸𝐵
The desired value for (𝝎 𝐾 ) is chosen as
𝐵
𝜶𝝎 = 𝑮−1
𝒚 ⋅ (−[𝑲𝒚
̂ 𝒚 ⋅ 𝝋𝒚 − 𝒇𝒚 + 𝒚̇ 𝑟𝑒𝑓 ).
𝑲𝒚,𝐼 ] ⋅ 𝑬𝒚 − 𝜣
(33)
Based on the choice of the Lyapunov function (29) and the intermediate control 𝜶𝝎 , the dynamics of the unachieved
portion 𝒁𝒚 results in
−[
𝒁̇𝒚 = [
𝑲𝒚
𝑲𝒚,𝐼 ] ⋅ 𝒁𝒚 + 𝑮𝒚 (𝜶𝝎,𝑓𝑖𝑙 − 𝜶𝝎 ) + 𝑮𝒚 𝒛𝝎
]
𝒛𝒚
(34)
Substituting (33) and (34) in (31) leads to the error dynamics
̃̇ 𝒚 = [−𝑲𝒚
𝑬
⏟𝑰
̃ ⋅ 𝝋𝒚
−𝑲𝒚,𝐼 ̃
𝑮
𝜣
] ⋅ 𝑬𝒚 + [ 𝒚 ] ⋅ 𝒆̃𝝎 + [ 𝒚
],
𝟎
𝟎
𝟎
(35)
̅𝒚
𝑲
̅ 𝒚 . Due to the missing time-scale separation, the cross-coupling term
which exhibits an eigen dynamics defined by 𝑲
𝑮𝒚 ⋅ 𝒆̃𝝎 , which acts as input to (35) cannot be neglected like in NDI control design schemes [16], [17], [18].
Substituting (35) in (30), applying the trace identity
𝒂𝑇 ⋅ 𝒃 = 𝑇𝑟(𝒃 ⋅ 𝒂𝑇 )
(36)
and equation (32) results in the Lyapunov time derivative
1 𝑇 𝑇
̃ 𝒚 (𝝋𝒚 𝑬
̂̇ 𝑇
̃ (𝑲
̅ 𝑷 + 𝑷𝒚 𝑲
̅ 𝒚) 𝑬
̃ +𝑬
̃ 𝑇𝒚 𝑷𝒚 [𝑮𝒚 ] 𝒆̃𝝎 + 𝑇𝑟 (𝜣
̃ 𝑇𝒚 𝑷𝒚 [ 𝑰 ] − 𝜞−𝟏
𝑉1̇ = 𝑬
𝒚 𝜣𝒚 )).
𝒚 𝒚
𝟎
2 𝒚 𝒚 𝒚
𝟎
Applying the projection-based ([19], [20]) adaptive update law
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(37)
̂̇ 𝑇𝒚 = Proj (𝜣
̂ 𝑇𝒚 , 𝜞𝒚 𝝋𝒚 𝑬
̃ 𝑇𝒚 𝑷𝒚 [ 𝑰 ])
𝜣
𝟎
(38)
in combination with the solution 𝑸𝒚 ∈ ℝ6×6 of the matrix Lyapunov equation
̅ 𝑇𝒚 𝑷𝒚 + 𝑷𝒚 𝑲
̅ 𝒚,
−𝑸𝒚 = 𝑲
(39)
leads to the indefinite Lyapunov time derivative
1 𝑇
̃ 𝒚 𝑸𝒚 𝑬
̃𝒚 + 𝑬
̃ 𝑇𝒚 𝑷𝒚 [𝑮𝒚 ] 𝒆̃𝝎 .
𝑉1̇ = − 𝑬
2
𝟎
(40)
In the second design step, the control layout is concluded by calculating the final control law 𝒖𝑐𝑚𝑑 . Within this step
the reaction deficit caused by the actuator dynamics is compensated analogous to the deficit caused by the command
̃𝝎
filter (23). The Lyapunov function (29) is extended by the compensated tracking error 𝒆̃𝝎 and the parameter error 𝜣
of the dynamics (16):
1
1
̃ 𝝎 𝜞−𝟏
̃𝑇
𝑉2 = 𝑉1 + 𝒆̃𝑇𝝎 𝑷𝝎 𝒆̃𝝎 + 𝑇𝑟(𝜣
𝝎 𝜣𝝎 ).
2
2
(41)
𝒖𝑐𝑚𝑑 is selected to render the time derivative of 𝑉2
1 𝑇
1
1
̃ 𝝎 𝜞−𝟏
̃̇ 𝑇
̃ 𝑸 𝑬
̃ +𝑬
̃ 𝑇𝒚 𝑷𝒚 [𝑮𝒚 ] 𝒆̃𝝎 + 𝒆̃̇𝑇𝝎 𝑷𝝎 𝒆̃𝝎 + 𝒆̃𝑇𝝎 𝑷𝝎 𝒆̃̇𝝎 + 𝑇𝑟 (𝜣
𝑉2̇ = − 𝑬
𝝎 𝜣𝝎 )
2 𝒚 𝒚 𝒚
2
2
𝟎
(42)
negative semi-definite. Including the input 𝒖𝑐𝑚𝑑 of the actuator subsystem (Section II.), the compensated tracking of
the body-rate dynamics is given by
̂ 𝝎 ⋅ 𝝋𝝎 + 𝜣
̃ 𝝎 ⋅ 𝝋𝝎 − 𝜶̇ 𝝎,𝑓𝑖𝑙 − 𝒛̇ 𝝎 .
𝒆̃̇𝝎 = 𝒇𝝎 + 𝑮𝝎 ⋅ (𝒖 − 𝒖𝑐𝑚𝑑 + 𝒖𝑐𝑚𝑑 ) + 𝜣
(43)
Using proportional feedback to stabilize the body-rate subsystem, the final control law and the dynamics of the
unachieved portion result as
−1
𝒖𝑐𝑚𝑑 = 𝑮−1
𝝎 ⋅ (−𝑲𝝎 𝒆𝝎 − 𝑷𝝎 [𝑮𝒚
̂ 𝝎 ⋅ 𝝋𝝎 + 𝜶̇ 𝝎,𝑓𝑖𝑙 )
̃ 𝒚 − 𝒇𝝎 − 𝜣
𝟎]𝑷𝒚 𝑬
(44)
and
𝒛̇ 𝝎 = −𝑲𝝎 ⋅ 𝒛𝝎 + 𝑮𝝎 ⋅ (𝒖 − 𝒖𝑐𝑚𝑑 ),
(45)
respectively. Substituting (44) and (45) in (43) results in the closed-loop error dynamics of the compensated tracking
error 𝒆̃𝝎 :
−1 [𝑮
𝒆̃̇𝝎 = −𝑲𝝎 𝒆̃𝝎 − 𝑷𝝎
𝒚
̃ 𝝎 ⋅ 𝝋𝝎 ,
̃𝒚 + 𝜣
𝟎]𝑷𝒚 𝑬
(46)
Substituting the error dynamics (46) in (42) and applying the trace identity (36) results in the time derivative of the
Lyapunov function
1 𝑇
1
̃ 𝝎 (𝝋𝝎 𝒆̃𝑇𝝎 𝑷𝝎 − 𝜞−𝟏
̂̇ 𝑇
̃ 𝑸 𝑬
̃ + 𝒆̃𝑇 ((−𝑲𝝎 )𝑇 𝑷𝝎 + 𝑷𝝎 (−𝑲𝝎 ))𝒆̃𝝎 + 𝑇𝑟 (𝜣
𝑉2̇ = − 𝑬
𝝎 𝜣𝝎 )).
2 𝒚 𝒚 𝒚 2 𝝎
̂ 𝝎 is updated via the projection-based update law
The parameter estimate 𝜣
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(47)
̂̇ 𝑇𝝎 = Proj(𝜣
̂ 𝑇𝝎 , 𝜞𝝎 𝝋𝝎 𝒆̃𝑇𝝎 𝑷𝝎 ).
𝜣
(48)
Therefore, applying (48) and the matrix Lyapunov equation:
−𝑸𝝎 = (−𝑲𝝎 )𝑇 𝑷𝝎 + 𝑷𝝎 (−𝑲𝝎 )
(49)
with the solution 𝑸𝝎 ∈ ℝ3×3 , the time derivative (47) is rendered negative semi-definite:
1 𝑇
1
̃ 𝑸 𝑬
̃ − 𝒆̃𝑇 𝑸 𝒆̃
𝑉2̇ = − 𝑬
2 𝒚 𝒚 𝒚 2 𝝎 𝝎 𝝎
(50)
Incorporating the unachieved portion of the actuator dynamics is in accordance with the theoretical framework of
CFAB. The proofs can be found in [7] and [8].
̃ 𝑖 = 𝟎),
Remark: Setting the learning rates 𝜞𝑖 , 𝑖 = 𝒚, 𝝎 to zero and considering the nominal, undisturbed model (𝜣
the standard CFB control law is obtained. Considering (35) and (43), the overall nominal error dynamics is given by
𝒆̃̇𝒚
[𝒆̃̇𝒚,𝐼 ] = [
𝒆̃̇𝝎
−𝑲𝒚
−𝑲𝒚,𝐼
𝑰
𝟎
𝑰
𝟎
−1 [𝑮
−1
−𝑷𝝎 𝒚 𝟎]𝑷𝒚 [ ] −𝑷𝝎 [𝑮𝒚 𝟎]𝑷𝒚 [ ]
𝟎
𝑰
𝑮𝒚
𝒆̃𝒚
𝟎 ] ⋅ [𝒆̃ ].
𝒚,𝐼
−𝑲𝝎
𝒆̃𝝎
(51)
The desired closed loop performance of the non-adaptive CFB controller is optimized in accordance to the missile’s
physical capabilities.
2. Anti-Windup CFAB (AWCFAB)
The idea of AWCFAB is to remove not only the reaction deficit caused by the command filters from the tracking
error 𝒆𝑖 , but also the reaction deficit between the actual state 𝒙𝑖+1 and its desired pseudo control 𝒙𝑖+1,𝑑 = 𝜶𝑖 . This
reaction deficit of the 𝑖-th step is denoted by 𝒘𝑖 and subtracted from the tracking error 𝒆𝑖 in an analogous way as
compensating for the unachieved portion (see (22) and (24)). Therefore, the new definitions of the compensated
tracking errors 𝒆̂𝑖 are given by
𝒆𝒚 − 𝒘𝒚
𝒚𝑃𝑓𝑖𝑙 − 𝒚𝑟𝑒𝑓 − 𝒘𝒚
𝒆̂𝒚
̂𝒚 = [ ] = [
𝑬
]=[
],
𝒆̂𝒚,𝐼
∫(𝒆𝒚 − 𝒘𝒚 )𝑑𝑡
∫(𝒚𝑃𝑓𝑖𝑙 − 𝒚𝑟𝑒𝑓 − 𝒘𝒚 )𝑑𝑡
𝐸𝐵
𝒆̂𝝎 = 𝒆𝝎 − 𝒘𝝎 = (𝝎 𝐾 ) − 𝜶𝝎,𝑓𝑖𝑙 − 𝒘𝝎 .
𝐵
(52)
(53)
Considering a quadratic Lyapunov function (29) as in C.1
1 𝑇
1
̃ 𝒚 𝜞−𝟏
̃𝑇
̂ 𝑷 𝑬
̂ + 𝑇𝑟(𝜣
𝑉1 = 𝑬
𝒚 𝜣𝒚 ),
2 𝒚 𝒚 𝒚 2
(54)
̂ 𝑖 are selected in order to render
(with 𝑷𝒚 = 𝑰) the pseudo control law 𝜶𝝎 , the reaction deficit 𝒘𝒚 , and the update law 𝜣
the time derivative
1 ̇𝑇
1 𝑇 ̇
̃ 𝒚 𝜞−𝟏
̃̇ 𝑇
̂𝒚𝑬
̂𝒚 + 𝑬
̂ 𝑬
̂ + 𝑇𝑟 (𝜣
𝑉1̇ = 𝑬
𝒚 𝜣𝒚 )
2
2 𝒚 𝒚
𝐸𝐵
(55)
of (54) negative semi-definite. Using the substitution (𝝎 𝐾 ) = 𝒆̂𝝎 + 𝜶𝝎,𝑓𝑖𝑙 + 𝒘𝝎 , which is based on (53), the error
𝐵
dynamics of the compensated outer layer variables:
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̂ 𝒚 ⋅ 𝝋𝒚 + 𝜣
̃ 𝒚 ⋅ 𝝋𝒚
𝒇 + 𝑮𝒚 ⋅ (𝒆̂𝝎 + 𝜶𝝎,𝑓𝑖𝑙 + 𝒘𝝎 + 𝜶𝝎 − 𝜶𝝎 ) − 𝒚̇ 𝑟𝑒𝑓 + 𝜣
̂̇ 𝒚 = [ 𝒚
𝑬
] − 𝑾̇𝒚
𝒆̂𝒚
(56)
With the pseudo control feedback law
𝜶𝝎 = 𝑮−1
𝒚 ⋅ (−[𝑲𝒚
̂ 𝒚 ⋅ 𝝋𝒚 ).
̂ 𝒚 − 𝒇𝒚 + 𝒚̇ 𝑟𝑒𝑓 − 𝜣
𝑲𝒚,𝐼 ] ⋅ 𝑬
(57)
and the dynamics of the reaction deficit
𝑮 (𝜶
− 𝜶𝝎 + 𝒆̂𝝎 + 𝒘𝝎 )
𝑾̇𝒚 = [ 𝒚 𝝎,𝑓𝑖𝑙
],
𝟎
(58)
the error dynamics (56) results in
̂̇ 𝒚 = [−𝑲𝒚
𝑬
⏟𝑰
̃ ⋅ 𝝋𝒚
−𝑲𝒚,𝐼 ̂
𝜣
] ⋅ 𝑬𝒚 + [ 𝒚
].
𝟎
𝟎
(59)
̅𝒚
𝑲
The update law of the estimated parameter 𝜣𝑖 has the same gradient-based structure as in (38):
̂̇ 𝑇𝒚 = Proj (𝜣
̂ 𝑇𝒚 , 𝜞𝒚 𝝋𝒚 𝑬
̂ 𝑇𝒚 [ 𝑰 ])
𝜣
𝟎
(60)
With the closed-loop error dynamics of this step (59) and the parameter update law (60), the Lyapunov function
̂ 𝒚:
derivative (55) is rendered negative semi-definite in the error variable 𝑬
1 𝑇
̂ 𝑬
̂
𝑉1̇ = − 𝑬
2 𝒚 𝒚
(61)
According to (41), the extension of the Lyapunov function in the second and last design step is chosen as a quadratic
̃ 𝝎:
function (𝑷𝝎 = 𝑰) in the control 𝒆̂𝝎 and the parameter error 𝜣
1
1
̃ 𝝎 𝜞−𝟏
̃𝑇
𝑉2 = 𝑉1 + 𝒆̂𝑇𝝎 𝒆̂𝝎 + 𝑇𝑟(𝜣
𝝎 𝜣𝝎 )
2
2
(62)
The final control 𝒖𝑐𝑚𝑑 is designed in order to render the time derivative
1 𝑇
1
1
̃ 𝝎 𝜞−𝟏
̃̇ 𝑇
̂𝒚 𝑬
̂ 𝒚 + 𝒆̂̇𝑇𝝎 𝒆̂𝝎 + 𝒆̂𝑇𝝎 𝒆̂̇𝝎 + 𝑇𝑟 (𝜣
𝑉2̇ = − 𝑬
𝝎 𝜣𝝎 )
2
2
2
(63)
of the Lyapunov function (62) negative definite. Substituting the body-rate dynamics (16) into the time derivative of
the compensated tracking error (53), the definition of the error dynamics used in (63) becomes
̂ 𝝎 ⋅ 𝝋𝝎 + 𝜣
̃ 𝝎 ⋅ 𝝋𝝎 − 𝜶̇ 𝝎,𝑓𝑖𝑙 − 𝒘̇𝝎 .
𝒆̂̇𝝎 = 𝒇𝝎 + 𝑮𝝎 ⋅ (𝒖 − 𝒖𝑐𝑚𝑑 + 𝒖𝑐𝑚𝑑 ) + 𝜣
(64)
By applying the nonlinear control law
̂ 𝝎 ⋅ 𝝋𝝎 + 𝜶̇ 𝝎,𝑓𝑖𝑙 )
̂ 𝝎 − 𝒇𝝎 − 𝜣
𝒖𝑐𝑚𝑑 = 𝑮−1
𝝎 ⋅ (−𝑲𝝎 𝒆
(65)
with the proportional error feedback of 𝒆̂𝝎 , the parameter update law
̂̇ 𝑇𝝎 = Proj(𝜣
̂ 𝑇𝝎 , 𝜞𝝎 𝝋𝝎 𝒆̂𝑇𝝎 ),
𝜣
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(66)
and the dynamics of the reaction deficit
𝒘̇𝝎 = 𝑮𝝎 ⋅ (𝒖 − 𝒖𝑐𝑚𝑑 ),
(67)
1 𝑇
1
̂ 𝑬
̂ − 𝒆̂𝑇 𝒆̂
𝑉2̇ = − 𝑬
2 𝒚 𝒚 2 𝝎 𝝎
(68)
the Lyapunov time derivative (63)
becomes negative semi-definite.
Remark: Both control layouts (CFAB and AWCFAB) account for the actuator and filter dynamics. AWCFAB
compensates for the filter dynamics already in the control error, which is fed back via the feedback law. CFAB leaves
the tracking error unchanged and accounts for the actuator and filter dynamics only in the cross-coupling term
̃𝒚.
𝟎]𝑷𝒚 𝑬
𝑷−1
𝝎 [𝑮𝒚
Due to the different insertion of the unachieved portion, the layout of the two loops in AWCFAB is completely
̃
independent of each other. In contrast to standard CFAB (CFB), the term 𝑷−1
𝝎 [𝑮𝒚 𝟎]𝑷𝒚 𝑬𝒚 accounting for the timescale property of the two loops – describing the interaction between the two dynamical layers – vanished due to the
modified feedback law and the introduction of the reaction deficit 𝒘𝑖 . This property of two completely independent
layers can also be seen by considering the closed-loop error dynamics of the non-adaptive Anti-Windup CFB
(AWCFB) control law:
𝒆̂̇𝒚
−𝑲𝒚
[𝒆̂̇𝒚,𝐼 ] = [ 𝑰
𝟎
𝒆̂̇
𝝎
𝒆̂𝒚
𝟎
𝟎 ] ⋅ [𝒆̂𝒚,𝐼 ].
−𝑲𝝎
𝒆̂𝝎
−𝑲𝒚,𝐼
𝟎
𝟎
(69)
The upper right and lower left entry is zero compared to (51), which means there is no dynamical interaction between
the outer and inner loop.
Besides the choice of the non-adaptive, baseline controller gains (𝑲𝒚 , 𝑲𝒚,𝐼 , 𝑲𝝎 ), the selection of the regressor
structures of 𝝋𝒚 and 𝝋𝝎 and the learning rates 𝜞𝒚 and 𝜞𝝎 play a crucial role in terms of closed-loop robustness and
performance for both control layouts. The learning rates 𝜞𝒚 and 𝜞𝝎 are selected based on physical considerations, in
the same way as presented in [12]. Based on the model used for control design (see Section III.B), a feasible choice
for the regressor vectors of inner and outer loop is shown exemplarily for the pitch channel in equation (70) and (71).
𝝋𝒚 = [0
𝝋𝝎 = [0
𝐶𝑀0
𝐶𝑧,𝛼 𝑍𝛼
𝐶𝑀0,𝛼
𝐶𝑀𝑞 ⋅ (𝑞𝐾𝐸𝐵 )𝐵
𝐶𝑧,𝛽 𝛽̇
𝑇
…] ∈ ℝ7×1 ,
(𝑟𝐾𝐸𝐵 )𝐵 ⋅ (𝑝𝐾𝐸𝐵 )𝐵
𝑇
…] ∈ ℝ7×1 .
(70)
(71)
This selection was motivated by identifying the predominant dynamic effects and the major uncertainty effects. Due
to the beneficial ratio between control effectiveness and damping in the roll channel, the proportional/integral action
̂ 𝒚, 𝜣
̂ 𝝎 are initialized with zero.
provided by the non-adaptive CFB/AWCFB is sufficient. The parameter estimates 𝜣
IV. Simulation Results
For all simulations, the filter cut-off frequency is chosen as 𝜔𝑖,𝑓𝑖𝑙 = 100[𝑟𝑎𝑑/𝑠] and the damping ratio is 𝜁𝑖,𝑓𝑖𝑙 =
0.8.
First, the effect of the anti-windup strategy is compared to the standard CFB design for a demanding STT maneuver,
forcing the fins to reach their saturation limits. The second evaluation considers the AWCFAB controller and its
capability of coping with large deviations from the nominal, assumed missile dynamics.
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A. Comparison between AWCFB and CFB
At the flight envelope point 𝑀𝑎 = 2.5, ℎ = 6𝑘𝑚, an absolute maximum acceleration of approximately 40𝑔 is
commanded to the missile autopilot. This command is filtered and reduced by the reference model to make it accessible
by the missile controller. Since the anti-windup capability of both controllers are considered, the adaptive layer is
𝐼𝑀𝑈
𝐼𝑀𝑈
switched off for this evaluation series. Figure 3 shows the three control variables 𝜙𝑣 , (𝑎𝐴,𝑦
) , (𝑎𝐴,𝑧
)𝐵 obtained by
𝐵
the two control approaches (dashed lines), the commanded signal and the signal shaped by the reference model (solid
lines). Besides the control variables, the second plot of Figure 3 depicts the overall fin deflection |𝜉| + |𝜂| + |𝜁| ≤
̇
𝛿𝑖,𝑙𝑖𝑚𝑖𝑡 = 25[𝑑𝑒𝑔] and its velocity |𝜉̇| + |𝜂̇ | + |𝜁 |̇ ≤ 𝛿𝑖,𝑙𝑖𝑚𝑖𝑡
= 572[𝑑𝑒𝑔/𝑠]. These scenarios clearly show the
superior robustness and tracking capability of the AWCFAB as compared to the standard CFB approach. Reaching
the fin limits at 𝑡 ≈ 1.7𝑠 in the case of the CFB approach leads to undesirable oscillations and therefore a prohibitive
tracking performance. Therefore, the CFB (CFAB) approach – even though it accounts for the unachieved portion
caused by the filter and the actuator dynamics – is not suitable for systems requiring the full exploitation of the system
input limits. The novel AWCFB approach tracks the reduced reference signal almost perfectly even in cases of hitting
the actuator limits. Since the nominal case (no uncertainties) is considered, the reference trajectory provided by the
nonlinear reference model is fully accessible by the missile system. In cases of deviations from the nominal, assumed
plant (see Table 1), the reference trajectory, which is based on nominal plant information, is beyond the physical
possible flight envelope. Nevertheless, the AWCFB autopilot enables a continuation of the missile’s mission within
the missile’s physical limits (see Figure 4).
B. Evaluation of the Adaption Capability of AWCFAB
In order to demonstrate the benefits of AWCFAB in terms of robust performance with respect to model
uncertainties, a demanding STT maneuver is commanded to the missile system. As initial condition for the simulation,
the same flight envelope point as in the evaluation series above is chosen: 𝑀𝑎 = 2.5, ℎ = 6𝑘𝑚. A clinical STT
maneuver consisting of step inputs simultaneously applied to both acceleration channels with an amplitude of 20𝑔,
while the roll angle 𝜙𝑣 should remain zero. The STT maneuver demands an overall acceleration of the missile system
that is close to the limit of the maximum trimmable acceleration at the considered operating point.
A Monte Carlo simulation was obtained and 32 representative uncertainty cases are selected to demonstrate the
increased tracking performance and robustness of the adaptive part. The uncertainty spectrum of Table 1 is combined
in a physically feasible way by selecting the 32 uncertainty combinations only at the corners of the uncertainty
hypercube. Figure 4 and Figure 5 depicts the 32 simulation runs for the AWCFB (upper plot) and the AWCFAB
(lower plot) autopilot design, respectively. The adaptive autopilot (Figure 5) shows an improved tracking performance
characterized by reduced overshoot, faster settling time, and an almost perfect steady-state tracking behavior in
trimmed flight.
Remark: Due to its lack of compensating actuator saturations, the Monte Carlo runs obtained with the CFAB/CFB
autopilot are omitted in this analysis.
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Figure 3. Comparison of CFB and AWCFB for a STT maneuver at 𝑀𝑎 = 2.5 and ℎ = 6𝑘𝑚.
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Figure 4. STT maneuver obtained with the non-adaptive autopilot (AWCFB) at 𝑀𝑎 = 2.5, ℎ = 6𝑘𝑚 for the
32 uncertainty cases considered.
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Figure 5. STT maneuver obtained with the adaptive autopilot (AWCFAB) at 𝑀𝑎 = 2.5, ℎ = 6𝑘𝑚 for the 32
uncertainty cases considered.
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V. Conclusion
In this paper, a CFAB and a novel AWCFAB autopilot are presented, which use a nonlinear reference model
introduced in [12]. Due to the demanding maneuvers of a tail-controlled air defense missile, an anti-windup autopilot
strategy was developed based on the CFB framework. The superior behavior of the AWCFAB, which compensates in
a different way for the actuator and filter reaction deficit, becomes obvious by a command sequence that requires the
full actuator capability. AWCFB leads to a safe continuation of the mission even in the case of actuator saturations.
Hitting the input constraints leads to instability in case of the CFB approach presented herein.
The benefit of the adaptive layer becomes obvious when the plant used for control design deviates from the real
missile dynamics. The adaptive controller exhibits an increased level of robustness and almost maintains the nominal,
desired closed-loop performance over the total uncertainty spectrum.
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