Design of Higher Order Sliding Mode Control Laws
for a Multi Modal Agile Maneuvering DCAV
N. Kemal Ure! and Gokhan Inalhan2
Istanbul Technical University, Faculty of Aeronautics and Astronautics.
34469 Maslak, Istanbul, Turkey
Abstract-In this work, we consider the design of a nonlinear
control system for an unmanned combat air vehicle for
executing agile maneuvers over the full flight envelope. From
the inspection of well known smooth aerobatic and combat
maneuvers, we see that complex maneuvers can be decomposed
into a specific set of different sub maneuvers to cover any
arbitrary flight maneuver. To control each sub mode an
inner/outer control loop approach with higher order sliding
mode controllers are developed. These controllers attain robust
tracking of maneuver profiles for nonlinear aircraft dynamics.
Resulting algorithms are applied to a high fidelity six degrees of
freedom F-16 fighter aircraft model. We show that final design
is capable of autonomously tracking the reference trajectories
in the presence of unmodeled dynamics, disturbances and nonminimum phase outputs.
I.
INTRODUCTION
With the increasing complexity of unmanned vehicle
operations in combat scenarios, there is a growing demand
on agile maneuvering from unmanned combat air vehicles
(UCAVs). Rather this be for evasive maneuvers or for attack
patterns, UCAVs are expected to operate in dense and often
threatening environments which require aggressive trajectory
planning and controls. Such trajectories in complex
environments will need to make use of the full agile
maneuvering capability of the aircraft (such as high "g" turns)
and its full envelope flight characteristics (such as high angle
of attack flight). In this work, we develop a multi modal
control scheme that allows the vehicle to perform such
aggressive maneuvers.
nonlinear sliding mode control system that overcomes
problems associated with non-minimum phase output tracking
and chattering.
Description of aircraft dynamics from hybrid system point of
view has been studied previously in [3], [4] .These works have
been successful in using the advantages of hybrid system
methodology in control of both single and multiple aircrafts.
However, these approaches did not include the full flight
envelope dynamics of the aircraft. Specifically, both mode
selection and controller design is strictly based on selected
maneuvers; therefore controllability is limited [3], [4] to these
predefined trajectories. In our work, we make use of
parameterized sub maneuvers which builds up complex
maneuver sequences. We show that it is possible to cover
almost any arbitrary maneuver and the entire flight envelope by
this approach.
For tracking of the maneuver sequences decomposed by this
method, linear control systems are not adequate as their
tracking capabilities are limited to trimmed or non-aggressive
trajectories. For this reason, we have to rely on nonlinear
control techniques to achieve tracking of nonlinear agile
maneuvers of the aircraft. Two of the applicable nonlinear
control techniques are feedback linearization [1], [2] and
sliding mode control [5]. Unfortunately, these techniques
cannot be applied directly to the trajectory control of aircraft
due to non-minimum phase nature of the controlled outputs
resulting in unstable internal dynamics. Early works such as
[6], [7] at this area neglected the non-minimum phase (NMP)
In general it is a very challenging task to describe the general effect on the control system design, but included it in
motion of an aircraft and design a single control law that simulations to show that its effect was negligible. However,
handles aggressive reference tracking. From the inspection of during aggressive maneuvering NMP effect becomes
the well known smooth aerobatic maneuvers and more significant due to high dynamic pressure and high angles of
complex combat maneuvers, we see that this task can be attack. An alternative way to overcome NMP effect is to
quantized by decomposing general maneuvers to maneuver perform a stable inversion technique. This is discussed in [8],
modes in which both system dynamics and control task is [9], [10] for both conventional and vertical take-off and landing
simplified. Arbitrary maneuvers can be generated by aircraft models. However, the stable inversion technique limits
sequencing of these modes and by selection of maneuver the full control bandwidth of the aircraft and diminishes the
parameters (expressed as modal inputs). The controller design maneuvering capability to attain stability. In our approach, we
is structured around this finite state automaton that spans the have followed an outer loop sliding mode control design to
full-flight-envelope maneuvers of a generic aircraft model. For overcome non-minimum phase effect on the system. Sliding
each mode of this automaton, we design a dual inner/outer loop mode controllers offers insensitivity (robustness) to matched
Research Assistant, Controls and Avionics Lab., [email protected]
Assistant Professor, Director of Controls and Avionics
Lab., [email protected]
1
2
1-4244-2386-6/08/$20.00 ©2008 IEEE
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disturbances with known upper bounds, and they can be
integrated with a dynamic compensator to overcome NMP
effect as shown on [11]. Main drawback of the sliding control
methods is the resulting chattering in the actuator signals due to
discontinuous terms in the control laws. This can be
problematic for the operation of the actuators specifically when
the aircraft is exposed to high gain noises. Boundary layer
solution [12] offers a low pass filter in the neighborhood of
sliding manifold to soften the effect of chattering. A more
elegant solution to the problem is through higher order sliding
modes [13], which moves the discontinuous term into higher
derivatives of the control. For this purpose, several algorithms
were developed by [14] and these algorithms were applied to a
real world aircraft pitch control problem. We make use of the
higher order sliding mode control laws at inner loop to provide
continuous signals to controllers and increase accuracy of the
controller.
The structure of the paper is as follows: The multi modal
control concept is presented in Section II. In Section III, we
provide the details of the aircraft model used in the control
system design and the simulations. Development of the control
laws for a specific mode is presented in Section IV. In the final
section, through numerical simulations, we demonstrate the
ability of the proposed control methodology and the control
system to achieve tight control of complex agile maneuvers.
II.
A.
MULTI MODAL CONTROL
Derivation flight modes and controllers
The basic idea of multi modal control hinges on the fact that
agile maneuvers can be quantized into distinct flight modes.
Analysis of aerobatic and combat maneuvers [20] reveals that
these modes can be used as building blocks to generate
complex maneuvers in agile flight.
Fig. 1: Common Combat (High Yo-Yo) and Aerobatic (Cuban Eight)
Maneuvers
As seen in Fig.1, the aerobatic maneuvers consists of loops in
both lateral and longitudinal directions, however a combat
maneuver usually consists of loop in a three dimensional plane.
There are also non turning flight segments, such as straight
level flight or climbing flight. The final maneuver segment
consists of rolling of the aircraft around its velocity axis.
To decide the set of modes to describe the general motion of
the aircraft, well known aerobatic and combat maneuvers are
analyzed [21]. Based on this analysis, we note that each
complex flight maneuver can be decomposed into simpler
maneuvers. The simplest maneuver segment is called maneuver
mode, and these modes can be sequenced to create more
complex maneuvers.
In addition to defining the modal sequence, as seen in Fig. 1,
maneuver parameters (named as modal inputs) should be
specified to obtain the complete maneuver specification. The
modal sequence and the modal inputs associated with each
mode can define an arbitrary maneuver of the aircraft in its
flight envelope. Under the light of above discussion we
describe the seven main modal blocks and their respective
parameters in Table 2. An extensive treatment of this analysis
and formalism can be found in [21].
Table 2: Maneuver Modes, Inputs and Controllers
In Table 2, each mode is defined by specific state constraints
that define each mode and by modal inputs that parameterize
the maneuvers of that mode. The first two modes (level flight
and climb/descent) in Table 2 are usually used in cruise
configuration. Roll mode acts as a transition mode and it is
used either for turning the direction of the lift vector to enter
into loops, or for inverting the aircrafts attitude. Longitudinal
and lateral loops are integral parts of agile maneuvering and
they can be found on almost any maneuver. To cover
trajectories in three dimensions we have defined a 3D mode,
which in many maneuvers corresponds to a coordinated
climbing-turning maneuver identified by either angular body
rates or wind axes Euler angles (subscript “w” refers to wind
axes in all the tables and the figures). The last particular mode
is the safety mode, which is used for preventing the aircraft
from stalling, or leaving the domain of a mode. This mode
basically regulates the aircraft back to level flight from
arbitrary initial attitude positions. Last column in table 2 refers
to controllers assigned to each mode which we will discuss in
section II.b.
Two constraints arise when building a motion alphabet from
this modal system description. First constraint is the maneuver
sequencing problem. Due to physical considerations, maneuver
execution cannot necessarily be arbitrary. For this, a set of
rules is reflected on a mode transition chart and this chart
describes which maneuver mode can be executed after another.
Second constraint is associated with modal inputs. Due to
aerodynamic, structural and actuator limitations, modal inputs
must lie inside the flight envelope (described partially by V-n
diagram) during the execution. Transition logic, domains and
trajectory acceptance conditions of the automata are formed by
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the constraints arising from the above dynamic transition
limitations. Overall, this framework reduces the complexity of
the motion control problem into mode control problem. This is
illustrated in Fig.2 for an Immelmann Tusrn. Details of the
complete hybrid system description including connection of
maneuvers (set of transition rules between modal blocks),
domain of feasible modal inputs (transformation of actuator
saturation limits to flight envelope limits), and maneuver
sequence and parameter extraction from a given flight path can
be found in [21]. In this paper, we focus on constructing set of
controllers to track given maneuver sequence.
B.
Switching Controllers
Although it is theoretically possible to design one controller
which tracks the inertial trajectory of the aircraft, it is much
more structured and performance wise feasible to design
specific control laws to each mode (or a set of similar modes).
After designing such a set of controllers, control of a complete
agile trajectory can be achieved by switching between these
controllers following the mode transitions. This idea is
illustrated in Fig. 2 in which three different output tracking
controllers are switched between to track the Immelmann Turn.
flight path, the effectiveness of this controller is limited. This is
based on the fact that C4 is an NMP attitude controller which
requires restrictions on Cartesian coordinates of the aircraft.
This is in comparison to the MP controllers which stabilize
both rotational and translational elements. Note that if every
mode is locally controllable by the set of output controllers
defined as above, then we can globally control any maneuver,
by decomposing it into sub modes and switch between
controllers at each transition step. For a formal analysis of
controller switching conditions, see [22].
III. AIRCRAFT MODEL
In this section, we review the nonlinear aircraft model used in
the control system design and describe the sub models used in
nonlinear flight simulations.
A.
State Equations
Following the six degrees of freedom rigid aircraft body
formulation from [18], we denote aircraft states as, velocity in
body axes VB = [U V W ]T (or equivalently VB = [VT α β ]T ), northeast-up Cartesian position coordinates RNEU
rates in body axes ω = [ P
Table 3: Tracked Output Sets for each controller
Choice of the control set as defined in Table 3 is based on both
the modal inputs and the mode constraints of each mode. For
example, the roll mode controller combines both modal input
integration of wind axis roll rate, and mode constraint of zero
sideslip angle. In Table 3, although C4 can be observed as an
ultimate controller which controls any maneuver given by a
h ⎤⎦ ,
angular
Q
R]
T
B
⎡1 tan θ sin φ
= ⎢0
Φ
cos φ
⎢
⎢⎣ 0 cos θ sin φ
Following the maneuver descriptions and their modal inputs
(maneuver parameters) in Table 2, Table 3 summarizes the
output tracking controllers assigned to each mode.
ep
and Euler angles Φ = [φ θ ψ ] .
Rotation matrix R ∈ SO(3) is used for axes transformations in
the usual 3-2-1 Euler angle notation.
Complete state equations describing the aircraft motion can be
represented under force, navigation, kinematic and moment
equations as follows.
m {VB + ωVB } = mgR (θ ) R (φ ) + FA + T (1.a)
R
= R (ψ ) R (θ ) R (φ ) V (1.b)
NEU
Fig. 2: Controller Switch Diagram for Immelmann turn
= ⎡⎣ nP
tan θ cos φ ⎤
(1.c)
− sin φ ⎥⎥ ω
cos θ cos φ ⎥⎦
I ω + ω I ω = M A (1.d)
Here M, F, T terms correspond to moment, force and thrust
terms of the aircraft (denoted with A) with B subscript denoting
the body axis. Note that, use of Euler angles is not convenient
for agile flight since the kinematic equation Eq. (1.c) becomes
singular for 90 degrees of pitch angle which happens regularly
during agile maneuvers. This singularity is avoided with use of
quaternion attitude description during simulations. Most of the
controller design is based on Euler angles, but the singularities
are avoided during control design process. Only quaternion
based controller design is for the safety mode. The above
equations are valid for almost every fixed wing conventional
aircraft and what makes the model specific to the aircraft
modeled (F-16 in our case) is the sub-models used in
simulation.
B.
Sub Models for Simulation
The F-16 sub-models used in simulation are briefly explained
as follows. Note that thrust vectoring version of this model has
been also used in [19] for nonlinear control design procedures.
Aerodynamic Model: Aerodynamic data (relationship
between the aerodynamic forces and control surface
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deflections) and physical properties are taken from the NASA
report [17] in tabular form. Atmosphere Model: To control
over the aircraft in full flight envelope, the change of air
density and therefore dynamic pressure during maneuvers is
achieved through the standard atmosphere model. Engine
Model: To simplify the model, we have assumed that a
separate engine control system was designed (which is a
complicated study on its own) and we can directly command
thrust that can be exerted on aircraft with a first order lag.
Actuator Models: All control surface deflections are modeled
with 0.495 seconds of lag; limits are +- 25, 21.5, 30 degrees
deflection and +-60, 80,120 deg/s rate limit for elevator,
ailerons and rudder respectively.
IV.
CONTROL SYSTEM DESIGN
Fig. 4: Control System Diagram
Following the switching controller formalism in Section II.b,
the design of low level controllers for each mode is a key step
to controlled agile maneuvering. This is based on the fact that
the controllability of maneuver sequences depends on local
convergence of each maneuver mode. Due to nonlinear nature
of agile maneuvers, we will particularly make use of sliding
mode techniques. This is inspired from the fact that each
maneuver mode can be cast as a hypersurface in state space
where the maneuver tracking is achieved. As an advantage of
hybrid system description, it is possible to design output
tracking controller specific to each mode (or a set of modes).
The control system design procedure includes an inner/outer
loop approach based on time/scale separation which is inspired
from [24]. The outer loop consists of flight path variables
which were used to express maneuvers in Section II. At the
outer loop, body angular rates are used as inputs to flight path
angles. Inner loop controller regulates the control surfaces to
track body angular rate profiles created by outer loop. Overall
schematic of control system is shown on Fig. 4.
Among the set of controllers presented in Table 3, we will
show the design steps for C2, (true velocity and wind axis Euler
angles) to illustrate a more challenging design case.
A.
Outer Loop
Aim of the controller is to track the output profile, so that;
(ψ w , θ w , φw , VT ) → (ψ wd , θ wd , φwd , VTd ) . In order to construct
the controller, output differential equations must be extracted
first. Equation for true velocity is easily derived from Eq. 1.a.
To derive the differential equations for wind axis Euler angles
kinematic relationship for transformation between body and
wind axes (2) can be used:
R(ψ w , θ w , φw ) = R(ψ , θ , φ ).R −1 (− β , α , 0) (2)
Differentiating Eq. (2) once and using Eq. 1.a for the
derivatives of aerodynamic angles and Eq. 1.c for the
derivatives of Euler angles, complete set of nonlinear equations
for output profiles can be shown to be;
⎡ VT ⎤
(3)
⎢ ⎥
⎢ψ w ⎥ = f ( Φ ,V , ω , F , ΔF (δ , δ , δ ) , δ )
w
B
A
A
e
a
r
T
⎢ θw ⎥
⎢ ⎥
⎣⎢ φw ⎦⎥
It is clear that system has relative degree one, because all of the
true inputs have already appeared in first differentiation.
However, force terms associated with control surface
deflections (shown by ΔFA (δ e , δ a , δ r ) in Eq. (3)) are the main
reason for NMP (or unstable internal dynamics) problem. If
control surfaces in Eq. (3) are taken into account, control laws
based on this approach destabilizes internal dynamics of the
system (Body Euler angles in this example). This is due to fact
that, main purpose of control surfaces is to produce moments,
not forces [6]. In order to overcome this problem, these forces
have been neglected in past works as pointed in introduction.
Our approach is not to neglect these forces but accept them as
disturbances to the system and compensate these disturbances
by the help of dynamic sliding mode control. Aim is to derive
feedback control laws for throttle and body angular rates to
robustly track velocity and wind axes Euler angles profile.
In the light of above discussion Eq. (3) is broken into several
functions (Exact analytical expressions have been avoided for
simplicity).
f w = f w1 (Φ,VB ) + ( B1 (Φ,VB ) + ΔB1 (Φ ,VB ))[δ T , ω ], + w(δ e , a , r ) (4)
In Eq. (4) f w1 (Φ,VB ) is associated with expressions which
contain Euler angles and velocity variables. Four by four
matrix B1 is the input matrix of the system and the perturbation
ΔB1 is for uncertainties at input modeling (it only signifies the
uncertainty in throttle – engine dynamics, since equations
related with body angular rates are exact). w(δ e ,a ,r ) is the
disturbances caused by control surfaces as indicated above.
Now the problem can be treated as a standard nonlinear sliding
mode control problem; Eq. (4) is re-written once again in the
standard linear affine form.
x = f ( x) + g ( x)u
(5)
f ( x) = fˆ + w, g ( x) = ( B + ΔB ), u = [δ , P, Q, R]T
1
1
T
Here we assume that,
(6)
w < α i , B1 ΔB1 < β i , i = 1,..., 4
and the bounds on this function can be easily estimated from
nonlinear simulations. Defining the sliding surfaces in state
space for each output, we obtain:
Si = ( yid − yi ) + ci ∫ ( yid − yi ) dt , i = 1, 2,3, 4
(7)
y1 = VT , y2 = ψ w , y3 = θ w , y4 = φw
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In Eq. (7) the integral terms are added as a Hurwitz polynomial
to the desired dynamics for to enhance robustness. Here the
control gains are cast as:
u = ueq + K Β1−1 sgn ( S ) , K = diag {ki } , i = 1,.., 4
(8)
u = Β −1 fˆ
eq
1
for which the switching gains are chosen to ensure that these
gains are the dominant terms in derivative of the sliding
surfaces. Specifically, this can be accomplished by inspecting
the Lyapunov function which can be given as:
{
}
1 T S S ,V = S T S = S T fˆ + w + gu
2
= S T yid − fˆ + w + g (ueq + K Β1−1 sgn ( S ) + Ce
V=
{
(
)
= S {μ + K ( I + B1ΔB1 ) sgn( S )}
If μ ≤ γ i , selecting sliding gains such that
i
ki > max
γ i + ηi
3
1 − ∑ βi
}
(9)
j =1
ensures that Lyapunov function decays to zero at finite time:
1
(11)
V = Si 2 , V = Si Si ≤ ηi Si
2
With this, the required body angular rate trajectories is
obtained, the last step is to design an inner loop control law to
find control surface deflections. Note that throttle input is
already obtained at this step.
B.
Inner Loop
We will mainly follow same procedure in inner loop design (in
the sense of choosing sliding surfaces and rendering them
invariant), but unfortunately switching control laws in (8)
cannot be directly implemented in the inner loop, due to
resulting chattering will be likely to saturate actuators and
cause damage to them. Control laws could be smoothed with
use of saturation function instead of signum function, but this
will degrade the performance. As an attractive alternative, we
will use higher order sliding modes (HOSM) to eliminate
chattering. Basic idea behind the HOSM is to keep higher
derivatives of the sliding surface to zero along with first
derivative. This results in moving the switching term inside the
derivative of the input, so that when integrated no
discontinuous term is present at actual input. HOSM control
laws also provide better accuracy and robustness compared to
conventional sliding mode methods [13].
We first consider the moment equation (1.d) and write it in a
similar form to Eq. (5):
x = f ( x) + g ( x)u
(11)
f ( x) = fˆ ( x ) + w, u = [δ , δ , δ ]T
e
a
Si = ( yi − yi d ) , i = 1, 2,3
(12)
y1 = P, y2 = Q, y3 = R
Note that for the HOSM design the constraints that have to be
satisfied are:
(13)
Si = Si = Si = 0
Defining the differentiating operator as
(14)
∂ (.) = ( ∂ / ∂x )(.)( f + gu )
And differentiating the sliding surfaces two times, we obtain:
(15)
Si = ∂∂Si + ( ∂Si / ∂u ) u
To accomplish the constraints in Eq. (13), the derivative of the
input term in Eq. (15) should become the dominant term. This
is achieved by the super-twisting algorithm [14] for which the
input is defined as:
u = ustatic + udynamic
(10)
, i = 1,..., 4
similarly, we obtain:
r
In order to put the inputs onto linear affine form, aerodynamic
tables were approximated with polynomial and trigonometric
functions with least squares fit. Disturbance terms accounts for
the uncertainty in the aerodynamic model, which is around
%10 for each aerodynamic coefficient gathered from wind
tunnel tests (Deviation of functional fits from tabular data can
also be count as an uncertainty). Defining the sliding surfaces
⎧⎪ −λ Si 0 p sgn( Si )
Si ≥ Si 0
ustatic = ⎨
p
Si < Si 0
⎪⎩−λ Si sgn( Si )
u ≥ u0
⎪⎧ −u
udynamic = ⎨
sgn
S
u
α
(
)
−
< u0
⎪⎩
i
(16)
In Eq. (16), undefined constants are derived from the bounds
on second derivatives of the nonlinear functions f and g. These
constants can be estimated from nonlinear simulations or they
can be tuned during test of the control system. Note that Eq.
(16) is in the SISO form, but it can be extended to MIMO form
with an invertible input transformation v = ζδ , ζ ∈ R3 x 3 which
decouples the system so that only one input vi appears at each
sliding surface equation. Then Eq. (16) is applied to each
surface separately and the true inputs are recovered from
invertible transformation. The resulting control law can be
written as:
(17)
δ = ζ −1 ( vstatic + vdynamic )
This completes the design of the inner loop control system.
V.
SIMULATION
To display the capacity of the control system, we will show
tracking of an agile combat maneuver inspired from [20]. First
the maneuver is inspected formally, and decomposed in to
modal block sequences as described in Section II. Later, mode
sequences are transformed to tracking profiles for sliding mode
controller. Simulation model that is used is the high fidelity F16 model described in Section III. The performance of the
control system and the key agility metrics are illustrated in
Figs. 5 and 6. During this specific maneuver, we observe that
the load factor also varies periodically around 8 “g”s, which is
almost impossible to overcome by a human pilot. Following
the performance of control systems from Fig. 7, we see that
inner loop had done its job in tracking body angular rates
almost perfectly. In addition, inputs didn’t saturate during the
maneuver (this is actually due to maneuver synthesis system
which selects feasible maneuver sets) and there is no chattering
in any channel due to HOSM. Note that the control system has
been successful for tracking a highly nonlinear agile maneuver
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in presence of uncertain effects which on the order of %30 for
aerodynamic coefficients in the given angle of attack regime.
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