BACKSTEPPING AND GAIN SCHEDULING TECHNIQUES

Anais do XX Congresso Brasileiro de Automática
Belo Horizonte, MG, 20 a 24 de Setembro de 2014
BACKSTEPPING AND GAIN SCHEDULING TECHNIQUES: COMPARATIVE
ASPECTS AIMING ITS INDUSTRIAL IMPLEMENTATION
Francisco J. Triveno Vargas∗, Erik O. Pozo Irusta†, Pedro Paglione†
∗
Embraer Defense and Security
Avenida Brigadeiro Faria Lima, 2170 - S˜
ao Jose dos Campos, SP 12227-901 - Brasil
Tel: +55 12 3927-1000, Fax: +55 12 3927-6600
†
Instituto Tecnol´
ogico de Aeron´
autica (ITA) - Departamento de Mecˆ
anica de Vˆ
oo
Pra¸ca Marechal Eduardo Gomes, 50 - S˜
ao Jose dos Campos, SP 12228-900 - Brasil
Tel: +55 12 3947-5968, Fax: +55 12 3947-5967
Emails: [email protected], [email protected], [email protected]
Abstract— In this paper is briefly mentioned a linear LQR control technique, when applied the specific
weighting algorithm. This is done as a preamble to the implementation of gain scheduling technique, already
established in the industry. Latter, the nonlinear backstepping technique is presented; some characteristics and
design methodology are highlighted. Finally, comparative aspects between both techniques are analyzed, this is
done taking into account the same controlled variables, the same reference inputs and the same control surfaces
(i.e. actuators). Aspects such as: ease of designing, control magnitude effort, tracking error, the coupling, reuse
of control structure for different aircraft projects, some certification aspects and other problems are considered
at simulation level. This paper highlights relevant aspects related to design of nonlinear backstepping technique
looking its industrial application.
Keywords—
1
aircraft control, nonlinear control, gain scheduling, backstepping, linear quadratic regulator.
Introduction
The gain scheduling is considered a standard
method to design Linear Time Invariant (LTI)
controllers for nonlinear systems. It also has
widespread and successful engineering, applications. In the aerospace industry, this technology was first used on military applications, see
(Leithead, 1999). Research of gain scheduling applications in civil aircraft and other areas developed gradually since then, examples of its implementation in aircraft control are (Gangsaas
et al., 2008), and (da Silva et al., 2011).
Gain scheduling is an attractive control strategy to deal with nonlinearities in aircraft. The
main idea of this methodology is to design a set of
LTI controllers for specific operating points over
the flight envelope and then to interpolate the
gains against the current value of the scheduling
parameters (flight conditions), instead of seeking
a single robust LTI controller for the entire operating range.
Backstepping technique constitutes an alternative to gain scheduling. Using backstepping,
the nonlinearities of the system do not have to
be cancelled in the control law. If a nonlinearity
acts in a stabilizing way, it may be retained in the
closed loop system see (Chen, 1996).
The backstepping technique appeared implicitly in several papers in the late 1980’s. However,
it received important attention after the works
of Professor Petar V. Kokotovic and coworkers
(Khalil, 1996).
Related to aircraft flight control applications,
previous nonlinear flight control designs were
typically based on feedback linearization, called
nonlinear dynamic inversion (NDI), see (Meyer
The design of flight control systems is a typical nonlinear control problem, due directly to the
changes in aircraft dynamics with flight conditions and aircraft configuration. For this reason,
a dynamic model that is stable and adequately
damped in one flight condition may become unstable or at least inadequately damped in another
one. In commercial aircraft, a lightly damped oscillatory mode may cause discomfort to passengers
or make difficult the control of the aircraft for the
pilot. For a combat aircraft, this condition may
lead to more critical situation once the aircraft is
inherently unstable due the maneuverability requirements and capability of attack.
These problems are overcome by using feedback control to modify the aircraft dynamics
which also bring along improvements in terms of
aircraft weight reduction, aerodynamic efficiency
and optimization of fuel consumption. These
changes are naturally leading the design of new
airplanes to embedded relaxed stability, boosting the use of feedback control laws (Holzapfel
et al., 2006). Such control laws are know in
the aeronautical either as Stability Augmentation
System (SAS) if is necessary to change the damping and the natural frequencies of aircraft modes
or Control Augmentation System (CAS) if the
purpose is to control the modes providing the pilot with a particular type of aircraft response.
In this paper, two strategies are used to design
and simulate longitudinal and lateral directional
controllers: the gain scheduling and backstepping
techniques.
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Belo Horizonte, MG, 20 a 24 de Setembro de 2014
et al., 1984), (Lane and Stengel, 1986), and
(Enns et al., 1994). Most importantly, Harkegard
and Glad (2000) introduced a backstepping design procedure for flight path angle control, and
Harkegard and Glad (2001), investigated backstepping as a new framework for a complete nonlinear flight control design.
More recently, important research of backstepping technique for aircraft flight control is being done at Delft University of Technology, where
(Sonneveldt, 2010), proposed the design of a stability and control augmentation system for a modern fighter aircraft using a nonlinear adaptive
backstepping method with handling qualities evaluation. Another work was produced at Aeronautics Institute of Technology (ITA) Brazil by
(Morales et al., 2011), in which a control system based on backstepping for a flexible medium
transport aircraft was developed. Finally, a survey of adaptive backstepping control and safety
analysis for modern fighter aircraft is presented in
(Oort, 2011).
The main contribution of this paper is the
project of Backstepping technique and the comparison with Gain Scheduling as an initial step
aiming an industrial implementation.
2
marized below.
1
(X + FT )
m
1
(1)
= pω − ru + g sin φ cos θ + Y
m
1
= qu − pυ + g cos φ cos θ + Z
m
u˙
= rυ − qω − g sin θ +
v˙
w˙
p˙ =
q˙
r˙
=
=
(c1 r + c2 p)q + c3 L + c4 (N + qHeng )
c5 pr − c6 (p2 − r2 ) + c7 (M − rHeng )(2)
(c8 p + c2 r)q + c4 L + c9 (N + qHeng )
φ˙
θ˙
=
p + tan θ(q sin φ + r cos θ)
=
ψ˙
=
p cos φ − r sin φ
cos φ
sin φ
+r
q
cos θ
cos θ
V˙ T
=
α˙
=
β˙
=
uu˙ − v v˙ + ww˙
VT
uw˙ − wu˙
u2 + w2
vV
˙ − v V˙
v 2 cos(β)
(3)
(4)
where
2
Γc1 = (Iy − Iz )Iz − Ixz
Mathematical Modelling
Γc2 = (Ix − Iy + Iz )Ixz
For his fidelity and availability, the aircraft model
used in this paper is the F-16 Fighter shown in
Figure 1 (Lewis and Stevens, 1992).
Γc3 = Iz
Γc4 = Ixz
Γc9 = Ix
Iz − Ix
Iy
Ixz
c6 =
Iy
1
c7 =
Iy
2
Γc8 = Ix (Ix − Iy ) + Ixz
c5 =
2
Γ = Ix Iz − Ixz
where u, v and w are velocity components in body
frame, θ, φ, ψ are attitude angles including pitch,
roll and yaw respectively, p, q and r are rolling,
pitching and yawing moments in body frame, α
is angle of attack, β is sideslip angle, VT is total
velocity, Heng is the angular moment of engine and
X, Y and Z are the aerodynamical forces and L,
M and N are aerodynamical moments.
The aerodynamic moments and aerodynamic
forces can be expressed as:
X = qSCXT (α, β, p, q, r, δ, . . . )
..
.
L = qSbClT (α, β, p, q, r, δ, . . . )
..
.
Figure 1: Reference systems and main aircraft
variables
2.1
where q is the aerodynamic pressure. CXT and
ClT are usually obtained from wind tunnel data
and flight tests. One example of aerodynamic coefficient are shown in Figure 2. Finally δe , δa and
δr are the elevator, aileron and rudder deflection
of aircraft control surfaces. Details can be consulted in (Huo, 2012).
Nonlinear model
The set of dynamic equations representing F-16
fighter model consists of 12 differential equations,
(i.e.) three velocity equations, three moment
equations, three kinematic equations and three
wind axis equations. These equations are sum-
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Anais do XX Congresso Brasileiro de Automática
Belo Horizonte, MG, 20 a 24 de Setembro de 2014
3
Gain Schedule Controller
The Gain Schedule procedure consists in applying
the LQT1 at linear models generated at different
operation points, what results in a look-up table of
gains. So, the controller’s gains are settled using
an interpolation between the current state of the
system and the equilibrium states.
3.1
The initial goal of aircraft control is to drive any
initial condition error to zero (regulation). One
way to do that is selecting the control input u(t)
to minimize the following performance index
Z
¢
1 ∞¡ T
x Qx + uT Ru dt
JLQR =
(6)
2 0
Figure 2: Lift coefficient of F-16
2.2
Linear models
where Q and R are symmetric positive semidefinite weighting matrices. The LQR 2 is one the
most fundamental applications of the Optimal
Control. In fact, this method determines simultaneously all the gains in a MIMO3 systems (Lewis
et al., 1993).
The LQR problem can be transformed into
an LQT problem through the change of variables.
Instead of regulating the system around an operation point, the new objective is to track a reference
input r(t).
A common practice in flight controls, when designing and analyzing control systems, is to linearize
the aircraft equations around an operating point.
Therefore, nonlinear models of aircraft (1)-(4) are
linearized for different operating points of its envelope, usually, parameterizing by airspeed and
altitude.
The longitudinal and lateral/directional axis
linear models obtained at different operation
points to design and analyze the gain scheduling
controllers are represented by standard state equation given by:
x(t)
˙
=
y(t) =
Ax(t) + Bδ(t)
Cx(t)
Design
(5)
where A ∈ R4×4 is the state matrix and C ∈
R4×4 is the output matrix. For longitudinal axis
x = [VT AS α θ q]T and the matrix B ∈ R4×1
has as to input the elevator deflection δe . For
lateral/directional axis x = [β φ p r]T and the
matrix B ∈ R4×2 has as to inputs the aileron and
rudder deflections [δa δr ].
2.3
Figure 4: Compensator and linear model aircraft
structure
Actuator models
The actuators models are represented by a transfer function, with deflection saturation and rate.
Figure 4 shown the model of actuators for elevator, aleiron and rudder.
In the control structure shown in Figure 4,
the dynamics of aircraft and the compensator are
given by:
x˙ =
Ax + Bu + Gr
y
Cx + F r
=
(7)
where the state x(t) ∈ Rn , control input u(t) ∈
Rm , the reference r(t) ∈ Rq and the output y(t) ∈
Rp .
Figure 3: Actuator of F-16 fighter
1 Linear
Quadratic Tracker
Quadratic Regulator
3 Multiple Inputs and Multiple Outputs
The actuators have a deflection saturation of
±25[o ] and ±60[o /s] of response velocity.
2 Linear
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The control input is given by:
u = −Ky = −KCx − KF r
For the lateral tracker controllers, a Proporcional Integral compensator for φ and proporcional compensator for r were considered. An SAS
for p was also implemented, resulting in the structure presented in Figure 6.
(8)
Using the equations (7) and (8) the close loop
system is:
x˙ = (A − BKC)x + (G − BKF )r
= Ac x + Bc r
(9)
where the gain K to be determined corresponds
to the optimal gain that minimizes the following
performance index:
J=
1
2
Z
∞
h
³
´ i
tk x
˜T P x
˜+x
˜T Q + C T K T KC x
˜ dt (10)
0
where x
˜ comes of the difference between x(t) and
steady state value x.
The value of (10) for a given value of K is
given by successively solving the nested Lyapunov
equations
0
=
g0 ≡ ATc P0 + P0 Ac + P
0
..
.
=
g1 ≡ ATc P1 + P1 Ac + P0
Figure 6: φ − r Controller
The matrices A, B and C are given by linearization of the F-16 model. These elements will
allow the application of the LQT method.
0 = gk−1 ≡ ATc Pk−1 + Pk−1 Ac + Pk−2
T
T
T
For the scheduling, the F16 fighter was
0 = gk ≡ Ac Pk + Pk Ac + k!Pk−1 + Q + C K RKC
trimmed
for VT (True Air speed) ranging from
(11)
350 ft/s to 600 ft/s, with steps of 50 ft/s. Linearized models were generated for each condition.
Then,
The LQT was applied at each of these models. An
1
extensive study was performed in order to select
(12)
J = tr(Pk X)
2
the appropriate weights of the cost function.
The gains of integrator and the states at each
A minimization routine can be used to find
equilibrium
point were stored in 7 one-dimensional
the optimal gains used (11) and (12) to evaluate
look-up
tables,
using VT as interpolation variable.
the performance index for a specified value of gain.
The Figure 7 shows how the longitudinal gains
varying with the True Air Speed.
3.2 Results
For the longitudinal tracker controller, a Integral
Compensator was considered. An SAS for angle
of attack and pitch rate was also implemented,
resulting in the structure presented in Figure 5.
Figure 7: Longitudinal gains scheduled gains
The Figure 8 shows how the lateral directional
gains varying with the True Air Speed.
Figure 5: α Controller
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Assumption 3 The control surface deflection
has not an effect on the aerodynamic force component b3 (x1 ).
Now, introducing the first virtual variable z1
and its derivative z˙1
= x1 − xd1
= b1 + b2 x2 − x˙ d1
z1
z˙1
(14)
(15)
Considering the following Candidate Lyapunov Function (CLF):
Figure 8: Lateral and directional gains scheduled
gains
4
=
V˙ 1
=
=
1 2
z
2 1
z˙1 z1
z1 (b1 + b2 x2 − x˙ d1 )
(16)
(17)
Hence, choosing a stabilizing feedback term,
the virtual control is:
Backstepping Technique
£
¤
xd2 = b−1
−k1 z1 − b1 + x˙ d1
1
Backstepping is a systematic, Lyapunov-based
method for nonlinear control design. The name
backstepping refers to the recursive nature of the
design procedure. The design procedure starts
at the scalar equation which is separated by the
largest number of integrations from the control
input and ’steps back’ toward the control input.
Each step an intermediate or ’virtual’ control law
is calculated and in the last step the real control
law is found (Sonneveldt, 2010).
4.1
V1
The second virtual variable is defined as:
z2
z˙2
The backstepping project can be viewed as two
time-scale approach because the fast states p, q
and r are used as control inputs for the slow states
α, β and φ (Lee and Kim, 2001).
Substituting all forces and moments into the
nonlinear equations (2)-(4) and rearranged to separate the slow states variables x1 = [α, β, φ]T ,
the fast states variables x2 = [p, q, r]T , the control variables u = [δe ; δr ; δa ]T and x3 = [θ, ψ]T is
obtained:
=
=
b1 (x1 , x3 ) + b2 (x1 , x3 ) x2 + b3 (x1 ) u
c1 (x1 , x2 ) + c2 x1 + c3 (x1 ) u
x˙ 3
=
d1 (x1 , x3 )
x2 − xd2
c1 + c2 x2 + c3 u − x˙ d2
=
=
(19)
(20)
Then, the following Lyapunov Function is introduced:
Design
x˙ 1
x˙ 2
(18)
V2
=
V˙ 2
=
1
1 T
z1 z1 + z2T z2
2
2
∂V2
∂V2
z˙1 +
z˙2
∂z1
∂z2
(21)
(22)
Substituting equations (14), (15), (19), (20)
and (18) in the equation (22) and performing operations to make negative definite, the following
control law is obtained:
u = c−1
˙ d2 − G)
3 (−k2 z2 + x
(23)
where G depends on the derivatives of xd2 and is
given by:
G
(13)
Bellow the most important assumptions
adopted in the design process:
∂xd2
[b1 + b2 x2 ] −
∂x1
£
¤
∂xd2
k1 x˙ d1 + x
¨d1
[d1 + d2 x2 ] − b−1
2
∂x3
= c1 + c2 x2 −
The equation (23) was implemented.
d
Assumption 1 The desired
°£ trajectory
¤° x1 =
d
d
d
d
d
d °
°
[α , β , φ ] is bounded as
x1 , x˙ 1 , x
¨1
≤ cd
where cd ∈ R is a known constant and k . k denotes the 2-norm of a vector or matrix.
4.2
Results
The controller structure is shown in Figure 9.
In order to obtain differential commands to
satisfy assumption 1 a third-order filter filter is
used in the reference. The controller design constants chosen are: k1 = 15, k2 = 8
Assumption 2 The magnitude of θ is bounded
as |θ| ≤ θm < π/2.
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Figure 9: Backstepping controller
5
Simulated Maneuvers
To compare the two control strategies presented,
two maneuvers based on inputs from pilot to the
attack angle and roll angle were defined. These
maneuvers allow change the speed aircraft within
the specified range. In both maneuvers the reference of sideslip angle β is zero and the time of
simulation corresponds to 30 seconds.
5.1
Figure 11: Attack angle α desired trajectory
First maneuver
The aircraft starts trimmed with airspeed equal to
590 ft/s, with xcg position equal to 0.28 and 15000
ft of altitude. The throttle are always kept at its
initially trim value. The aircraft needs to track
the roll angle φ reference as shown in the Figure
10. The angle of attack α reference corresponds
to trim value equal to 2.95 degrees as shown in
Figure 11.
Figure 12: Roll angle φ
The Figure 13 shown the attack angle α response. In this figure we can see that both controllers follow the constant reference.
Figure 10: Roll angle φ desired trajectory
5.2
Results
Figure 13: Attack angle α
The Figure 12 shown the result of roll angle φ for
both controllers. In this figure it is possible to verify that the backstepping response in blue is faster
than gain scheduling response in red. Additionally the gain schedule response present a overshoot
than backstepping when compared with backstepping response.
The Figure 14 shown the sideslip β angle response, as expected.
The elevator deflection for both techniques is
illustrated in the Figure 15, in this result is verified
that the deflections are similar in magnitude, but
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Figure 17 shown the rudder deflection, the
magnitude is equal to 0.3 degrees.
Figure 14: Sideslip angle β
it is important to highlight a problem of oscillation
at the beginning of the simulation.
Figure 17: δr Rudder deflection
5.3
Second maneuver
The aircraft starts trimmed with airspeed equal
to 580 ft/s, with xcg position equal to 0.32 and
15000 ft of altitude. The throttle are always kept
at its initially trim value. The aircraft needs to
track the attack angle α reference as shown in the
Figure 18. The roll angle φ reference corresponds
to trim value equal to 0 degrees.
Figure 15: δe Elevator deflection
The aileron deflection is illustrated in the Figure 16, in this result is verified the difference in the
magnitude between both techniques. The backstepping is two degrees greater than gain scheduling technique .
Figure 18: Attack angle α desired trajectory
5.4
Results
Figure 19 shown the roll angle response, as waited
is zero.
The Figure 20 shown the result of attack angle α for both controllers. It is possible to verify
that the backstepping response is faster than gain
scheduling response. Additionally, in both controllers do not exist a overshoot.
The Figure 21 shown the sideslip β angle response, as expected it is near zero.
Figure 16: δa Aileron deflection
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Figure 19: Roll angle φ
Figure 22: δe Elevator deflection
Figure 20: Attack angle α
Figure 23: δa Aileron deflection
Finally, Figure 24 illustrate the rudder deflection, equal than the first maneuver its magnitude
is near zero.
Figure 21: Sideslip angle β
The elevator deflection is illustrated in the
Figure 22, in this result is verified the difference
in the magnitude between both techniques. The
backstepping is seven degrees greater than gain
scheduling technique.
Figure 23 shown the aileron deflection, as
waited, in this maneuver the magnitude is near
of zero.
Figure 24: δr Rudder deflection
6
Conclusions
From the results, it can be seen that the nonlinear
technique, with knowledge of the aircraft model,
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was able to achieve a significantly better response
for attack and roll angle control. The simpler design process (only two gains to adjust for the entire
flight envelope) coupled with the global stability
from Lyapunov theory are two very desirable characteristics of the backstepping technique.
It is important to note that the same Backstepping designed controller could be used in another aircraft by simply substituting the chosen
model and readjusting the two gains. For the gain
scheduling approach, on the other hand, the new
model would have to be linearized and new gains
would have to be found for each of the chosen operation conditions.
To implement the technique of Backstepping
in real time, it is necessary the precise knowledge
of the aerodynamic coefficients, therefore, this is
possible only after a matching campaign, which
would allow the use of coefficients tables coming from the wind tunnel experiments and flight
tests. Another alternative is to estimate these coefficients using the signals coming from the available sensors on the aircraft, this should take care
errors estimation. Since from the point of certification view, it can be mentioned that nonlinear techniques are already certified in military aircraft, but civil certification processes are not defined still. This should happen when non-linear
techniques are flown and subject to official certification.
Harkegard, O. and Glad, S. (2001). Flight control
design using backstepping, NOLCOS 2001,
St. Petersburg, Russia.
Holzapfel, F., Heller, M., Weingartner, M., Sachs,
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Huo, Y. (2012). Model of F-16 Fighter Aircraft,
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Khalil, H. K. (1996). Nonlinear Systems, Prentice
Hall Inc.
Lane, S. H. and Stengel, R. (1986). Flight
control design using nonlinear inverse dynamics, American Control Conference, 1986,
pp. 587–596.
Lee, T. and Kim, Y. (2001). Nonlinear adaptive
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