Waypoint Tracking of Unmanned Aerial Vehicles Using Robust 2 /H H∞ Controller
A. Baudkoobeh1 and M. Farrokhi.2
Iran University of Science and Technology, Tehran, 16846
Abstract: This paper investigates the use of 2 /H H∞ robust controller in tracking
trajectories for Unmanned Aerial Vehicles (UAVs). First, a trajectory is generated off-
line by solving a trajectory optimization problem, which incorporates the mission
constraints and the dynamic model of the UAV. Next, a controller is designed using the
2 /H H∞ robust control method to track the reference trajectory on-line while
preserving the mission constraints. The control design process involves two steps: a
convenient system identification and systematic robust controller design. Simulation
results show effectiveness of the proposed controller in comparison with the optimal
preview controller, which has gained more popularity recently in navigation of UAVs
to track a reference path despite wind disturbances and measurement noises.
Keywords: Unmanned aerial vehicles, Path tracking, Robust control, Waypoint
tracking
I. Introduction
NE of the viability requirements for Unmanned Aerial Vehicles (UAVs) is the low
altitude flight. This issue is defined as the terrain following problem. The first stage in
the terrain following is the trajectory optimization, which can be solved by either direct
or indirect methods [1]. Indirect methods invoke necessary conditions of optimality to
obtain solutions. Alternatively, numerical solutions can be employed via the nonlinear
searching of the discrete space, known as direct methods. The solution of the trajectory
1 Department of Electrical Engineering, [email protected] 2 Department of Electrical Engineering, and Center of Excellence for Power System Automation and Operation, [email protected].
O
optimization problem generates a vector of open-loop controls and optimal state
trajectories.
A general way to make a UAV to fly along a reference trajectory is to design guidance
algorithms in the horizontal and vertical planes separately. In this case, the vertical
guidance is for terrain following or to achieve the height trajectory while the horizontal
guidance is often designed for threat avoidance, terrain masking, or arriving at the target
location at the desired time [2]. Following the trajectory optimization stage, it is
necessary to design a feedback controller to track the reference trajectory on-line. This
feedback controller must be robust against disturbances (e.g. winds), measurement
noises, and modeling errors. In this manuscript, after introducing different layers of a
terrain following problem for a UAV, the use of 2 /H H ∞ robust controller is examined
to track the reference path. However, before the controller design, the UAV system
should be modeled properly.
Today, there are several techniques, which address system identification and
controller design for UAVs [3]. A comprehensive effort towards identifying UAV
models is currently being studied at the Texas A&M University [4, 5]. Balakrishnan and
Wang have used the Texas A&M model to develop a gain-schedule controller [6, 7].
Many techniques for the controller design are developed in literatures such as the
feedback linearization [8], the sliding-mode controller [9], the Linear Quadratic
Regulator (LQR) [2], and the model predictive controller [10, 11]. While these control
techniques sometimes can be proven to work (i.e., stabilize the system or provide
acceptable performances), they can be quite conservative, especially when uncertainties
in the model and disturbance signals are considered.
A number of papers tackle only longitudinal problems (e.g. the height, the forward
position, and the pitch angle). The focus of this research is on the waypoint tracking in
the vertical plane. An optimal preview controller for the longitudinal flying in the
vehicle guidance problem is defined and solved in [12] using the LQR framework and
state-space augmentation techniques similar to those first used by Tomizuka [13, 14]. In
this method, the vehicle uses a preview controller to track a pre-computed optimal
trajectory. A tracking error of just 20m is reported for a preview length of 1.6km with
600 preview points and a nominal speed of 272 m/s. Farooq et al. [15] extend this work
to the output feedback case. Li et al. [16] consider another full information longitudinal
guidance problem, where the authors design an optimal version of the preview
controller to track the reference trajectory. Tracking of longitudinal trajectories is
considered by Paulino et al. in [17] and [18] for an underwater autonomous vehicle and
a helicopter, respectively. In all these researches, the optimal preview control is proved
to work for the tracking problem. Cohen and Shked [19] have focused on H ∞ version
of the preview control. They present the linear discreet time H ∞ solution for the output
tracking. Hazell and Limbeer [20, 21] have cast the problem in a unified framework
where the continuous and discreet time 2H and H ∞ solution are derived using calculus
of variations.
The controller design methodology in this paper addresses a systematic robust
controller design problem for a longitude trajectory-tracking controller. Existing system
identification techniques are integrated with the controller design process. The
contribution of this research are three folds: 1) describing a total system architecture for
modular layers of the terrain following problem for the UAV, 2) a systematic robust
controller design procedure for the nonlinear UAV model, and 3) a systematic method
for selecting the robust controller weighting function.
This paper is organized as follows. Section II describes the system architecture for
the modular layer of UAVs in the terrain following flight. Sections III and IV present
the system identification and the robust tracking design method, respectively. Section V
explains application of the derived strategy to a nonlinear UAV system. Section VI
reviews the optimal preview controller. Section VII illustrates Simulation results of the
proposed robust controller and the preview controller. Finally, section VIII provides
discussions and draws conclusions.
II. Problem Statement and Method of Solution
As shown in Fig. 1, the overall system architecture for a UAV consists of five
layers [22]: the Waypoint Path Planner (WPP), the Dynamic Trajectory Smoother
(DTS), the Trajectory Tracker (TT), the Longitudinal and Lateral Autopilots, and the
UAV.
The WPP generates waypoint paths (straight-line segments), which can be
determined according to the Digital Terrain Elevation Data (DTED), the UAV’s
location, the target location, and threat sites. The DTS produces a smooth and feasible
time-parameterized reference trajectory through the waypoints. The TT generates the
velocity command CV , the heading command Cψ , and the altitude command Ch to the
autopilot based on the desired trajectory. The autopilot, then, uses these commands to
control the elevator ( )e tδ , the aileron ( )a tδ , the rudder ( )r tδ , and the throttle ( )t tδ of
the UAV [23]. For the UAV equipped with the standard autopilot, the resulting
autopilot models of UAV are assumed first order for the heading and second order for
the altitude [24].
Fig, 1: System Modular Architecture
In the reminder of this paper, the focus is on the longitude autopilot design for
holding the reference altitude based on the assumption that the UAV is equipped with
sensors, which can provide the actual altitude height of the terrain beneath the flight.
Hence, the control-oriented goal is to design the altitude controller. To this end, a robust
controller for the nonlinear UAV system will be designed. The design process combines
a common technique for the system identification and a controller synthesis process into
a complete systematic procedure. The process is broken into two distinct parts: system
identification and controller synthesis. First, a mathematical nonlinear system for the
fixed-wings UAV is considered here. Then, using the system identification methods, a
series of linear models around the operation points will be extracted. The average of
these linear systems is selected as the nominal plant. The upper bound of deviations
between the other models and the nominal plant determines the uncertainty function. In
other words, this uncertainty is the error between the nonlinear system and the nominal
model in a bound around the operation point. This nominal plant and the uncertainty is
then used to design a 2 /H H∞ controller for the tracking problem. Another important
issue for selecting the 2 /H H∞ controller is that performance of the close-loop system
depends on the selection of performance weighting functions that is often difficult to
choose. However, general guidelines for their development are presented. The technique
selected in this paper has been developed by Franchek [25] where the weighting
functions are selected such that time domain tolerances are enforced.
The close-loop tracking performance specifications include constraints for the
control effort and for the allowable tracking error. The constraints for the time domain
control input about the nominal effort can be written as
( ) 0u t tκ≤ ∀ > , (1)
while the tracking deviation constraint is
( ) 0e t tδ≤ ∀ > , (2)
where e(t) is the tracking error of the close-loop system and κ and δ are positive
constants. The specific goal in this paper is to design a feedback controller that meets
close-loop performance specifications given in Eqs. (1) and (2).
III. System Identification
To design a robust controller, the linear dynamic model of the plant will be
estimated using input–output data of the system. Nonlinear plant will be stimulated
using chirp signals (other frequency–rich signals could also be selected) with
frequencies around the operation frequency of the flying vehicle. The resulted input–
output data are used to estimate parameters of linear models. Different criteria such as
minimum norm of the error, the frequency and time response of model are used to select
the fittest linear models at each operation conditions. These estimations create a family
of experimental linear models representing different operation conditions. Additive or
multiplicative uncertainty can be selected to capture uncertainties of the plant.
Assuming multiplicative uncertainties, the uncertain plant model is
( )2 0ˆ( ) 1 ( ) ( ),P s W s P s= + ∆% (3)
where 0 ( )P s is the selected nominal plant, ∆ is an allowable variable stable transfer
function satisfying
1≤∆∞
, (4)
and 2ˆ ( )W s is a fixed stable transfer function, which may be determined by [26]
20
( ) ˆ1 ( ) ,( )
P j W jP j
ωω ω
ω− < ∀ (5)
Further discussion on this subject may be found in [26]. The Uncertainty weighting
function provides the upper norm bound on system uncertainties. It is advantageous to
select 0 ( )P s such that it minimizes the uncertainty weighting of the plant, ultimately
leading to a less conservative design.
The next section presents synthesis of a robust controller for the equivalent
linear system.
IV. Tracking Controller 2 /H H∞ Weighting Function Selection
Presented in this section is the development of performance weights for the
tracking control problem. Consider the tracking problem with the multiplicative
uncertainty shown in Fig. 2. In this figure, ( )RG s represents the reference dynamics
(assumed stable), u(s) is the scalar controller output variation about a nominal effort,
R(s) represents a reference command for the altitude h , e(s) is the scalar error variation
of the system about zero, q(s) and p(s) are uncertainties in the input and output,
respectively, K(s) is the controller, and Y(s) is the scalar output of the system. The plant,
as described by Eq. (3), is assumed proper without any hidden unstable modes. The
following theorems describe selection of performance weights to enforce the SISO
tracking control effort and tracking deviation constraints, respectively. Fig. 3
incorporates performance weights into the feedback structure for the tracking problem
[27].
Fig. 2: Feedback tracking system with multiplicative unstructured uncertainty
Fig. 3: Feedback structure for tracking with uncertainty
Theorem 1. Assume the feedback system in Fig. 2 has zero initial conditions and
2( )RG s RH∈ and 0 2 2ˆ( )(1 ( ) )P s W s RH+ ∆ ∈ are proper without any hidden unstable
modes. Then, ( )u t κ≤ for t > 0 when r(t) is a step input of the altitude provided that
2 0( ) ( ) 1,W j T jω ω∞
< (6)
where 2 ( )W s is a stable and minimum phase transfer function (not necessarily proper)
satisfying
2 20
ˆ ( ) ˆ( ) ( ) ,( )
RhG jW j W jP j
ωω ω
κ ω≥ + (7)
where h h= Λ in which Λ is a scaling constant, which justifies the enforcement of
time domain specifications through frequency domain amplitude constant and 0 ( )T s is
the complimentary sensitivity function of the nominal system
00
0
( ) ( )( )1 ( ) ( )
K j P jT jK j P j
ω ωω
ω ω=
+. (8)
Proof. Appling change of variables, the proof is similar to the proof given by Franchek
for selection of the weighting function of the regulation control H∞ [25]. W
Theorem 2. Assume the feedback system in Fig. 2 has zero initial condition and
2( )RG s RH∈ and 0 2 2ˆ( )(1 ( ) )P s W s RH+ ∆ ∈ are proper without any hidden unstable
modes. Then, ( )e t δ≤ for t > 0 when r(t) is a step input of the altitude provided that
1 0 2 0( ) ( ) ( ) ( ) 1,W j S j W j T jω ω ω ω∞
+ < (9)
where
1
ˆ ( )( ) ,RhG jW j ωω
δ= (10)
2 ( )W s and 0 ( )T s are defined in Theorem 1, and 0 ( )S s is the sensitivity function of the
nominal system
00
1( ) .1 ( ). ( )
S jK j P j
ωω ω
=+
(11)
Proof. Applying change of variables, the proof is similar to the proof given by Franchek
for selection of the weighting function of the regulation control H∞ [25]. W
Corollary: If the control effort specification is infinite (i.e. κ = ∞ ), then
2 2ˆ( ) ( )W s W s≥ and 2 ( )W s would only depend on the uncertainty weighting for all
frequencies.
Proof: According to (7) and under conditions of Theorem 1, the weighting function
2 ( )W s depends on the uncertainty weighting 2ˆ ( )W s , the reference dynamic and
magnitude ˆ ( )RhG s , actuator saturation κ , the plant dynamic 0 ( )P s , and the scaling
constant Λ . If the control effort specification is selected relatively large, then the proof
is straightforward.
The weighting function 1( )W s emphasizes the most important frequencies in the
reference command. Therefore, the shape of 1( )W s is similar to the reference dynamics
( )RG s . This requires that the 2 /H H∞ controller to minimize the energy between the
reference command and tracking deviations at those frequencies where the reference-
command dynamics contains significant energy.
V. Implementation of Robust Altitude Controller Design
A. Nominal Plant and Uncertainty Function of UAV
A model of the UAV dynamic is required for the altitude controller design. To this
point, a nonlinear mathematical representation of the UAV system is needed. In this
work, the method used to derive the motion equation of fix wings UAV closely follows
that of the reference [28]. The model is developed based on some simplifications. The
first simplification is to decouple the velocity equation. It is assumed that the velocity is
controlled independently and the thrust may be regulated to follow a prescribed velocity
profile using the autopilot, as the velocity profile might need to be tightly controlled due
to tactical requirements. More precisely, the velocity affects the turn radius and hence,
the pitch acceleration demand. The second simplification is to remove the downrange
effects ( xyR ) from equations. This is because it is more convenient the independent
variable in differential equations to be the downrange rather than the time. The
downrange is a monotone variable and is a suitable choice of the independent variable.
The time-to-downrange transformation is simple and is implemented by dividing each
equation by the downrange rate [12]. This simplification reduces the state dimension
and the state vector. After considering several Cartesian coordinate systems for the
UAV modeling, the following nonlinear UAV model in the state space can be derived:
1
0
3
0
0 0
( )( / ) cos
( / )cos(sin ) cos ( / ) cos sin
tan sin ta
q r r p
q
q r p p
p l l q l r l l r l p i qrq m m q m p m h m g vr n n q n r n p n p i pq
q p z h g vy p r g v
p q r
h
β βα α
α α α α
β α
α
β β αα β θ
β αα β α θβ β α α α θ ϕϕ θ ϕθ
+ + + + + − + − + − + + + + − − + + − = + + − +
+ +
&&&&&
&&
000
000
0n cos 0 00
cos sin 0 00cos( )sin( ) 0 00
ra
e
raa
rr
rae
llm
nnz
yy
q rv
δδ
δ
δδ
δ
δδ
δδδ
θ ϕφ φ
φ θ α
+
−
−
(12)
where , , , ,a a a a a a q q a e e a el l l m m m m m m m m m zδ δ δ δ δ δα α= + ∆ = + ∆ = + = +& & &
.n n nδα αδ αδ α= + ∆ The state and input vectors are
[ ] [ ],T Ta r ex p q r h uα β φ θ δ δ δ= =
where ,a rδ δ , and eδ denote the deflections of the aileron, the rudder, and the elevator,
respectively. To control the longitudinal motion of the vehicle in vertical plane, the vehicle is controlled using the fin elevator eδ , which generates a net torque. The torque
generates lift by developing an appropriate angle of incidence α. Therefore, By selecting
eδ as the input and h (the altitude) as the output and setting other two inputs (i.e. aδ
and rδ ) to zero, the linear relation between the input and output to design an altitude
controller is verified. Even though only the longitudinal motion of the vehicle (i.e. , , ,h qθ α ) is considered here, the interaction of other channels with the model is not
neglected (i.e. the effects of , , ,p r β φ on the elevation channel has also been taken into
account). Equation (12) is non-minimum phase and unstable. To identify the system, a Banded-Input-Banded-Output (BIBO) system is required. By selecting 0.002k = − as the feedback gain, linear identification of the BIBO close-loop system can be assured.
As it was mentioned in Section III, chirp signals with frequencies from 0.1 Hz to 5 Hz (as shown in Table I) are applied as frequency-rich inputs to the close loop-system. The results implicated by the Box Jenkins (BJ) [29] linear model yields acceptable estimation of the actual dynamic of the system. The norm of error between the BJ models and the actual system is illustrated in Table I.
The frequency response of these models is shown in Fig. 4. It is clear from the frequency responses that a time delay exists in the system. This delay is associated with the non-minimum phase nature of the aerial vehicle. The delay may be incorporated into the nominal plant by the Padé approximation. However, the Padé approximation would increase the order of the nominal plant and therefore, a higher-order controller would be needed. In this paper, the delay will be incorporated into the uncertainty weighting function as discussed by Dole et al. in [26]. The selected nominal plant transfer function is
0 3 2
8 22.16( )2 51.2 34.5 112.5
sP ss s s
+=
+ + + (13)
Table 1: The Norm of error between the actual model and the linear BJ model at different
frequencies
5 3 2 1 0.5 0.4 0.3 0.2 0.1 f BJ43311 BJ22111 BJ21141 BJ21141 BJ21141 BJ31140 BJ22221 BJ33241 BJ33331 Model
15.41 17.42 17.60 13.6 23.15 32.69 38.61 32.97 54.81 Error
Fig.4: system identification of nominal linear plant
Nonlinearities and uncertainties must be incorporated into the model. For this design,
multiplicative uncertainty is selected to capture system nonlinearities and the system
delay. Multiplicative uncertainty may be written as shown in Eq. (3) with the condition
stated in Eq. (4). The time delay incorporated into the uncertainty weighing may be
determined by expression given in Eq. (5) (Doyle et al. [26]). The experimental
multiplicative uncertainty for this system is shown in Fig. 5 and in equation form as
2
2 2
(0.25 0.7033 0.4688)( 20)( )(0.002 0.09 1)( 20)
s s sW ss s s
+ + −=
+ + + (14)
Fig.5: Uncertainty of the nominal plant
B. Weighting Function Selection
Now the nominal plant and uncertainty function have been identified, the 2 /H H∞
altitude controller may be synthesized. The block diagram of the closed-loop altitude
controller can be written in the form shown in Fig. 2 and with the proper selection of
weighting functions, transformed into the form shown in Fig. 3.
The control orientated goal is to design a tracking controller which maximizes the
system responsiveness to a reference path while minimizes the control effort. Therefore,
Theorems 1 and 2 are implemented to determine the proper weighting function while
specification for the time domain tracking deviation (δ ) is set to a relatively large
value. The specification for the time domain control effort (κ ) depends on the actuator
deflection and is selected equal to 0.6 rad., which is assumed to be the maximum
allowable deflection of the elevator.
In order to implement Theorems 1 and 2, the reference dynamic ( )RG s must be defined
first. The reference dynamic for this application are selected to capture the reference
command dynamic in a slightly larger frequency range than the plant would respond,
i.e.
2
1( )( / 50 1)RG ss
=+
(15)
The maximum value of h (i.e. the maximum allowable altitude above the terrain
according to the flight mission) was found to be 120. Hence, According to (10) the final
weighting function is
1 2
0.012( )( / 50 1)
W ss
=+
(16)
According to the Corollary, 2 ( )W s can be selected as
20.25( )
/ 150 0.02sW s
s+
=+
(17)
Now, the 2 /H H∞ robust controller can be designed based on the nominal plant,
the uncertainty function, and weighting functions. The controller is developed
simultaneously by minimizing 2H and H∞ performances of the system and considering
the constraint on close-loop poles. The H∞ performance enforces robustness to the
model uncertainty and to frequency-domain specifications such as the bandwidth and
the low-frequency gain. The 2H performance is useful to handle stochastic aspects of
the system such as measurement noises and random disturbances. In addition,
constraints on closed-loop poles help to avoid fast dynamics and high-frequency gain in
the controller that in turn facilitates its digital implementation [30]. The resulted
controller is
4 3 2
5 4 3 2
0.0001618 s 0.0003583 s 0.003951 s 0.0002789 s 9.8e 006( )s + 3.493 s + 28.07 s + 32.59 s + 11.65 s
k s − − − − − −= (20)
Implementation results of the controller on the nonlinear model are presented in
Section VII.
VI. Designing Optimal Preview Controller
In this section, the methodology of designing an optimal preview controller, which
has found more popularity recently in the control of flying vehicles in terrain following
flight, is briefly reviewed [12-21]. Then, the results will be compared with results of the
proposed robust controller. To implement an optimal preview controller, it is necessary
to linearize the nonlinear model of UAV in Eq. (12) at one operating point only. The
linearization procedure yields the following system matrices:
0 0
0 0
0 0 0 0 0 00 0 0 0 0 0
0 0 0 0 0 00 1 0 0 0 0 1 0
, ,sin 0 cos 0 0 0 0 0 0
1 0 0 0 0 / 0 0 0 00 1 0 0 0 0 0 0 0 00 0 cos( ) 0 0 0 cos( ) 0 0 1
p q r
q
p q r
l l l lm m m m
n n n nz z
A B Cy
g v
v a v
β
α α δ
β
α δ
βα α
α
= = = −
− − −
, 0
T
D =
(18)
For the operating point, it is assumed that the altitude is equal to 60 m and the velocity
is equal to 0.52 mach (~172m/s); other parameters are given in Table II. It can be easily
verified that the linearized system is controllable as well as observable. The linearized
system matrices (A, B, C, D) then are discredized using the zero-order-hold method with
the sampling time of 0.01 sec. Equations of the discrete-time system can be expressed as
( 1) ( ) ( )( ) ( ),
x k Ax k Bu ky k Cx k
+ = +=
(19)
where A, B, and C are the discrete-time system matrices and D = 0.
As it was mentioned before, the goal of this paper is to track a desired height profile.
This requirement is formulated by the following infinite horizon cost function:
0
1 ( ) ( ) ( ),2
T T
kJ e k Qe k u Ru k
∞
=
= +∑ (20)
where ( ) ( ) ( )de k y k y k= − is the tracking error and ( )dy k is the reference signal;
matrices Q and R penalize the tracking error and the control energy, respectively.
Matrices of the discrete-time system can be augmented by a command generator
system, which models the preview part of the system as
( 1) ( ) ( )( ) ( ),
d d d d d
d d d
x k A x k B w ky k C x k
+ = +=
(21)
where
0 1 0 ... 0 00 0 1 ... 0 0
, 00 ... ... ... 10 ... ... ... 0 1
( )1( 1)0( 2), .0
( 1)0
d d
d
d
dd d
d p
A B
y ky ky kC x
y k N
=
+
+ = = + −
M M M O MM
MM
(22)
The state of the command generator ( )dx k is composed of sampled values of the
reference signal over the preview horizon of length Np. Matrix dA implements a shift
register operation. The input applied to the command generator system is the reference
signal at the end of the preview horizon ( )d py k N+ . In order to formulate the optimal
preview control problem in a standard LQG form, it is convenient to augment the plant
and the preview system to get
( 1) 0 ( ) 0( ) .
( 1) 0 ( ) 0 dd d d d
x k A x k Bu k w
x k A x k B+
= + + + (23)
Therefore, matrices of the augmented plant and the preview system are
( ) 0 0, , , .
( ) 0 0a a a dad d d
x k A Bx A B B
x k A B
= = = =
(24)
The tracking error can now be written as
( ) [ ] ( ) ( ).d a a ae k C C x k C x k= − = (25)
With these definitions, the cost function can be expressed in terms of the augmented
system as
0
1 ( ) ( ) ( ) ( ),2
T Ta a a
kJ x k Q x k u k Ru k
∞
=
= +∑ (26)
The problem is a discrete linear quadratic regulator problem and the standard theory
[31] can be invoked to obtain the solution as
1 2
1
( )( ) ( ) [ ]
( )
( ) ,
ad
Ta aT
a a
x ku k Kx k K K
x k
K MB SAM B SB R −
= − = − −
=
= +
(27)
where the steady-state Riccati equation is used to obtain S as
.T Ta a a a a a aS A SA Q A SB MB SA= + − (28)
The use of the augmented system induces partitioning of S. If S is partitioned as
11 12
21 22
S SS
S S
=
, (29)
Then, S11 corresponds to the plant-only part of the system and S12 corresponds to the
preview part of the system; S22 is the solution of the plant-only Ricatti equation. The S21
partition can be obtained through a simple linear recursive equation. The gain matrices
K1 and K2 can be easily shown (by expanding (27)) to be 1
1 11 111
2 11 12
( )( )
T
Td
K B S B R BS AK B S B R BS A
−
−
= +
= + (30)
The gain K1 is just the standard LQR gain for the plant-only system and is thus
independent of the preview system. The gain sequence 2K is multiplied by ( )dy k i+ for
0,1, , 1pi N= −… since ( )dx k is composed of Np units of reference signals into the
future. To compute 2K , the S12 partition is needed. This can be obtained by expanding
(29). The details are omitted here, but it can be easily shown that the S12 partition can be
reduced to the following linear relationship [12]:
12 12 12 ,TC d aS S A Q= Φ + (31)
where Φc is the closed-loop transition matrix, which is related to S11, and 12 aQ is the
upper-right partition of Qa. If the ith column of S12 is denoted by S12i ( 1, , pi N= … ) then,
121
12212
12 1
0
.
p
T
d
N
SSS A
S −
=
M (32)
Matrix 12 aQ can be split into columns Q12ia for 1, , pi N= … and simplified as
12 121 122 12[ ]pa a a N aQ Q Q Q= … (33)
TdC QC= − (34)
[ ]1 0 0TC Q= − … (35)
0 0TC Q = − … (36)
By substituting (36) and (32) into (31), the partition S12 can be expressed as 1
12 ( ) , 1,..., .T i Ti C pS C Q i N−= − Φ = (37)
Expression (37) is a recursive solution used to obtain the S12 partition. Once S12 is
computed, the preview gain sequence can be calculated using (30). The amount of the
required preview depends on the closed-loop poles, which are the eigenvalues of CΦ .
For fast closed-loop poles, the preview horizon can be reduced. The poles of CΦ are
always inside the unit circle and the preview gain will decay to zero. After three to five
closed-loop time constants, there is little benefit in increasing the preview length
further.
The simulations were implemented on the nonlinear model, which includes actuator
dynamics with angle limits of 0.6 rad. Tuning of the preview length pN and the
weighting matrices Q and R are required. A preview length of 600 samples, Q = 50.0
and R = 10.0 is used in the simulations. These selections of weightings results in closed-
loop poles at 0.43, 0.81 0.09j± , 0.37± , and 0.99 0.06j± , which are all within the unit
circle. Further discussion on this subject can be found in [12] and [21].
VII. Simulation Results The results of implementing both controllers on the nonlinear model of UAV are
presented in this section. A hard limiter with saturation band of 0.6 rad. is used for the
maximum elevator deflection in the UAV and white noise with noise power of 0.9 is
consider as the measurement noise. Discrete wind gust model [32] is selected as the
disturbance applied to w-axes in the bode frame with start time 100 sec, gust length of
120 m, and gust amplitude of 10 m/sec. Fig 6(a) and (b) show the reference trajectory
and the tracked trajectory without disturbance and noise for the optimal preview
controller. The reference trajectory is a repeating sequence of stairs. Although such
trajectories may not exist in actual circumstances, nonetheless, the goal of such
assumption is testing the designed controller in worst-case scenarios. In the next
scenario, the wind disturbance and the measurement noise is applied to the model. As
shown in Fig. 7, the optimal preview controller is very sensitive to the external
disturbance and noise and it not only yields intolerable error but also needs considerable
time to damp disturbances. However, the result of the proposed 2 /H H∞ robust
controller in Fig. 8 shows the effectiveness of this strategy to track the reference
trajectory and to attenuate external disturbances. Fig. 9 shows that the maximum
tracking error is less than 20 m while the control effort is appropriate.
Fig. 10 shows a generic trajectory (the piecewise linear path) resulted from solving the
optimal trajectory problem as the desired trajectory. The waypoints are shown as small
triangles. In order to produce a smooth trajectory, the Radial-Basis-Function (RBF)
interpolation method [33] is used here. It is important to note that the trajectory is
projected on a 2D plane since only the altitude of the UAV is controlled in this paper.
The controller enables the UAV to track this smooth reference trajectory with a
negligible error.
VIII. Conclusion
In this paper, a terrain following control method for UAVs was described. The
main goal was designing a longitudinal autopilot to track the reference trajectory. To
this end, an 2 /H H∞ robust controller was designed. The design process is two fold: the
system identification and the robust controller design. The proposed controller
methodology was successfully applied to vertical waypoint tracking for a nonlinear
UAV model. Simulation results showed effectiveness of the proposed strategy as
compared with the optimal preview controller to control the UAV to track the reference
waypoints despite wind disturbances and measurement noises.
(a)
(b)
Fig 6: (a) Tracking performance of the preview controller (b) Error between the reference trajectory and the tracked trajectory
Fig 7: Tracking performance of the preview controller
against wind disturbance and measurement noise
Fig 8: Tracking performance of the proposed robust controller
against wind disturbance and measurement noise
Fig 9: Control effort and tracking error of the proposed robust controller against wind disturbance and measurement noise
Fig. 10: Open-loop trajectory resulted from solving the trajectory optimization and the tracked trajectory by the altitude hold autopilot
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