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, ABuadkoobeh@ee.iust.ac.ir 2 Department of Electrical Engineering, and Center of Excellence for Power System Automation and Operation, farrokhi@iust.ac.ir.

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

IX. References [1] R. W. Beard, T. W. McLain, M. Goodrich, and E. P. Anderson, “Coordinated

target assignment and intercept for unmanned air vehicles,” IEEE Trans. Robot. Autom., vol. 18, No. 12, pp. 911–922, 2002.

[2] I. H. Wang and T. W. Hwang, “Horizontal waypoint guidance design using optimal control,” IEEE Trans. Aerospace and Electronic Systems, vol. 38, no. 3, 2002.

[3] M. Fravolini, G. Campa, and M. Napolitano, “Modelling and performance analysis of a machine vision-based semi-autonomous aerial refueling,” Int. J. of Modelling, Identification and Control, vol. 3, no. 4, pp. 357-367, 2008.

[4] J. Valasek and W. Chen, “Observer/Kalman filter identification for online system identification of aircraft,” Journal of Guidance Control and Dynamics, vol. 26, no. 2, pp. 347-353, 2003.

[5] M. Marwaha, J. Valasek, and P. Singla, “GLOMAP approach for nonlinear system identification of aircraft dynamic using flight data,” In Proc. AIAA Atmospheric Flight Mechanics Conference, pp. 1-19, Hawaii, 2008.

[6] F. Wang and V. Balakrishnan, “Improved stability analysis and gain-scheduled controller synthesis for parameter-dependent systems,” IEEE Trans. Automatic Control, vol. 47, no. 5, pages 720-734, 2002.

[7] V. Balakrishnan, “Robust gain-scheduled nonlinear control design for stability and performance,” ONR UACV technical report, Purdue University, 2001.

[8] Y. Jiang, D. Necsulescu, and J. Sasiadek,” Robotic unmanned aerial vehicle trajectory tracking control,” In Proc. 8th Int. IFAC Symp. Robot. Control, vol. 8, part 1, pp. 190-197, Italy, 2006.

[9] A. Ferrara and C. Vechio, “Second-order sliding mode control of a platoon of vehicles,” Int. J. of Modelling, Identification and Control, vol. 3, no. 3, pp. 227-285, 2008

[10] L. Singh and J. Fuller, “Trajectory generation for a UAV in urban terrain using nonlinear MPC,” In Proc. Amer. Control Conf., vol. 3, pp. 2301–2308, California, 2003.

[11] M. Saffarian and F. Fahim, “Non-iterative nonlinear model predictive approach applied to the control of helicopters’ group formation,” Robotics and Autonomous Systems, in Press, 2008.

[12] A. Farooq and D. Limebeer, “Path following of optimal trajectories using preview control,” In Proc. 44th IEEE Conference on Decision and Control, pp. 2787–2792, Spain, 2005.

[13] M. Tomizuka, The Optimal Finite Preview Problem and its Application to Man-Machine Systems, PhD Thesis, Massachusetts Institute of Technology, Cambridge, MA, 1973.

[14] M. Tomizuka, “Optimal continuous finite preview problem” IEEE Trans. Automat. Contr., vol. 20, no. 3, pp. 362–365, 1975.

[15] A. Farooq, D. Limebeer, and P. J. MacBean, “Optimal trajectory tracking using an output feedback preview controller,” In Proc. IEE Conference on Target Tracking, pp. 91-98, Birmingham, 2006.

[16] D. Li, D. Zhou, Z. Hu, and H. Hu., “Optimal preview control applied to terrain following flight,” In Proc. IEEE Conference on Decision and Control, vol. 1, pp. 211–216., Florida, 2001.

[17] N. Paulino, R. Cunha, and C. Silvestre, “Affine parameter-dependent preview control for rotorcraft terrain following flight,” AIAA Journal of Guidance, Control and Dynamics, vol. 29, no. 6, pp.: 1350–1359, 2006.

[18] N. Paulino, C. Silvestre, R. Cunha, and A. Pascoal, “A bottom-following preview controller for autonomous underwater vehicles,” IEEE Trans. Contr. Syst. Technol., in press, 2008.

[19] A. Cohen and U. Shked, “Linear discreet time Hinf optimal tracking with preview,” IEEE Trans. Automat. Contr., vol. 42, no. 2, pp. 270–276, 1997.

[20] A. Hazell, D. Limbeer, “An efficient algorithm for discrete-time Hinf preview control,” Automatica, vol. 44, no. 9, pp. 2441-2448, 2008.

[21] A. Hazell, Discreet-time optimal preview control, PhD Thesis, Imperial College University of London, 2008.

[22] D. Kingston, R. Beard, T. McLain, M. Larsen, and W. Ren. “Autonomous vehicle technologies for small fixed wing UAVs,” AIAA J. Aerospace, Computing, Information, and Communication, vol. 2, pp. 6549–6559, 2003.

[23] W. Ren and R. Beard, ”Trajectory tracking for unmanned air vehicles with velocity and heading rate constraints”, IEEE Trans. Contr. Syst. Technol., vol. 12, no. 5, pp. 706-716, 2004.

[24] A. W. Proud, M. Pachter, and J. J. D’Azzo, “Close formation flight control,” J. Guidance, Control, and Dynamics, vol.24, no.2, pp. 1379-1404, 2001.

[25] M. Franchek, “Selectimg the performance weights for mu and H-infinity synthesis method for SISO regulation system,” Trans. of the ASME, vol. 118, pp. 126-131, 1996.

[26] J. Doyle, B. Francis, and A. Tannenbaum, Feedback Control Theory, Macmilian Publishing Co., 1992.

[27] A. Ingram, A. Franchek, V. Balakrishnan, and G. Surnilla, “Robust SISO Hinf controller design for nonlinear system”, Contr. Eng. Practice, vol. 13, pp. 1413-1423, 2005.

[28] D. Mclean, Automatic Flight Control Systems, 1st Ed., Prentice Hall, New Jercy, 1991.

[29] K.J. Astrom, B. Wittenmark, Adaptive Control, 2nd Ed., Addison-Wesley Publishing Co., 1995.

[30] C, Scherer, P. Gahinet, and M. Chilali, “Multiobjective output-feedback control via LMI optimization”, IEEE Trans. Autom. Contr., vol. 42, no. 7, pp. 896-911, 1997.

[31] B.D.O. Anderson, J.B. Moore, Optimal Control: Linear Quadratic Methods. Prentice-Hall International, New Jersey, 1989.

[32] U.S. Military Specification MIL-F-8785C, 5 November 1980. [33] S. Chen, C.F.N. Cowan, and P.M. Grant, “Orthogonal least squares learning

algorithm for radial basis function networks,” IEEE Trans. Neural Nets., vol. 2, no. 2, pp. 302-309, 1991,