Preliminary Results on UAV Path Following

Using Piecewise-Affine Control

Samer ShehabDepartment of Mechanical and Industrial Engineering

Concordia University1515 St. Catherine W., EVS2.111Montreal, QC H3G 2W1, Canada

Email: sshehab@me.concordia.ca

Luis RodriguesDepartment of Mechanical and Industrial Engineering

Concordia University1515 St. Catherine W., EV4.243Montreal, QC H3G 2W1, Canada

Email: luisrod@encs.concordia.ca

Abstract— The path following problem for a simplifiedmodel of an Unhabited Aerial Vehicle (UAV) in longi-tudinal motion is investigated using a piecewise-affine(PWA) control law and a Lyapunov based controller designtechnique. The search for the parameters correspondingto the controller and to a piecewise quadratic Lyapunovfunction is formulated as an optimization problem subjectto linear and bilinear matrix inequality constraints. Sim-ulation results will show the effectiveness of the proposedmethodology.

I. INTRODUCTION

For systems with nonlinearities, linear controllers can

be designed if the nonlinear dynamics are linearized

around a certain operating point. The linear controllers

are then designed to stabilize the system while working

around the operating point [1]. This obviously limits

the operation of the system to a small region around

the operating point. However, in most missions of a

UAV many aggressive maneuvers with sudden increases

in variables such as pitch angle might be necessary.

Therefore, for UAV missions, controller design cannot

be handled effectively by linear techniques. In contrast to

linear models, PWA models offer a global approximation

to a nonlinear system. The basic idea is that the whole

state space is partitioned into several regions, each of

which has its own affine (linear with offset) model.

PWA models can thus be used as a good approximation

to complex systems involving nonlinearities. This ap-

proximation is in fact exact for many cases of practical

interest since a wide variety of nonlinearities in physical

systems are actually piecewise-affine. For instance, the

dead-zone phenomena in DC motors (and hydraulic

actuators) and the characteristics of a saturated linear

actuator are piecewise-affine. To the best of the authors’

knowledge, PWA controllers have never been used in

aircraft systems. In fact, previous work on advanced

continuous-time autopilot design has concentrated on

other techniques such as adaptive control, neural net-

works control and gain scheduling.

The design of autopilots for high-performance aircraft

operating over a wide range of speeds and altitudes

was one of the primary motivations for active research

on adaptive control in the early 1950s [2]. This makes

adaptive control one of the techniques that has long

been used in aircraft control systems. However, adap-

tive control may exhibit local instability and complex

nonlinear behavior when adequate process information

is not supplied to the parameter estimator [3]. Moreover,

large transients occur when the controller is switched on

and the parameters have not yet converged to its desired

values. Because of the high transients, input saturation

can occur, which degrades performance and can even

affect stability. Additionally, the design and analysis of

nonlinear adaptive systems is difficult and can lead to

relatively expensive solutions in computational terms.

Researchers in artificial neural networks (ANN) argue

that most of the problems just mentioned in adap-

tive control can be mitigated using ANN ([4], [5],

[6]). Despite their potential in many pattern recognition

applications, neural networks require a great deal of

computational effort in order to achieve a good set of

final weights. In addition, neural networks are sometimes

criticized because of their lack of repeatability when

computing the weights for a given problem given the

high dependence of the final weights on the initial values

of the weights. Once the weights are found, another

drawback of neural networks for control is that no

proof of stability is available. Furthermore, the relations

between the input variables and the output variables are

not developed by engineering judgment. This inherent

”black-box” nature of the operation of neural networks

causes many engineers to be reluctant of relying heavily

on the results from a system they cannot truly understand

nor have intuition to modify.

Another popular method used in flight control and

other systems for handling parameter variations is gain

scheduling [7], [8]. In this method, a feedback controller

with continuously scheduled gains is designed to meet

Proceedings of the2005 IEEE Conference on Control ApplicationsToronto, Canada, August 28-31, 2005

MB6.2

0-7803-9354-6/05/$20.00 ©2005 IEEE 358

the performance requirements for the corresponding

model. In gain scheduling techniques it is often nec-

essary to assume that both the scheduling parameters

and their time rate of change are measured because of

stability issues. However, this assumption occurs seldom

in practice.

Related to autopilot design, the problem of path

following for autonomous vehicles has received a sig-

nificant attention in the past decade. For example, in

[9] trajectory tracking of UAVs is addressed using

the tracking-error model presented in [10] where the

equations of motion are expressed with respect to a

fixed reference frame. Our approach here will however

be significantly different. It will use the path parame-

terization method suggested in [11] to transform the

problem coordinates to an error space. This error space

is formed by the distance between the UAV and a

reference point to be tracked on the desired path and

the velocity heading error. In [11], Soeanto et al. have

proposed a parameterization method that allows the rate

of progression of a virtual target along the path to be

an extra design parameter. This extra degree of freedom

overcomes singularity problems that may arise when the

position of the virtual target is defined by the projection

of the actual vehicle on that path, as it was done in [12].

Furthermore, global convergence of the actual vehicle

trajectory to the desired path can be achieved using the

method from [11].

Based on what was explained in the previous para-

graphs, the objective of this paper is to synthesize a PWA

state feedback autopilot for a path following mission

of a UAV. PWA controllers are scheduled controllers

that have the advantage that no assumption on the time

rate of change of the scheduling parameters is neces-

sary. Instead of continuously scheduling the gains, PWA

controllers switch among a finite discrete set of gains.

Lyapunov theory will be used to formulate the design

of the controller as an optimization problem subject to a

set of linear matrix inequality (LMI) and bilinear matrix

inequality (BMI) constraints. The paper is divided into

six sections. In section II, the parameterized model of

the path following problem is presented. Section III

then gives a description of the PWA longitudinal path

following dynamical model of the UAV. Section IV will

be dedicated to the description of a Lyapunov based

synthesis methodology for PWA controllers. In section

V the methodology from section IV is then applied to

the UAV path following problem in longitudinal motion

and some simulation results are presented. Finally, the

conclusions are stated.

Fig. 1. Path parameterization description

II. PATH FOLLOWING KINEMATICS

This section is based primarily on the work described

in [11]. Figure 1 shows a UAV and a trajectory to be

followed in the x-z vertical plane. Point P is the origin

of the Serret-Frenet frame {F}, which moves along the

desired path of the vehicle. Point Q is the position of the

vehicle. It can either be expressed in the inertial frame

{I} by the vector q = [x 0 z]T , or in the moving frame

{F} by the vector r = [x1 0 z1]T . The heading angle

of the vehicle’s velocity and the orientation of the Serret-

Frenet frame are represented by θv and θc, respectively.

Both these angles are measured with respect to the xaxis of the inertial frame. Denoting by s the signed

curvilinear abscissa of P along the path, define cc(s)as the path curvature and t as the tangent vector at point

P on the path. The inertial velocity of point Q expressed

in {F} is

F

I R

(dqdt

)

I

=(

dpdt

)

F

+(

drdt

)

F

+ (ωc × r)F , (1)

where the rotation matrix from {I} to {F} is

F

I R =

⎡⎣

cos θc 0 − sin θc

0 1 0sin θc 0 cos θc

⎤⎦ . (2)

The inertial velocity of P expressed in {F} is

(dpdt

)

F

= s (t)F =[s 0 0

]T, (3)

the inertial velocity of Q in {I} is

(dqdt

)

I

=[x 0 z

]T, (4)

while the velocity of Q in {F} is

(drdt

)

F

=[x1 0 z1

]T. (5)

359

Fig. 2. UAV in longitudinal motion

Since θc = cc(s)s, the cross product term of (1) is

(ωc × r)F =

⎡⎣

0cc(s)s

0

⎤⎦ ×

⎡⎣

x1

0z1

⎤⎦ =

⎡⎣

cc(s)sz1

0−cc(s)sx1

⎤⎦ . (6)

Combining and rearranging equations (2)–(6), equa-

tion (1) becomes

x1 = x cos θc − z sin θc − cc(s)sz1 − s (7)

z1 = x sin θc + z cos θc + cc(s)sx1 (8)

Assuming that the vertical component w of the ve-

locity is zero (see next section) the inertial velocity of

point Q is [xz

]= u0

[cos θv

− sin θv

], (9)

where u0 is the forward speed of the UAV. Defining now

θ = θv − θc and substituting (9) into (7) and (8) yields

θ = q − cc(s)sx1 = −cc(s)sz1 + u0 cos θ − s (10)

z1 = cc(s)sx1 − u0 sin θ

where q = θv is the pitch rate of the UAV.

III. PWA PATH FOLLOWING DYNAMICS

A UAV in longitudinal motion is shown in Figure 2.

This motion is controlled by the elevator which when

deflected creates a moment about the y-axis. In order to

simplify the dynamical model of the UAV, the following

assumptions will be made:

1) The aircraft is a rigid body with constant mass.

2) The vertical component w of the velocity is zero1

Taking these assumptions into account and applying

Newton-Euler’s law of rotational motion yields

q =xeCeu

20

Jyδe (11)

1This is not the case for a general trajectory, but it is rather a firstsimplifying assumption for this research.

where δe is the elevator deflection, Ce = 12CLh

ρSh,

where CLhis the tail lift coefficient, ρ is the air density

and Sh is the tail area, xe is the moment arm from the

elevator center of pressure to the center of mass (CM),

and Jy is the moment of inertia of the UAV around

the pitch axis. Augmenting (10) with (11), and taking

x = [q θ x1 z1]T as the state vector, the complete

dynamical model is

⎡⎢⎢⎣

q

θx1

z1

⎤⎥⎥⎦ =

⎡⎢⎢⎣

0 0 0 01 0 0 00 0 0 −ccs0 0 ccs 0

⎤⎥⎥⎦

⎡⎢⎢⎣

qθx1

z1

⎤⎥⎥⎦ +

⎡⎢⎢⎣

0−ccs

u0 cos θ − s−u0 sin θ

⎤⎥⎥⎦

+

⎡⎢⎢⎣

xeCeu20

Jy

000

⎤⎥⎥⎦ δe (12)

The moment Me in Figure 2 is caused by δe, which is

taken as the system input. To find a PWA approximation

of these dynamics, the sine and cosine functions must

be approximated by PWA functions. To do this, the state

space is partitioned based on the state variable θ into the

following regions or cells

Ri ={x ∈ �4|θ ∈ (θi, θi+1)

}; i = 1, . . . , 11.

θj ={−π,− 3π

4 ,− 2π3 ,−π

2 ,−π4 ,− π

24 , π24 , π

4 , π2 , 2π

3 , 3π4 , π

},

for j = 1, . . . , 12. Each cell is thus constructed as the

intersection of a finite number pi of half spaces and

can be expressed as

Ri = {x | Eix > 0}, (13)

where x =[xT 1

]Tand Ei ∈ IRpi×(n+1). For

example,

E1 =[

0 1 0 0 π0 −1 0 0 − 3π

4

].

The resulting approximate dynamics will then be a PWA

system of the form

x(t) = Aix(t) + ai + Biu(t), for x(t) ∈ Ri, (14)

where x(t) ∈ IRn and u(t) ∈ IRm. Matrices Ai ∈IRn×n, ai ∈ IRn and Bi ∈ IRn×m are constant within

each Ri. The polytopic cells, Ri, i ∈ I = {1, . . . ,M},

partition the state space IRn and Ri ∩ Rj = ∅, i �= j,

where Ri denotes the closure of Ri. Any two cells shar-

ing a common facet will be called level-1 neighboring

cells. Let Ni = { level-1 neighboring cells of Ri}. A

parametric description of the boundaries can be obtained

as

Ri ∩Rj ⊆ {Fijs + fij | s ∈ IRn−1}, (15)

360

for i = 1, . . . ,M , j ∈ Ni, where Fij ∈ IRn×(n−1) is

a full rank matrix and fij ∈ IRn. For example, for the

boundary between regions R1, R2 we get

F12 =

⎡⎢⎢⎣

1 0 00 0 00 1 00 0 1

⎤⎥⎥⎦ , f12 =

⎡⎢⎢⎣

0− 3

4π00

⎤⎥⎥⎦ .

Furthermore, it is assumed that Ri can be outer approx-

imated by a (degenerate) quadratic curve εi

εi = {x|xT Six > 0}. (16)

One possible configuration for Si is [13]

Si = ETi ΛiEi, (17)

where Λi ∈ IR(pi+1)×(pi+1) is a matrix with nonnegative

entries and

Ei =

⎡⎣

01×n 1

Ei

⎤⎦ ∈ IR(pi+1)×(n+1). (18)

In this case, it is said that Si defines a bounding ellipsoidfor region Ri.

IV. LYAPUNOV-BASED CONTROLLER SYNTHESIS

For PWA systems of the form (14), this section will

review the PWA synthesis algorithm developed in [14],

which will then be applied to the UAV example in

Section V. The goal is to stabilize the equilibrium point

xcl for system (14) by designing a PWA state feedback

control signal

u = Kix, for x(t) ∈ Ri (19)

where

Ki =[Ki ki

], (20)

and −KLim ≺ Ki ≺ KLim, with KLim being a vector

of upper bounds for the entries of Ki, i = 1, . . . ,M .

Replacing (19) into (14) yields

˙x(t) = (Ai + BiKi)x(t), for x(t) ∈ Ri (21)

where Ai ∈ IR(n+1)×(n+1) and Bi ∈ IR(n+1)×m are

Ai =[Ai ai

0 0

], Bi =

[Bi

0

]. (22)

The controller will be designed by searching for a

piecewise-quadratic candidate control Lyapunov func-

tion which is continuous at the boundaries and defined

in ∪Mi=1Ri as

V (x) =M∑i=1

βi(x)Vi(x), (23)

where

βi(x) ={

1, x ∈ Ri

0, x ∈ Rj , j �= i(24)

for i = 1, . . . ,M . The expression for the candidate

Lyapunov function in region Ri can be written as

Vi(x) =[x1

]T [Pi −Pixcl

−xTclP

Ti ri

] [x1

]= xT Pix

(25)

where Pi = PTi > 0, Pi ∈ IRn×n, ri ∈ IR and therefore

Pi = PTi ∈ IR(n+1)×(n+1). The next subsections will

revise the constraints to be imposed on the controller

and the Lyapunov function.

A. Constraints on the Controller

1) Continuity: In order to enforce continuity of the

control input, the following constraint should be consid-

ered [13]:

(Ki − Kj)Fij = 0, for j ∈ Ni, (26)

where

Fij =[Fij fij

0 1

].

B. Constraints on the Lyapunov function

1) Continuity: Using the boundary description (15),

continuity of the candidate control Lyapunov function

across the boundary between regions Ri and Rj is

enforced by [13]

FTij (Pi − Pj)Fij = 0, for j ∈ Ni. (27)

2) Positive definiteness: The candidate control Lya-

punov function is positive definite if it satisfies the

inequality

Vi(x) > 0, ∀x ∈ Ri, x �= xcl. (28)

Using the polytopic description of the cells (13) and

the S − procedure [15], it can be shown that sufficient

conditions for satisfying the above inequality for each

region Ri are the existence of Pi, with Pi > 0, and Zi

with nonnegative entries satisfying [13]

Pi − ETi ZiEi > 0. (29)

3) Decreasing over time: This is equivalent to

dV

dt< 0. (30)

It can similarly be shown that sufficient conditions for

satisfying the above inequality for each region Ri are

the existence of matrix Λi with nonnegative entries

satisfying [13]

Pi(Ai+BiKi)+(Ai+BiKi)T Pi+ETi ΛiEi < 0. (31)

361

C. Desired Closed-Loop Dynamics

We assume that using linear control theory, a local

controller can be designed to achieve the desired closed-

loop dynamics in the region where the closed-loop

equilibrium point is located, Ri� . Consider the dynamics

of the system in this region

x(t) = Ai�x(t)+ai�+Bi�u(t), for x(t) ∈ Ri� . (32)

Introducing a new variable z(t) = x(t) − xcl, we have

z(t) = Ai�z(t) + Ai�xcl + ai� + Bi�u(t). (33)

We assume that there exists a vector ki� which satisfies

Bi�ki� + Ai�xcl + ai� = 0. (34)

Thus, using the control input

u(t) = Ki�z(t) + ki� , (35)

the closed-loop dynamics in region Ri� are now linear:

z(t) = (Ai� + Bi�Ki�)z(t). (36)

The matrix gain Ki� can then be designed using linear

control methodologies to satisfy desired design objec-

tives. To find a Lyapunov function for the linear con-

troller, the following Linear Matrix Inequalities (LMIs)

should be solved

find Pi� > 0Pi�(Ai� + Bi�Ki�) + (Ai� + Bi�Ki�)T Pi� < 0.

(37)

The affine controller and the quadratic Lyapunov func-

tion for the region holding the equilibrium point are

Ki� =[Ki� ki�

]

Pi� =[

Pi� −Pi�xcl

−xTclP

Ti� xT

clxcl

]. (38)

D. Uniformity of the Closed-Loop Dynamics

The linear controller in the region holding the equilib-

rium point was designed to satisfy requirements locally.

The closed-loop dynamics of the system in this region

can serve as a reference model for closed-loop dynamics

in other regions. In the proposed method, we try to

minimize the upper bound of the difference between

the closed-loop dynamics of all regions and that of

the region holding the equilibrium point. This can be

formulated as minimizing β > 0 satisfying

‖Ai + BiKi − (Ai� + Bi�Ki�)‖ < β. (39)

E. Synthesis Algorithm

1) Design a local linear controller for (36) by choos-

ing a controller gain Ki� for region Ri� , with ki�

fixed by (34).

2) Solve (37) to find a quadratic Lyapunov function

for region Ri� .

3) Given xcl, fix Pi� , Ki� and ki� , and solve

min βs.t. (26), (27), (29), (31), (38), (39),

β > 0, Pi = PTi > 0, Zi � 0, Λi � 0,

−KLim ≺ Ki ≺ KLim,for i ∈ I = {1, . . . ,M}, i �= i�,

(40)

where i� is the index of the region Ri� con-

taining the equilibrium point and � and ≺ mean

component-wise inequalities.

Remark 1: The solution to this problem will be thePWA controller that minimizes the difference of theclosed-loop dynamics of all regions to the closed-loopdynamics of the region containing the equilibrium point.In this sense, the resulting PWA controller effectivelyfeedback linearizes the system. �

Remark 2: The constraints of the synthesis problem(40) include a set of Bilinear Matrix Inequalities (BMIs).BMIs are nonconvex constraints and this makes themhard to solve. Several numerical algorithms have beenproposed to solve BMI problems locally and the one usedhere is implemented in the software package PENBMI[16]. �

V. EXAMPLE: UAV PATH-FOLLOWING

Having formulated the optimization problem in (40),

we will now solve it for a case study on UAV path

following. We will assume as before that s = u0 is

constant. For a circular path of radius R, cc = R−1 is

also constant. The goal of the controller is to drive θ, x1,

and z1 to zero to be able to follow this circular path. u0

is taken to be 20m/s, Jy = 3.5Kgm2, Ce = 0.5Kg/m,

xe = 2m, s = 20, and cc = 0.005. The PWA controller

362

parameters are found after solving (40) to be

K1 =[−0.2637 −1.1866 −0.0706 0.0037

],

K2 =[−0.2577 −1.2049 −0.0773 0.0262

],

K3 =[−0.2547 −0.9914 −0.0929 0.0595

],

K4 =[−0.2479 −1.1766 −0.1246 0.0774

],

K5 =[−0.2397 −1.4519 −0.1772 0.1644

],

K6 =[−0.2387 −1.3520 −0.1758 0.1891

],

K7 =[−0.2402 −1.5511 −0.0906 0.1310

],

K8 =[−0.2466 −0.8629 −0.0228 0.0756

],

K9 =[−0.2467 −0.9451 0.0369 0.0282

],

K10 =[−0.2479 −1.0543 0.0480 0.0078

],

K11 =[−0.2508 −0.9033 0.0448 −0.0091

],

m1 = 0.1162, m2 = 0.0726, m3 = 0.5187,

m4 = 0.2270, m5 = 0.0105, m6 = 0.0239,

m7 = 0.0635, m8 = −0.4711, m9 = −0.3425,

m10 = −0.1140, m11 = −0.4699.

The simulation of the UAV performing a loop in the lon-

gitudinal plane is depicted in Figure 3. The initial state

value used in the simulation is x0 = [0.2 0.5 10 10]and the initial UAV position is (−50, 0, 0) in the inertial

reference frame.

Fig. 3. Simulation: Loop Following by UAV

VI. CONCLUSIONS

This paper has presented a new control methodol-

ogy for UAV path following that synthesized a PWA

state feedback controller. PWA functions were used to

approximate nonlinearities that appear in the model of

the system. Possible extensions of the work done in

this paper include output feedback as well as studying

the effectiveness of the PWA controller in the pres-

ence of actuator saturation and plant parameter un-

certainty. More general trajectories for UAV missions

with nonzero vertical velocity component will also be

addressed in future work.

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