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THEStS

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This is to certify that the

thesis entitled
INDIRECT ADAPTIVE OUTPUT FEEDBACK CONTROL
presented by
Sridhar Seshagiri
has been accepted towards fulfillment

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1/98 cJClRC/DatoDuopfiS—p.“

INDIRECT ADAPTIVE OUTPUT FEEDBACK
CONTROL

By

Sm'dhar S eshagim’

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

MASTER OF SCIENCE

Electrical Engineering

1998

ABSTRACT
INDIRECT ADAPTIVE OUTPUT FEEDBACK CONTROL
By

Sridhar Seshagz'rz'

An adaptive output feedback control scheme for the output tracking of a class of
continuous-time nonlinear plants is studied. The method uses parameter projection,
control saturation, and a high-gain observer to achieve semi-global uniform ultimate
boundedness. First, an application to the longitudinal control of a platoon of non-
identical vehicles is discussed. A nonlinear model is used to represent the vehicle
dynamics of each vehicle within the platoon. The model depends linearly on unknown
parameters which belong to a known compact set. In contrast to previous work, the
number of measured quantities is kept to a minimum. The efficacy of the proposed
method is demonstrated through simulations. Next, an application to the control of
unknown nonlinear systems using an RBF neural network is discussed. The RBF
network is used to adaptively compensate for the plant nonlinearities. The network’s
weights are adjusted using a Lyapunov-based scheme. It is shown that by using
adaptive control in conjunction with robust control, it is possible to tolerate larger

approximation errors resulting from the use of lower-order networks.

To my family

iii

ACKNOWLEDGMENTS

I would like to express my deepest appreciation to the following people for having
made the completion of this thesis possible :

To my advisor Dr. Hassan K. Khalil, for his insight, guidance and invaluable
assistance, without which this thesis would not have been possible. His dedication to
the subject and personal integrity have served as an example and an inspiration to
me.

To all my other instructors at MSU, for their skillful instruction in the classroom,
but more importantly, for their constant encouragement and friendship.

To my friends, for having made my stay here the memorable experience that it
has been.

And to my family, for years of support, patience and understanding.

iv

TABLE OF CONTENTS

LIST OF FIGURES vii
1 Introduction 1
1 . 1 Background .................................. 1
1.2 Organization ................................. 3
2 Robust Adaptive Tracking Control 5
2.1 Problem Statement .............................. 5
2.2 Control Design ................................ 7
2.2.1 State Feedback ............................... 8
2.2.2 High-gain Observer ............................. 12
2.3 Tracking Error Convergence ......................... 14
2.4 A Robustness Property ............................ 19
2.5 Robust Output Tracking ........................... 2O
3 Longitudinal Control of a Platoon of Vehicles 24
3.1 Introduction .................................. 24
3.2 Longitudinal Vehicle Model ......................... 26
3.3 Control Objective and Design ........................ 27
3.4 Simulations .................................. 31
3.4.1 Simulation 1 ................................ 31
3.4.2 Simulation 2 ................................ 32
4 Indirect Adaptive Contol Using RBF Neural Networks 42
4.1 Introduction .................................. 42
4.2 Problem Statement .............................. 43
4.3 Function Approximation Using Gaussian Radial Basis Functions ..... 44
4.4 Control Design ................................ 45
4.4.1 Small Reconstruction Error ........................ 47
4.4.2 Reconstruction Error With a Known Bound ............... 53
4.5 Simulations .................................. 54
4.5.1 Simulation 1 ................................ 54
4.5.2 Simulation 2 ................................ 55
4.5.3 Simulation 3 ................................ 57
5 Conclusions 66
APPENDICES 68

A Matlab programs for simulations in Chapter 3 68

A.1 run.m ..................................... 68
A2 runn.m ..................................... 69
A3 projm ..................................... 76
B Matlab programs for simulations in Chapter 4 77
B.1 trainf.m .................................... 77
B.2 nno.m ..................................... 78
B.3 slotsimom ................................... 79

vi

3.1
3.2
3.3
3.4
3.5
3.6
3.7
3.8
3.9

4.1
4.2

4.3

4.4
4.5

4.6

LIST OF FIGURES

A platoon of N+1 vehicles .........................
Velocity, acceleration and jerk profiles for the leader. ...........
Perfect knowledge of vehicle parameters, D,(t) = 0. ............

Vehicle parameters unknown, control based on nominal values, D,(t) = 0.

Vehicle parameters unknown but adapted for, D,(t) = 0. .........
(a) d11(t), (b) d12(t), (c) d13(t), (d) d14 (t); d2i(t) = 0. .........
D,(t) 75 0, robustifying component not included. .............
D,(t) 34$ 0, robustifying component included, /\ = 0.9. ...........
D,(t) 7e 0, robustifying component included, A = 0.2. ...........

The reference signal and its derivatives ....................
(a)State feedback [1 = 0.5 (b) Output feedback, 6 = 10-3, p = 0.5 (c)
Output feedback 6 = 10““, n = 0.5 (d) Output feedback 6 = 10“, p =
0.1 .....................................
(a)State feedback [I = 0.5 (b) Output feedback, 6 2 10"“, p = 0.5 (c)
Output feedback 6 = 10‘4, u = 0.5 ((1) Output feedback 6 = 10“, p =
0.1 .....................................
(a) F(y,3'/) (b) F(y, y) — 01Tf(y,y). ....................
(a) No adaptation for weights, no robust control (b) Only adaptation for
weights (c) Only robust control (d) Adaptation for weights and robust
control. ..................................
Effect of network size on the controller’s performance. . .g ........

vii

Chapter 1

Introduction

1 . 1 Background

Adaptive control of nonlinear systems has matured as an exciting area of research over
the last few years. Early efforts in the design of adaptive controllers for nonlinear
systems focused on the state-feedback problem and resulted in a systematic design
procedure called adaptive backstepping [16]. An extension to the more challenging
output feedback problem for the case when the system nonlinearities depended only
on the output was next made. This problem was first solved under restrictive struc-
tural and growth conditions on the nonlinearities [14, 15]. Subsequently, the growth
restrictions were removed [17], but the structural restriction remained: the output
nonlinearities were not allowed to precede the control input. The removal of this
structural restriction by Marino and Tomei in [31] was a breakthrough in adaptive
nonlinear output feedback control. Their work addressed the problem of design-
ing a global adaptive output feedback tracking control for single-input single-output

1

2

(SISO) nonlinear systems that are linear with respect to the input and an unknown
constant parameter vector. This was achieved by merging the filtered transforma—
tions of [29] and [30] with the adaptive backstepping scheme of [16] and using a novel
compensation of the estimation error efl'ects. The scheme however suffered from the
drawbacks of overparametrization inherited from the original adaptive backstepping
procedure and restriction to the unnormalized gradient update law. These drawbacks
were addressed in the work of Krstic and Kokotovic [27], which pr0posed three new
adaptive schemes to achieve minimal parametrization and to remove the restriction
of unnormalized gradient update.

Indirect adaptive control design for systems representable by input-output mod-
els was done by Khorasani in [26]. The starting point in [26] was a class of SISO
nonlinear systems represented in state-Space form. Under the assumption of certain
rank conditions, the technique of output prolongation [12] was used to convert the
model to an equivalent input-output representation. The scheme however required
the derivatives of the output to be available for feedback. In the work of Khalil
[24] this requirement was relaxed by the use of a high-gain observer to estimate the
derivatives of the output. The system under consideration in [24] was SISO, input-
output linearizable, minimum-phase, and modeled by an input-output model of the
form of an nth-order differential equation. By combining results from [9, 40, 41] with
Lyapunov-based adaptive design [18, 32], he designed a semiglobal controller that
achieved asymptotic output tracking for reference signals which were bounded and
had bounded derivatives up to the nth order. The design was simpler than traditional

ones since it did not use filtering or error augmentation ideas. It was simply a state

3

feedback controller with a linear observer. A similar control design was also presented
in [23].

An important drawback of the result of [24] was the requirement of persistence
of excitation (PE) not only for parameter convergence but even for tracking error
convergence. This is unusual in adaptive control results where tracking error conver-
gence is shown without PE. This drawback was addressed in [2]. This improvement
over the result of [24] was made possible by changing the analysis approach. In [24],
convergence was proved by showing that, under state feedback and the PE condition,
the set of zero tracking error and zero parameter error is an exponentially stable
invariant set. Then, singular perturbation analysis was used to Show that this same
property is recovered under output feedback for sufficiently small 6. This idea does
not work in the lack of PE because the set of zero tracking error and zero param-
eter (or partial parameter) error is not exponentially stable. In [2], the closed-loop
system under output feedback is analayzed directly and various Lyapunov functions
are combined to form a composite Lyapunov function that shows tracking error and

partial parameter convergence.

1 .2 Organization

The rest of this thesis is organized as follows. Chapter 2 is a brief overview of the
technique of [2]. In Chapter 3, an application to the longitudinal control of automated
vehicles is presented. The controlled vehicle is assumed to be capable of measuring (or

estimating) necessary dynamical information from the vehicle immediately in front of

4

it by its onboard sensors. The computer in the vehicle processes the measured data
and generates proper throttling and braking actions to follow the vehicle in front at
a safe distance. The property that the spacing error for a controlled vehicle can be
regulated is referred to as local stability [38]. An important control objective in the
longitudinal control problem is that of asymptotic stability of a platoon or string of
cars following one another. A platoon is said to be asymptotically stable if there
are no slinky-type effects [37] within the platoon, i.e, there are no amplifications
in the deviations of vehicle spacings from their steady state values from the front
to the end of the platoon. It is well-known that for the case where the vehicle
following is not cooperative,i.e, information is not exchanged with other vehicles,
velocity dependent spacing rules can guarantee asymptotic platoon stability. In our
work, we use the “constant time headway” spacing rule [8, 39]. Simulations are
presented for the case of a platoon of four cars following a leader. Good references
on the longitudinal vehicle control problem can be found in [1], [6],[20] and [36]. In
Chapter 4, we study the application of the technique of [2] to the adaptive control of
unknown nonlinear systems using RBF networks. The design is developed for systems
represented by input-output models and RBF networks are used to approximate the
system’s nonlinearities. The weights of the networks are adapted using a Lyapunov-
based scheme. Simulations are presented to demonstrate the need for adaptation,
the role of the robustifying component and the effect of the network’s size (number
of Gaussian nodes) on the tracking performance. Discussions and a summary are

presented in Chapter 5.

Chapter 2

Robust Adaptive Tracking Control

2.1 Problem Statement

Consider a single-input—single-output nonlinear system represented globally by the

nth-order differential equation

if") = f0(°) + Zfzi f.(-)0.- + [900 + Zf=1 gi(')0i]u(m) (2'1)

where u is the control input, y is the measured output, y“) denotes the ith derivative
of y, and m < n. The functions f,- and g,- are known smooth nonlinearities which may

depend on y, y“), ..., y("‘1), u, um, ..., rim—1); i.e.,
fi() : fi(yay(l)1 ' ' .,y(n—l),u,u(1),. ' - ,u(m—l)), 0 S 2 S P and

93(.) : 93(y7y(1), ' ' ’ 3 y(n_l), 2“) “(1), ' ‘ ' 7u(m-l)), O S i S p

5

6

The constant parameters 01 to 0,, are unknown, but the vector 6 = [61,...,6,,]T
belongs to Q, a known compact convex subset of B”. By augmenting a series of m
integrators at the input side of the system, the extended system can be represented
by a state space model. The states of these integrators are 2.1 = u, 22 = u(1),up to
2m 2 um”) and v = aim) is the control input of the extended system. Taking $1 = y,

2:2 = y“), up to 113,, = y("‘1) yields the extended system model

it = $141, ISiSn—l
in = fo(a:,z)+6Tf(:r,z)

+ [90(23, 2) + 0Tg(:r:, z)]v

 

(2.2)
it = Zi-l-la lszsm_l
2... = v
y = 931
J
where
x : [$1,...,In]T, z:[zl,...,2m]T
f : [f1,...,fp]T, g=[91a---,9plT

Assumption 1 lgo(a:, z) + 0Tg(:r, z)| 2 k > 0 V :1: E R", z E R'" and 0 E 91, where

521 is a compact set that contains 9 in its interior.

Assumption 1 ensures that (2.2) is input-output linearizable by full state feedback

for every 0 E Q. Using the results of [5], it can be shown that there exists a global

7

diffeomorphism, possibly dependent on 9,

fl: IL"
= dé‘ Ta, 2)

C T1659 Z)
with T1(0, 0) = 0, which transforms the last m state equations of (2.2) into

C=F(C.$,9)

This, together with the first n state equations of (2.2), defines a global normal form.
The objective is to design an adaptive output feedback controller which guarantees
boundedness of all state variables and tracking of a given reference signal y,, where
y, is bounded, has bounded derivatives up to the nth-order, and yin) is piecewise

continuous.

2.2 Control Design

The controller design is done in two steps. First a state feedback controller that en-
sures boundedness of all signals and yields zero steady-state tracking error is designed.
This same controller is used in the output feedback case with the states replaced by
estimates provided by a high-gain observer (HGO). The control is saturated outside a

compact region of interest to protect the system from peaking induced by the HGO.

2.2.1 State Feedback

We design an adaptive state feedback controller so that the output y tracks the given

reference signal yr. Define

6.- = y“‘”-y§“”, 132572
e = [81, e2, ..., 6an

Mt) = [y(t). y“)(t). y‘"‘”(t)lT

yr“) = [yr(t), iii—”(75), ..., yin-”(Dir

3230) = [yr(t), 319W), y§""”(t), yf")(t)lT

and let Y and YR be any given compact subsets of R" and Rn“, respectively, such

that 37(0) 6 Y and y3(t) 6 YR V t 2 0. We rewrite (2.2) as

e = Ame +b{Ke + fo(e +yr,z) +0Tf(e+yr,z)
+ [go(e + yr, 2) + 6Tg(e + 3)., z)]v — 31W} . (2.3)

2' 2 A22 +b2u

where (A, b) and (A2, b2) are controllable canonical pairs that represent chains of n

and m integrators, respectively, and K is chosen such that Am = A — bK is Hurwitz.

Assumption 2 The system C = F (C ,y,,0) has a unique steady-state solution C.

Moreover, with C =: C — (— the system
5: F(C + cle + 32., a) — new) “‘3 ac”, at. C, a) (2.4)

has a continuously difierentiable function V1(t, 6), possibly dependent on 9, that sat-
isfies 1
villi“2 S V1(t, 5) S n2||5||2

61/1 3V1 ~ - ” 2 “
— —‘T < —
(it + 8C F2((,e,y,.,C,9) _ n3||C|| +n4llCl|||ell

where 771, 772, 773 > 0, and 774 2 0 are independent of y. and 0.

The steady-state response of a nonlinear system is introduced in [19, Section 8.1].
Basically, it is a particular solution towards which any other solution of the system
converges, as time increases. The inequalities satisfied by V1 imply that such conver-
gence is exponential. They also imply that (2.4), with e as input, is input-to—state
stable. Consequently, the zero dynamics of (2.2) are exponentially stable and (2.2) is

minimum phase.

Let P = PT > 0 be the solution of the Lyapunov equation PAm + AgP = —Q

where Q = QT > 0, and consider the Lyapunov function candidate

v = eTPe + lz-éTr-lé (2.5)

 

1Unless specified otherwise, I] . M denotes the Euclidean norm.

10

where F = FT > 0, d = d — 6, and d is an estimate of 0 to be determined by the
parameter adaptation law. The derivative of V along the trajectories of the system

is given by

V = -eTQe + (WP-16.7 + 2eTPb{fo(e + yr, 2:) + 0Tf(e + yr, 2)

+ Ke -— yin) + [go(e + yr, z) + 0Tg(e + yr, 2)]u}

 

Taking
”U _ —Ke+yi-n) —f0(€+yraz) _éTf(e+yraz)
go(e + yr, 2) + éTg(e + yr, 2)
gt. w(eazryR9é) (26)
we can rewrite the expression for V as V = —eTQe + dTI‘ '1[d — I‘d] where

<25 = 2eTlDblfle + yr, 2) + 9(6 + yr, (2)1146, 2, 3712, 9)] = 05(6, z, ya, 9)
The parameter adaptation law is chosen as in [24], i.e.,
5: Proj(é, <15)

where Proj(d, ab) = I‘d) for d E Q and is modified outside (2 to ensure that

éTr-1[é — rqs] g 0 (2.7)

11

and 6(t) belongs to a compact set 96 for all t Z 0, where (21 D 06 D $2. This can be
achieved by standard adaptation laws with smoothed parameter projection to ensure

that Proj(0, (25) is locally Lipschitz. As an example, consider the case when S2 is the

convex hypercubeQ={6 | a,- 30, sh}, 1323p}. Let
95={0|ai—630,Sb,+6},1323p}

where 6 > 0 is chosen such that {26 C 521, and choose F to be a positive diagonal

A

matrix. In this case the projection Proj(6l, d) is taken as

f

711452, if ai S at S be 01‘
iré.>b.and¢.300r
ifé,<a.and¢,2o

[Prowl cm].- = 4 (28)

723 [1+ (bi ‘éi)/6] (bi, If 63' > b,' and (bi > 0

 

t 7“ [1+(éi- 00/6] ¢n if d, < a.- and (b,- < 0

Inequality (2.7) ensures that V S 0. Therefore, e(t) and d are bounded for all t Z 0.
Since )7, is bounded, we conclude that z(t) is bounded, which implies, in view of
Assumption 2, that z(t) is bounded. With all signals bounded, we conclude that e(t)

—)Oast-—)oo.

12
2.2.2 High-gain Observer

To implement the state feedback adaptive controller using output feedback, we need to
estimate 6; there is no need for estimating 2 as it is already available (the state of the
integrators at the input side). With the goal of recovering the performance achieved

under state feedback, we use the same high-gain observer used in [24], namely,

61' = éi+1+ai(el-é1)/e‘, ISZSTI—l

(2.9)

8,; = an(el—é1)/e"

where e is a small positive parameter to be specified. The positive constants a,- are
chosen such that the roots of s"+als"—1+- - -+cr,,_ls+01n = 0 have negative real parts.
To implement the control using output feedback the state e in w’ and 45 is replaced
by its estimate e. We assume that all initial conditions are in a given compact set;
in particular, 6(0) 6 Q, e(0) E E0, and 2(0) 6 Z0, where E0 and 20 are compact sets.
The sets E0 and Z0 can be chosen large enough to cover any given bounded initial
conditions, but once they are chosen we cannot] allow initial conditions outside them.
Let c1 = maxeeg0 eTPe, C2 = maxoenflem %(d — 6)TF‘1(9— 0) and c3 > c1 + C2. Then
e(t) E E (if {eTPe 3 03} for all t 2 0. Let Z be a compact subset of R” such that

Z0 is in the interior of Z and 2

2(0)€Zo and e(t)€EVtZ 0 => z(t)€ZVtZO. (2.10)

 

2The set Z can be determined using the Lyapunov function V; of Assumption 2. The basic idea
is to choose cz large enough that the set {V1 3 ex} is positively invariant, and then determine the
corresponding set in the z-coordinates.

13

Let S 2 max |w(e, z, 323, 9)] where the maximization is taken over all

e E E1 déf eTPe 3 c4}, 2 E Z, 373 E Ymd E {2.5, where c4 > c3.

Define the saturated function it” by

 

10(8) 27 yRa 9)
S

$309, ZayR,é) = S sat<

where sat(-) is the saturation function.

By taking {,- = (e,- — é,)/e"“, 1 g i S n and 6 = [(1, . . . ,én]T, the closed-loop

system is represented by the standard singularly perturbed form

é = Ame + b{Ke + M) + 6770)

+ (go(-) + 9Tg(-))¢3(-) - 21$")}
2 = A22 + 52W”
(2.11)

0 : Pr0j(éa ¢())

eé = (A — HC)€ + eb{fo(-) + 6Tf(-)

 

+ (900 + 6Tg(-))W(-) - W} I

where C = [1,0,...,0], H : [a1,...,an]T, (A — HC) is Hurwitz, and é = e - DE

where D is a diagonal matrix with 6"“ as the ith diagonal element.

14
2.3 Tracking Error Convergence

Tracking error convergence is proved in [2]. For the sake of completeness we outline
the proof. The first step in showing tracking error convergence is to confirm that for
any initial conditions in the given compact set, all signals of the closed-loop system
(under output feedback) are bounded. First it is shown that there exist constants c5,

as > 0 such that the set 3
R. = {{v 3 03} n {(5 e Qg}}X{V1$ 05} x {vé 3 cats?)

is positively invariant for sufficiently small 6, where V): = {T156 and P = PT > 0
is the solution of the Lyapunov equation 15(A — HC) + (A — HC)TP = —I. Then,
using the difference in speeds between the slow and fast variables and the fact that
V): S —(1/2e)||§||2 outside {V5 3 c6} it is shown that the trajectory enters the set R,
during the time interval [0,T(c)] and remains thereafter, where T (e) —-) 0 as e —+ 0.
From that time on, the control saturation is not effective and the closed-loop system

is given by

e = Ame—béTau) +A(-)

0 = 1“,,(0, a) [ (2.12)

C : 172(6789327‘7696)

 

.5 = (A — HC){ — came) + Ke] + eA(-) J

 

3Note that the set {V1 3 as} could be time-dependent. See [25, Section 3.4] for the use of
time-dependent sets in the analysis of nonautonomous systems.

15

where

macs) -—- Proj(é,¢(é,z,ya,é))
e(t) = ft)+9(-)w(é(t).z(t).yR(t),é(t))
At) = b{K(e—é)+(fo—fo)
+9T(f-f)+(go-éo)v+9T(g-§)v}
f(-) = f(é+y.,z)

if) = 9(é+3’r,2)

Define w, by w,(t) déf f(yr,2) + g(y,,z)u(0, 2,323,150 where Z is the steady state

solution of the zero dynamics, determined uniquely from (- = T1(y,., Z).

Assumption 3 There exists a constant nonsingular matrix S, possibly dependent on

0, such that Sw,(t) = [w,1(t) 0]T where w” is persistently exciting.

The possibility that wr is persistently exciting or that w, = 0 is not excluded. Using
the transformation 5"1 to transform 6 into VS” 2 [61319;] , the equation for e and

0 can be rewritten as

('3 = Ame — béTs—lszu. + béT(w. — 110+ A(-),
91 PIP

62 P2P

16
Define

f() : f(yra 2), §() : 9(yr12)1 gO() : 90(yra 2)

12%) : W0, ‘5ny) 6), £4) 2 VJ“), Z,yR,é), ¢() : 1/J(é, ZayRaé)

It can be shown that e and 61 satisfy

e A... —bgw,?1 e A.(-)
. = + (2.13)
él 2111911)”pr O a] Ae(°)
where
1‘1 F2 - _ . - .
S‘Tr‘s-l = , As(') = A(-) + b9T[(f — f) + (W - 5110)]
F3 F4

Aef') = [Flp — 2l‘1gw,1bTPe] and K91 > 9f) = (90 + 9T§)/(§o + 5T9) > K92

for some positive constants Kgl and K92 independent of 6. Since f, go, 9 and 1b are

Lipschitz functions in their arguments, we have

llAs(')ll S 61||e|| + 52||€H + 53||C~||, llAe(-)ll S 54||e|| (2-14)

for some 6,- 2 0, i = 1, .., 4. Consider the system

é Am ‘bngi e

. (2.15)
61 2F19wrleP 0 01

17

Using well known results from adaptive control theory (see for example [25, Section
134]) and the fact that wrl is persistently exciting and Q() is bounded from below,
it can be shown that (2.14) has an exponentially stable equilibrium point at the
origin. Then, from the converse Lyapunov theorem, there exists a Lyapunov function
V2(t, e, 91) whose derivative along (2.13) satisfies

V2 S —<5.5ll€ll2 — 66ll0~lll2 + 57|l€|lll5|l + 58ll9~1|lll§ll
(2.16)

+ égllell2 + 610l|6||||51l|+ 6n||€l|l|€|l + 612||é1||||5||

for some positive constants 65 and 66 and some non-negative constants 6,- ,7 g i S 12.

The derivative of Vg with respect to
65' = (A — HC)g — ebglewrl — eKe + 5A, (2.17)

satisfies

. 1 .. -
V: S glléll2 + 73ll91||||€l| + 74||6||||€|| + 75|lC||||€|| + nlléll2 (2-18)

for some non-negative constants *7,- ,3 S i g 6. Construct the Lyapunov. function

candidate

W=aV+flV1+V2+Vg (2.19)

where a > 0 and 6 > 0 will be chosen later. Using the inequality

V s —lclllell2 + k2||6||l|€||

18

together with Assumption 2, (2.16), and (2.18), it can be Shown that the derivative

of W with respect to (2.12) satisfies

- . .
Hell Ilell
. ”a” “an
W s — M (2.20)
urn urn
L Iléll J . Hall J

 

 

 

 

where M is given by

akl + (55 — 59 —510/2 (-/3714 — 510/2 (—ak2 - 57 — ”Ml/2

 

 

—510/2 56 —<512/2 (“58 — Val/2
M =
(‘3774 — (511V? ‘512/2 5773 —75/2
L (‘Ok2 — 57 — 70/2 (—68 — 73)/2 ‘75/2 “/0“ ’76

By choosing a and 6 large enough and 6 small enough, M can be made positive

definite. Hence, by [25, Theorem 4.4], we conclude that

[“6” llfhll Hill llflllT -+ 0, as t —+ co

19
2.4 A Robustness Property

In this section we recall from [2] robustness of the adaptive controller to unknown

bounded disturbance. Consider a perturbation of (2.1):

P

3100 = fof') + : f.-(-)0.- + [900 + Zgi(')0ilu(m) + d(‘) (2-21)

i=1

where d() is a disturbance term of the form
d(t, x, z, u, 9) = df(t, x, z, 9) + dg(x, z, 9)v
The error equation (2.3) becomes

«2 = Ame + We + M) + 6TH.) + (90(‘) + 0Tg(-))v

+ d(t, e + 32., 2,1), 9) — yiw}

Suppose the disturbance d satisfies “d(t, e+y., 2, 7,12%), 9)“ 3 d1 Vt 2 0, e E E, y. E
Y, z E Z, and E E R". Suppose further that for sufficiently small d1 Assumptions 1
and 2 hold uniformly in d and the set Z has the pr0perty (2.10) for all d. Using the
same controller of the previous section, it can be shown that, for all (e, 9, 6,6) 6 R,,

the derivative of V satisfies

V g —eTQe + klc + kddl (2.22)

20

for some kd > 0. For every 6 < 6“ = k(c3 — c2)/k1 and d1 < 191(6‘ — e)/kd, V < 0 on
V 2 (:3. Hence, as in the previous case, it can be shown that there is d} = dI(c) such
that for all all < d}, all variables are bounded and (2.22) is satisfied for all t Z T(e),
where T(e) —> 0 as e —+ 0. Therefore, the mean-square tracking error is of order
0(6 + d1).

If Assumption 3 is satisfied in the ideal case d = 0, then it can be shown that, in

the presence of d, the derivative of W with respect to the closed-loop system satisfies

. 5 5 -

w 5 fine“? - 361101112 + add. (2.23)

for some ed > 0. Since all signals are bounded, the mean-square tracking error and

the mean-square error of 91 are of order 0(d1). From [2], if it), is persistently exciting,
then W satisfies

Wg—mW+Qw an)

for some kw > 0, which shows that all variables, including the parameter error 9,

converge to a ball centered at the origin, whose size is of the order of 0 (fl) .

2.5 Robust Output Tracking

We introduce an additional robustifying control component to make the mean-square
tracking error arbitrarily small, irrespective of the bound on the disturbance d, pro-
vided this bound is known. Once again, we consider the perturbed system (2.19)

and assume that Assumptions 1 and 2 are satisfied uniformly in d(). Moreover, we

21

assume that the set Z has the property (2.10) in the presence of d. Let

_ (fl)_ _”T
u: Ke+yr f0(e+yr,z) 9f(e+yr,z)+u1 (2.25)

go(e + 32., z) + éTg(e + yr, 2)

 

We will choose the robustifying component 221 using the Lyapunov redesign technique,

e.g., [25, Section 13.1]. Assume that

||d(t,x,z,u,9)|| g p(e,z) + kvllulll, 0 3 kn <1

where p and kv are known. Take 77(e, z) 2 p(e, z) and define s = 2eTPb,
- (135.) ' £7 for n(e,2)|8| Z M

¢.(e, z) = (2.26)

(”l-kw) ~fi for n(e,z)|s| < u

and 4

—KC + y?!) — f0(€ + yr, 2). — éTf(e + yr) 2) + Alb-(9,2)

. (2.27)
go(e + yr, 2:) + 9Tg(e + yr, z)

19(6) 2, yR: é) =

 

The adaptive controller is taken as u = ibs(é, 2, ya, 9). Using Lyapunov redesign ideas

and the same adaptation law as before, it can be Shown that

V g —eTQe + free + § (2.28)

 

“This definition of ti) replaces (2.6) for the current case. Quantities defined in terms of 11), like (11,
S, if), and w, are now defined in terms of the new 111. Notice, however, that w, remains the same
because 1b, vanishes at e = 0.

22

for all (e, 9, C, 6) E R3, for some he 2 0. Repeating the argument described in Section
2.3, it can be shown that there exist 6" > 0 and u“ > 0 such that for all 0 < e < 6*
and 0 < u < u“, all state variables are bounded and there is time T(e), with T(c) —)
0 as e —> 0, such that (e, 9, 6,5) 6 R3 for all t 2 T(e). Consequently, (2.28) is satisfied
for all t Z T(e). Therefore, the mean-square tracking error is of order 0(u + e) where
the design parameters u and 6 can be made arbitrarily small.

If Assumption 3 is satisfied in the ideal case d = 0, inequalities similar to (2.23)
and (2.24) can be shown in the current case. The right-hand side of such inequalities
will have a term proportional to the disturbance upper bound despite the presence of
the robustifying control component. Thus, such analysis does not reveal an advantage
for the robustifying control. The only advantage we can Show is the fact that the
mean square tracking error can be made of the order 0(u + 6).

Finally, in the ideal case d = 0, the controller with the robustifying component
recovers the tracking-error convergence property of Section 2.3, provided Assumption
2.3 is satisfied. This can be seen by noting that in the ideal case W satisfies an
inequality similar to (2.28) with M replaced by M1, where M1 equals M except for
the constants '74, 76, 65, 66 and 67. The argument of Section 2.3 can be repeated to
Show that M1 is positive definite.

The robustness results of this section and the preceding one have potential ap-
plication to adaptive control of nonlinear systems using neural networks or other
nonlinear function approximators. Consider a system whose input-output model is of
the form y(") = F () + G(-)u(m). Using neural networks, the nonlinear functions F ()

and G'() can be approximated to any desired tolerance. In the special case of linear-

23

in-the-weights neural networks, as in radial basis function networks, the functions F
and G can be represented by F(-) = f;1h,-(-)Vi + 61(-) , G() = :21 h,(-)W.- + 62(-)
for some weights V, and W,. It follows that the system can be represented in the form
(2.21) with d = 61 + rig-um). The discussion of adaptive control using RBF networks

is taken up in Chapter 4.

Chapter 3

Longitudinal Control of a Platoon

of Vehicles

3.1 Introduction

The subject of design and analysis of various longitudinal control laws for automated
highway systems (AHS) has been studied extensively since the late 1960’s. The goal
is to significantly increase the traffic capacity of existing highways through vehicle
and roadway automation. Furthermore, since many of today’s automobile accidents
are caused by human error, automating the driving process may actually increase
highway safety. In such a system, vehicles will be driven automatically with onboard
lateral and longitudinal controllers. The lateral controller will be used to steer the
vehicle around corners, make lane changes, and perform additional steering tasks.
The longitudinal controller will be used to maintain a steady velocity if the vehicle is
traveling alone (conventional cruise control) or follow a lead vehicle at a safe distance

24

25

(vehicle following). In this chapter, we discuss the application of the adaptive con-
trol technique of Chapter 2 to the vehicle following problem. A Simplified nonlinear
longitudinal powertrain model is used for designing the controller. The vehicle pa-
rameters are partially known or completely unknown and are adapted for. We assume
that the following measurements are available to the vehicle’s sensors (i) the relative
distance 1 between the controlled car and the car in front of it and (ii) the forward
velocity of the controlled car. The other quantities of interest, namely the relative
velocity, relative acceleration and the acceleration/ deceleration of the controlled car,
are estimated from the measured quantities. The idea of replacing measured quanti-
ties by their estimates has also been used in earlier works. For example [37] mentions
the possibility of “direct computation” of relative velocity and acceleration using the
measured value for the relative distance. However, in [37] (i) the control objective is
different from the one we consider here, (ii) the model used is a simplified one where
all parameters are assumed exactly known and there are no disturbances and (iii) no
analysis is presented for the case where estimates are used in feedback. Similarly, [1]
uses an estimate of the leading vehicle’s acceleration in the control. However, the mea-
sured quantities still include (in addition to the relative distance and the controlled
vehicle’s velocity) the relative velocity between the controlled and leading vehicles,

and the acceleration / deceleration and propulsion force of the controlled vehicle.

 

1Referred to as the intervehicle spacing in the next section.

26
3.2 Longitudinal Vehicle Model

A widely proposed stategy for effectively increasing traffic throughput on existing
highways through automation is to group the controlled vehicles into tightly spaced
vehicle group formations called platoons [43]. A configuration of a platoon of N+1
vehicles is shown in Fig 3.1. The lead vehicle is numbered 0 and the ith follower
(henceforth referred to as the ith vehicle) is numbered i. L,- denotes the length of the
ith vehicle and x. its position. Let 6,- : xi-) — x,- - L,- for i = 1,2, , N. 6,- is the

intervehicle spacing between the (i — 1)th and ith vehicles. In developing a model

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

. Xi , xi-l
7—,. I—’.
| I
1 Vi I Vi-i
L———> |-——>
N 1 H 0
Li
<——>

Figure 3.1: A platoon of N+l vehicles

for the system, we assume that the road surface is horizontal and that all vehicles
travel in the same direction at all times. From Newton’s Second Law, the relationship
between the acceleration of the ith vehicle, its prOpulsion force, and the drag forces

acting on it can be derived as

mix}- : f,‘ — kdifli‘iz — d + d1i(t) (3.1)

27

where m, is the mass of the vehicle, 2'.- its acceleration, f,- the propulsion force, [£0,515,-2
the aerodynamic drag force, d a nominal constant mechanical drag and d1i(t) the
resultant of the external disturbances (such as wind gust,...etc.) The propulsion
system which represents the engine dynamics of the vehicle can be modelled as a first

order system [1]

f.- = -17(-f. + u.) + can) (32)

T:

where 7',- denotes the vehicle’s engine time-constant, u,- is the throttle/brake input
and d2i(t) is a disturbance term (possibly due to engine transmission variations,
etc.) This model differs from the one in [1] in that both the engine time-constant
and the mechanical drag term are independent of the vehicle’s velocity. However,
we note that the effects of neglecting this dependence can be incorporated into the
disturbance terms d1i(t) and (12:. (t) The constants kdi, m,- and r.- are unknown but

belong to known compact subsets of R1.

3.3 Control Objective and Design

The dynamics of the ith vehicle maybe described by the state vector [6,-, ’05, f,]T, where
u, = x,- is the ith vehicles’s velocity. With this choice of state variables, (3.1) and

(3.2) maybe rewritten as

5r = vi—i - ’01,
a. = (f.- — aim? — d + 2711(2)) /m,- (3.3)
ft = (—f:' +ui)/Ti +d2ift)

28

for l g i ;<_ N. The control objective is to design u,- in such a way that the intervehicle
Spacing 6,- tracks a desired reference. It is well known (see for example [36, 38]) that
for the case where the desired intervehicle spacing is constant, asymptotic platoon
stability can be guaranteed only if the lead vehicle is transmiting its velocity and
acceleration to all other vehicles in the platoon. This approach yields stable platoons
with small intervehicle spacings at the cost of introducing and maintaining continuous
intervehicle communication with high reliability and small delays. In [8], it is shown
that platoon stability can be recovered in a non-cooperative or autonomous operation
if a speed dependent spacing policy is adopted, which incorporates a constant time
headway in addition to the constant distance. This takes the form 641' 2 Av,- + A0,
where 642. is the desired intervehicle spacing and A and A0 are suitably chosen positive
constants. The introduce more spacing between the ith and (i — 1)th vehicles as the
velocity of the ith vehicle increases, which intuitively makes sense. Following [39], we
set A0 to zero, which basically allows for the minimum desired distance between two
adjacent vehicles to be zero provided the vehicle that is following has zero velocity.
With this choice, we‘define the plant output as gm 2 6, — Am. The control objective
is thus to regulate yp'. to zero. Differentiating the output twice and making use of

(3.3), the following error equation is obtained

zip,- = 5}- + 0.T[F.-(v.-, 23,-) + Gm] + D.(t) (3.4)

where 9,- : [kdi/mi,1/r,,kdi/(m,-r,-),1/(m,~r,-)]T, F,() = [2Auiui,)\u,-,Avi2,x\d]T, G =

[0,0, 0, —/\]T and D,(t) = —/\(d11./’r.-+d.1i+d2i)/m,-. From the knowledge of the intervals

29
in which kdi, m, and T,- lie, it is possible to calculate the compact subset of R4 to which
9,- belongs. In particular, suppose that ka, 6 [ML/€327], m,- 6 [mf",mf"], and 7‘,- 6
[7,“, 71M ] Then

m M 1 1 m M 1

kd k.
mX] dtmlelMM’mmm

m
mm m,- r,’" m ”rim

 

 

 

def
06 9: mM’ XT[T”—-"—’]X [an-M’ ]CR4

17"!

By defining Ypi = [ypv (MP, it is possible to rewrite (3.4) as
Yl’i = AmYpi + b{KYPi + 6i + 0iT[E(’Ui, ’01) + Gui] + Di(t)},

where (A, b) is a controllable canonical pair that represents a chain of two integrators
and K is chosen such that Am 2 A — bK is Hurwitz. To estimate 5,, (i,- and u}, we
use two high-gain observers, driven by 6,- and 2),- respectively. Denote the estimates
by 6:, i5:- and ii, respectively. The high-gain observers are described by the following

equations

191'. 2 W21. + ,61(9i — wli)/e
2'11. = zgi + a1(v,- — 21,-)/6
192,- = 103,- + 5292' — w1,-)/£2
] and 22:. = 02(1). — zli)/62
793,- = 33(51 — “(Uni/63

A A
C

6i = “121-, 6i :

vi = 22,-

 

where e > 0 and the 0’s and 9’s are chosen such that the roots of 32+013+a2 = 0 and

A

53 + [6132 + 52.5- + 93 = 0 have negative real parts. Let y“: 6— AUhY p,- "it/pi, 9p,-lT

and assume that an upper bound on the term D,(t) is known. Then the control u.- is

30

designed as
.1 ~T . ..
—6i_9i17iia.i_KY-+r-
u,- = (“f ) ”I ”1 (3.5)
9,- G

 

A

where 9,- is an estimate of 9,- and um. is a robustifying component designed using
the Lyapunov redesign technique,e.g., [25, Section 13.1]. The control u,- is saturated
outside a compact set of interest to prevent the peaking induced by the high-gain
observers [9]. 2 The parameter adaptation law is chosen as in Chapter 2.
Proceeding along the lines of Chapter 2, it is possible to show ultimate bounded-

ness of the spacing deviation error. We use the Lyapunov function candidate

T ~T - ~
v, = YpiPYpi + to, r 19,-,

where P 2 PT > 0 is the solution of the Lyapunov equation PAm + AgP = —I, 9,- =
9,- —9,- the parameter estimation error and P 2 FT > 0 the “adaptation gain”. Though
the proof in Chapter 2 was done for the single-output case, an extension to the multi-
output case is not very difficult, and has been addressed, for example, in [28]. It is
worth mentioning that the proof in Chapter 2 guarantees only the boundedness of
ypi and ypi. To argue boundedness of 6,, (ii, if], u,- and 2),, we first assume that there
exist achievable bounds on the leading vehicle’s velocity uo [39] and acceleration no
[8]. Noting that

All] +111 2 U0 — 9P1

and that gym is bounded and A > 0, we see that ul is bounded. Extending this

 

2For the purpose of simulations, the control is saturated at a value slightly higher than the
observed value under state feedback.

31

argument inductively Shows that u,- is bounded for all i. Since 9p.- = 6,- — Am, bound-
edness of 6, follows. Furthermore, since each u, is bounded, so is 5,- : 12,--1 — 21,-. From

9,, = d,- — Aui, each u,- is bounded. And finally, since 6,- : 6,-1 — 25,-, each (i,- is bounded.

3.4 Simulations

In this section, we present two sets of Simulations for a platoon of five cars. In all
simulations, we assume that all vehicles are initially travelling at a velocity of 15
m/s. The lead vehicle’s velocity, acceleration and jerk profiles are shown in Fig 3.2.
We assume that m, E [1100, 1550]kg, T, E [0.15,0.25]s, kdr E [0.1,0.5]N32/m2 and
d = 100N. These values arechosen to be the same as or close to the ones in [1, 39].
For the first set of simulations, we use a value of A = 0.9 and for the second A = 0.2.

The particular values for A are explained in some detail below.

3.4.1 Simulation 1

The value of A = 0.9 is based on the California rule of thumb, [8, 39], which suggests
an intervehicle spacing of one vehicle length for every 10 mph. Assuming an average
vehicle length of 4 m, this translates to a value of A = 0.9 In all simulations, we

assume the following values for the vehicle parameters, m1 = 1300, T1 = 0.16, 117,11 2

0.3, m2 = 1400, 1'2 = 0.22, [€42 = 0.35, m3 = 1200, 7'3 2 0.18, (€43 = 0.2, m4

1350, r4 = 0.24 and [€44 = 0.45. For the first simulation, we assume perfect knowledge
of the vehicle parameters and that d1, (t) and d2, (t) are identically zero. Fig 3.3 shows

the velocity and acceleration profiles for the following vehicles, the spacing deviation

32

errors and their relative positions relative to the leader . 3 Though not clear from
the figure, the spacing deviation error does not exceed 1.5 cm in magnitude. 4 Fig
3.4 is for the case where the vehicle parameters are unknown and nominal values
are used in the control. The disturbance terms are still identically zero. In Fig 3.5,
the vehicle parameters are unknown but are adapted for. Compared to the previous
case, the spacing deviation errors Show a marked decrease. Fig 3.7 is for the case
where d2, is still identically zero, but d1, /m,- is as shown in Fig 3.6. The disturbance
profiles are similar to, though not identical to the ones in [1]. In particular, they
are “smooth” functions of time. No robustifying control is used in this case. Fig
3.8 is for the case where a robustifying control is used. The spacing deviation error
shows a marked decrease in this case. The spacing deviations do not exceed 1.6 cm
in magnitude. The above results compare favourably with the results of [39, 1]. It is
worth mentioning however, that the spacing policy in [1] is different from the one we
adopt here. The spacing deviation errors reported above are also of the same order
of magnitude as in [36, 20], where the spacing deviation errors are between 1 and 10
cm. However, as with [1], the results are not directly comparable owing to differences

in the vehicle model and / or the spacing policy adapted.

3.4.2 Simulation 2

The California rule of thumb takes into account human reaction times and delays

[8]. In automatic vehicle following, human delays are eliminated and we can afford to

 

3This is assumed to be the distance from the front of the car to the rear of the leader. We assume
that L1 = 3.9, L2 = 4 and L3 = 3, all values being in meters.
4We assumed a non-zero initial value for the estimation error 6,- — 6,.

33

have a smaller time headway without affecting safety. In [8], based on a worst case
stopping scenario, where the lead vehicle is assumed to be at full deceleration and the
following vehicle is at full acceleration at the instant the stop maneuver commences,
a value of A in the range of 0.1 to 0.2 is obtained. For this simulation, we assume
A = 0.2. For brevity, we present simulation results only for the case corresponding
to Fig 3.8 of the previous set of simulations, i.e, d1i(t) is not zero and a robustifying
component is used. Fig 3.9 Shows the results for this case, and the decrease in the

relative distances from the leader as we move down the platoon is clear.

34

 

 

 

 

50 I I r I I T
45— ' ' -
gw- . —
v35- ..... ‘
.?-:‘ :
830.. ..... ....................
:25- ............. . ........................... ,H,H-.2.----H72HH.HMH_-H,H_
20- .1 ...... _.
15 l L 41 1 l I
0 20 40 60 80 100 120

 

Acceleration (m/sz)

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

0.05 _r I r r r r I l i l
..,A i Z ' i f
E : : ; i
V o- [-—_J .. ........
{- ‘ : ‘ .
o . . . . .
7 L ' L .
_005 I l l l I I l l I l
0 1O 20 3O 4O 50 60 70 80 90 100 110
t r
0.5.. ’ ............................................ .4
on" f
m 0.. ............................................................................................. ..
E
E_0.5_ ............... ..,
..,
_1._ ............ A. ............................ _.
l l
110 115 120 125

Figure 3.2: Velocity, acceleration and jerk profiles for the leader.

35

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Vehicle velocity (in/s) Vehicle acceleration (hi/$2)
.I .
. ,l
35 . . .. . . .....
. f . [
25.. . .. ,. ......... .......
_ i _1.5 .................
20.... .. ...... .(
' ' _2 .........
15 * —2.5 -
0 50 100 0 50 100
10 x “3'4 Separation error (m) Position relative to platoon leader (m)
’ I 160 l. .. ................. ...... d _,
_ gar; 5 3 — Car1 f y I (
' ar : f ----Car2: : I (
---Car3 ; : 140" ""CBI‘3TWH " “2;!‘1
--Car4 g 3 --Car45 /'. ]

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 3.3: Perfect knowledge of vehicle parameters, D,(t) = 0.

36

Vehicle velocity (m/s) Vehicle acceleration (we?)
45 1 . i 2

 

 

35*

. d~
-. .1 5“,

 

30. _ ‘, _ ......

J

I
—L
o
01

f

 

. Y~3"7.3Vv‘4vvivgi‘v.v&v'Ivv ; ‘ ’*
l. j 1

20» -----

 

 

 

 

 

 

15 ’ ‘ -2.5 ‘ ‘
O 50 100 0 50 100

 

Separation error (m) Position relative to platoon leader (m)

 

 

 

 

'. . ] 160"" ‘ ............... .., ...... I‘J
1""— CEIIW " ' I — Car1; i
Car2i ‘ . Car27 ' ,1
0'8....._ Cara ............... ...... H 140.. __. Cara ............... _"'/."
-- Car4g ; l -- Cent} ’3 l
0.6" ....... l4 I // I
‘ ‘ . 120 ................ ................. :......‘.‘—l
0.4 ................ .. --< - l‘
. . l . .

 

 

 

 

 

 

02-... ........ i: ............ ...... i 100r~~~r~~~~

0.. 805 ..... K ............ .’.. . ..... 3:.)

60 ...... .,.--.'.'.-.-.' ‘L 4

-o.4

-O.6

 

-0.8

 

 

 

 

 

_1 A 1 1 I
0 50 100 0 50 100

 

 

Figure 3.4: Vehicle parameters unknown, control based on nominal values, D,(t) = 0.

37

Vehicle velocity (m/s) Vehicle acceleration (111/52)
45 . T 2 . .

 

 

 

 

20).. .. _. ......... ...... E

 

 

 

 

 

 

15 i ‘ —2.5 i i
o 50 100 o 50 100

Separation error (m) Position relative to platoon leader (m)

I I T I

 

 

 

 

. ...... . ..... ....... 4 H _Car1l ’\

_ .. .. .. -~--Car2 ..... § ......... .,'--Cr . "
ml "'Car3 3 ’4°' --C:r3§ ’5’ i

 

 

 

 

 

 

120 ......... . ....... .......... /. ...... ...... :4

1m], ...... ..,-I.'.....

—0.01

 

—0.02[. .............. ................. ....... H ,‘
4103] ~ .......... _______ i

_004.l . ............... ....... .,

 

 

 

 

 

 

-o.05 '1 L
0 so 100

 

Figure 3.5: Vehicle parameters unknown but adapted for, D,(t) = 0.

38

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

39

 

 

 

 

 

 

 

 

 

 

 

 

 

 
 
  

 

 

 

 

 

 

Vehicle velocity (m/s) Vehicle acceleration (tn/$2)
45 .' T 2 r r
40...... ..... ;.
f 1
35 ............... : .............. I.
l
30-~ *
25. ............... ...... ........ .
20 ....... :.. ....... a
15 . 1.
O 50 100
Separation error (m) Position relative to platoon leader (m)
0.2 . . . r
’ : 160" ....... a
— Car1 , — Car1 ; 1 [Fl
._. Cars; 140”.-. Car3.i ............... ;'-""'f
i I [

 

 

 

 

 

 

 

 

 

Figure 3.7: D,(t) 76 0, robustifying component not included.

Vehicle velocity (in/s)
45 i .
4O - a
' i
35 - . _
l
30 > .
r
25 _ ............... _,
20 - j ...... «
15 * ‘
0 50 100
Separation error (m)
0.015 .' r
— Car 1
0.01 . ._ . Car 3 ....... ..

0.005 I.

-0.01 r

-0.015
0

 

 

 

 

 

 

 

 

 

 
 

 

 

 

 

 

 

 

40

160 ................................. [a
_ Car1 I“
140»... _ Cars ............ ’ ’- - 'J
- - Car4 / l
120 ......................... /./ ............ ‘1

 

Vehicle acceleration (m/s?)

 

 

 

 

 

50 100

Position relative to platoon leader (m)

 

 

 

 

 

 

 

 

 

Figure 3.8: D,(t) 75 0, robustifying component included, A = 0.9.

41

Vehicle velocity (m/s) Vehicle acceleration (in/52)
45 . r

 

 

40).. ......... 1]
35~- . ’

 

3o). .......... _05 ................. ................ i ...... ..

25.. ...... . . ......... ..,.- [

 

 

 

 

 

 

 

15 A r _3 i L
0

Separation error (m) Position relative to platoon leader (in)
0.01 i r 50 r .
— Car1 :
45.... Wcarz ...... ..... ’\,.(
--- Cars : , l
40..... --Car4 ............... -'-_-'- .._,...\2

 

 

 

0.008 ‘-

 

 

 

0.004

0.002

 

-0.002

 

—0.004

 

 

 

-0.006

-0.008

 

 

 

 

 

 

 

 

-0.01
0

Figure 3.9: D,(t) 5f 0, robustifying component included, A = 0.2.

Chapter 4

Indirect Adaptive Contol Using

RBF Neural Networks

4.1 Introduction

In recent years, the analytical study of adaptive nonlinear control systems using uni-
versal function approximators has received much attention (See [33] for references).
Typically, these methods use neural networks as approximation models for the un-
known system nonlinearities [3, 7, 10, 21, 22, 33, 34, 35, 42].

In section 5 of chapter 2, a mention was made of the potential application to
adaptive control using neural networks. In this chapter, we investigate the use of
a radial basis function (RBF) network for the purpose. From a mathematical per-
spective, RBF networks represent just one family in the class of linear in the weight
approximators. This class includes, among others, splines, wavelets, certain fuzzy sys-
tems and CMAC (Cerebellar Model Articulation Controller) networks (See [10, 11]

42

43

for references).

4.2 Problem Statement

We consider a Single-input—single—output nonlinear system represented globally by the
nth-order differential equation y(") = F () + G(-)u(’") where u is the control input,
y is the measured output, (-)(‘) denotes the ith derivative of (), and m < n. The
functions F and G' are smooth functions of y, y“), ..., y("“1), u, u“), ..., u‘m‘l). In

particular,

F() = Pit/.31"), - - -,y("“”,u, a“), - - -,u(’"”’),
G() = C(y1 31(1),. - ' i Elm—1),“, 11(1), ' ' ' l ”(m-1))

We augment a series of m integrators at the input side of the system and represent the
extended system by a state space model. The states of these integrators are 21 = u,
22 = u“), up to 2m : u(""1) and we set 12 = u("‘) as the control input of the extended
system. Taking x1 = y, x2 = y“), up to x7, = y("‘1) yields the extended system model

‘

531‘ = $i+1, ISiSn—l

2,, = F(x,z)+G(x,z)u

 

2i = 2121.1, IS 2 S m — 1 l (4'1)
im = u
9 = $1

wherexz [2:1, ..., xn]T, z: [z], ...,zm]T.

44

Assumption 4 [C(x, z)| 2 k, > 0 V x E R" and z E R’".

Assumption 4 ensures that (4.1) is input-output linearizable by full state feedback.

As before, it can be shown that there exists a global diffeomorphism,

x x
= déf T(x, z)

C T1032)

with T1(0,0) = 0, which transforms the last m state equations of (4.1) into C =
H (C, x). This, together with the first n state equations of (4.1), defines a global normal
form. The objective is to design an output feedback controller which guarantees that
the output y and its derivatives up to order n-l track a given reference signal y, and
its corresponding derivatives, while keeping all the states bounded. The reference y,
and its derivatives up to order n are assumed to be bounded and yin) is assumed to

be piecewise continuous.

4.3 Function Approximation Using Gaussian Ra-

dial Basis Functions

The control design presented in this chapter employs an RBF neural network to
approximate the functions F() and G() over a compact region of the state space.
RBF networks are of the general form F (-) 2 9T f (-), where 9 E R” is a vector of
adjustable weights and f () a vector of Gaussian basis functions. Their ability to

uniformly approximate smooth functions over compact sets is well documented in

45

the literature (see [35] for references). In particular, it has been shown that given a
smooth function F : f2 l——> R, where Q is a compact subset of Rm” and c > 0, there
exist a Gaussian basis function vector f : Rm” l——> RP and a weight vector 9* 6 RD
such that [F(x) — 9’Tf(x)| _<_ e V x E Q. The quantity F(x) — 9"Tf(x) déf d(x) is
called the network reconstruction error.

The optimal weight vector 9* defined above is a quantity required only for ana-
lytical purposes. Typically 9* is chosen as the value of 9 that minimises d(x) over (2,
that is,

9* = arg gggfigg IF (:6) - 0Tf(x)|} (4-2)

The choice of the Gaussian network parameters used in our control design is
motivated by the discussion in [35, Section III]. The basis functions are located on a
regular grid that contains the subset of interest of the state space. The update law

for the weight vector 9 is derived in the next section.

4.4 Control Design

In this section, we first design an adaptive output feedback controller under the
assumption that the network reconstruction errors are “small”. Next, the condition
of small reconstruction errors is relaxed by adding a robustifying control component
to make the mean-square tracking error arbitrarily small. The design of the output
feedback controller is done in two steps: first, a state feedback controller is designed;
then, the states are replaced by their estimates provided by a high-gain observer. We

start with the following representation for the functions F (-) and G(-), valid for all

46

x E Y and z E Z, where Y and Z are compact sets defined later in the chapter.

F(x, 2) :2 9;Tf(x, z) + dp(x, z), C(x, z) = 9;Tg(x, z) + dG(x, z) (4.3)

Assumption 5 The vectors 9} and 9; belong to known compact subsets

Q, C R”1 and 99 C R“.

Typically, some off-line training is done to obtain values 9m and 990 that result
in “good” approximations of the functions F and C over Y x Z. This can be ac-
complished, for example, by the standard gradient descent technique 1 [13]. The sets
S2, and {29 are then chosen judiciously as compact sets that contain 9,0 and 990. If
we denote the “optimal” reconstruction errors that result from the use of the vectors
9} and 9; by d};(-) and d3(-) respectively, then, in view of the off-line training, it is
reasonable to expect that our choice of the sets Q; and fig will result in reconstruction
errors dp(-) and dG(-) that are comparable to d};() and da() respectively. Notice also
that it is simply possible to choose the sets 0, and fig arbitrarily large. However, this
would be undesirable from the viewpoint of using parameter projection. The fixed
optimal weights 9} and 9; in (4.3) are replaced by their time varying estimates 9; and

99, that are adapted during learning. The network approximations associated with

these weights are denoted by F and C respectively.

Assumption 6 |G(-)| 2 k2 > 0 V x E Y, z E Z and 99 6 {29, where 529 is a compact

set that contains $29 in its interior.

 

1For our purposes, a variant of Matlab’s solverb function was used, see Appendix B

47

4.4.1 Small Reconstruction Error

Under the assumption of small reconstruction errors, we design an adaptive contoller
so that the output y tracks the given reference signal y... Let e, y(t), y,(t) and yR(t)
be defined as in Section 2.2.1 of Chapter 2 and Y0 and YR be any given compact
subsets of R" and R“1 respectively, such that y(0) 6 Y0 and yR(t) 6 YR V t Z 0.

We rewrite (4.1) as

A

e = Ame + b{Ke + 9}Tf(e + yr, 2) + 93Tg(e + yr, z)u
+d(e+yr,2,9f,9g) _ 9191)} l (4'4)

Z 2 A22 +02?)

 

where 2 d(e+yr, 2, 9f, 99) = dp(-)+dG(-)u, (A, b) and (A2, b2) are controllable canon-
ical pairs that represent chains of n and m integrators, respectively, and K is chosen

such that Am = A -— bK is Hurwitz.

Assumption 7 The system C = H (C, 37,) has a unique steady-state solution C. More-
over, with C = C — C the system

C : H(C—+§ie+yr) _ H(C-iyr)
(4.5)

d-i‘f H(€7 8, yr) 5)

 

2The dependence on 9 f and 99 comes through 0. See (4.7).

48

has a continuously diflerentiable function V1(t,C~) that satisfies 3

mlléll2 S 139.5) S n2||§||2

av. av, - ~ _ - 2 -
— '—.:' < -
a, + at H(C,e,y.,<) _ nsllcll +174||C|H|e||

where 771,1)2, 713 > O, and 774 2 0 are independent of 37,.

Assumption 7 implies that (4.5), with e as input, is input-to-state stable. Conse-
quently, the zero dynamics of (4.1) are exponentially stable and (4.1) is minimum

phase.

State Feedback

Let P : PT > 0 be the solution of the Lyapunov equation PAm + AgP = -Q where
Q = QT > 0, and consider the Lyapunov function candidate

V = 31%: + @1716, + $3316, (4.6)

where9,=9f—9;, 99:99—95”; and F;=Ff>0and Fg=FZ>0aregainstobe
specified later. Using (4.4) the derivative of V along the trajectories of the system is
given by

V = —eTQe + 97F;19f + 9TI‘;199 + 2eTPb{ 9;Tf(e + yr, z)

9

+ 9;Tg(e + yr, Z)U + d(e + yrs 2, éfa 69) + K6 — yin)}

 

3Unless otherwise specified, || - M denotes the Euclidean norm.

49

Taking

—Ke+ Inl—Fe+y.,z,é e « -
“U 2 El ( A f) (i=r1/J(C,Z,y3,0f,09) (4.7)
G(e+y,.,z,9g)

 

we can rewrite the expression for V as
V = —eTQe + b"; 1719‘, — r,¢,] +é§r;1[ég — r9459] + 2eTPb d() (4.8)
where

if = 2eTbe(e + 32., z), at = ZeTPb9(e + 32., zit/’0

Let Of be a compact subset of R”1 that contains (if in its interior. Define

r = diag[I‘,,I’9], n = n, x n, and {2 = 0, x 0,

The parameter adaptation law is chosen as in Chapter 2. By our design of the
RBF network, we seek to impose a bound on d(-) over a compact subset of RM“.
With that goal, we first assume that e(O) and 2(0) belong to known compact subsets
E0 C R" and Z0 C Rm and let c1 = maerE0 eTPe. Choose c4 > c1 and define

E d-—e-f {eTPe g c4} and Y déf {e + yrle E E, y, 6 YR}. Let Z be a compact subset of

50

Rm such that Z0 is in the interior of Z and
z(0)€Zoande(t)€EVtZ 0 => z(t)€ZVtZO.

The RBF networks are used to approximate F () and G () over the compact set Y x Z.

Let (25 be as in Chapter 2. Define Q6; and {259 by (25 = 95!. x 959 and let

02 = max %(9; — 9})TI’;1(9f — 9;),

o;en,,é,en,,

(é, — 9;)Trg-1(ég — 9;).

c3 = max %

agengflgemg
The adaptation gains I’, and F9 are chosen large enough to ensure that c4 —c1 > c2+c3.
This is different from Chapter 2 where the adaptation gain is not required to be large.
This is because, in Chapter 2, the parameter vector 9 has some physical meaning
and the compact set 0 to which it belongs is known apriori. In particular, the
definition of the set E implicitly involves the set Q. In the present case however,
the compact sets f2] and {29 to which the optimal weights 9; and 9; of the neural
network belong themselves depend on the set E, because the approximation of F
and G is done over the set E x Z. Hence the set E has to be defined prior to, and
consequently, independent of the sets Q, and fly. This requires making the adaptation
gains large. Let d1 = max||d(e + y... Z, 9,, 99)“, where the maximization is done over

all e e E, y. e Y,z e Z, 6, e 05, and (i, e 059. From (4.8) and (2.7), we have

V g —eTQe + kddl Ve E E, where kd = mglx2]]e]] ]]P b]] (4.9)

51
If d1 < d” = k(c4 — c3 — c2)/kd , where k 2 M then V < 0 on {V = c4} (1 95.

Amuzr:(P) ’

Thus the set {V S c4} (1 Q; is positively invariant for all d1 < d‘. Inside this set,
8 E E. As long as e E E, 2 will remain in Z. Thus the trajectory (e, z, 9) is trapped
inside the set R, = {e E E} x {z E Z} x {9 E (25}. Hence all the states are bounded

and from (4.9), the mean-square tracking error is of the order 0(d1).

Output Feedback

To implement the controller deveIOped in the previous section using output feed-
back, we replace the states 6 by their estimates é provided by a high gain observer
(HGO). The control is saturated outside a compact region of interest to prevent the
peaking induced by the HGO. We assume that 91(0) 6 Q; and 99(0) 6 99. Let
S 2 max]w(e,z,y3,9f,9g)] where the maximization is taken over all e E E1 (Li-f

{eTPe S c5}, 2: E Z, 32,, 6 YR, 9, E 95f, 99 E 5259 where 05 > c4. Define the

saturated function W by

 

We, 2.3% éfiég))

¢8(€,Z,y12,9f,9g) = S sat ( S

where sat(-) is the saturation function. The output feedback controller is taken as
u = ib’(é, z, 323, 9,, 99). The HGO used to estimate the states is the same one used in

Chapter 2, i.e,
6,‘ '2 éi+1+%(€1—él), 1$i_<_n—1

(4.10)

6n = %rli'(el_él)

52

where c > 0 is a design parameter that will be specified shortly. The positive constants
a, are chosen such that the roots of s” + ms"—1 + - - - + an_ls + an 2 0 have negative
real parts. Let C,- = (e,—é,-)/e"", 1 S i S n, f = [51, . . . ,Cn]T and V5 2 {TI-DC, where
R 2 FT > 0 is the solution of the Lyapunov equation F(A—HC)+(A—HC)TF = —I.
Boundedness of all signals of the closed-loop system can be proved by an argument
similar to the one in the state feedback case. First, it is not difficult to show that for
all (e, 9) E {V S c4}fl (25, there exist constants c6, c7 > 0 such that the sets {V1 S as}
and {V5 S 676.2} are positively invariant. Next, using the results of Section 2.4, for
all (e,9, C, E) belonging to the set R = {{V S 04} (1 {25} x {VI S c6} x {Vg S C762},

the derivative of V satisfies

V S —eTQe + kg: + kddl, where k, > 0. (4.11)

Hence for all

 

 

k(C4—63—C2) ande<e‘=k(C4—C3—CQ)

d d“ =
1 < 2k.) 21:.

(4.12)

V < 0 on {V 2 c4} (1 (25 and the set R is positively invariant. Using the difference
in speeds between the slow and fast variables and the fact that Vg S —(1/2c)]]€]]2
outside {V5 S C762} it can be shown that the trajectory enters the set R during the
time interval [O,T(e)], where T(e) —> 0 as e —) 0. Hence, as in the previous case, for
sufficiently small d and c, all the states are bounded and (4.11) is satisfied for all

t 2 T(e). Hence the mean-square tracking error is of the order 0(6 + d1).

53

4.4.2 Reconstruction Error With a Known Bound

As in Section 2.5, we design an additional robustifying control component to make
the mean-square tracking error arbitrarily small, irrespective of the bound on the

disturbance d(-), provided this bound is known. Let

_ —Ke+y£") -F(€ +yriZ,éf) +111
e(e+y.,z,é,)

 

Assume that

||d(')l| 5 1003.4?) + kvllvilli 0 S k, <1

where p and k, are known. Take n(e, z) 2 p(e, z) and define s = 2eTPb,

f

 

’ 44635.10 for nlczilsi 2 .1
Wm) =<
2 e,z ‘
l ‘ 23—4.) 7‘: for nlcziisl < h

and

. . —Ke+ (n)_fie+y,.,z,é + rm
40,212.20»: 9 - ( all M >
G(6+yr.z,0g)

 

The adaptive controller is taken as u = w’(é, 2,32,10,69). As in Chapter 2, it can be
shown that El 6* > 0 and u‘ > 0 such that V 0 < c < 6‘ and 0 < u < u‘, all signals
are bounded and the mean-square tracking error is of the order 0(6 + u), where the

design parameters 6 and u can be made arbitrarily small.

54

4.5 Simulations

In this section, three simulations are presented to illustrate the points made in the
earlier sections. The programs for the simulations are written in Matlab, using the
Neural Network toolbox. In the first simulation, we Show the effect of changing
various design parameters on the tracking error. In the second, we attempt to justify
the need to adapt for the network’s weights. Lastly, we demonstrate the effect of the
network’s size on the controller’s performance. The plant used in all these simulations

is the same one used in [35, 42], namely

9 = F(y,y) + G(y)u, where
. , , 2
szn(47ry) (sin(7ry)) and

4ny «y
C(y) = 2 + sin(37r(y — 0.5))

 

F(y.z'/) = 16

4.5.1 Simulation 1

The plant output is required to track a reference signal y, that is the output of

a low -pass filter with transfer function (1 + s/10)‘3, driven by a unity amplitude

square wave input with frequency 0.4 Hz and a time average of 0.5. The reference

and its derivatives are shown in Fig 4.1. As can be seen, the set 313 can be taken as

[0, 1] x [—3, 3] x [—25, 25]. Since m = 0, there is no need to augment integrators at the
~ def

system’s input. Let Y z [—1, 1] x [—3, 3]. We use 2 RBF networks to approximate

the functions F (y, y) and G(y) over Y. The networks have 48 Gaussian nodes with

55

variance 4 o2 = 47r spread over a regular grid that covers Y. Off-line training is done
to obtain weights 9 f0 and 990 that result in “optimal” approximations of the functions
F and G. However, the reconstruction errors are still quite large in this case, at
some points being comparable to the value of the function. Based on the values of
9h, and 990, the sets Q; and 99 in Assumption 4 are taken as [9fo — 01,910 + 0.1]
and [990 - 0.1, 990 + 0.1], where the addition and subtraction are done componentwise.
The adaptation gains I’; and F9 are taken for simplicity as 1031. The values of the
various design parameters are 6 = 0.001, n = 40 and k, = 0.7. The initial condition
x(0) is taken as (-0.5,2.0). Fig 4.2(a) shows the tracking error for the state feedback
case with p = 0.5, Fig 4.2(b) is the output feedback case with c = 10‘3, Fig 4.2(c)
with 6 reduced to 10”, and Fig 4.2(d) with u reduced to 0.1. Fig 4.3 shows the
corresponding control inputs. The simulation illustrates several points : (a) by using
a robustifying component, it is possible to obtain reasonable performance even with
networks that give large reconstruction errors; (b) as c is decreased, we recover the
performance obtained under state feedback; and (c) an n-fold decrease in it results
in approximately an n-fold decrease in the tracking error. Thus, by decreasing u, we

can meet more stringent requirements on the tracking error.

4.5.2 Simulation 2

The initial weights obtained by off-line training may not be close to their optimal

values. This might, for example, be the case when the off-line training is done (based)

 

4See [35] for a definition of this term in relation to RBF networks

56

on a nominal model that differs considerably from the actual one. For definiteness,

suppose the function F (y, y) is any one of the functions

 

my ,1) Z ,1 sin-(440 — 42)) (more — k3)))2

49(9 — k2) 7r(9 — k3)

where k, 6 [15,17], kg 6 [——0.5,0.5] and kg 6 [—1, 1] and that a nominal model is the

one used before, that is,

 

sin(47ry) (Sin(7ry))2

Fnom 1. = 1 .
(y y) 6 47w 7ry

For simplicity, we take C (y) = 1. Further, the reference signal is taken as yr 2 0.4.
This time, we use an RBF network with 192 Guassian nodes to “construct” the
function F (), with the parameters of the network chosen as before. Based on the
nominal model, we do off-line training to obtain initial estimates 9b and 990. The
choice of the set Q, is crucial. It is chosen in a way which gaurantees that any of
the functions F(-) mentioned above can be reasonably approximated 5 by some 0f
in (2,. By this, we mean that, forevery possible choice of k1, 1:2 and k3, there exists
0” E Q; such that ]F(-) — 9fo(-)] ~ IFnom(-) — 9fon(-)] V 31,3) 6 Y. For the purpose
of simulation, the values of k1, kg and k3 are taken to be 17, 0.4 and 0 respectively.
This choice ensures that with the “nominal” weights 9 fo, the reconstruction error is
quite large in the region of the state space where the reference lies. Fig 4.4 shows

the function F (y, y) and the error F () — 9foT f () that results from using the nominal

 

5This might require making Q, larger than what it would have been if we had assumed that the
actual and the nominal models are identical.

57

weights. The values of the parameters used in the design are I} = 1031, 6 = 0.001, e 2
10‘4, 77 = 20, k, = 0 and ,u = 0.2. The initial condition 22(0) is taken as (0.9, —2.75).
Fig 4.5(a) Shows the tracking error for the case when there is no adaptation for the
weights, that is, F, = 0 and no robust control component, Fig 4.5(b) for the case
when the weights are adapted but there is no robust component, Fig 4.5(c) for the
case when the weights are not adapted but there is a robust component, and Fig

4.5(d) for the case when the weights are adapted and a robust component is used.

The following points are noteworthy. In the first case the tracking error is quite
large because we simply do a crude cancellation of the network nonlinearity based on
a nominal model. When we start adapting for the weights, the difference between
the function F and its estimate F provided by the network decreases and hence the
tracking error also decreases. However, even with the network providing its “best”
approximation, there is a residual error. In the case where we simply use robust
control, the performance shows an improvement over the first case and is almost
comparable to the error in the second case. Finally, in the case where we do both
adaptation and robust control, the network reconstruction error decreases and the
robust component handles this smaller error better. Thus the tracking error is the

smallest in this case.

4.5.3 Simulation 3

In Section 4.4.2 we saw that decreasing p results in a decrease in the mean-square

tracking error. While theoretically u can be made as small as we want, it is not

58

always possible in practice to do so. This is because, in many practical applications,
the system contains high-frequency unmodelled dynamics. Decreasing it implies a
“high-gain like” feedback inside the layer [s] < ’5’ which might result in the excitation
of the unmodelled dynamics. In this section, we assume that ,u cannot be made smaller
than 0.1, fix it at this value and examine the controller’s performance as the network
size is varied. To be able to do this, we first need to define a “suitable” measure
of the network’s performance. For a given network, let e1,e2, - - - , en denote ultimate

6 corresponding to initial conditions x01,x02, - - -,x0,,.

bounds on the tracking error
We take the mean-square error ([2? to be a measure of the network’s performance.
We use the same plant used in the previous simulation, with initial estimates for the
weights based on the nominal model. The reference y, is chosen as 0.4+0.lsin(t). We
compare the performance of three networks, having 64, 100 and 144 Gaussian nodes
respectively. The networks are used to construct F on Y = [—1, 1] x [—1, 1]. For each
network, four sets of initial conditions for the state x are used, (-0.9,—0.9), (-0.9,0.9),
(09,-0.9) and (09,09) and the mean-square error is evaluated. Fig 4.6 summarizes
the results of the simulation. The dashed 'line shows the mean-suare error for the case
when only robust control is used, the dotted line for the case when only adaptation
is used and the solid line for the case when both adaptation and robust control are
used. As can be seen, the error is almost constant in the first case. Thus, by using
only robust control, we cannot hope to decrease the error beyond a certain point. In

the second case, the mean-square error decreases as the network size increases. This

is not surprising because increasing the network’s size increases its approximation

 

6As observed by simulation

59

capabilities. Since the error is of the order 0(6 + d), it decreases as the size of the
network increases. This suggests that as requirements on the tracking error become
more stringent, it becomes necessary to increase the size of the network. Lastly, the

performance in the last case is the best of the three cases.

60

 

 

 

 

4.5

.5

3

2.5

I
I
1.5

T
i
0.5

  

 

 

 

 

 

 

 

 

 

 

-20*"""""

_
w
_chm 3:283. ozaméoo Em 0232.3 200.3

30
20
0
0

Figure 4.1: The reference signal and its derivatives.

61

Tracking errors

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

o i i E i i l 1 i i
10 11 12 13 14 15 16 17 18 19 20
--(d)--

 

Figure 4.2: (a)State feedback p = 0.5 (b) Output feedback, 6 = 10‘3, u = 0.5 (c)
Output feedback 6 = 10'4, u = 0.5 (d) Output feedback 6 = 10‘4, u = 0.1

62

Control inputs

 

 

 

 

 

__(a)__

 

 

 

 

 

 

_(c)...

 

 

 

 

 

Figure 4.3: (a)State feedback u = 0.5 (b) Output feedback, 6 = 10‘3, u = 0.5 (c)
Output feedback 6 = 10““, u = 0.5 (d) Output feedback 6 = 10‘4, u = 0.1

63

F()

Function

   

             

\\
’1

  

\\ .
the

 

 

F()-0‘ f()

Error

 

 

(3) F041) 0)) F (31, 3'1) - 9rTf(y, 9)-

Figure 4.4

64

Tracking Errors for 4 Different Cases

 

 

. . y g 0.5 . f . .
0.4... - 0.45“ . 1,, . lllllll VVVVVVV ‘

_ . I g ,4. ........ .f ...... g ________ .
02 ..... . ..... ‘E"'““ 0 : . E

' ' I 035 , .. ..... ........
0.3 ........ ' ...... ........ ....... 1
025 ........ I ..... .
0.15 I ' V l

0.1 >

0.05 "

 

 

 

 

 

 

 

 

 

 

 

0-5 T T f f 0'5 ' r T ._
0.45 ........ ........ ...... ....... ...... . O.'”45“'”"" “

o4. ....... ., 0'4

0.35
0.35

0.3
0.3»
0.25.
0.25- 3 3 E E

0.2
0.15

0.15 0.1

0.1 ' 0'05

 

 

 

 

 

 

 

0.05

 

-lci— —(d)-

Figure 4.5: (a) No adaptation for weights, no robust control (b) Only adaptation for
weights (c) Only robust control (d) Adaptation for weights and robust control.

65

Effect of Network Size on Controller's Performance

 

 

 

 

 

 

    

 

 

 

0-2 I I f I I I I I
’3 (640.1928) 2 E i i ------- adaptive
2 , : : : : -— robust
. g —— robust&adaptive

0.18”"'" Q ....... ....

(N,e) : N - Number of nodes, 9 '- Mean—square tracking error
0.16_ ......... ' ........ _.
0.14_.. ............................................ ...
§0.12— -‘
G}
O)
.E
x
O
.5?
2 01.. . .............................. _.
(B
3
U
‘1’
C
(U
0
2008- ....................................................................................... ..
0.06._\. .......................................... —
-.3 (10000405)

Q04_W .................. x ...... ......... .1
(64.0.0273) (14490274)
...—.3-—__:__—_3____s..___2_———2—————3"4”—---.+..-._§

: : : (109.0-0271) : : : (14400252)
0.02.— ......... .......... ......... .......... : .......... ......... ...
(64.0.0097) f : 3 _
' (10.0.0.0035) g ; (14409022)
0 i l i T 1 L i ifi“
60 70 80 90 100 110 120 130 140 150

Number of Gaussian Nodes

Figure 4.6: Effect of network size on the controller’s performance.

Chapter 5

Conclusions

We have studied the application of an adaptive control technique to two different
problems, (a) the longitudinal control of a platoon of vehicles and (b) output feedback
control of nonlinear systems using RBF neural networks.

For the platoon problem, good performance has been achieved in the presence of
parameter uncertainities and unknown time-varying disturbances. The main contri-
bution of the method is the use of high-gain observers to reduce the number of sensor
measurements. In particular, we do not require direct measurement of the relative
velocity or acceleration between the controlled and leading vehicles or the controlled
vehicle’s acceleration. A drawback in the present work is the use of the same model
for the controller design and in the simulation, i.e, we do not consider imperfections
in the plant such as time delays involved in actuators and sensors etc. This drawback,
while present in the work of [8],[39], has been addressed in [20]. Phture work on this
problem must therefore include these imperfections. Another possible area of work
is the study of robustness of the proposed method with respect to a large class of

66

67

commanded maneuvers (such as when a vehicle exits the lane or when two platoons
merge to form a single platoon) in the multi—platoon scenario.

The second problem we have studied is adaptive output feedback control of non-
linear systems represented by input-output models. The objective of the design is to
achieve good tracking performance in the absence of known system dynamics. RBF
networks are used to approximately construct the system nonlinearities. The recon-
struction errors of the networks are not required to be small, thus allowing for the use
of lower-order networks. Simulations are done which illustrate the effect of changing
various design parameters and of the network size on the controller’s performance.
Possible future work involves investigating the effect of unmodelled dynamics on the
controler’s performance and the use of nonlinear-in-the-weights networks. The latter
especially holds promise because it is known that nonlinear networks have inherently
“better” approximating capabilities in comparison to linear-in-the—weight networks
[4]. In particular, for sigmoidal networks, it is shown that with the basis functions
“tunable”, the integrated square error can be made of the order 0(1/n), n being the
number of basis functions (or hidden layer weights), whereas, for the case where the
basis functions are fixed, the approximation error is typically of the order 0(1/n1/d),
d being the dimension of the input space. Thus nonlinear networks can achieve more
compact representations of nonlinear functions compared to linear networks, espe-

cially when the dimension of the network input is high.

 

APPENDICES

Appendix A

Matlab programs for simulations in

Chapter 3

A.1 run.m

ZZZXZZZZZZZZZZ%XXZZZZZZZZZZZZZZZZX%%%%%%%%%%%%%%%%%%%%Z%%%

Z This is the main program, it uses the model simrun.md1 %

7. which in turn calls the H-file runn.m - 7.
ZZZZZZZZZXZXZZZZXZZZ2ZZXXZ2Z1ZZZZZZZZZZZZZXZZZZZZZZZZXZXZZ

clear all; tf=125; [t,x]=sim(’simrun’,tf);

subplot(2,2,1)
plot(t,out(:,1).’-’,t,out(:,4).’:’,t,out(:,7),’-.’,t,out(:,10),’--’);
grid; axis([0 125 15 45]); title(’Vehic1e velocity (m/s)’)
subplot(2,2,2)
plot(t,out(:,2),’-’,t,out(:,5),’:’,t,out(:.8),’-.’,t,out(:,11),’--’);

68

69

grid; axis([0 125 -3 2]); title(’Vehicle acceleration (m/s“2)’)
subplot(2,2,3)
plot(t,out(:,3),’-’,t,out(:,6),’:’,t,out(:,9),’-.’,t,out(:,12),’--’);
grid; axis([0 125 -0.001 0.002]); title(’Separation error (m)’)
subplot(2,2,4)

L1=-out(:,3)+0.2*out(:,1); L2=L1+3.9-out(:,6)+O.2*out(:,4);
L3=L2+4.0-out(:,9)+O.2*out(:,7); L4=L3+3.8-out(:,12)+O.2*out(:,10);
plot(t,L1,’-’,t,L2,’:’,t,L3,’-.’,t,L4,’--’);

grid; axis([0 125 O 50]); title(’Position relative to platoon leader (m)’)

1egend(’Car 1’, ’Car 2’,’Car 3’, ’Car 4’)

A.2 runn.m

function [sys,xo,str,ts]=runn(t,x,u,flag)
ZZZZZZZZZXX7%Z%%%%Z2%ZZZZZZZZZZXXXZXZZZXZXZKZXZZZZZZZZZZZZZXZXXXZZZZZZZZXZ
% This file is used in calculating by the simrun.mdl file and specifies Z
Z all the relevant equations.The assumed definition of delta differs (in Z
2 sign) from that in Chapter 3. As a result, some signs differ from the Z
% ones mentioned in Ch 3,for eg, =delta+1ambdatv and not delta-lambdatv Z
ZZXZXZZZZZXXZZZZXZZZZZZXZZZXZXXZXZZZZXXZXZXZZZZZZZZZXZXZZZZXXZXZZZZZXZXZZZ
if f1ag==0,

Z x=[delta v f delta“ deltadot“ deltaddot‘ v“ vdot‘ thetas] Z

x01=[-3 15 167.50 -2 0 0 15 0 -0.0001 -4.5 -0.0005 0.0030] ;

xo2=[-3 15 178.75 -2 0 0 15 0 -0.0003 -5.0 -0.0008 0.0040] ;

70

x03=[-3 15 145.00 -2 0 0 15 0 -0.0002 -4.2 -0.0010 0.0050] ;
xo4=[-3 15 201.25 -2 0 0 15 0 —0.0004 —5.5 -0.0004 0.0045] ;
xo=[xol x02 103 104]; sys=[12*4,0,3*4,0,0,0,0];

else

X ----- define constants and ’reference’ ----- %
lambda=0.2; epsilon=le-3; Gamma=1e-4; de1=1e-3;

eta=4.5; kv=0; mu=0.1; d=100;

K=[3 4]; P=[1.1167 0.1667;0.1667 0.1167]; bb=[0 1J’;
a=[-0.00046;-20/3;-0.003;0.0026]; b=[-0.000064;-4;-0.00025;0.0061];
if t<=30,

wddot=-1/90; wdot=(30-t)/90; w=15+t/3-t‘2/180;

elseif t<=40,

wddot=0; wdot=0; w=20;

elseif t<=70,

wddot81/60; wdot=(t-40)/60; w=20+(t-40)‘2/120;

elseif t<=100,

‘wddota-1/60; wdot=0.5+(70-t)/60; w=15+5*t/3-(t‘2/120)-(190/3);
elseif t<=110,

wddot-O; wdotIO; w=35;

elseif t<-115,

wddot=0.4; wdot=0.4*(t-110); w=35+(t-110)‘2/5;

elseif t<=120,

wddot=-1; wdot=117-t; w=-6802.5+117*t-t‘2/2;

else

71

wddot=0.6; vdot=-3+O.6*(t-120); w=397.5-3*t+0.3*(t-120)“2;

end
Z ---- define states, exogenous inputs and derivatives ---- Z
m1=1300; A_ph01=0.30; tau1=0.16; m2=1400; A_pho2=0.35; tau2=0.22;
m3=1200; A_ph03=0.20; tau3=0.18; m4=1350; A_pho4=0.45; tau4=0.24;
thlnomls-A_ph01/m1; th1nom2=-A_pho2/m2;
th1nom3=-A_pho3/m3; th1nom4=-A_pho4/m4;
th2nom18-1/tau1; th2nom2=-1/tau2; th2nom3=-1/tau3; th2nom4=~1ltau4;
th3nom1=-A_phol/(mittaul); th3nom2=-A_pho2/(m2*tau2);
th3nom3--A_pho3/(m3ttau3); th3nom4=-A_pho4/(m4*tau4);
th4nom181/(mlttau1); th4nom2=1/(m2*tau2);
th4nom3=1/(m3*tau3); th4nom4=1/(m4*tau4);
thnom1=[th1nom1;th2nom1;th3nom1;th4nom1];
thnom2=[th1n0m2;th2nom2;th3nom2;th4nom2];
thnom3-[th1n0m3;th2nom3;th3nom3;th4nom3];
thnom4=[th1nom4;th2nom4;th3nom4;th4nom4];
deltal-x(1); v1=x(2); f1=x(3); delta2=x(13); v2=x(14); f2=x(15);
delta3-x(25); v3=x(26); f3-x(27); delta4=x(37); v4=x(38); f4=x(39);
y1=delta1+lambdatv1; y2=delta2+lambda¢v2; %
y3=delta3+lambdatv3; y4=de1ta4+1ambdatv4;
delta_hat1=x(4); v-hat1=x(7); delta_hat2=x(16); v_hat2=x(19);
delta_hat3=x(28); v_hat1=x(31); delta_hat4=x(40); v_hat2=x(43);
th1=[x(9);x(10);x(11);x(12)]; th2=[x(21);x(22);x(23);x(24)];

th3=[x(33);x(34);x(35);x(36)]; th4=[x(45);x(46);x(47);x(48)];

72

x4dot1=x(5)+6*(deltal-x(4))/epsilon;
x4dot2=x(17)+6*(delta2-x(16))/epsilon;
x4dot3=x(29)+6*(delta3-x(28))/epsilon;
x4dot4=x(41)+6*(delta4-x(40))/epsilon;
x5dot1=x(6)+11*(deltal-x(4))/(epsilon“2);
x5dot2=x(18)+11*(delta2-x(16))/(epsilon‘2)3
x5dot3=x(30)+11*(delta3-x(28))/(epsilon“2);
deot4=x(42)+11*(delta4-x(40))/(epsilon“2);
16d0t1=6*(de1ta1-x(4))/(epsilon‘3);
x6dot2=6*(delta2-x(16))/(epsilon“3);
16d0t3=6*(delta3-x(28))/(epsilon“3);
16d0t4=6*(de1ta4-x(40))/(epsilon‘3);
x7dot1=x(8)+4*(v1-x(7))/epsilon;
x7dot2=x(20)+4*(v2-x(19))/epsilon;
x7dot3=x(32)+4*(v3-x(31))/ep8ilon;
x7dot4=x(44)+4*(v4-x(43))/epsilon;
18d0t1=3*(v1-x(7))/(epsilon‘2);
x8dot2=3*(v2-x(19))/(epsilon‘2);
18d0t3=3*(v3-x(31))/(ep8110n“2)3
x800t4-3*(v4—x(43))/(epsilon“2);

if t<=30,

d1130; d12=0; d13=0; d14=0;

elseif t<=45

d11-0.45*(1-exp(-1.2*(t-30))); d12=0.44t(1-exp(-1.2*(t-30)));

d13=0.

elseif

d11=0

d12=0

d13=0

d14=0

elseif

d11=0

d12=0.

d13=0.

d14=0.

elseif

d11=0

d12=0

d13=0

d14=0.

else

73

40*(1-exp(-1.2*(t-30))); d14=0.35*(1-exp(-1.2*(t-30)));

t<=55

t<=85

.45*(1-exp(-30))-0.
44*(1-exp(-30))-0.
40*(1-exp(-30))-0.

35*(1-exp(-30))-0.

t<=100

.45*(1-exp(-30))-0.
.44*(1-exp(-30))-0.
.40*(1-exp(-30))-0.

35*(1-exp(-30))-0.

45*(1-exp(-10));
44*(1-exp(-10));
40*(1-exp(-10));

35*(1-exp(-10));

45#(1-exp(-10))-0.
44*(1-exp(-10))-0.
40*(1-exp(-10))-0.

35*(1-exp(-10))-0.

d11=0.45*(1-exp(-30))-0.45*(1-exp(-10))+0.

d11=d11-0.40*(1-exp(-1.1*(t-85)))3

d12=0.44*(1-exp(-30))-0.44*(1-exp(-10))+0.

d12=d12-0.35*(1-exp(-1.1*(t-85)))3

d13=0.40*(1-exp(-30))-0.40*(1-exp(-10))+O.

d13=d13-0.30*(1-exp(-1.1*(t-85)));

d14=0.35*(1-exp(-30))-0.35*(1-exp(-10))+0.

.45*(1-exp(-1.2*(t-30)))-O.45*(1-exp(45-t));
.44*(1-exp(-1.2*(t-30)))-0.44*(1-exp(45-t));
.40*(1-exp(-1.2*(t-30)))-0.40*(1-exp(45-t));

.35*(1-exp(-1.2*(t-30)))-0.35*(1-exp(45-t));

40*(1-exp(-1.

35*(1-exp(-1.

30*(1-exp(-1.

28*(1-exp(-1.

40*(1-exp(-0.

35*(1-exp(-0.

30*(1-exp(-0.

28*(1-exp(-0.

1*(t-85)));

1*(t-85)));

1*(t-85)));

1*(t-85)));

8*(t-100)));

8*(t-100)));

8*(t-100)));

8*(t-100)))3

74

d14=d14-0.28*(1-exp(-1.1*(t-85)));

end

vdot1=(-A_ph01*v1“2-d+f1)/m1+d11; vdot2=(-A_pho2*v2“2-d+f2)/m2+d12;
vdot3=(-A-pho3*v3‘2-d+f3)/m3+d13; vdot4=(-A_pho4*v4‘2-d+f4)/m4+d14;
deltadot1=v1-w; deltadot2=v2~v1; deltadot3=v3-v2; deltadot4=v4-v3;
deltadhat1=x(5); deltadhat2=x(17); deltadhat3=x(29); deltadhat4=x(41);
deltaddhat1=x(6); deltaddhat2=x(18); deltaddhat3=xf30); deltaddhat4=x(42); i
vdothat1=x(8); vdothat2=x(20); vdothat3-x(32); vdothat4=x(44);
ydothat1=de1tadhat1+1ambdatvdothat1; ydothat2=deltadhat2+lambdatvdothat2;
ydothat3=deltadhat3+lambdatvdothatB; ydothat4=deltadhat4+lambdatvdothat4;
Yhat1=[y1;ydothat1]; Yhat2=[y2;ydothat2];

Yhat3=[y3;ydothat3]; Yhat4=[y4;ydothat4];

$1=2*Yhat1’*P*bb; 8282*Yhat2’*P*bb; 83=2*Yhat3’*P*bb; s4=2*Yhat4’*P*bb;
f1hat1=2*lambda*v1*vdothat1; flhat2=2tlambda*v2*vdothat2;
f1hat3=2¢lambdatvStvdothat3; f1hat4=2*lambdatv4tvdothat4;
f2hat1=1ambda¢vdothat1; f2hat2=lambdatvdothat2;

f2hat3=lambdatvd0that3; f2hat4-lambdatvdothat4;

f31=lambdatv1‘2; f32-1ambda*v2“2; f33=1ambda*v3“2; f34=1ambda*v4‘2;
f4common=-lambda*d; G=[0;0;0;1ambda];
Fhat1=[f1hat1;f2hat1;f31;f4common]; Fhat2=[f1hat2;f2hat2;f32;f4common];
Fhat3=[f1hat3;f2hat3;f33;f4common]; Fhat4=[f1hat4;f2hat4;f34;f4common];
if etatabs(sl)>=mu, vr1=-(eta*81)/((l-kv)*abs(81));

else vr1=-((eta‘2)*sI)/((1-kv)*mu); end

if etatabs(82)>=mu, vr2=-(eta*s2)/((1-kv)*ab8(82));

75

else vr2=-((eta‘2)*s2)/((1-kv)*mu); end
if etatabs(83)>=mu, vr3=-(eta*s3)/((l-kv)¢abs(83));
else vr3=-((eta‘2)*s3)/((1-kv)*mu); end
if eta*abs(s4)>=mu, vr4=-(eta*s4)/((l-kv)*abs(84));
else vr4=-((eta‘2)*s4)/((1-kv)*mu); end
u1=(-deltaddhat1-th1’*Fhat1-K*Yhat1)/(th1’tG);
u2=(-deltaddhat2-th2’*Fhat2-KtYhat2)/(th2’*G);
u3=(-de1taddhat3-th3’*Fhat3-KtYhat3)/(th3’*6);
u4= ('deltaddhat4-th4 ’ *Fhat4-K'0'Yhat4) / (1:114 ’ *G) 3
if abs(u1)>5000, u1=sign(u1)*5000; end
if abs(u2)>5000, u2=sign(u2)*5000; end
if abs(u3)>5000, u3=sign(u3)*5000; end
if abs (u4) >5000 , u4=sign(u4) *5000; end
fdot1=(-f1+u1)/tau1; fdot2=(-f2+u2)/tau2;
fdot3=(-f3+u3)/tau3; fdot4=(-f4+u4)/tau4;
Psi1=si*(Fhat1+G*u1); Psi2=s2*(Fhat2+G*u2);
Psi3=s3*(Fhat3+G*u3); Psi4=s4*(Fhat4+G*u4);
if abs(f1ag)== ,
sysll=[de1tadot1 vdotl fdotl]; sys12=[deltadot2 vdot2 fdot2];
sysi3=[deltadot3 vdot3 fdot3]; sysl4=[deltadot4 vdot4 fdot4];
sys21=[x4dot1 x5dot1 x6dot1 x7dot1 x8dot1];
sys22=[x4dot2 deot2 x6dot2 x7dot2 18d0t2];
sys23=[x4d0t3 x5dot3 deot3 x7dot3 deot3];

sys24=[x4dot4 x5dot4 x6dot4 x7dot4 x8dot4];

76

for i=1:4,
sy331(i)=proj(th1(i),Psil(i),a(i),b(i),Gamma,del);
sy832(i)=proj(th2(i),Psi2(i),a(i),b(i),Gamma,de1);
sys33(i)=proj(th3(i),Psi3(i),a(i),b(i),Gamma,del);
sys34(i)=proj(th4(i),Psi4(i),a(i),b(i),Gamma,del);
end
sysl=[sy311 sys21 sysBl]; sys2=[sy312 sys22 sy832];
sys3=[sy813 sys23 sys33]; sys4-[sysl4 sys24 sys34];
sys=[sysl sys2 sysB sys4];
elseif abs(f1ag) == .
sys=[v1;vdot1;y1;v2;vdot2;y2;v3;vdot3;y3;v4;vdot4;y4];
end

end

A.3 proj.m

function Sdot=proj(Theta,Phi,a,b,Gamma,de1ta)
11=Theta>=a; 12=Theta<=b; 133(11)&(12);

14=Theta>b3 15=Phi<=0; 16=(14)&(15);

17=Theta<a; 18=Phi>=0; 19=<17)&(18);

if (13)|(16)I(19) Sdot=Gamma*Phi;

elseif (“l2)&("1c5) Sdot=Gamma*(1+(b-Theta)/delta)*Phi;
elseif (”11)&("18) Sdot=Gamma*(1+(Theta-a)/de1ta)*Phi;

end

Appendix B

Matlab programs for simulations in

Chapter 4

B.1 trainf.m

ZZZZXZZZZZZZZZZZZZZZXZZZZZZZZZZZZZZXXZZZZZZZZZZZZZZZZZXZXXZZZZZZZX
% This program is to obtain initial estimates for the weights of Z
Z a network with 48 Gaussian nodes.The program for G is similar. %
%%%%%%%%%%%%%%%%1ZZZZ%%%%%%%%%%%%Z%%%%%%Z%%%%ZZZZZXZZZXXZZZZZZXZZZ
for i=1:48, a(i)=-(3/4)+0.5*mod(i-1,4); end
for i=1:48, b(i)=-(11/4)+0.5*f100r((i-1)/4); end
W1f=[a;b]; Blf=2tpi.*ones(1,1ength(H1f));
Po=[]; y=[]; yd=[]; To=[]; k1=16; k2=0; k3=0;

for i=1z48,

y(i)=-(3/4)+0.5*mod(i-1,4); yd(i)=-(11/4)+0.5*floor((i-1)/4);

77

78

if y(i)==k2, f1=1;

else f1=[sin(4*pi*(y(i)-k2))]/[4*pi*(y(i)-k2)]; end

if yd(i)==k3, f2=1;

else f2=[sin(pi*(yd(i)-k3))]/[pi*(yd(i)-k3)]; end
To=[To,k1*f1*(f2‘2)];

end
Tp=[100 0.0048]; Po=[Y3Yd]; [W2f,B2f] = mysolverb(Po,To,H1f’,2*pi,Tp);
Z mysolverb.m is similar to matlab’s solverb.m with minor modifications.
Z While in solverb.m the RBF’s ’Gaussian wts’ are chosen by the program
Z depending on the training samples, in mysolverb.m, we use apriori

Z fixed basis wts and determine only the linear-weights (ie, mainly
Z use only the solvelin part of solverb.)

save weightsfs.mat Hlf Blf w2f 82f;

B.2 nno.m

ZZZZZZZ%%X%%%%%%%%%%%%ZZ%%ZZZZZZX%%%%%%%%%%%%ZZXXZZZZ%%%Z%Z%
Z This is the main program,it uses the model simsloto.mdl, %
X which in turn calls the matlab M-file slotsimo.m %
722%Z%XZZXZ1%X7%ZX2%XZZZXXZZXZZZXZXXXZZZZXZXZZZZXZZZZZXZZXZZ
clear; clc; load weightsfs; load weightsg;

ar=w2r-0.1; br=w2r+0.1; ag-H2g-0.1; bg=H2g+0.1;

xo=[0 O 0 -0.5 2 0 0 H2f H2gJ’; tf=[0 20];

deltasle-S; Gamma-1e3;

79

[t,x]=sim(’simsloto’,tf);

e=simout(:,1); v=simout(:,2);

subplot(2,1,1); plot(t,simout(:,1)); grid;

xlabe1(’ Time in seconds’) tit1e(’Tracking error e’)
subplot(2,1,2); hold on; plot(t,simout(:,2)); grid;

x1abe1(’ Time in seconds’); title(’Control input v’)

B.3 slotsimo.m

Z This program contains all relevant equations Z
ZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZZ
function [sys,xi,str,ts]=slotsimo(t,x,u,f1ag,xo,H1f,W2f,Blf,
B2f,af,bf,ng,W2g,Blg,B2g,ag,bg,Gamma,delta)

if f1ag== ,

xi=xo; sys=[7+length(H2f)+1ength(w2g),0,2,0,0,0,0];
else

P=[1.1167 0.0833;0.0833 0.1167]; bb=[0 1]’; K=[3 2];
eta=40; kv=0.7; mu=0.5; epsilon=0.001; vmax=60;
yr=x(1); yrd=x(2); yrdd=x(3);

y=x(4); yd=x(5); H2f=x(8:(7+1ength(W2f)))3
H2g=x(8+1ength(w2f):7+1ength(W2f)+length(V23));
e1=y-yr; e2=yd-yrd; e1hat=x(6); e2hat=x(7)/epsilon;
e=[e1 e2]; s=2*e*P*bb; U=[y e2+yrd]’;

[Phif,Fhat]=simurb(U,H1f’,B1f’,H2f’,B2f);

80

[Phig,Ghat]=simurb(U,H1g’,Blg’,w2g’,BZg);
if eta*abs(s)>=mu, vr=-(eta*s)/((1-kv)*abs(s));
else vr=-((eta‘2)*s)/((1-kv)*mu); end
if y==0, f1=1; else f1=[sin(4*pi*y)]/[4*pi*y]; end
if yd==0, f2=1; else f2=[sin(pi*yd)]/[pi*yd]; end
F=16*f1*(f2“2); G=2+sin((3*pit(y-0.5)));
v=-K*e’-Fhat+yrdd+vr/Ghat;
if abs(v)>vmax v=sign(v)*vmax; end
ydd=F+v*G; Psif=s*Phif; Psig=s*Phig*v;
x6dot=(x(7)+1*(e1-e1hat))/epsilon; x7dot-6*(e1-e1hat)/epsilon;
if abs(flag)== ,
for i=1:length(W2f).
W2fdot(i)=proj(W2f(i),Psif(i).af(i),bf(i),Gamma,de1ta);
end
for i=1:length(U2g).
H2gdot(i)=proj(fl2g(i),PsigCi),ag(i),bg(i),Gamma,de1ta);
end
u=0.5*sign(sin(0.8*pi*t))+0.5;
sys=[x(2);x(3);-1000*x(1)-300*x(2)-30*x(3)+1000*u;yd;ydd;
x6dot;x7dot;w2fdot’;H2gdot’];
elseif abs(flag) == , sys=[e1,v];
end

end

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