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6/01 cJCIRC/DatoDue.p65-p.15

REGULATION OF NONLINEAR SYSTEMS USING
CONDITIONAL INTEGRATORS

By
Abhyudai Singh

A THESIS

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

MASTER OF SCIENCE
Department of Electrical Engineering

2004

ABSTRACT

REGULATION OF NONLINEAR SYSTEMS USING
CONDITIONAL INTEGRATORS

By
Abhyudai Singh

Regulation of nonlinear systems using conditional integrators is studied. Previ-
ous work introduced the tool of conditional integrators that provide integral action
inside a boundary layer while acting as stable systems outside, leading to improve—
ment in transient response while achieving asymptotic regulation in the presence of
unknown constant disturbances or parameter uncertainties. The approach, however,
is restricted to a sliding mode control framework. This thesis extends this tool to
a fairly general class of state feedback control laws, with the stipulation that we
know a Lyapunov function for the closed-loop system. Asymptotic regulation with
improvement in transient response is done by using the Lyapunov redesign technique
to implement the state feedback control as a saturated high-gain feedback and in-
troducing a conditional integrator to provide integral action inside a boundary layer.
Improvement in the transient response using conditional integrators is demonstrated

with an experimental application to the Pendubot.

To my family

iii

ACKNOWLEDGMENTS

I would like to express my deepest appreciation to the following people:

To my advisor Dr. Hassan K. Khalil for his guidance and insight. It has
been my pleasure, and indeed privilege to have worked with him. His pure, unselfish
and honest passion for the subject will always serve as an example and inspiration

for me.

To my grandparents (V. K. S. Chaudhary, G. B. Singh and K. D. Singh),

who instilled in me an appreciation for honesty and hardwork.

And to my parents (Yatindra and Neeta Singh), who have supported me, en-

couraged me, and believed in me.

iv

TABLE OF CONTENTS

LIST OF FIGURES .............................. vii
1 Introduction ................................ 1
2 Regulation using Conditional Integrators ................ 4
2.1 Problem Statement and Controller design ........... 4
2.2 Class of Systems ......................... 8
2.3 Asymptotic Regulation ...................... 14
2.4 Performance ............................ 23
3 Output Feedback of Minimum-Phase Systems ............. 32
3.1 Problem Statement ........................ 32
3.2 Partial State Feedback Design .................. 36
3.3 Output Feedback Design ..................... 37
3.4 Comparison of Controllers .................... 42
4 Application to the Pendubot ....................... 47
4.1 The Pendubot ........................... 47
4.2 Mathematical Model ....................... 49
4.3 The Equilibrium Manifold .................... 51
4.4 Controlling The Pendubot .................... 52
4.5 Hardware Description ....................... 56

4.6 Observer design .......................... 56

4.7 Addition of Uncertainty ..................... 57
4.8 Integral action ........................... 59
4.9 Conditional Integrators ...................... 59
5 Conclusion ................................. 62
APPENDICES ................................. 64
A Controller Design ............................. 64
A.1 Linear Controller ......................... 64
A2 Traditional Integral Action .................... 65
A3 Conditional Integrator ...................... 66
BIBLIOGRAPHY ............................... 67

vi

2.1

2.2

2.3

3.1

3.2

4.1

4.2

4.3

4.4

LIST OF FIGURES

Two-link robot. .............................. 11

Plot of error ||x1(t) — 1:}(t)|| for systems (2.38) and (2.39) with p = 1
(solid), )1 = .1 (dashed) and plot of s = Hy + OH (dashed) for p = 1,
with initial conditions 1'1(O) = 1‘{(0) = 0,:r2(0) = 13(0) = —20 and
0(0) 2 0 .................................. 30

Plot of error llx1(t) — :r’l‘(t)|| for systems (2.40) and (2.41) with ,u = 1
(solid), p = .1 (dashed) and plot of s = My + all (dashed) for ,u = 1,
with initial conditions 1:1(0) = x’f(0) = 0,:r2(0) 2 135(0) 2 ~15 and
0(0) = 0 .................................. 31

Plots of 3:1(t), 222(t) and 173(t) for the controller design (3.20) (dashed)
and the controller design using sliding mode control (solid) ...... 45

Plots of u1(t) and u2(t) for the controller design (3.20) (dashed) and
the controller design using sliding mode control (solid) ........ 46

Front and side view drawing of the Pendubot. ............. 48
The pendubot arm at a: Controllable position, b: Uncontrollable position 53

Trajectories for angle ql for different balancing controls, linear con-
troller with no disturbance (solid), linear controller with constant dis-
turbance of 1.6V in the control (dotted), linear state feedback integral
controller with constant disturbance of 1.6V in the control (dashed) . 58

Trajectories for angles ql for different balancing controls with distur-
bance of 1.6V in the control, traditional integral action (solid), con-
ditional integrator with u=0.25 (dashed), conditional integrator with
,u=0.15 (dotted) ............................. 60

vii

4.5 Trajectories for angles qg for different balancing controls with distur-
bance of 1.6V in the control, traditional integral action (solid), con-
ditional integrator with #2025 (dashed), conditional integrator with
#2015 (dotted) .............................

viii

CHAPTER 1

Introduction

When a system is subjected to unknown constant disturbances or parameter uncer-
tainties that cause a shift in the equilibrium point, we need to use integral action
to achieve robust asymptotic regulation. The traditional approach for introducing
integral action is to augment integrators with the system and design feedback
control to stabilize the augmented system [4], [7], [8], [9], [10], [12]. The integrators
are introduced in such a way that they create an equilibrium point at which the
regulation error is zero. Hence, stabilizing the equilibrium point ensures asymptotic
regulation. In this approach, achieving asymptotic regulation usually happens at
the expense of degrading the transient response. A new approach of conditional
integrators has been recently introduced in [14]. The work [14] deals with continuous
implementation of sliding mode control and introduces conditional integrators that
provide integral action inside a boundary layer of the sliding surface, while acting
like a stable system outside it. The striking feature of the results of [14] is that
the state of the conditional integrator is always of the order of p (the width of the

boundary layer) and as u —> O the trajectories of the closed-loop system approach the

trajectories obtained under ideal sliding mode control without integral action, which
provides analytical confirmation of the property that integral action is introduced

without degrading the transient response.

In this thesis we extend the tool of conditional integrators beyond the sliding
mode control framework. Towards that end, we consider a stabilizing, locally
Lipschitz, state feedback control law that stabilizes the origin of a nonlinear system
and assume that we know a Lyapunov function for the closed-loop system. Then,
we perturb the system with a matched uncertainty that causes a shift in the
equilibrium point. To recover asymptotic regulation of the state, we introduce
integral action via a conditional integrator. The key idea is to use Lyapunov redesign
to implement the state feedback control as a saturated high-gain feedback and then
introduce the conditional integrator to provide integral action inside a boundary
layer. We prove analytically that, for sufficiently small )1, conditional integrators can
recover asymptotic regulation of the state to the origin in the presence of matched
uncertainty and, under certain assumptions, global asymptotic regulation can be

achieved.

We also show that, for a certain compact set of the initial states, as the width of
the boundary layer approaches zero, trajectories of the system with conditional
integrator approach those of a system with no integral action. Hence, degradation in
transient response due to integral action can be attenuated by tuning the width of

the boundary layer without sacrificing asymptotic regulation. In the afore—mentioned

results the goal is to regulate the state of the system to zero despite the constant
disturbance or parameter uncertainty. In many regulation problems, however, such
requirements could be restrictive and it might be sufficient to regulate the states to
a disturbance-dependent equilibrium point at which the regulation error is zero. In
Chapter 3, we consider such regulation for a class of minimum-phase, input-output
linearizable, nonlinear systems which is basically the same class studied in [14], aside
from some differences in the technical assumptions. As in [14], we study output
feedback control using high-gain observers, but, unlike [14], we do not use sliding
mode control. Instead, we use a saturated high—gain feedback implementation of the

linearizing feedback control.

Finally, in Chapter 4, we demonstrate the improvement in transient response

with conditional integrators by an application to the Pendubot.

CHAPTER 2

Regulation using Conditional

Integrators

2.1 Problem Statement and Controller design

Consider a nonlinear system represented by
i‘ = f(;L') + G(2:)u (2.1)

where u E Rm is the control input, :1: E R" is the state vector, C(13), f (:13) are,
possibly unknown, sufficiently smooth functions in a domain X C R", with {0} E X,
and f (0) = 0. Suppose there are, possibly unknown, locally Lipschitz function I/)(:13),
with 2M0) = 0, and C2 Lyapunov function V(:r), with %’-(0) = 0, such that the

following assumptions are satisfied.

Assumption 1:

8V , ,

EEUQ0+GUWMOLS-WCU, ViEk’ 92)
for some continuous positive definite function IV(1:).

Assumption 2: %G(r) can be expressed as %¥G(r) = hT(:z:)L(:r) where
h(:r) is a known continuous function with h(0) = 0 and L(2:) is a, possibly unknown,
continuous function satisfying LT(:r) + L(;r) Z 2A1 and‘ ||L(:r)|| g k, \7’ :1: E X where

k2A>0

Assumption 3: The square matrix M (I) = %G(a:) has the property that

M(0) -I- IIIT(0) is positive definite.
Let Q = {V(:r) 3 Cl} C X be a compact set and

AP = A... { [M ‘0) +21” Tm” } k, = ||M(0)|| (2.3)

 

where Ami,,{.} stands for the smallest eigenvalue of a positive definite matrix. Suppose
system (2.1) is perturbed by a sufficiently smooth matched uncertainty 6(1‘) that

causes a shift in the equilibrium point, i.e, 6 (0) 75 0. The perturbed system can be

 

"II-ll = ||-||2

written as

1:, = fia')+G(:r)lu+6(w)l

f(1‘) + Gil-me + emu + G(I)l6(:r) — 212(1)]. (24)
Let 5(13) 2 0 be a known continuous function such that
“5(1?) - MI)” E x303), V :1: 6 9-
Choose a locally Lipschitz function a(:1:) such that, V :2: 6 (2,
am) 2 mix) + a0. (2.5)

where k0 = max {§, E3} 2 1 and do > 0. We make the following local assumption
P

on the functions G(:r),f(r),6(:r),h(:r),M17) and a(:z:).

Assumption 4: There exist non-negative constants k1 to 1:6 such that

”5(1‘) - 5(0) - 111(3)“

l|a($)-a(0)|| S ksll’l($)ll+ks W(I)

I/\

k1||h(l‘)|| + k2 W0?)

|/\

k3]]h(;r)|| + k4 W(;t:)

 

”gum + Game}

in some neighborhood of :1: = 0.

If VV(3:) 2 kw||1r||2,kw > 0, near the origin, then any locally Lipschitz func-
tion can be bounded by [cm/WW) for some kc > 0. Define the vector saturation
function ap(y) by

«L if l|y||21
my): “y” (2.6)

y if Ilyll S 1.

It is not hard to verify that the saturated high-gain feedback control

it = —a(;1:)<,0 G) , ,1 > 0, (2.7)

where y = h(:c), achieves practical stabilization in the sense that, within a finite time
interval, :r(t) reaches a neighborhood of the origin whose size is a class IC function of
p. Hence, by reducing II, we can reduce the ultimate bound on 2:. However, in reality,
we cannot make the ultimate bound arbitrarily small because it would require it to be
arbitrarily small; hence, inducing chattering, excitation of unmodeled fast dynamics,
and other well-known problems. To achieve asymptotic regulation without forcing ,u

to be arbitrarily small, we introduce a conditional integrator by modifying (2.7) to

 

u = —a(:1:)cp (31 + I”) , (2.8)

where a is the output of the conditional integrator

 

. + 0
a=—70+u<p(y #7 ), 7>0. (2.9)

Because ||t,9(y)|] S 1, V y E R", it can be verified that ]|0(t)|| S p/7 for all t Z 0

provided ”0(0)“ 3 ,u/y, as from (2.9)

 

on, = _,,..,T,, + ”an, (m)
p
,u
S -7'||0||2 +H|I0|| S 0, V H0“ 2 5. (2-10)
Inside the boundary layer {lly + 70H 3 it}, (2.9) reduces to
. + 0
0 = —70+# (y #7 > = 31, (2-11)

which provides integral action.

2.2 Class of Systems

We will now Show that Assumptions 1-4 are satisfied for a class of feedback linearizable

systems. Consider the system

51 = 65
git—1 = {:i (212)
gr = 91(596) + Zdu(€,0)uj, IS i _<_ m
j=1
yi = git

where u = [111. . -- ,um]T E U E Rm is the control input, E =
IGV", 1113'”, fl,"- ,€;';’an E X E R" is the state vector, with {0} E X,9 6
O E R” is a vector of unknown constant parameters, and the functions b,(€, 0) and
dU-(ffl) are locally Lipschitz in their arguments over the domain of interest, with
b,(0, 6) = 0, V 6 E 9. We assume that the decoupling matrix D(€, 9) : {do-(E, 0),1 S

i g m, 1 g j g m} is nonsingular V 6 E X and V 9 E 9. System (2.12) can be written

as

6' = A€+B[b(€,0)+D(£,6)ul

A5 + Blbit, 6) + D(€, EDD—1c, é)D<£, 6M.

where

A = block diag[A1, . . . ,Am], (2.13)
0 1 0
0 0 1 0
At = i
0 O 1
0 0
- d rini

 

 

 

 

r0-
0
B = block diag[Bl,...,Bm], Bi: 5 , (2.14)
0
b1(€,9)
b(€.9) = s ,
[bm(€,6)_

 

 

1 _<_ i S m, and 6 is a nominal value of 6. It can be seen that Assumption 1 is satisfied

with

me) = D“(€,9)[—b(€,9) — Kt]. V(€) = tTPt, W(€) = #5,

where the matrix K is chosen such that A—BK is Hurwitz and P = PT is the solution
of the Lyapunov equation P(A— BK) + (A— BK)TP = —I. As D‘1(€, 6)D(€, 6) = I
for 6 = 6 it must be true that for some perturbation of 6 around 6,

ID“(€, é)D(£,9)I 2 [0-.(5,(;)D(5,9)]T > 0. (2.15)

 

Let (2.15) be satisfied for 6 E 91 C O. For the single input case, condition (2.15)
implies that the scalars D(§, 6) and D(£, 6) have the same sign. With (2.15), Assump-

tion 2 will be satisfied with WE, a“) = ifiglspgfl) and L(§, 0) = D-1(g,é)D(g,9).

10

 

Figure 2.1. Two-link robot.

As V(§) is quadratic and rank[D(0, 6)] = m

021/05)

T A T
D (can? 8,,

30(5, (9) > 0.

e=o

 

Using similar arguments as used for (2.15), it can be shown that there exists a set 92 C
9 such that the matrix D1 (6, 6) = DT(£, 6)BT§2—¥§9BD(€, 6)[D‘1(§, 6)D(£, 6)] [5:0
satisfies

2

 

> 0 (2.16)

for 6 E 92. Inequality (2.16) satisfies Assumption 3. Finally, Assumption 4
is also satisfied as W(£) is quadratic. Hence, Assumption 1-4 will be satisfied

VQEGIHGQCG.

Example : The two-link robot shown in Figure 2.1, can be modeled [13] by the

11

following equation

M(qfii + C(q,(1')<i+ Dq’ + 9((1) = u,

where q is a two—dimensional vector of generalized coordinates representing joint po-
sitions, u is a two-dimensional control (torque) input. M (q) is a symmetric inertia
matrix, which is positive definite for all q E 1?.2 and is given by
(11 + 2a4 cos q2 a2 + (14 cos q?
M (q) = ’
a2 + a4 cos qg a3
where (11,. . . ,a4 are positive constants. The term C(q,cj)q’ accounts for centrifugal

and coriolis forces and is given by

‘92 _(91 + 92

r. )j

 
 

C(q, (1)4 = a4 sin (12

[ (Ii 0

The term 04 accounts for viscous damping , where D is a positive semidefinite matrix.
However, in this example we will neglect damping and take D = 0. The term g(q),

which accounts for the gravity forces, is given by

I
b1 COS Qt + 52 003(91 + (12)
QIQ) = ,
b2 COS(CI1 + (12)
where b1 and b2 are positive constants. These constants a1, . . . ,a4, b1 and b2 depend

on masses, moment of inertia, lengths of the two links, acceleration due to gravity

12

and the load kept on the robot. The nominal values corresponding to no load on the

robot are

a, = 200.01, 52 : 23.5, 53 = 122.5, a, = 25, 13, = 784.8, 132 = 245.25.

The state space equation for the two link robot is given by

‘11
. " _1 A _1 . . 41
:r = Ax + B{M (q)M(q)M (aliu - C(q, q)q - 9(q)l}, 5!: = ,
C12
42

.— .I

 

 

where A and B given by (2.13) and (2.14) respectively with {rt 2 2, 1 S 2' S 2} and

£11 + 264 cos (12 61.2 + 64 cos qg

M(q) =

(22 + 6,4 COS q2 (23

It can be seen that Assumption 1 is satisfied with

$03) = [C ((1,4)4 + 901) - M (q)K 33], V(r) = ITPI, W(r) = -10:vTx,

15.7074 0.1548 —0.4792 0.0103
0.1548 0.4273 0.0074 0.0143
—0.4792 0.0074 18.4157 0.1014

0.0103 0.0143 0.01014 0.3423

L .4

 

 

13

32.492 12.0791 —2.8711 —0.4700

—2.9927 ——.4903 49.5071 14.9209

Due to an unknown load, the actual system parameters are perturbed to
01 = 259.7, 0.2 = 58.19, a3 = 157.19, a4 : 56.25, bl = 1030.1, b2 2 551.8125.
It can be verified that for this perturbation, V q E R2,

[Maw-10>] + [Mew—liq)? > 0

A A T
[M‘1T(q)BPBTM‘1(q)]+ [M-1T(q)BTPBM-1(q)] > 0.
Hence, Assumptions 2 and 3 are satisfied with
5%) = 2:1:TPBM‘1(q), La) = M(q)M-1(q).

Finally, Assumption 4 is also satisfied as W(a:) is quadratic.

2.3 Asymptotic Regulation

In this section we prove that, for sufficiently small ,0, the system (2.4), (2.8), (2.9)

has an asymptotically stable equilibrium point at which a: = 0.

Theorem 1: Suppose Assumptions 1-4 are satisfied. Then, there exists ,u" > 0 such

14

that V p e (0,p*]

a) The system (2.4), (2.8), (2.9) has an asymptotically stable equilibrium point at
(a: = 0, 6 = —L—6(0)) and Z = {V(:r) 3 c1} x {”0“ g p/y} is a subset of the region

0(0)?

of attraction.

b) If (2.5) and Assumptions 1, 2 hold globally, V(:r) is radially unbounded,

and
a(:r) S kgW(:rt) + cg, V :1: E R" (2.17)

for some 139,09 > 0, then, V (:r(0),0(0)) E R" x {“0“ g ,u/y}, limt_.c,,D :c(t) = 0.

Proof : a) The proof is divided into three parts. First we show that all tra-
jectories of the system (2.4), (2.8), (2.9) starting in the set E reach a positively
invariant set 2,, = {V(:i:) g p(p)} x {”0” 5 p/cy} in finite time, where p is a class IC
function. Then we show that the trajectories in the set 2,, enter the boundary layer
{Hy + 70]] S p} in finite time. Inside the boundary layer, we show that the system

has an asymptotically stable equilibrium point at which :5 = 0.

To show that all trajectories starting in 2 enter 2,, in finite time, we calcu-

15

late V for (1:,0) 6 2.

 

 

 

8 -
%lf(:v) + G(r)ui'(r)l + %G<:c)u + 8—:G(r)[0(:r) — we]
—W<a~) — yTa(r)L(r)so (y 43") + yTL(:r)l6(x) — we)
— "J: — “'O’TO'J‘ :1: y+70 an: UT :1: y+~ya
WI) 0+,) rM<w( # )+ (h L<w( # )

-70TL(I)I5(I) - it"(fll + (y + 70)TL($)I5(17) — Will-

 

Ir ll(y+i0)ll 2 )1. NM) = (”+7") Using VIIUII s M,LT(I) +120) 2

2/\I, ||L(:1:

P ||(y+7all'

)II S k and (2.5) we have

S -W(I) - A0t(l‘)l|(y + 7'0)“ + kB(I)II(y + 70')“ + kIOI-T) + (Willa

g —W(:r.) + my, 01: 1:135 kla‘fl‘) + [3(3)]

If Humans» Wei—”FM and

V g —W(:r) + k[a(:r.) + 2;3(1-)],i (2.18)
5 —W(:1:)+ 02p, (12 = mgxflaw) + 26613)] 2 (11.

Thus, V (1:,0) E E, V g —W(r) + 02p. Hence, from ([11], Theorem 4.18), for

sufficiently small )1, the set E is positively invariant and all trajectories starting in

2 enter a positively invariant set 2,, = {V(:z:) S p(p)} x {”0“ g p/cy} in finite time,

where p is a class IC function.

16

Next, we show that the trajectories reach the boundary layer {Hy + 70” S p}

in finite time. For (33,0) 6 2“, we have

 

 

y' = $00) + Cir-mm] + M(0)[u — 1W) + 6(a)]
+[.M(1') — AI(0)][u — 0(1) + 6(1)]. (2.19)
From (2.8) and (2.9),
(y + 70) = -a(;z:)]l/I(0)ip (y + 70) + N(:z:) + III(0)[6(1:) — tb(;r)] — 720
+7099 (y + 7a) ,

where

Nix) = (3,300) + C(rrwirfl + Wine) — M(0)l[u - WI) + 6(0).

As we are already inside the set 2,, and N (0) = 0, [|N(:r) + 70]] can be bounded by

a class IC function p102), i.e.

 

+~0
]|N($)+wso (y u, )-720 59101)-

 

 

Let v1=guy + 70“?-

V1 = (y + 70)T(3) + 767)

|/\

 

-a(:v)(y + 7'0)TM(0)<.0 (y 7") + (y + )a)TMi0)i6<x) — we)

17

+p1(u)ll(y + 2'0)”-

 

u ||(y+‘)’0||

If My + 70]] 2 ,u, 99 (Ii—+33) = (“I”) and, using (2.3) and (2.5),

 

 

v, s -Ap[air)—%0ix)—p1]f)]Hy+wu
s —),, [a0 - “(6] lliy + 70)].

Thus, for sufficiently small )1, all trajectories inside 2,, would reach the boundary
layer {]]y + 70]] S p} in finite time. Inside the boundary layer, the system (2.4),

(2.8), (2.9) is given by

 

 

, + '0
:c = fir) + G(x)w(r) — aix)Gia:) (y I,” ) + 00000:) — i200)
0' = —70’+'Ll,(y+70)_
,u
We can see that (a: = 0, 0 = a—(‘éfiMOD is an equilibrium point. Defining 0 = 0 — 0

and if (1:) = 6(1) — 6 (0), the system inside the boundary layer can be written as

:i: = f(:v) + Gixwix) + air) {—air) (35,719) — 3%iaix) — aiml]

+G<x)i8ix) — win) (220)

(2.21)

Qt
II
‘2

18

Define the Lyapunov function
(y + 76)T(y + 76),

where b and c are positive constants to be chosen. Calculating V2 along the trajectories

of the system (2.20) and (2.21), we obtain

. . 6h
V2 = V + b0Ty + C(y + 70)T($:i: + 73/). (2.22)

Using Assumptions 1, 2 and 4 yields

V = QKIfixHGiIhC'ifll+yTLix){-a(r)(

 

8:1:
+yTL($)]5($) — W33”

0 :75) 5(0)

I/\

/\ 0'
-W(117)- a(l‘);|lyll2 - 0(IlyTLixlfi'; + k7|lyl|2 + ksllyllv W(I)
for some non-negative constants k7, [03. We also have

(1% + w) = gym + Gin-am] — $31)“ny + )6)

8:1:
—M<x)%iaix) - 0(0)] + Mix)i8ia:) — we) + 7y.

Since :1: E {V($) S p(p)},a(:1:) and ]]M(:r)]] can be bounded by constantsfMoreover,

since M (0) + M T(0) is positive definite, there is a positive constant Cg S c1 such that

19

M(:1:) + MT(:1:) is positive definite for all :1: E 91 = {V(1:) S c2} C 9. Let
A, = 2123} 1mm {M(as) + 11%)} . (2.23)
For sufficiently small 11, {V(;r) S p(p)} C Q], hence
Mp0+1fl0022nr vxe{vu)gpmn. 020
Using (2.24) and Assumption 4 we obtain
04

.. 3h. :1: ~
(0 + 10)T (as: + 11/) S -—!;-)-/\qll(y + 10)ll2 + kn llyllx/ W(:v)

+k12l|3I| v W($) + knllyll2 + klollgllllyll,

where k10,k11,k12,k12 Z 0. Therefore,

 

 

 

. W a: A W :1:
v2 3 _ 2()—a($);l]y]]2+(k7+Ck12)II3/II2+(k8+6k11)IIyII\/W(x)T 2H
~T 0(1‘.) , .~ 2 * W T a
+ba y—c # Aqlliy+w)ll +Ck12||0||\/ 01-61(1)?! WW;
+ckloll8'llllyll.

Taking b = 2ch'yO—ffl, we can express the upper bound on V2 as

v. s -W1(:r)-W2(:r)—Wa(:r),

20

where

 

 

 

 

””1: 0mm
Wliar) = 2] ) +1114?) — k7 — 6km) — (1.. + cku)llyII\/W(r)
W . /\ ~ ~ . .
W2(:r) = 2:1) + COm1n‘12jH0H2 — ck12||0]]\/W (1'), 01mm 2 IDEI‘IIIOCE)
, 0(1: c ~ 20A 7/) 11 +ck 11 ~
Mix) — p){c11111112+§)21.1101r— m‘ " 2‘] m) 1011in

and p2 is a class [C function such that
llaix) - a(0)]! 5 p201), V :1: E {V(-’r) _<_ 1001)}.

We can see that both I'Vl(;1:) and Wg(:r) will be positive definite for sufficiently small

11. For W'3(:1:) to be positive definite, we need

2

c221
_,\ _
27q>4

 

7k + flan/1201) + 6161011)]2

amin

 

Taking c = k/Aq ensures that V2 will be negative definite for sufficiently small 11;
consequently the equilibrium point (:1: = 0,0 = filth—76(0)) will be asymptotically

stable for sufficiently small 11.

b) Inequality (2.18) is valid V (:r, 0) E R" x {”0“ S 11/7}. Using (2.5) and

(2. 17) we obtain

V S -lV(:1:) + 3ka(:1:)/1

21

S —-[1 — 3pkgk]lV(:r) + 3kcgp.

Hence, as in part a), for sufficiently small 11, all trajectories starting in
(:r,0) E {V(:1:) S c1} x {]]0]] S p/y} enter 2,,. Since V(:1:) is radially un-
bounded, C, can be chosen to include any 2(0) 6 R" in the set {V(:1:) S c1}.
Analysis inside the set 2,, remains the same as in part a). Thus, for sufficiently small

11,V a:(0) E R" and ”0(0)“ S p/7,:1:(t) —> 0 as t -—> 00.

Remark: In Theorem 1 we assumed ]]0(0)]| S 11/7. However, we can relax

this requirement. Given |]0(0)|| S Isa/7, kg 2 11, then, from (2.9),

2
0% < gnaw, VlloHZ-fi.
,

Hence, from ([11], Theorem 4.18), 0(t) reaches the set {”0“ S 211/7} in finite time.

For ]]0(0)]| S lea/7 we can repeat the derivation preceding (2.18) to show that
V S —W(:1:)+ 02kg, V21: 6 (2.

Thus, from ([11], Theorem 4.18), there exists a positively invariant set S = {V(:1:) S
p3(k,,)}, where p3 is a class [C function, such that trajectories starting outside 8 reach
it in finite time. If k0 is small enough such that S C Q, then 50(0) 6 51 => a:(t) E
O, V t Z 0. Once [[0]] reaches the set {“0” S 2,11/7} the proof of Theorem 1 can be

repeated for the set 51 x {“0“ S 211/7}.

22

2.4 Performance

In this section we show that the conditional integrator does not degrade the transient
response of the system, in the sense that, as 11 —) 0, the trajectory of the closed loop
system under saturated high-gain feedback with conditional integrator (2.8) approach
those of the closed-loop system under saturated high-gain feedback without integrator
(2.7). The closed-loop system under (2.8) is represented by (2.4), (2.8), (2.9) while

the closed-loop system under (2.7) is given by

1:: = fix‘)+G(r*)6(I‘)—a(:v*)G(x‘)so (31—) (2.25)

where y" = h(2:*). In both cases, the trajectories eventually enter a boundary layer,
which is {lly + 70]] S p} for (2.4), (2.8), (2.9) and {]]y*]] S 11} for (2.25). Showing
closeness of trajectories is relatively easy when trajectories of both systems are either
outside or inside their respective boundary layers. The tricky part is to keep track
of the closeness of trajectories as trajectories enter and leave the boundary layers
since the entry and exit times will be different for the two systems. Note that, while
trajectories eventually settle inside the boundary layers, there could be a period of
time when trajectories go in and out for a finite number of times. For convenience,
we restrict our analysis to a case where once the trajectories enter the boundary
layers, they cannot leave. This will require us to limit the compact set of analysis

and to change the choice of the function 0(12) in the feedback control law (2.7) or (2.8).

23

Recall from the proof of Theorem 1 that Q, = {V(:r) S 02} C Q is a com-

pact set such that M T(:1?) + M (11:) is positive definite in {21. Let A,, be defined by

(2.23) and
kq = ggfllM (13)”-

Let 61(1) be a known continuous function such that

 

 

[$002) + Gixwixn

The choice of a(:1:) is changed to

a(:1:) 2 max {IE/3(2), :—:B(r) + )I-qfldflf + 00.

S 161(13): V 33 E 91-

(2.26)

The choice (2.26) is more conservative than (2.5); hence, the conclusions of Theorem 1

hold for all 23(0) E (21. For all (:17, 0) in the positively invariant set (I, x {]]0]] S 11/7},

the derivative of V, = élly + 70]]2 satisfies

VI = (y + 70W?) + 7(3)

31+?”

 

S -a(-’L‘)(y + 7'0)TM(IE)99 (
Outside the boundary layer {Hy + 70]] S 11}, we have

. 2
V1 3 _1, [50.712] III/+10”-
q

24

) + War) + 1.00:) + 211110 + w)”.

Hence, for sufficiently small p, all trajectories starting inside 9, x {“0” S ,u/7} will
reach the boundary layer {Hy + 70]] S p} in finite time and stay there for all future

time. We can arrive at a similar conclusion for the system (2.25) using V2 = %Hy"]|2.

To compare the equations of the two systems inside the boundary layers, it is
convenient to recognize that high-gain feedback creates slow and fast dynamics,
which can be represented in the singularly perturbed form. The following two
assumptions are used in analyzing the closeness of trajectories inside the boundary

layers.

Assumption 5: There exists a diffeomorphism T(:r) such that

 

0
amcix) — ,
8:1:
I
V II: E 91.
21
The change of variables 2 = = T(:1:) transforms the system (2.1) into
22
21 = ff(21122)
232 = fQI(21, 22) 'I' u

25

and y = h(:1:),=T-1(z) can be written as y = hI(z1, 22)

Assumption 6: 0 = hI(zl, 22) has a unique solution 22 = 0(4) for all 2: E Q, and

the system

2‘1 = ff(21,v(31))

has an exponentially stable equilibrium point at z, = 0.

Theorem 2: Suppose Assumptions 1-6 are satisfied and 0(2) satisfies (2.26).
Let 2:(t) and 0(t) be the state of (2.4), (2.8), (2.9) with 2:(0) E 01, |]0(0)H S 11/7, and
:1:"‘(t) be the state of (2.25) with :1:"(0) E 91. Suppose that H:r(0) — :r"(0)H = 0(a).

Then, Ill‘ft) - $‘(tlll = 001), V t Z 0-

Proof : Let t, be the first time one of the two systems reaches its boundary
layer. For all t S t1, trajectories of both the systems are outside their respective

boundary layers and the systems are represented by

d = —70 + #99 (ii?)
(2.27)
as = f(x) + Ginair) — a(I)G(sr) (fig,
1:: = f(2:‘) + G(:1:*)6(:1:*) — a(2:*)G(:1:") (iii) . (2-28)

26

As [[0]] = (9(p), the difference in the state equations for :i: and :1,“ is 0(p) for all
t S t1. From ([11], Theorem 3.4), H2:(t) — 23*(t)|| 2 C(11), V t S t1. From, [[0]] 2
C(11), and |]:r(t1) — 23*(t1)|| = 0(p) it can be seen that if trajectories of one system
reach its boundary layer, the trajectories of the other system will be in some 0(a)
neighborhood of its boundary layer. As derivatives of both %]](y + 70)]]2 and %]]y"‘]|2
are strictly negative outside their boundary layers, uniformly in p, it must be true that
trajectories of the other system also reach its boundary layer in time t2 = t, + 0(11).
Hence, H2:(t) —2:“(t)H 2 C(11), Vt E [t1, t2]. For allt > t2, trajectories of both systems

(2.27) and (2.28) are inside their respective boundary layers and can be represented

by
0' = 11
(2.29)
x = fix)+Gix)6iz)—aix)0ix)(11:31)
.13“ = f(:1:*) + G'(:1:")6(2:*) — a(:1:‘)G(:1:*) (9;) . (2-30)
With the change of variables 2 = 21 = T(1:) and 2* = z; = T(:1:"), (2.29)
22 z;

and (2.30) can be written in the singularly perturbed forms

0 = hI(zl, 22)
ii = ff(21,22) (2'31)
#22 = u[f§(z1,z2) + (“(21, 22)] - OI(21,Z2)IhI(21, 22) + 70]

27

it = flizi,z5) (2 32)

I

11255 = 110301.23) + 5*(21‘123H - 0*(21, ZS)hI(Zi'» 23)

where 6I(zl,z2) = 6($)|,=T_1(z),aI(z1,z2) = a(:r)|x=T—1(z) and hI(21,22) —
h(:1:)|,___.T—1(,). Let :52 = v1(z1,0) be the unique solution to hI(21,22) + 70 = 0.

The slow and fast models of (2.31) and (2.32) are given by

0=—70

(2.33)
21 = ff(21)U1(21,0))
% = ‘0I(21,Z2)[hl(21,22)+70] (2.34)
a = ffiz;,vl(z;)) (2.35)
$3 = —a*iz:,z;)h*iz1,za), (2.36)

where 7 -_—-_ t/p. The differences between the right hand sides of the slow models
(2.33), (2.35) and the fast models (2.34), (2.36) are C(11). Furthermore, from As-
sumptions 3 and 6 it can be verified that systems (2.33), (2.34), (2.35), and (2.36) are
exponentially stable. Hence, from ([11], Theorem 9.1) and Tikhonov theorem ([11],

Theorem 11.2), H2:(t)—:1:“(t)H = 0(11), Vt 2 t2. Thus, ]]:r(t)—2:*(t)]] = 001), Vt 2 0.

Example: Consider the second-order system

1131 = —331-$2

(2.37)

332 = $131+?!)

28

Assumption 1 is satisfied with 111(2) = —.’I}2, V(2:) = (1r,2 +1022)/2, W(;r) = —:1:12 —2:22
and Assumption 2 is satisfied with h(:r) = 2:2,L(:1:) = 1. It can be easily verified
that JIJT(:1:) + M (1') is positive definite V a: E R2 and Assumptions 3-6 are also
satisfied. A constant matched uncertainty 6(1‘) 2 1 is added to the control 11 to shift
the equilibrium point from the origin. To recover asymptotic stability of the origin,
system (2.37) is augmented with the conditional integrator (2.9) and from (2.8) and

(2.26), the control it is given by

 

u = —(2]2‘2| + [11] + 2.2) sat ($2,: 0).

The closed-loop system can be written as

0 : -—70 + ,u sat (5?)
1°31 = ”1‘1 — 172
(2.38)
2:2 2 11:1 + 11 +1
u = —(2I:C2I+|1:1]+ 2.2) sat(52f£)

We will now demonstrate through simulation that as ,11 —2 0, trajectories of the

system (2.38) approach the trajectories of the closed-loop system

11:; = —2:[' — x;
at; = 21+ 11 +1 - (239)
u = —(2]:r.§] + [33;] + 2.2) sate-,5)

Simulation was run for the initial conditions 21(0) 2 201(0) 2 0,:1:2(0) = 23(0) = -—20

29

 

 

 

 

 

 

 

 

 

 

 

 

0.3 , 3
0 g ———————————
I
I
-5r I 4
I
I
U I
«10+’ 1
I
I
-154
I
I
—o.2 . . -20 ‘ . . .
2 4 6 8 10 O 2 4 6 8 10

Figure 2.2. Plot of error ]];r,(t) — r{(t)|| for systems (2.38) and (2.39) with p = 1
(solid), )1 = .1 (dashed) and plot of s = Hy + 0]] (dashed) for p = 1, with initial
conditions 251(0) 2 21(0) 2 0,$2(0) = 1135(0) = —20 and 0(0) = 0

and 0(0) = 0, for different values of )1. It can be seen from Figure 2.2 that the
trajectories of (2.38) enter its boundary layer and stay there for all future time, and
that they approach those of (2.39) as p —) 0. However, if a(:1:) is given by (2.5), as

was taken in Theorem 1, in place of (2.26) then the control 11 would be

 

U = —(].’L‘2] + 2.2) sat (I2 + a) .
,U.

The closed-loop systems (2.38) and (2.39) would be

0 = —70 + )1 wag?)

33‘ : —:1:1—:1:2 (2 40)

2:2 = 231+u+1

u = —(]2:2]+2.2)sat(£2¥)

30

 

 

 

 

 

 

 

 

 

 

 

Figure 2.3. Plot of error ]]:1:1(t) — :rf(t)|| for systems (2.40) and (2.41) with p = 1
(solid), 11 = .1 (dashed) and plot of s = Hy + 0]] (dashed) for p = 1, with initial
conditions 201(0) 2 :r‘,’(0) = 0,:122(0) = 23(0) = —15 and 0(0) 2 0

and

if; = —:1:'{ — at;
a}; = II+u+1 . (241)
u = ——(];r.§] + 2.2) sat(—,f)

For similar initial conditions 231(0) = 271(0) 2 0432(0) = 2:3(0) = —20 and 0(0) 2 0, it
can be seen from Figure 2.3 that the trajectories of (2.40) enter and leave its boundary
layer once before settling there; yet the trajectories of (2.40) still approach those of

(2.41) as p —» 0, a property that is not guaranteed by Theorem 2.

31

CHAPTER 3

Output Feedback of

Minimum-Phase Systems

In this section we apply the conditional integrator design of Section 2 to output
feedback regulation of a class of minimum-phase, input-output linearizable systems.
Aside from some technical differences, this is the same problem treated in [14], but
the controller design presented here is different from the “continuous” sliding mode

controller of [14].

3. 1 Problem Statement

Consider the MIMO nonlinear system

i: = f(:L‘,w)+G(1:,w)u (3.1)

y = l(:1:,w) (3.2)

32

where u E Rm is the control input, :1: E R" is the state vector, y E Rm is the
measured output and the disturbance input 111 belongs to a compact set W C R‘.
The functions G(2r,w), f(:1:,w),l(:r,w) are sufficiently smooth functions in :r on a
domain X C R" and continuous in w for w E W. We want to regulate y(t) —> 0 as

t —> 00. We make the following Assumptions about (3.1), (3.2).

Assumption 7: For each w E W there is a unique pair ($33,113,) that de-

pends continuously on w and satisfies

0 = f($s,,w)+G(x,,,w)uss (3.3)

0 = (($33,111). (3.4)

Subtracting (3.3) from (3.1), we have

.1: = fix.w)+Gix,w)iu—u..1, (3.5)

where f(:r,w) = f(:1:,w) — f(:1:3,,w) + [C(x,w) — G(:r,,,w)]u,,. Hence, the regulation
problem reduces to the stabilization of the equilibrium point :1: = 2:8,. The system

(3.5) is in the form (2.4) with us, as the matched uncertainty 6.

33

Assumption 8: For all w E W there exists a mapping

which is a diffeomorphism over X onto its image X" x XC) that transforms (3.5), (3.2)

into the normal form

7'1 = 001.9111)
4' = AC+B{b(n,(,w)+D(n,(,w)[-u—1183]} (3.6)
y = CC

and maps the equilibrium point 2:5, into (173,, 0) with n and C belonging to the sets
X,, C R""" and X4 C R’" respectively. The m x m matrix D(77, C, 111) is non-singular
for all (77, C, 111) E X,7 x XC x W. The r x 7* matrix A and the r x m matrix B are given

by (2.13) and (2.15), respectively. The m x 7‘ matrix C is given by

C = block diag[C1,...,Cm], (3.7)

CE =3 1 0 ... ... O , C38)

lxri

where 1 S i S m and r = r1 + . . . + rm. The triple (A, B, C) represents 711 chains of

integrators.

Conditions for the existence of such a change of variables that transform (3.5), (3.2)

into the normal form (3.6) are discussed in ([8], Chapters 5 and 9)

34

Assumption 9: For all (n, C, 10) E X" x XC x W,

Div. 9 w) + 07in, c, w) > 0 (3.9)

BTPBD(77,,, 0, w) + DT(n.,,0, w)BTPB > 0 (3.10)

where P = PT is the solution of the Lyapunov equation P(A — BK) + (A— BK)TP =

—Q in which Q is a positive definite matrix and A — BK is Hurwitz.

In [14], Assumption 5, the decoupling matrix D(17,C,w), is required to be of

the form

Din. c. w) = rin.<.w)Ai<,w), (3.11)

where A is a known nonsingular matrix and I‘ = diag[71,--- ,7m] with
7,-(.) _>_ 70 > 0, 1 2 1' 2 m, for all (7),C,w) E X,7 x X4 x W and some posi-
tive constant 70. In the case of (3.11), without loss of generality a new control input
can be defined as v = A(C,w)u and with respect to the input 11, the decoupling

matrix D(77, C, 10) = F(77,C, 111) will satisfy (3.9).

Assumption 10: There exists a Lyapunov function Vr(77, 10) where 77 = 17 — 77,, and

35

a compact set Xw C X,, x X4 x W such that V (77, C, 10) E Xw

8‘4 77:“) T ‘—
_éfi_>,1(,,,g,w) s —klslln2ll+klsl|0||||C|l

for some constants 1715 > 0,1016 2 0.

With 05(7),C,w) being locally Lipschitz in its arguments, if 17 = 7),, is an expo-
nentially stable equilibrium point of 77 = 05(7), 0, w), then, the existence of such
Lyapunov function is ensured by the converse Lyapunov theorem ([11], Theorem
4.16). In [14], Assumption 4, exponential stability of 17 = ¢(n,0,w) is required but
the upper bound on QI/Qg—Z‘fldn, C, 10) does not need to be quadratic in ”CH and [[77]]

over the compact set Xw.

3.2 Partial State Feedback Design

It can be seen that V (7),C,w) E Xw, Assumption 1 is satisfied with 1/1(17,C,w) =
D407 + n..,<,w)[—biv1 + n..,<,w) — KC]. and Viacw) = Maw) + k17CTPC,
where kn is chosen such that kn > leg/43:15, and that the function W(77,C,w) is
quadratic in [[C H and [[77]]. From (3.9) it can be seen that Assumption 2 is satisfied
with hT(C) = 2k17CTPB and L(77, C,w) = D(17 + 773,, C, 111). Inequality (3.10) satisfies
Assumption 3 and as W(77,C,w) is quadratic in [[C H and [[77]] Assumption 4 is also

satisfied.

36

Let 5(C) be a known function such that V (7), C, 111) E Xw,

llusa+v(fi,C,W)H S fi(C)- (3-12)

Then, from (2.8), a partial state feedback control is taken as

 

- T
u = -0'(C)so (QWB :C + 7a) , 14 > 0, (3.13)

where 0(C) and 0 are given by (2.5) and (2.9), respectively. Ffom Theorem 1 it can be
seen that for the closed-loop system (2.9), (3.6), (3.13), for sufficiently small 11, C —+ 0

as t —) 00. Thus, lin1,_.00 y(t) = 0.

3.3 Output Feedback Design

The controller (3.13) cannot be implemented as a state feedback controller because
the partial states C depends on the unknown vector 10 through the change of variables
Tw(:1:). However, it can be implemented as an output feedback controller that uses

the high-gain observer

3 = AC+ H(y — CC) (3.14)

37

 

to estimate C, where A is given by (2.13) and the observer gain H is chosen as

a
21
6

1111400“ .

H = block diag[H1,...,Hm], H,- =

 

 

 

L (n - rixl

in which 6 is a positive constant and the positive constants a; are chosen such that

the roots of

s"+a'ls"_1+---+a:.,=0

are in the open left-half plane, for all 1' = 1, . -- ,m. The output feedback control is
given by
E: AE+Hiy-CE)
(I = “10471180 (2kivBt‘Pfi70) , (3.15)
u = -a(<)<p (#231735 )

38

If the closed-loop system under the state feedback control (3.13) has an exponentially
stable equilibrium point at the origin then, for sufficiently small 6, the closed-loop
system under the output feedback control (3.15) will have an exponentially stable
equilibrium point [2]. If the equilibrium point is asymptotically, but not exponentially,

stable we need to place an upper bound on the modeling error

60(77) (1 w) : b(”1<1u') + D(771Ca UJ)[’U. _ ”33]-

In the latter case, for each 111, the following properties hold. The closed-loop system
under state feedback can be transformed into the form ([11], Theorem 4.7)

if: Alf] + 91071517”)
, (3.16)

C = 1426+ 9207,5110)

where A, has all its eigenvalue with zero real parts and A2 is Hurwitz. Also ([11],
Theorem 8.1), guarantees the existence of a continuously differentiable center man-
ifold C = 7107,10) for all [[77]] S dw, for some dw > 0. From ([11], Corollary 8.2) the

origin of the reduced system

1‘1: Am + glifi, 71(7), w), w) = gait), w) (317)

is asymptotically stable and the converse Lyapunov theorem ([11], Theorem 4.16)

guarantees the existence of a continuously differentiable function V3(77, 10) defined on

39

B,—,(0, r), a ball of radius 7 around 17 = 0, such that V 17 E B,,(0, 7),

8V
3539007112) 3 -ao(||77|I),

where 00 is a class IC function, possibly dependent on 10.

For the above properties to hold uniformly in w E W, we make the following

assumptions.

Assumption 11: The closed-loop system under state feedback can be trans-
formed into the form (3.16), where A, has all its eigenvalue with zero real parts and

A2 is Hurwitz, uniformly in w E W.

Assumption 12: There exist a continuously differentiable center manifold
C = 7107,10) for all [[17]] S dw, for some dw > 0, such that origin of the system (3.17)

is asymptotically stable, uniformly in w E W.

Assumption 13: There exist a continuously differentiable function V3(17,w) defined

on B,—,(0, r), a ball of radius 1" around 17 = 0, such that V 17 E B,—,(0, r),V w E W,

6V
a—ggoiw) s woman),

40

where 010 is a class [C function, possibly dependent on 10.

Let 6107,10) be the projection of the modeling error onto the center manifold,

i.e.

60(771C1w) : 610111)) + 62(771C1w)1

where 62(17, 0,111) = 0. Now we make the following assumption on the functions

Vsiaw) and 5.07.41).

Assumption 14: V 17 E B,—,(0, r), V w E W

an s clasinnn),

 

 

6V
“6111.101 5 coatinmi) and ]]———

where a, b < 1 are some positive constants such that a + b = 1, c0 2 0 and C, > 0.

Under Assumption 14, from [2], the closed loop system under output feedback

will have an asymptotically stable equilibrium point for sufficiently small 6.

Furthermore, from [2], if ’R is the region of attraction under state feedback
control, then for any compact set M in the interior of 7?. and any compact set

Q Q R’ of initial estimate C (0), the set M x Q is included in the region of attraction

under output feedback control for sufficiently small 6, and trajectories under output

41

feedback approach those under state feedback as e —) 0.

3.4 Comparison of Controllers

In this section we compare the controller design presented in Section 3.2 to that of

[14] through an example.

Example: Consider the two-input, third-order system

\
331 = I2
.12 = 01:03 + 11,
£173 = 021‘? + U2 I 1 (318)
331
y =
173 I

 

where a, and (12 are unknown constants with [(11] S 1 and [02] S 1. It can be seen

that Assumption 1 and 2 are satisfied with

—a,2:§ — 2:1 - 22:2 1:1 + .132

wins) = . him) = , (3.19)
—a2I¥ — 1‘3 5001133

V(:1:) = 1.52:,2 + 0.51:22 + 231332 + 250.13%, W(:1:) = —:r12 — 1:22 — 500.17%, L(:1:) = 1.

From (3.12) and (3.19) a function a(:1:) can be chosen as

 

01(2) 2 \/(]2:1[+2[:r2[+:r§)2+(:rf+[$3[)2+2

42

and from (2.8) the control 11 can be taken as

11 = “1 = —a 2: h(:1:) + 0)
i )2 (———, (3.20)
a = —o + W (m) ,

01
where 0 =

02

Now we will design a “continuous” sliding mode controller as presented in
[14]. The first step in a sliding mode design is to specify a sliding surface on which

the sliding motion occurs. The sliding surfaces are taken as

81 = I1+I2+01

32 2 1:3 +02,

where 0,- is the output of

31'

0',- = —0, +11,- sat ( ), 0,-(0) E [—p,-,p,], 1S 1 S 2
p

l

and 11,- are positive constants to be chosen.

43

From ([14], (12)), a control 11 can be taken as

111 (1% + [172] + 20) sat (11)
u = = — ,1 1 . (3.21)
112 (er + 1) sat (ii)
Simulation was run for the following initial conditions 5131(0) = —3,:1:2(0) =

—15,1‘3(0) = 1,01(0) = 0, and 02(0) 2 0 with p = 111 : p2 = .1. The difference in
performance of controllers (3.20) and (3.21) can be seen in Figure 3.1. The controller
(3.20) regulates the states 2:, and 11:3 faster as compared to the sliding mode controller

(3.21). The control inputs u, and 11.2 are shown in Figure 3.2.

From this example its clear that the controller design of Section 3.2 is a good

alternative to the sliding mode controller design presented in [14] and can give better

performance in some cases.

44

 

 

x2

-10-

 

 

 

 

 

—15 ‘ .

 

 

 

 

 

— _I— — —

0.2 0.3 0.4 0.5
t

C)

 

 

 

Figure 3.1. Plots of x1(t),x2(t) and 2:3(t) for the controller design (3.20) (dashed)
and the controller design using sliding mode control (solid)

45

 

 

 

 

 

 

 

 

 

 

 

 

50 . . T - 0
40 . / 1| . -5
I -10
30] |
I I -15
'5 20) ‘24 -20
I I
IOI I -25 ]
I
' [ -3o
0.
I I ’ .- "’35 '1
/
-10 ‘ ‘ ‘ ‘ —40 ‘
0 1 2 3 4 5 0 1 2 3 4 5
t t
a) b)

Figure 3.2. Plots of 111(t) and 112(t) for the controller design (3.20) (dashed) and the
controller design using sliding mode control (solid)

46

CHAPTER 4

Application to the Pendubot

We illustrate the improvement in the transient response using conditional integrators

by application to the Pendubot.

4. 1 The Pendubot

The Pendubot is an electro-mechanical system consisting of two rigid aluminum
links interconnected by a revolute joints. The first joint is actuated by a DC-motor
while the second joint is unactuated, thus making the Pendubot an under actuated

mechanism.

The Pendubot is in some ways, similar to the inverted pendulum on a cart,
where the linear motion of the cart is used to balance the pendulum. The Pendulum
uses instead the rotational motion of the first link to balance the second link. In this
regard, the Pendubot is also similar to the more recent rotational inverted pendulum,

invented by Professor Furuta of the Tokyo Institute of Technology [6]. In the

47

 

 

 

 

 

 

 

 

 

 

 

 

 

[HI moron "— OPTICALENCODER1
LII—IL , I] ' '—
C: ‘ f _ ——3
TABLE
uum
L 1 £4": OPTICAL sarcoma:
|
I .
UNKZ

 

 

 

L1 .
Figure 4.1. Front and side view drawing of the Pendubot.

rotational inverted pendulum, the axis of rotation of the pendulum is perpendicular
to the axis of rotation of the first link. The Pendubot has both joint axes parallel,
which results in some additional rotational coupling between the degrees of freedom.
This additional coupling, which is not found in either the linear inverted pendulum
or the rotational inverted pendulum makes the Pendubot more interesting and more
challenging from both a kinematics and a dynamic stand points. For example, in
both the linear inverted pendulum and the rotational inverted pendulum, the Taylor
series linearization around any operating point results in a controllable linear system.
Moreover, the linearized model (A, B, C) is the same at all points. In the Pendubot,

the linearization is operating-point-dependent; in other words, the linearization

48

(A,B,C) changes at each configuration and there are even special configurations

where the linearization is uncontrollable.

The Pendubot possesses many attractive features for control research and edu-
cation. It can be used to investigate system identification, linear control, nonlinear
control, optimal control, learning control, robust and adaptive control, fuzzy logic
control, intelligent control, hybrid and switching control, gain scheduling and other
control paradigms. One can program the Pendubot for swing up control, balancing,
regulation, tracking, identification, gain scheduling, disturbance rejection and friction
compensation to name just a few of the applications. Some of these applications are
described in [18], [15] and [16]. The maker of the Pendubot is Mechatronics Systems,

Inc. Figure 4.1 shows the front and side View of the Pendubot.

4.2 Mathematical Model

The equation of motion of the Pendubot can be found using Lagrangian dynamics

[17]. In matrix form the equation is

D((1)('1' + 001,411+ 9(9) = u (41)

49

where

C(q,1j)

and

61 + 92 + 293005012) 92 + 93005IQ2)

62 + 63COS(q2) 02

p

—93 Sill (1241 —93 Sillf‘hWIi + (12)

63 Sin(‘]2)(fl 0

b

p-

649 c0s(q1) + 659 COS((11 + £12)

959 C08((11 + 02)

 

711,131+ m21¥+ 11, 62 = 7112132 + 12,
m2lc2

the total mass of link one

the length of link one

the distance to the center of mass of link 1

93 = m2lllc21

C11 7'

92 0

64 = mllcl + "1.211

the moment of inertia of link one about its centroid.

the total mass of link two

the distance to the center of mass of link two

the moment of inertia of link two about its centroid

the acceleration due to gravity

50

The Pendubot parameters are 61 = 0.0308, 62 = 0.016, 63 = 0.0095, 64 = 0.2087 and

65 = .063. The matrix D(q) is invertible for all q E R2; hence, the state equations

can be written as

1'1:

.171:

2,:

.2332

£134:

D‘1(q)lu - C(q, (1)4 - 9(a)]
91,132 = 91.13 = 92,334 =92
$2
('11
1‘4

92.

4.3 The Equilibrium Manifold

For each constant value of T the Pendubot will have a continuum of equilibrium

configurations. Since at equilibrium we q, = q, = q2 = q2 = 0, we have

94 c08(<11) + 95 C08(611 + 92) =

O bl‘l

95 008011 + (12) =

and the Pendubot will balance at

91

92

- cos’1 (i)
64.9

7f
= —— =1a
n2 q]? n 7

51

4.4 Controlling The Pendubot

The Pendubot control strategy developed for the Pendubot in [18], [5] is divided
into two parts: a balancing control which balances the Pendubot about the desired
equilibrium point, and a swinging control that swings the Pendubot up from the

downward configuration to the desired configuration.

Balancing Control

The balancing problem may be solved by linearizing the equation of motion about an
operating point using Taylor series expansion and designing a linear state feedback
controller. As we saw before, the Pendubot has an equilibrium manifold which is
a continuum of balancing positions. The linearized system becomes uncontrollable
at q, = 0, 7r as illustrated in Figure 4.2 which shows controllable and uncontrollable

positions of the arm.

Swing Up Control

The problem of swinging the Pendubot up from the downward position to the

inverted position is an interesting and challenging nonlinear control problem.

From the equation of motion we have

d1191 + (11292 + hl + 951 = T (4-2)

(12191 + (12292 + ’12 + 952 = 0, (43)

52

 

 

“—

q2

 

 

Ob":

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Figure 4.2. The pendubot arm at a: Controllable position, b: Uncontrollable position

where

 

dll d1?
13(0) =
(121 (122
. hl (3)1
C(20) = . 9(1)):
I12 (1'52

Solving for (jg from (4.3) and substituting in (4.2). we obtain

.. (1 2(12 (1 2112 1 (1 2992
‘1‘ (‘111 _ :11) + (1” _ .11 H + (0‘ 7 :1 H 2 7'

 

 

 

Taking the control variable T as

d d d h d
min-i Wi i::)+ii- 1:?)

 

 

 

results in

91 = U1

612292 + I12 + 952 —d21U1

The control 11, is taken as

v1: kp(qf - <11) + kd((If - (11)

to track some reference positions 7" = q‘li, r -_- q? = 0 and the positive constants 13,, and

kd are the control gains. Defining the tracking errors as

81: (Iii-91, €2=9f’91

771: 92, 772:92

the system (4.4) can be rewritten as

é1=€2

62 = —kp€1—kde2

54

01:92

, 1 d . .
112 = ——ihz+¢2)——33kpiqf—ql)+kdiqi—ql)
6122 d2?

yzel.

The tracking error part can be written as

where e = [81, €2]T, k, and kd are chosen to make E Hurwitz. On the sliding surface

6 = 0, the dynamics are given by

1'11 = 112 (4.5)

1
772 = —d—(h2+¢2) (4.6)
22

which represents the zero dynamics with respect to the output y = 61. We see
from (4.5) and (4.6) that the zero dynamics are just the dynamics of the unactuated
arm, which has a periodic orbit. While the error e(t) converges to zero, the steady-
state behavior for the first link converges exponentially to q‘,‘ and the second link
oscillates about (—7r, 0). The swing up control job then is to excite the zero dynamics
sufficiently by the motion of link 1 such that the pendulum swings close to its unstable
equilibrium. When the pendulum is close to the desired equilibrium, the controller
is switched from a partial feedback linearization controller to the linear balancing

controller.

55

4.5 Hardware Description

The Pendubot consists of two rigid aluminum links of length 14 in and 8 in. Link
one is directly coupled to the shaft of a 90 V permanent magnet DC motor mounted
to the end of the table. The motor mount and bearing support the entire system.
Link one includes the bearing housing for two joints. The shaft extends out in both
directions of the housing, allowing coupling to the second link and to an optical
encoder mounted on link one. The design gives both links full 3600 of rotational

motion. The optical encoders resolution is 1250 pulse/rev.

All the control computations are performed in Pentium PC with a D/A card
and encoder interface card. Using the software routines supplied with the Pendubot,

the control algorithm are programmed in C.

4.6 Observer design

As the optical encoders only measure the positions of the links, i.e., the angles q, and

(12, we estimate the angular velocities q, and q2 using a nonlinear high-gain observer

2 = A31? + 8006,11) + H(y — 05:)

where

<i>o = D“(q)[u - C(q, (1)4 - 9(a)]

56

 

and

 

 

 

 

[0100 00
0000 10
A: ,8: ,
0001 00
0000 01
1000 35,00
C: ,HT=
0010 00%;),

The observer parameter 6 > 0 was chosen to be .008, to recover the performance

under state feedback.

4.7 Addition of Uncertainty

The Pendubot was stabilized at angles q, = 75° and (12 = 15° using the control strat-
egy developed for the Pendubot in Section 4.4, which applies first a swing-up control
to swing the Pendubot from the downward configuration to its desired equilibrium po-
sition and then applies a balancing controller to do the balancing. A linear balancing

controller

11 = 16.4615(:1:, — 1.3090) + 3.12871:2 +16.242(:r.3 - .2618) + 2.06581:4 + 0.5296

57

 

 

q1

 

 

 

)—

 

_2 l l 1

Figure 4.3. Trajectories for angle q, for different balancing controls, linear controller
with no disturbance (solid), linear controller with constant disturbance of 1.6V in the
control (dotted), linear state feedback integral controller with constant disturbance
of 1.6V in the control (dashed)

with 51:, = q,,:1:2 = (71,273 = Q2 and :54 = 42 was obtained using pole placement
techniques and linearized nonlinear equations of the Pendubot from [3]. The details
of the controller design is presented in Appendix A. The linear controller regulated
the states to desired values as shown in Figure 4.3. Matched uncertainty in the form
of a constant disturbance of 1.6 Volt was added to the control leading to an offset in
the motor torque and shifting of the equilibrium point causing steady-state error as

shown in Figure 4.3.

58

4.8 Integral action

Traditional integral action was introduced in the balancing controller by augmenting
the states with integrators such that at the equilibrium point of the new augmented
system the steady-state error is zero. The linear state feedback integral controller is

given by
u = 51.5023(2:1 — 1.3090) + 8.60222 + 40.0770(:1:3 — .2618) + 5.184024 + 23.3882é,

where é is the augmented state. The performance of this controller in the presence
of constant disturbance in the motor torque can be seen in Figure 4.3. Although
the integral controller regulates the states to desired values, the transient response

becomes more oscillatory.

4.9 Conditional Integrators

Conditional integrators were introduced to both improve the transient response and
also achieve zero steady-state error. As the linear controller designed above achieves
exponential stability of the Pendubot in the absence of disturbances, Assumption 1
is satisfied with quadratic Lyapunov function V(x) and quadratic positive definite
function W(:1:). With both Lyapunov function and state equation being known, As-
sumption 2 is satisfied with L(:1:) = 1. Finally, it can be verified from the linearized

nonlinear equations of the Pendubot in [3] that Assumption 3 and 4 are also satisfied.

59

 

 

 

 

 

 

 

_2 l l 1 I

Figure 4.4. Trajectories for angles q, for different balancing controls with disturbance
of 1.6V in the control, traditional integral action (solid), conditional integrator with
[120.25 (dashed), conditional integrator with ,11=0.15 (dotted)

The balancing controller is given by the saturated high-gain feedback

11 = —3 sat (w—U)
11

y : 3.5[—.811(:r?1 — 1.3090) — 0.14522 — .8045(173 — .2618) — 0.11:4],

where the conditional integrator state 0 is given by (7). As can be seen from Fig-
ure 4.4 and Figure 4.5, the controller achieves zero steady-state error and improves
the transient response with decreasing ,11. Note that the apparent discontinuity in the

three figures corresponds to the point of switching from the swing-up controller to

the balancing controller.

60

 

   

 

 

 

Figure 4.5. Trajectories for angles q2 for different balancing controls with disturbance
of 1.6V in the control, traditional integral action (solid), conditional integrator with
117-025 (dashed), conditional integrator with p=0.15 (dotted)

61

CHAPTER 5

Conclusion

We presented conditional integrators as a tool for achieving asymptotic regulation
of nonlinear systems subject to constant disturbances or parameter uncertainties,
without compromising the transient response. We considered a fairly general
class of nonlinear systems that can be stabilized by state feedback control and
showed that, in the presence of matched parameter uncertainties that cause a
shift in the equilibrium point, the system can be augmented with conditional
integrators to recover asymptotic regulation of the state to the origin. We showed
that the conditional integrator does not degrade the transient response, in the
sense that as the width of a boundary layer approaches zero, trajectories of the

system with conditional integrator approach those of a system with no integral action.

We also considered regulation of a class of minimum-phase, input-output lin-
earizable, nonlinear systems where instead of regulating the states to the origin, they

are regulated to a disturbance-dependent equilibrium point at which the regulation

62

error is zero. Output feedback control was implemented using high-gain observers.
Towards the end, the performance of traditional and conditional integral control
were demonstrated experimentally on the Pendubot. Asymptotic regulation with

improved transient response was achieved with conditional integrators.

63

Appendix A

Controller Design

A. 1 Linear Controller

The linearized nonlinear dynamic equation of the Pendubot about the operating point

q, = 75° and Q2 2 15" is obtained from the Matlab code provided in [1]. The linearized

equation is
i: = .45: + B[11 — 0.5296]

where (I: = [$1 — 1.3090,:6bil'2 — .2618,T2]T and

0 1 0 0 [ 0
63.2865 0 —23.8152 0 44.0722
A = , B: (A 1)
0 0 0 1 0
—60.3877 0 102.9146 0 —82.6285

 

 

 

 

64

The controller gain matrix K was found using the place command of Matlab. The
poles of the closed-loop system were placed at —10.15 + 7.661, —10.15 — 7.661, —8.3

and —4.2.

A.2 Traditional Integral Action

Traditional integral action was introduced by augmenting the states with 6 = :13, —

1.3090. With the augmented state, the linearized equation is

2 = A2? + B[11 — 0.5296]

where :1: = [2:1 — 1.3090,231,:1:2 — .2618,:1':2, é]T and

 

 

 

 

[ 0 1 0 0 0- I 0 -
63.2865 0 —23.8152 0 0 44.0722
A = 0 0 0 1 0 , B: 0
—60.3877 0 102.9146 0 0 —82.6285
. 1 0 0 0 0 j [ 0 .

The controller gains were found using the place command of Matlab. The poles of the
closed-loop system were placed at —23.67, -6.76 + 5.18441, —6.76 — 5.18441, —7.1618

and —4.88.

65

A.3 Conditional Integrator

Let P be the solution of the Lyapunov equation

(A — BK)TP + P(A — BK) = —Q,

where A, B are as given in (A1), controller gain matrix K is chosen as in Appendix

Al and the positive definite matrix Q is taken as

.0206 0 0 0
0 .0206 0 0
Q =
0 0 .0206 0
0 0 0 .0206
- 1

 

 

Then, for the closed-loop system under the linear controller, Assumptions 1-4 are

satisfied with

V(:1:) = :cTPz, W(:t:) = —:1:TQ:1:, L(:1:) = 1, h(:r.) = 2BTP2:

and control 11 is taken as (2.8).

66

BIBLIOGRAPHY

[1] Pendubot Model P-2 User’s Manual, Mechatronic System, Inc., Champaign, IL.

[2] AN. Atassi and H.K. Khalil. A separation principle for the stabilization of a
class of nonlinear systems. IEEE Trans. Automat. Contr., 44:1672-1687, 1999.

[3] DJ. Block. Mechanical design and control of pendubot. Master’s thesis, Univer-
sity of Illinois, Urbana Champaign , IL, 1991.

[4] CI Byrnes, F.D. Priscoli, and A. Isidori. Output Regulation of Uncertain Non-
linear Systems. Birkhauser, Boston, 1997.

[5] I. Fantoni, R. Lozano, and M.W. Spong. Energy based control of the pendubot.
IEEE Trans. Automat. Contr., 45:725—729, 2000.

[6] K. Furuta and M. Yamakita. Swing up control of inverted pendulum. In In-
ternational Conference on Industrial Electronics, Control and Instrumentation,
Kobe, Japan, 1991.

[7] J. Huang and W.J. Rugh. Stabilization on zero-error manifolds and the nonlinear
servomechanism problem. IEEE Trans. Automat. Contr., 37:1009—1013, 1992.

[8] A. Isidori. Nonlinear Control Systems. Springer-Verlag, New York, 3rd edition,
1995.

[9] A. Isidori and GI. Byrnes. Output regulation of nonlinear systems. IEEE Trans.
Automat. Contr., 35(2):131—140, 1990.

[10] H.K. Khalil. Universal integral controller for minimum phase nonlinear systems.
IEEE Trans. Automat. Contr., 45(3):490—494, 2000.

[11] H.K. Khalil. Nonlinear Systems. Prentice Hall, Upper Saddle River, New Jersey,
3nd edition, 2002.

67

 

[12] NA. Mahmoud and H.K. Khalil. Asymptotic regulation of minimum phase non-
linear systems using output feedback. IEEE Trans. Automat. Contr., 4121402—
1412, 1996.

[13] L. Sciavicco and B. Siciliano. Modeling and Control of Robot Manipulators.
McGraw—Hill, New York, 1996.

[14] S. Seshagiri and H.K. Khalil. Robust output feedback regulation of minimum-
phase nonlinear systems using conditional integrators. 2005. To appear in Au-

tomatica.

[15] M. Spong. Energy based control of a class of underactuated mechanical systems.
In IFAC World Congress, San Hancisco, CA, 1996.

[16] M. Spong and L. Praly. Control of underactuated mechanical systems using
switching and saturation. In Proc. of the Block Island Workshop on Control
Using Logic Based Switching, Springer-Verlag, 1996.

[17] M. W. Spong and M. Vidyasagar. Robot Dynamics and Control. Wiley, New
York, 1989.

[18] M.W. Spong and DJ. Block. The Pendubot: A mechatronic system for control
research and education. In Proc. IEEE Conf. on Decision and Control, pages
555—556, New Orleans, LA, 1995.

68

   

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