WIWlmNWlHllHHHlHUNIHWlHIHlHIUHUI

   

THESE

 

LERA?‘ A my
fi—uunzi '

0
L44.
[Moan Swat-a
'-"3'b: —-2‘-
L-h-vs'l . I

‘.e]! sflL

 

 

This is to certify that the

thesis entitled

0N RGBUST‘ TRACKING IN UNCERTAIN SYSTEMS
-A VARIABLE STRUGTURE APPROACH

presented by
Seung-Bok Choi

has been accepted towards fulfillment
of the requirements for

”.5- degree in M119..—

[W
jaw

/____..

7
(‘

 

Date 0 {/If/ Pg

0-7639 MS U i: an Waive Action/Equal Oppofium‘ty Institution

 

 

 

RETURNING MATERIALS:
)V1531_] Place in book drop to
LJBRARJES remove this checkout from

a— your recorc. FINES win
be charged if book is

 

 

returned after the date
stamped below.

 

 

 

ON ROBUST TRACKING IN UNCERTAIN SYSTEMS

- A VARIABLE STRUCTURE APPROACH

By
Seung-Bok Choi

A THESIS

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

MASTER OF SCIENCE

Department of Mechanical Engineering

1986

ABSTRACT

ON ROBUST TRACKING IN UNCERTAIN SYSTEMS
- A VARIABLE STRUCTURE APPROACH

BY

Seung-Bok Choi

A simple tracking controller design applicable for both linear
and nonlinear uncertain dynamical systems based on variable structure
system(VSS) theory is given. The control design employs so called
"sliding conditions" to guarantee asymptotic tracking for any ar-
bitrary initial conditions lying off of sliding surfaces. To assure
the existence of sliding modes for multi-input systems, we study the
domains of attraction for sliding modes in terms of Lyapunov
stability theory. It is shown that the robustness of the control
scheme to input uncertainties is achieved primarily by assuming a
form of matching conditions. The important properties of gradient
vectors of the sliding surfaces will also be highlighted. The design
methodology is straightforward and requires little computational
effort. To illustrate this, several numerical examples are
presented, and the method is applied to the control of a three-
degrees-of-freedom manipulator subjected to variable payloads and

external torque disturbances.

Dedicated to my parents

ii

ACKNOWLEDGEMENTS

I would like to express my sincere gratitude to Dr. Suhada
Jayasuriya, my major professor, for his expert guidance,genuine
wisdom and continuous encouragement throughout the course of this
work, and also for his painstaking review of this manuscript. Sincere
appreciation is extended to professors R. Rosenberg and M. Gandhi
for their valuable advice on my presentation.

I, of course, owe a great debt to my lovely wife Yeon-Ook and
son Min-kyu for being a wonderful family.

Special thanks go to graduate students C.D.Kee, M.S.Suh, and my

friends at MSU for making my graduate career a successful experience.

iii

LIST OF

CHAPTER

II.

III.

TABLE OF CONTENTS

page

FIGURES ................................................. v
INTRODUCTION ............................................ 1
SLIDING MODES IN DISCONTINUOUS DYNAMIC SYSTEMS .......... 4
2.1 Mathematical description ........................... 4
2.2 Conditions for the existence of a sliding mode ..... 7
2.2.1 For single-input systems ....................... 7
2.2.2 For multi-input systems ........................ 11
CONTROLLER FORMULATIONS ................................. 17
3.1 A class of single-input systems ................... 17
3.1.1 Linear dynamical systems ....................... 17
3.1.2 Nonlinear dynamical systems .................... 22
3.2 A class of multi-input systems ' .................... 24
3.2.1 Linear dynamical systems ....................... 24
3.2.2 Nonlinear dynamical systems .................... 29

3.3 Gradient vectors of sliding surfaces and approximat-

ions of discontinuous control laws ................. 33

IV. AN APPLICATION TO A ROBOTIC MANIPULATOR ................. 39

V. CONCLUSIONS ............................................. 54
LIST OF REFERENCES .............................................. 56

iv

FIGURE

10
11
12
13
14
15
16
17

18

LIST OF FIGURES

page
The construction of fo(x,t) .............................. 6
Sliding mode domain ....................................... 8
Sliding mode motion due to a scalar control function ...... 10
Sliding mode motion due to two control functions ......... 13
State portrait of two dimensional VSS(sliding mode) ...... 14
State portrait of two dimensional VSS(no sliding mode) ... 15
State trajectories for cl-O.S, 100, 500 ................. 35
Construction of the boundary layer ....................... 37
An interpolation of u(t) in the boundary layer ........... 38
Three-degrees-of-freedom manipulator ..................... 40
State trajectories under no load(discontinuous) .......... 46
State trajectories under full load(discontinuous) ........ 47
Discontinuous control efforts under no load .............. 48
Discontinuous control efforts under full load ............ 49
State trajectories under no load(continuous) ............. 50
State trajectories under full load(continuous) ........... 51
Continuous control efforts under no load ................. 52
Continuous control efforts under full load ............... 53

CHAPTER I

INTRODUCTION

In recent years,increasing attention has been given to control
designs that utilize the theory of variable structure systems (VSS)
as described in [2,3] : that is a class of systems with discontinuous
feedback control. Young[4],Ryan[5], and Slotine and Sastry[6]
proposed variable structure controllers for multi-input systems by
introducing a single surface of control discontinuity for each input.
Young and Ryan used the so called hierarchy of control method. The
basic idea here is that the system states are forced to the surfaces
of control discontinuity sequentially. Slotine and Sastry on the
other hand used sliding conditions to drive the system states to all
the switching surfaces simultaneously. It is interesting that the
continuous controller proposed by Corless and Leitmann[9] appears to
be a variable structure controller in the limit as its saturation
function parameter tends to zero.

One of salient features of VSS is that the so-called sliding
mode may occur on the switching surfaces independently or on the
intersection of several switching surfaces. During the sliding mode
motion, the system remains insensitive to disturbances and parameter
variations producing perfect tracking of the desired trajectories.

However, the ideal sliding mode does not occur in practice due to the

2

ever present non-idealities such as delays,small time constants,
hysteresis,etc. These non-idealities cause the trajectories to
chatter along the sliding surfaces resulting in the generation of
undesirable high frequency components. Thus, in real systems the
discontinuous controls should be replaced by smooth
approximations[6,7] so that the actual trajectories will be in the
vicinity of the sliding surfaces.

The technique used in this thesis to construct variable struc-
ture controllers is based on a special way of imposing the sliding
conditions. With a knowledge of Filippov's solution concept for
differential equations with discontinuous right-hand sides[l], we
exploit the condition for the existence of a sliding mode. For the
special case of scalar control, a sliding mode exists, if at a point
on a sliding surface, the directions of motion along the state
trajectories on either side of the surface tend towards the surface.
In the multi-input case, the sliding mode occurs, if the condition
for the existence of a sliding mode is met on each of the discon-
tinuity surfaces associated with corresponding inputs. To
substantiate the domain of sliding mode on the intersections of
sliding surfaces for the multi-input case, we consider the sliding
mode in terms of Lyapunov stability theory. The latter approach makes
it possible to realize sliding modes along individual surfaces or
along the intersection of such surfaces for multi-input systems
considered in this study. The design procedure based on these ideas
is straightforward and extremely simple compared with [4,6].
Moreover, the methodology is quite general and may be applied to

various dynamical systems having bounded uncertainties.

3

In the synthesis of the above mentioned controllers , it is
necessary to choose appropriate values of gradient vectors of sliding
surfaces to guarantee asymptotic tracking in view of the internal
characteristics of a system. The approach developed so far in select-
ing the parameters of the gradient vectors is essentially a trial and
error one and no analytical method is yet available for the proper
choice of the gradient vectors. In this thesis the importance of the
gradient vectors is highlighted by presenting a specific example.
Suggestions are made for the resolution of this issue and will remain
an interesting area for further investigation.

The layout of the thesis is as follows. In chapter 2 we review
the properties of the sliding mode in discontinuous dynamic systems
including the mathematical framework proposed by Filippov[l] and
conditions for the existence of a sliding mode. Chapter 3 illustrates
the procedures for the design of feedback controllers for both single
and multi-input systems in the presence of parameter variations and
disturbances. The control scheme is then applied to the control of
a robotic manipulater handling variable payloads in chapter 4, fol-

lowed by our conclusions in chapter 5.

CHAPTER II

SLIDING MODES IN DISCONTINUOUS DYNAMIC SYSTEMS

2.1 Mathematical description

We begin by constructing a mathematical framework to analyze
differential equations with discontinuous right-hand side associated
with our discontinuous control laws to be developed. A solution
concept for such differential equations has been developed and com-
pared with other solution concepts by Filippov[1].

We review this concept by considering the vector differential

equation
i(t)-f(X.t) (1)

where sznXR 4 Rn satisfies the following:

Assumption:

The function f(x,t) is a real-valued measurable function,

defined almost everywhere in an open domain QCIRnfl

Further, for
all compact D<: Q there exists integrable M(t) such that
IIf(x,t)IIS M(t) almost everywhere in D, where II-II denotes any

norm .

5

Qgfiini;ign(solution concept for equation (1)):

The vector function x(t), defined on the interval [to,t1] is
called a solution of (1) if it is absolutely continuous on [to,t1]

and if for almost all te[to,t1]

name 590 ”QC Conv f(B(x(t),5)-N) (2)
r\

where pN-O

denotes the intersection over all sets N of Lebesgue

measure zero and Conv refers to the convex hull of a set.
Geometrically, (2) means that at every point of the discontinuity
surface the velocity vector characterizing the solution belongs to a
minimal convex closed set containing all values of f(x) when x
covers the entire 6-neighbourhood of the point under consideration
(with 6 tending to zero) except a zero measure set.

We now apply this solution concept to our equation (1) to deter-
mine the phase velocity along the sliding mode. Let the regions G-
and G+ in the space xl,. . . .,xn be separated by a switching
surface(smooth function) 5 as shown in Figure 1. Suppose that
there exist limiting values f-(x,t) and f+(x,t) of f(x,t) for any

constant t when s is approached from G- and GT Let ffi and f;

be the projections of the vectors f' and f+ on the normal to the

surface s directed from G- and GT Then, when x(t)es, and f§>0

o
and f;<0, we may construct the phase velocity f (x,t) as in Figure

1. By the definition given above, at each point of the switching

surface the end of the velocity vector must belong to the segment

6

joining the points f'(x,t) and f+(x,t), i.e., its end point is the

intersection point of the plane tangential to the surface and the

straight line connecting the ends of the vector f-(x,t) and f+(x,t).

Thus, we define the sliding mode equation as
o
x(t)-f (x,t) (3a)

f°(x,t)-af++(l-a)f- (050.51) (3b)

where a is a parameter depending on the directions and magnitudes
of f-, f+ and the gradient of the switching surface 3. On the other

0
hand, the projection of the f on the normal to the 5 must be
equal zero so that the trajectory slides along 5 once it hits 3.

This fact is referred to as ideal sliding. Computing a with grad

o
s-f -0 gives the equation

   
 
 

phase velocity fO

\ tan ntial
\
“\ surface

discontinuity
surface (s)

 

0
Figure l. The construction of f (x,t)

grad s-f' grad s-f+

+ -
grad s.(£'-£+) f ' grad s-(f'-f+) f (4)

 

 

x(t)-fo(x,t)-

which is known as Filippov's continuation equation.

Note that to substantiate the validity of equation (4) in
studies of actual control systems with discontinuous controls,
various non-idealities such as delays, and hysteresis are recognized.

These are discussed in detail in [2].

2.2 Conditions for the existence of a sliding mode
2.2.1 For single-input systems

With this mathematical background, we turn our attention to a
dynamical system of a general type with a discontinuous scalar con-

trol described by the equation
i(t)-f(X.t.u) (5)

where f:RpxRxR*Rn,xeRpand ueR. VSS are characterized by discon-
tinuous control which changes structure on reaching a set of

switching surfaces. The control has the form

u(x,t)- [u+(x,t) , s(x)>O

u-(x,t) , s(x)<0

(6)

where u+(x,t), u-(x,t) and s(x) are certain continuous functions

and uffl u: Since we intend to have a sliding mode in the system,
conditions for the existence of a sliding mode should be found

analytically. Such conditions for a sliding mode to occur on a

8
switching surface may be stated in a number of ways. In this study
state concepts are employed to describe such conditions for the
general system (5). From the solution concept of equation (3) in
2.1, it follows that a sliding mode exists if there are domains of

non-zero measure on the surface s(x)-O where the projections of the

vectors f+-f(x,t,u+) and f--(x,t,u-) on the surface gradient are of
opposite sign and are directed towards the surface(Figure 2).

Analytically, these conditions can be expressed as

lim+s<0 and lim_s>0 (7)
s40 5+0

or equivalently

ss<0. (8)

   
    

s>O

sliding mode
domain

 

 

Figure 2. Sliding mode domain

9
Once in the domain of attraction the describing point is assumed
to move to the interior of the domain until the boundaries are
reached followed by motion along the boundary. This motion will be
termed ideal sliding which should be regarded as the motion that
results from a limiting process with all the non-idealities tending
to zero.

The condition (8) is referred to as the local sliding condition,
since it is sufficient to guarantee that trajectories starting from
initial conditions close to s converge to s and slide along 3.
Without loss of generality, the global sliding condition can be

extracted from (8) in view of stability, i.e.,

sé<-1(ISI) (9)
where 1(-) is a continuous function of class K defined in [10].
The condition (9) implies that all initial conditions lying off s
will converge to s and then slide along 5. Also condition (8) and
(9) guarantee that trajectories originating on s will remain on 5.
Note that once in the sliding mode, the system (5) satisfies the

equation

s(x)-O and S(x)-O. (10)
Thus, the system has invariant properties yielding a motion which is
independent of diturbances and parameter variations.
To illustrate concepts described above, consider the scalar

system

x1(t)-x2(t) - (lla)

x2(t)--a(t)x1(t)+x2(t)+u(t)+d(t) (llb)

10
where a(t) is a time-varying parameter and d(t) is an external
disturbance whose precise value is unknown but bounded. Defining

the switching function s(x)-cx1+x2 yields the discontinuous control

+
u_ [u , cx1+x2>0 (12)

u', cx1+x2<0
where c>0 and u+# u: The line 5-0 is the surface on which the
control has a discontinuity. It can be easily shown that the state
x(t) reaches the switching line s-O in a finite time if u+ and

u- (see Figure 3) are chosen appropriately. The state x(t)

crosses the switching line and enter the domain s<0 resulting in

 

the value of u being altered from u+ to u: The proper choice

x
112

"x

l

s>0

S<O s=cx +x =0
1 2

 

Figure 3. Sliding mode motion due to a scalar control function

ll
of control makes the state trajectory immediately recross the switch-
ing line and enter the domain s>O, i.e., this yields the sliding
mode motion. For infinitely fast switching logic, the state is

forced to remain in a neighbourhood of a switching line by the con-

trol which oscillates between the values uf and u: Then, we have a
new dynamic system during the sliding mode motion described by

s-cx1+x2-O (13)
which is quite independent of the original system, and stable since
c>O. The sliding condition for this system can be established by

appropriately selecting u+ and u- so that

s--a(t)x1(t)+(c+l)x2(t)+u+(t)+d(t) < o, for s>O

s--a(t)x1(t)+(c+l)x2(t)+u-(t)+d(t) > O, for s<O

for expected values of a(t) and d(t).

2.2.2 For multi-input systems

We now consider multi-input systems described by

i<c)-F(x,t,U) (14)

where anpxRme»RP, XeR“

and UeR? Analogous to what was done for
VSS with scalar control each component of control is assumed to
undergo a discontinuity on an appropriate surface in the state

space,i.e.,

(15)

u [u:(x,t) , si(X)>O

1' u;(X,t) , si(X)<O

12

where s1 is a switching surface associated with a corresponding

u . The design problem then consists of choosing the continuous

functions uI,

u; and the vector SeRm with the functions si(X)
as components. For the scalar control case, we have a transparent
geometric interpretation(Figure 2) of the conditions for the exist-
ence of a sliding mode, but for the vector control case the
conditions can be hardly described in geometrical terms of mutual
positioning of state velocity vectors. Since introducing one switch-
ing surface associated with one control permits us to consider the

sliding conditions as in the scalar case,we can construct the follow-

ing sliding conditions from (8) and (9)

s ai< 0 (16a)

i

and

sisi< -1(Isi|). (16b)

A possible trajectory of the system (l4),(15) is given in
Figure 4 if the sliding conditions (16) are met. Case A implies
that there exists a sliding mode on the intersection of two sliding
surfaces, not necessarily separate sliding modes on the switching
surfaces. This type of sliding mode is a new phenomenon and an
interesting subject for investigation. Case 8 indicates the pos-

sibility of a sliding mode occuring on the switching surface 51.

Note that in the final stage of motion the phase point is forced to
move along the intersection of the two discontinuous surfaces in both

cases .

l3

 

 

 

 

 

 

 

 

Figure 4. Sliding mode motion due to two control functions

Let us further investigate case A in terms of Lyapunov stability
theory. As is usually done in stability theory, we can try to find
sufficient conditions for a certain domain S of the intersection of
discontinuous surfaces in the state space of the system (l4),(15).
Utkin[2] established these conditions in a theorem. This theorem
says that for the intersection of discontinuous surfaces of the
system(l4),(15) to be a sliding mode, it is sufficient that for all
X belonging to a certain domain S of the sliding mode there exists a
continuously differentiable function V(S,X,t) such that V(S,X,t)>0
with SiO, V(O,X,t)-O with arbitrary X and t, and its total
time derivative along the trajectories of (14) and (15) be negative

everywhere, except the discontinuous surfaces on which this function

14
is not defined. To show that the requirement of continuous differen-
tiability is essential for finding the conditions for a sliding mode
to exist, we now consider two simple examples. First consider the
example of a system with two dimensional control whose motion projec-

tions on a plane 31, s: are described by the equations

s,--2 sgn 51+3 sgn sz

s,--3 sgn 51-2 sgn 82

where
1, if si>O
sgn si- 0, if si-O.
-1, if 8 <0
i
32:

 

gas

 

Figure 5. State portrait of two dimensional VSS(sliding mode)

15
Select a positive definite function V in the form

V-|81|+ I82l-
Then its time derivative

is negative everywhere except at 51-0. Consequently, the function

V and U have opposite signs and from the state portrait of Figure
5 it follows that there exists a sliding mode at the origin for
arbitrary initial conditions.

For the second example, consider the system whose motion projections

are on a plane 51» 52 described by the equations
a,— -5 sgn s1 - 2 sgn $2

s2- -sgn $1 + sgn $2.

 

AAAAAAAAAAAA
wvvvvvvvvv'

 

 

 

Figure 6. State portrait of two dimensional VSS(no sliding mode)

16
We select the positive definite function V as

V ' '51] + '52]-

Its time derivative

V - -4 -3 sgn sls2

is negative everywhere except at 51-0. However, from the state

portrait of Figure 6 it follows that there is no sliding mode

despite the difference in signs of the functions V and U, i.e.,
this example shows that the knowledge of signs of the piecewise
differentiable function and its derivative is not sufficient to
ascertain the existence of a sliding mode.

Thus, establishing the existence of a sliding mode is the most
important portion in the design of a discontinuous controller by
using the theory of VSS. In the next chapter, we formulate the
discontinuous control laws that are based on the concepts discussed

in this chapter.

CHAPTER III

CONTROLLER FORMULATIONS

3.1 A class of single-input systems

3.1.1 Linear dynamical systems

Let us start with a linear time-varying single-input control
system described by
x(t)-A(t)x(t)+b(t)u(t) (17)
where x(t)eRp, u(t)eR and

A(t)- . b(t)-

:H-n-ooo
x-C»--<D

arc-coco
arc-now
x-cxouaca

The control problem is to get x(t) to track a desired trajectory
1

xd(t) which belongs to the class of continuously differentiable(C )

functions on [to,m). In other words, the controller should force

the tracking error to zero asymptotically for any given initial

states. Thus we define the tracking error a(t) as

e<t> 9 x(t)-xd<c) (18)

where e(t)eRn, xd(t)eRn , and also define the sliding surface s in

17

18

the error state space by

s(e(t)) 9 Ce(t) - o (19)
where C is a lxn row vector with constant elements of the form

[c1, - - . . ,c 1]. We see that the tracking error e(t)40 for

n-1’
any given initial conditions, provided (19) is asymptotically
stable. Thus it remains only to obtain a control u(t) so as to
cause the trajectory x(t) to slide along the surface defined by
(19). This can be achieved by satisfying the sliding condition
(9). It can be easily shown that the sliding condition (9) is always

satisfied by letting the time derivative of s be

s - -k sgn s (20)
where k is a suitably selected positive constant. Now, using (17)-
(20) we construct a discontinuous control u(t) that satisfies the

sliding condition (9), with the obvious assumption Cb(t)#O
u(t) - [Cb(t)]1[-k sgn s-CA(t)x(t)+de(t)]. (21)

The design procedure is straightforward and requires little computa-
tional effort. The discontinuous control law (21) guarantees that all
state trajectories starting off the sliding surface converge to it
and those starting on the surface remain on it for all future time.
Therefore, the feedback control yields that e(t)+0 as ten for any
given initial conditions. Note that u(t) is discontinuous across 3
and that control discontinuity increases as the value of k

increases.

19
we next add an unknown but bounded disturbance to the system

(17), namely we consider

x(t) - A(t)x(t)+b(t)u(t)+h(t)d(t) (22)

where d(t)eR, and h(t)é[0, . . . , O, *1? It is obvious that with

the exact same structure of controller (21) with the value of k
properly chosen in accordance with the magnitude of the disturbance
assures the existence of a sliding mode of the system (22) resulting
in asymptotic tracking. To illustrate the suggested technique, con-
sider a second order single-input system in the presence of bounded

external disturbance described by the equation

x(t)-[g i]x(t) +[g]u(t) +[2]d(c) (23)

where d(t)e[61,62]. The problem is to get x(t) to track a desired

trajectory xd(t) which belongs to C1 on [to,w). The sliding
surface is chosen from (19) as

s - (xl-xld)+(x2-x2d). (24)
Thus, from (21) we obtain the discontinuous controller

u(t) - [-k sgn s-2x1-(l+t)x2+x1d+x2d] (25)
which yields the sliding mode motion of system (23), so long as

k>max(|61l,l6zl). Of course, with a small value of k enough to

compensate process noise and measurement noise we may guarantee
asymptotic tracking of the system (23) in the absence of d(t). Note
that the control discontinuity increases with the strength of distur-

bance to be compensated for.

2O

Assume now that equation (22) is replaced by
x(t) - A(t)x(t)+AA(t)x(t)+b(t)u(t)+h(t)d(t) (26)
where A(t), b(t), h(t) are as in (22) and

68 C C C Q C a
1 6 n an

such that each entry of last row 6a1 is bounded as ais Sais §i<w,

for i-l,- - . . , n. Let us define that

 

 

F .
Afi-

Lao 1 O O O O O aOn‘ an
A A _ (:3 _

La1 O O O O O O and rlxn

where a°i is an average value corresponding to Gai and ai-ai-aoi.

It can then easily be shown that the sliding condition (9) is always

satisfied by letting the time derivative of s be
a - C(AA(t)-Ao)x(t)-(k+ICAx(t)I) sgn s +Ch(t)d(t) (27)

where C is a row vector ensuring a stable sliding surface (19), I-

 

denotes absolute value, and k is a parameter to be tuned according
to the magnitude of the disturbance. A simple calculation shows that

the sliding mode motion of system (26) will occur if we use the

feedback control law
u(t) - [Cb(t>Ilt-<k+|ch(c)l> sgn s - c<A<t)x<t>

+on(t))+de(t)]. (23)

21
Thus, this controller produces asymptotic tracking for arbitrary
initial conditions. Note again that the control discontinuity in-
creases in order to compensate parameter uncertainties and external
disturbances.
By a minor modification of the foregoing procedure, it can be

extended to systems of the form

x(t) - (A(t)+AA(t))x(t)+(b(x,t)+6b(x,t))u(t)+h(t)d(t). (29)
In order to deal with input possessing uncertainty, we make the
following assumption.
Assumption(matching condition for single input):

There exist a Caratheodory function 6p(x,t)eR, and a con-
tinuous function ¢(x,t)eR+ such that, for all (x,t)eRnXR,

6b(x,t)-b(x,t)6p(x,t), I8p(x,t)ls ¢(x,t)<l (30)
where '0] denotes absolute value.
Since |5p(x,r)|<1, we have l+6p(x,t)>0 and ((l+6p(x,t)/(l-

¢(x,t))zl. We now prOpose control u(t) of the form
u(t) - [Cb(x,t)]1[-(k+|CA(t)x(t)-de(t)I+|CAx(t)I) sgn s

' CAoX(t)] / [1-¢(X.t)]. (31)
From which, we obtain

ss - s[Cx(t)-de(t)] - s[CA(t)x(t)+CAA(t)x(t)+(l+6p(x,t))
Cb(x,t)u(t)+Ch(t)d(t)-de(t)]
- [(CA(t)x(t)-de(t))s-¢ICA(t)x(t)-de(t)IIsl] + [(CAA(t)

x(t)-$Con(t))s-¢ICAx(t)IIsl] + [Ch(t)d(t)s-¢k|s|]

22

where $—(l+6p(x,t))/(l-¢(x,t)). It is easy to see that the control-
ler (31) satisfies the sliding condition if k is chosen sufficiently

large to compensate for the external disturbance.

3.1.2 Nonlinear dynamical systems

We now consider a nonlinear time-varying single-input control
system. The control design methodology for nonlinear systems follow
along the same lines of linear systems. Therefore, we just formulate
discontinuous feedback controllers which guarantee asymptotic track-
ing without verifying sliding conditions. Let the dynamic system be
represented by

x(t) - f(x,t)+b(t)u(t) (32)

where x(t)eRn, u(t)eR and

f(x,t) 9 [x2(t),o - - - -.xn(t>.g<x.t>1T

b(t) e [0,. o o o o,o’*]T.

Then we have
u(t) - [Cb(t)]1[-k sgn s-Cf(x,t)+de(t)]. (33)
Assume next that system equation (32) is replaced by

x(t) - f(x,t)+b(x,t)u(t)+h(t)d(t) (34)
where d(t) is an unknown external disturbance bounded as

d(t)e[61,62] and h(t) is in (22). Then we obtain

23
u(t) - [Cb(x,t)]1[-k sgn s-Cf(x,t)+de(t)] (35)
where k>max(Ch(t)I61I,Ch(t)|62l).

Now let us consider parameter variations of the system (34) so

that the system dynamics are described by

x(t) - f(x,t)+Af(x,t)+b(x,t)u(t)+h(t)d(t) (36)
where
A n T
Af(x,t) - [0,0 - . - -,1§16aivi(x,t)]lxn
such that each 6a1 is unknown but bounded as ais Sais §i<w.
Defining

A n T
f0(xot) - [0). . . . .’1§1801vi(x,t)]lxn

- A n - 'r
f (x, t) - [0, . . . . . 9 i§181v1(x1 t) ]1Xn

where aoi is an average value of the corresponding 6ai and ai-

31

-a0. yields the robust control law
1
u(t) - [Cb(x,t)]1[-(k+lcf(x,t)I) sgn s—C(f(x,t)
+fo(x,t))+cxd(t)] (37)

where k is as defined in (35) and I-I denotes absolute value. Now
we consider an application of this controller to a second order

nonlinear system given by

x2 0 0 O
x(t)- 2x: + 81(t)x: + 1 u(t) + 1 d(t) (38)

24

where a1(t)e[l,5] and d(t)e[61,82]. The sliding surface is chosen
from (19) as

s - c1(x1-x1d)+(x2-x2d) (39)

1
where c1>0 and the desired trajectory x e C [to,w). Thus, from

d
(37)
3 2 3
u(t) - [-(k+|2x1[) sgn s-(c1x2+2x2+3x1)+c1x1d+x2d] (40)

where k>max(|51|,162l).

In order to retain asymptotic tracking in the presence of input

possessing uncertainty we consider the

x(t) - f(x,t)+Af(x,t)+(b(x,t)+6b(x,t))u(t)+h(t)d(t). (41)

Assuming the matching condition (30) the control is given by
u(t) - [Cb(x,t)]1[-(k+]Cf(x,t)-de(t)|+|Cf(x,t)|) sgn s
- crocx.t)1 / [1-¢<x.t)1 (42)

where k is proportional to the magnitude of external disturbance
proposed for system (41). Next we generalize the above design proce-
dures to a large class of multi-input time-varying systems.

3.2 A class of multi-input systems

3.2.1 Linear dynamical systems

We discussed that in vector control cases if the conditions for

the existence of a sliding mode are met on each of the sliding

25
surfaces, the generation of the sliding mode can be represented by
the cases A and B in Figure 4. Thus the problem is to design
discontinuous controllers which satisfy sliding condition (16).
Let us consider a linear time-varying multi-input system in the

presence of external disturbances given by
f((t) - A(t)X(t)+B(t)U(t)+H(t)D(t) (43)

where X(t)eRp, U(t)eRm and disturbances D(t)eR£. The nxn matrix
A(t) has m diagonal blocks each of which is in the phase canonical
form, whereas the off-diagonal blocks have nonzero entries only in
the last row of each block. B(t) and M(t) have m and fl
diagonal block vectors respectively, which have only the last entries
and also the off-diagonal block vectors have entries in the last row

of each block. By way of notation, define

x<c> 9 [x1<t>.- - . - -.xm<t>1T
A T

U<c> - [u1(t>.o - - - -.um<t>1
A T

u(t) - [d1<t).- - - - -.d,(t)1

and
A T
xd(t) - [xld(t>.- - - - -.xmd<t>1
1
where desired trajectories Xd(t)eRn, and each x1d(t) e C [to,w).

Since we want to design control laws that make each xi(t) track a

corresponding desired trajectory xid(t)’ define tracking errors

E(t)eRn as

B(t) 9 [e1(t),o - - - menu->1T - X(t)-xd(t) (44)

26

where ei(t)-xi(t)-x1d(t), for i-l,- . . -,m.
The components of U(t) undergo discontinuities on m planes.

Thus, the set of sliding surfaces ScRm is defined in the error
state space as

5(r) 9 [s1,- - - - -,sm]T 9 GB(t)-O (45)

where si-Ciei’ and G is an an constant matrix whose each row

C. is a an gradient vector of the function 5 We are now in

1 i'
the same situation as we were in the single input case, that is, we
have to check the sliding conditions (16) for each surface to con-

struct control ui(t) forcing the trajectory xi(t) to slide along
the surface 51 thus yielding ei(t)+O as t+w for any arbitrary

initial conditions. It is easy to verify that the sliding conditions

(16) are always satisfied by selecting the time derivative of S as

s - -K sgn s + GH(t)D(t) (46)

A

where K [k1,o - - - -,km]T such that each k1 is greater than

the magnitudes of the corresponding elements of GH(t)D(t). From
(43)-(46) we can obtain discontinuous control laws U(t) that
satisfy the sliding condition (16) resulting in asymptotic tracking
for all desired trajectories. These feedback controllers take the

form
U(t) - [GB(t)]1[-K sgn s- GA(t)X(t)+ cxd(e)] (47)

where K is an appropriately chosen parameters accounting for the
process noise,measurement noise and disturbances. It is obvious

that GB(t) should be invertible. Implications of singularities of

27
GB(t) can be found in [2]. To illustrate the design procedure,

let us consider the system

x o 1 o o o o o o

. 1 - 2 2 o 1 o

X(t)- - 3 o 3 1 X(t)+ 3 o U(t)+ 0 o D(t) (48)
*2 -1 1 1-1 0 1 o 1

where disturbances are bounded as d1(t)e[61,62] and d2(t)e[63,64].

The objective is to get xi(t) to track the corresponding desired

1
trajectory xid(t)eC [t°,w). From (45) we select sliding surfaces as

'31 - c11(x1-x1d)+(x1-x1d)
L82 c21(x2-x2d)+(x2-x2d)

ID

(49)

 

where c

11 should be positive to make stable sliding motions.

Thus, using (47) the control laws can be constructed as

u (t) -k sgn s +X1-(cu+2)X2-X3-2X‘+c11).{1 +R2
U(t)_ 1 _ 1 1 . d. d (50)
“2(t) -k28gn 52+X1-X2-X3-(C21-1)X‘+C21X3d+x‘d

where k1>max(|6,|,|6,|) and k2>max(|63|,|5,|). We note that
u1(t) contains only 51 and u2(t) does 52 only. However, as we

defined B(t) in (43), it is not necessary that S should be

decoupled for each ui, i. e. , ui may include 51 and sj (ifij).

For example, by changing matrix B in (48) as

O 0
g -

P‘CDNJ
rac>»:

we have corresponding feedback cotrollers

28

61(t) u1(t)-u2(t)
U(t) ' '2 ' -u1(t)+2u2(t) (51)

where u1(t) and u2(t) are as in (50). Both (50) and (51)

guarantee the existence of a sliding mode on the intersection of s1
and 52.
We now consider the case where equation (43) is replaced by
i(t) - A(t)X(t)+AA(t)X(t)+B(X,t)U(t)+H(t)D(t) (52)

where parameter uncertainties AA(t) have m diagonal blocks which
have elements only in the last row and also off-diagonal blocks have

elements only in the last row of each block such that each non-zero

A

element is unknown but bounded as a .5 6a. 5 a .<m. Let us define

that

A

A0 an matrix with entries such that each entry aoij is

an average value of corresponding element of the matrix
AA(t)

A e nxn matrix with entries such that each entry 5 - a

11 ii
- a°ij'

Similar to the scalar case, we choose the time derivative of each

component of S to meet the sliding condition (16), that is,
3 - G(AA(t)-A°)X(t)-(K+IGAX(t)I) sgn s + GH(t)D(t) (53)

where K is in (47) and l-I denotes absolute value of each
component. The feedback control laws which will generate sliding

mode motion of the system (52) are formulated as

29
U(t) - [GB(X,t)]1[-(K+IGAX(t)I) sgn s - G(A(t)X(t)
+AOX(t))+Gxd(t)]. (54)

Note that the discontinuity of each control component increases as

the magnitudes of parameter uncertainties and external disturbances

increase.

We now extend the results to systems of the form

x(t) - (A(t)+AA(t))X(t)+(B(X,t)+AB(X,t))U(t)+H(t)D(t). (55)
We assume the following matching condition which is quite similar to

the single input case.

Assumption(matching condition for multi-input case):
There exist Caratheodory functions AP(X,t)cRpxm, and a con-
tinuous function ¢(X,t)eR+ such that, for all (x,t)eRan,

AB(X,t) - B(X,t)AP(X,t) , IIAP(X,t)IIs a(x,t)<1 (56)
where II-II denotes any induced norm. With this assumption it can
be easily shown that the sliding conditions (16) are satisfied if we

construct the discontinuous control laws as
U(t) - [GB(x,t)jl[-(x+|cA(t)X(t)-cid(t)|+|cAX(t)|) sgn s

' GAoX(t)] / [1-¢(X.t)]- (57)

3.2.2 Nonlinear dynamical systems

Since the same design concepts developed for linear multi-input
systems apply to nonlinear multi-variable systems, we only do

costruct the discontinuous control laws that guarantee asymptotic

3O
tracking for arbitrary initial conditions. For dynamic systems

described by
i(t) - F(X,t)+B(X,t)U(t)+H(t)D(t) (58)
where X(t)eRp, U(t)eRm, D(t)eR£, B(X,t) and H(t) are as in (52), and
F(X,t) é [f1(X,t),- . - . ~,fm(X,t)]T

such that each fi(X,t) has the form of

A T
f1(X.t) - [X21.- ° ° - . xp1'31(x't)]1xp . p < n.

We have
U(t) - [GB(X,t)]1[-K sgn s-cr(x,t)+cxd(t)] (59)

where K should be determined in accordance with the strength of
disturbances and measurement noise.

Add now parameter uncertainties to the system (58), i.e.,

x(t) - F(X,t)+AF(X,t)+B(X,t)U(t)+H(t)D(t) (60)

where
A T
AF(X,t) - [6f1<X.t).- - - - -.6fm<x.t>1
such that each 6fi(X,t) has the form of

A n T
6fi(x,t) - [0,. . - . -,j§16ajv5(x,t)]1xp

where 6a '3 are unknown but bounded as

J

< 6a.S §j<m. Let us

83' J
define

Fo(xot) é [f01(x9t)1. . . . .of0m(xrt)]T

ID

F (x,t) [£1(x.c).- - - - -.fm(x.t>1T

31

where foi(X,t) has the form

n

f01(xrt)é [0" ° ° ° ° j- E1303 vj(x t)]1xp

such that aoj is an average value of the corresponding 6a. of

6f1(X,t), and fi(X,t) is of the form

£101.09 10,- - - - - ._j zléj vj(x tnlxp

such taht a - aj - a°j° Then, we have

1
U(t) - [GB(X,t)]1[-(X+|GF(X,t)I) sgn s -G(F(X,t)
+ F0(X,t)) + GXd(t)] (61)

where l-I denotes absolute value of each component. To illustrate

this design technique, we consider the following example represented

by
2 X2 0 O O O O
X(t)- 2X 1+X2+2X1 X cosX3 + 2a 1(t)X (t) +3 8 U(t)+ 3 g D(t)
X1-2X2-XX‘cosX1 a2(t)X3 O 1 O l
(62)

where a1(t)e[-2,8], a2(t)e[0,lO], d1(t)e[51,62] and d2(t)e[63,64].
Since the problem is to get x1(t) to track x1d(t) and x2(t) to

track x2d(t), from (45) we select sliding surfaces as

s _ [s1 _ c11<x1'x1d)+(*1'*1d)
32 c21(x2—x2d)+(x2-x2d)

L

ID

.d>

 

32

I‘D-(D

A

A C11 1 O O E
- O 0 C21 1 G (t) (53)

1
where 011 is in (49) and xid(t)eC [to,w). Then, from (61) the

(D00
NNHH

controllers become

u1(t) - -(k1+llOX4I) sgn s -c11X2-2X1-2X1X4cosX3

l
s
u2(t) - -(k2+|5X3[) sgn 52-c21X4-X1+2X2+X‘cosX1

-5x,+c2,i,d+i,d (64b)
where k1>max(]6ll,l62|), k2>max(|63l,l84l) and cil should be

chosen according to the qualitative characteristics of the system

which will be discussed later in this chapter.

For the input possessing uncertainties of the system (60)

described as

X(t) - F(X,t)+AF(x,t)+(B(X,t)+AB(X,t))U(t)+H(t)D(t) (65)

we have feedback controllers
U(t) - [GB(X,t)]1[-(K+|GF(X,t)-GXd(t)I+IGF(X,t)I) sgn s
' GFo(X.t)] / [1-¢(X.t)] (66)
which guarantee asymptotic tracking for desired trajectories belong-

1
ing to C on [to,w).

33

3.3 Gradient vectors of sliding surfaces and approximations of

discontinuous control laws

We now discuss the importance of gradient vectors of sliding
surfaces. Since the matrix GB affects the rate of convergence to
the sliding surface, the matrix C should be chosen according to the
intrinsic characteristics of the system such as unmodelled high
frequency of the actual system and the desired eigenvalues to be
located. For linear time-invariant systems, there are some analyti-
cal methods to specify G by using geometric notions[ll] and
minimizing quadratic functionals[12]. However, in general it is hard
to select the optimal G in nonlinear time-varying systems, espe-
cially in uncertain dynamical systems. Here we take up one simple
single input system to show the importance of the gradient vector.

Consider a system described by

s 2

x(t)-a(t)x(t)+2x(t)+u(t)+d(t) (67)
where parameter uncertainty a(t)e[l,5] and external disturbance
d(t)e[-4,4]. We choose the desired trajectory as

xd(t) - t

1
which belongs to C. By defining

[X, it] A [X1,X2] o [xd'id] 9' [X1d:xzdl

34

we define the sliding surface from (19) as
8 - 61(X1-X1d)+(X2-X2d)
where c1 is an element of the gradient vector of a sliding surface.

After some manipulations, following (37) we obtain the discontinuous

control law

3 2 s
u(t)-[-(5+I2x1I) sgn s-c1x2-2x2-3x1+c1x2d+x2d]. (68)

Figure 7 shows the resulting state trajectories for c1-0.5, 100

and 500 respectively, on application of the controller (68). It is

clear that asymptotic tracking is not produced for the value c1-0.5

even though this value satisfies a sliding condition (8) for the
system (67). This implies that for the given desired trajectory the

value of cl-O.5 is not enough to cope with the speed of convergence

to the sliding surface. The speed of convergence depends on un-
modelled high frequency dynamics of the actual system and the

eigenvalues of the desired system. Simulations reveal that c1-0.5

is enough to force the states to the sliding surface if the desired

trajectory is xd(t)-sin(t) for system (67). Another interesting
observation is made from Figure 7. When we use cl-lOO, it takes

2.3 seconds to get asymptotic tracking, whereas 4 seconds are

needed to hit the sliding surface when we choose cl-SOO. These
indicate the possible existence of an optimal c1 value that depends

on the desired trajectory. A study of this aspect may be worth

considering in future work.

35

 

 

 

 

 

 

6 [
c1=0.5
23-4r
a“ X t)
13
:2.
xd(t)
0 1 1 J; l
O 1 2 3 4
' TIME [SEC]
6r
c1=IOO
13 4*
V'U
X
,: X(t)
J.)
x 2»
t
O 1 de() 1 1
O l 2 3 4
TIME [SEC]
6r
,\ c =500
e 4- 1
go
. x t)
13
7:
2 r
xd(t)
00 i 2 3 4‘ 5
TIME [SEC]

Figure 7. State trajectories for c1-0.5, 100, 500

36

As we mentioned in the introduction, the application of discon-
tinuous control laws in practice is not desirable due to chattering
which is undesirable both in itself and in the fact it generates a
high frequency signal component resulting in destruction of the
plant. Thus, the discontinuous controller should be approximated by a
smooth one which will preserve the properties of the discontinuous
one, and in addition not generate undesirable high frequency signals
and excessive control efforts. The basic idea is to find continuous
control law within a small boundary layer neighbouring the sliding
surface by smoothing out the discontinuity in the control law. Then
the boundary layer becomes a modified sliding surface to which
trajectories starting outside the boundary layer converge. Thus this
is achieved by choosing the discontinuous control law outside the
boundary layer, and then interpolating the control law inside the
boundary layer. Figure 8 shows the construction of the boundary layer
in the case of second order systems. By defining the sliding surface
as
8+
s

8.06
s+c¢

IDID

(69)

where e is the boundary layer width, c is an element of C in (19),

we may define the boundary layer by

B(t) é (xl s+< 0 and s'> 0 ). (70)

We choose control u(t) outside B(t), i.e.,

{s+>0 or s'<0).

37

x0

boundary
\1

layer

 

 

 

 

 

 

 

Figure 8. Construction of the boundary layer

Then, we have the sliding conditions s+s+< 0 and s's' < 0, since

from (69) c+ - s - a? Now it remains to specify continuous con-

troller inside B(t). The Urysohn's lemma[l3] says that there
exists at least one continuous interpolation between u+(t) and

u-(t). A simple interpolation of controller in the boundary layer is
shown in Figure 9. This amounts to replacing sgn s by sat(s/ce)
inside the boundary layer. For instance, the discontinuous control

law (40) then becomes
3 2 8
u(t)-[-(k+|2x,|) sat(s/c,e)-(c,x,+2x2+3x,)+c,r,d+r,d] (71)

where sat(a) is defined by

sat(a)-a , Ialsl
58““) ' sat(a)-sgn (a) , Iol>l ' (72)

38
Note that the tracking error decreases as the boundary layer
width 5 decreases. In other words, we trade off tracking accuracy
against the generation of chattering in the state trajectory by
approximating the discontinuous control law. The selection of an
optimal e is not resolved yet. Some of the important properties

of this trade off are quantified in [7].

 

Figure 9. An interpolation of u(t) in the boundary layer

CHAPTER IV

AN APPLICATION TO A ROBOTIC MANIPULATOR

The development of modern industrial robots and manipulators
calls for robustness of performance with regard to variable loads,
torque disturbances, as well as other task specifications. In general
the system dynamics of industrial robots can be represented by equa-
tions of the form (65). We shall illustrate the applicability of our
methodology to these robots by designing controllers for a three-
degrees-of—freedom manipulator. By construction, it will be shown
that our sliding mode feedback controllers are robust to variable
payloads and time-varying torque disturbance.

Consider the three-degrees-of—freedom manipulator of Figure 10.
The manipulator has one rotational and translational joint in the
(x,y) plane, and the arm can be lifted along the vertical z-axis
which constitutes the third degree of freedom. The kinetic equations
of this configuration follow directly from an application of
Lagrange's equations(see [14]). By assuming normalized unit mass and
unit length of the arm and upright column, and neglecting the gravity

force, we obtain the following dynamic equations.

39

4O

 

 

f(t)-r(t)é(t)--§z%:fi; d(t) +‘Ilfi' F1(t) (73a)
" -2(1+M)r(t)+l§__ . 1
- ' 2
a(t) (5/6)-r(t)+(l+M)r(t) r(t)a(t)+(5/6)-r(t)+(l+M)r(t) T(t)
+d(t) (73b)
0. 1
z(t)-'i;fi' F2(t) (73c)

where an unknown but bounded external torque disturbance d(t) and

a variable load M bounded as 05Mm s M SM are imposed to

in max

demonstrate the robustness of our control scheme.

 

 

/// vertical z-axis
t
F (t) F2( )

Figure 10. Three-degrees-freedom manipulator

41

We introduce state variables

x - Ix..x2.x3.x..x..X.1T- [r(t>.t<t).o<t).é<t>.z<t>.z<t>)§4a)
and inputs
T T
U - [u1,u2,u3] - [F1<c>.T<c).F2<t>1 (vab)

to obtain the following problem statement which is of the form (65).

 

 

 

 

 

 

 

 

 

 

 

 

Thus
'x," o '000‘ ”000‘ I0“
x,xi Af2(X,t) 1 o o u Ablo o o
. x, o o o o u o o o :1 o
X(t)- x,x, + Af4(X,t) + o 1 0 n2 + o AbZO 2 + 1 d(t) (75)
x, o o o o 3 o o o ‘13 o
_o,_ o ,[001‘ bOOAb3‘ [0,
where
Af x —1-—— 2
2‘ 't) ' '2(1+M) X4
2 (76a)
Af4(x.t) _ -(1+M)X1-X1(1+2M):(l/6) x2x,
(5/6) -X1+(1+M)X1
M
Ab1 - 7 1+M
1 6 6 2
Ab2 - + x1- §1<1+Ml (76b)
5- 61(,+6x1 (1+M)
_li_
Ab3 - ' 1+M '
The system (75) can be rewritten in the form
x(t)-r(x, t)+AF(X, t)+(B+AB(X, t) )U(t)+hd(t). (77)

The objective of the design is to force r(t), a(t) and z(t) to

track asymptotically the desired trajectories rd(t), 0d(t) and

zd(t) respectively. We choose desired trajectories as

42

rd(t) - O.5+0.3sin(t) m (78a)
0d(t)[- -7o:+50°(1-ccs(t)), tS3.l4 sec (78b)
- 30 , t>3.l4 sec
zd(t) - O.4+0.3cos(t) m (78c)
and define
xd(t) é [desxzdoxsd,x4dox5dtx6le
9 [r (c) r (c) a (t) 6 (c) z (c) z (t)]T (79)
d ' d ' d ' d ' d ' d '

From (45) we select sliding surface as

s - s2 - 7(x3-x3d)+(x,-x,d) . (30)
This gives the matrix
5 l O O O O
G - O O 7 l 0 O . (81)
O O O O 5 1

Matching condition (56) is shown to hold by considering

Ab1 0 O
AP(X,t) - O Ab2 O (82)
O 0 Ab3

and for which it is readily shown that

I]AP(X,t)II1- IIAP(X,t)IIm- maxIAbiI.

Thus, from (76b) we see that if the M -M . is sufficiently
max min

small, there exists a ¢(X,t) such that é(X,t)<l. Now we choose

M

minfo K8. M ax-O'G K3. and ¢(X,t)-¢-0.8. According to the design

m
procedure given by (60), we obtain the following results after some

algebraic manipulations

43

 

cs - 13x3 (83a)
2 .
GF(X,t)-GXd(t) - n2(X,t) - 7x,+x2x,-7x,d-x,d (83b)
2
fol(x,t) 2'( l/2.6)X4
GFo(X,t) é f02(X,t) - ”1°3x1'21'6x1+(1/6) xzx, (83c)

1.3x,-x,+(5/6)
£,3(x,t) o

f1(X,t) f01(X.t)
GF(X,t) é f2(X,t) - f02(X,t) . (83d)
f3(xrt) f03(xrt)

Now, from (66) we can construct sliding mode feedback cotrollers as

ul -(k1+|nl(X,t)|+|fl(X,t)I) sgn sl-f01(X,t)

__l__ -
u2 - 1_¢ -(k2+|n2(X,t)|+|f2(X,t)I) sgn 32-f02(X,t) . (84)
u3 -(k3+|n3(X,t)|) sgn $3

The value of k1 should be chosen according to the magnitude of

the external disturbances, measurement noise and process noise. We
assume the torque disturbance is bounded as d(t)e[-O.3,0.3] N-m.

For simulations k k

1, 2 and k3 are chosen to be 1, 2 and 1

respectively. And we chose the initial conditions as

X(O)-[6O cm,O cm/sec,-86 deg,O deg/sec,60 cm,0 cm/sec]T.
Figures 11 and 12 show the results of asymptotic tracking for the
desired trajectories described by (78) under no load and full load
when we use controllers (84). The converging time to the sliding

surface may be decreased by increasing the value of k1 and ¢.

However, the discontinuities of the control effort increase propor-

tionately causing the generation of higher chattering which is

44
undesirable. Under full payload, after hitting the sliding surface,

the simulation results show tracking errors to be within 0.02 cm,

0.060 and 0.05 cm for r(t), d(t) and z(t) respectively.
Figures 13 and 14 show the discontinuous control histories under no
load and full load. As we mentioned earlier, it is not desirable to
use these controllers in practice. So we approximate these discon-
tinuous feedback control laws by continuous ones inside the boundary

layer. To do this, we replaced sgn s in (84) by sat(ai). Then,

 

1
the new controls become
ul -(k1+ln2(X,t)I+If1(X,t)I) sat (e,)-£,1(x,t)
1 ..
u2 - 1-¢ -(k2+|n2(X,t)|+If2(X,t)I) sat (02)-f02(X,t) (85)
u3 -(k3+ln3(X,t)|) sat(03)

where

01 - $1 / (561)

02 - $2 / (7e2)

03 - s3 / (5E3).
The selection of ti depends on the strength of the discontinuities
of control efforts. We chose 61-0.005, 52-0.Ol and 63-0.Ol for

this simulation. Figures 15 and 16 show asymptotic tracking under no
load and full load when we employ continuous feedback controllers

(85). Under full payload, after converging to the sliding surface,

0
the simulation shows tracking precisions of 0.08 cm, 0.07 and

0.07cm for r(t), a(t) and z(t) respectively. As we expected the
tracking errors have increased. These tracking errors may be reduced

by decreasing the '5. Figures 17 and 18 reveal that the control

‘1

45
efforts of approximated controllers under no load and full load do
not indicate explicitly the discontinuities seen in Figures 13 and
14. Simulation results also indicate that larger control efforts are
needed to drive the system states to the sliding surface. However,
once on this surface smaller control efforts are needed to maintain

tracking.

H

00'

r(t)

\J
U1
I

(t) [Cm]

4n
c3

rd(t)

r(t),r

N
U1
l

 

I l

2 3
TIME [SEC]

 

O
O
p.
U1

50-

 

. 9d(t) [DEG]

9d(t)

9(t)
61
O

9(t)

‘1000 1 2 3 4

TIME [SEC]

 

 

U1L

80- ( )
Zdt

(t) [Cm]

'20 z(t)

Z

2(t).

20-

l l L ._J

0 1 2 3 4 5
TIME [SEC]

 

C

Figure 11. State trajectories under no load(discontinuous)

47

100'

r(t). rd(t) [Cm]

 

 

 

 

 

 

 

 

 

0 l l L l __l
O 1 2 3 4 5
TIME [SEC]
sor
E
51 O
73
‘13
CD
‘:‘-50
ES
-100 1 n J J 1
O 1 2 3 4 5
TIME [SEC]
80-
zd(t)
§ 60
:3 z(t)
‘10
N 40.
13
I:
20-
0 n L 1 L #1
0 1 2 3 4 5
TIME [SEC]

Figure 12. State trajectories under full load(discontinuous)

48

 

 

 

 

 

 

100'
50 *
E
i} 1
‘1. 0’ x
In 1
It}
0
a: 1
8 -so'1
—1oo 1 . r. . -
O l 2 3 4 5
TIME[SEC]
100
H 50
e .
53
{-4 1 .
Ill
0'
“o‘
['9 -504
-100 1 1 1 L 1
O 1 2 3 4 5

TIME[SEC]

 

FORCE F2(t) [N]

l I l I

o 1 2 3 4 5
TIME[SEC]

b

-20

Figure 13. Discontinuous control efforts under no load

49

 

 

100
E
3
tho-1
a: J
U
a:
O
k.
-50
_100 1 J 1 1 J
0 1 2 3 4 5
TIME [SEC]
100

TORQUE T(t) [N.m]

 

 

 

 

 

 

 

“100 J 1 l l 1
0 1 2 3 4 5
TIME [SEC]
20
-10
E.
’13
:«o ~
In]
0
ES
“‘10-
-20 1 L 1 L J
O 1 2 3 4 5

TIME [SEC]

Figure 14. Discontinuous control efforts under full load

50

 

 

 

 

 

 

100~
[S 75
3
QUSO
1:
7:
25
o 1 1 . 1 .
o 1 2 3 4 5
TIME [SEC]
50-
2?
EE
5—1 0 "
:3
23o
. 9 (t)
i? -50 ~
25
9(t)
-100 L , - . +J
o 1 2 3 a 5
TIME [SEC]
80'

2;

2(t). gd(t) [Cm]
O

20

 

 

o 1 1 1
0 1 2 3

TIME [SEC]

(>7-
UI

Figure 15.. State trajectories under no load(continuoqs)

100'

\1
UI

End
0

r(t). 1' (t) [Cm]

N
Ul

 

51

 

1 l ‘ 1 ‘ J

 

 

 

 

 

 

0 o 1 2 3 4
TIME [SEC]
5%-
E?
E
C. o.
13
V'U
CD
. 9 (C)
13-50. ‘1
E;
9(t)
_100 1 1 4 1 1
0 1 2 3 4 5
TIME [SEC]
80 . zd(t)
25360
3 2(t)
NFZO-
E
” 20F
0 1 I l 1 I
0 1 2 3 4 5
TIME [SEC]

Figure 16. State trajectories under full load(continuous)

52

 

 

 

 

 

 

 

 

 

 

 

 

20-
E
H
In
(:1
:3 -2o-
0
In
-400 1 2 3 a_ 5
TIME[SEC]
40-
E.“
520-
E?
H
‘3
8 0V #‘
H
'200 1 2 3 a 5
_ TIME[SEC]
20'
15-
E
E; 10»
N
Li.
S 5’
n6
0
In
0 L/Ififlfl—Efi ——-
'50 . 1 2 3 a s
TIME[SEC]

Figure 17. Continuous control efforts under no load

53

25F\\
0 1.

I
N
U!

FORCE F1(t) [N]
'E

I
‘1
U

V

L

1 l

o 1 2 3 4 5
TIME [SEC]

 

J.
92

6O

40 -

20 '

TORQUE T(t) [N.m]

 

 

 

 

TIME [SEC]
20

3—1
0 U1

UI

FORCE F202 [N]

 

 

 

 

O

 

IK
l l k

l 2 3 4 5
TIME [SEC]

 

I
U1

_

0

Figure 18. Continuous control efforts under full load

CHAPTER V

CONCLUSIONS

Based on sliding conditions, variable structure control laws for
both single and multi-input systems in the presence of uncertainties
were derived. The variable structure controllers provide compensa-
tions that eliminate dynamic interactions by introducing sliding
modes. By ensuring sliding mode motion on the switching surfaces,
the robustness to parameter variations and disturbances was achieved.
Compared to the state of the art of involved compensations, the
proposed control strategies are much easier to construct and require
less knowledge of the physical parameters of the systems since only
inequality of sliding conditions are to be satisfied in the design
process.

It is obvious that the sliding mode controller generates a
discontinuous control signal that changes sign rapidly similar to
pulse amplitude signals. To eliminate adverse effects of such con-
trol efforts on physical hardware, we have approximated
discontinuous controllers by smooth ones inside the boundary layer.
It was shown by simulation that the proposed sliding mode feedback
controllers are very effective for an industrial manipulator handling
variable payloads. It will be useful to apply these control schemes

to more complicated robot manipulators.

54

55

We selected gradient vectors C boundary layer widths e and

i’ i
gains k1, which are vital for the success of our methodology in an
ad-hoc way. As we have seen, these values affect tracking accuracy

and the magnitude of control discontinuities. Formally selecting
these parameters in an optimal manner remains an open research
issue. Further research should include the effects of process and
measurement noise on the sliding mode control laws, the sensitivity
to inaccuracy in the implementation, the relaxation of matching
conditions, and the design of observers arising when some of the

states are not available.

[1]

[2]

[3]

[4]

[5]

[6]

[7]

[3]

[9]

[10]

[11]

LIST OF REFERENCES

Filippov,A.F.,"Differential equations with discontinuous
right-hand side," Am. Math. Soc. Trans., vol.42, pp.l99-23l,
1964.

Utkin,V.I., din Mode and Their A lication in Variable
Structure Systems, MIR Publishers, Moscow, 1978.

Utkin,V.I., "Variable structure systems with sliding modes,"
IEEE Transactions on Automatic Control, vol. AC—22, no.2,
pp.212-222, April 1977.

Young,K.K.D., "Controller design for manipulator using theory
of variable structure systems," IEEE Transactions on System,
Man and Cybernetics, vol.8, no.2, pp.lOl-lO9, Feb 1978.

Ryan,E.P., "A variable structure approach to feedback regula-
tion of uncertain dynamical systems," Int. J. Control,
vol.38, no.6, pp.1121-1134, 1983.

Slotine,J.J. and Sastry,S.S.,"Tracking control of nonlinear
systems using sliding surfaces,with application to robot
manipulator," Int. J. Control, vol.38, no.2, pp.465-492,
1983.

Slotine,J.J., "Sliding controller design for nonlinear sys-
tems," Int. J. Control, vol.40, no.2, pp.421-434, 1984.

Corless,M.J., Leitmann,G. and Ryan,E.P., "Tracking in the
presence of bounded uncertainties," 4th Int. Conf. on Control
Theory,Cambridge University, September 1984.

Corless,M.J. and Leitmann,G., "Continuous state feedback
guaranteeing uniform ultimate boundness for uncertain dynamic
systems," IEEE Transactions on Automatic Control, vol.AC-26,
no.5, pp.1l39-1l43, 1981.

Vidyasagar,M., Nonlinear Systems Analysis, Prentice-Hall,
Englewood Cliffs, N.J., 1978.

El-Ghezawi,0.M.E., Zinober,A.S.I. and Billings,S.A.,
”Analysis and design of variable structure system using a
geometric approach," Int. J. Control, vol.38, no.3, pp.657-
671, 1983.

56

[12]

[13]

[14]

57

Utkin,V.I. and Yang,K.D., "Methods for constructing discon-
tinuity planes in multidimensional variable structure
systems," Automation and Remote Control, vol.39, pp.l466-
1470, 1978.

Guillemin,V. and Pollack,A., Differential Topology, Prentice-
Hall, Engliwood Cliffs, N.J., 1976.

Freund,E., "Fast nonlinear control with arbitrary pole-
placement for industrial robots and manipulators," Int. J. of
Robotics Research, vol.1, no.1, pp.65-78, 1982.