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1994
specific techniques for solving
this
bilinear Sylvester equation, will
be
presented in detail elsewhere.
IV.
CONCLUSION
In
this
note, a geometric theory has
been
presented for output stabilization and eigenstructure assignment by output feedback for linear multivariable systems. We emphasized the importance of
C,
A,
B)invariant subspaces, which constitute openloop informa tion, in revealing the possible structure of the closedloop system under output feedback. We
also
proposed a computationally efficient technique for the characterization of candidate “outputstabilizable subspaces.” Necessary and sufficient conditions in terms of the
C,
A,
B)
invariance property were presented for stabilization and eigenstruc
ture
assignment by output feedback. We provided the
links
between the proposed geometric theory and the existent algebraic results. We proposed a
bilinear generalized Sylvester equation,
the solutions of which completely characterize the structure of the closedloop system under the action
of
output feedback. We have also shown how differ ent descriptions of
this
bilinear equation lead to different necessary and sufficient conditions for the output stabilization problem. Finally, we present a preview of possible ways to solve the proposed bilinear equation. Therefore, for the eigenstructure problem, the solutions to a bilinear matrix equations have to
be
studied. ACKNOWLEDOMENT The authors would like to thank the reviewers for their helpful comments, suggestions, and criticisms that contributed to the overall presentation of
this
note.
REFERENCES
F. M. Brasch and J.
B.
Pearson, “Pole assignment using dynamic compensators,”
ZEEE
Tms.
Automat.
Contr.,
vol.
AC15,
pp.
348351,
June
1970.
J. W. Brewer, “Kronecker products and matrix calculus in system theory,”
ZEEE
Trans.
Circuits
Sysr.,
pp.
772785,
Sept.
1978.
R.
W. Brockett and
C.
I.
Bymes, “Multivariable Nyquist criteria, root loci, and pole placement:
A
geometric viewpoint,’’
ZEEE Trans.
Automar.
Contr.,
vol.
AC26, 271284,
Feb.
1981.
E.
I.
Davison, “On pole assignment in linear systems with incomplete state feedback,”
ZEEE
Trans. Auromat.
Contr.,
vol.
AC15,
pp.
348351,
June
1970. “A
note on pole assignment in linear systems with incomplete state feedback,”
ZEEE
Trans. Automar. Contr.,
vol.
AC16,
pp.
9899,
Feb.
1971.
E. J. Davison and
S.
G.
Chow,
“An
algorithm for the assignment of closedloop poles using output feedback in large linear multivariable systems,”
ZEEE
Trans.
Auromat.
Contr.,
vol.
AC18,
pp.
7475,
Feb.
1973.
E. J. Davison and
S.
H.
Wang,
“On
pole assignment in linear multivari able systems using output feedback,”
ZEEE
Trans.
Automat.
Contr.,
vol.
AC20,
pp.
516518, Aug. 1975.
L.
R.
Fletcher and
J.
F.
Magni, “Exact pole assignment by output feedback Part
1,”
Znr.
J.
Cow.,
pp.
19952007, 1987.
C.
Giannakopoulos and N.
Karcanias,
“Pole assignment of strictly proper and proper linear systems by constant output feedback,”
Znt.
J.
Contr.,
R. Herman and
C.
F.
Martin, “Applications of Algebraic Geometry to System TheoryPart
I,”
ZEEE
Trans.
Automat.
Contr.,
vol.
AC22,
pp.
1925, 1977.
H.
Kimura, “On pole assignment by gain output feedback,”
ZEEE
Trans.
Automar.
Contr.,
vol.
AC20,
pp.
509416, Aug. 1975.
H. Kimura, “On pole assignment by output feedback,”
Znt.
J.
Contr.,
vol.
28,
no.
1,
pp.
1122, 1978.
pp.
543565, 1985. “A
further
result on the problem
of
pole assignment by gain output feedback,”
ZEEE
Trans.
Automat.
Conrr.,
vol.
AC22,
pp.
458463,
June
1977.
B.H.
won and M.J. Youn, “Eigenvalueeigenvector assignment by output feedback,”
ZEEE
Trans.
Auromar. Contr.,
vol.
AC32,
pp.
417421,
May
1987.
J. F. Magni,
“On
exact pole placement by output feedback
3,”
Znt.
J.
Contr.,
vol.
45,
pp.
20212033, 1987.
J. F. Magni and
C.
Champetier,
“A
general framework for pole place ment assignment algorithms,” in
Pm.
7rh
CDC,
ustin,
TX,
ol.
3,
G.
S.
Miminis,
“A
direct algorithm for pole assignment of time invariant linear systems using output feedback,” in
Proc
25rh
CDC,
985,
pp. P. Misra
and
P.
V.
Patel, “Numerical algorithms for eigenvalue assign ment by constant and dynamic output feedback,”
ZEEE
Trans.
Automat.
Conrr.,
vol.
AC34,
pp.
579588,
June
1989.
J.
M.
Schumacher, “Compensator synthesis using
C,
A,
B)pairs,”
ZEEE
Trans.
Automat,
Contr.,
vol.
AC25,
pp.
11331138,
Dec.
1980.
“Regulator synthesis using
C,
,
B)pairs,”
ZEEE
Trans.
Automar.
Conrr.,
vol.
AC27,
pp.
12111221,
Dec.
1982.
S.
Srinathkumar, “Eigenvaludeigenvector assignment
using
output feed back,”
IEEE
Trans.
Auromat.
Conrr.,
vol.
AC23,
pp.
7981,
Feb.
1978. A.
I.
Vardulskis,
“A
sufficient condition for
n
specified eigenvalues to be assigned under constant output feedback,”
ZEEE
Trans.
Automat.
Contr.,
vol.
20,
pp.
428429,
June
1975.
W.
M.
Wonham,
Linear Multivariable Control:
A Geometric Approach.
New York: SpringerVerlag,
1984. 1988,
pp.
22232229. 541547.
A
New Method
of
Switching
Surface
Design
for
Multivariable Variable Structure
Systems
JuJang
Lee
and Yangsheng
Xu
AbstractIn
this
paper,
we
discuss
the problem
of
the
switchjng
surface
for
a
multivariable variable
structure
system. We
6rst
propose a new
switching
surface by which robustness
of
the controlled system to disturbances
is
provided.
Based
on
the
switching
surface, we develop control input
to
drive the system
to
track the desired
trajectory.
This
control input
does
not
produce the chattering that normally
occurs
when
a
conventional control input
is
used.
We apply the proposed method
to
a
multivariable
linear
system and
discuss
the simulation results
to
show the computational procedure and
its
effectiveness.
I.
INTRODUCTION
The Variable Structure Control (VSC) is a special class of nonlinear control method characterized by a discontinuous control action which changes a structure upon reaching a switching surface
a(x(t))
=
0.
For Variable Structure Control System (VSCS), the system is time invariant in the sliding mode, and thus the motion is independent
of
parameters and disturbances. The representative point of the system is constrained
to
move along
a
predetermined switching surface
[3],
[5],
[
11
Therefore, design of the switching surface completely determines the performance of the system. With a conventional switching surface, the sliding mode occurs after the system reaches the switching surface. Before the system reaches the switching surface, however, the system is sensitive to Manuscript received July
21, 1991;
revised September
14, 1992.
The authors
are
with The Robotics Institute, Carnegie Mellon University, IEEE
Log
Number
9214019.
Pittsburgh,
PA 15213. 00189286/94 04.00
0
1994
IEEE
IEEE
TRANSACTIONS
ON
AUTOMATIC CONTROL,
VOL.
39, NO.
2,
FEBRUARY 1994
0
1
0
0
e..
0
0
0
1
0
...
0
0
0
0 0
...
1
.. .. ..
7
a1
a2
a2
a4
...
an

415
parameter variations and disturbances. This produces discontinuous trajectory and causes a chattering which is particularly undesirable when the actuator mechanism may
be
damaged by rapid switching
[2].
Therefore, a new method to design control input is necessary. In
this
note, we propose a new timevarying switching surface for
a
multivariable variable structure system which ensures the robustness of the controlled system to disturbances. Based on the switching surface, we develop a control input to drive the system in tracking the desired trajectory.
II.
VARIABLE
TRUCTURE
ONTROL
OR
SCALAR INEAR
SYSTEMS
A.
Switching
Sudace
Design
It is known that if a linear system represented by a triple
(A,
B,
)
is completely controllable, then the system can
be
reduced by a nonsingular transformation to an equivalent controllable form
[4],
and it is often advantageous to work with such an equivalent system.
A
scalar VSCS is usually represented in a controllable canonical form
[4]].
Consider a scalar linear system
i(t)
=
Az(t)
+
Bu(t)
and assume the coefficient matrices A and
B
have the following forms:
n
u(z(t))
=
Cz(t)
=
Ccizi,
c;
=
constant,
cn
=
1.
(2)
For
convenience, throughout this note, the arguments
t
and
z
are
sometimes omitted when no confusion is likely to arise. In the sliding motion,
a(z(t))
emains zero, and thus
i=l
n1
n
=
 22
i=l
(3)
The resultant sliding mode
i
=
x,+l,
=
1,
2,.,n

n1 in1
=
~c*zt
4)
i l
depends only on the parameter
c,.
Using this switching surface, when the system does not reach the switching surface, the system is sensitive to parameter variations and disturbances and dependent on parameters of
a,
and
bi.
Only when the switching surface has been reached is the system robust to the parameter variation and disturbance.
To
overcome this disadvantage, we propose a new switching surface which provides robustness of control through whole control intervals. Consider the following system:
2,
=
zl+l,
=
1,
2,..,n

1
n
Ca, t)z,
+
f(t)
+
u(t)
(5)
where
u(t)
s an input,
f(t)
s a disturbance,
a,(t)
is
the
system parameter which
is
timevarying,
or
unknown. In what follows, we denote
a,(t)
by
az
for brevity.
,=1
A
model system can
be
described by
i
zz+1,
=
1,
2,...
,n

n
in1
=
CC~X~,
,
=
constant.
(6)
,=1
The response of the system
6)
depends only on the constant parameter cI If the controller makes the system
5)
follow the model system
(6),
ndependently of parameter variations and disturbances, the robustness of the controller is ensured.
To
this
end, we define a new switching surface which drives the actual system
(5)
to follow the model system
6).
\
a(z(t))
z

lt clzl
2z2

.
.

,z,)dt

zn 0),
t
>
0.
7)
If the control input guarantees the sliding mode, then the switching surface remains zero. That is,
zn
=~(c1z1~2~~.c~z~)dt+z~(O),
>0.
(8)
In this way, the system response of
(5)
follows the model system
6)
exactly. Using
this
new switching surface, the sliding mode occurs from the initial time, and thus the proposed switching surface guarantees the robustness
of
the control.
B.
Control
Input
Design
Based on the switching surface designed, the control scheme to guarantee the sliding mode can
be
developed. When the system reaches the sliding mode from either side of the surface near the switching surface, the state remains on the switching surface
a(z(t))
0
for some finite time. The condition for the sliding motion on the switching surface
is
stated as follows: lim
&(z(t))
0,
lim
&(z(t))
0
UO+
U0
or,
equivalently,
dz(t)P(z(t))
0
in the neighborhood of
o(z(t))
=
0.
The following control input
u(t)
was proposed by
[3]:
n
u(t)
=
Eqzz,
6
sgn
(u(z(t)))
*=1
where for
o(z(t))z;
>
0
for
a(z(t))z,
0,
9)
Differentiating
2)
with respect to time and substituting it into
(5)
yields
416
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,
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1994
From
(2), (lo), (ll),
we obtain
a(z(t))&(z(t))
C(cIl

,(t)

bQs)a(z(t))zz
n
*=1
+[f(t)

t)b.
sgn
(44t)))]a(z(t)).
12)
To
satisfy the existence condition, we obtain the gain constants:
1)
a(z(t))z,
>
0
a,
>
maxt
(l/b(t))(cll

,(t)), 2)
a(z(t))zZ
0
P
<
mint
(l/b(t))(ctl

a,(t)), 3)
a 4t))[f t)

b t)S
sgn(a(4t)))l
<
0
>
maxt
(l/b(t))fmax.
The control structures described
so
far are designed to drive the system state into an ideal sliding mode. Whether this can be achieved relies on idealizations. Because of delays in switching, hysteresis, and other factors that
are
not considered in dynamic modeling, the ideal sliding mode may not
be
easily obtained in practice. Moreover, the system state remains only within a neighborhood of
a
switching surface, and continuously passes back and forth
across
the switching surface. This behavior is known
as
chattering, which is particularly undesirable for the actuator mechanisms which may be damaged by the rapid switching.
To
overcome the chattering problem,
a
smoothing control scheme was proposed by
[2],
one of which can be represented by
DP(0I
Fig.
1
Proposed
control
scheme when
b(t)
=
1.
where
DP(t)
=
xAa,z;
+
f t),
Au*(t)
=
Au(t)

DPu(t).
i=l
In general,
DP t)
is
an unknown timevarying function with an unknown structure. In the following, we address the problem of uncertainty cancella tion. Using the switching surface
a(z(t)),
he system
(5)
is govemed by
&(z(t))
DP t)
+
Au*(t)
=
DP t)
+
(Au(t)

DPu t)).
(17)
The auxiliary control input
DPu t)
is defined in the following where
ueq
is the continuous equivalent control input,
UN
is the discontinuous part of
(lo),
L
is the feedback gain vector, and 6 is a small positive constant.
M
can
be
selected to ensure the existence condition for the maximum unknown bounded factor. Now, we present a control scheme based on the switching surface proposed previously.
Theorem:
Consider the system
(5),
(6),
and the switching surface
(2),
and assume that
a
continuous function
DP t)
represents the plant uncertainty due to the unknown disturbance and unmodeled dynamics. We define a continuous function
Au(t)
as
Au(t)
=
Kpa(z(t))
(13)
where
Kp
s positive. Then the proposed switching surface
(7)
and control input
(13)
guarantee the sliding mode of the control system.
Proof:
From
(l),
(2),
(4),
S),
the derivative of the switching surface,
a(z(t)),
an be obtained
&(z(t))
Clkl
+
C2kZ
+
.
+
kn
=
ClzZ+cZz3
+ '+Cn1Zn

alzl+ azzz
+
...
+
anzn)
+
f t)
+
~(t)

an

nl)rn
+
f t)
+
u(t).
=
a121

a2

c1)~ (14)
The timevarying parameters
a;
can
be
divided into two parts: constant part
a,,
and timevarying part
Aa,;
i.e.,
a,=a,,+Aa;,
i=l,2,...,n
(Aa;)~,, a;
5
(Aar)max.
=
DP(t)
+
(Au(t)

DPu(t))

Au(t)

DPu t))
=
DP t)
+
(A.(t)

A.(t)) (18)
in which
t
denotes
t

dt)
dt
is a sufficient small value). If
DP t)
and
Au(t)
are
continuous functions, i.e.,
DP t)
DP t)
and
Au(t)
E
Au(t),
DPu t)
%
DP(t).
(19)
&(z(t))
Au(t). (20)
Therefore, from (17), we have
A
block diagram of the proposed control scheme is shown in Fig.
1.
Equation
(20)
results in
a
t))&
z
))
=
Au(
)a
z
)).
(21) Au(~)
Kpu(z(t)),
Kp
>
0
(22) (23)
If control input
Au(t)
is defined as follows: then
u(z(t))&(z(t)) Kp[a(z(t))y
0.
This
proves the existence condition
of
the sliding mode
of
the control system.
Corollary:
Consider the following system:
kz
=
zt 1,
i
=
1, 2,..,n

n
in
=
Casz,
+
f(t)
+
b(t)u(t>
(24)
8=1
6(z(t))
=
DP(t)
+
Au*(t) (16)
where
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1994
~
417
WO
Fig.
2. Proposed
control scheme
when
b(to)
=
b
+
Ab(t).
and
1
bo
Au*(t)
=
(Au(t)

DPu t)).
If
DP(t),
Au(t).
and
Ab(t)
are
continuous functions, then
&(z(t))
Y
Au(t).
Prooj?
If the switching surface
a(s(t))
s given by
(2),
the derivative of
a(c(t))
s determined by
n
n1
(s(t))
Cais,
+
Ccizz+1
+
f(t)
+
b(t)
~(t).
25) t=l i=l
It is noted that
ue,(t)
can be written as
1
bo
eq(t)
=
[~Io~I
(azo

I)ZZ
+ +
Uno

ni)~,]
26)
thus
(25)
becomes
n
&(~(t))
CAatI,
+
f(t)
Ab(t)
ueq(t)
+
b(t). Au*(t)
i=l
=
DP t)
+
b(t) Au*(t)
=
DP t)
+
1
+
Au(t)

DPu(~)].
(27)
[
The auxiliary control input
DPu(t)
s defined as
DPu(t)
=
&(s(t))
Au(t)

DPu(t))
=
(Au(t)

DPu(t)
1
+
::)I
+
DP(t)

(Au(t)

DP t))
=
(Au(t)

Au(t))
+
(Au(t)

DPu(t))
+
DP(t).
b(t)
.
bo
Using
(28)
and the following relation:
DPu(t)
=
b(z(t*))

Au(t)

DPu(t'))
(29)
where
t'
denotes
(t

dt),
b(z(t))
can
be
written as
&(z(t))
[(Au(t)
DP t))

Au(t)

Au(t))
+
DP(t)]
+
[(Au(t)
DP(t))

Au(t)

Au(~))] b(t)
bo

(Au(t)

DP(t'))

(Au(t'j

Au(t)]
.+...
+
l)nl
[(Au(t( '))
DP(t( ')))
bb(t)
bo

Au(t(n2))
Au(t(nl)
Ab(t)
.
Ab(t(nl))
b2t
+
(l) [(AU(t )
DP(t '))

Au(t ')

Au(t ))]
This implies that
(s(t))
an
be
approximated by
Au(t),
.e.,
b(z(t))
S
Au(t).
(31)
This condition is the same
as
20).
A
block diagram of the control structure is illustrated in Fig.
2.
111.
VARIABLE
mum
CONTROL
FOR
MULTIVARIABLE INEAR
YSTEMS
A.
System Model
The multivariable linear system to
be
considered in this section is
=
As
+
Bu
(32)
where
I
E
R
and
U
E
R
represent the state and control vectors, respectively.
To
develop a variable structure control, we specify a particular canonical form for the system. This form is closely related to the controllable canonical form for a multivariable linear system, and is similar to that used by
[6].
If the matrix
B
has full rank
m,
then there exists an orthogonal
n
x
n
transformation matrix
T
such that
TB=
[L]
where
Bz
is an
m
x
m
nonsingular matrix.
A
set of new state variable
y
can
be
transformed from
2
.e.,
y
=
TI.
Let us define the state vector,
y
E
R
as
Y
=
[Yl
yzIT.
By using new state variable
y.
the system
(32)
can
be
written
1
=
Ally1
+
Aiz~z
z
=
Aiz~i Azzyz
+
Bzu
(33)
where
y1
and
y~
are vectors formed by the first
n

m)
and last
m
components of the vector
y,
respectively, and
A,,
are block matrices with corresponding dimensions.
B.
Conventional Methods
for
Switching Sulface
that The conventional switching surface
u(z(t))
an be defined such
(34)
where
CI
and
CZ
are
m
x
n

m)
and
m
x
m
matrices. In the sliding mode,
a(z(t))
=
0
and from
(34),
we obtain
a(s(tj)
=
Cz(t)
=
CT'y(t)
=
[Cl
CZI
.
[Yl(tjY2(t)lT
Yz(t)
=
CT'Civi(t)
=
Fyi(t),
Vt
>
t,
(35)
where
F
=
CF'C1
E
R
and
t,
represents the time at which the sliding mode is reached.
This
indicates that the evolution of
yz t)
in the sliding mode is linearly related to that of
yl(t).
From
(33)
and
(33,
an ideal sliding mode is govemed by
Yl(t)
=
(A11

A12F)?/l(t)
=
Ae,yl(t) (36)
which indicates that the design of a stable sliding mode (i.e.,
y
+
0
as
t
00,
requires the determination of the
gain
matrix
F
such that
A,,
has
n

m
lefthand halfplane eigenvalues.
This
can
be
achieved by modifying a standard design method of a linear feedback control for a linear system. However, since
F
does not uniquely determine
C,m2
degrees of freedom remain due to
Ci
=
CzF.
(37)
The simplest method to determine
C
from
F,
as suggested by Utkin and Yang
(1978),
is to force
CZ
=
1,.
In
this
way, we have
C
=
[Ci
CZ] [Flm].
(38)
418
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0.00
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2.00
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.
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s.20
0.10
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0.00 0.00 600.00
1200.00
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2400.00
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so.
00
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~
0.00
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1EOO.00
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ms)
00
600.00
1200.00
lSOO.00
1490.00
3000.00
Time(
ms)
Fig.
3.
(a)
State
q(t)
and
sz t)
vs. time
(state
feedback controller).
(b)
Control
ul(t)
and
u~(t)
s.
time (state feedback controller).
(c)
State
~(t)
and
sz(t)
vs.
time
(proposed controller).
d) Control
~(t)
nd
uz(t)
vs. time
(proposed
controller).
C. Switching Surjihce Design
When
the sliding mode
is
achieved,
o(y(t))
equals zero, i.e., (42) Consider the following system:
where
All,
A 2
are
constant matrices, and the others are the unknown timevarying matrices.
y
E
R ,
E
R ,
and
f(t)
E
R
represent the
state
variables, control inputs, and unknown disturbances, respec tively.
y1
E
R 
and
yz
E
R
are
the substate variables. We denote the matrices formed by the nominal value of each element of the matrices
Azl,Azz,
and
B2
by
Azlo,Azzo,
and
Bzo.
Consider the model system
Jo
Thus,
the system (39) exactly follows the system
40).
D. Control Design
input for the system. From
(41),
we have Based on the switching surface proposed, we can derive control
If
the timevarying matrices
A21(t), A22(t),
BZ
can
be
written in
the
following
forms:
&(t)
=
B20
+
&(t)
44)
where
Azlo,
Azzor
Bzo
are
time independent, and
Az~ t),
&(t),
&(t)
epresent the timevarying part,
in
a manner similar to
16)
or
(24),
then (45) where
yL
E
R
,
;
E
R  ,
and
y;
E
R
are
the state and substate variables. We define the switching surface
as
follows:
u Y ~))
[g1g2***omlT
=
YZ

t(Czlyl

C22~2)dt
z(0).
(41)
~(t)
ueq(t)
Au*(t)