574
IEEE TRANSACTIONS ON AUTOMATIC
CONTROL,
VOL.
AC-30. NO.
6,
JUNE
1985
Technical Notes
and
Correspondence
Robust
Stability
of
Systems
with
Integral Control
MANFRED
MORARI
Abstract-A
number of necessary and sufficient conditions are de-
rived, which must be satisfied by the plant d.c.
gain
matrix of a linear
time invariant system in order for an integral controller to exist for which
the closed loop system
is
stable. Based on these results, the robustness
of
integral control systems
is
analyzed, i.e., the family of plants is defined
which are stable when controlled with the same integral controller.
Conditions for actoator/sensor failure tolerance of systems with integral
control are
also given. Finally, parallels are
drawn
between the results of
this
paper and the bifurcation theory of
nonlinear
systems.
INTRODUCTION
Process control, and in particular, chemical process control, is
characterized by open-loop stable and sluggish processes, severe model-
ing problems, and the ovemding need for reliability, robustness, and
good steady-state performance of the control system, i.e., negligible
offset.
In order to reduce the system sensitivity at
o
=
0
to a small value,
controllers with integral action are typically employed in all important
situations. Therefore, the modeling requirements for the design of
controllers with integral action, their robustness in the event of plant
changes, and their tolerance
to
actuator and/or sensor failure are of
significant practical interest.
Unless stated otherwise, we will assume throughout the paper that the
plant is an open loop stable, linear, time-invariant system. Let
G(s)
denote
the plant transfer matrix. We will assume that the plant is functionally
controllah!s [l], i.e., that the right inverse of
G(s)
exists, because only
then it
i:,
possible to install controllers with integral action
on
all the
outputs.
i'oi
simplicity in notation but without
loss
of generality, we will
restrict
C?
-)
to be a square matrix relating
n
inputs to
n
outputs.
We wil use the following notation:
C+
is
the open right half and
C-
the open left half complex plane; A
u
is
the matrix A with the ith row and
thejth column removed, and
Aj(A)
and det
(A)
are thejth eigenvalue and
the determinant of the matrix A, respectively.
INTEGRAL CONTROLLABILITY
The basic control system configuration
is
shown
in
Fig. 1. Here
G(s)
and
C(s)
are the transfer matrices of the plant and the dynamic
compensator, respectively, both of which are assumed to
be
strictly
stable. Throughout the paper
K
=
diag(k,
k,).
For
this
section
K
=
kl
where
Z
is the identity matrix and
k
is a positive constant. We define
H(s)
=
G(s)C(s).
Note that
H(s)
can
be
improper. For realizability of the
controller, only
C(s)/s
has to be proper. In this paper we would like to
address the following questions. What are the requirements on
H(s),
or
equivalently, how does the compensator
C(s)
have to
be
designed, for a
positive
k
to exist for which the closed loop system is stable? How tolerant
is a control system of
this
type to plant changes and actuator andor sensor
failures? A necessary condition for a positive
k
to exist is provided
by
the
following theorem.
Theorem
I:
Assume that
H(s)
is a proper rational transfer matrix.
There exists a
k
>
0 such that the closed loop system in Fig.
1
with
K
=
kl
is stable only if det
(H(0))
>
0.
and November
20, 1984.
This paper is
based
upon
a prior submission
of
March
Manuscript received September
1,
1983;
revised May
7,
1984.
September
17,
1984,
31,
1983.
This
work was supported
by
the National Science Foundation under Grant
CPE-8115022
and by the Dtparment
of
Energy under Contract DE-AC02-80ER10645.
Technology, Pasadena, CA
91125.
The author
is
with the Department
of
Chemical Engineering, California Institute
of
Fig.
1.
Proof:
The characteristic equation for the closed loop system of Fig.
1 is given by
&(s)
.
det
Z+H(s)
-
=O
where
$(s)
is the open loop characteristic polynomial of
H(s).
Express
H(s) as H(s)
=
N(s)d-
I(s)
where
d(s)
is the common denominator of the
elements of
H(s)
and N(s) is a polynomial matrix. Equation (1) can then
be
expressed
as
(
:>
(1)
_.
'(')
det
(sd(s)Z+
kN(s))
=
0
d(s)
(2)
where
4(s)
and d(s) are stable polynomials with all coefficients positive.
Upon expansion of the determinant, this expression becomes
_.
(s"d"(s)
+
.
.
.
+
k"
det
(N(0)))
=O.
4s)
(3)
If
H(s)
is proper, the coefficient of the highest power of
s
in
(2)
will be
the coefficient of the highest power of
S
in
d(s)
which is positive. The
closed
loop
system will be stable only if
all
the coefficients in det
(sd(s)l
+
kN(s))
are positive. The constant coefficient is det
(kN(0))
and
therefore, for closed loop stability, it is required that det
(N(0))
>
0
and
det
(H(0))
>
0.
Q.E.D.
Note that system where
H(s)
is improper can be stable for some
k
>
0
even when det
(H(0))
<
0. For
SISO
systems, the condition in Theorem
1 becomes necessary and sufficient.
Theorem
2:
Assume that
h(s)
is a proper rational transfer function.
There exists a
k
>
0
such that the closed loop system in Fig.
1
is stable if
and only if
h(0)
>
0.
Theorems 1 and
2
strengthen a result by Sandell and Athans
171
.
It
is
our objective to derive sufficient stability conditions for other than just
2
x
2
systems. It is also practically useful to restrict the range of
k
somewhat. For this purpose we will introduce the following definition.
Definition
I:
The open-loop stable system
H(s)
is called
integral
controllable
if there exists a
k*
>
0
such that the closed loop system
shown Fig. 1 with
K
=
kl
is stable for all
k
satisfying
0
<
k
5
k*
and
exhibits zero tracking error for all asymptotically constant inputs.
It is important to note that we exclude conditionally stable systems in
this
definition. There could
be
a
k
=
k'
>
0
for which the system in Fig.
1
is stable. But unless
k'
can be made arbitrarily small, the system is not
integral controllable according to our definition. Conditionally stable
systems which are only stable for high gains
k
are undesirable from a
practical point of view. We will discuss
this
issue
in
more detail later.
The following theorem is the main result of this paper and forms the
basis of some of the subsequent theorems on robustness and failure
tolerance.
Theorem
3:
The system H(s) is integral controllable
if
all the
eigenvalues of
H(0)
lie in
C+
. The system is not integral controllable if
any of the eigenvalues lie in
C-
.
Proof: Let the Nyquist D-contour be indented at the origin to the
right to exclude the pole of
l/sH(s)
at the origin. The system
will
be
closed loop stable if none of the characteristic loci (CL)
[2]
encircles the
0018-!T286/85/0600-0574$01.~
0
1985 IEEE
JEEE
TRANSACTIONS
ON
AUTOMATIC CONTROL, VOL. AC-30,
NO.
6, JUNE
1985
575
point
(-
l/k,
0).
For integral controllability it is sufficient that the
CL
intersect the negative real
axis
only at finite values.
An
intersection at
(-
03,
0)
could only occur because of the pole of
H(s)/s
at the origin.
Along the indentation, the
small
semicircle with radius
E
around the
origin, the
CL
can
be
described by
1
XJ{H(0))
*
-
e-’q
--c&s-
;
j=
1, n
E
2=
-2
(4)
for small
E.
Let
Aj(H(0))
=
r,eaj;
then the expression can
be
rewritten as
?r?r
j=l,
n.
(5)
The
CL
do not cross the negative real
axis
if
-
?r
<
6,
-
4
<
T
or
-
~/2
<
6,
<
~/2
which means
X,(H(O))EC+,
j
=
1,
n.
Similar arguments show that there will
be
an intersection at
(-
w,O)
if
hj(H(O))d-
for any
j.
Q.E.D.
Theorem 3 says nothing about systems for which the eigenvalues of
H(0)
lie in the closed right half-plane
(RHP)
and include eigenvalues
on
the imaginary axis (not at the origin).
I
A comparison of Theorem 3 with Theorems
1
and
2
shows that
conditional stability without integral controllability is only possible if an
even number of eigenvalues of
H(0)
is in
C-
.
If the number of
eigenvalues of
H(0)
in
C-
is
odd, the closed loop system
is
unstable for
all positive gains
k.
In particular, if the steady-state gain
h(0)
of a single-
input-single-output system
is
negative, it is unstable for
all
positive gains.
It would be useful to know how to design the compensator
C(s)
such
that
the system is
integral
controllable.
A
possibility
is
10
choose
C(s)
such that
C(0)
=
crG(0)’;
(a
>
0)
or
C(0)
=
G(O)-’,
that is, to
completely “decouple” the system at the steady state. In practice, we
often like to reduce the complexity of
C(s)
and to restrict its structure, for
example,
to the following form:
C(0)
=
PD, where D
is
a diagonal
matrix of constants and Pa permutation matrix.
This
form of
C(0)
would
imply that a set of
single-input-single-output
controllers can be used.
Guadarbassi
et
ai.
[5] prove that such a compensator always exists when
det
(H(0))
#
0.
As
we will show in the next section, such systems are
often
not “failure tolerant.”
FAILURE TOLERANCE
Obviously, for any actuator or sensor failure, the system shown in Fig,
1
is unstable because
of
the integral mode. The problem can be
remedied by placing the controller in the failure loop
on
“manual.”
In
Fig.
1
where
K
=
diag
(k,,
.
*
*
,
kn),
this
corresponds
to removing one
integrator and
setring one of the elements of
K
to zero.
In
such a situation
it is desirable that, without readjustment to the other parts of the control
system, stability is preserved.
If an actuator fails and only
(n
-
1) actuators are operating, only
(n
-
1)
variables
can
be
controlled in an offset-free manner.
Thus,
any actuator
failure requires that one controlled variable be left uncontrolled. For
simplicity in notation, we will assume that output
yj
is left uncontrolled
when the actuator of uj fails. The following theorems, derived directly
from Theorem
1,
state necessary conditions for stability in the event
of
actuator or sensor failure.
Theorem
4:
Assume that H(s) is rational and proper, and that there
exists a
k
>
0
such that the closed loop system in Fig.
1
is
stable for
K
=
kl
(det
(H(0))
>
0). If
sensor
j
fails and loop
j
is
taken off line
(kj
=
0),
then the system
will
be
stable only if det
(H(0)j)
>
0.
Theorem
5:
Assume that H(s) is rational and proper, and that there
exists a
k
>
0
such that the closed loop system
in
Fig.
1
is stable for
K
=
kl
(det
(H(0))
>
0).
If actuatorj fails, variable
yj
and loop
j
are taken
off
line
(k,
=
0),
then the system will be stable only if det
(G(O)JJC(O)u)
>
0.
Summarizing, we can say that if,
upon
removal of an actuator and/or
sensor, the sign of the determinant of the d.c. gain matrix changes sign,
the whole control system
has
to
be
redesigned to maintain stability. Every
I
We
are
grateful
to
Dr.
N.
Schiavoni
for
pointing
out
this
fact
to
us.
effort should
be
made to design the compensator
C(s)
such that these
problems are avoided. Instability in the event of sensor failure
can
be
easily prevented by a steady-state decoupler
C
=
G(O)-l. Then
det
(If@))
=
1
and det (H(0))Jj)
=
1.
No
such simple scheme exists to
avoid stability problems associated with actuator failure. Of special
interest
is the case when the structure of the compensator
C(s)
is
“decentralized,” that
is,
one input-output pair is controlled separately
from the rest. It
turns
out that the relative gain
array
(RGA) [3] provides
some information in this respect.
Let the elements of
G(0)
be denoted by go and the elements of
G(O)-l
by
go.
Define the matrix
M
with the elements
m,
=
g&.
(6)
M
is called the RGA and enjoys widespread use in process control as an
interaction measure, despite its empirical derivation.
M
can be easily
shown to be invariant under input and output scaling of
G
and to satisfy
m,=l
,=I,
n
i=
I
j=
I
Theorem
6:
If
mjj
<
0, then for any compensator
C(s)
with the
properties
a)
G(s)C(s)
is proper
b)
c,,
=
CQ
=
0
Vi
#
j
(yj
affects
uj only, uj
is
affected
by
y,
only)
and any
k
>
0,
the closed loop system shown in Fig. 1 with
K
=
kl
has at
least one of the following properties:
a)
the
closed loop system is unstable
b) loop
j
is
unstable by itself, i.e., with all the other loops opened
c) the closed loop system is unstable as loop
j
is removed.
Proof:
Because
mu
is
invariant under input and output scaling, we
have for any diagonal compensator
C(0)
m,=(-l)’+jg,
det
(G(0)J‘)
det
(G(O))
If
m,
<
0,
then
one
or three of the terms
in
(9)
is negative. For
property a), det
(H(0))
<
0; for property b),
hi
<
0;
for property c), det
This theorem can be interpreted in two ways. Let
us
assume first that
loop
j
is to
be
designed independently of the others. Then Theorem
6
implies that if
loop
j
by itself is stable and if all the other loops with the
loop
j
removed are stable
[b)
and c) are not met], then the closed loop
system
musr
be unstable. Thus, it is
impossible
to design loop
j
independently of the others.
On
the other hand, let
us
assume that for a particular
C(s)
there exists a
k
>
0 such that the closed loop system is stable. Then either loop
j
is
unstable by itself
or
the system becomes unstable when loop
j
fails, or
both.
Thus,
the system is extremely failure sensitive.
There are two ways around
this
problem. One could sacrifice the single
loop structure of loop
j,
e.g., introduce a steady-state decoupler. This will
avoid the problems of sensor failure, as was argued previously. The other
possibility is to look for an alternate pairing of manipulated and controlled
variables. Trivially, because of the properties of the RGA, for
2
X
2
systems there
is
always a pairing such that
m,,
=
m2*
>
0.
However,
examples show that for
3
X
3
and larger systems, there might be
no
pairing for which all the
m4’s
are positive, that is, there does not exist a
fault tolerant decentralized single-loop control structure.
Finally, it is worth emphasizing again that
mjj
<
0
are sufficient but not
necessary for the properties of Theorem
6.
For
3
x
3 and larger systems,
all properties of Theorem
6
might hold even when
m,
>
0.
So far, only sufficient conditions for instability have been derived.
Using the newly introduced idea of integral controllability (Definition l),
(Hjj(0))
<
0.
Q.E.D.
576
IEEE
TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30,
NO.
6,
JUNE 1985
sufficient conditions for stability can be stated. Let us first rigorously
define failure tolerance.
Definition
2:
The system shown
in
Fig. 1 isj-sensor failure tolerant
(j-
SFT)
if both
the
complete system and the reduced system with the jth
sensor removed
(k,
=
0)
are
integral controllable.
Again we have to assume that the sensor failure has been recognized
and that the faulty sensor has been removed from service. j-SFT is a very
rich system property. The controller of a
j-SFT
system can always be
tuned such that the closed loop system will remain stable when sensor
j
fails. After failure, all the inputs are used to control the remaining outputs
and
the
control quality might very well degrade, but without any
controller adjustments, stability
will
be preserved.
Just
as
in the previous discussion, we
will
assume that output
yl
is left
uncontrolled when the actuator of
u,
fails.
Definition
3.-
The system shown in Fig. 1 is j-actuator failure tolerant
(j-AFT)
if both the complete system and the reduced system with the
jth
actuator and the jth sensor removed
are
integral controllable.
The following theorems, which follow directly from Theorem
3,
specify the conditions for sensor and actuator failure tolerance.
Theorem
7:
The system shown in Fig. 1 with
H(s)
rational isj-SFT if
all
the
eigenvalues
of
H(0)
and
H(0)a are
in
CT
.
It is not
j-SFT
if any of
the eigenvalues of
H(0)
or
H(0Y’
are in
C-
.
Theorem
8:
The system shown in Fig.
1
with
H(s)
rational is
j-AFT
if
all the eigenvalues of
H(0)
and
G(O)UC(O)J’
are
in
C+,
It
is not
j-AFT
if
any of the eigenvalues of
H(0)
or
G(O)’-C(O)jl
are in
C-
.
Except for 2
X
2 systems, the RGA gives no information on
SFT
and
AFT.
Theorem
9:
Let
G(s)
be a
2
x
2 system. If
mj,(C)
>
0,
then there
exists
a
diagonal compensator
C(s)
such that
H(s)
is
1-SFTIAFT
and 2-
SFT/AFT.
Moreover, any 2
x
2
system can always
be
brought into a
form such that
mjj
>
0
by a permutation of
the
inputs.
Proof:
The necessity follows from Theorem
6.
The sufficiency
can
be
proved as follows:
mll=rnx=-
hIlh22
det
(H(0))
’
There always exists a diagonal compensator
C
such that
hll
>
0
(2-SFT/
AFT) and
hz
>
0
(1
-
SFT/AFT).
Therefore,
m,,
>
0
implies
det
(H(0))
>
0.
The eigenvalues
of
H(0)
are
the roots of
hZ
-
(hll
+
hz3h
+
det
(H(0))
=
0.
For this second-order polynomial, det (H(0))
>
0
and
hll
+
hZ2
>
0
implies that all the eigenvalues of
H(0)
are
in the RHP. H(s) is therefore
integral controllable. Moreover, when
m,,
<
0,
define
LA
that is, exchange the system inputs. Then
mll(G’)=m12(G)=
1
-m,,(G)>O.
Q.E.D.
ROBUSTNESS
The model
of
a plant is never perfect, and therefore, it is important that
the control system is not
only stable for
the
nominal plant but
also
for a
family of plants in some “neighborhood” of the nominal plant. We would
like to investigate the
robusr stability
of plants with integral control.
This
property is enjoyed by a family of plants
6
if there exists a single
compensator
C
which makes
all
the members of the family integral
controllable. Let
the
transfer matrix of any member of the family be
denoted by
G(s)
and the transfer matrix of the nominal plant by
Go@).
We
may define for each plant in the family the matrix function
II(s)=
G(s)G,’(s).
(10)
The function
n(s)
can be interpreted as a multiplicative perturbation of the
nominal plant. Then we have the following result.
Theorem 10:
Suppose that the family
6
of plants satisfies the following
assumptions.
a) Each plant in
6
is open-loop stable.
b) There exists a fixed square matrix
N
such that for each plant
the
matrix
n(0)N
has
all
its eigenvalues in a bounded region
in
C‘.
Then there exists a single compensator
C
and a single
k
>
0
such that
each plant in
6
is stable with the control figuration shown in Fig.
1.
Proof:
Theorem
10
is an immediate consequence of Theorem
3
when
C
=
G,-,(O)-’N.
Note that when the nominal plant
Go(s)
is
also a
member of
the
family, as is usually the
case,
then all the eigenvalues of
N
also
have to be
in
the
open right half-plane.
Theorem
II:
Assume that each plant in
6
is open loop stable. Then
there exists a single compensator
C
and a single
K
such that each plant in
6
is stable with the control confi,mation shown in Fig.
1
only if
det
(G(0))
has the same
sign
for
all
plants in
6.
This result is disturbing, because quite frequently systems are ill-
conditioned, and small uncertainties in the parameters
can
change the sign
of
the determinant.
It would almost not be worthwhile to state Theorem
10
because it is so
similar to Theorem
3,
were it not for a striking resemblance with a result
obtained by Kwakernaak.
Theorem
12
14)
Suppose that the family
6
of plants satisfies the
following assumptions.
a) Each plant in
6
is finite dimensional, is a nonsingular perturbation
of the nominal plant such that
n,
=
limisi-a
n(s)
exists,
has
the
same
number of transmission zeros
as the nominal plant, and is stabilizable and
detectable.
b) The transmission zeros of each plant
all
lie in a bounded region in
C-.
c)
There exists a fixed square
matrix
N
with all its eigenvalues in
C+
such that for each plant, the matrix
IIJV
has
all
its eigenvalues in a
bounded region in
C+
.
Then there exists a single controller that stabilizes
the
control system
for each plant in the family
6.
Let us analyze the similarities and differences between Theorems
10
and
12.
Theorem
12
puts no restriction on the pole location of the
different plants (the plants can be unstable), but requires the zeros to be in
C-
.
Theorem
10
puts no restriction on the zero location of
the
different
plants (the plants
can
be nonminimum phase), but requires the poles to be
in
C-.
Theorem
12
puts a restriction on the asymptotic behavior of the
characteristic loci for
w
-+
m;
the
CL
of
all
the plants in
6
together with
the compensator have
to
approach the origin from the same half plane.
Theorem
10
puts a restriction on
the
asymptotic behavior of the
CL
as
w
+
0;
it is required that the
CL
of all the plants in
6
together with the
compensator do not cross the negative
real
axis
in
the
limit.
Let the plant
be
a single-input-single-output system
GO)
=
kx(W4JS)
(1
1)
where
4
is the plant characteristic polynomial,
x
a monic polynomial,
and
k
a scalar constant. Then Theorem
12
requires that for each plant
in
6
the
constant
k,
Le., the gain at very high frequencies, has the same
sign.
For single-input-single-output systems, Theorem
10
requires the gain at
very low frequencies (d.c. gain) of all the plants in
6
to have the same
sign.
Thus, Theorems
10
and 12 complement each other in an interesting
manner and can be regarded as dual to each other.
CONCLUSION
A variety of results relating to the stability and robustness of linear and
nonlinear systems with integral controllers
has
been derived. It is most
significant that the conditions which have to
be
satisfied for the controller
design to be feasible can all be obtained from steady-state information
about the plant. Several issues remain unresolved.
Of
particular impor-
I
am
indebted
to
Prof.
Kwakemaak
for
pointing
out
this
resemblance.
IEEE
TRANSACTIONS ON AUTOMATIC CONTROL, VOL. AC-30, NO. 6, JUNE 1985
577
tance is the question of how many restrictions can be placed on the
structure of the compensator, which makes a system integral controllable
and robust. Any restrictions imply a simplified control structure and are,
therefore, practically significant.
Morari
[6]
has discussed the implications of the results derived here for
nonlinear sy_stems. He has shown that for systems with input multiplici-
ties,
det
(G)
of the linearized system changes sign. The resulting
robustness problems cannot be removed with linear compensators.
ACKNOWLEDGMENT
Financial support from the National Science Foundation and the
Department of Energy is gratefully acknowledged. We are
thankful
for
criticism from
C.
A.
Desoer
and
N.
Schiavoni.
REFERENCES
[l]
H.
H.
Rosenbmk, Computer Aided Control System Design. New York:
Academic, 1974.
[2]
1.
Postlethwaite
and
A. G.
J.
MacFarlane A Complex Variable Approach
to
the
Analysis
of
Linear MuItivariable Feedback Systems. New York: Springer-
[3] E.
H. Bristol, “On a new measure
of
interaction for multivariable process
Verlag, 1979.
[4]
H.
Kwakernaak. “A condition
for
robust stabilizability,” Syst. Contr. Lett., vol.
control,”
ZEEE
Trans. Automat. Contr.. vol. AC-11, pp. 133-134, 1966.
[SI
G.
Guardabassi, A. Lccatelli, and N. Schiavoni,
“On
the initialization problem in
the parameter optimization
of structurally constrained industrial regulators,”
[6] M. Morari, “Robust stability of systems with integral control,” in
Proc.
ZEEE
Large Scale Syst., vol.
3,
pp. 267-277, 1982.
Conf. Decision Contr.,
San
Antonio,
TX.
1983, pp. 865-869.
[7]
N. Sandell and M. Athans,
“On
type
L multivariable linear systems,’’
Automatica, vol. 9, pp. 131-136, 1973.
2,
pp.
1-5.
1982.
Improved Measures
of
Stability Robustness
for
Linear
State Space Models
RAMA
KRISHNA
YEDAVALLI
Abstract-In
this paper,
the
aspect
of
“stability robustness” of linear
systems
is
analyzed
in
the
time
domain.
A
bound
on
the structured
perturbation
of
an asymptotically stable
linear
system
is
obtained
to
maintain stability using a Lyapunov matrix equation solution.
The
resulting
bound
is
shown
to
be an improved bound over the
ones
recently
reported in the literature.
Also,
special
cases
of
the nominal system matrix
are
considered, for which the bound is given
in
terms
of the nominal
matrix, thereby, avoiding the solution
of
the Lyapunov matrix equation.
Examples given include comparison
of
the
proposed approach with the
recently reported results.
NOMENCLATURE
R“
=
Real
vector space of dimension
a
p[
.]
=
Spectral radius of the matrix
[
.]
u[
*]
=
Singular values of the matrix
[a]
=
The largest of the modulus of the eigenvalues of
[e]
supported
by
the
Air
Force
Office
of
Scientific Research
and
by
Air
Force
Systems
Manuscript received June
20,
1984; revised August 13, 1984. This work was
Command under Grant 83-0139
and
Contract F33615-84-K-3606.
The author
is
with the Department of Mechanical Engineering, Stevens Institute
of
Technology, Hoboken,
NJ
07030.
A[-]
=
Eigenvalues of the matrix
[-I
I[ *]I
=
Modulus
manix
=
Matrix with modulus entries
Vi
=
For all
i
[*Is
=
Symmetric part of a matrix
[e]
I. INTRODUCTION
The problem of maintaining the stability of a nominally stable system
subjected to perturbations
has
been of considerable interest to researchers
for quite some time
[1]-[5].
The recent published literature on
this
“stability robustness” analysis can
be
viewed from two perspectives,
namely i) frequently domain analysis and ii) time domain analysis. The
analysis in the
frequency domain
is carried out using the singular value
decomposition
[6]-[8],
where the nonsingularity of a matrix is the
criterion in developing the robustness conditions. Barrett
[SI
presents a
useful
summary
and comparison of the different robustness tests
available, with respect to their conservatism. Bounds are obtained by
Kantor and
Andres
[9]
in the frequency domain using eigenvalue and
M
matrix analysis. On the other hand, the
time domain
stability robustness
analysis
is
presented using Lyapunov
stability
analysis starting from
Barnett and Storey
[2],
Bellman
[
11, Davison
[
101
(in the context of robust
controller design), and Desoer
et ai.
[
111, among others. Despite the
availability of considerable analysis
in
the time domain stability conditions
in
the above references,
explicit
bounds on the perturbation of a linear
system to maintain stability have been reported only recently by Patel,
Toda, and Sridhar
[12],
Patel and
Toda
[13],
and
Lee
[14].
In
[13],
bounds are given for “highly structured perturbations”
as
well as for
“weakly structured perturbations” (according to the classification given
by Barrett
[8]),
while Lee’s condition
[14]
treats “weakly structured
perturbations.” Highly structured perturbations are those for which only a
magnitude bound on individual matrix elements is known for a given
model structure. Weakly structured perturbations are those for which only
a spectral norm bound for the error is known.
In
this
paper, we consider the time domain analysis.
A
new
mathematical result is presented
[15],
which when extended
to
the result
of Patel and Toda
[13],
provides an improved upper bound for highly
structured perturbation. Then some special cases of the nominally stable
matrix are cqnsidered,
for
which the bound is given in terms of the
nominal matrix, thereby avoiding the solution of the Lyapunov equation.
Examples presented include a comparison with the approaches of Patel
and Toda
[13].
II.
STABILITY
ROBUSTNESS
MEASURES
IN
THE
TIME
DOMAIN
FOR
LINEAR
STATE SPACE
MODELS
Robustness Measures Due
to
Patel and Toda
In
[
131, Patel and Toda consider the following state space description of
a dynamic system:
x(t)
=
Ax(0
+
EMt)
=
(A
+
E)x(t)
(1)
where
x
is the n-dimensional state vector
(R”),
A
is
an
n
X
n
time
invariant, asymptotically stable matrix, and
E
is
an
n
X
n
“error”
matrix. However, in a practical situation, one does not exactly know the
matrix
E.
One may only have knowledge of the magnitude of the
maximum deviation that can
be
expected in the entries
of
A. In
this
case of
highly structured perturbation, the entries of
E
are such that
IEVI
IE
(2)
where
e
is the magnitude of the maximum deviation.
For
this
situation, it is shown in
[13]
that the system of
(1)
is stable
if
0018-9286/85/0600-0577$01.00
0
1985
IEEE