Nghiên cứu ortega1993

IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38,

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[231
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NO. 11, NOVEMBER 1993

REFERENCES
V. L. Kharitonov, “Asymptotic stability of an equilibrium position of a family of systems of linear differential equations,” Difjerential’nye Uracneniya, vol. 14, pp. 2086-2088, 1978.
B. R. Barmish, “Invariance of the strict Hunvitz property for
polynomials with perturbed coefficients,” IEEE Trans. Automat.
Contr., AC-28, pp. 935-937, 1984.
N. K. Bose and E. Zeheb, “Kharitonov’s theorem and stability
test of multidimensional digital filters,” IEE Proc. G, vol. 133, pp.
187-190, 1986.
A. C. Bartlett, C. V. Hollot, and H. Lin, “Root locations of an
entire polytope of polynomials: It suffices to check the edges,”
Math. Contr. Signals. Syst., vol. 1, pp. 61-71, 1988.
B. R. Barmish, “A generalization of Kharitonov’s four-polynomials
concept for robust stability problems with linearly dependent
coefficients,” IEEE Trans. Automat. Contr., vol. 34, pp. 157-165,
1989.
H. Chapellat and S. P. Bhattacharyya, “A generalization of
Kharitonov’s theorem: Robust stability of interval plants,” IEEE
Trans. Automat. Contr., vol. 34, pp. 306-311, 1989.
Y. C. Soh, “Stability of an entire polynomials,” Inf. J. Contr.,
vol. 49, pp. 993-999, 1989.
M. Fu and B. R. Barmish, “Polytopes of polynomials with zero in a
prescribed set,” IEEE Trans. Automat. Contr., vol. 34, pp. 544-546,

1989.
I. R. Petersen, “A new extension to Kharitonov’s theorem,” IEEE
Trans. Automat. Contr., vol. 35, pp. 825-828, 1990.
A. Cavallo, G. Celentano, and G. de Maria, “Robust stability
analysis of polynomials with linearly dependent coefficient perturbations,” IEEE Trans. Automat. Contr., vol. 36, pp. 380-384, 1991.
I. R. Petersen, “A class of stability regions for which a
Kharitonov-like theorem holds,” IEEE Trans. Automat. Contr.,
vol. 34, pp. 1111-1115, 1989.
Y. C. Soh, “Strict Hunvitz property of polynomials under coefficient perturbation,” IEEE Trans. Automat. Contr., vol. 34,
pp. 629-632, 1989.
Y. K. Foo and Y. C. Soh, “A generalization of strong Kharitonov’s
theorem to polytopes of polynomials,” IEEE Trans. Automat. Contr.,
vol. 35, pp. 936-939, 1990.
-,
“Kharitonov’s regions: It suffices to check a subset of
vertex polynomials,” IEEE Trans. Automat. Contr., vol. 36,
pp. 1102-1105, 1991.
M. B. Argoun, “Frequency domain conditions for the stability of
perturbed polynomials,” IEEE Trans. Automat. Contr., vol. 32, pp.
913-916, 1987.
S. Dasgupta, P. J. Parker, B. D. 0. Anderson, F. J. Kraus, and
M. Mansour, “Frequency domain conditions for the robust stability of linear and nonlinear dynamical systems,” IEEE Trans. Circ.
Syst., vol. 38, pp. 389-397, 1991.
B. D. 0. Anderson, E. I. Jury, and Mansour, “On robust Hunvitz
polynomials,” IEEE Trans. Automat. Confr., vol. 32, pp. 909-913,
1987.
M. B. Argoun, “On the stability of low-order perturbed polynomials,” IEEE Trans. Automat. Contr., vol. 35, pp. 180-182, 1990.
F. Kraus, B. D. 0. Anderson, and M. Mansour, “Robust stability
of polynomials with multilinear parameter dependence,” Int. J.
Contr., vol. 50, pp. 1745-1762, 1989.

B. R. Barmish and Z. C. Shi, “Robust stability of a class of
polynomials with coefficients depending multilinearly on perturbations,” IEEE Trans. Automat. Contr., vol. 35, pp. 1040-1043, 1990.
J. Ackermann, H. Z. Hu, and D. Kaesbauer, “Robustness analysis:
A case study,” IEEE Trans. Automat. Contr., vol. 35, pp. 352-356,
1990.
K. H. We1 and R. K. Yedavalli, “Invariance of strict Hunvitz
property for uncertain polynomials with dependent coefficients,”
IEEE Trans. Automat. Contr., vol. 32, pp. 907-909, 1987.
L. R. Pujara, “On the stability of uncertain polynomials with
dependent coefficients,” IEEE Trans. Automat. Contr., vol. 35, pp.
756-759, 1990.
N. K. Bose, “A system theoretic approach to stability of sets of
polynomials,” Contemporary Math., vol. 47, pp, 25-34, 1985.
N. K. Bose and Y. Q. Shi, “A simple general proof of Kharitonov’s
generalized stability criterion,” IEEE Trans. Circults Syst., vol. 34,
pp. 1233-1237, 1987.

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[26] K. S. Yeung and S. S. Wang, “A simple proof of Kharitonov’s
theorem,” IEEE Trans. Automat. Contr., vol. 32, pp. 822-823, 1987.
[27] H. Chapellat and S. P. Bhattacharyya, “An alternative proof of
Kharitonov’s theorem,” IEEE Trans. Automat. Contr., vol. 34, pp.
448-450, 1989.
[28] F. R. Gantmacher, Theory of Matrices, vol. 11, New York Chelsea,
1964.

Nonlinear Control of Induction Motors: Torque
Tracking with Unknown Load Disturbance
Romeo Ortega, Carlos Canudas,

and Seleme I. Seleme
Abstrecf-In a recent note Ortega and Espinosa [SI presented a
globally stable controller for torque regulation of a complete induction
motor model with partial state feedback, i.e., no assumption of flux
measurement. The result was established under the assumptions that
both the desired and load torques are constant, that the former does not
exceed certain bounds which depend on the systems natural damping,
and that the motor parameters are known. In the present contributions
we extend these results in several directions. First, by “adding mechanical damping” to the closed-loop system we relax the upper bound
condition on the desired torque. Second, we use a new controller structure that allows us to treat the case of time-varying desired torque.
Finally, a new estimator is proposed to handle time-varying (linearly
parameterized) unknown loads.

I. PROBLEM
FORMULATION
We consider in this note the classical dq model [lo] of the
induction motor

(1.1)

(1.2)
with generated torque

Manuscript received March 14, 1992; revised August 21, 1992.
R. Ortega is with Genie Informatique, Universite de Technologie de
Compiegne, BP 649-60206, Cedex, France. He was a Visiting Professor
at the Department of Electrical Engineering, McGill University, Montreal, Canada, when this work was completed.
C. Canudas is with the Laboratoire D’Automatique de Grenoble,
ENSIEG, BP 46, 38402, Saint Martin D’Heres, France.
S. I. Seleme is on leave from Faculdad de Engenharia de Joinville,

UDESC, Brazil and is currently with the Laboratoire D’Automatique, de
Grenoble, ENSIEG, BP 46, 38402, Saint Martin D’Heres, France. His
work was supported in part by CNPq/CEFI, Brazil-France.
IEEE Log Number 9208709.

0018-9286/93$03.00 0 1993 IEEE


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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993

TABLE I

where we have defined the signal vectors

u l , u 2 ,and u 3

w,-primary frequency, are the control inputs, x
is the state vector and y , is a load torque. All symbols are
explained in Table I.
It has been shown in [8] that a more convenient form for (1.11,
which reveals the work less forces acting on the motor, can be
obtained directly from the total energy function and Lagrange's
equations as

(1.6)
The matrices D , R , C , 5 and M are given by

5

D

diag{D,, c J }

[o, o,o, o
E

-

I R S x 5 ,R

L

e

~E

I ~R ~

I

~

diag{R,, 16) E I R j X 5 ,

(1.7)

dX)

+

L . [ ~ , x 2L

, , ~ - ~( L, , ~ +
, L , , ~ , ) 0,
, 0i7

with 0, the 2 x 2 identity matrix and
matrix

List of Symbols
p = ( d / d i ) = Derivative opertator
R , = Stator Resistance
R, = Rotor Resistance
L , = Stator Inductance
L , = Rotor Inductance
L,, = Mutual Inductance
J = Rotor Inertia
b = Motor Damping
l ' = Number of Pole Pairs
x, = 1 ; = I , = d-component of stator current
x2 = I' = I , = q-component of stator current
x -- j L- I , - d-component of rotor current
xq = i; = I, = q-component of rotor current
xj = w, = Rotor angular speed
U , = Vj = d-component of stator voltage
u 2 = V; = q-component of stator voltage
u 3 = w, = Primary frequency
y = Generated torque
y,. = Load torque

wr = ci3 - l'x5 = Slip frequency
= Observed value of I,, i = 1;..,4

J, the 2 x 2 antisymmetric
(1.8)

Notice that the matrix C is skew-symmetric.
The control problem can now be formulated as follows. Consider the induction motor model (1.3), (1.6) with states x, control
variable U , disturbance 5 and regulated signal y . Assume (see
point 1) of the discussion below).
A.l) Stator currents x , , x 2 and rotor speed x 5 are available
for measurement.
A.2) Motor parameters are exactly known.
A.3) Load torque can be linearly parameterized as

Discussion:
1) The following comments regarding the assumptions are in
order:
A.l) is the only realistic situation for practical applications.
A.2) is a very strong assumption since it is well known that,
e.g., the rotor resistance changes considerably in operation. Our
contention is that the fact that our scheme does not rely on
nonlinearity cancellations makes this assumption less stringent
(see [81). Further, in the known load case, we are able to
establish exponential stability of the scheme. Thus, a robustness
margin to parameter variations is expected. As is well known in
adaptive control, this property still holds under suitable excitations assumptions.
A.3) is a more realistic assumption than constant load
torque. It is well known that bearings and viscous forces vary
linearly with speed, while large fluid systems as pumps and fans

have loads proportional to the square of the speed. Thus, we
propose a torque load of the form
y,

=

(0,

+ O,x:)sgn(xi) + ozxs.

(1.12)

Clearly the assumption of bounded x 5 for all bounded generated torques y restricts the values of 8. Other prior knowledge
can be used to select the vector 4.
A.4) we believe is a reasonable, pratical assumption.
2) A brief review of the literature follows. The problem of
torque regulation assuming full state measurement was studied
y1. = O W t )
(1.9) using linearization techniques by [3] for a model neglecting the
mechanical dynamics, that is, x s = const. [7] proposes an adapwhere 0 E IRq is an unknown constant vector and
contains tive version of the feedback linearization scheme of [5] to
measurable signals. Further, 0 and I#J are such that for all address the speed control problem with measurable state, unbounded y the solution of (1.2) yields a bounded x , .
known parameters and constant load torque. In [ l l ] sliding
A.4) Desired torque yd is a differentiable function with known mode techniques are used for partial state-feedback velocity
first derivative.
control. [4] established local stability of a scheme designed using
Under these conditions, design a control law that will ensure backstepping, which is recent Lyapunov-based stabilization technique, for the velocity control problem with flux observer. [8]
(1.10)
lim ( y - y d ) = 0
provided the first solution to the torque regulation problem with

partial state feedback and unknown constant load for constant
with all internal signals bounded. Further, we want our con- desired torques which satisfy an upperbound determined by the
troller to attain asymptotically field orientation [6], i.e.,
motor mechanical damping. An adaptive scheme to handle un(1.11) certain rotor resistance is also presented in that note, but
lim ( L , , X , + L , x , ) = 0.
r+=
requires measurement of rotor signals. T o the best of our
I#J


_____

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IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993

knowledge, the problem of combining parameter adaptation and

K, is a nonlinear gain and K,, is a time-varying gain chosen as

flux observation remains open. Torque control is also studied in
[ l ] using some ideas of energy minimization and simple linear
P D control laws.
3) Our interest in this paper for torque control is motivated by
the unquestionable potential of A C drives in robotics applications. This potential stems from the fact that A C drives supersede D C drives in nowadays technology due to its simpler
construction, reliability and lower cost. The first proof of stability of robot motion controllers with induction motors was recently reported in [2]. It is not clear to the authors how this
problem can be addressed with a controller that regulates the
motor speed instead of the torque.
4) In many applications speed regulation is required. The
design methodology of [8] also applies cerbatim for this problem.

Actually, as will become clear from the developments below,
most of the difficulty in our study comes from the nonlinear
dependence of torque uis a cis the state variables, a problem
that is clearly absent in speed control.

Kp(xdj)

with 0 <

E

L$c

=

- R,

(2.8)

< R,. The feedforward terms -yl(t),-y2(t)are given by

and the slip frequency

U,

is
(2.12)

Finally, the estimate of the load torque is obtained as

11. MAINRESULT

6%

jL=

Proposition 2.1: Consider the induction motor model (1.21,
(1.6) under assumptions A.l)-A.4) and the bounded desired
generated torque y , with known derivative yd. Let the controller
be defined as follows:

2

-x:j
2€

where

(2.13)

6 is updated with
6 = -g+(xj - x d 5 ) , 6(0) = Bo

E

IRq, g > 0

(2.14)

Under these conditions,

(2.15)

lim ( y - y d ) = 0
f + X

(2.2)

with all internal signals bounded. Further, the field orientation
objective (1.11) is attained and the observer states asymptotically
converge to their true values.

111. PROOFOF MAINRESULT
First, we proceed to define the error equations. T o this end,
let
n
e = x -xd
(3.1)
where

where xd5 is the controller state which satisfies

xd' p [x,,, X d 2 ,

Xd3, Xd4, Xdjl E

IR5

(3.2)

plays the role of desired values for x , see [8] for further explanations. We will choose

with
xl.

xd,

=

p

X&

=

-

=

const.

(3.3)

P > 0 a desired value for the stator current d component
and 9, are the estimates of the currents x 3 and x4

P,

respectively, and they are obtained from the nonlinear observer

-1

Lsr p Y d .

(3.6)

Notice that if x = x d we have y = y d . Further, the choices of
x d 2 and x d 4 ensure
LsrXd2

+ Lrx,, = 0

(3.7)

which reflects our objective of attaining field orientation.
We will also define a state observation error
-

A

I k I - I
and a parameter estimation error
ijL

(2.7)

6-

(3.8)

0.

(3.9)

In terms of the error signals (3.1) we can write (1.6) as

De

+ Ce + ( R + K ) e = 4

(3.10)


IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993

1678

where K a diag{K K p , 0, 0, Kpb)
pi
perturbation vector with components
?= U1

E

I R 5 x 5 and IC, is a

+ ws(LSxd2 + LsrXd4)

+ uxd5(Lsx2+ L s r x 4 )

-

whose derivative, taking into account the skew-symmetry of
CA - F and C, looks like

+ K ) e - f T R , i + e T S ( x d , ) f : = -zTQz
and the matrix
where we have defined z [eT,iTIT
H,

+ K,e,

R,P

(3.11)

=

-eT(R

QA[R+K
-ST/2

Noting that R
definite iff

-JXdsU

-

- ubx,,

L'L,
(LsX2 f LsrX4)PC'+ ( L s x l + LsrX3)-

PGYd

+ KpbeS - ueTO

(3.15)

where we have used, where convenient, (3.3143.6). Now, if we
replace the control law in the expressions above we get

-Uxd5Lsri4

(3.16)

CXd5LsrI3

(3.17)

*2 =

(3.18)

0

UL

(3.19)

sr

The closed-loop system equations can be written in compact
form as

De

3

x, EP~
i =, 1;..,4

3

y ~ y ? , (from(l.3))

D e i + (CA - F ) J = - R e f

(3.21)

e = -g+e,

(3.22)

where, to get (3.21) we have used the fact that the first four
equations of (1.6) can be written using the notation (2.6), (2.7) as

D,I

I:[

+ [ C A ( u 3 , x 5+) R,]I = M I

0

0

0
0
0

0
0
0

xd5J2

0
0

0
0

1

(3.23)

E

I R 5 x 4 (3.24)

The system is fully described by the state equations (2.4),
(3.20)-(3.22). T o study the stability of (3.20)-(3.221, consider the
quadratic function

11
H, L - I T D p i + -eTDe
2
2

+ - c0 %2g

2

0

(3.25)

(3.28)

-

(from A.3)

xs

xdi E P ~
(since e5 EY_)

= e , ~ E (from
P (3.20)
~
and (3.21)

-

respectively)

e,i+o

+ Ce = - ( R + K ) e + S ( x d 5 ) i +[O, 0 , 0 , 0 , ~ $ 3 1 ~where the last implication fol!ows
(3.20)

> 0.

It is easy to check that (2.81, (2.9) ensures (3.28). NO?, from
2 0 and H , I -A,,,,,(Q)ll~11? we conclude that' e,I and 0
are bounded, and further e and I are square integrable. Convergence to zero of the torque tracking and observation errors can
be established with the following chain of implications (Zdenotes the set of essentially bounded functions):

3

*3 = *4 =

+ K)-'S

H,

x d , ,e, E Z ~
i =, 1;..,4
=

(3.27)

Re

we see that this matrix is positive

1
Re - - S T ( R
4

(3.14)

l/ls

+ K > 0,

-s'21

(3.26)

ast+m

from square integrability and
uniform continuity of e and I .
Internal stabilitv follows from boundedness of the state vector

IV. SIMULATION
RESULTS
The performance of the control scheme of Proposition 2.1 was
investigated by simulations. The numerical values of the fourpole squirrel-cage induction motor used in [8] were chosen, that
is R, = .687C12,R , = .842R, L , = 84mH, L , = 85.2mH, L,, =
81.3mH, J = .03Kgm2 and b = 0.01Kgm2s-'. We present here
simulations of torque sinusoidal change with load torque as
given in (1.12). The motor is initially in stand-still with zero
iFitial conditions. The initial conditions of the observer are
I ( 0 ) = [21.87,- 11, 85, 0, - 11.29IT, and the estimator initial
values are zero. The values of the torque load parameters are
0 = [2.75,0.15,0.003]T.At time t = .6 the load torque parameters
are changed to 0 = [5.5, 0.25, 0.004IT.
Fig. 1 shows the response of the generated and the desired
torque. Fig. 2 shows the rotor currents and its observed values
transients. The rotor speed and it corresponding reference are
shown in Fig. 3. Load torque and its estimated value are illustrated in Fig. 4. Notice that good load estimation is achieved in
spite of the rapid changes of the actual load torque during the
rotor speed zero crossover and at t = 0.6. Finally, the appied
voltages at the stator end are presented in Fig. 5.
'Notice that these two conditions insure exponential convergence
when the load is known.


IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993

torque (N.m)

1679

speed (rads)

20

I

time (s)

time (s)

Fig. 3. Rotor speed, x s , and its reference, xSd,

Fig. I. Torque response with respect to a reference signal.

current (A)

torque (N.m)

201
10.

0

0.2

current (A)

0.4

0.8

(a)

i
time (s)

-30
0.2

time (s)
(b)

0.4

0.6

0.8

1

time (s)
Fig. 4. Load torque, y,, and its estimated value, j L .

Fig. 2. Transient of the rotor current compo-nents on the (d-q-axes
frame and its pbserved values: (a) id, = x 3 and id, = P,, (b) i,, = P, and

i,,

= x,

and i,

=

2,.
V. CONCLUDING
REMARKS

1) As discussed in [8] a key step in the derivation of the
control law and the state observer is the selection of a suitable
representation to obtain the skew-symmetric property of C and
CA - F . As explained in that note, see also [9], this is tantamount to identifying the workless forces of the system. Also the
definition of the controller dynamics x d s follows directly from
the design methodology of [8].
2) Three are the modifications of the controller of [8] introduced here. First, the selection of the desired values for the
current allows us to solve the output regulation problem posed

in that paper for time varying desired torques. Notice that this
choice is more consistent with the field orientation philosophy
[6] since here we require the q coordinate of the rotor flux to be
zero all the time. Second, the inclusion of the term KpbeS in
(2.4) allows us to inject mechanical damping to the closed loop
hence relaxing the magnitude condition on the desired torque of
[8]. Finally, by allowing the load torque to be time varying,
though linearly parameterized, i.e., (I .9), we considerably extend
the realm of application of our control scheme.
3 ) The proposed control law is very simple to implement and
tune. The controller is always well defined, even in startup. This
in contrast with most existing schemes where the control calculation may cross through singularities during the transients.

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IEEE TRANSAClTONS ON AUTOMATIC CONTROL, VOL. 38, NO. 11, NOVEMBER 1993

voltage (V)
60.

On Invariant Polyhedra of Continuous-TimeLinear
Systems

40.

E. B. Castelan” and J. C. Hennet

20.

Abstract-This note presents some conditions of existence of positively
invariant polyhedra for linear continuous-time systems. These conditions are first described algebraically, then interpreted on the basis of
the system eigenstructure. Then, a simple state-feedback placement
method is described for solving some linear regulation problems under
constraints.

0.
0
0

1

0.2

0.4

0.6

0.8

i1
time (s)

voltage (V)
“2

2o01

-400
0

//
1

0.2

0.4

0.6

0.8

1

time (s)
(b)

Fig. 5. The input voltages with respect to the (d-q)-axes frame: (a)
U I = Vd,, (b) U Z
VqS.

4) The main open problem that remains to be solved is the
case when we combine motor parameter estimation with state
ObSeNerS.
5) As pointed out in [8], the design technique used here
applies directly to the fixed frame motor model of 141, 151, [7].

[31
[41
[51

[61

REFERENCES
C. Canudas and S. Seleme, “An energy minimization approach to
induction motor control,” LAG Int. Rep., 1992.
C. Canudas, R. Ortega, and S. Seleme, “Robot motion control
using AC drives,” in h o c . IEEE ICRA Conf, Atlanta, GA, USA,
1993, May 3-7.
A. Deluca, “Design of an exact nonlinear controller for induction
motors,” IEEE Trans. Automat. Contr., vol. 34, no. 12 pp.
1304-1307, 1989.
I. Kanellakopoulos, Adaptive control of nonlinear systems, Ph.D.

dissertation, Univ. of Illinois, Aug. VILV-ENG-91-2233, DC-134,
1991.
Z. Krzseminski, “Nonlinear control of induction motor,” in Proc.
10th IFAC World Congress, Munich, 1987, pp. 349-354.
W. Leonhard, “Microcomputer control of high-performance dynamic AC drives: A survey,’’ Automatica, vol. 22, no. 1, pp. 1-19,
1986.
R. Marino, S. Peresada, and P. Valigi, “Adaptive partial feedback
linerization of induction motors,” IEEE Trans. Automat. Contr.,
vol. 38, no. 2, pp. 208-221, 1993.
R. Ortega and G. Espinosa, “A controller design methodology for
systems with physical structures: Application to Induction Motors,”
in Proc. IEEE CDC, Brighton, UK, 1991; also in Automatica, vol.
29, no. 3, pp. 621-633, 1993.
R. Ortega and M. Spong, “Adaptive motion control of rigid robots:
A tutorial,” Automatica, vol. 25, no. 6, pp. 877-888, 1989.
S. Seely, Electromechanical Energv Concersion. New York McGraw-Hill, 1962.
V. J. Utkin, “Sliding mode control design principles and applications to electric drives,” IEEE Trans. Ind. Electr., vol. 40, no. 1, pp.
23-36, 1993.

I. PRELIMINARY RESULTS
ON POSITIVE INVARIANCE AND
POLYHEDRAL
SETS
Any locally stable time invariant dynamical system admits
some domains in its state-space from which any state-vector
trajectory cannot escape. These domains are called positively
invariant sets of the system. If a system is subject to constraints
on its state vector and can be controlled, the purpose of a
regulation law can be to stabilize it while maintaining its statevector in a positively invariant set included in the admissible
domain. Under a state feedback regulation law, this design

technique can also be used to satisfy constraints on the control
vector, possibly by transferring these constraints onto the statespace. The existence and characterization of positively invariant
sets of dynamical systems is therefore a basic issue for many
constrained regulation problems. T o analyze the desired properties of a closed-loop time invariant linear system under a linear
state feedback, it suffices to study the “autonomous” model:
i ( t ) =Ax(t),

x(t)

E

SH”,

A E % “ * “ , t 2 0.

(1)

Definition 1: Positive Inuariance. A nonempty set Cl is a
positively invariant set of system (1) if and only if for any initial
state xo E R,the complete trajectory of the state vector, x(t),
remains in R. Or, equivalently, fi has the property e A t R E Cl V t .
Definition 1 is general and the set Cl can for example be a
bounded polyhedron, a cone or a vectorial subspace. In the last
case, positive invariance is equivalent to the well-known property of A-invariance of subspaces [lo].
Definition 2: Convex Polyhedron. Any nonempty convex polyhedron of tli” can be characterized by a matrix Q E !Ti‘*“ and
a vector p E 9tr, r E M - (0}, n E M- (O}. It is defined by:
R[Q,p ] = {X E 91”; QX 5 p } .
By convention, equalities and inequalities between vectors
and between matrices are componentwise.
Without loss of generality, it can be assumed that the set of

inequality constraints defining R[Q,p ] is nonredundant. Let Q,
be the ith row-vector of matrix Q , and p , the ith component of
vector p. Then (see [8]), there exists a one-to-one corresponManuscript received March 14, 1992. This work was supported in part
by CAPES, Brazil.
E. B. Castelan is on leave from LCMI/EEL/UFSC, Florian6polis,
Brazil. He is now with the Laboratoire d‘Automatique et d’halyse des
SystSmes du CNRS, 7, avenue du Colonel Roche, 31077 Toulouse
France.
J. C. Hennet is with Laboratoire d’Automatique et d’Analyse des
Systkmes du CNRS, 7, avenue du Colonel Roche, 31077 Toulouse
France.
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