A GUARANTEED STATE BOUNDING ESTIMATION FOR

UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING

HYBRID AUTOMATA

Nacim Meslem, Nacim Ramdani and Yves Candau

CERTES EA 3481 Universit

´

e Paris 12-Val de Marne, 61 av. G. de Gaulle, Cr

´

eteil, France

INRIA Sophia Antipolis - M

´

editerran

´

ee - Antenne de Montpellier (LIRMM)

161 rue Ada - 34392 Montpellier cedex, France

Keywords:

State estimation, non linear systems, bounded error context, non linear differential equations, interval analysis,

guaranteed numerical integration, hybrid automata.

Abstract:

This work is about state estimation in the bounded error context for non linear continuous time systems. The

main idea is to seek to estimate not an optimal value for the unknown state vector but the set of feasible values,

thus to characterize simultaneously the value of the vector and its uncertainty. Our contribution resides in the

use of comparison theorems for differential inequalities and the analysis of the monotonicity of the dynamical

systems with respect to the uncertain variables. The uncertain dynamical system is then bracketted between

two hybrid dynamical systems. We show how to obtain this systems and to use them for state estimation with

a prediction-correction type observer. An example is given with bioreactors.

1 INTRODUCTION

State estimation with continuous dynamic systems is

recognized as problem of great importance in prac-

tice. Indeed, to apply advanced methods for the con-

trol or the diagnosis of dynamical systems one often

needs to compute on-line their internal state. Gener-

ally the direct measurement of this state by means of

sensors may not be available for various reasons such

as physical, practical, economic, . . . etc. However, it

is possible to carry out this task by software sensors,

i.e. observers or estimator which can provide on line

an estimate of the real state system, under certain con-

ditions of observability (Hermann and J.Krener, 1977;

Hermann, 1963).

In fact, there are always uncertainties in the math-

ematical models used for characterizing the system

under study. Consequently, the classical approaches

for building observers are insufﬁcient (Dochain,

2003). Thus, a new approach was developed recently

in a deterministic set-membership context, which

aims to reconstruct all the state trajectories which are

consistent with both the uncertain models and the un-

certain measurements.

This approach can be used easily when mea-

surements are available at discrete time. It is of

prediction-correction type: (i) The prediction phase

consists in computing a guaranteed over (conserva-

tive) approximation of the reachable state space gen-

erated by the uncertain system, (ii) and the correction

phase consists in removing from this over approxima-

tion all the part which are not consistent with feasible

measurements domains, each time a measurement is

available.

In the literature, several geometrical forms are

used to implement this set-membership approach with

linear systems. For example, parallelotopes (Chisci

et al., 1996), ellipsoidal (Chernousko, 2005) and

zonotopes (Combastel, 2005).

With nonlinear systems, guaranteed numerical in-

tegration method for the ordinary differential equation

(ODE) (Nedialkov, 1999) based on intervals Taylor

models (Moore, 1966) was used recently to solve this

estimation problem (Ra

¨

ıssi et al., 2004). Generally,

the wrapping effect (Moore, 1966; Nedialkov, 1999)

associated with the intervals representation of uncer-

tain variables limits considerably their use in order to

deal with practical cases. Thereafter, in order to cir-

cumvent the wrapping effect, the authors of (Kieffer

and Walter, 2006) used the M

¨

uller’s existence theo-

rem (M

¨

uller, 1926; Walter, 1997) as a tool for deriv-

ing a guaranteed enclosure for an uncertain dynami-

32

Meslem N., Ramdani N. and Candau Y. (2008).

A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME SYSTEMS USING HYBRID AUTOMATA.

In Proceedings of the Fifth International Conference on Informatics in Control, Automation and Robotics - SPSMC, pages 32-37

DOI: 10.5220/0001490400320037

Copyright

c

SciTePress

cal system between two deterministic dynamical sys-

tems. The main difﬁculty resides in the deﬁnition of

the bracketing systems. Our contribution is in the con-

tinuation of these work since we show how to build

deterministic hybrid dynamical systems as bracketing

systems and thus make it possible to use the M

¨

uller

theorem with a larger class of nonlinear dynamical

systems.

This work is organized as ﬂow. In the second

section we present the context and the main ideas of

set-membership estimation. Then, we will recall the

M

¨

uller’s theorem in section 3. We introduce the hy-

brid bracketing approach for uncertain dynamical sys-

tems in section 4. Finally we illustrate our method on

a model drawn from bioreactors domain in section 5.

2 PROBLEM STATEMENT

2.1 Context

Let us consider the uncertain continuous dynamic sys-

tem (1) where uncertainties are naturally represented

by bounded intervals with a priori known bounds,

˙

x(t) = f(x,u,[p],t)

y(t) = g(x,u,[p],t)

x(t

0

) ∈ [x

0

] ⊂ D

(1)

where t ∈[t

0

,T ], f ∈C

k−1

(D ×U ×[p]), D ×U ×[p] ⊆

R

n+n

u

+n

p

is an open set; n, n

u

, m and n

p

are the di-

mension of respectively the state vector x, the input

vector u, the output vector y and the parameter vec-

tor p. The functions f : D ×U ×[p] → R

n

and h :

D ×U ×[p] → R

m

are possibly nonlinear. The initial

state x

0

is assumed to belong to a prior known set [x].

We assume that measurements y

j

of the output vec-

tor are available at sampling times t

i

∈ {t

1

,t

2

,...,t

n

}

in I = [t

0

,t

n

T

]. Note that the sampling interval needs

not be constant. The measurement noise is a dis-

crete time signal assumed additive and bounded with

known bounds. Denote E

j

a feasible domain for out-

put error at time t

j

: the feasible domain for model

output at time t

j

is then given by

Y

j

= y

j

+ E

j

(2)

Under these considerations, estimating the state

vector x consists in determining an upper approxima-

tion of the set X(t) of all acceptable state trajectories

X(t) =

x(t) | (∀t ∈ I

˙

x(t) = f(x,u,[p],t))

∧

(∀t

j

∈ {t

1

,t

2

,...,t

nT

},

x(t

j

) ∈ (g

−1

(Y(t

j

),u,[p]) ∩[x

j

]))

(3)

j

x

1j

+

+

x

()

f

=xx

()

1

1j

gy

−

+

Figure 1: Prediction and correction phases.

2.2 Principle: Prediction-correction

Method

Figure 1 shows the principle of the prediction and

correction phases for this set-membership approach,

between two successive time measurement indexes t

j

and t

j+1

. Indeed, by using one of the guaranteed nu-

merical simulation methods for uncertain ODEs, the

prediction phase computes an guaranteed over enclo-

sure [x

j+1

]

p

for all solutions of (1) at the time t

j+1

with (t

j

, [x

j

]) as initial conditions.

x(t

j+1

;t

j

,[x

j

]) ⊆ [x

j+1

]

p

. (4)

The correction phase uses set inversion, consis-

tency techniques and intervals analysis (Jaulin et al.,

2001) to characterize the reciprocal image [x

j+1

]

inv

at time t

j+1

of the admissible measurement domain

Y

j+1

by the output model g

[x

j+1

]

inv

= g

−1

([y

j+1

],[p]) (5)

where [y

j+1

] = Y

j+1

.

Thereafter, it contracts the predicted state inter-

vals vector by comparing it with the reciprocal image

[x

j+1

]

inv

and eliminating the inconsistent state vectors

[x

j+1

]

c

= [x

j+1

]

inv

∩[x

j+1

]

p

(6)

Thus for the next measurement the predication

phase will be initialized by [x

j+1

] = [x

j+1

]

c

. In fact,

by repeating these two phases each time a new mea-

surement is available, one improves considerably pre-

cision of the guaranteed over approximation of the set

X(t).

The algorithm below shows the process for this

set-membership estimation approach

A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME

SYSTEMS USING HYBRID AUTOMATA

33

Algorithm : Prediction Correction estimation

1. Input: ([x

0

],[p], f,g, [y

1

],. .. ,[y

nT

])

2. t

j

= t

0

; [x

j

] = [x

0

];

3. while (t

j

< t

nT

) do

4. {t

j+1

,[x

j+1

]

p

}=Validated Integration([x

j

],[p],t

j

);

5. [x

j+1

]

inv

= g

−1

([y

j+1

],[p]);

6. [x

j+1

]

c

= [x

j+1

]

inv

∩[x

j+1

]

p

;

7. [x

j+1

] = [x

j+1

]

c

8. j = j + 1;

9. end

10. Output: X(t).

3 M

¨

ULLER’S THEOREM

In this section, we introduce an approach for brack-

eting an uncertain dynamical systems when both the

initial state and parameter vectors are deﬁned by

boxes (intervals vector). The main idea consists in

building a lower and an upper dynamical system

which involve no uncertainty and enclose in a guar-

anteed way, the all state trajectories generated by

the original uncertain system. This approach relies

on comparison theorems for differential inequalities

(Smith, 1995; Hirsch and Smith, 2005), and in partic-

ular the work of M

¨

uller (M

¨

uller, 1926; Marcelli and

Rubbioni, 1997).

Theorem:(M

¨

uller, 1926; Kieffer and Walter, 2006)

Consider the dynamical system

˙

x(t) = f(x,p,u(t)), (7)

where function f is continuous over a domain T de-

ﬁned by

T :

ω(t) ≤ x(t) ≤ Ω(t)

p ≤ p ≤ p

t

0

≤t ≤t

n

T

(8)

Functions ω

i

(t) and Ω

i

(t) are continuous over [t

0

,t

n

T

]

for all i and satisfy the following properties

1. ω(t

0

) = x

0

and Ω(t

0

) = x

0

2. the left derivatives D

−

ω

i

(t) and D

−

Ω

i

(t) and the

right derivatives D

+

ω

i

(t) and D

+

Ω

i

(t) of ω

i

(t)

and Ω

i

(t) are such that

∀i, D

±

ω

i

(t) ≤ min

T(t)

f

i

(x,p,t) (9)

∀i, D

±

Ω

i

(t) ≥ max

T(t)

f

i

(x,p,t) (10)

where T(t) is the subset of T(t) deﬁned by

T

i

:

x

i

= ω

i

(t)

ω

j

(t) ≤ x

j

≤ Ω

j

(t), j 6= i

p ≤ p ≤ p

(11)

and where T(t) is the subset of T(t) deﬁned by

T

i

:

x

i

= Ω

i

(t)

ω

j

(t) ≤ x

j

≤ Ω

j

(t), j 6= i

p ≤ p ≤ p

(12)

Then for all x

0

∈[x

0

,x

0

], p ∈[p,p], system (1) admits

a solution x(t) that stays in the domain

X :

t

0

≤t ≤t

n

T

ω(t) ≤ x(t) ≤ Ω(t)

(13)

and takes the value x

0

at t

0

. If, in addition, for all p ∈

[p

0

,p

0

], function f(x, p,t) is Lipschitzian with respect

to x over D then this solution is unique for any given

p.

Finally, an enclosure for the solution of (7) is

given by

∀t ∈ [t

0

, t

n

T

], [x](t) = [ω(t), Ω(t)] (14)

The main difﬁculty is to obtain suitable bracketing

functions ω(t) and Ω(t) in the general case. However,

when the components of f are monotonic with respect

to each parameter and each state vector component,

it is quite easy to deﬁne these systems (Kieffer et al.,

2006), while avoiding possible divergence that may

occur when both upper and lower components of the

parameter/state vector appear simultaneously in the

same expression of the components of the bracketing

systems (Ramdani et al., 2006).

Rule 1 - Use of monotonicity property (Kieffer et al.,

2006)

In order to build the upper system, i.e. the one which

yields the upper solution Ω(t), one can replace in the

formal expression of f

i

, x

i

by Ω

i

, x

j

( j 6= i) by Ω

j

if

∂ f

i

∂x

j

≥ 0 or by ω

j

if

∂ f

i

∂x

j

≤ 0 and p

k

by p

k

if

∂ f

i

∂p

r

≥ 0 or

by p

k

if

∂ f

i

∂p

k

≤ 0. The components of the lower sys-

tem, i.e. the one which yields the lower solution ω(t)

are derived by reversing monotonicity conditions.

Obviously ω(t) and Ω(t) are in general, solutions

of a system of coupled differential equations, i.e.

˙

ω(t) = f(ω,Ω,p,p,t)

˙

Ω(t) = f(ω,Ω,p,p,t)

ω(t

0

) = x

0

Ω(t

0

) = x

0

(15)

which involves no uncertain quantity. Therefore in-

terval Taylor models such as the one presented in (Ne-

dialkov, 1999) can be used for efﬁciently solving (15).

Indeed when these methods are used for solving dif-

ferential equations with no uncertainty, they are usu-

ally able to curb the pessimism induced by the wrap-

ping effect, even over long integration time.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

34

4 HYBRID BRACKETING

SYSTEM

Now, one address the case of uncertain dynamical

systems (1), for which the signs of the partial deriva-

tives ∂ f

i

/∂p

k

and ∂ f

i

/∂x

j

change along the integra-

tion time interval [t

0

,t

n

T

]. In such a case, the uncertain

system (1) admits an enclosure over each time inter-

val where functions f

i

are monotonic with respect to

variables p

k

and x

j

. Therefore both upper and lower

bounding systems are deﬁned by piecewise nonlinear

ODEs and can thus be regarded as hybrid dynamical

systems. Thus, they can be modeled by an hybrid au-

tomaton(Alur et al., 1995).

So to compute a guaranteed enclosure of reach-

able state space generated by the uncertain system (1),

we will built hybrid system of l continuous dynamic

modes which satisﬁed locally conditions imposed by

rule 1

M = {M

1

,M

2

,...,M

l

} (16)

and which given in state space representation by

(

˙

ω(t) = f

M

i

(ω,Ω,p,p,t)

˙

Ω(t) = f

M

i

(ω,Ω,p,p,t)

for i = 1,...,l (17)

Hence, the evolution of this hybrid system is con-

trolled by the sign changes of the partial derivatives

∂ f

i

/∂p

k

and ∂ f

i

/∂x

j

which represents the guard con-

ditions which authorize the transitions between the

continuous bracketing modes. Thus, for a given ini-

tial conditions the execution of this hybrid automata

makes it possible to obtain a guaranteed upper ap-

proximation of the reachable state space of the con-

tinuous time system (1).

Example:

Let us consider the following system

˙x(t) = f (x,[p]). (18)

According to the sign of ∂ f /∂p, the system (18)

admitted two possible bracketing modes

i f ∂ f /∂p ≤0

then M

1

=

˙

Ω(t) = f

M

1

(Ω, p

)

˙

ω(t) = f

M

1

(ω, p)

(19)

else M

2

=

˙

Ω(t) = f

M

2

(Ω, p)

˙

ω(t) = f

M

2

(ω, p).

(20)

and its hybrid bracketing automata is represented in

the ﬁgure 2

0

∂

>

∂

f

p

M

1

M

2

0

∂

≤

∂

f

p

Figure 2: Hybrid automata.

Fou

t

Fin, Sin

X

,

S

Figure 3: General representation of a bioreactor.

5 APPLICATION

5.1 Model

Generally, the mathematical model of biotechnolog-

ical processes is difﬁcult to establish with accuracy,

that is due to the living behavior of the bacteria rep-

resented by a complex poorly known function of the

bioreactor state. In this section we consider a simple

model where only one population of bacteria is taken

into account. In this context, to describe the state of

the bioreactor, two state variables are necessary, the

ﬁrst one represents the bacteria concentration called

biomass and denoted X, the second one represents the

substrate concentration, denoted S. Thus the model

below shows the evolution of the biomass by consum-

ing the polluting body S

X(t) = µ(S)X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S)

(21)

where µ(s) is the growth rate of biomass modeled by

the Haldane law:

µ(S) = µ

0

S

S + k

s

+ S

2

/k

i

(22)

with the uncertain bounded parameter µ

0

µ

0

≤ µ

0

≤ µ

0

.

A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME

SYSTEMS USING HYBRID AUTOMATA

35

This bioreactor is fed by a solution containing sub-

strate in concentration S

in

which is not exactly mea-

sured

S

in

(t) ≤ S

in

(t) ≤ S

in

(t)

and we suppose that the biomass is accessible to mea-

surement

y(t) = X(t).

5.2 Building Hybrid Bracketing System

For this application, according to the sign of the

derivative of µ with respect to S,

sign(

dµ(S)

dS

)

> 0 if S <

√

k

s

k

i

, ∀S ∈[S]

≤ 0 if S ≥

√

k

s

k

i

, ∀S ∈[S]

ambiguous if

√

k

s

k

i

∈ [S]

(23)

system (21) allows three possible modes for the

bracketing. The ﬁrst mode, M

1

= 1, corresponds to

the intervals time when this derivative is negative

(M

1

= 1)

X(t) = µ(S)X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S)

X(t) = µ(S)X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S)

(24)

and the second mode, M

2

= 2, is linked to intervals

time where this derivative is positive

(M

2

= 2)

X(t) = µ(S)X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S)

X(t) = µ(S)X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S).

(25)

Finally, the third mode M

3

= 0 is associated to case

when the sign of this derivative is ambiguous. In this

case either one uses guaranteed integration methods

based on interval Taylor models to bracket (21), or if

this is possible, one ﬁnds a trivial bracketing for µ(s).

For example,

µ

0

S

S + k

s

+ SS/k

i

≤ µ(S) ≤ µ

0

S

S + k

s

+ SS/k

i

.

Hence, one propose the below differential equations

system for the third mode M

3

= 0

(M

3

= 0)

X(t) = µ

0

S

S+k

s

+SS/k

i

X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S)

X(t) = µ

0

S

S+k

s

+SS/k

i

X −αDX

S(t) = −k

1

µ(S)X + D(S

in

−S).

(26)

Now, function Validated Integration in algo-

rithm Prediction Correction Estimation selects on

line the local bracketing mode according to the sign

of

dµ(.)

dS

and then uses guaranteed numerical integra-

tion methods for ODE based on intervals Taylor mod-

els for solving (24), (25) or (26).

5.3 Results of Simulation

The data considered in this example are as fol-

lows: α = 0.5, k = 42.14, k

s

= 9.28mmol/l, k

i

=

256mmol/l, µ

0

∈ [0.64,0.84], X

0

∈ [0 >,10], S

0

∈

[0,100], S

in

(t) ∈ ([62,68] + 15 cos(1/5t)),

D(t) =

2 si 0 ≤t ≤ 5

0.5 si 5 ≤t ≤ 10

1.14 si 10 ≤t ≤ 20,

and the feasible measurement domain

Y(t

j

) = [0.98y

m

(t

j

),1.02y

m

(t

j

)]

with a constant measurements time step t

j+1

−t

j

= 2.

The red continuous lines curves in ﬁgures 4 and

5 show the guaranteed enclosure of all the possible

state trajectories of (21) which are compatible with

the model and its uncertainties and the acceptable do-

main of the discrete measurements signal. The dis-

continuous blue curves represent the real state of (21)

which corresponds to the following values of the un-

certain parameters and the initial state: µ

0

= 0.74,

X

0

= 5, S

0

= 40 and S

in

(t) = 65 + 15 cos(1/5t).

So ﬁgure 6 shows the actual commutations be-

tween the three bracketing modes as obtained during

the simulation period. This represents the evolution of

the discrete component of the hybrid automata used to

bracket the state ﬂow generated by the uncertain sys-

tem (21).

As a conclusion, guaranteed numerical integration

methods for ODE based on the interval Taylor mod-

els fail to give non divergent enclosures after few inte-

gration step because of the large magnitude in uncer-

tainty in both parameter vector and initial state vector.

In addition, rule 1 is not applicable over all the simu-

lation period because the sign of the partial derivative

∂

˙

X

∂S

changes with time. Hence, our hybrid bracketing

method is an important alternative to solve set mem-

bership estimation problems of this kinds.

Figure 4: The both real and estimated evolution the

biomass.

ICINCO 2008 - International Conference on Informatics in Control, Automation and Robotics

36

Figure 5: The both real and estimated evolution the sub-

strate.

Figure 6: Indices of the three bracketing modes used over

the simulation period.

6 CONCLUSIONS

In this communication, we wanted to show that by us-

ing the M

¨

uller’s theorem and by analyzing the mono-

tonicity of the uncertain dynamical system with re-

spect to both the uncertain variables and parameters,

one is able to solve the state membership estimation

problem for a large class of uncertain dynamical sys-

tems. Indeed, the method presented makes it possi-

ble to circumvent the propagation of the pessimism

due to the wrapping effect which generally causes the

divergence of guaranteed numerical integration meth-

ods based on interval Taylor models when used with

uncertain ODEs. In the future, we wish to extend this

approach for systems with higher dimension and also

to hybrid dynamical systems.

REFERENCES

Alur, R., Courcoubetis, C., Halbwachs, N., Henzinger, T.,

Ho, P.-H., Nicollin, X., Olivero, A., Sifakis, J., and

Yovine, S. (1995). The algorithmic analysis of hybrid

systems. Theoretical Computer Science, 138:3–34.

Chernousko, F. (2005). Ellipsoidal state estimation for dy-

namical systems. Nonlinear analysis.

Chisci, L., Garulli, A., and Zappa, G. (1996). Recur-

sive state bounding by parallelotopes. Automatica,

32:1049–1055.

Combastel, C. (2005). A state bounding observer for un-

certain nonlinear continuous-time systems based on

zonotopes. In 44th IEEE Conference on decision and

control and European control conference ECC 2005.

Dochain, D. (2003). State and parameter estimation in

chemical and biochemical processes: a tutorial. Jour-

nal of process control, 13:801–818.

Hermann, R. (1963). On the accessibility problem in control

theory. In in Int. Symp. on nonlinear differential equa-

tions and nonliear mechanics, pages 325–332, New

York. Academic Press.

Hermann, R. and J.Krener, A. (1977). Nonlinear control-

lability and observability. Transactions on Automatic

Control, AC-22:728–740.

Hirsch, M. and Smith, H. (2005). Monotone dynamical sys-

tems. In Canada, A., Drabek, P., and Fonda, A., ed-

itors, Handbook of Differential Equations, Ordinary

Differential Equations, volume 2, chapter 4. Elsevier.

Jaulin, L., Kieffer, M., Didrit, O., and Walter, E. (2001).

Applied interval analysis: with examples in parame-

ter and state estimation, robust control and robotics.

Springer-Verlag, London.

Kieffer, M. and Walter, E. (2006). Guaranteed nonlinear

state estimation for continuous-time dynamical mod-

els from discrete-time measurements. In Proceedings

6th IFAC Symposium on Robust Control, Toulouse.

Kieffer, M., Walter, E., and Simeonov, I. (2006). Guaran-

teed nonlinear parameter estimation for continuous-

time dynamical models. In Proceedings 14th IFAC

Symposium on System Identiﬁcation, pages 843–848,

Newcastle, Aus.

Marcelli, C. and Rubbioni, P. (1997). A new extension of

classical m

¨

uller theorem. Nonlinear Analysis, Theory,

Methods & Applications, 28(11):1759–1767.

Moore, R. (1966). Interval analysis. Prentice-Hall, Engle-

wood Cliffs.

M

¨

uller, M. (1926). Uber das fundamentaltheorem in

der theorie der gew

¨

ohnlichen differentialgleichungen.

Math. Z., 26:619–645.

Nedialkov, N. (1999). Computing rigorous bounds on the

solution of an initial value problem for an ordinary

differential equation. PhD University of Toronto.

Ra

¨

ıssi, T., Ramdani, N., and Candau, Y. (2004). Set mem-

bership state and parameter estimation for systems de-

scribed by nonlinear differential equations. Automat-

ica, 40(10):1771–1777.

Ramdani, N., Meslem, N., Ra

¨

ıssi, T., and Candau, Y.

(2006). Set-membership identiﬁcation of continuous-

time systems. In Proceedings 14th IFAC Symposium

on System Identiﬁcation, pages 446–451, Newcastle,

Aus.

Smith, H. (1995). Monotone dynamical systems: An intro-

duction to the theory of competitive and cooperative

systems. Ams. providence, ri edition.

Walter, W. (1997). Differential inequalities and maximum

principles: Theory, new methods and applications.

Nonlinear Analysis, Theory, Methods & Applications,

30(8):4695–4711.

A GUARANTEED STATE BOUNDING ESTIMATION FOR UNCERTAIN NON LINEAR CONTINUOUS TIME

SYSTEMS USING HYBRID AUTOMATA

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