Post on 26-Jun-2020
RobustHybridStateEs1ma1on usingIntervalMethods
!NacimRAMDANI
UniversityofOrléans,PRISMEEA4229,Bourges!
DoctorateThesisofMoussaMAIGA(UnivOrléans2015)co-supervisedwithLouiseTRAVE-MASSUYES(LAAS-CNRS,Toulouse)
!SWIM2016@ENSLyon
Outline
n Hybrid dynamical systems
n Set membership estimation
n Hybrid reachability based approach
n Example
n Research directions
2
Hybrid Cyber-Physical Systems
n Interaction discrete + continuous dynamics
n Safety-critical embedded systems
n Networked autonomous systems3
Hybrid Cyber-Physical Systems
nModelling → hybrid automaton (Alur, et al. 1995)
l Non-linear continuous dynamics l Nonlinear guards sets l Nonlinear reset functions l Bounded uncertainty
Continuous dynamics
Discrete dynamics
l
x ∈ Inv(l)
l′
x′ ∈ Inv(l′)
e : g(x) ≥ 0
x′ ∈ Flow(l′, x′)
x′ = r(e, x)
x ∈ Flow(l, x)
x ∈ Init(l)
4
5
nExample : the bouncing ball
Hybrid Cyber-Physical Systems
initial conditions
discrete transition jump
Hybrid Cyber-Physical Systems
6
nExample : the bouncing ball
-1
0
1
2
3
4
5
6
-12 -10 -8 -6 -4 -2 0 2 4 6 8 10
Discrete transitions
Initial conditions
freefallContinuous transtions
X
V
Free fall Continuous transitions
Initial conditions
Discrete transitions
Estimation of Hybrid State
nModelling → hybrid automaton l Nonlinear … l Bounded uncertainty
!
nHybrid State Estimation → reconstruct system variables l Switching sequence l Continuous variables l Hybrid solution trajectory tube
Author's personal copy
N. Ramdani et al. / Nonlinear Analysis: Hybrid Systems 4 (2010) 263–278 275
0 500 1000 1500 2000 2500 3000
x 2
time (s)
no uncertaintywith uncertainty
0
50
100
150
200
250
300
Fig. 3. Time history of the x2 component of the reachable set of (52) as obtained with Theorem 2, with an initial domain for state vector of size 100%. Thecurve labelled ‘no uncertainty’ corresponds to no uncertainty in the parameter vector (CPU time= 38.26 s PIV 2GHz) and the one labelled ‘with uncertainty’corresponds to the presence of uncertainty in the parameter vector (CPU time = 38.58 s PIV 2 GHz).
switc
hing
seq
uenc
e fo
r up
per
brac
ketti
ng s
yste
mt
18000
20000
22000
24000
26000
28000
30000
32000
34000
36000
38000
0 50 100 150 200 250 300
Fig. 4. Switching sequence for the hybrid automaton which drives the upper bounding system for (53).
the latter, i.e. using functions �i⇧(.) defined in Rule 2, the upper bounding systems are obtained by replacing parametercomponents in each algebraic expression of fi� either by their upper or by lower bound, form-typemodes, or by using wholeparameter uncertainty domain, for s-type modes. Since there are 10 partial derivatives to monitor and 3 possible values forthe parameter components (lower bound / upper bound / whole uncertainty interval), the set Q of discrete modes contains310 elements, and we merely use a word of ternary digits of length 10, to number the modes. Note however, that not all ofthem may be activated.
Fig. 4 shows the switching sequence for the hybrid automaton which derives the upper component-wise bounds of thereachable set of (53), as generated by algorithm Hybrid-Upper-Bounding. Some modes are active on very short timeintervals. Fig. 5 magnifies the switching sequence around t = 60 s. In fact, such modes are s-type modes which are usuallyactive only over one or two integration time intervals.
The automatonwhich derives the lower component-wise bounds is obtained in a similarmanner. The switching sequencefor this automaton is shown in Fig. 6.
Note that both initial state vector and parameter vector are taken uncertain with large uncertainties. Fig. 7 shows thetime history of the x12 component of the reachable set. Obviously, even for very large parameter boxes the hybrid bracketingmethod successfully computes the reachable set.
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Outline
n Hybrid dynamical systems
n Set membership estimation
n Hybrid reachability based approach
n Example
n Research directions
8
Set Membership Estimation
n Unknown but bounded-error framework
Set membership estimationContinuous-time systemsHigh dimensional system
ContextState-of-the-art
Set-membership or bounded error approach(Belforte et al., 1990) ; (Milanese et al., 1996) ;(Vicino and Zappa, 1996) ;(Walter et al., 1990) ; (Norton et al., 1994, 1995)
f(p)
n
ys Optimisation de J(e(p))
e(p)
ys p1
p2
Régions de confiance
f(p)
n
Set membersip algorithm
Y p1
p2
Solution set
Y
Hypothesis
Uncertainties and errors are bounded with known prior bounds
A set of feasible solutions
S = {p � P|f(p) � Y} = f�1(Y) ⇥ P
Nacim Ramdani et al. Set-membership identification . . .
Set Membership Algorithm
9
n State estimation with continuous systems l Interval observers ‣ (Moisan, et al. 2009), (Mazenc & Bernard, 2010),
(Meslem & Ramdani, 2011), (Raïssi, et al., 2012), (Combastel, 2013), (El Thabet, et al. 2014), (Efimov, et al. 2015) !
‣ Monotonicity
‣ Change of coordinates
‣ LMI ….
‣ Ensure practical stability
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Set Membership Estimation
n State estimation with continuous systems l Prediction - Correction / Filtering approaches ‣ (Raïssi et al., 2005), (Meslem, et al, 2010),
(Milanese & Novara, 2011), (Kieffer & Walter, 2011) … !
‣ Reachability + Set inversion
‣ Forward backwardconsistency
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Set Membership Estimation
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Set Membership Estimation
n Set inversion. Parameter estimation l Branch-&-bound, branch-&-prune, interval contractors …
(Jaulin, et al. 93) (Raïssi et al., 2004)
n State estimation with Continuous systems l Interval observers l Prediction-correction / Filtering approaches
n State estimation with Hybrid systems l Piecewise affine systems (Bemporad, et al. 2005) l ODE + CSP (Goldsztejn, et al., 2010) l Nonlinear case (Benazera & Travé-Massuyès, 2009) l SAT mod ODE (Eggers, Ramdani, et al., 2012) l Reachability-based (Maïga, Ramdani, et al. 2015).
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Set Membership Estimation
Outline
n Hybrid dynamical systems
n Set membership estimation
n Hybrid reachability based approach
n Example
n Research directions
14
n Predictor-Corrector approach for hybrid systems
15
Reachability based approach
n Predictor-Corrector approach for hybrid systems
16
Reachability based approach
l
x ∈ Inv(l)
l′
x′ ∈ Inv(l′)
e : g(x) ≥ 0
x′ ∈ Flow(l′, x′)
x′ = r(e, x)
x ∈ Flow(l, x)
x ∈ Init(l)
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Reachability based approach
17
Reachability based approach
guard
mode = 1
mode = 2
reset mapping
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Reachability based approach
X1(t0)
guard
mode = 1
mode = 2
reset mapping
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Reachability based approach
X1(t0)ϕ(., t0, X(t0))
guard
mode = 1
mode = 2
reset mapping
17
Reachability based approach
X1(t0)ϕ(., t0, X(t0))
guard
mode = 1
mode = 2
reset mapping
[Maïga, Ramdani, Travé Massuyes & Combastel, IEEE TAC 2016].
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Reachability based approach
X1(t0)ϕ(., t0, X(t0))
guard
mode = 1
mode = 2
reset mapping
[Maïga, Ramdani, Travé Massuyes & Combastel, IEEE TAC 2016].
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Reachability based approach
Y(t1)
X1(t0)ϕ(., t0, X(t0))
guard
mode = 1
mode = 2
reset mapping
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Reachability based approach
Y(t1)
X1(t0)
X1+(t1)
guard
mode = 1
mode = 2
reset mapping
X2+(t1)
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Reachability based approach
Y(t1)
X1(t0)
X1+(t1)
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
reset mapping
X2+(t1)
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Reachability based approach
Y(t1)
X1(t0)
X1+(t1)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
X2+(t1)
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Reachability based approach
Y(t1)
X1(t0)
X1+(t1)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
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Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
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Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
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Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.)
g2-1(.)
mode = 1
mode = 2
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))
guard g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard
Y(t3)
g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard
Y(t3)
g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard
Y(t3)
g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard
Y(t3)
g1-1(.) g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.) g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)
17
Reachability based approach
Y(t1)
Y(t2)
X1(t0)
X1+(t1)
X1+(t2)
X1(t1)=X1+(t1)∩g1-1(Y(t1))X1(t2)=X1+(t2)∩g1-1(Y(t2))
guard
Y(t3)
g1-1(.) g1-1(.) g1-1(.)
g2-1(.)
mode = 1
mode = 2
g2-1(.) g2-1(.)
X2(t1)=X2+(t1)∩g2-1(Y(t1)) X2(t2)=X2+(t2)∩g2-1(Y(t2))
X2(t3)=X2+(t3)∩g2-1(Y(t3))
X1(t3)=∅
reset mapping
t1 {q=1, X1(t1)} ∪ {q=2, X2(t1)}
t2 {q=1, X1(t2)} ∪ {q=2, X2(t2)}
t3 {q=2, X2(t3)}
Reconstructed Hybrid Solution State Trajectory:
X2+(t1)t4 …
Filter out mode q=1 at time t3
nReachable set !
!l Set integration
l Interval Taylor methods l Bracketing enclosures
Continuous Reachability
nReachable set !
!l Set integration
l Interval Taylor methods l Bracketing enclosures
Continuous Reachability
Continuous Reachability
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Continuous Reachability
nGuaranteed set integrationl … with interval Taylor methods.
‣ (Moore, 66) (Lohner, 88) (Rihm, 94) (Berz, 98) (Nedialkov, 99)
l … with interval Taylor models. ‣ (Chen, 2012)
l also via interval Runge Kutta.‣ (Alexandre dit Sandretto & Chapoutot, 2015)
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Continuous Reachability
nGuaranteed set integrationl … with interval Taylor methods.
‣ (Moore, 66) (Lohner, 88) (Rihm, 94) (Berz, 98) (Nedialkov, 99)
l … with interval Taylor models. ‣ (Chen, 2012)
l also via interval Runge Kutta.‣ (Alexandre dit Sandretto & Chapoutot, 2015)
nComparison theorems for differential inequalitiesn Monotone systems
‣ (Ramdani et al., 2010)
n Muller’s theorem‣ (Kieffer et al. 2006) (Ramdani, et al. 2006), (Ramdani, et al. 2009)
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Hybrid Reachability Computation
nGuaranteed event detection & localization l An interval constraint propagation approach
l(Ramdani & Nedialkov, Nonlinear Analysis Hybrid Systems 2011)
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Hybrid Reachability Computation
nGuaranteed event detection & localization l An interval constraint propagation approach
l(Ramdani & Nedialkov, Nonlinear Analysis Hybrid Systems 2011)
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Interval Taylor Methods
Guaranteed set integration with Taylor methods(Moore,66) (Eijgenraam,81) (Lohner,88) (Rihm,94) (Berz,98) (Nedialkov,99)
x(t) = f (x,p, t), t0 ≤ t ≤ tN , x(t0) ∈ [x0] , p ∈ [p]
Time grid → t0 < t1 < t2 < · · · < tN
[xj ] [xj+1]
actual solution x⋆
a priori [xj ]
Analytical solution for [x](t), t ∈ [tj , tj+1]
[x](t) = [xj ] +k−1!
i=1(t − tj)i f [i ]([xj ], [p]) + (t − tj)k f [k]([xj ], [p])
N. Ramdani (INRIA) Hybrid Nonlinear Reachability 11 / 28
Hybrid Reachability Computation
nGuaranteed event detection & localization l An interval constraint propagation approach
l(Ramdani & Nedialkov, Nonlinear Analysis Hybrid Systems 2011)
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Hybrid Reachability Computation
nGuaranteed event detection & localization l An interval constraint propagation approach
l(Ramdani & Nedialkov, Nonlinear Analysis Hybrid Systems 2011)
20
nGuaranteed event detection & localization l An interval constraint propagation approach
l(Ramdani & Nedialkov, Nonlinear Analysis Hybrid Systems 2011)
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Hybrid Reachability Computation
nDetecting and localizing events l Improved and enhanced version. A faster version.
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
22
Hybrid Reachability Computation
nDetecting and localizing events l Improved and enhanced version. A faster version.
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
22
Hybrid Reachability Computation
23
nDetecting and localizing events l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
Hybrid Reachability Computation
24
nDetecting and localizing events l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)
Hybrid Reachability Computation
25
nDetecting and localizing events l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)Bouncing ball in 2D.
Hybrid Reachability Computation
-1
0
1
2
3
4
5
6
2 4 6 8 10 12 14
x2
x1
25
nDetecting and localizing events l Improved and enhanced version
l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)Bouncing ball in 2D.
Hybrid Reachability Computation
-1
0
1
2
3
4
5
6
2 4 6 8 10 12 14
x2
x1
-1
-0.5
0
0.5
1
1.5
2
2.5
3
3.5
-6 -3 0 3 6
py
px
26
nDetecting and localizing events l Improved and enhanced version
l Impact of uncertainty on sliding mode control (Maïga, Ramdani, Travé-Massuyès, Combastel, IEEE TAC 2016)
Hybrid Reachability Computation
26
nDetecting and localizing events l Improved and enhanced version
l Impact of uncertainty on sliding mode control (Maïga, Ramdani, Travé-Massuyès, Combastel, IEEE TAC 2016)
Hybrid Reachability Computation
27
Solution trajectory tube
Mean value form + Lohner’s QR transformation method
is a particular zonotope
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Zonotope of minimum size enclosing the intersection of a zonotope and a strip
zonotope support strip
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Hybrid Reachability based Predictor Corrector approach
Outline
n Hybrid dynamical systems
n Set membership estimation
n Hybrid reachability based approach
n Example
n Research directions
30
31
nHybrid Mass-Spring l Velocity-dependent damping. Mode switching driven by velocity.
Parameter identification
32
nHybrid Mass-Spring l Unknown initial mode.
l CPU time approx. 1m20s
State Estimation
Mode
State space
Position
Velocity
32
nHybrid Mass-Spring l Unknown initial mode.
l CPU time approx. 1m20s
State Estimation
Mode
State space
Position
Velocity
32
nHybrid Mass-Spring l Unknown initial mode.
l CPU time approx. 1m20s
State Estimation
Mode
State space
Position
Velocity
33
nHybrid Mass-Spring l Unknown initial mode.
l CPU time approx. 1m20s
State Estimation
Position Velocity
Outline
n Hybrid dynamical systems
n Set membership estimation
n Hybrid reachability based approach
n Example
n Research directions
34
35
Research directions
35
Research directions
n Contractors for hybrid dynamical systemsl To build upon a hybrid reachability approach
n Push forward set membership estimation l SM hybrid state estimation of nonlinear hybrid systems
35
Research directions
n Contractors for hybrid dynamical systemsl To build upon a hybrid reachability approach
n Push forward set membership estimation l SM hybrid state estimation of nonlinear hybrid systems
n Address SM estimation with controlled samplingl Event- & Self-triggered SM hybrid state estimation of nonlinear hybrid systems
35
Research directions
n Contractors for hybrid dynamical systemsl To build upon a hybrid reachability approach
n Push forward set membership estimation l SM hybrid state estimation of nonlinear hybrid systems
n Address SM estimation with controlled samplingl Event- & Self-triggered SM hybrid state estimation of nonlinear hybrid systems
n Combine with decision makingl Application to actual hybrid systems, in robotics, smart buildings, personalized medicine …
36
Focused References
n N. Ramdani and N. S.Nedialkov, Computing reachable sets for uncertain nonlinear hybrid systems using interval constraint propagation techniques, Nonlinear Analysis: Hybrid Systems, 5(2), pp.149-162, 2011.
n A.Eggers, N.Ramdani, N.S.Nedialkov, M.Fränzle, Set-Membership Estimation of Hybrid Systems via SAT Mod ODE. in IFAC SYSID 2012. pp.440-445
n M. Maïga, N. Ramdani, L. Travé-Massuyès, C. Combastel, A CSP versus a zonotope-based method for solving guard set intersection in nonlinear hybrid reachability, Mathematics in Computer Science, pp.407-423, 2014.
n M. Maïga, N. Ramdani, L. Travé-Massuyès, C. Combastel, A comprehensive method for reachability analysis of uncertain nonlinear hybrid systems, IEEE Trans. Automatic Control, to appear in 2016.
n M. Maïga, N. Ramdani, L. Travé-Massuyès, Robust fault detection in hybrid systems using set-membership parameter estimation, in IFAC SafeProcess 2015, Paris.
n M. Maïga, N. Ramdani, L. Travé-Massuyès, Bounded-error state estimation for uncertain non-linear hybrid systems with discrete-time measurements}, Automatica, Submitted May 2016