Robust Hybrid State Es1ma1on using Interval Methods · Robust Hybrid State Es1ma1on using Interval...

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


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


initial conditions

discrete transition jump


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 Example


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

10


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

11


12


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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Outline




n Example


14

n Predictor-Corrector approach for hybrid systems

15

Reachability based approach

n Predictor-Corrector approach for hybrid systems

16


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)

17


17


guard

mode = 1

mode = 2

reset mapping

17


X1(t0)

guard

mode = 1

mode = 2

reset mapping

17


X1(t0)ϕ(., t0, X(t0))

guard

mode = 1

mode = 2

reset mapping

17


X1(t0)ϕ(., t0, X(t0))

guard

mode = 1

mode = 2

reset mapping

[Maïga, Ramdani, Travé Massuyes & Combastel, IEEE TAC 2016].

17


Y(t1)

X1(t0)ϕ(., t0, X(t0))

guard

mode = 1

mode = 2

reset mapping

17


Y(t1)

X1(t0)

X1+(t1)

guard

mode = 1

mode = 2

reset mapping

X2+(t1)

17


Y(t1)

X1(t0)

X1+(t1)

guard g1-1(.)

g2-1(.)

mode = 1

mode = 2

reset mapping

X2+(t1)

17


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)

17


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)

17


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)}


X2+(t1)

17


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)}


X2+(t1)

17


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)}


X2+(t1)

17


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)}


X2+(t1)

17


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)}


X2+(t1)

17


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)}


X2+(t1)

17


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)}


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


19


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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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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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




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nDetecting and localizing events l Improved and enhanced version. A faster version.

l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)

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nDetecting and localizing events l Improved and enhanced version



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l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)Bouncing ball in 2D.


-1

0

1

2

3

4

5

6

2 4 6 8 10 12 14

x2

x1

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l(Maïga, Ramdani, et al., IEEE CDC 2013, ECC 2014)Bouncing ball in 2D.


-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

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l Impact of uncertainty on sliding mode control (Maïga, Ramdani, Travé-Massuyès, Combastel, IEEE TAC 2016)


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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 Example


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nHybrid Mass-Spring l Velocity-dependent damping. Mode switching driven by velocity.

Parameter identification

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nHybrid Mass-Spring l Unknown initial mode.

l CPU time approx. 1m20s

State Estimation

Mode

State space

Position

Velocity

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nHybrid Mass-Spring l Unknown initial mode.

l CPU time approx. 1m20s

State Estimation

Position Velocity

Outline




n Example


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

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Research directions



n Address SM estimation with controlled samplingl Event- & Self-triggered SM hybrid state estimation of nonlinear hybrid systems

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Research directions



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 …

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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

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