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Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
A Novel Thick Ellipsoid Approach for Verified Outer and Inner
State Enclosures of Discrete-Time Dynamic Systems
19th IFAC Symposium System Identification:
Learning models for decision and control — SYSID (online)
July 15, 2021
Andreas Rauh, Luc Jaulin
Lab-STICC (Robex)
ENSTA Bretagne, Brest, France
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 1/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Contents
Motivation: Propagating bounded domains when analyzing uncertain dynamic systems
Thick ellipsoids as a means to quantify worst-case overestimation due to nonlinearities
Basic algorithm: Mapping of thick ellipsoid domains
Application to a recursive simulation of discrete-time dynamic systems
ILinear system models
INonlinear system models
IExtension: Splitting and merging of ellipsoids
Conclusions and outlook on future work
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 2/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Wrapping Effect of Interval Computation (1)
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1x1
x2
x1
x2
1
−1
x2
−1 1
k= 0 k= 1 k= 2
Linear set of state equations
xk+1 =Akxk
Example:
[x0] = [−1 ; 1]
[−1 ; 1]
Ak=A=1
2√21 1
−1 1
⇒Rotation of 45◦
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 3/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Wrapping Effect of Interval Computation (2)
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1x1
x2
x1
x2
1
−1
x2
−1 1
k= 0 k= 1 k= 2 Traditional interval arithmetic
evaluation:
x1=A x0
x2=A x1
.
.
.
xk+1 =A xk
⇒Exponential growth of the enclosing
interval boxes
⇒Wrapping effect
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 4/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Wrapping Effect of Interval Computation (3)
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1
x2
x1x1
x2
x1
x2
1
−1
x2
−1 1
k= 0 k= 1 k= 2 Modified system matrix:
x1=A x0=˜
A0x0
x2=A˜
A0x0=˜
A1x0
.
.
.
xk+1 =A˜
Ak−1xk=˜
Akx0
˜
Ak=A˜
Ak−1
⇒Significant reduction of the wrapping
effect for linear systems
⇒Elimination of the wrapping effect for
point matrices A
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 5/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Discrete-Time Systems with Time-Invariant Interval Parameters (1)
xk+1
pk+1=gk(xk,pk,uk, k)
pk
uk: given open-/ closed-loop control
Problem
Determine state enclosures at each time step kfor a given finite time horizon
Solution Approach 1
Recursive computation of state intervals
[xk+1] = gk[xk],[pk],uk, k:open-loop control
[xk+1] = gk[xk],[pk],uk([xk],[pk]) , k:closed-loop control
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 6/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Discrete-Time Systems with Time-Invariant Interval Parameters (2)
xk+1
pk+1=gk(xk,pk,uk, k)
pk
uk: given open-/ closed-loop control
Problem
Determine state enclosures at each time step kfor a given finite time horizon
Solution Approach 2
Computation of state intervals [xk+1]by coordinate transformations for reduction
of overestimation caused by the wrapping effect, e.g. linear transformations
[xk] = Tk·[˜
xk] =⇒[˜
xk+1] = T−1
k+1 ·gkTk·[˜
xk],pk, . . .
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 7/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Discrete-Time Systems with Time-Invariant Interval Parameters (3)
Solution Approach 3
Idea: Subdividing and merging of state intervals representing [xk+1]for tight
enclosures of complexly shaped regions
1Consistency tests by inverse mapping of state equation
xk=¯
gkxk+1,pk, . . .
x0
k+1⊂[xk+1 ]subinterval of forward computation
2Interval Newton methods for state equations where inverse mapping cannot
be calculated analytically
3Merging of subintervals in case of small overestimation of the union of the
merged subintervals
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 8/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Discrete-Time Systems with Time-Invariant Interval Parameters (4)
x2,k+1
x1,k
x2,k
x2,k+1
x1,k+1
x1,k+1
Subdivision into
interval boxes
Natural interval evaluation
(without optimization)
Consistency test
by inverse mapping Result of forward computation
of subdivided interval boxes
Xk
Application of optimized
Xk+1
interval techniques
xk+1 =gk(xk,pk,uk, k) for all xk∈ Xk
xk=¯
gk(xk+1,pk,uk, k)
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 9/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Discrete-Time Systems with Time-Invariant Interval Parameters (4)
Solution Approach 4
Computation of state variables [xk+1]by explicit replacement of [xk],[xk−1],
. . . , [x2],[x1]in terms of the initial state [x0]and all parameter uncertainties
[x0],[x1], . . . , [xk−1],[xk], i.e.,
[xk+1] = gk gk−1. . . g1g0([x0],[p0],u0,0) ,[p1],u1,1. . . !
Evaluation by mean-value rule, monotonicity tests, and global optimization
Solution Approach 5
Choice of alternative enclosure representations (zonotopes, Taylor models,
ellipsoids)
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 10/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Solution Representation by Means of Thick Ellipsoids (1)
µ1,k x1,k
x2,k
µ2,k µ2,k+1
µ1,k+1 x1,k+1
x2,k+1
EI
k
EO
k
Ak
xk+1 =f(xk)
EO
k+1
EI
k+1
Ak+1
Definition
((E)) = A ∈ P (Rn)EI⊆ A ⊆ EOwith
EI=nx∈Rn(x−µ)TρΓ−TρΓ−1(x−µ)≤1o,
EO=nx∈Rn(x−µ)T(ρΓ)−T(ρΓ)−1(x−µ)≤1oand 0≤ρ≤ρ
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 11/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Solution Representation by Means of Thick Ellipsoids (2)
µ1,k x1,k
x2,k
µ2,k µ2,k+1
µ1,k+1 x1,k+1
x2,k+1
EI
k
EO
k
Ak
xk+1 =f(xk)
EO
k+1
EI
k+1
Ak+1
Thick ellipsoid operators ∈{+,−,·, /, ∪,∩}
(A ∈ ((A)) ,B ∈ ((B))
C=AB =⇒ C ∈ ((A)) ((B))
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 12/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Solution Representation by Means of Thick Ellipsoids (3)
µ1,k x1,k
x2,k
µ2,k µ2,k+1
µ1,k+1 x1,k+1
x2,k+1
EI
k
EO
k
Ak
xk+1 =f(xk)
EO
k+1
EI
k+1
Ak+1
Thick ellipsoid function evaluation f:Rn7→ Rm
(A ∈ ((A))
B=f(A)=⇒ B ∈ ((B)) = ((f)) ((A))
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 13/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Recursive Enclosure Technique (1)
⊕=
x2,k
x1,k
µ1,k
µ2,k
x1,k+1
µ1,k+1
µ2,k+1
x2,k+1
˜
x2,k+1
˜
x2,k
˜
x1,k ˜
b1,k
˜
b2,k
˜
x1,k+1
1
Sx
xk+1 =f(xk)
EI
k+1
[xk]
ρ∗
k1 + ρ∗
k
Sb,k
h˜
bki
˜
bk(Sx)
˜
xk+1 =˜
xk+˜
bk(˜
xk)
((E))k=EI
k=EO
k
f(((E))k)
EO
k+1
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 14/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Recursive Enclosure Technique (2)
Alternatively, (16) can be expressed as ˜
xk+1 ∈ Sx⊕ SO
b,k,
where ⊕is the Minkowski sum of two sets and
˜
bO,k (˜
xk)∈ SO
b,k := {˜
xk| k˜
xkk ≤ ρO,k}.(19)
Therefore, ˜
xk+1 belongs to a ball of radius 1 + ρO,k as
illustrated in Fig. 2. According to (13) and (16), the
equality
˜
xk+1 =ρ−1
k·Γ−1
k·A−1
k·(xk+1 −µk+1) (20)
holds. Substituting this equality into (17) yields the outer
boundary EO
k+1 of the thick ellipsoid according to (7)–(8).
For the second part of the proof, set
˜
xk=ρ−1
k·Γ−1
k·(xk−µk) (21)
instead of (14). Then,
˜
xk+1 =˜
xk+˜
bI,k (˜
xk) (22)
with ˜
bI,k (˜
xk) given in (11) yields
˜
bI,k (˜
xk)∈ SI
b,k := {˜
xk| k˜
xkk ≤ ρI,k}(23)
with
ρI,k = arg min
ρ∈R+nρ∈R+
˜
bI,k (˜
xk)
≤ρ , ∀˜
xk∈ Sxo.
(24)
The maximum ellipsoid in the interior of the solution set
that is parallel to the outer domain boundary EO
k+1 results
from the Minkowski difference SI=Sx SI
b,k. Applying
Eq. (20) after the replacement of ρkby ρkleads to EI
k+1 as
an inner approximation of SI. Summarizing both EI
k+1 and
EO
k+1 into the single thick ellipsoid ((E))k+1 , cf. (7), completes
the proof. 2
⊕=
x2,k
x1,k
µ1,k
µ2,k
x1,k+1
µ1,k+1
µ2,k+1
x2,k+1
˜
x2,k+1
˜
x2,k
˜
x1,k ˜
b1,k
˜
b2,k
˜
x1,k+1
1
Eq. (20)
Sx
xk+1 =f(xk)
EI
k+1
[xk]
ρ∗
k1 + ρ∗
k
Sb,k
h˜
bki
˜
bk(Sx)
˜
xk+1 =˜
xk+˜
bk(˜
xk)
Eqs. (14), (21)
((E))k=EI
k=EO
k
f(((E))k)
EO
k+1
Fig. 2. Recursive computation of ellipsoidal state enclo-
sures (simplified for ((E))k=EI
k=EO
k, leading to Sb,k =
SO
b,k =SI
b,k,˜
bk=˜
bO,k =˜
bI,k and ρ∗
k=ρO,k =ρI,k in
the illustration).
Remark 4. Due to the fact that the matrices Γkare
typically not symmetric in the recursive evaluation of the
ellipsoidal simulation procedure, the following algorithm
is based on a Cholesky-like decomposition of the shape
matrix ΓkΓT
kof an ellipsoid. The factorized representation
of the shape matrix simplifies the implementation of a
recursive simulation procedure, see line 3 in Algorithm 1.
Remark 5. Eqs. (11) and (13) require that the Jacobian
Akat the midpoint of the thick ellipsoid ((E))kis invertible.
If this is not the case, the shape matrix can be approxi-
mated in practice by using a representative slope within
the domain of interest or with the help of the matrix square
root of the covariance matrix cov(f(Xk)), where
Xk= [µk−∆1,k . . . µk−∆n,k µk+∆1,k . . . µk+∆n,k]
(25)
with ∆i,k,i∈ {1, . . . , n}, are determined in analogy to the
sigma points of an Unscented Kalman Filter according to
∆i,k =√n·Γi,k with Γk= [Γ1,k . . . Γn,k ].(26)
3.2 Implementation of the Algorithm
The implementation of the thick ellipsoid simulation pro-
cedure is summarized in Algorithm 1.
Algorithm 1: Recursive simulation approach
input : f (xk), {µk,Γk, ρk, ρk}
output: {µk+1,Γk+1 , ρk+1, ρk+1 }
1µk+1 =f(µk)
2Ak=∂f
∂xk(µk)
3Γk+1 =Ak·Γk
4ρI,k ←Compute norm depending on ρk
5ρO,k ←Compute norm depending on ρk
6ρk+1 = (1 −ρI,k )·ρk
7ρk+1 = (1 + ρO,k )·ρk
Algorithm 2: Compute norm
input : µk,Γk,Γk+1, ρι∈ {ρk, ρk}
output: ρι,k
1[xι,i,k] = µi,k +
ρι·ΓT
i,k
·[−1 ; 1] , i ∈ {1, . . . , n}
2[xι,k] = [xι,1,k ]×. . . ×[xι,n,k ]
3[Jf,ι] = h∂f
∂xki([xι,k])
4h˜
bι,ki=Γ−1
k+1 ·[Jf,ι]·Γk−I·[−1 ; 1]×n
5ρι,k = sup n
h˜
bι,ki
o
For the computation of ρI,k and ρO,k in the lines 4 and 5
of Algorithm 1 according to Theorem 3, it is necessary to
evaluate the Euclidean norm of the vectors ˜
bI,k (˜
xk) and
˜
bO,k (˜
xk) over the respective domains EI
kand EO
k. For that
purpose, both (11) and (13) are evaluated in Algorithm 2
with the help of interval analysis. For that purpose, we
exploit ˜
bI,k (0) = 0and ˜
bO,k (0) = 0and perform the
interval evaluation by a centered form representation.
According to Rauh and Jaulin (2021), this is given by
˜
bι,k ∈"∂˜
bι,k
∂˜
xk#([˜
xk]) ·[˜
xk], ι ∈ {I,O},with
(27)
∂˜
bι,k
∂˜
xk
(˜
xk) = Γ−1
k·A−1
k·∂f
∂xk
(Eι
k)·Γk−I
=Γ−1
k+1 ·∂f
∂xk
(Eι
k)·Γk−Iand (28)
∂f
∂xk
(Eι
k)∈∂f
∂xk([xι]) ,(29)
Alternatively, (16) can be expressed as ˜
xk+1 ∈ Sx⊕ SO
b,k,
where ⊕is the Minkowski sum of two sets and
˜
bO,k (˜
xk)∈ SO
b,k := {˜
xk| k˜
xkk ≤ ρO,k}.(19)
Therefore, ˜
xk+1 belongs to a ball of radius 1 + ρO,k as
illustrated in Fig. 2. According to (13) and (16), the
equality
˜
xk+1 =ρ−1
k·Γ−1
k·A−1
k·(xk+1 −µk+1) (20)
holds. Substituting this equality into (17) yields the outer
boundary EO
k+1 of the thick ellipsoid according to (7)–(8).
For the second part of the proof, set
˜
xk=ρ−1
k·Γ−1
k·(xk−µk) (21)
instead of (14). Then,
˜
xk+1 =˜
xk+˜
bI,k (˜
xk) (22)
with ˜
bI,k (˜
xk) given in (11) yields
˜
bI,k (˜
xk)∈ SI
b,k := {˜
xk| k˜
xkk ≤ ρI,k}(23)
with
ρI,k = arg min
ρ∈R+nρ∈R+
˜
bI,k (˜
xk)
≤ρ , ∀˜
xk∈ Sxo.
(24)
The maximum ellipsoid in the interior of the solution set
that is parallel to the outer domain boundary EO
k+1 results
from the Minkowski difference SI=Sx SI
b,k. Applying
Eq. (20) after the replacement of ρkby ρkleads to EI
k+1 as
an inner approximation of SI. Summarizing both EI
k+1 and
EO
k+1 into the single thick ellipsoid ((E))k+1 , cf. (7), completes
the proof. 2
⊕=
x2,k
x1,k
µ1,k
µ2,k
x1,k+1
µ1,k+1
µ2,k+1
x2,k+1
˜
x2,k+1
˜
x2,k
˜
x1,k ˜
b1,k
˜
b2,k
˜
x1,k+1
1
Eq. (20)
Sx
xk+1 =f(xk)
EI
k+1
[xk]
ρ∗
k1 + ρ∗
k
Sb,k
h˜
bki
˜
bk(Sx)
˜
xk+1 =˜
xk+˜
bk(˜
xk)
Eqs. (14), (21)
((E))k=EI
k=EO
k
f(((E))k)
EO
k+1
Fig. 2. Recursive computation of ellipsoidal state enclo-
sures (simplified for ((E))k=EI
k=EO
k, leading to Sb,k =
SO
b,k =SI
b,k,˜
bk=˜
bO,k =˜
bI,k and ρ∗
k=ρO,k =ρI,k in
the illustration).
Remark 4. Due to the fact that the matrices Γkare
typically not symmetric in the recursive evaluation of the
ellipsoidal simulation procedure, the following algorithm
is based on a Cholesky-like decomposition of the shape
matrix ΓkΓT
kof an ellipsoid. The factorized representation
of the shape matrix simplifies the implementation of a
recursive simulation procedure, see line 3 in Algorithm 1.
Remark 5. Eqs. (11) and (13) require that the Jacobian
Akat the midpoint of the thick ellipsoid ((E))kis invertible.
If this is not the case, the shape matrix can be approxi-
mated in practice by using a representative slope within
the domain of interest or with the help of the matrix square
root of the covariance matrix cov(f(Xk)), where
Xk= [µk−∆1,k . . . µk−∆n,k µk+∆1,k . . . µk+∆n,k]
(25)
with ∆i,k,i∈ {1, . . . , n}, are determined in analogy to the
sigma points of an Unscented Kalman Filter according to
∆i,k =√n·Γi,k with Γk= [Γ1,k . . . Γn,k ].(26)
3.2 Implementation of the Algorithm
The implementation of the thick ellipsoid simulation pro-
cedure is summarized in Algorithm 1.
Algorithm 1: Recursive simulation approach
input : f (xk), {µk,Γk, ρk, ρk}
output: {µk+1,Γk+1 , ρk+1, ρk+1 }
1µk+1 =f(µk)
2Ak=∂f
∂xk(µk)
3Γk+1 =Ak·Γk
4ρI,k ←Compute norm depending on ρk
5ρO,k ←Compute norm depending on ρk
6ρk+1 = (1 −ρI,k )·ρk
7ρk+1 = (1 + ρO,k )·ρk
Algorithm 2: Compute norm
input : µk,Γk,Γk+1, ρι∈ {ρk, ρk}
output: ρι,k
1[xι,i,k] = µi,k +
ρι·ΓT
i,k
·[−1 ; 1] , i ∈ {1, . . . , n}
2[xι,k] = [xι,1,k ]×. . . ×[xι,n,k ]
3[Jf,ι] = h∂f
∂xki([xι,k])
4h˜
bι,ki=Γ−1
k+1 ·[Jf,ι]·Γk−I·[−1 ; 1]×n
5ρι,k = sup n
h˜
bι,ki
o
For the computation of ρI,k and ρO,k in the lines 4 and 5
of Algorithm 1 according to Theorem 3, it is necessary to
evaluate the Euclidean norm of the vectors ˜
bI,k (˜
xk) and
˜
bO,k (˜
xk) over the respective domains EI
kand EO
k. For that
purpose, both (11) and (13) are evaluated in Algorithm 2
with the help of interval analysis. For that purpose, we
exploit ˜
bI,k (0) = 0and ˜
bO,k (0) = 0and perform the
interval evaluation by a centered form representation.
According to Rauh and Jaulin (2021), this is given by
˜
bι,k ∈"∂˜
bι,k
∂˜
xk#([˜
xk]) ·[˜
xk], ι ∈ {I,O},with
(27)
∂˜
bι,k
∂˜
xk
(˜
xk) = Γ−1
k·A−1
k·∂f
∂xk
(Eι
k)·Γk−I
=Γ−1
k+1 ·∂f
∂xk
(Eι
k)·Γk−Iand (28)
∂f
∂xk
(Eι
k)∈∂f
∂xk([xι]) ,(29)
Propagation of the thick ellipsoid based on a mapping of the ellipsoid midpoint and forecasting
the new shape matrix after a linearization of the system model. The stretch parameters ρk+1
and ρk+1 are determined by verified interval methods without costly online optimization.
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 15/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Numerical Simulations (1)
Linear model of a controlled wheel suspension system (T= 10−3)
xk+1 =xk+T·
010
−200 −15 −400
0 0 −200
xk+
0
0
8
x2,k
State variable x1State variable x2State variable x3
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 16/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Numerical Simulations (2)
Nonlinear Brusselator model (T= 0.2), N= 8 subdivisions in each coordinate
xk+1 =xk+T·
1 + x2
1,kx2,k −2.5x1,k + 0.5x2
3,k
1.5x1,k −x2
1,kx2,k
−0.5x2
3,k
State variable x1State variable x2State variable x3
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 17/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Conclusions and Outlook on Future Work
Definition of thick ellipsoid state enclosures
Guaranteed inner and outer solution enclosures for discrete-time systems
Applicable to linear and nonlinear systems including bounded parameter uncertainty
Quantification of the reliability of linearization-based stochastic estimators (Extended
Kalman Filter)
Specialized implementation for quasi-linear system models
Use for the implementation of predictor-corrector type state estimators
Verification of stability properties of discrete-time systems
Extension to continuous-time dynamic systems
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 18/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Conclusions and Outlook on Future Work
Definition of thick ellipsoid state enclosures
Guaranteed inner and outer solution enclosures for discrete-time systems
Applicable to linear and nonlinear systems including bounded parameter uncertainty
Quantification of the reliability of linearization-based stochastic estimators (Extended
Kalman Filter)
Specialized implementation for quasi-linear system models
Use for the implementation of predictor-corrector type state estimators
Verification of stability properties of discrete-time systems
Extension to continuous-time dynamic systems
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems 18/18
Discrete-Time Systems Thick Ellipsoids Thick Ellipsoid Mapping Examples Conclusions
Related publications
A. Rauh; L. Jaulin: A Computationally Inexpensive Algorithm for Determining Outer and Inner Enclosures of Nonlinear Mappings of Ellipsoidal Domains,
International Journal of Applied Mathematics and Computer Science AMCS, Vol. 31, No. 3. 2021. In print.
A. Rauh; A. Bourgois; L. Jaulin: Union and Intersection Operators for Thick Ellipsoid State Enclosures: Application to Bounded-Error Discrete-Time State
Observer Design, Algorithms, 14(3), 88. 2021.
A. Rauh; A. Bourgois; L. Jaulin: Verifying Provable Stability Domains for Discrete-Time Systems Using Ellipsoidal State Enclosures, SNR 2021. Under
review.
A. Rauh; A. Bourgois; L. Jaulin; J. Kersten: Ellipsoidal Enclosure Techniques for a Verified Simulation of Initial Value Problems for Ordinary Differential
Equations, ICCAD 2021. Under review.
Thank you for your attention!
A. Rauh, L. Jaulin: A Novel Thick Ellipsoid Approach for Verified Outer and Inner State Enclosures of Discrete-Time Dynamic Systems
