These are slides of my presentation at the 62nd IEEE Conference on Decision and Control (CDC) in Singapore, on Dec. 2023. In this work, we synthesized interval observers to simultaneously estimate states and unknown inputs (attacks) in nonlinear discrete-time systems. The considered systems are subject to distribution-free (set-valued) noise and disturbance, and are compromised by adversarial or malicious false data injections on their sensors and actuators. We provide sufficient conditions for stability and optimality of the designed observers. The proposed framework has several applications in resilient estimations and control, attack mitigation, and input reconstruction in cyber-physical systems.

1. Resilient State Estimation via Input and State
Interval Observer Synthesis
Mohammad Khajenejad, Zeyuan Jin, Thach Ngoc Dinh and
Sze Zheng Yong
Department of Mechanical and Industrial Engineering
Northeastern University, Boston, USA
Email: s.yong@northeastern.edu
62nd IEEE Conference on Decision and Control, Singapore
Dec. 13-15, 2023

2. Cyber-Physical Systems Under Attack
CPS are subject to attacks, malicious
behaviors, unknown inputs
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 2 / 21

3. Data Attack Resiliency
Research Question
Can we simultaneously obtain guaranteed estimates of states and
unknown inputs (adversarial signals)?
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 3 / 21

4. Resilient Observer Design; State and Input Estimation
xk+1 = f (xk ) + Wwk + Gdk ,
yk = h(xk ) + Vvk + H dk
|{z}
unknown input
no prior ‘useful’ knowledge or
assumption or known bounds
on the dynamics of dk
Problem (Simultaneous Input and State Observer)
Design stable and optimal interval-valued input and state estimator
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 4 / 21

5. Resilient Observer Design; State and Input Estimation
xk+1 = f (xk ) + Wwk + Gdk ,
yk = h(xk ) + Vvk + H dk
|{z}
unknown input
no prior ‘useful’ knowledge or
assumption or known bounds
on the dynamics of dk
Problem (Simultaneous Input and State Observer)
Design stable and optimal interval-valued input and state estimator
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 4 / 21

6. A Glance at Related Work
Resilient Estimation
Linear Systems
▶ [Yong.ea.2018]: residual-based estimation for stochastic systems
▶ [Nakahira.Mo.2018],[Khajenejad.Yong.2019]: set-membership
resilient estimation with multiple-model approach
Nonlinear Systems
▶ [Kim.ea.2018], [Chong.ea.2020]: only sensors are attacked
▶ [Khajenejad.Yong.2021]: both sensors and actuators are
compromised
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 5 / 21

7. Preliminaries: Decomposition Functions
Definition (DT Decomposition Functions (Yang.ea.2019))
x+
t = f (xt, wt): a DT system, f : Z → Rn
fd : Z × Z → Rn: a DT-MMDF with respect to f , if
▶ fd (z, z) = f (z)
▶ ẑ ≥ z ⇒ fd (ẑ, z′
) ≥ fd (z, z′
)
▶ ẑ ≥ z ⇒ fd (z′
, ẑ) ≤ fd (z′
, z)
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 6 / 21

8. Preliminaries: Mixed-Monotone Decompositions
f (x) = Hx
|{z}
linear remainder
+ g(x)
|{z}
JSS mapping
, Hij = J
f
ij ∨ Hij = Jf
ij
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
2
2.5
3
H⊕
x − H⊖
x ≤ Hx ≤ H⊕
x − H⊖
x
g(xc ) ≤ g(x) ≤ g(xc )
max
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
≤ Hx + g(x)
| {z }
f (x)
≤ min
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 7 / 21

9. Preliminaries: Mixed-Monotone Decompositions
f (x) = Hx
|{z}
linear remainder
+ g(x)
|{z}
JSS mapping
, Hij = J
f
ij ∨ Hij = Jf
ij
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
2
2.5
3
H⊕
x − H⊖
x ≤ Hx ≤ H⊕
x − H⊖
x
g(xc ) ≤ g(x) ≤ g(xc )
max
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
≤ Hx + g(x)
| {z }
f (x)
≤ min
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 7 / 21

10. Preliminaries: Mixed-Monotone Decompositions
f (x) = Hx
|{z}
linear remainder
+ g(x)
|{z}
JSS mapping
, Hij = J
f
ij ∨ Hij = Jf
ij
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-2
-1.5
-1
-0.5
0
0.5
1
1.5
2
-1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1
-1
-0.5
0
0.5
1
1.5
2
2.5
3
H⊕
x − H⊖
x ≤ Hx ≤ H⊕
x − H⊖
x
g(xc ) ≤ g(x) ≤ g(xc )
max
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
≤ Hx + g(x)
| {z }
f (x)
≤ min
H∈H
{H⊕
x − H⊖
x + g(xc )}
| {z }
fd (x,x)
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 7 / 21

11. Preliminaries: Embedding Systems
x+
= f (x, w)
| {z }
original n-dimensional system
x+
x+
=
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
| {z }
lifted 2n-dimensional embedding system
Proposition (State Framer Property [Khajenejad.Yong.2021])
xt ≤ xt ≤ xt, ∀t ≥ 0, ∀w ∈ W.
Proposition (Tight Decompositions [Khajenejad.Yong.2021])
if f (z) = µ(z) + Hz such that µ is JSS, then:
µd,i (z1, z2) = µi (Di
z1 + (In − Di
)z2),
δµ
d ≜ µd (z, z) − µd (z, z) ≤ Fµ(z − z)
Di = diag(max(sgn(J
µ
i ), 01,nz )), Fµ ≜ 2 max(Jf − H, 0p,nz )−Jf +H
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 8 / 21

12. Preliminaries: Embedding Systems
x+
= f (x, w)
| {z }
original n-dimensional system
x+
x+
=
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
| {z }
lifted 2n-dimensional embedding system
Proposition (State Framer Property [Khajenejad.Yong.2021])
xt ≤ xt ≤ xt, ∀t ≥ 0, ∀w ∈ W.
Proposition (Tight Decompositions [Khajenejad.Yong.2021])
if f (z) = µ(z) + Hz such that µ is JSS, then:
µd,i (z1, z2) = µi (Di
z1 + (In − Di
)z2),
δµ
d ≜ µd (z, z) − µd (z, z) ≤ Fµ(z − z)
Di = diag(max(sgn(J
µ
i ), 01,nz )), Fµ ≜ 2 max(Jf − H, 0p,nz )−Jf +H
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 8 / 21

13. Preliminaries: Embedding Systems
x+
= f (x, w)
| {z }
original n-dimensional system
x+
x+
=
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
f d ([x⊤ w⊤]⊤, [x⊤ w⊤]⊤)
| {z }
lifted 2n-dimensional embedding system
Proposition (State Framer Property [Khajenejad.Yong.2021])
xt ≤ xt ≤ xt, ∀t ≥ 0, ∀w ∈ W.
Proposition (Tight Decompositions [Khajenejad.Yong.2021])
if f (z) = µ(z) + Hz such that µ is JSS, then:
µd,i (z1, z2) = µi (Di
z1 + (In − Di
)z2),
δµ
d ≜ µd (z, z) − µd (z, z) ≤ Fµ(z − z)
Di = diag(max(sgn(J
µ
i ), 01,nz )), Fµ ≜ 2 max(Jf − H, 0p,nz )−Jf +H
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 8 / 21

14. Preliminaries: Affine Outer Approximation
Proposition (Affine Abstraction [Singh.ea.2018])
f (·) : B ≜ [x, x] ⊂ Rn → Rm, VB: set of
vertices.
(AB, AB, eB, eB, θB) ∈ argmin
θ,A,A,e,e
θ (1)
s.t Axs + e + σ ≤ f (xs) ≤ Axs + e − σ,
(A − A)xs + e − e − 2σ ≤ θ1m, ∀xs ∈ VB,
Then, Ax + e ≤ f (x) ≤ Ax + e, ∀x ∈ B.
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 9 / 21

15. Resilient Interval Observer Synthesis
G :
(
xk+1 = f (xk ) + Wwk + Gdk ,
yk = h(xk ) + Vvk + Hdk
Problem (Input and State Interval Observer)
Synthesize framers xk , xk , dk , dk such that
states inputs are framed: xk ≤ xk ≤ xk , dk ≤ dk ≤ dk
framers are uniformly bounded
design is optimized
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 10 / 21

16. Resilient Interval Observer Synthesis
G :
(
xk+1 = f (xk ) + Wwk + Gdk ,
yk = h(xk ) + Vvk + Hdk
Problem (Input and State Interval Observer)
Synthesize framers xk , xk , dk , dk such that
states inputs are framed: xk ≤ xk ≤ xk , dk ≤ dk ≤ dk
framers are uniformly bounded
design is optimized
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 10 / 21

17. Observer Overview
xk+1 = f (xk ) + Gdk + Wwk ,
yk = h(xk ) + Hdk + Vvk
Key Insights:
dk ⇔ d1,k d2,k ,
yk ⇔ z1,k z2,k
auxiliary state: γk ≜ Λ(I − NC2)xk
▶ unaffected by dk by choosing
N = G2(C2G2)†
▶ Λ ≜ A−1
g with Ag being
affine approximation slope of
xk + Nψ2(x)
Λ(I − NC2)(f (x) − G1Sh1(x)) =
Ax + ρ(x)
| {z }
mixed-monotone decomposition
L(z2,k − C2xk − ψ2(xk ) − V2vk ) = 0
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 11 / 21

18. Observer Overview
xk+1 = f (xk ) + G1d1,k + G2d2,k + Wwk ,
z1,k = h1(xk ) + Ξd1,k + V1vk
z2,k = C2xk + ψ2(xk )
| {z }
mixed-monotone decomposition
+V2vk
Key Insights:
dk ⇔ d1,k d2,k ,
yk ⇔ z1,k z2,k
auxiliary state: γk ≜ Λ(I − NC2)xk
▶ unaffected by dk by choosing
N = G2(C2G2)†
▶ Λ ≜ A−1
g with Ag being
affine approximation slope of
xk + Nψ2(x)
Λ(I − NC2)(f (x) − G1Sh1(x)) =
Ax + ρ(x)
| {z }
mixed-monotone decomposition
L(z2,k − C2xk − ψ2(xk ) − V2vk ) = 0
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 11 / 21

19. Observer Overview
xk+1 = f (xk ) + G1d1,k + G2d2,k + Wwk ,
z1,k = h1(xk ) + Ξd1,k + V1vk
z2,k = C2xk + ψ2(xk )
| {z }
mixed-monotone decomposition
+V2vk
Input
Framers
dk−1,dk−1
Auxiliary
State
Framers
γk
, γk
State
Framers
xk , xk
Start x0, x0
Measurement
yk
k ← k + 1
k = 1
Recursive
algorithm:
Key Insights:
dk ⇔ d1,k d2,k ,
yk ⇔ z1,k z2,k
auxiliary state: γk ≜ Λ(I − NC2)xk
▶ unaffected by dk by choosing
N = G2(C2G2)†
▶ Λ ≜ A−1
g with Ag being
affine approximation slope of
xk + Nψ2(x)
Λ(I − NC2)(f (x) − G1Sh1(x)) =
Ax + ρ(x)
| {z }
mixed-monotone decomposition
L(z2,k − C2xk − ψ2(xk ) − V2vk ) = 0
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 11 / 21

20. 3-Step Recursive Observer
assumption: C2G2 has a full column rank
d1,k = S(z1,k − h1(xk ) − V1vk ), S = Ξ−1
d2,k = (C2G2)†C2(xk+1 − f (xk ) + G1Sh1(xk ) + G1S(V1vk − z1,k ) − Wwk )
dk = E1d1,k + E2d2,k
⇓
Step 1: Input Framer Computation
dk−1 = Φ⊕xk − Φ⊖xk + κd (xk−1, xk−1) + Az z1,k−1
+A⊕
v v − A⊖
v v + Φ⊖w − Φ⊕w,
dk−1 = Φ⊕xk − Φ⊖xk + κd (xk−1, xk−1) + Az z1,k−1
+A⊕
v v − A⊖
v v + Φ⊖w − Φ⊕w,
Φ ≜ E2(C2G2)†C2, Av ≜ (ΦG1 − E1)SV1, Az ≜ (E1 − ΦG1)S
κ(x) ≜ (ΦG1 − E1)Sh1(x) − Φf (x)
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 12 / 21

21. 3-Step Recursive Observer



























γk ≜ Λ(I − NC2)xk = xk − Λ(N(z2,k − V2vk ) − ϵk )
N = G2(C2G2)† → cancel out the effect of dk
L(z2,k − C2xk − ψ2(xk ) − V2vk ) = 0
˜
f (x) ≜ Λ(I − NC2)(f (x) − G1Sh1(x)) = Ax + ρ(x)
| {z }
mixed-monotone decomposition
⇒
γk+1 = (A − LC2)γk + ρ(xk ) − Lψ2(xk ) − V̂ vk + Ŵ wk − Dϵk + ẑk
⇓
Step 2. Auxiliary State Propagation
γk+1
= (A − LC2)⊕γk
− (A − LC2)⊖γk + ρd (xk , xk )
+D⊖ϵ − D⊕ϵ + L⊖ψ2,d (xk , xk ) − L⊕ψ2,d (xk , xk )
+V̂ ⊖v − V̂ ⊕v + Ŵ ⊖w −Ŵ ⊖w + ẑk ,
γk+1 = (A − LC2)⊕γk − (A − LC2)⊖γk
+ ρd (xk , xk )
+D⊖ϵ − D⊕ϵ + L⊖ψ2,d (xk , xk ) − L⊕ψ2,d (xk , xk )
+V̂ ⊖v − V̂ ⊕v + Ŵ ⊕w − Ŵ ⊖w + ẑk ,
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 13 / 21

22. 3-Step Recursive Observer
xk = γk + Λ(N(z2,k − V2vk ) − ϵk )
⇓
Step 3. State Framer Computation
xk = γk
+ ΛNz2,k + Λ⊖ϵ − Λ⊕ϵ + (ΛNV2)⊖v − (ΛNV2)⊕v,
xk = γk + ΛNz2,k + Λ⊖ϵ − Λ⊕ϵ + (ΛNV2)⊖v − (ΛNV2)⊕v,
by construction and properties of decomposition functions:
(
xk ≤ xk ≤ xk
dk ≤ dk ≤ dk
, for all L.
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 14 / 21

23. H∞-Optimal State and Input Observer Design
to design L to satisfy stability and optimality
Error Dynamics
ex
k+1 ≜ xk − xk = ≤ (|A − LC2| + Fρ + |L|Fψ2
)ex
k + |Ŵ |δw
+(|Va − LVb| − |A − LC2||ΛNV2| + |ΛNV2|)δv
+(|Λ| + |Da − LDb| − |A − LC2||Λ|)δϵ,
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 15 / 21

24. H∞-Optimal State and Input Observer Design
Theorem (ISS H∞-Observer Design)
f , h have bounded Jacobians, C2G2 has full column rank
if ψ2(x) ̸= 0 then Ag is invertible
(P∗ ≻ 0, η∗ 0, Γ∗ ≥ 0) is a solution to
min
{η,P,Γ}
η
s.t.




P PÃ − ΓC̃ PB̃ − ΓD̃ 0
∗ P 0 I
∗ ∗ ηI 0
∗ ∗ ∗ ηI



≻0, (P, Γ) ∈ C, −P ∈ Mn
then the observer is ISS and optimal with L = (P∗)−1Γ∗
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 16 / 21

25. H∞-Optimal State and Input Observer Design
Theorem (ISS H∞-Observer Design)
f , h have bounded Jacobians, C2G2 has full column rank
if ψ2(x) ̸= 0 then Ag is invertible
(P∗ ≻ 0, η∗ 0, Γ∗ ≥ 0) is a solution to
min
{η,P,Γ}
η
s.t.




P PÃ − ΓC̃ PB̃ − ΓD̃ 0
∗ P 0 I
∗ ∗ ηI 0
∗ ∗ ∗ ηI



≻0, (P, Γ) ∈ C, −P ∈ Mn
then the observer is ISS and optimal with L = (P∗)−1Γ∗
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 16 / 21

26. Different Choices for the Set C
(i) C={(P, Γ) | P
A Va Da
−Γ
C2 Vb Db
≥ 0}, if:
à = A + Fρ, C̃ = C2 − Fψ2
,
B̃ =
Va + (I − A)|ΛNV2| |Ŵ | Da + (I − A)|Λ|
,
D̃ =
Vb − C2|ΛNV2| 0 Db − C2|Λ|
.
(ii) C = {(P, Γ) | Γ
C2 Vb Db
≥ 0}, if
à = |A| + Fρ, C̃ = −C2 − Fψ2
,
B̃ =
|Va|+(I−|A|)|ΛNV2| |Ŵ | (I−|A|)|Λ|+|Da|
,
D̃ =
C2|ΛNV2| − Vb 0 C2|Λ| − Db
.
(iii) C = {(P, Γ) | PA − ΓC2 ≥ 0}, if:
à = A + Fρ, C̃ = C2 − Fψ2
, D̃ =
−V2 0 0
,
B̃ =
|Λ(I−NC2)G1SV1|+|ΛNV2| |Ŵ | |Λ|
.
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 17 / 21

27. Simulation Results: A Three-Area Power Station
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 18 / 21

28. Takeaway
leveraged mixed-monotone decompositions and affine
outer-approximations for resilient estimation
designed state and input interval observers
the design is correct-by-construction
derived sufficient conditions for stability and optimality
future work:
▶ alternative designs for minimizing L1 gain
▶ continuous-time and hybrid systems
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 19 / 21

29. References
L. Yang, O. Mickelin, and N. Ozay. “On sufficient conditions for mixed monotonicity”. IEEE Transactions
on Automatic Control, 64(12):5080–5085, 2019
M. Khajenejad, Z. Jin, and S.Z. Yong. “Interval observers for simultaneous state and model estimation of
partially known nonlinear systems”. In American Control Conference, pages 2848–2854. IEEE, 2021.
M. Khajenejad and S. Z. Yong. “Simultaneous mode, input and state set-valued observers with
applications to resilient estimation against sparse attacks”. In IEEE Conference on Decision and Control
(CDC), pages 1544–1550, 2019.
S.Z. Yong, M. Zhu, and E. Frazzoli. “Switching and data injection attacks on stochastic cyber-physical
systems: Modeling, resilient esti- mation, and attack mitigation”. ACM Transactions on Cyber-Physical
Systems, 2(2):9, 2018.
Y. Bar-Shalom, T. Kirubarajan, and X.R. Li. “Estimation with Applications to Tracking and Navigation”.
John Wiley Sons, Inc., New York, NY, USA, 2002.
Y. Nakahira and Y. Mo. “Attack-resilient H2 , H∞ , and ℓ11 state estimator”. IEEE Transactions on
Automatic Control, 63(12):4353–4360, 2018.
J. Kim, C. Lee, H. Shim, Y. Eun, and J.H. Seo. “Detection of sensor attack and resilient state estimation
for uniformly observable nonlinear systems having redundant sensors”. IEEE Transactions on Automatic
Control, 64(3):1162–1169, 2018.
M.S. Chong, H. Sandberg, and J.P. Hespanha. “A secure state estimation algorithm for nonlinear systems
under sensor attacks”. In IEEE Conference on Decision and Control, pages 5743–5748, 2020.
M. Milanese and C. Novara. “Set membership identification of nonlinear systems”. Automatica,
40:957–975, 2004.
Z.B. Zabinsky, R.L. Smith, and B.P. Kristinsdottir. “Optimal estimation of univariate black-box Lipschitz
functions with upper and lower error bounds”. Computers Operations Res., 30(10):1539–1553, 2003.
Z. Jin, M. Khajenejad, and S.Z. Yong. “Data-driven model invalidation for unknown Lipschitz continuous
systems via abstraction”. In American Control Conference (ACC), pages 2975–2980. IEEE, 2020.
K.R. Singh, Q. Shen, and S.Z. Yong. “Mesh-based affine abstraction of nonlinear systems with tighter
bounds”. In IEEE Conference on Decision and Control (CDC), pages 3056–3061, 2018.
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 20 / 21

30. Thank you!
Questions?
M. Khajenejad, Z. Jin, T.N. Dinh, S. Z. Yong Resilient State Estimation Dec., 13, 2023 21 / 21