An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against

Cyberattacks: Application to Vehicle Cruise Control

Milad Farsi

, Shuhao Bian

, Nasser L. Azad

, Xiaobing Shi

and Andrew Walenstein

Department of Systems Design Engineering, University of Waterloo, Waterloo, Canada

BlackBerry Limited, Waterloo, Canada

Keywords:

Resilient Control, Robust Control, Model Predictive Control, Denial of Service Attack.

Abstract:

We propose a novel framework for designing a resilient Model Predictive Control (MPC) targeting uncertain

linear systems under cyber attack. Assuming a periodic attack scenario, we model the system under Denial of

Service (DoS) attack, also with measurement noise, as an uncertain linear system with parametric and additive

uncertainty. To detect anomalies, we employ a Kalman ﬁlter-based approach. Then, through our observations

of the intensity of the launched attack, we determine a range of possible values for the system matrices, as

well as establish bounds of the additive uncertainty for the equivalent uncertain system. Leveraging a recent

constraint tightening robust MPC method, we present an optimization-based resilient algorithm. Accordingly,

we compute the uncertainty bounds and corresponding constraints ofﬂine for various attack magnitudes. Then,

this data can be used efﬁciently in the MPC computations online. We demonstrate the effectiveness of the

developed framework on the Adaptive Cruise Control (ACC) problem.

1 INTRODUCTION

Resilient control refers to the capability of a control

system to maintain stable and optimal performance

despite cyber-attacks (Sandberg et al., 2022), distur-

bances, uncertainties, and faults. Traditional con-

trol systems assume ideal conditions, leading to per-

formance degradation or failure during unexpected

events. Resilient controllers enhance robustness and

adaptability, especially against cyber-attacks in criti-

cal systems. In the context of modern vehicles vul-

nerable to cyber threats, successful attacks can cause

loss of control, safety compromises, and harm to pas-

sengers (Ju et al., 2022). Resilient control techniques

detect, mitigate, and recover from cyber-attacks, pre-

serving vehicle functionality and safety in adverse

conditions.

Denial of Service (DoS) as one of the well-known

cyber attacks have become increasingly prevalent in

today’s digital landscape that can gravely affect mod-

ern vehicle systems (Biron et al., 2018). These attacks

aim to disrupt or disable the targeted system’s services

or resources, making them unavailable to legitimate

users. Therefore, different techniques are employed

in the literature to mitigate potential damages caused

by such attacks. Game theory provides a frame-

work for modeling strategic interactions and decision-

making processes during cyber attacks (Gupta et al.,

2016; Huang et al., 2020). Moreover, event-triggered

control methods have been popular considering their

advantages in cyber-physical systems, including vehi-

cle control (Xiao et al., 2020; Wu et al., 2022).

Robust Model Predictive Control (RMPC) as a

subcategory of Model Predictive Control (MPC) tech-

niques is a powerful control framework that excels

at handling uncertainty and disturbance in real-world

applications (Bemporad and Morari, 2007). The in-

herent ability of RMPC to explicitly account for un-

certainties makes it particularly well-suited for com-

plex systems operating in dynamic environments.

RMPC encompasses various approaches to handle un-

certainties and disturbances in control systems. Min-

Max RMPC approaches formulate the control prob-

lem as a min-max optimization, where the objective

is to minimize online the worst-case performance sub-

ject to constraints (Raimondo et al., 2009). However,

these techniques can involve overly expensive com-

putations. Tube-based RMPC constructs an invariant

set, known as the robust tube, that captures the possi-

ble system trajectories considering the uncertainties

(Langson et al., 2004; Sakhdari and Azad, 2018).

By formulating the optimization problem within this

tube, the system stability and constraint satisfaction

are obtained. In (Mayne et al., 2005), the authors

674

Farsi, M., Bian, S., Azad, N., Shi, X. and Walenstein, A.

An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against Cyberattacks: Application to Vehicle Cruise Control.

DOI: 10.5220/0012190900003543

In Proceedings of the 20th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2023) - Volume 1, pages 674-682

ISBN: 978-989-758-670-5; ISSN: 2184-2809

Figure 1: A view of the Adaptive Cruise Control (ACC) problem.

address RMPC problem in the presence of bounded

disturbances for constrained linear discrete-time sys-

tems.

In (Aubouin-Pairault et al., 2022), to address the

resilience issue in maintaining system operation un-

der repeated DoS attacks, the concept of µ-step Ro-

bust Positively Invariant (µ-RPI) sets is introduced.

These sets aim to restrict the impact of attacks, en-

suring that any deviation from nominal operation re-

mains limited in time and/or magnitude. Although

such approaches offer different perspectives on ro-

bust control design and enable the handling of uncer-

tainties and disturbances efﬁciently, they may result

in rather conservative and computationally expensive

solutions.

Constraint-tightening techniques involve itera-

tively reﬁning the constraints in an optimization prob-

lem to enforce robustness (K

ohler et al., 2018). In

(Bujarbaruah et al., 2021), the authors demonstrated

that by selecting appropriate terminal constraints and

employing an adaptive horizon strategy, constraint

tightening may not necessarily result in excessively

conservative behavior where its Region of Attraction

(ROA) can be as large as 98% of the tube-based tech-

niques, such as (Langson et al., 2004). More impor-

tantly, it can run 15x faster.

Building upon this RMPC approach, in this paper,

we develop an efﬁcient resilient control framework

against a class of cyber-attacks that can be potentially

employed in real-time as an alternative to the current

MPC implementations. Our approach involves the

computation of a set of uncertain models that encom-

passes different levels of the strength of DoS attacks,

as well as accounting for potential noise and unmod-

eled dynamics. Through an iterative scheme, we em-

ploy the Kalman ﬁlter to detect the occurrence of an

attack. Subsequently, in the control loop, we estimate

the intensity of the attack and adaptively evaluate the

control based on the models pre-computed for differ-

ent circumstances speciﬁcally. Hence, our contribu-

tions include: obtaining an overapproximate model

with additive and parametric uncertainties based on a

practical problem formulation, developing a resilient

control framework that can be employed as an exten-

sion of (Bujarbaruah et al., 2021) against DoS, and

validating it on the Adaptive Cruise Control (ACC)

problem, illustrated in Fig.1.

The rest of the paper is organized as follows.

In Section 2, we formulate the problem. Section 3

presents the resilient control framework and outlines

the algorithms designed based on the obtained re-

sults. In Section 4, we summarize a commonly used

anomaly detection method. In Section 5, we present a

case study to validate and compare the proposed ap-

proach in the simulation environment.

Notations. We denote n-dimensional Euclidean

space by R

, and the space of positive reals with the

subscript as R

. We further denote by |X | the abso-

lute value of a variable X , where for non-scalars, it

represents the component-wise absolute value. ||x||

denotes the 2-norm of a vector x ∈ R

. A diago-

nal square matrix A with elements a

,... ,a

on the

diagonal is shortened as A = diag([a

,. .. ,a

]). We

use the upper script as x

for discrete-time signals,

and x

k| j

represents the estimation of x

at the time j.

diag(x

,..., x

) denotes a diagonal matrix of the el-

ements x

,..., x

. The set G[a, b] represents a grid

with the bounds a and b.

2 PROBLEM FORMULATION

In this section, considering a class of systems, we for-

mulate the effect of the DoS attack. Accordingly, we

deﬁne the problem of interest.

Consider the following system

˙x = Ax + ∆(x) + Bu (1)

where x ∈ D ⊂ IR

and u ∈ U ⊂ IR

are respectively

the state and control input, and take values on the

compact convex sets D and U. System matrices are

given as A ∈ IR

n×n

, and B ∈ IR

n×m

. Furthermore, the

unmodeled dynamics are given by ∆ : D → IR

Assumption 1. Elements of ∆(x) are assumed to be

a Lipschitz continuous function of the state, i.e. ∃η ∈

such that we have

|∆

) − ∆

)| ≤ η

∥x

− y

∥,

for any x

∈ D, where i ∈ {1,. .. ,n}.

An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against Cyberattacks: Application to Vehicle Cruise Control

675

While more general classes of dynamical systems

do exist, the speciﬁc formulation we adopt in this

study enables efﬁcient analysis of a wide range of en-

gineering problems, encompassing domains such as

robotics, automotive, power systems, and more. This

formulation can be also applied to the ACC system,

which is the focus of our investigation in this paper.

In the subsequent section, we will proceed to model

the impact of a DoS attack on this particular system.

2.1 Attack Model

Regarding the fact that the attacks are implemented in

the cyber layer, one needs to take into account the in-

teractions in the discrete space. Therefore, let us con-

sider the Euler approximation of (1) under actuator

attack and with measurement noise as the following

(

t+1

= A

+ ∆(x

)τ + Bτu

+ d

, t /∈ α

t+1

= A

+ ∆(x

)τ + d

, t ∈ α

(2)

= C(I + χ

+ ε

, t ∈ {0, 1,. .. } (3)

where τ is the sampling time and A

= Aτ + I and

= Bτ are discretized system matrices. Moreover,

we assume that we can measure all the states, i.e.

C = I. Then, the term d

lumps together the dis-

cretization error and some bounded input disturbance.

Moreover, we denote the set of time steps during

which the DoS attack is active using α ∈ S

, where

represents the set of all possible sequences of at-

tacks.

Moreover, to ensure a rather practical model of

the problem we take also into account the effect of

noise. Therefore, the measurements are given by

∈ IR

for steps t ∈ {0,1,. .. } that are affected by

noise. The vector ε

∈ IR

and the diagonal matrix

χ = diag([χ

,. .. ,χ

]) ∈ IR

n×n

are the bounded addi-

tive and multiplicative noises affecting the measure-

ments, i.e |ε

| ≤

ε ∈ IR

and |χ

| ≤

χ, where

χ is a di-

agonal matrix with positive values. The distributions

of the noises applied are not necessarily uniform. In

fact, the formulation can accommodate other distribu-

tions, such as truncated Gaussian noise.

2.2 Objective

Given the deﬁnition of the system under DoS attack,

we can deﬁne the constrained control problem which

is solved at each time step t in the rolling horizon fash-

ion for the horizon length of N as

∗

= min

(.)

t+N−1

∑

k=t

+ u

) + y

subject to (2), (3),

∈ D, and u

∈ U, (4)

for all sequences of attacks α ∈ S

, noises, and dis-

turbances within their sets of deﬁnitions. The objec-

tive deﬁned includes stage and terminal costs, respec-

tively.

Addressing the uncertainties inherent in the

model, it becomes apparent that a direct approach to

solving the optimization problem is not viable. In

light of this, the subsequent section explores an alter-

native technique that can effectively handle the prob-

lem by transforming it into the standard form, incor-

porating parametric and additive uncertainties. This

approach capitalizes on the existence of efﬁcient tech-

niques speciﬁcally designed to tackle such formula-

tions.

3 RESILIENT FRAMEWORK

In this section, we present the components required

for establishing the proposed resilient control in de-

tail. Having the model of the attack deﬁned, we ﬁrst

derive an equivalent uncertain model that facilitates

efﬁcient analyses. Second, we present the tightening-

based solutions for addressing this problem. Finally,

we summarize the entire framework by presenting

two algorithms that encapsulate the proposed resilient

control approach.

3.1 Equivalent Uncertain Model

The following lemma provides regulation for the

lumped disturbance present in the model.

Lemma 1. The disturbance term d

is bounded by

d ∈ IR

Proof. Considering the Assumption 1 and compact

domains, it can be shown that the local truncation er-

ror resulting from the discretization remains bounded

for all (x

) ∈ D×U. Moreover, according to our as-

sumption, the system may be prone to some bounded

input disturbance. Therefore, the lumped disturbance

is also bounded by some

d ∈ IR

Assuming a periodic DoS attack, as a well-known

class of attacks (Cetinkaya et al., 2019), we can

rewrite the system by averaging both modes of (2) as

t+1

= A

+ ∆(x

)τ + B

(1 − ω

) + d

, (5)

where ω

∈ [0, 1] takes continuous values in this

closed interval representing the intensity of the DoS

attack.

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

676

Assumption 2. The attack signals ω

is bounded and

the estimated values of the upper bounds are known

at each time step, i.e. ∃

ω ∈ IR

such that |ω

| ≤

ω for

k ∈ {0,...,N − 1}.

Assumption 2 automatically holds for the type of

attack considered and the problem formulation where

a worst case of ω

values is given by 1. However, in

practice, based on the estimations of the attack inten-

sity, smaller values than 1 may be considered for

ω at

each step t.

In what follows, we investigate how the measure-

ments deviate from the predictions given by the nom-

inal dynamics

F(x

) = A

+ B

Therefore, consider

t+1

−

F(x

)

= (I + χ

t+1

+ ε

t+1

− A

− B

= (I + χ

t+1

)



+ ∆(x

)τ + B

(1 − ω

) + d



+ ε

t+1

− A

− B

= χ

t+1



(I + χ

t+1

)(1 − ω

) − I



+ (I + χ

t+1

)



∆(x

)τ + d



+ ε

t+1

= χ

t+1

+ (χ

t+1

− (I + χ

t+1

)ω

+ (I + χ

t+1

)∆(x

)τ + (I + χ

t+1

+ ε

t+1

(6)

where we used (5) in the derivations. Starting with

the ﬁrst two terms, let us deﬁne the convex polytopic

sets Π

and Π

as below that contain the uncertainty

corresponding to A

and B

matrices of the nominal

dynamic, respectively,

= conv({χ

t+1

|χ

t+1

∈ χ

}),

= conv({(χ

t+1

− (I + χ

t+1

)ω

|χ

t+1

∈ χ

∈ {0,

ω}}), (7)

for all vertices χ

given by the extreme values of χ

t+1

Regarding that

χ is diagonal, χ

can be easily calcu-

lated.

In the subsequent step, the remaining terms are

treated as additive uncertainty. It is important to high-

light that, in order to obtain speciﬁc bounds for each

system dynamic speciﬁcally, component-wise calcu-

lations are employed, rather than considering a norm-

based approach. In this regard, the non-scalar bounds

deﬁned and the Lipschitz constants in Assumption 1

facilitate these computations. Hence, we aim for a

bound using equation (6) that yields

t+1

− (

F(x

) + χ

t+1

+ (χ

t+1

− (I + χ

t+1

)ω

= |(I + χ

t+1

)∆(x

)τ + (I + χ

t+1

+ ε

t+1

≤ |(I + χ

t+1

)∆(x

)τ| + |(I + χ

t+1

| + |ε

t+1

≤ |(I + χ

t+1

)τ||∆(x

)| + |(I + χ

t+1

| + |ε

t+1

≤ |(I + χ

t+1

)τ|η∥x

∥ + |(I + χ

t+1

| + |ε

t+1

≤ |(I +

χ)τ|∥x

∥η + (I +

χ)

d +

ε, (8)

where we used Assumption 1 to derive the last two

steps. This provides a bound for the remaining terms

while one can use max

∈D

(∥x

∥) to bound ∥x

∥. However,

it may not offer a useful bound if η is not small. As an

alternative, we can set different values for the bounds

based on the current value of x

, instead. In this case,

considering that we do not measure x

exactly, we can

employ the measurements through (3) to obtain

∥x

∥ = ∥(I + χ

t+1

)

−1

∥∥(y

− ε

t+1

)∥

≤ ∥(I + χ

t+1

)

−1

∥(∥y

∥ + ∥ε

t+1

∥)

≤ ∥(I −

χ)

−1

∥(∥y

∥ + ∥

ε∥). (9)

We summarize the computations by utilizing a

discrete-time linear model that incorporates paramet-

ric and additive uncertainty. This model serves as an

overapproximation of the continuous-time dynamics

(1) in the presence of a DoS attack and uncertainty,

t+1

= (A

∆

+ (B

∆

, (10)

where |

| ≤

δ with

δ = |(I +

χ)τ|∥(I −

χ)

−1

∥(∥y

∥ + ∥

ε∥)η

+ (I +

χ)

d +

ε, (11)

∆

∈ Π

and

∆

∈ Π

, with deﬁned Π

and Π

(7).

3.2 Resilience via Constraint Tightening

In this section, we summarize the constraint tighten-

ing technique employed for solving the RMPC prob-

lem. Accordingly, we deliver the resilient framework

proposed using also the results obtained in the previ-

ous section.

Regarding the model in (10), although utilizing

ﬁxed bounds can be effective for addressing slight

model uncertainty and noise effects, it may not ad-

equately capture the impact of DoS attacks, which

is considered the major source of uncertainty in the

model. To address this, we propose a more adaptable

approach that can accommodate various strengths of

attacks while maintaining satisfactory system perfor-

mance. Moreover, as previously suggested, the Lip-

schitz values may be large leading to large additive

bounds in (11) that can be also addressed by a similar

approach.

An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against Cyberattacks: Application to Vehicle Cruise Control

677

In order to overcome these limitations, let us de-

ﬁne different quantities of bounds for ω

and ∥y

∥

as [ω]

and [∥y∥]

that are taken from a set of grid

points, i.e. ([ω]

,[∥y∥]

) ∈ G[0,

ω] × G[0,sup(∥y∥)]

for j = 1, .. ., N

and i = 1,. .. ,N

, where N

and

are the numbers of grid points for each dimen-

sion. Then, by online observations, one needs to en-

sure that the conditions ω

≤ [ω]

and ∥y

∥ ≤ [∥y∥]

hold by choosing a suitable q from the set of indices

{1,. .. ,N

× N

By employing a similar scheme as (Bujarbaruah

et al., 2021), we consider an adaptive prediction hori-

zon where at each time step, we solve the problem

for different horizon lengths N

∈ {1, .. ., N}, and pro-

ceed with the one with the least cost. However, there

is a key distinction in our approach as we take into

account a collection of uncertain models, which are

deﬁned by different bounds corresponding to varying

levels of attack intensity. Accordingly, we apply one

of the following two approaches in handling the un-

certainties depending on the value of N

• Case N

= 1: Accordingly, the robust MPC prob-

lem is exactly solved for a horizon length of one.

For this purpose, assuming that there exists a feed-

back gain K

such that (A

∆

) + (B

∆

is stable for all

∆

∈ Π

and

∆

∈ Π

, we can

construct the terminal sets X

as the maximal ro-

bust positive invariant set for x

t+1



∆



, with q ∈ {1, .. ., N

× N

Therefore, in addition to the state and control in-

put constraints, we need to satisfy the condition



∆

) + (B

∆



∈ X

(12)

in the optimization problem, where X

is a convex

set deﬁning the terminal set for the model given

by the index q. It should be noted that, ∆

and

| ≤

are characterized by the quantities [ω]

and [∥y∥]

for given index q, according to rela-

tions (7) and (11).

• Case N

> 1: Given the computational intensity

of the method employed in the previous case for

multi-step predictions, an alternative approach is

taken. Bounds are utilized to over-approximate

system uncertainty, rather than precise calcula-

tions by using the technique found in (Goulart

et al., 2006). This allows for the treatment of all

uncertainties as a net-additive component, utiliz-

ing a more constructive technique. The adoption

of this approach aims to mitigate the computa-

tional burden while still effectively accounting for

system uncertainties.

The presented resilient framework can be effectively

implemented in two parts. In the ﬁrst part, the model

and uncertainty bounds are processed to character-

ize the constraints. These computations are con-

ducted ofﬂine in advance which facilitates the prepa-

ration of constraints. By performing these com-

putations beforehand, the constraints can be readily

available for subsequent utilization. In Algorithm

1, which presents the ofﬂine procedure, we grid the

space G[0,

ω] × G[0,sup(∥y∥)] to obtain different val-

ues [ω]

and [∥y∥]

. Then, we use (7) and (11) to

calculate the corresponding bounds for all q.

Data: System matrices,

χ,

ε, domain and

control constraints ← D and U

Result:

∆

, and

Terminal sets X

∀([ω]

,[∥y∥]

)

and weights Q

% Number of grid points

and N

← positive integers ;

% Grid points

List

← linspace(0,

ω,N

);

List

← linspace(0, sup(∥y∥),N

);

%Using relation (7):

Calculate

∆

;

for ([ω]

,[∥y∥]

) in ω

List

×Y

List

%Using relation (7):

Calculate

∆

;

% Using relation (11):

Calculate

;

% Employing (Bujarbaruah et al., 2021):

Calculate the terminal sets X

;

Calculate Q

;

end

Algorithm 1: Ofﬂine computations of the bounds and state

constraints.

The second part of the implementation involves

the utilization of the pre-determined constraints in an

online optimization-based control approach. During

the online phase, these prepared constraints are incor-

porated into an optimization framework to generate

real-time control actions by exploiting (Bujarbaruah

et al., 2021). For this purpose, we obtain an esti-

mation of the intensity of an ongoing attack using

an anomaly detection method and choose applicable

[ω]

and [∥y∥]

. By integrating their corresponding

constraints into the optimization process, the control

approach ensures that the system operates within de-

sired limits while effectively addressing uncertainties.

This procedure is summarized in Algorithm 2.

Remark 1. The ofﬂine computations enable the ef-

ﬁcient characterization of constraints, resulting in

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

678

computational time savings during online implemen-

tation. Furthermore, using a hybrid technique em-

ploying the two cases according to N

facilitates a re-

sponsive optimization process, thereby potentially en-

abling realtime resilient control actions in real-world

applications.

Data: A

, B

∆

, and

∀([ω]

,[∥y∥]

∀([ω]

,[∥y∥]

) and weights Q

Result: u

%initialization

N ← positive integer;

for t = 0, 1,. .. do

Measure y

;

Detect attack;

Estimate

based on attack detected;

Choose ([ω]

,[∥y∥]

) based on

and y

;

for N

= 1, 2,. .. ,N do

Solve RMPC for

∆

, X

;

end

Apply u

: arg min

∗

;

end

Algorithm 2: Online computation of resilient control value.

4 ANOMALY DETECTION

To implement RMPC, we need to observe the inten-

sity of the launched attack. Therefore, the Kalman

ﬁlter (Bai et al., 2017; Bai and Gupta, 2014) tech-

nique is utilized to detect the launched attack. Ac-

cordingly, the system states are estimated, and the re-

sulting residuals are employed to detect the attack.

The residual at tth step is deﬁned as

r = y

− y

t|t−1

(13)

s.t.

r = y

−C ˆx

t|t−1

. (14)

Then, according to (Mo et al., 2010; Miao et al.,

2014), we can determine if the system is under attack.

(

t /∈ α, if r

−1

r ≤ T,

t ∈ α, if r

−1

r > T,

(15)

where T ∈ IR

is the threshold of the corresponding

residue, to be tuned by the user. Moreover, P repre-

sents the covariance matrix of the residue r, and T is

the threshold. According to (15), we can determine

whether the system is under attack.

In the next section, the detection results are pre-

sented in more detail. For improved results, robust

Kalman ﬁlters (Elsisi et al., 2023) can be exploited

alternatively.

5 SIMULATION RESULTS

To demonstrate the efﬁcacy of the proposed approach

under DoS attacks, we validate Algorithm 1 and 2 on

the ACC problem. Moreover, we present the details of

the detection procedure and discuss some comparison

results. All the simulations are conducted in Matlab

on the Windows operating system with the hardware

conﬁguration of AMD Ryzen 9, 16-Core, 3.40 GHz,

and 64GB of RAM.

As shown in Fig. 1, ACC system consists of two

vehicles, one of which is the ego vehicle and the other

is the lead vehicle. The control objective deﬁned for

the ego vehicle is to maintain the distance from the

lead vehicle while satisfying the state and control con-

straints.

To describe the model, we employ the state vari-

ables x = [δd,δv, ˙v

]

deﬁned as the followings:

• δd is deﬁned as the distance error which is the

difference between the actual distance d and the

desired distance d

from the lead vehicle, i.e δd =

d − d

• δv denotes the velocity difference between the

lead vehicle v

and the ego vehicle v

• ˙v

represents the acceleration of the ego vehicle.

According to (Takahama and Akasaka, 2018; Al-

Gabalawy et al., 2021), the longitudinal dynamics of

the ego vehicle is given as

˙v

= A

+ B

= C

(16)

where a

is the traction force of the vehicle converted

to acceleration

= −

eng

, B

= −

eng

, and C

= 1.

(17)

Furthermore, T

eng

is the constant of acceleration

of the engine, and K

eng

is the gain of the engine.

In addition, the reference distance is deﬁned with

respect to the velocity of the ego vehicle using

= T

+ d

(18)

where T

is the constant time headway, and d

is the

safety clearance when the lead vehicle comes to a full

stop. However, for simplicity, it is assumed that the

lead vehicle has some positive velocity, hence, d

= 0

can be used safely.

According to (16), (18), and the deﬁned state vec-

tor x, we can obtain the discrete-time model as (2)

with A

= I +





0 1 −T

0 0 −1

0 0 A





τ, B









τ and

C =





1 0 0

0 1 0

0 0 1





An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against Cyberattacks: Application to Vehicle Cruise Control

679

For more details, we refer the readers to (Taka-

hama and Akasaka, 2018; Al-Gabalawy et al., 2021).

Also, similar to these works, we set the values of the

parameters T

, T

eng

, and K

eng

, as shown in Table 1.

Table 1: Parameters of ACC model.

eng

1.6 0.46 sec 0.732

5.1 Detection

In the simulated attack scenario, we generate a peri-

odic attack characterized by varying active lengths for

each period. Fig. 2 illustrates the attack signal proﬁle

that initiates at t = 2 seconds.

To detect this attack, we employ the Kalman

ﬁlter approach, with noise covariances Q

diag(0.01,0.01,0.01), R = diag(0.01,0.01,0.01), the

sampling time t

sample

= 0.01, and the initial covari-

ance matrix P

init

= diag(0.1,0.1,0.1). The residuals

obtained from the detection process are also depicted

in the same ﬁgure, showcasing the impact of the at-

tack events. By appropriately setting threshold val-

ues, we are able to successfully detect the attack, as

demonstrated in Fig. 2. It is important to note that

while some false positive and negative cases may oc-

cur, the Kalman ﬁlter detector is effective in detecting

DoS attacks in the majority of instances.

The bottom graph in Fig. 2 shows the results for

the estimation of attack intensity, i.e.

, together

with the exact signal ω

, which is obtained based on

the true attack signal. In fact, we use a backward-

moving average of the attack signal to generate such

an estimation and it illustrates how effectively one can

reconstruct a signal representing the attack intensity.

By comparing the estimated attack strength

with

the exact signal, we observe a close correspondence

between the two. This demonstrates the ability of the

estimation process to capture and reconstruct the fea-

tures of the attack signal that can be used to establish

the resilient controller accordingly.

5.2 Control

For validation, we apply the resilient control approach

proposed on the ACC system speciﬁed. For the con-

trol computations, we consider a discretized system

τ = 0.2 sec. To characterize the objective function

(4), we set Q = diag(10,10,10), R = 2, and N = 5

for all simulation if not mentioned otherwise. More-

over, the state and control are constrained within in-

tervals [−100,−100,−100]

< x

< [10, 100,100]

and −20 < u

< 20, respectively.

0 2 4 6 8 10 12 14 16 18 20

0.5

DoS Attack

0 2 4 6 8 10 12 14 16 18 20

-0.02

-0.01

0.01

Residual

Attack Launch

0 2 4 6 8 10 12 14 16 18 20

0.5

Attack

Detected

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

0.5

Estimation

Exact

Estimated

Figure 2: The result of anomaly detection together with the

estimation of the strength of attack ω

For the preparation of constraints, we employ the

proposed Algorithm 1 in Matlab, with N

= 10, N

3,|χ| ≤ diag([1,1, 1])×10

−2

,|ε| ≤ [0.1,0.1,0.1]

, and

η = [1, 1,1] × 10

−2

Having the uncertainty bounds and the terminal

sets calculated in our deposit, Algorithm 2 can be ex-

ecuted within the control loop to handle the attack

proﬁle discussed in the previous section. In Fig. 3,

we present the evolutions of the ACC system under

attack, controlled by the proposed resilient controller.

The control plot demonstrates that the control signal

becomes zero when a DoS attack is initiated, as it

cannot be transmitted to the vehicle. Therefore, the

control strategy must efﬁciently compensate for this

absence of control during attack periods.

Velocity

(m/sec

0 2 4 6 8 10 12 14 16 18 20

-20

-10

Position(m)

0 2 4 6 8 10 12 14 16 18 20

(m/sec)

0 2 4 6 8 10 12 14 16 18 20

-10

-5

Acceleration

)

0 2 4 6 8 10 12 14 16 18 20

-10

Control

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

0.5

Attack

Figure 3: Details of the resilient control responses against

DoS attack.

Regarding the system states, including relative po-

sition, velocity, and acceleration, the primary objec-

tive of the control is to regulate the system to the ori-

gin, represented by x = [0,0,0]

. Remarkably, de-

spite the occurrence of the DoS attack on the sys-

tem, all the states effectively converge to zero starting

ICINCO 2023 - 20th International Conference on Informatics in Control, Automation and Robotics

680

from a random initial condition with the utilization of

the proposed resilient control strategy. These results

clearly indicate the effectiveness and resilience of the

proposed control approach in managing the impact of

DoS attacks on the ACC system.

5.3 Comparison Results

For a better demonstration of the effectiveness of the

proposed resilient control, we also apply the standard

MPC in the same attack scenario and ACC system

conﬁguration. Therefore, we employ CasADi opti-

mization library (Andersson et al., 2018) in Matlab.

Time (sec)

(m/sec

)

0 5 10 15 20

-100

-50

Position(m)

a) MPC

0 5 10 15 20

-10

Velocity(m/sec)

0 5 10 15 20

Time (sec)

-5

Acceleration

0 5 10 15 20

-100

-80

-60

-40

-20

b) Resilient Control

0 5 10 15 20

-20

-10

Figure 4: Comparison between MPC and the proposed re-

silient control for a set of initial conditions.

In Fig. 4, we illustrate the comparison between

the proposed resilient control and MPC in terms of the

convergence of system trajectories. For this result, we

consider a set of initial conditions, then we show the

mean and upper/lower bound of the trajectories using

the line curves and the shaded areas, respectively. It is

evident that the proposed method results in faster con-

vergence while MPC fails to effectively regulate the

system trajectories to zero. The performance can be

also quantitatively compared by keeping track of the

cost function over time, i.e. the summation in (4), for

both techniques. These values of cost can be shown

as two signals as in Fig. 5. In Table 2, the numerical

values of costs are reported by averaging for all initial

conditions. Accordingly, in the simulated scenario,

the mean cost value shows about 38% improvement

for the proposed approach in comparison to standard

MPC.

Table 2: Comparison of mean cost values obtained.

MPC Resilient Control Improvement

9.2163 × 10

5.7300 × 10

38%

Computational Time. Finally, as a comparison of

the computational complexity of the proposed ap-

proach, we illustrate the runtime results for both pre-

sented and MPC techniques in Fig. 6. We run the pro-

posed technique in two different conﬁgurations with

N = 1 and N = 5. By comparing these results, the ba-

sic MPC runs faster as expected since it solves a less

complicated problem. However, the runtime results

for both settings of the proposed technique are com-

parable to MPC, making it a potential candidate for

replacing non-resilient controllers in real-world im-

plementations. This is mostly because the main part

of the computations is done ofﬂine using Algorithm

0 2 4 6 8 10 12 14 16 18 20

Time (sec)

Cost

MPC

Resilient Control

Figure 5: In this graph, by keeping track of the cost func-

tion for the proposed resilient control and MPC, we com-

pare the performance, where the proposed method clearly

outperforms by resulting in a lower control cost.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45

Runtime (sec)

100

Density

MPC

Resilient Control N =5

Resilient Control N =1

Figure 6: We compared the runtime results for both tech-

niques: the basic MPC and two different conﬁgurations of

the proposed method with N = 1 and N = 5.

5.4 Conclusion

This study proposed a novel framework for design-

ing a resilient MPC system to handle uncertain linear

systems under periodic DoS attacks. The DoS attack

was modeled as an uncertain parameter-varying sys-

tem with additive disturbance, and the Kalman ﬁlter

was used for anomaly detection. An optimization-

based resilient algorithm was developed using a ro-

An Efﬁcient Resilient MPC Scheme via Constraint Tightening Against Cyberattacks: Application to Vehicle Cruise Control

681

bust constraint-tightening MPC approach. We imple-

mented the approach to the ACC problem, showcas-

ing its effectiveness in mitigating the impact of pe-

riodic attacks and ensuring system stability. Overall,

the study provided a solution to enhance the resilience

of control systems in the presence of DoS attacks. In-

corporating robust attack detection methods and ex-

tending the framework to encompass various types

of attacks can be potentially promising for future re-

search.

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