INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
VOL.
NO.
NOVEMBER 2023
Optimal Control of an Uncertain Linear Networked Control System Under Denial of Service Attacks Tua Agustinus Tamba AbstractAiController design based on networked control system (NCS) framework combines sensing, control, and actuation tasks using shared channel of communication network.
Such a use of communication channel often makes the NCS design more challenging due to the presence of system uncertainties, limitation on available computational/communication systems resources, as well as possible occurrences of data transmission failures or cyber This paper develops mathematical grounds for dynamic event-triggered control design method for an uncertain linear NCS with matched uncertainties whose communication channel undergoes cyber attaks in the form of Denial-of-Service (DoS).
The DoS attacks are assumed to halt the execution of control update tasks that was scheduled by the dynamic event-triggered control scheme.
Under such possible occurrences of DoS attacks, this paper bounds the allowable duration of the DoS presence and derives suitable control signals which can guarantee the closed loop NCS remains stable.
Index TermsAiOptimal Control.
Uncertain Linear Networked.
Networked Control System.
I NTRODUCTION
NCS typically uses a network of computers and dedicated communication system to regulate the data interactions/exchanges between the plant, sensor, controller and actuator elements .
This coupling between physical, computational and communication components often makes the NCS implementation becomes more challenging.
Over the last two decades, significant studies and researches on NCS design are continuously done to improve NCSAo resource utilization efficiency as well as robustness to cyber attacks.
To improve the efficiency of NCSAo resource utilization, a new control scheduling approach based on a static eventtriggered scheme (SETS) was proposed recently .
, .
The SETS is essentially an aperiodic sampling strategy which updates the control signals only when some predefined conditions .
on the systems are satisfied.
When compared to the commonly practiced periodic update schemes, the SETS-based method is shown to be more resource-aware as it utilizes the computational resource more efficiently .
, .
With regard to the used communication systems, previous studies on NCS design have commonly focused on examining the effects of communication system constraints .
uch as transmission delay, data quantization, or packet drop.
on the closed loop NCS stability and performances .
and references therei.
Such studies on NCS communication constraints are usually done under the assumption that the Tua Agustinus Tamba is with Department of Electrical Engineering.
Parahyangan Catholic University.
Bandung 40132.
West Java.
Indonesia email: ttamba@unpar.
Manuscript received February 27, 2023.
accepted August 4, 2023.
used communication channels are predesigned such that their models and bounded uncertainties are known .
, .
In recent years, increased attentions have been given to explore and address related issues on NCS security to cyber This new research trend is particularly driven by real life observation of NCS applications which demonstrate their vulnerability to unpredictable cyber attacks .
, .
, .
, .
, .
One of such attacks is the well-known denial-of service (DoS) attack which can reduce or even diminish the timeliness of data transmission between different elements of the NCS.
When coupled with possible uncertainties on the NCS model, the occurrence of DoS attacks can significantly deteriorate both the performance and stability of NCS.
These thus suggest the need for NCS design frameworks which can ensure not only the efficiency of resource utilization but also the resiliency towards possible cyber attacks .
, .
, .
This paper presents mathematical grounds of a dynamic event-triggered scheme (DETS) for the design of uncertain linear NCS which is subjected to DoS attacks.
The considered model uncertainty is assumed to be of matched uncertainty type .
, while the DoS attack is modeled as in .
, .
which only assumes limited information about attacksAo duration and frequency.
This paper presents DETS formulation of the optimal control solution to uncertain linear systems as developed in .
, and then derives an update scheme for control signals that can guarantee the input-to-state stability (ISS) property of the closed loop system.
In particular, the derived control signal and its update scheme illustrate the impacts of DoS attacksAo presence/absence on the ISS property of the NCS.
Notations: I denotes the set of nonnegative integers.
R and 0 , respectively, are the sets of real and nonnegative real numbers, while Rn is Euclidean space of dimension n.
OuxOu is the norm of a vector x OO Rn .
(M ) and (M ), respectively, are the maximum and minimum of the eigenvalues (M ) of matrix M .
M 0 means matrix M is positive semidefinite.
F (A) OO K means F (A) belongs to class K function such that it is continuous, strictly increasing, and F .
= 0.
F (A) OO KO
means F (A) is of class K function which further satisfies F () Ie O as Ie O.
For a function F .
, t > 0.
F .
Oe ) denotes the limit of F (E ) as E Ny t from the left, such that F .
Oe ) := F .
when F is continuous at t.
II.
P ROBLEM S ETUP AND F ORMULATION
We first recall the model of an uncertain linear NCS with its corresponding optimal controller, and then presents the closed loop control problem to be considered in this paper when DoS attacks occurrences are taken into consideration.
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
VOL.
NO.
NOVEMBER 2023
An Uncertain Linear NCS Model We consider an uncertain linear NCS model below.
= A.
Bu.
, x.
= x0 , .
with state x.
OO Rn and input u.
OO Rm .
B is the input matrix, and the system matrix A.
with uncertain parameter p OO P satisfies the matched uncertainty below .
A for a nominal parameter p0 OO p, the pair {A.
0 ).
B} is A given some matrices R.
S 0, there is a matrix .
such that: T .
O S.
Also, for some known p0 :
Oe A.
0 ) = B.
The matched uncertainty property allows .
to be written as xN.
= A.
0 )x.
Bu.
The controls of NCS .
can be done using LQR design approach to ensure the closed loop asymptotic stability for all p OO P .
In particular, a feedback control signal of the form u.
= K O x.
can be searched through the minimization of the following cost function.
Z O
O (S Q) x.
O .
Ru.
) dt, .
in which S := inf {S : .
O .
O S}.
In this regard, an optimal control law for .
is of the following form u.
= OeROe1 B O P x.
= K O x.
Fig.
NCS schematic of .
under DoS attacks.
nth DoS with duration En during which the communication between controller and system actuator is compromised.
During such a duration, the system actuator is assumed to implement the last succesfully received control signal.
By the definitions of such n .
Dn and tn , the time duration .
, .
can be partitioned into the following elements:
A The set E.
denoting the period up to time AytAy at which the controllerAeactuator communication exists, i.
:= .
, .
\ OnOOI Dn .
with P 0 satisfies an algebraic Riccati equation below.
P O A.
0 ) AO .
0 )P S Q Oe P BROe1 B O P = 0.
Consider the NCS setup of .
Ae.
as shown in Fig.
Assume the control signal .
is generated using a zero-order hold sampler and then transmitted through a communication channel to the plant/actuator.
Let .
i }iOOI with t0 := 0 denotes the sequence of control signal update times.
Then for two successive update times, the control signal .
satisfies u.
= K O x.
i ).
OAt OO .
i , ti 1 ), .
such that the closed loop NCS model .
can be written as xN.
= A.
0 )x.
BK O x.
i ) B.
Define the error e.
between the values of NCS states at the last control update time and at the current time t below.
= x.
i ) Oe x.
The control signal .
is implemented during o.
The set E.
denoting the instances of DoS occurrences when controllerAeactuator communication do not exist:
:= OnOOI Dn O .
, .
Problem Formulation OAt OO .
i , ti 1 ).
Using .
, model .
can be rewritten for all t OO .
i , ti 1 ) as xN.
= (A.
0 ) BK O ) x.
BK O e.
The objective of this paper is to study the stability property of NCS model .
when its communication channels are subjected to DoS attacks .
Fig.
For this purpose, our analysis will use a DoS attack model developed in .
Let .
}nOOI be the sequence of times when DoS attacks Define Dn := .
, n 1 ) as the time interval of the .
The control signal at each element of E.
= K O x ti.
where the subscript i.
is defined as:
Oe1, if .
= OI i.
:= sup .
OO I : ti OO o.
) , otherwise .
It can be seen that .
essentially defines the latest control signal received and updated by the actuators.
It is also assumed that the DoS sequence .
n }nOOI satisfies inf En > 0, nOOI and that the following holds on E.
for a constant > 0:
Condition .
basically assumes that the occurrence of DoS is regular .
the occurrence frequency is finite and non-Zen.
, while .
sets the DoSAo slow-on-the-average property with average dwell-time parameter of T .
Given NCS model .
and DoS characteristics .
, we aim to bound the allowable time duration of DoS presence as well as suitable control signals that can stabilize the closed loop NCS.
In .
, this problem was first examined for LTI |E.
| O INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
VOL.
NO.
NOVEMBER 2023
systems of the form xN.
= Ax.
Bu.
using SETS .
with control update time logic as follows: t0 = 0, and ti 1 = inf t OO R t > ti O Ex.
O Qx.
Oe 2x.
O P BK O e.
Oe )] O .
with E OO .
, .
Choosing a Lyapunov function V .
) = x.
O P x.
with P 0, .
showed that LTI systems with control update logic .
that are attacked by DoS satisfying .
will remain GAS if the conditions below hold:
1 > E2 , and .
E > 1
2 2 .
Oe E2 ) .
with 1 = (Q), 2 = 2OuP BK O Ou, 1 = (P ), 2 = (P ), and Q 0 satisfies (A BK O )O P P O (A BK O ) = OeQ.
The work in .
extends the results in .
for uncertain linear NCS of the form .
using update logic in .
, while .
examined similar problem as in .
but using a DETS that was proposed in .
as the control update time logic.
This paper essentially further develops the results in .
, .
, .
by addressing DoS-resilient control design problem for uncertain linear NCS of the form .
using DETS framework.
The inclusion of system uncertainties in NCS model and analysis makes the proposed method more realistic/applicative.
Furthermore, as shown in .
, the use of DETS also provides a sequence of control update times with longer inter-sampling intervals and therefore ensures more efficient utilization of the available NCS resources.
The DETS that is used in this paper assumes the following sequence of control update times .
: t0 = 0, and ti 1 = inf.
OO R | t > ti O .
[Ex.
O Qx.
Oe Oe 2x.
P BK e.
)] O .
where the dynamic variable .
satisfies equation below.
= OeI.
Ex.
O Qx.
Oe 2x.
O P BK O e.
Properties of Dynamic Event-Triggered Scheme Lemma 1 below shows that the dynamic variable .
in DETS .
is always non-negative.
Such a property will be useful for deriving the ISS conditions of NCS .
Lemma 1.
Consider the DETS in .
Then for all t OO .
O), the following inequalities hold for .
(Ex.
O Qx.
Oe 2x.
O P BK O e.
) Ou 0 .
Ou 0 Proof.
We first show that condition .
in Lemma 1 is true.
The construction of the DETS update times logic .
implies that the following holds for all t OO .
O).
Ex.
O Qx.
Oe 2x.
O P BK O e.
Oe ) Ou 0 .
By noting that e.
Ou e.
Oe ) holds for all t OO .
O), then .
implies that the following also true.
(Ex.
O Qx.
Oe 2x.
O P BK O e.
) Ou 0, .
= 0 .
This paperAos objective is to examine required conditions under which the closed loop NCS .
maintains an ISS property when using DETS .
and is subjected to DoS attacks.
Definition 1 formalizes such an ISS concept.
Definition 1.
Dynamical systems of the form .
is said to satisfy the ISS property if for all x.
:= x0 OO Rn , there exist an ISS Lyapunov function V .
) : Rn y R Ie R and functions 1 , 2 .
OO KO of class KO such that:
1 Oux.
Ou2 O V .
) O 2 Oux.
Ou2 , .
VN .
) O OeI (Oux.
(Oue.
M AIN R ESULTS
This section presents this paperAos main results regarding properties of DETS .
and their use to derive conditions for ISS of NCS .
when DoS attacks are present.
which is statement .
in Lemma 1.
Now we show .
by examining .
when = 0 and = 0.
If = 0, .
Ou 0 Next, if = 0, then .
becomes Ex.
O Qx.
Oe 2x.
O P BK O e.
Ou Oe .
Based on .
, the dynamics of .
becomes N.
Ou Oe (I 1/) .
, .
Ou 0.
By the comparison lemma .
, then .
implies that .
Ou .
eOe(I 1/)t , .
Ou 0 which proves .
Ou 0 as in statement .
of Lemma 1.
Next.
Lemma 2 shows that the inter-execution time .
i 1 ) of DETS .
is greater than that of the SETS in .
Lemma 2.
Let tsi 1 be the next .
th control update time of SETS .
, and tdi 1 be the next .
th control update time of DETS .
Then, it holds that: tdi 1 Ou tsi 1 .
Proof.
Assume for the moment that tsi 1 Ou tdi 1 .
Then the SETS update logic .
implies the following must hold.
Ex.
di 1 )O Qx.
di 1 ) Oe 2x.
di 1 )O P BK O e.
di 1 ) > 0 Now, consider the DETS update logic .
for two cases of Firstly, if > 0, then the update time logic .
and the non-negativity of .
showed in Lemma 1 imply:
0 Ou .
di 1 ) Ex.
di 1 )O Qx.
di 1 ) Oe 2x.
di 1 )O P BK O e.
di 1 ) .
Ou Ex.
di 1 )O Qx.
di 1 ) Oe 2x.
di 1 )O P BK O e.
di 1 ) .
Since Ou 0, then .
will hold only if .
below is true.
Ex.
di 1 )O Qx.
di 1 ) Oe 2x.
di 1 )O P BK O e.
di 1 ) < 0.
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
VOL.
NO.
NOVEMBER 2023
Note that .
, so tdi 1 Ou tsi 1 should instead be true.
Secondly, if = 0, the DETS update logic in .
implies t 1 O 0, and so the .
dynamics in .
becomes N.
O 0 Ne Ex.
O Qx.
Oe 2x.
O P BK O e.
O 0.
Again, .
contradicts the assumption in .
and therefore tdi 1 Ou tsi 1 should instead be true as stated in Lemma 2.
Lastly, we present in Lemma 3 below an upper bound for the functional of the error e.
when DoS attacks are present.
Lemma 3.
Consider NCS .
under DETS control update time logic .
Assume the NCS is subjected to DoS attacks with properties as in .
Ae.
Then for all t OO E.
, the inequality below holds regarding the error e.
22 Oux.
OuOue.
Ou O (E1 22 ) Oux.
)Ou2 22 Oux.
Ou2 Ou.
)Ou .
= x.
) ) Oe x.
On the other hand, the construction of control update time logic .
implies the following also holds for all t OO E.
) P BK e.
) O Ex.
) Qx.
) .
) .
Consequently, we can write the norm of .
as below.
2OuP BKOuOux.
)OuOue.
)Ou O E(Q)Oux.
)Ou2 Ou.
)Ou 22 Oux.
)Ou Oux.
) ) Oe x.
n )Ou O E1 Oux.
)Ou2 Ou.
)Ou 22 Oux.
)OuOux.
) )Ou O 22 Oux.
)Ou2 E1 Oux.
)Ou2 Ou.
)Ou .
As a result, the following functional relationship can be obtained based on the error e.
definition in .
O P BK O e.
= 2x.
O P BK O x.
) ) Oe x.
By taking the norm of .
, we then have that:
O 2OuP BKOuOux.
n )OuOuxti.
) Ou NCS Stability Analysis: DoS Attacks are Absent We first derive required conditions to ensure ISS property of the uncertain linear NCS .
when DoS attacks are absent on the communication channels.
Proposition 4 below states such conditions.
Proposition 4.
Consider the uncertain linear NCS in .
Assume the DETS control update logic .
is used.
Then the closed loop NCS .
is GAS when DoS is absent.
Proof.
For P 0, consider the quadratic Lyapunov function V .
) = x.
O P x.
for NCS .
Thus:
where 1 = (P ), 2 = (P ).
On the state trajectories of .
, the time derivative VN of V .
) can be written as VN = Vx {[A.
0 ) BK O ] x.
BK O e.
} , = Vx [(A.
0 ) BK O ) x.
] Vx BK O e.
Vx B.
where Vx := (OCV .
Using .
, .
can be rewritten as VN = Oex (S Q K O RK) x 2x.
O P BK O e.
Oe 2x.
O K O R.
, = Oex (S Q K O RK K O R.
O .
RO K) x 2xT P BK O e, .
Now add xT .
T .
to and substract it from the right hand side of .
We can then write the following.
VN = Oex.
(S Q K O RK) x.
O P BK O e.
Oe 2x.
O K O R.
, = Oex S Oe T .
Q K O RK K O R.
O RO K O .
O R.
x 2x.
O P BK O e.
, = Oex.
(S Oe .
O R.
) Q
(K .
) R (K .
) x.
O P BK O e.
, = Oex.
O x.
O P BK O e.
, 2OuP BK O OuOux.
OuOue.
Ou O 2OuP BKOuOux.
OuOuxti.
) Ou 2OuP BKOuOux.
Ou2 In subsections i-B-i-C, we use the DETS properties derived in this subsection to characterize conditions which will guarantee the ISS property of uncertain linear NCS .
in the absence or presence of DoS attacks.
1 Oux.
Ou2 O V .
) O 2 Oux.
Ou2 .
Proof.
On the one hand, note for the error e.
that the following can be written for all t OO E.
O Oe Oux.
Ou2 2OuP BK O OuOux.
OuOue.
Ou, .
2OuP BKOuOux.
Ou2 The substitution of .
allows us to write .
as 22 Oux.
OuOue.
Ou O 22 Oux.
)Ou2 E1 Oux.
)Ou2 Ou.
)Ou 22 Oux.
Ou2 .
O (E1 22 ) Oux.
)Ou2 22 Oux.
Ou2 Ou.
)Ou which is as stated in the lemma.
The proof is completed.
where = () in which = S Oe T .
Q (K .
) R (K .
When there are no DoS attacks, the DETS control update time logic .
imply that the following holds.
(Ex.
O Qx.
Oe 2x.
O P BK O e.
) Ou 0.
Taking the norm of .
and substituting it into .
allow us to write .
into the following.
VN .
) O Oe Oux.
Ou2 E(Q)Oux.
Ou2 Ou.
Ou, .
O OeO1 V .
) Ou.
Ou2 .
INTERNATIONAL JOURNAL OF COMPUTING SCIENCE AND APPLIED MATHEMATICS.
VOL.
NO.
NOVEMBER 2023
which satisfies Definition 1 in which I := O1 = ( Oe E1 )/1 and = 1/.
This thus shows the ISS property of the closed loop NCS .
when DoS attacks are absent.
NCS Stability Analysis: DoS Attacks are Present Next, we examine the stability of NCS .
when DoS attacks are present.
During such DoS occurrences .
e OAt OO E.
defined in .
), the DETS control update time logic .
cannot be carried out.
In this regard, one instead may examines the upperbound of the error functional term 2OuP BK O OuOux.
OuOue.
Ou of the time derivative VN .
This can be done using the result in .
of Lemma 3.
More specifically, the substitution of .
allows us to write the time derivative in .
as follows.
VN O Oe Oux.
Ou2 2OuP BK O OuOux.
OuOue.
Ou O Oe Oux.
Ou2 (E1 22 ) Oux.
)Ou2 22 Oux.
Ou2 Ou.
)Ou .
O .
Oe )Oux.
Ou (E1 22 ) Oux.
)Ou Ou.
)Ou2 Note on one hand that if Oux.
)Ou O Oux.
Ou, .
becomes VN O (E1 Oe 42 )Oux.
Ou2 Ou.
Ou2 .
O O2 V .
) Ou.
Ou2 with O2 = (E1 Oe 42 )/1 .
On the other hand, inequality .
below instead will hold if Oux.
)Ou Ou Oux.
Ou.
VN O O2 V .
n )) Ou.
Ou2 .
Using the obtained results in .
, .
, and .
Theorem 5 below states the conditions which will guarantee the ISS property of the closed loop NCS .
for all time t Ou 0.
Theorem 5.
Consider the uncertain NCS .
with DETS control update time logic .
Assume the NCS is being attacked by DoS phenomenon with properties as in .
Ae.
Then the closed loop NCS .
maintains ISS property for any DoS satisfying .
with a constraint of the form
O1 O2
in which O1 and O2 are as in .
, respectively.
Proof.
Define nOe1 = 0.
EnOe1 = 0.
Then .
allows us to write the following for all t OO .
Oe1 EnOe1 , n ).
V .
) O eOeO1 .
Oe.
En )) V .
Oe1 EnOe1 )) Ou.
Ou2O .
O1 Similarly, we have the following from .
for all t OO Dn .
V .
) O eO2 .
Oen ) V .
)) Ou.
Ou2O , .
O2 Now note that .
Ae.
| = t Oe |E.
Thus for all t Ou 0, one may combines .
as follows.
V .
) O eOeO1 .
| eO2 |E.
| V .
0 ) nOu.
Ou2O O eOeO1 t e(O2 O1 )|E.
| V .
0 ) nOu.
Ou2O .
in which n := max (/O1 , /O2 ).
Substituting the constraint on DoS occurrences into .
, we have that V .
) O eOeO1 t e(O1 O2 )( .
/T )) V .
0 ) nOu.
Ou2O O e(O1 O2 ) eOe(O1 Oe(O1 O2 )/T )t V .
0 ) nOu.
Ou2O .
Using property .
of V .
), we may write from .
that 2 (O1 O2 ) Oe(O1 Oe O1 O ) Oux Ou2 Oux.
Ou O Ou.
Ou2O 1 Now note that an inequality of the form a2 b2 O .
2 holds for any pair of real numbers a > 0 and b > 0.
Using this fact on .
, we may infer the following inequality.
2 (O1 O2 ) Oe[O1 Oe( O1 O )] 2t Oux Ou Oux.
Ou O Oe(Oux0 Ou,.
Ou.
OuO 1 e(Ou.
OuO ) O Oe (Oux0 Ou, .
e (Ou.
OuO ) Notice that e(A) in .
is of class KO .
In order for .
to satisfy the second ISS condition in Definition 1 .
the ISS property of NCS .
), then (Oux0 Ou, .
should also be of class KO which can be ensured if condition .
in the theorem is satisfied.
The proof is completed.
Condition .
in Theorem 5 suggests that the ISS property of DoS-attacked uncertain NCS in .
is dependent on both the frequency and individual duration of DoS occurrences.
Consequently, such a condition can also be viewed/used as a measure of the NCSAo resiliency to DoS attacks which occurr in the NCSAo communication channels.
IV.
C ONCLUSION
This paper has presented mathematical grounds for DETSbased optimal control design approach to maintain an ISS property of a class of NCS which satisfies the matched uncertainty condition and undergoes DoS attacks.
Under the assumption that the DoS attacks occur in a regular manner, this paper uses LyapunovAos stability analysis method to derive conditions for the proposed DETS-based control that will produce a sequence of control update times which can preserve the ISS property of the closed loop NCS.
Future works will be directed toward examining and implementing the proposed resilient optimal control design method in practical and real life NCS applications under uncertainties.
ACKNOWLEDGMENT