Model-based Predictive Security Control for Discrete Switching Systems Under Deception Attacks

Authors: jianghan  xu
lili  li
mengjie  li
xiaowei  zhao
Venue: IEEE_org_ICIST_2024_Conference
Published: 2024-08-02 09:26:40

Abstract: This paper focuses on model prediction-based security control for discrete switching systems under deception attacks. An event triggering mechanism based on the optimal prediction is designed to shorten the asynchronous time between the subsystem and the controller, which helps the system to adjust in time when the prediction deviates, and reduces the impact of the attack on the system performance. A new quadratic cost function is proposed to find a smoother and more stable control strategy; a subsystem switching rule based on the optimal prediction state is constructed to select the optimal subsystem switching and optimise the overall performance of the system; and a sufficient condition for asymptotic stabilisation of the closed-loop system under the above event-triggering mechanism and optimal safety control strategy is given.

Main paper: 
 i. introduction: 
 Networked switching systems describing multi-modalisation models have received increasing attention as the complexity of control objects in various types of real industrial systems continues to rise [1] . Due to the intertwining of control and network, the two main challenges are resource scheduling under communication constraints and attack prevention in open environments. To cope with the resource scheduling problem, the event-triggering mechanism has become an effective solution [2] [3] [4] . Also due to the openness of the network, the switching system communication network frequently suffers from malicious attacks, seriously interfering with the normal operation of the system [5] .Model predictive control shows significant advantages in solving the problem of switching system security control under attack [6] . However, when the switching system is subjected to deception attacks, the controller modes are tampered with several times and are not consistent with the actual system modes, which can lead to a complex and difficult optimization process for model predictive control. In order to solve this problem, new index coefficients are introduced into the quadratic cost function, which can portray the dynamic behaviour of the system more accurately by considering the rate of increase and decrease of the Lyapunov function [7] , [8] investigated mean residence time-based model predictive control for switched linear systems with time-varying delays in data transmission. [9] developed a model predictive control framework for the co-optimization of switching sequences and control inputs of a switched linear system under continuous residence time constraints with guaranteed recursive feasibility and stability. [10] investigated observer-based model predictive control for discrete switching systems under event-triggered mechanisms and denial-of-service attacks. So as to better reflect the stability requirements of the system in the optimization problem, and alleviate the problems caused by modal mismatch to a certain extent. However, since the switching rules in this paper rely on hysteresis state switching, it is still an urgent challenge to design a new quadratic cost function using model prediction methods to optimize the overall performance of the system and enhance the security for the complex problem of multiple tampering of the controller modes in discrete switching systems under deception attacks.This article specifically addresses the optimal model prediction-based security control problem for discrete switch-ing systems under deception attacks. In contrast to prior studies, four distinct contributions are outlined below: 1) Designing the event triggering mechanism based on optimal prediction, the system can dynamically adjust the triggering strategy when the prediction deviates based on the comparison result between the prediction error and the preset threshold. 2) Designing a new quadratic cost function allows the controller to trade-off control performance and switching cost, avoiding costly switching and switching that may seem beneficial in the short term but may damage system performance or stability in the long term, and thus finding a smoother and more stable control strategy that reduces system fluctuations caused by switching. 3) Designing system switching rules based on the optimal prediction state helps to select the optimal subsystem during switching, thus optimizing the overall performance of the system and improving the safety of the system. 4) The complex asynchronous switching behavior between the controller and the system is analyzed under the multiple designs mentioned above and stability criteria are given. 1 : Diagram of model predictive optimal security control strategy for discrete-time switched system under deception attacks.ˆ( | ) u k k  ˆ( | ) x k k  Observer-based Model Predictive Control ˆ( | ) u k l k + ( ) s x k ( ) s k  ( ) x k ˆ( | ) x k l k + ( ) x k ( ) k  ˆ( | ) x k k  ( ) a k  Fig.As portrayed in Fig. 1 , the DSSs are described asEQUATIONwhere x ⊆ R n x , u ⊆ R n u , y ⊆ R n y are state, control input, regulated output. A σ , B σ , C σ are known appropriate matrices. The system state x(k) and switching signal σ(k) are sent to the event triggering mechanism based on optimal prediction, which determines whether the sampling data and the subsystem switching signal satisfy the triggering conditions and then transmitted to the controller side through the network. When deception attack occurs, both x(k s ) and σ(k s ) are tampered with and subject to network-induced delay. If the attack occurs multiple times, the healthy subsystem switching signal σ(k) can be tampered with multiple times, which can lead to complex asynchronous switching behavior. The observer measures the system state x(k) for transmission to the model predictive controller, after the quadratic cost function to obtain the optimal control signal û * (k|k) and the corresponding optimal system state x * (k|k). The control inputs û * (k|k) are transmitted to the actuator for use by the system, and x * (k|k) is used for the design of ETM and switching rules.Since data is transmitted to the controller over network, it is vulnerable to transmission delays and Deception attacks. Define τ s as the transmission delay at the s-th triggering instant.τ ≤ τ s ≤ τ, τ = max s∈N + {τ s }, τ = min s∈N + {τ s }. Besides, the ι-th attack interval is expressed asΥ 1 ι = [d ι , d ι ) and sleeping interval is Υ 2 ι = [d ι , d ι+1 ). d ι = d ι + ℓ a ι with duration ℓ a ι , ι ∈ N + . Attacker injects an attack signal x a (k) into data x(k) transmit- ted in the network, altering it to x(k) = x(k) + µ x (k)x a (k). If k ∈ Υ 1 ι , µ x (k) = 1, otherwise µ x (k) = 0.For Υ 1 ι , the healthy switching signal σ(k) in the sensor-to-controller channel is tampered with by an erroneous switching signal σι (k). Without loss of generality, Deception attacks must meet the assumed limits on duration and frequency [11] .Deception AttacksIn Fig. 1 , the ETM is designed asEQUATIONwhere k1 s+1 = min r∈N + {δ s,r | f (δ s,r ) > 0} is the error detection condition, f (δ s,r ) = e Tx (δ s,r )Φ 1 σ(δ s,r ) e x (δ s,r )γ σ(δ s,r ) x T (δ s,r )Φ 2 σ(δ s,r ) x(δ s,r ), e x (δ s,r ) = x(δ s,r )x(k s ) is the error of current state x(δ s,r ) and triggering state x(k s ). δ s,r = k s + r is the r-th sampling instant after s-th triggering instant, weighted matrices Φ 1 σ(δ s,r ) > 0, Φ 2 σ(δ s,r ) > 0. and the threshold 0 < γ σ(δ s,r ) < 1. k 2 s+1 = min r∈N + {δ s,r |[ε(δ s,r )] T ε(δ s,r ) > θ} is the optimal prediction parameter error detection condition, ε(δ s,r ) = x(δ s,r )x * (δ s,r ) is the error of current state x(δ s,r ) and optimal prediction state x * (δ s,r ). θ > 0 is the given threshold value. k 3 s+1 = min r∈N + {δ s,r |σ(δ s,r ) σ(δ s,r-1 )} is the modality matching condition, when modalities are distinct at adjacent sampling instants, an event triggering is arranged at δ s,r . k 4 s+1 = k s + ℓ is the attack parameter condition, ℓ is the lower bound of the attack sleeping time. By introducing a lower limit on the duration of sleeping to limit the upper limit on the instant of two triggerings, it is guaranteed that at least one triggering will exist during the attack sleep to achieve synchronisation of the subsystem and controller modes.For a detailed timing analysis, signal holding intervalEQUATIONℑ s,r =        [δ s,r + τ s , δ s,r+1 + τ s ), r = 0, • • • , rs -1, [δ s,r s + τ s , k s+1 + τ s+1 ), r = rs , with rs = min r∈N + {r|δ s,r+1 + τ s ≥ k s+1 + τ s+1 }. Then, a piecewise time-delay function η(k) = k -δ s,r satisfying η = τ ≤ η(k) ≤ η = τ + 1.Under the optimal prediction-based event triggering mechanism (2) with deception attacks, the system switching signal can be described as σ(k) = σ(kη(k)) + µ σ (k)σ a (k), k ∈ ℑ s,r , system state can be described as x(k) = x(k -η(k)) - e(k) + µ x (k)x a (k), k ∈ ℑ s,r .Based on the received x(k) and σ(k), the full-dimensional state observer is constructed asEQUATIONwhere x(k) ∈ R n x is the state vector of the observer, û(k) ∈ R n u is the observer control input vector, L ϕ(k) is the observer gain matrix of appropriate dimensions to be determined, σ = σ(k) ∈ P calculated by the following hysteresis state switching ruleEQUATIONwhereEQUATIONwhere x(k + l|k) is the predicted state at instant k + 1 based on instant k, û(k + l|k) is the control input at instant k + 1 based on instant k, designed asEQUATIONBased on (5)(6), the model predictive control optimises the control input sequenceEQUATIONwhereJ ∞ = ∞ l=0 J ∞ (k + l|k) J ∞ (k + l|k) = || x(k + l|k)|| 2 S x + ||û(k + l|k)|| 2 S û + || x(k + l|k)|| 2 N iϕ 1 ,iϕ 2 (8) with || x(k+l|k)|| 2 S x = xT (k+l|k)S x x(k+l|k), ||û(k+l|k)|| 2 S û = ûT (k+ l|k)S û û(k +l|k) and || x(k +l|k)|| 2 N iϕ 1 ,iϕ 2 = xT (k +l|k)N iϕ 1 ,iϕ 2 x(k +l|k)denote state cost, input cost and switching cost, respectively. S x > 0 and S u > 0 are symmetric weight matrices for the given state and control inputs, respectively. ū is the size of the constraint.Solve the optimisation problem ( 7) by applying only the first control input û * (k|k) = û(k|k) in the optimally obtained control signal sequence U(k) to the system (1).The optimal control input û * (k|k) is substituted into the state observer based prediction model ( 5) to obtain the optimal prediction stateEQUATIONThe switching rule based on the optimal prediction state is designed as follows:EQUATIONIts initial value σ(∂) = arg min j∈P {[ x * (∂)] T P jϕ x * (∂)| x * (∂) ∈ Ω jϕ }, ∂ ∈ [-η, 0) and Ω jϕ = { x * (k|k)|[ x * (k|k)] T (P jϕ -P iϕ + N jϕ,iϕ ) x * (k|k) < 0i ∈ P\ j}, Ω jϕ,iϕ = { x * (k|k)|[ x * (k|k)] T (P jϕ - P iϕ + N jϕ,iϕ ) x * (k|k) < 0i ∈ P\ j}, N jϕ,iϕ is an indeterminate matrix, Ω jϕ,iϕ is boundary of Ω jϕ and p j=1 Ω jϕ ∈ R n x .Defining an error vector ε(k) = x(k)x * (k|k), the state vector is augmented asX(k) = [ x * (k|k) T ε T (k)] T , k ∈ ℑ s,r .Substituting equation ( 6) into the switching system (1) and the optimal prediction state (9), respectively, and combining the hysteresis state switching rule (4) and the switching rule based on the optimal prediction state (10), the closed-loop system can be represented asX(k + 1) = Λ 1 ϑ X(k) + Λ 2 ϑ X(k -η(k)) + Λ 3 ϑ e x (k) + Λ 4 ϑ x a (k) (11)whereϑ ∈ {σ σ|σ, σ ∈ P}, X(∂) = ⌢ X(∂), ∂ ∈ [-η, 0), Λ 1 ϑ = A σ + B σ K σ -L σC σ 0 A σ -A σ + (B σ -B σ)K σ + L σC σ A σ Λ 2 ϑ = L σC σ L σC σ -L σC σ -L σC σ Λ 3 ϑ = -L σC σ L ϕ C ϕ Λ 4 ϑ = µ x (k)L σC σ -µ x (k)L σC σ Without lossof generality, it is assumed that the j-th and ith subsystems are activated within the neighbouring switching intervals [κ q-1 , κ q ) and [κ q , κ q+1 ). Under the influence of deception attacks and network delay, the change of modal signal (σ, σ) on any interval [κ q , κ q+1 ) can be classified into the seven cases in Table I , depending on the activation interval Υ 1 ι of the ι-th attack covering different subsystem or controller switching instants. For k ∈ [κ q , κ q+1 ), κq is the controller switching instant after network delay for the subsystem switching instant κ q .Case A. For k ∈ [κ q , κ q+1 ), ι-th attack activation interval Υ 1 ι override subsystem switching instant κ q , dι < κ q < d ∼ ι < κ q+1 , ι ∈ N + . As the deception attack tampers with the switching signal, it causes the subcontroller mode σ to switch from j to σι at dι . Under the switching rule (10) based on the optimal prediction state, (σ, ϕ) changes from ( j, σι ) to (i, σι ) after the switching instant κ q . Under the influence of the attack sleeping interval triggering condition in (2), (σ, ϕ) becomes synchronised mode (i, i). Therefore, (σ, ϕ) = (i, σι ) and (σ, ϕ) = (i, i) hold on the intervals κ q , d ∼ ι) andd ∼ ι , κ q+1 ). Case B. For k ∈ [κ q , κ q+1 ), ι-th attack activation interval Υ 1 ι override subsystem switching instant κ q , dι < κ q < d ∼ ι < dι+1 < • • • < dι+ι q < d ∼ ι+ι q < κ q+1 ι ∈ N + .As the deception attack tampers with the switching signal, it causes the subcontroller mode σ to switch from j to σι at dι . Under the switching rule (10) based on the optimal prediction state, (σ, ϕ) changes from ( j, σι ) to (i, σι ) after the switching instant κ q . Under the influence of the attack sleeping interval triggering condition in (2), (σ, ϕ) becomes synchronised mode (i, i). At the end of the attack, (σ, ϕ) is synchronised to (i, i) by the event-triggering mechanism (2) after d ∼ ῑ. Therefore, (σ, ϕ) = ( j, σι ) and (σ, ϕ) = (i, σι ) hold on the intervals dι , κ q ) and κ q , d ∼ ι).ι ⊆ [κ q , κ q+1 ) override controller switching instant κq . Under the switching rule (10) based on the optimal prediction state, (σ, ϕ) changes from ( j, j) to (i, j) after the switching instant κ q . As the deception attack initiates and ends triggering a sub-controller switching, (σ, ϕ) undergoes a consecutive asynchronous switching from (i, j) to (i, σι ), (i, i) at switching moments dι , d ∼ ι . Thus, (σ, ϕ) = (i, j) on the interval (κ q , dι ) and (σ, ϕ) = (i, σι ) and (σ, ϕ) = (i, i) hold on the intervals ( dι , d ) and (d∼ ι , κ q+1 ), respectively. Case D. Attack interval Υ 1 ι ⊆ [κ q , κ q+1 ) override controller switching instant κq . κ q < dι < d ∼ ι < • • • < dι+ι q < d ∼ ι+ι q < κ q+1 , ι q ∈ N + .Under the switching rule (10) based on the optimal prediction state, (σ, ϕ) changes from ( j, j) to (i, j) after the switching instant κ q . Under the deception attack, the controller mode ϕ changes from i to σῑ to i after the sub-controller switching instants dῑ and d ∼ ῑ. Thus, (σ, ϕ) = (i, j) on the interval (κ q , dι ) and (σ, ϕ) = (i, σῑ ) and (σ, ϕ) = (i, i) hold on the intervals ( dῑ , d ∼ ῑ) and (d ∼ ῑ, dῑ+1 ), respectively. ῑ ∈ {ι,ι + 1,• • • , ι + ι q }, dι+ι q +1 = κ q+1 . Case E. Attack interval Υ 1 ι ⊆ [κ q , κ q+1 ) override controller switching instant dῑ . ῑ ∈ {ι,ι + 1, • • • , ι + ι q }, ι q ∈ N + .Under the influence of the hysteresis state switching rule (4), the controller mode ϕ changes from j to i after the sub-controller switching instant κq . Under the deception attack, the controller mode ϕ changes from i to σῑ to i after the sub-controller switching instants dῑ and d ∼ ῑ. Thus, (σ, ϕ) = (i, j) on the interval κ q , κq ) and (σ, ϕ) = (i, σῑ ) and (σ, ϕ) = (i, i) hold on the intervals ( dῑ , d ∼ ῑ) and (d ∼ ῑ, dῑ+1 ), respectively. ῑ ∈ {ι,ι + 1,• • • , ι + ι q }, dι+ι q +1 = κ q+1 .Case F. There is no controller switching instant dῑ induced by the ῑ-th attack during the attack interval Υ 1 ῑ ⊆ [κ q , κ q+1 ), ῑ ∈ {ι,ι+1, • • • , ι+ι q }, ι ∈ N + . Under the influence of the hysteresis state switching rule (4), the controller mode ϕ changes from j to i after the sub-controller switching instant κq . Therefore, (σ, ϕ) = (i, j) and (σ, ϕ) = (i, i) hold on the intervals (κ q , κq ) and (κ q , κ q+1 ) respectively.Case G. For k ∈ [κ q , κ q+1 ), Attack interval Υ 1 ι ⊆ [κ q , κ q+1 ) override sub-system switching instant κ q+1 , κq < dι < κ q+1 < d ∼ ι , ι ∈ N + . Under the influence of the hysteresis state switching rule (4), a synchronised switching from (i, j) to (i, i) occurs in (σ, ϕ) after the sub-controller switching instant κq . After the sub-controller switching instant dι , due to the deception attack ϕ is tampered from i to σι . Therefore, (σ, ϕ) = (i, j) and (σ, ϕ) = (i, i) hold on the intervals (κ q , κq ) and (κ q , dι ), and (σ, ϕ) = (i, σι ) hold on the interval [ dι , κ q+1 ).Through the analysis of Cases A-G, the optimal security control problem based on model predictions for system (1) is transformed into control objectives: (i) The model predictive control design ( 7)-( 8) is transformed into an optimisation problem with a quadratic cost function that ensures the asymptotic stability of the closed-loop system (11) . (ii) Incorporating constraint ||u(k + l|k)|| ≤ ū as a linear matrix inequality into an observer-based model predictive control scheme.Theorem 1: Given positive parameters η, η -, γ σ < 1, model prediction control weight matrix S x > 0, S û > 0 and ς θ, θ < 0, minimising an upper bound β on J ∞ (k) to ensure that the closed-loop system ( 11) is asymptotically stable, if forthcoming conditions are met:(i) There exist matricesEQUATIONEQUATION)(ii) There exist indeterminate matrice of appropriate dimensions N θ, θ satisfyingEQUATIONEQUATIONEQUATION(iii) There exist scalar β > 0 satisfyingEQUATIONEQUATION)The matrixΨ ϑ = {φ ϱ 1 ,ϱ 2 ϑ }, (ϱ 1 , ϱ 2 = 1, • • • , 6) is composed of sub-blocks: φ 1,1 ϑ = (Λ 1 ϑ ) T (P ϑ + Rϑ )Λ 1 ϑ -Rϑ Λ 1 ϑ -(Λ 1 ϑ ) T Rϑ -P ϑ + Rϑ + Q 0 ϑ -R 0 ϑ + ÎT S x Î + ÎT K T ϕ S û K ϕ Î + p υ=1 ς θ, θ P θ -P θ + N θ, θ + N θ, θ, Î = 1 0 , φ 1,2 ϑ = (Λ 1 ϑ ) T (P ϑ + Rϑ )Λ 2 ϑ -Rϑ Λ 2 ϑ , φ 1,3 ϑ = R 0 ϑ , φ 1,5 ϑ = (Λ 1 ϑ ) T (P ϑ + Rϑ )Λ 3 ϑ -Rϑ Λ 3 ϑ , φ 1,6 ϑ = (Λ 1 ϑ ) T (P ϑ + Rϑ )Λ 4 ϑ -Rϑ Λ 4 ϑ , φ 2,2 ϑ = (Λ 2 ϑ ) T (P ϑ + Rϑ )Λ 2 ϑ -2R 1 ϑ + M 1 ϑ + (M 1 ϑ ) T + γ σ Z T Φ 2 σ Z, Z = 1 1 , φ 2,3 ϑ = R 1 ϑ -(M 1 ϑ ) T , φ 2,4 ϑ = R 1 ϑ -M 1 ϑ , φ 2,5 ϑ = (Λ 2 ϑ ) T (P ϑ + Rϑ )Λ 3 ϑ , φ 2,6 ϑ = (Λ 2 ϑ ) T (P ϑ + Rϑ )Λ 4 ϑ , φ 3,3 ϑ = -Q 0 ϑ + Q 1 ϑ -R 0 ϑ -R 1 ϑ , φ 3,4 ϑ = M 1 ϑ , φ 4,4 ϑ = -Q 1 ϑ -R 1 ϑ , φ 5,5 ϑ = (Λ 3 ϑ ) T (P ϑ + Rϑ )Λ 3 ϑ -Φ 1 ϑ , φ 5,6 ϑ = (Λ 3 ϑ ) T (P ϑ + Rϑ )Λ 4 ϑ , φ 6,6 ϑ = (Λ 4 ϑ ) T (P ϑ + Rϑ )Λ 4 ϑ , Rϑ = η - 2 R 0 ϑ + ( η -η - ) 2 R 1 ϑ ,(ϑ, σ, ϕ) ∈ {(i, i, i), (i j, i, j), (i σῑ , i, σῑ )}, υ ∈ {i, j, σῑ } ( θ, θ) ∈ {( j σι , i σι ), ( j, i j), (i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)},( θ1 , θ1 ) ⊆ ( θ, θ), ( θ1 , θ2 ) ⊆ ( θ, θ), (iϕ 1 , iϕ 2 ) ∈ {(i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)} ⊆ ( θ, θ),( jϕ, iϕ) ∈ {( j σι , i σι ), ( j, i j)} ⊆ ( θ, θ),P θ = Pθ 0 0 0 , P θ = Pθ 0 0 0 , N θ, θ = Nθ , θ 0 0 0 . The matrix Ω ϑ = {ζ π 1 ,π 2 ϑ }, (π 1 , π 2 = 1, • • • , 4) is composed of sub-blocks:EQUATIONϑ Proof. To fulfil control objectives (i), first, the selection of the Lyapunov candidate function is made:EQUATIONwhereV P ϑ (k) = X T (k)P ϑ X(k), V Q ϑ (k) = k-1 ρ=k-η - X T (ρ)Q 0 ϑ X(ρ) + k-1-η - ρ=k-η X T (ρ)Q 1 ϑ X(ρ), V R ϑ (k) = η -1 ρ=-η - k-1 w=k+ρ Γ T (w)R 0 ϑ Γ(w) + ( η -η) -1-η - ρ=-η k-1 w=k+ρ Γ T (w)R 1 ϑ Γ(w), Γ(w) = X(w + 1) -X(w)Taking difference of (19) along the trajectory of closed-loop system (11) yieldsEQUATIONw=k-η Γ T (w)R 1 ϑ Γ(w).Applying Lemmas from [12] to the summation terms in ∆V R ϑ (k), it follows from condition ( 13) that-η - k-1 w=k-η - Γ T (w)R 0 ϑ Γ(w) ≤ -             k-1 w=k-η - Γ T (w)             T R 0 ϑ             k-1 w=k-η - Γ T (w)             ≤ -X T (k)R 0 ϑ X(k) + 2X T (k)R 0 ϑ X(k -η - ) -X T (k -η - )R 0 ϑ X(k -η - ) (20) -( η -η - ) k-η - -1 w=k-η Γ T (w)R 1 ϑ Γ(w) ≤ -X T (k -η - )R 1 ϑ X(k -η - ) + 2X T (k -η - )(R 1 ϑ -M 1 ϑ )X(k -η(k)) + 2X T (k -η - )M 1 ϑ X(k -η) + X T (k -η(k))[-2R 1 ϑ + M 1 ϑ + (M 1 ϑ ) T ]X(k -η(k)) + 2X T (k -η(k))(-M 1 ϑ + R 1 ϑ )X(k -η) -X T (k -η)R 1 ϑ X(k -η) (21)The error judgement function can be rewritten asEQUATIONwhere Φ2 σ = Z T Φ 2 σ Z, Z = 1 1 . Then, substituting ς θ, θ < 0 into (4) and (10) 22) into ∆V ϑ (k), according to condition (12) , thatyields p υ=1 ς θ, θ X T (k)(P θ -P θ + N θ, θ)X(k) ≥ 0. Substituting them with (20)-(∆V ϑ (k) ≤ ∆V p ϑ (k) + ∆V Q ϑ (k) + ∆V R ϑ (k) + γ σ X T (k -η(k)) Φ2 σ X(k -η(k)) -e T x (k)Φ 1 σ e x (k) + xT (k)S x x(k) + ûT (k)S û û(k) + xT (k)N iϕ 1 ,iϕ 2 x(k) + p υ=1 ς θ, θ X T (k)(P θ -P θ + N θ, θ)X(k) ≤ XT (k)Ψ ϑ X(k) < 0 (23) where X(k) = [X T (k)X T (k -η(k))X T (k -η - )X T (k - η)e Tx (k)x T a (k) T ] T ( θ, θ) ∈ {( j σι , i σι ), ( j, i j), (i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)} (iϕ 1 , iϕ 2 ) ∈ {(i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)} Second, discussed (19) on interval (κ q , κ q+1 ).Case A. On the interval (κ q , d ∼ ι), (σ, ϕ) = (i, σι ). On the interval (d ∼ ι , κ q+1 ), (σ, ϕ) = (i, i). According to (23) yieldsV ii (κ - q+1 ) ≤ V ii (d ∼ + ι ), k ∈ (d ∼ ι , κ q+1 ) (24)For controller switching instant d ∼ ιaffected by deception attack parameters, according to (4) yieldsV ii (d ∼ + ι ) = V i σι (d ∼ - ι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ) (25) On the interval [κ q , d ∼ ι), according to (23) yieldsEQUATION)Combining ( 14) and ( 24)-( 26), it can be deduced thatV ii (κ - q+1 ) ≤ V ii (d ∼ + ι ) = V i σι (d ∼ - ι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι)≤ V i σι (κ + q ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι)≤ V i σι (κ + q ) + X T (κ q )N i σι ,i X(κ q ) (27) Case B. On the interval (κ q , d ∼ ι), (σ, ϕ) = (i, σι ). On theEQUATIONFor controller switching instant d ∼ ι+ι q affected by deception attack parameters, according to (4) yieldsEQUATION)On the interval ( dι+ι q , d ∼ ι+ι q), according to (23) yieldsEQUATION)For controller switching instant dι+ι q caused by deception attacks, we getEQUATIONCombining ( 14)-( 15) and ( 28)-(31), it can be deduced thatV ii (κ - q+1 ) ≤ V ii (d ∼ + ι+ι q ) = V i σι+ιq (d ∼ - ι+ι q ) + X T (d ∼ ι+ι q )N i σι+ιq ,i X(d ∼ ι+ι q ) ≤ V i σι+ιq ( d+ ι+ι q ) + X T (d ∼ ι+ι q )N i σι+ιq ,i X(d ∼ ι+ι q ) = V ii ( d- ι+ι q ) + X T ( dι+ι q )N i,i σι+ιq X( dι+ι q ) + X T (d ∼ ι+ι q )N i σι+ιq ,i X(d ∼ ι+ι q ) ≤ V ii ( d- ι+ι q ) + X T ( dι+ι q )(N i,i σι+ιq + N i σι+ιq ,i )X( dι+ι q ) ≤ V ii ( d- ι+ι q ) + X T ( dι+ι q )N i,i X( dι+ι q ) ≤ V ii ( d- ι+ι q ) (32) Repeating the above derivation process on the interval k ∈ [ dι+1 , κ q+1 ) yields V ii (κ - q+1 ) ≤ V ii ( d- ι+ι q ) ≤ • • • ≤ V ii ( d- ι+1 ) (33) On the interval [d ∼ ι, dι+1 ), according to (23) yieldsEQUATIONFor controller switching instant d∼ ιaffected by deception attack parameters, according to (4) yieldsEQUATIONCombining ( 15)-( 16) and ( 33)-( 35), it can be deduced thatEQUATIONCase C. On the interval [κ q , dι ), (σ, ϕ) = (i, j).On the interval [ dι , d ∼ ι ), (σ, ϕ) = (i, σι ). On the interval [d ∼ ι , κ q+1 ), (σ, ϕ) = (i, i). Analyses on interval [d ∼ ι, κ q+1 ) are the same as in case A, according to (24)-(25) yieldsV ii (κ - q+1 ) ≤ V ii (d ∼ + ι ) = V i σι (d ∼ - ι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ), k ∈ [d ∼ ι , κ q+1 ) (37) On the interval [ dι , d ∼ ι), according to (23) yieldsEQUATIONFor controller switching instant dι caused by deception attacks, we getEQUATIONOn the interval [κ q , dι ), according to (23) yieldsEQUATIONCombining ( 14)-( 15) and ( 37)-( 40), it can be deduced thatV ii (κ - q+1 ) ≤ V ii (d ∼ + ι ) = V i σι (d ∼ - ι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ) ≤ V i σι ( d+ ι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ) = V i j ( d- ι ) + X T ( dι )N i j,i σι X( dι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ) ≤ V i j (κ + q ) + X T ( dι )N i j,i σι X( dι ) + X T (d ∼ ι )N i σι ,i X(d ∼ ι ) ≤ V i j (κ + q ) + X T ( dι )(N i j,i σι + N i σι ,i )X( dι ) ≤ V i j (κ + q ) + X T (κ q )N i j,i X(κ q ) (41) Case D. On the interval [κ q , dι ), (σ, ϕ) = (i, j). On the interval [ dι , d ∼ ι ), (σ, ϕ) = (i, σι ). On the interval [d ∼ ῑ, dῑ+1 ), (σ, ϕ) = (i, i), ῑ ∈ {ι,ι + 1, • • • , ι + ι q },dι+ι q +1 = κ q+1 . Analyses on interval [ dι+1 , κ q+1 ) are the same as in case B.On the interval [ dι , κ q+1 ), combining (33)-( 35) and ( 38)-(39) yieldsEQUATIONOn the interval [κ q , dι ), according to (23) yieldsEQUATIONAccording to condition (17), it implies that the upper bound of the quadratic cost function J ∞ is β. Up to this point, the asymptotic stability of the closed-loop system (11) is proved and the control objective (i) is achieved.To complete the proof of control objective (ii). Firstly, for the control input constraint U, the invariant ellipsoid is constructed asS = {X| ||X T P ϑ X|| ≤ β} (59) implies that max k {|û(k)| 2 } = max k {|K σ x(k)| 2 } =max k {|K σ ÎX(k)| 2 } =max k {|K σ ÎP -1 2 ϑ | 2 }{|P 1 2 ϑ X(k)| 2 } ≤ β|K σ ÎP -1 2 ϑ | 2 =βK σ ÎP -1 ϑ ÎT K T σ (60)At this point, the control input constraints are proved and the control objective (ii) is achieved.Next, we handle nonlinear terms in ( 12) and ( 14). Theorem 2: Given positive scalars η, η -, γ σ < 1, model prediction control weight matrix S x > 0, S û > 0 and ς θ, θ < 0. There exist matricesP ϑ > 0, Q ϑ > 0, R ϑ > 0, Φ 1 σ > 0, Φ 2 σ > 0 and M 1 ϑ with (ϑ, σ, ϕ) ∈ {(ii, i, i), (i j, i, j), (i σῑ , i, σῑ )}, ῑ ∈ {ι,ι + 1, • • • , ι + ι q }, scalar β > 0 and indeterminate matrix N θ, θsatisfying Ψϑ < 0 (61) Ωϑ < 0 (62)Then, β is an upper bound for J ∞ (k) and the closedloop system (11) is asymptotically stable. The matrix Ψϑ= { φρ 1 , ρ2 ϑ }, ( ρ1 , ρ2 = 1, • • • , 11) is composed of sub-blocks: φ1,1 ϑ = -P ϑ + Rϑ + Q 0 ϑ -R 0 ϑ + ÎT S x Î + ÎT K T ϕ S û K ϕ Î + N θ, θ + p υ=1 ς θ, θ(P θ -P θ + N θ, θ), φ1,3 ϑ = R 0 ϑ , φ1,7 ϑ = (Λ 1 ϑ ) T , φ1,11 ϑ = RT ϑ , φ2,2 ϑ = -2R 1 ϑ + M 1 ϑ + (M 1 ϑ ) T + γ σ Z T Φ 2 σ Z, φ2,3 ϑ = R 1 ϑ -(M 1 ϑ ) T , φ2,4 ϑ = R 1 ϑ -M 1 ϑ , φ2,8 ϑ = (Λ 2 ϑ ) T , φ3,3 ϑ = -Q 0 ϑ + Q 1 ϑ -R 0 ϑ -R 1 ϑ , φ3,4 ϑ = M 1 ϑ , φ4,4 ϑ = -Q 1 ϑ -R 1 ϑ , φ5,5 ϑ = -Φ 1 σ , φ5,9 ϑ = (Λ 3 ϑ ) T , φ6,10 ϑ = (Λ 4 ϑ ) T , φ7,7 ϑ = φ8,8 ϑ = φ9,9 ϑ = φ10,10 ϑ = λ 2 [4(P ϑ + Rϑ ) -Rϑ ] -2λI, φ11,11 ϑ = 4λ 2 Rϑ -2λI, Rϑ = η - 2 R 0 ϑ + ( η -η - ) 2 R 1 ϑ , Î = 1 0 , Z = 1 1 , (ϑ, σ, ϕ) ∈ {(i, i, i), (i j, i, j), (i σῑ , i, σῑ )},υ ∈ {i, j, σῑ }, ( θ, θ) ∈ {( j σι , i σι ), ( j, i j), (i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)}, ( θ1 , θ1 ) ⊆ ( θ, θ), ( θ1 , θ2 ) ⊆ ( θ, θ), (iϕ 1 , iϕ 2 ) ∈ {(i σῑ , i), (i, i σῑ ), (i j, i σι ), (i j, i)} ⊆ ( θ, θ), ( jϕ, iϕ)∈ {( j σι , i σι ), ( j, i j)} ⊆ ( θ, θ)P θ = Pθ 0 0 0 , P θ = Pθ 0 0 0 , N θ, θ = Nθ , θ 0 0 0 . The matrix Ωϑ = { ζ π1 ,π 2 ϑ }, (π 1 , π2 = 1, • • • , 4) is composed of sub-blocks: ζ1,1 ϑ = -N θ, θ, ζ1,5 ϑ = (Λ 1 ϑ ) T , ζ2,6 ϑ = (Λ 2 ϑ ) T , ζ3,7 ϑ = (Λ 3 ϑ ) T , ζ4,8 ϑ = (Λ 4 ϑ ) T , ζ5,5 ϑ = ζ6,6 ϑ = ζ7,7 ϑ = ζ8,8 ϑ = 4λ 2 N θ, θ -2λI.The other block matrices in Ψϑ and Ωϑ are 0 in the appropriate dimensions.Proof:Applying Lemma from [13] yields -[4(P ϑ + Rϑ ) -Rϑ ] -1 < λ 2 [4(P ϑ + Rϑ ) -Rϑ ] -2λI and -4N -1 θ, θ < 4λ 2 N θ, θ -2λI.Then, with the help of lemma [12] , Ψ ϑ < 0 can be deduced from Ψϑ < 0, and Ω ϑ < 0 from Ωϑ < 0. That is, (61)-(62) ensures that ( 12) and ( 14) hold. The proof is complete.In this section, the feasibility of the proposed contribution is verified by a NUMV example [1] . Select parameters η -= 0.02, η = 0.2, γ x 1 = 0.2, γ x 2 = 0.1 and model prediction control weight matrix S x = 0.01, S û = 0.1. The sampling period is 0.1 s. By solving conditions (61)-( 62), the controller gain and event-triggering parameters can be calculated as EQUATIONEQUATIONEQUATIONEQUATION                        Fig. 2 illustrates the depicted deception attack. Under the designed optimal security control policy based on model prediction, the optimal state trajectories and control inputs are plotted in Fig. 3 and Fig. 7 , respectively. Fig 4 illustrates the system state trajectory without optimal prediction parameter error detection, comparing with the optimal state trajectory in Fig. 3 , it can be clearly seen that the event-triggering mechanism based on the optimal prediction can converge faster and smoother, and cope with the prediction deviation effectively. Fig. 5 and Fig. 6 show the state trajectories and control inputs under observation-dependent state switching signals, respectively. Fig. 8 and Fig. 9 compare the subsystem switching signals of the optimal state-dependent design with the subsystem switching signals of the observed state-dependent design. The subsystem switching in Fig. 8 occurs at 4.1s, 10.6s, 18.7s, 29.2s, 36.8s and 47s for a total of six switching behaviours. While the number of subsystem switching occurs more frequently in Fig. 9 for the subsystem switching signals designed by relying on the observed states, which shows that the optimal state-based switching rule designed in this paper effectively avoids unnecessary switching, especially when the system state fluctuates near the switching boundary. Combined with the comparison of state trajectories, the advantages of the predictive state-based switching rule are verified to help select the optimal subsystem during switching and optimise the overall performance of the system thereby improving the safety of the system. In this paper, we study the security control problem of discrete switching system based on model prediction under spoofing attack. An event triggering mechanism based on optimal prediction is designed to shorten the asynchronous time between subsystems and controllers, which helps the system to make timely adjustments when the prediction deviation occurs, and reduces the impact of the attack on the performance of the system; a new quadratic cost function is proposed to take into account the extra cost of modal switching of the controllers in solving the optimal security control policy, and to avoid unnecessary and frequent switching, and at the same time, find a smoother and more stable control policy. A smoother and more stable control strategy is found to reduce the system fluctuation caused by switching; a subsystem switching rule based on the optimal prediction state is designed to select the optimal subsystem switching to optimise the overall system performance; sufficient conditions for asymptotic stability of the closed-loop system under the above event-triggering mechanism and optimal safety control strategy are given, and the validity of the proposed method is verified by the unmanned ship mass-switching dynamic positioning system model. 
 
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Reviews: review: title: title: This paper focuses on model prediction-based security control for discrete switching systems under deception attacks. An event-triggering mechanism based on the optimal prediction is designed to shorten the asynchronous time between the subsystem and the controller, which helps the system adjust when the prediction deviates and reduces the impact of the attack on the system performance. The referee thinks it contains some publishable materials and it is worthy of publishing after some revisions.
main_review: main_review: 1. What are the significant differences between this study and previous studies? The author needs more explicit emphasis.
2. The legend of Figure 9 should not overlap with the image, and the authors are requested to correct it.
3. The disadvantage of the proposed method and future work must be described in conclusion.
title: title: The obtained result is valuable and can be accepted after responding the following comments.
main_review: main_review: This article focuses on the development of a model-based predictive security control strategy for discrete switching systems under deception attacks. This article presents a comprehensive framework for predictive security control of discrete switching systems under deception attacks. The proposed method not only improves the security and stability of the system but also provides theoretical insights and practical guidelines for the design of such systems.The obtained result is valuable and can be accepted after responding the following comments. 
 (1) In the introduction, it is not enough to state the current work. It should be expended and reconstructed. (2) In the simulation section, more analysis can be added to better explain the main results of this paper, that's not enough. (3) There are many typos and grammar errors. The authors should have a native English speaker or software packages to perform the editing check. (4) The conclusion of the article suggests using the present perfect tense for description
title: title: accept
main_review: main_review: This paper considered the model-based predictive control for discrete switching systems under deception attacks. The theory is correct and can be accepted after responding the following comments.
(1)	In the introduction, it is not enough to state the current work. It should be expended and reconstructed.
(2)	In the simulation section, more analysis can be added to better explain the main results of this paper, that's not enough.
(3)	There are many typos and grammar errors. The authors should have a native English speaker or software packages to perform the editing check.