Robust distributed model predictive control of connected vehicle platoon against DoS attacks

2.1. Communication graph

Considering that there is one leading vehicle and $$ N $$ following vehicles. We represent the communication interactions among vehicles by using an undirected communication graph^[32]. For the graph $$ \xi $$, the adjacency matrix $$ {{\rm{{\cal A}}}_N} $$ is defined as $$ {{\rm{{\cal A}}}_N} = [{a_{ij}}] \in {R^{N \times N}}{\rm{ }}\, (i, j = 1, ..., N) $$, $$ {a_{ij}} = 0 $$ denotes the $$ i $$-th following vehicle cannot obtain communication from the $$ j $$-th following vehicle; otherwise, $$ {a_{ij}} = 1 $$. It is assumed that $$ {a_{ii}} = 0 $$. The Laplace matrix is defined as $$ {\rm{{\cal L}}_N} = [{l_{ij}}] \in {R^{N \times N}}\, (i, j = 1, ..., N) $$, where $$ {l_{ij}} = - {a_{ij}}(i \ne j) $$ and $$ {l_{ii}} = \sum\limits_{m = 1, m \ne i}^N {{a_{im}}} $$. We define $$ {\rm{{\cal P}_N}} = diag\{ {b_1}, \cdots , {b_N}\} $$, $$ {b_i} = 0 $$ denotes the $$ i $$-th following vehicle cannot obtain communication from the leading vehicle; otherwise, $$ {b_i} = 1 $$.

2.2. Vehicle model

The longitudinal dynamics model of a vehicle is nonlinear in practice. The vehicle model has been linearized in many articles by using feedback linearization techniques to simplify the analysis^[33,34]. In this paper, it is assumed that the longitudinal dynamics of the $$ i $$-th following vehicle and the leading vehicle are given as follows ^[35,36]:

where

with $$ {p_i}(t) $$, $$ {v_i}(t) $$, $$ {a_i}(t) $$, and $$ {u_i}(t) $$ being the position, velocity, acceleration, and control input of the $$ i $$-th following vehicle, respectively. $$ {p_0}(t) $$, $$ {v_0}(t) $$, and $$ {a_0}(t) $$ are the position, velocity, and acceleration of the leading vehicle, respectively. $$ {\sigma _i} $$ and $$ {\sigma _0} $$ denote the engine inertia time constant for the $$ i $$-th following vehicle and the leading vehicle. In this paper, we consider the case that all vehicles have the same mathematical models.

By discretizing systems Equations (1) and (2), we can obtain the following discrete-time systems^[33]

where $$ T $$ is the sampling period, and $$ \sigma > 0 $$ is the vehicle engine inertia time constant.

2.3. DoS attacks

In this paper, it is assumed that all communication channels are blocked, and all following vehicles cannot receive any real-time data when DoS attacks occur. Define $$ {\left\{ {{k_j}} \right\}_{j \in {\rm{{\cal N}^+}}}} $$ be the moment when DoS attacks occur, and $$ {\tau _j} $$ is the duration of the $$ j $$-th DoS attacks, where $$ {\tau _j} \ge 0 $$. Define the activated time interval at $$ j $$-th DoS attacks as^[37]

Define $$ \Psi \left( {\omega , k} \right) $$ and $$ \Gamma \left( {\omega , k} \right) $$ as the time interval sets that the DoS attacks are active and inactive in the time interval $$ \left[ {\omega , k} \right] $$, respectively. $$ {{\Psi _{tot}}\left( {\omega , k} \right)} $$ and $$ {{\Gamma _{tot}}\left( {\omega , k} \right)} $$ represent the lengths of $$ \Psi \left( {\omega , k} \right) $$ and $$ \Gamma \left( {\omega , k} \right) $$, respectively. Define $$ \phi $$ as the DADR in the time interval $$ \left[ {\omega , k} \right] $$, where $$ \phi = \frac{{{\Psi _{tot}}}}{{k - \omega }} $$. The descriptions corresponding to this statement are as follows:

Then, two assumptions about the frequency and duration of DoS attacks are given. $$ n\left( {\omega , k} \right) $$ is the number of DoS attacks occurring in the time interval $$ \left[ {\omega , k} \right] $$.

Assumption 1 (DoS attacks frequency): There exist positive constants $$ \kappa \in R{_{ \ge 0}} $$ and $$ {\tau _D} \in R{_{ > 0}} $$, satisfying

for all $$ \omega , k \in {R_{ \ge 0}} $$ with $$ k > \omega $$.

Assumption 2: For the duration of DoS attacks, there exist positive constants $$ \eta \in R{_{ \ge 0}} $$ and $$ {T_a} \in R{_{ > 1}} $$, satisfying

for all $$ \omega , k \in {R_{ \ge 0}} $$ with $$ k > \omega $$. We assume that $$ \eta + \frac{{k - \omega }}{{{T_a}}} $$ is the maximum total duration of DoS attacks in the time interval $$ \left[ {\omega , k} \right] $$.

Remark 1: The behavior of DoS attacks has been studied in many articles ^[37–39]. However, in practical applications, it is difficult to determine the accurate statistical parameters of DoS attacks for controller design. Considering that the energy of attackers is limited, we model DoS attacks by the frequency and duration parameters under Assumption 1 and Assumption 2. Such a modeling approach can also be found in Ref.^[37,40], which can capture a wide range of different types of DoS attacks. Therefore, Assumption 1 and Assumption 2 are physically meaningful.

Remark 2: There are two methods to control the system when DoS attacks occur. One sets the system control input to zero, and the other keeps the system input of the last value. When DoS attacks occur, the hold input method can be used to update the control input with past data in the buffer, but it can also cause the time delay phenomenon. It is worth noting that both of the above methods are applicable^[41]. In our work, the zero-input method is used.

2.4. Problem description

To control the CVP system, the following control protocol is introduced ^[21]:

where $$ {k_p} $$, $$ {k_v} $$, and $$ {k_a} $$ are the local feedback gains of the system. $$ {d_{ij}} $$ is the ideal distance between the $$ i $$-th and the $$ j $$-th following vehicle. $$ {d_{i0}} $$ is the ideal distance between the $$ i $$-th following vehicle and the leading vehicle.

Define the tracking error as

where $$ {\bar p_i}(k) $$, $$ {\bar v_i}(k) $$, and $$ {\bar a_i}(k) $$ represent the position error, velocity error, and acceleration error of the actual and ideal state of the $$ i $$-th following vehicle at the sampling time $$ k $$, respectively. Let $$ K = [{k_p}, {k_v}, {k_a}] $$ represent the controller gain.

Then the protocol Equation (5) can be written as

where $$ {e_i}(k) = {\left[ {\begin{array}{*{20}{c}} {{{\bar p}_i}^T(k)}&{{{\bar v}_i}^T(k)}&{{{\bar a}_i}^T(k)} \end{array}} \right]^T} $$.

The main goal of this paper is to maintain the ideal spacing, speed, and acceleration between the following vehicle and the leading vehicle. So, the desired state equation of the $$ i $$-following vehicle is

where $$ \Delta x = \left[ {\begin{array}{*{20}{l}} {{d_{i0}}}&0&0 \end{array}} \right] $$.

The actual state equation of the $$ i $$-th following vehicle is

Through Equation (7) and Equation (8), the tracking error of the $$ i $$-th following vehicle is obtained as follows:

Let $$ e(k) = {\left[ {\begin{array}{*{20}{c}} {{e_1}^T(k)}&{{e_2}^T(k)}& \cdots &{{e_N}^T(k)} \end{array}} \right]^T} $$ and $$ u(k) = {\left[ {\begin{array}{*{20}{c}} {{u_1}^T(k)}&{{u_2}^T(k)}& \cdots &{{u_N}^T(k)} \end{array}} \right]^T} $$, we can get

Due to the fact that no data can be received by those vehicles when the DoS attack occurs, we now introduce a signal $$ \delta (k) \in \left\{ {0, 1} \right\} $$ to describe whether the DoS attacks occur or not. Then the system can be described as

Let $$ {{\rm{{\cal H}}}_N} = {{\rm{{\cal L}}}_N} + {{\rm{{\cal P}}}_N} $$. It follows from Equation (11) that the system under DoS attacks is essentially a switched system with two subsystems. Thus, Equation (11) can be written as

Define $$ {{\rm T}_j} $$ and $$ {{\rm T}_j}^ - $$ as the $$ j $$-th switching instant of $$ {\delta (k)} $$ and the instant immediately before reaching $$ {{\rm T}_j} $$. The following lemma and definition will be used in the derivation of the main results.

Lemma 1^[42]: Since the matrix $$ {\cal H}_N $$ is a symmetric non-singular matrix, there exists an orthogonal matrix $$ M $$, satisfying

where $$ {\lambda _i}(i = 1, 2, ..., N) $$ are the eigenvalues of $$ {\rm{{\cal H}}_N} $$.

Definition 1^[43]: If there exist constant $$ \rho \in (0, + \infty ) $$ and $$ \varsigma \in (0, 1) $$ such that

is true, then the system Equation (12) is said to be exponentially stable, and $$ \varsigma $$ is the exponential decay rate.

3.1. Stability analysis

The main theorems in this section are given as follows to demonstrate that system Equation (12) can be exponentially stable with the designed controller Equation (5). Define the total number of switches in the time interval $$ \left[ {0 , k} \right] $$ as $$ n = n(0 , k) + {n_1} $$. If the system switches to the case of no DoS attacks in the end, then $$ {n_1} = n(0 , k) $$; otherwise, $$ {n_1} = n(0 , k) - 1 $$.

Theorem 1: Considering the parameters of DoS attacks as in Assumption 1 and Assumption 2. For the given positive scalars $$ \alpha \in (0, 1) $$, $$ \beta \in (0, + \infty ) $$, and $$ \mu \in (1, + \infty ) $$, if there exists a positive constant $$ \theta\in (1, + \infty ) $$ and symmetric positive definite matrices, $$ {P_0} $$ and $$ {P_1} $$, such that

hold, then the system (12) can be ensured to be exponentially stable with an exponential decay rate of $$ {\theta ^{ - \frac{{(\varphi - 2)}}{2}}}, \varphi > 2 $$, where $$ {\lambda _{max}} $$ is the maximum eigenvalue of $$ {\rm{{\cal H}}_N} $$.

Proof: When $$ k \in \Gamma \left( {\omega , k} \right) $$, by left- and right-multiplying Equation (18) by $$ {e_i}^T(k) $$ and its transposition, respectively, one can see that

Then we have

Let $$ \varepsilon (k) = (M^T \otimes {I_N})e(k) $$, with the help of Lemma 1, Equation (23) can be written as

Thus, it is obtained that

Now left- and right-multiplying Equation (25) by $$ {e}^T(k) $$ and its transposition, respectively, yields

Considering the Lyapunov function Equation (15), Equation (26) can be written as

When $$ k \in \Psi \left( {\omega , k} \right) $$, by left- and right-multiplying Equation (19) by $$ {e_i}^T(k) $$ and its transposition, respectively, one can see that

Obviously, Equation (28) can guarantee that

Similar to the derivation of Equation (27), Equation (29) can be written as

Thus, the conditions Equation (18) and Equation (19) can guarantee that

Considering the conditions Equation (20) and Equation (21), one has

According to Equation (31) and Equation (32) and by applying the iterative method to the time interval $$ [0, k] $$, we can obtain

The conditions Equation (16) and Equation (17) guarantee that

Let $$ c = {\mu ^{2\kappa }}{(\frac{{1 + \beta }}{{1 - \alpha }})^\eta } \in R {_{ > 0}} $$, Equation (34) can be written as

where $$ a = {\lambda _{\min }}({P_{\delta (k)}}) $$ and $$ b = {\lambda _{\max }}({P_{\delta (0)}}) $$. $$ {\lambda _{\min }}({P_{\delta (k)}}) $$ is the minimum eigenvalue of $$ {P_{\delta (k)}} $$ and $$ {\lambda _{\max }}({P_{\delta (0)}}) $$ is the maximum eigenvalue of $$ {P_{\delta (0)}} $$.

Then it yields

From Equation (36) and Definition 1, the system (12) is exponentially stable with an exponential decay rate of $$ {\theta ^{ - \frac{{(\varphi - 2)}}{2}}} $$. This ends the proof.

It follows from Theorem 1 that for given $$ \alpha $$, $$ \beta $$, $$ \mu $$ and DoS attacks parameters $$ \kappa $$, $$ \tau _D $$, $$ \eta $$, $$ T_a $$, we have

With the help of Definition 1, the exponential decay rate $$ \varsigma $$ can be written as

Remark 3: In the previous section, a quantitative relationship between DoS attack parameters and the exponential decay rate has been established. It can be seen that both the frequency and duration of DoS attacks have an impact on the exponential decay rate, which affects the performance of the system. For example, as the duration of DoS attacks increases, the exponential decay rate becomes larger.

3.2. Upper bound of DADR

As it is well known, if the total duration of a DoS attack is infinite, the system cannot be stable. Therefore, it is necessary to derive the upper bound of DARA for the system (12). To achieve this, we present the stability theorem as follows based on the DARA:

Theorem 2: For the given positive scalars $$ \alpha \in (0, 1) $$, $$ \beta \in (0, + \infty ) $$, and $$ \mu \in (1, + \infty ) $$, if the upper bound of DARA is $$ \phi < {\phi _{\max }} = \frac{{ - \frac{{2\ln \mu }}{{{\tau _D}}} - \ln (1 - \alpha )}}{{\ln \left( {\frac{{1 + \beta }}{{1 - \alpha }}} \right)}} $$, and if there exists a positive scalar $$ \theta\in (1, + \infty ) $$ and symmetric positive definite matrices, $$ {P_0} $$ and $$ {P_1} $$, such that the inequalities Equation (18)-Equation (21) hold, then the system (12) is exponentially stable.

Proof: Similarly to the analysis in Theorem 1, we can obtain

Similarly to Equation (35), we can obtain

By substituting $$ \phi < {\phi _{\max }} = \frac{{ - \frac{{2\ln \mu }}{{{\tau _D}}} - \ln (1 - \alpha )}}{{\ln \left( {\frac{{1 + \beta }}{{1 - \alpha }}} \right)}} $$ to Equation (40), we can obtain that $$ \frac{2}{{{\tau _D}}}\ln \mu + \phi \ln (\frac{{1 + \beta }}{{1 - \alpha }}) + \ln (1 - \alpha ) < 0 $$ and $$ 0 < {e^{\frac{1}{2}\left[ {\frac{2}{{{\tau _D}}}\ln \mu + \phi \ln (\frac{{1 + \beta }}{{1 - \alpha }}) + \ln (1 - \alpha )} \right]}} < 1 $$. From Definition 1, the system (12) is exponentially stable. This completes the proof.

3.3. Controller design

Based on Theorem 1, we now present the robust model predictive control algorithm. The state feedback control gain will be obtained by solving an optimization problem in the form of linear matrix inequalities (LMIs) at each sampling moment. The following theorem gives the design of the controller.

Theorem 3: Considering a CVP system consists of the plant Equation (12) and the controller Equation (5). Let $$ e(k) = e(k|k) $$ be the state of the system (12) measured at sampling time $$ k $$. DoS attack parameters satisfy conditions Equation (16) and Equation (17). For the given constant scalar $$ {u_{\max }} $$ and positive scalars $$ 0 < \alpha < 1 $$, $$ \beta > 0 $$, $$ \mu > 1 $$, $$ \theta > 1 $$, and $$ \varphi > 2 $$, if there exist matrices, $$ Q_0 $$, $$ {{Q_1}} $$, and $$ Y $$, and a positive scalar $$ \gamma>0 $$ such that

subject to

where $$ {Q_0} = \gamma {P_0}^{ - 1} $$, $$ {Q_1} = \gamma {P_1}^{ - 1} $$, and $$ Y = KQ_0 $$. If there is a solution to the above linear minimization problem, then the state feedback matrix $$ K $$ with sampling moment $$ k $$ can be obtained by the following equation:

and the system (12) can be ensured to be exponentially stable with an exponential decay rate of $$ {\theta ^{ - \frac{{(\varphi - 2)}}{2}}} $$.

Proof: We aim to design a state feedback control law at each sampling time that minimizes an infinite horizon global objective function. First, we define the infinite horizon local objective function for the $$ i $$-th following vehicle in the CVP as

where $$ {e_i\left( {k+t|k} \right)} $$ and $$ {u_i\left( {k+t|k} \right)} $$ represent the predictions of state and control input at sampling time $$ k $$, and $$ W $$ and $$ R $$ are state and control weighting matrices, respectively.

Adding up the infinite horizon local objective function of all following vehicles as the global objective function, so the global objective function is

Left- and right-multiplying Equation (43) by $$ diag\{ {Q_0}^{ - 1}, I, I, I\} $$ and its transposition, respectively. By Schur complement, one has

Left- and right-multiplying Equation (52) by $$ {e_i}^T(k) $$ and its transposition, respectively, yields

Then, it can be derived

where $$ {\Pi _1} = {I_N} \otimes A + \Lambda \otimes BK $$. Let $$ \varepsilon (k) = (M^T \otimes {I_N})e(k) $$, and with the help of Lemma 1, Equation (54) can be written as

where $$ {\Pi _2} = {I_N} \otimes A + {\rm{{\cal H}}_N} \otimes BK $$. Obviously, Equation (55) guarantees that

Left- and right-multiplying Equation (56) by $$ {e^T}(k + t|k) $$ and its transposition, respectively, yields

which leads to

where $$ {{\Pi _3} = {e^T}(k + t|k)\left( {{I_N} \otimes W} \right)e(k + t|k)} $$ and $$ {{\Pi _4} = {u^T}(k + t|k)\left( {{I_N} \otimes R} \right)u(k + t|k)} $$. Therefore, the condition Equation (43) can guarantee that Equation (58) hold. Then it can be obtained from Equation (58) that

Let Equation (59) be an iterative summation from $$ t = 0 $$ to $$ t=\infty $$, one has

In order to make the robust performance objective function finite, we define $$ e(k+\infty |k) = 0 $$, so Equation (60) can be written as

By Schur complement, Equation (61) can be rewritten as

By substituting $$ {Q_0} = \gamma {P_0}^{ - 1} $$ to Equation (62), it can be seen that Equation (42) and Equation (61) are equivalent.

The control input of any following vehicles in CVP control is constrained. For the $$ i $$-th following vehicle, satisfying $$ {\left\| {{u_i}(k)} \right\|^2} \le {({u_{\max }})^2} $$

where

By using Schur complement, the condition Equation (44) can guarantee that Equation (63) holds.

Left- and right-multiplying Equation (45) by $$ diag\{ {{{Q_0}^{ - 1}}}, I\} $$ and its transposition, respectively, it is obtained that

By substituting $$ {Q_0} = \gamma {P_0^{ - 1}} $$ into the previous equation, Equation (64) can be written as

Left- and right-multiplying Equation (46) by $$ diag\{ {{{Q_1}^{ - 1}}}, I\} $$ and its transposition, respectively, yields

By substituting $$ {Q_1} = \gamma {P_1^{ - 1}} $$ into the previous equation, Equation (66) can be written as

Left- and right-multiplying Equation (47) by $$ diag\{ {{{Q_0}^{ - 1}}}, I\} $$ and its transposition, respectively, yields

By substituting $$ {Q_1} = \gamma {P_1^{ - 1}} $$ and $$ {Q_0} = \gamma {P_0^{ - 1}} $$ into the previous equation, Equation (68) can be written as

Left- and right-multiplying Equation (48) by $$ diag\{ {{{Q_1}^{ - 1}}}, I\} $$ and its transposition, respectively, yields

By substituting $$ {Q_1} = \gamma {P_1^{ - 1}} $$ and $$ {Q_0} = \gamma {P_0^{ - 1}} $$ into the previous equation, Equation (70) can be written as

With the help of Theorem 1, one can see that the hold of conditions Equation (45), Equation (46), Equation (47), Equation (48), Equation (16), and Equation (17) can guarantee that the system (12) is exponentially stable with an exponential decay rate of $$ {\theta ^{ - \frac{{(\varphi - 2)}}{2}}} $$. The controller gain can be obtained by solving an optimization problem based on the form of LMIs and $$ K = Y{{Q_0}^{ - 1}} $$. This completes the proof.

Authors' contributions

Made substantial contributions to the research, idea generation, algorithm design, and simulation and wrote and edited the original draft: Zeng H

Performed critical review, commentary, and revision and provided administrative, technical, and material support: Ye Z, Zang D, and Lu Q

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Conflicts of interest

All authors declared that there are no conflicts of interest.

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Copyright

© The Author(s) 2023.