马永胜 ,车伟伟 ,邓 超

(1.青岛大学自动化学院,山东省工业控制技术重点实验室,山东青岛 266071;2.南京邮电大学先进技术研究院,江苏南京 210000)

1 Introduction

Recently,the control problem of direct-current(DC) microgrids has attracted vast attentions due to the wide applications,such as electric vehicles and electric aircrafts.In DC microgrids,the power electronic converter loads are usually considered as constant power loads(CPLs),and play an important role.Since the effects of the negative impedance of CPLs,some nonlinear control methods have been proposed to deal with them.A novel composite nonlinear controller is designed to stabilize the CPLs in [1].An adaptive backstepping controller design method is proposed for DC microgrids to feed nonideal CPLs in[2].

It is worth mentioning that the power of CPLs is generally unknown.The knowledge of the power can be obtained by using the states of DC microgrids,but they are not always measurable.So,estimating the states of DC microgrids is necessary.The extended Kalman filter is used to estimate the power of CPLs based on the measuring capacitor voltages of DC microgrids in[3].The nonlinear model predictive controller design method based on the pseudo-extended Kalman filter is proposed in[4].Additionally,in order to save communication resources,the event-triggered control is widely used [5-6].The event-triggered design problem of physical layer and cyber layer for the isolated microgrid is studied in[5].

It is noted that the aforementioned researches assume that the wireless network for data transmission is perfect.But there will be network attacks,faults and other factors in the actual wireless network.Therefore,several research works have been contributed to this problem [7-13].In [10],the problem of the resilient filter design for power systems under denial-of-service(DoS)attacks is studied.In[11],the distributed resilient secondary control framework is established to alleviate the influence of unbounded attacks.In [12],the eventtriggered attack-resilient control problem for DC microgrids is studied.In[13],the performance of DC microgrids using the state feedback controller is investigated under DoS attacks.However,the actual states of DC microgrids are used to design the controller in [12]and[13].Generally speaking,it is difficult to obtain the actual states of DC microgrids due to the internal states of DC microgrids are not always measurable.Therefore,it is necessary to design the observer to estimate the unmeasurable states of DC microgrids.

Inspired by the formerly mentioned problem,this paper studies the observer-based event-triggered control problem for DC microgrids with DoS attacks,which still lacks relevant study to the best of the authors’knowledge.The main contributions of this paper are summarized as follows:

1) Due to the fact that not all the states of DC microgrids can be measured directly,an observer is designed in this paper to estimate the unmeasurable states of DC microgrids,which is different from the existing state feedback controller design results for DC microgrids[12-14].Then,a novel event-triggered controller design method using the observer states is proposed for DC microgrids under DoS attacks,in which the problem of the controller design can be converted into a convex problem by proposing an improved separation method.Additionally,the Zeno behavior is eliminated in the designed event-triggered mechanism by using the contradiction method.

2) Unlike the existing results [15-16] on DoS attacks,in which the upper bound of DoS attacks tolerance (the frequency and the total duration of DoS attacks)is related to the eigenvalue of the Lyapunov matrix,while the DoS attacks tolerance obtained in this paper is independent of it.

The remainders of this paper are as follows.The system modeling and problem formulation are provided in Section 2.The security analysis is presented in Section 3.The design method of the observer-based eventtriggered controller is given in Section 4.In Sections 5,a practical microgrid example is shown to testify the effectiveness of the designed controller.Finally,conclusions are presented in Sections 6.

The notations used in this paper are defined as follows.The asterisk∗represents the symmetry term in the matrix.Irepresents the identity matrix.H ＞0 indicatesHis a positive definite matrix.diag{.}stands for the diagonal matrix.N represents the positive real number.M Ndenotes the set belongs to the setMbut not the setN.D+represents the upper right-hand derivative.

2 System modeling and problem formulation

2.1 DC microgrids modeling

The DC microgrids can be decoupled intoN+1 subsystems withNCPLs and an energy storage system(ESS).The dynamic of thejth CPL can be obtained as

wherexj(t)[xj1(t)xj2(t)]Twithxj1(t)andxj2(t)being the inductor current and the capacitor voltage of thejth CPL,respectively.xs(t)is the state of the ESS.yj(t)is the measurement output,and

withrj,lj,cjandpjbeing the filter resistance,inductance,capacitance and the power of thejth CPL.hj(xj(t))is a nonlinear function ofxj(t).

The dynamic of the ESS is obtained as

wherexs(t)andbeing the inductor current and the capacitor voltage of the ESS,respectively.Vdcis the DC voltage,and

withrs,ls,csandys(t)being the same as defined in(1).

The following nonlinear system of DC microgrids can be obtained by combining with(1)and(2):

where

and

To solve the nonlinear problem,the coordinate transformation is used to move the equilibrium point of the original system to the coordinate origin.Then,one can obtain that the following equivalent system of DC microgrids[14]:

Correspondingly,the output equation of(4)can be obtained as

Remark 1It is noted that the purpose of the coordinate transformation is to facilitate the handling of the nonlinear termmentioned in (5),which provides the help for the stability analysis of DC microgrids in Section 3.

2.2 Event-triggered controller design

It is worth pointing out that the capacitor voltage is available in the DC microgrids,while the inductor current is not available.To estimate the unavailable inductor current of DC microgrids,the observer is designed as

Based on the observer state,the controller is designed as

whereKrepresents the control gain matrix.

Obviously,the above designed controller is continuously updated.In order to reduce the number of the controller updates,the event-triggered method is introduced to update the control signal.Suppose that the set of the event-triggered instant is determined as{tk,k0,1,...}.Then,the event-triggered function is designed as

The control signal will be updated at the eventtriggered sequence{tk},which can be obtained by the following event-triggered condition:

Based on the event-triggered mechanism (10),the control inputis rewritten as

According to(4),(7)and(11),the following closedloop system can be obtained:

2.3 DC microgrids modeling under DoS attacks

The time interval without DoS attacks is given by

The upper bound of two consecutive inter-event time is defined asΔ*,i.e.,Δ*It is noted that the event-triggered mechanism does not work in[0,Δ*),i.e.,the control input does not update in[0,Δ*).In addition,during the time of DoS attacks,the control input also does not update,which is similar to the former case.Therefore,the duration of“actual effective”DoS attacks include[0,Δ*)and the actual DoS attacks duration.

Then,the total time interval of DoS attacks can be obtained as follows:

The total valid communication time interval can be represented as

Assumption 1Denotingna(t0,t)as the amount of DoS attacks occurring in [t0,t),then there existsFa(t0,t)＞0 defined the frequency of DoS attacks over[t0,t)such that

Assumption 2Defining|Ξa(t0,t)|as the total time interval of DoS attacks during[t0,t),then the total duration of DoS attacks satisfies the following constraint:

whereτa＞1 andΞ0＞0 are constants to be determined.

Remark 2Assumptions 1 and 2 have been used in[15-18].In Assumption 1,it specifies the frequency of DoS attacks.Assumption 2 implies that the total duration of DoS attacks is constrained by the certain fraction of time.Besides,the role ofΞ0is to consider the case that DoS attacks exist at the start time.

2.4 Problem formulation

For nonlinear system (4) under DoS attacks,the goal of this paper is to design a resilient observer-based event-triggered controller to guarantee(4)is asymptotically stable under DoS attacks,that is,

Lemma 1[17] If the functionV(t)＞0 satisfies

3 Stability analysis

The following theorem is the main result that guaranteeing the studied problem can be solved.

Theorem 1For known control gainK,observer gainGand constantsa1＞0,a2＞0,(0,a1),θ ＞0 andβ ＞0.The studied problem can be solved,if there exist positive definite matrixUdiag{P,Q}and positive constantΔ*such that the following inequalities hold:

and DoS attacks satisfy

where

with

ProofThe proof is divided into two parts:the stability analysis and the excluding of the Zeno behavior.

Part-I(The stability analysis): Define the Lyapunov function asV(t)zT(t)Uz(t) withUdiag{P,Q}.If there are no DoS attacks in DC microgrids,i.e.,1,the derivative ofV(t) is calculated as

Using the Young’s inequality,one has

withβbeing a positive constant andFdiag{F1,F2,...,Fi,...,FN,0}withFidiag{0,1}.

It is noted that the event-triggered condition satisfies the inequalitymT(t)m(t) ≤θ2eT(t)CTCe(t).Then,one can get the following inequality:

According to(21),one has

If there are DoS attacks in DC microgrids,i.e.,2.Similar to the situation without DoS attacks in DC microgrids,one can obtain the following inequality according to(20)and(22):

and the boundedness of all signals can be guaranteed according to Part-I.

From Part-I,one can obtain thate(t)is convergent,which means that there exist positive constantsκandλsuch that‖e(t)‖≤κe-λ(t-tk)‖e(tk)‖holds.Sincem(tk)0,then

whereφ(tk)Υ+κ‖GC‖‖e(tk)‖andΔ(tk)＞0 according to(45)in[19].

Combining with the event-triggered function (9)and(33),one has

Then,the lower bound of the inter-event intervaltk+1-tkis

Then,one can obtainτ≥2∊0.Invoking thattk ＞t*-∊0,k≥N0,one hastk+1≥tk+τ≥t*+∊0,which contradicts the fact thattk+1≤t*,k≥N0.Similar to the condition0,the conclusion of the condition‖A‖0 can be drawn.Therefore,there is no Zeno behavior in the event-triggered mechanism.

Remark 3The DoS attacks tolerance is related to several positive constantsa1,a2,andΔ*according to Theorem 1.However,in[15-16],the DoS attacks tolerance is related to the eigenvalue of the Lyapunov matrix.Thus,compared with[15-16],the DoS attacks tolerance constraint is relaxed.

4 Controller and observer design

and DoS attacks satisfy(23)and(24),where

Furthermore,the observer and controller gains are calculated byKZW-1andGQ-1Y,respectively.

where

Obviously,it can be seen thatΓ1＜0,Ξ6+δ1P ＜0 andΓ3＜0,Ξ8+δ1P ＜0 ensure inequalities(21) and (22) holding,respectively.It is obvious thatΓ1,Ξ6+δ1PandΓ3,Ξ8+δ1Pare nonlinear matrix inequalities.In order to transform them into LMIs,the following arrangement will be done.

Using the Schur complement Lemma and multiplying left and right with the matrix diag{P-1,P-1,P-1}forΓ1lead to

where

Further,Γ5can be rewritten asΓ5Γ6+Γ7,where

Γ6＜0 andδ2P-1-P-1P-1＜0 guaranteeΓ5＜0.In addition,it is obvious thatδ2P-1-P-1P-1＜0 is equivalent toδ2I -P-1＜0.

Multiplying left and right with the matrix diag{P-1,P-1}forΓ3leads to

where

Then,using the Schur complement Lemma and definingξβ-2,WP-1andZKWresult in(37)and(38).Besides,LMIs(39)can be obtained by using the Schur complement Lemma and definingYQGforΞ6+δ1PandΞ8+δ1P.

It is worth pointing out that the lower boundof the robustness parameterβcan be computed by using the optimization algorithm mentioned in[14].Additionally,a large robustness parameter can be obtained by optimizingξ(ξβ-2),which implies inequality(35)holding.

Remark 4Different from the observer-based controller design method in [20] and [21],in which the control input matrix is required to be full column rank.By using the improved separation method,the full column rank constraint of the control input matrix is removed in this paper.

5 Simulation

Selecting the initial state as[0.7-0.2-0.3 0.45 0.9-0.1]Tand ˆx(t0)[-5.5 4 5-7-6 4]T.

It can be seen from Fig.1 that the nonlinear DC microgrids using the proposed observer-based eventtriggered controller is asymptotically stable under DoS attacks.Fig.2 plots the inter-event intervaltk+1-tkof the control input.It can be seen that the designed eventtriggered mechanism can effectively reduce the number of the controller updates.

Fig.1 State trajectories of DC microgrids under DoS attacks

Fig.2 Event-triggered interval

In addition,Table 2 is provided to further illustrate the effectiveness of the proposed method compared with the traditional control method.As can been seen from Table 2 that the estimation error using the proposed method is less than the traditional control method,which implies that the DC microgrids using the proposed method have better performance under DoS attacks.

Table 1 _Parameters with two___C_P_L_

Table 2 Norm-2 of the estimation error

6 Conclusions

The observer-based event-triggered control problem is investigated for DC microgrids with DoS attacks in this paper.A novel event-triggered controller using the observer states is designed to stabilize the DC microgrids under DoS attacks,in which the controller design problem can be converted into a convex problem by proposing an improved separation method,and the Zeno behavior is also eliminated.Finally,a practical example is given to illustrate the effectiveness of the proposed method.