Fuzzy Resilient Control of DC Microgrids with Constant Power Loads Based on Markov Jump Models

Abstract

This paper addresses the fuzzy resilient control of DC microgrids with constant power loads. The DC microgrid is subject to abrupt parameter changes which are described by the Markov jump model. Due to the constant power loads, the DC microgrid exhibits nonlinear dynamics which are characterized by a T-S fuzzy model. According to the parallel distributed compensation principle, mode-dependent fuzzy resilient controllers are designed to stabilize the resultant T-S fuzzy Markov jump DC microgrid. The “resilient” means the controller could cope with the uncertainty caused by the inaccurate execution of the control laws. This uncertainty is governed by a Bernoulli distributed random variable and thus may not occur. Then, the mean square exponential stability is analyzed for the closed-loop system by using the mode-dependent Lyapunov function. Since the stability conditions are not convex, a design algorithm is further derived to calculate the fuzzy resilient controller gains. Finally, simulations are provided to test the effectiveness of the proposed results.

1. Introduction

In the past few years, the microgrid (MG) has attracted more and more attention due to its good capability of penetrating renewable power sources into the modern power grid [1,2,3,4]. For example, [2] discussed the sliding mode control of MG. The MG structure provides a simpler management procedure and brings both economical and technical advantages [5,6,7,8,9]. There are mainly two kinds of MG structure, i.e., the alternating current (AC) MG and direct current (DC) MG. Up to now, many results have been reported for the research of AC MG since the conventional power network is based on the AC system [10]. To fully utilize different renewable power sources subject to DC couplings, DC MG is becoming more and more popular. For example, in [3,4], stability issues were addressed for DC MG. Compared with AC MG, DC MG shows higher efficiency in connecting both various electric loads and energy storage systems. What is more, DC MG also avoids the harmonics and reactive power that do harm to the electric quality [11]. The DC MG usually consists of energy sources, DC/AC loads, power electronic converters, and filters.

In the DC MG community, the stabilization of DC MG with constant power loads (CPLs), which refer to power electronic loads consuming constant power, is an important topic [12,13]. Due to the negative impedance feature, the CPLs may lead to instability of the whole MG system. To tackle this problem, great efforts have been made in the open literature to relieve the unexpected effects caused by CPLs. For example, under the assumption that the nonlinear terms induced by CPLs are quadratically bounded, the authors of [14] proposed a linear controller to stabilize the DC MG by solving a Lur’e problem. However, the system uncertainties are not taken into account in the work of [14]. State feedback control is applied in [15] to generate the stabilizing inject current that is fed to all CPLs. In view of the nonlinearity of DC MG with CPLs, the backstepping method is used to deal with the nonlinearity of CPLs in the absence of noise in [16]. On the other hand, fuzzy rough theory can effectively describe real-world situations [17]. For example, Takagi–Sugeno (T-S) fuzzy model is powerful in modeling nonlinear systems [18]. By turning the nonlinear system into a set of linear subsystems through via fuzzy rules and membership functions, the global stabilization of nonlinear systems can be achieved using the linear control theory [19]. On the other hand, linear matrix inequalities (LMIs) have been proven to be a powerful tool in the stability analysis of DC MG with CPLs [14,20,21]. For detailed comparison, a comparative table has been provided in

Table 1. Comparison between our method and the related literature.

	[14]	[20,21]	Ours
Modelling of abrupt changes	No	No	Yes
Handling of CPL nonlinearity	Quadratic bounded	T-S fuzzy	T-S fuzzy
Using resilient control	No	Yes	Yes
Type of gain uncertainty	None	Deterministic	Random
Using mode-dependent control	None	None	Yes

. This motivates us to study the LMIs-based T-S fuzzy control of DC MG.

For a DC MG in practical situations, the system parameter/structure may change abruptly because of unexpected physical environment changes and stochastic parameter variations [22]. How to deal with this problem, i.e., establish an appropriate model taking the possible parameter/structure changes into account, becomes the first priority in the study of stabilization of LMI-based T-S fuzzy control. The Markov jump model is a good candidate for reflecting the above phenomenon [23,24,25]. The Markov jump model is made up of several subsystems, which jump to each other from time to time according to a rule named Markov chain. With this model, an operation mode subject to certain system parameters/structures can be represented by a subsystem, and the system parameter/structure changes are described by the jumps between subsystems [26]. Owing to this property, the Markov jump model has received much attention in the research area and is also widely used in plenty of applications such as electronic circuits and DC motors. For example, ref. [27] studied the state estimator design for repeated scalar nonlinear systems with Markov jumping parameters. In [28], the fault estimation issue was investigated for MJSs using a mode-dependent intermediate variable. Fuzzy state estimation is addressed in [29] for a class of Markov jump system via the probabilistic event-triggered strategy. It should be noted that the model uncertainty is a critical issue in system modeling [30,31,32,33]. However, in most of the existing results, the DC MG is assumed to be free of parameter uncertainties. This assumption is not practical, since the parameters of the system may be uncertain and vary along with time. There are three kinds of uncertainties, including (1) filters’ parameters, (2) ESS parameters, and (3) CPLs’ power value. To remove this assumption, we introduce the Markov jump model into the characterization of the DC MG. The existing DC MG is apparently a special case of the resultant model with only one mode.

By now, a large number of studies are available for the DC MG with CPLs. For instance, referring to the $H_{\infty }$ disturbance attenuation idea, ref. [34] represented the DC MG by viewing the perturbation of the CPLs’ current and input voltage as external disturbances and designed an $H_{\infty }$ controller to cut down the disturbance effect. Unfortunately, the proposed controller in [34] is not easy to conduct in practice due to the high-order polynomials induced by the small-signal model. However, it should be pointed out that most of the existing results are achieved by assuming an ideal implementation of control laws. In fact, due to computational delays, the round-off errors, finite word lengths, quantization effects, inaccuracies, and uncertainties are inevitable in practice during the controller implementation. To cope with this phenomenon, the so-called resilient control strategy has been developed in the past few years [35,36,37,38,39], which tries to decrease the adverse effect caused by uncertain feedback gains and thus improve the control performance. Recently, some inspiring work has been completed to deal with the uncertainty of implementing the control law. Based on the T-S fuzzy method, ref. [20] designed a fuzzy non-fragile controller to make sure the system performance is not violated even in the case of system and controller uncertainties. In spite of this work, the resilient control of DC MG has not been fully investigated, especially in the case of Markov jump modes and T-S fuzzy modeling.

Based on the above discussions, this paper focuses on stability analysis and stabilization control of DC microgrids subject to abrupt changes using a fuzzy resilient control strategy. The significant contributions are underlined as follows.

(i)

A continuous-time Markov jump DC microgrid fuzzy system is established, which accounts for both the abrupt parameter changes and the nonlinear dynamics of multiple CPLs. The resultant hybrid system shows more engineering backgrounds and thus improves the model fitness.

(ii)

By fully using the information of fuzzy rules, system modes, and random controller gain perturbations, exponential stability conditions are derived for the Markov jump DC microgrid with CPLs. The relationship between system stability and the affecting factors is thus presented clearly for the designers.

(iii)

A fuzzy resilient control algorithm is developed by transforming the non-convex stability analysis conditions into traceable ones. Afterward, the resultant mode-dependent fuzzy resilient control law not only handles the nonlinearity of the DC microgrid but also maintains robustness against the random gain perturbations.

Notations. $\mathrm{He}\left\{A\right\}=A^T+A$. ${\rho }_{max}\{·\}$ and ${\rho }_{min}\{·\}$ are the maximal and minimal eigenvalue of “·”, respectively. $E\{·\}$ is the expectation of event “·”.

2. Problem Formulation and Preliminaries

2.1. Typical Architecture of DC MG

Typically, a DC microgrid shown in Figure 1 includes an energy storage system (ESS), a DC source, and Q CPLs. The ESS, DC source, and CPLs are respectively used to inject current, maintain the DC bus, and attain constant power from converters. First, the voltage and current states of the DC microgrid are sampled periodically. Then, the controller receives plant states to generate a control command. Due to the uncertainties of implementing the control law, the controller output may undergo random perturbations.

2.2. Markov Jump Models for CPLs and DC Subsystem

Using the Kirchhoff voltage/current laws and the Markov jump models, the jth CPL is represented as follows:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{i}}_{L,j}=-{\displaystyle \frac{v_{C,j}}{L_j^{{\theta }_t}}}-{\displaystyle \frac{r_{L,j}^{{\theta }_t}}{L_j^{{\theta }_t}}}{i}_{L,j}+{\displaystyle \frac{v_{C,s}}{L_j^{{\theta }_t}}},\hfill \\ {\dot{v}}_{C,j}=-{\displaystyle \frac{P_j^{{\theta }_t}}{C_j^{{\theta }_t}{v}_{C,j}}}+{\displaystyle \frac{i_{L,j}}{C_j^{{\theta }_t}}},\hfill \end{array}\right.\end{array}$$

(1)

where $L_j^{{\theta }_t},C_j^{{\theta }_t}$ and $r_{L,j}^{{\theta }_t}$ ($j=1,2,\cdots ,Q$) denote the inductance, capacitance, and resistance of the jth CPL subsystem operating at mode ${\theta }_t$, respectively. $v_{C,j}$ and $i_{L,j}$ refer to the inductor current and capacitor voltage, respectively. ${\displaystyle \frac{P_j^{{\theta }_t}}{v_{C,j}}}$ is a voltage-controlled current source, which denotes the jth CPL with a constant load power $P_j^{{\theta }_t}$. Here, the abrupt changes of the jth CPL are reflected by a continuous-time Markov chain ${\theta }_t$. Note that all the possible values of ${\theta }_t$ belong to a finite set named $M\triangleq \{1,2,\cdots ,m\}$. The jumping rule between two consecutive modes is described by

$$Pr\{{\theta }_{t+\mathrm{\Delta }t}=b|{\theta }_t=a\}=\left\{\begin{array}{c}{\phi }_{ab}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\mathrm{if}a\ne b\hfill \\ 1+{\phi }_{aa}\mathrm{\Delta }t+o\left(\mathrm{\Delta }t\right),\mathrm{if}a=b\hfill \end{array}\right.$$

(2)

where ${\phi }_{ab}$, $a,b\in M$ are the transition rates subject to (i) ${\sum }_{b=1}^m{\phi }_{ab}=0$, and (ii) ${\phi }_{ab}\ge 0$ for $a\ne b$. Additionally, $\mathrm{\Delta }t>0$ and ${lim}_{\mathrm{\Delta }t\to 0}{\displaystyle \frac{o\left(\mathrm{\Delta }t\right)}{\mathrm{\Delta }t}}=0$. Based on the above definitions, a matrix $\mathrm{\Phi }=\left[{\phi }_{ab}\right]$ is introduced to express the mode-to-mode transition rates.

Similarly, the DC subsystem is described by the following Markov jump models:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{i}}_{L,s}=-{\displaystyle \frac{v_{C,s}}{L_s^{{\theta }_t}}}-{\displaystyle \frac{r_{L,s}^{{\theta }_t}}{L_s^{{\theta }_t}}}{i}_{L,s}+{\displaystyle \frac{V_{dc}}{L_s^{{\theta }_t}}},\hfill \\ {\dot{v}}_{C,s}=-{\displaystyle \frac{{\sum }_{j=1}^Q{i}_{L,j}}{C_s^{{\theta }_t}}}-{\displaystyle \frac{i_{es}}{C_s^{{\theta }_t}}}+{\displaystyle \frac{i_{L,s}}{C_s^{{\theta }_t}}},\hfill \end{array}\right.\end{array}$$

(3)

where $L_s^{{\theta }_t},C_s^{{\theta }_t}$, and $r_{L,s}^{{\theta }_t}$ denote the inductance, capacitance, and resistance of the DC source subsystem, respectively. $i_{L,s}$ and $v_{C,s}$, respectively, stand for the inductor current and capacitor voltage. $V_{dc}$ denotes the voltage of the DC source, and $i_{es}$ represents the injection current of the ESS.

2.3. Overall Markov Jump DC Microgrid

Take $x_j\left(t\right)={[i_{L,j}^T,v_{C,j}^T]}^T$, $x_s\left(t\right)={[i_{L,s}^T,v_{C,s}^T]}^T$, ${\kappa }_j(x_j\left(t\right)={\displaystyle \frac{1}{v_{C,j}}}$ and set ${\theta }_t=a\in M$ for simplicity. Then, we obtain from (1) and (3) that

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{\dot{x}}_j\left(t\right)={\overline{A}}_{j,a}{x}_j\left(t\right)+{\overline{A}}_{js,a}{x}_s\left(t\right)+d_{j,a}{\kappa }_j\left(x_j\left(t\right)\right),j=1,2,\cdots ,Q\hfill \\ {\dot{x}}_s={\overline{A}}_{s,a}{x}_s\left(t\right)+\sum _{j=1}^Q{\overline{A}}_{cn,a}{x}_j\left(t\right)+b_{es,a}{i}_{es}+b_{s,a}{V}_{dc},\hfill \end{array}\right.\end{array}$$

(4)

where

$$\begin{array}{cc}\hfill & {\overline{A}}_{j,a}=\left[\begin{array}{cc}-{\displaystyle \frac{r_{L,j}^a}{L_j^a}}& -{\displaystyle \frac{1}{L_j^a}}\\ {\displaystyle \frac{1}{C_j^a}}& 0\end{array}\right],{\overline{A}}_{js,a}=\left[\begin{array}{cc}0& -{\displaystyle \frac{1}{L_j^a}}\\ 0& 0\end{array}\right],\hfill \\ \hfill & d_{j,a}=\left[\begin{array}{c}0\\ {\displaystyle \frac{P_j^a}{C_j^a}}\end{array}\right],b_{s,a}=\left[\begin{array}{c}{\displaystyle \frac{1}{L_s^a}}\\ 0\end{array}\right],{\overline{A}}_{s,a}=\left[\begin{array}{cc}-{\displaystyle \frac{r_{L,s}^a}{L_s^a}}& -{\displaystyle \frac{1}{L_s^a}}\\ {\displaystyle \frac{1}{C_s^a}}& 0\end{array}\right],\hfill \\ \hfill & {\overline{A}}_{cn,a}=\left[\begin{array}{cc}0& 0\\ -{\displaystyle \frac{1}{C_s^a}}& 0\end{array}\right],b_{es,a}=\left[\begin{array}{c}0\\ -{\displaystyle \frac{1}{C_s^a}}\end{array}\right].\hfill \end{array}$$

Set $x\left(t\right)={[x_1^T\left(t\right),\cdots ,x_Q^T\left(t\right),x_s^T\left(t\right)]}^T$, $\overline{\kappa }\left(x\left(t\right)\right)={[{\kappa }_1^T\left(x_1\left(t\right)\right),\cdots ,{\kappa }_Q^T\left(x_Q\left(t\right)\right)]}^T$, it follows from (4) that

$$\dot{x}\left(t\right)={\overline{A}}_{a}x\left(t\right)+B_{es,a}{i}_{es}\left(t\right)+D\overline{\kappa }\left(x\left(t\right)\right)+B_{s,a}{V}_{dc}$$

(5)

where

$$\begin{array}{cc}\hfill & {\overline{A}}_a=\left[\begin{array}{cccc}{\overline{A}}_{1,a}& \cdots & 0& {\overline{A}}_{1s,a}\\ ⋮& \ddots & ⋮& ⋮\\ 0& \cdots & {\overline{A}}_{Q,a}& {\overline{A}}_{Qs,a}\\ {\overline{A}}_{cn,a}& \cdots & {\overline{A}}_{cn,a}& {\overline{A}}_{s,a}\end{array}\right],B_{es,a}=\left[\begin{array}{c}0\\ ⋮\\ 0\\ b_{es,a}\end{array}\right],\hfill \\ \hfill & B_{s,a}=\left[\begin{array}{c}0\\ ⋮\\ 0\\ b_{s,a}\end{array}\right],D_a=\left[\begin{array}{ccc}{d}_{1,a}& \cdots & 0\\ ⋮& \ddots & ⋮\\ 0& \cdots & d_{Q,a}\\ 0& \cdots & 0\end{array}\right]\hfill \end{array}$$

Let $x_0$ and $v_{C,j,0}$ be the equilibrium point of the voltage $v_{C,j}$ and the whole DC microgrid, respectively. To ease the analysis, we shift the equilibrium point of (5) to the origin using a coordinate change about an operating point. Then, the overall Markov jump DC microgrid whose equilibrium point locates at the origin is rewritten as follows:

$$\dot{\overline{x}}\left(t\right)={\overline{A}}_a\overline{x}\left(t\right)+B_{es,a}{\overline{i}}_{es}\left(t\right)+D\overline{\kappa }\left(\overline{x}\left(t\right)\right)$$

(6)

where $\overline{x}\left(t\right)=x\left(t\right)-x_0={[{\overline{x}}_1^T\left(t\right),\cdots ,{\overline{x}}_Q^T\left(t\right),{\overline{x}}_s^T\left(t\right)]}^T$, ${\overline{x}}_j\left(t\right)=[{\overline{i}}_{L,j}^T$, ${\overline{v}}_{C,j}^T{]}^T$, $j=1,\cdots ,Q$, ${\overline{x}}_s\left(t\right)={[{\overline{i}}_{L,s}^T,{\overline{v}}_{C,s}^T]}^T$, $\overline{\kappa }\left(\overline{x}\left(t\right)\right)={[{\kappa }_1^T\left({\overline{x}}_1\left(t\right)\right),\cdots ,{\kappa }_Q^T\left({\overline{x}}_Q\left(t\right)\right)]}^T$, ${\kappa }_j\left({\overline{x}}_j\left(t\right)\right)={\displaystyle \frac{{\overline{v}}_{C,j}}{v_{C,j,0}({\overline{v}}_{C,j}+v_{C,j,0})}}$.

2.4. Fuzzy Modeling of the Markov Jump DC Microgrid

As can be seen in (6), the overall MG system comprises Q nonlinear terms (i.e., ${\kappa }_j\left({\overline{x}}_j\left(t\right)\right)$).

Referring to [40], an MG with several CPLs can be turned to an MG with one equivalent CPL. So, for the sake of simplicity, the procedure of an MG with one CPL will be discussed in the following. To calculate the TS fuzzy model, one needs to model the nonlinearities of the model (6). Note that only one nonlinear term, i.e., ${\kappa }_1\left({\overline{x}}_1\left(t\right)\right)$, exists in the dynamic model.

For this single CPL, define the region $-{\overline{v}}_{C,1,m}\le {\overline{v}}_{C,1}\le {\overline{v}}_{C,1,m}$ where ${\overline{v}}_{C,1,m}>0$ is a known scalar. It is obtained that

$$W_{min}\le {\displaystyle \frac{{\kappa }_1\left({\overline{x}}_1\left(t\right)\right)}{{\overline{v}}_{C,1}}}\le W_{max}$$

(7)

where

$$\begin{array}{cc}\hfill & W_{min}={\displaystyle \frac{1}{v_{C,1,0}({\overline{v}}_{C,1,m}+v_{C,1,0})}},\hfill \\ \hfill & W_{max}={\displaystyle \frac{1}{v_{C,1,0}(-{\overline{v}}_{C,1,m}+v_{C,1,0})}},\hfill \end{array}$$

In order to derive the T-S fuzzy model, the sector nonlinearity approach is employed to represent the nonlinear term ${\kappa }_1\left({\overline{x}}_1\left(t\right)\right)$, by the following equivalent T-S model in the pre-defined local region:

$${\kappa }_1\left({\overline{x}}_1\left(t\right)\right)=w_1{W}_{min}{\overline{v}}_{C,1}+w_2{W}_{max}{\overline{v}}_{C,1}$$

(8)

where $w_1$ and $w_2$ are the fuzzy basis functions defined as follows:

$$\begin{array}{cc}\hfill & w_1={\displaystyle \frac{W_{max}{\overline{v}}_{C,1}-{\kappa }_1\left({\overline{x}}_1\left(t\right)\right)}{\mathrm{\Delta }W{\overline{v}}_{C,1}}},\hfill \\ & w_2={\displaystyle \frac{{\kappa }_1\left({\overline{x}}_1\left(t\right)\right)-W_{min}{\overline{v}}_{C,1}}{\mathrm{\Delta }W{\overline{v}}_{C,1}}},\hfill \\ \hfill & \mathrm{\Delta }W=W_{max}-W_{min}\hfill \end{array}$$

Based on (6) and (8), the T-S fuzzy Markov jump DC microgrid is given as follows:

$$\dot{\overline{x}}\left(t\right)=\sum _{i=1}^2{w}_i({\overline{A}}_{i,a}\overline{x}\left(t\right)+B_{es,a}{\overline{i}}_{es}\left(t\right))$$

(9)

where

$${\overline{A}}_{i,a}=\left[\begin{array}{cc}{\widehat{A}}_{i,a}& {\overline{A}}_{1s,a}\\ {\overline{A}}_{cn,a}& {\overline{A}}_{s,a}\end{array}\right],{\widehat{A}}_{1,a}=\left[\begin{array}{cc}-{\displaystyle \frac{r_{L,1}^a}{L_1^a}}& -{\displaystyle \frac{1}{L_1^a}}\\ {\displaystyle \frac{1}{C_1^a}}& {\displaystyle \frac{P_1^a}{C_1^a}}{W}_{min}\end{array}\right],{\widehat{A}}_{2,a}=\left[\begin{array}{cc}-{\displaystyle \frac{r_{L,1}^a}{L_1^a}}& -{\displaystyle \frac{1}{L_1^a}}\\ {\displaystyle \frac{1}{C_1^a}}& {\displaystyle \frac{P_1^a}{C_1^a}}{W}_{max}\end{array}\right].$$

3. Closed-Loop System Design

3.1. Fuzzy Resilient Controller Design

In order to stabilize the system (9) in the case of uncertain gain perturbations, the following mode-dependent fuzzy resilient controller is designed:

$${\overline{i}}_{es}=\sum _{i=1}^2{w}_i(K_{i,a}+\alpha \left(k\right)\mathrm{\Delta }K)\overline{x}\left(t\right)$$

(10)

where $K_{i,a}$ is the mode-dependent fuzzy controller gain. $\alpha \left(k\right)$ is a Bernoulli distributed variable with $E\left\{\alpha \left(k\right)\right\}=\overline{\alpha }\in [0,1]$, where $\overline{\alpha }$ is supposed to be known. $\mathrm{\Delta }K$ denotes the gain perturbation caused by the uncertainty of implementing the control law. Usually, this perturbation is bounded and is subject to the following assumption:

Assumption 1.

The controller gain perturbations are assumed to be norm-bounded, i.e., the following relation holds:

$${\mathrm{\Delta }}^{T}K\mathrm{\Delta }K\le {\delta }^{2}I$$

(11)

where $\delta >0$ is the given upper bound of uncertain gain perturbations.

Remark 1. 

Assumption 1 bounds the uncertainty of controller gains during the execution of control laws. Thus, the fluctuation of controller gains can be mathematically modeled to facilitate the analysis and synthesis. Since the upper bound instead of the detailed dynamics is assumed, different kinds of controller gain fluctuations can be tested, provided Assumption 1 is satisfied. Additionally, the parameter δ, which reflects the uncertainty degree, is adjustable, which helps the designer to analyze the maximal allowable controller gain uncertainty. Note that Assumption 1 should be used together with Lemma 1 such that the uncertain terms are eliminated during the controller gain computation.

Remark 2. 

Note that the gain perturbation $\mathrm{\Delta }K$ may not happen if the control law is implemented ideally. Thus, it may bring conservatism to assume $\mathrm{\Delta }K$ occurs definitely. To avoid this shortcoming and model the intermittent phenomenon of $\mathrm{\Delta }K$, a random variable $\alpha \left(k\right)$ obeying the Bernoulli distribution is introduced in our controller design. When $\alpha \left(k\right)=1$, the gain perturbation occurs. Otherwise, the controller gain is exactly implemented.

Remark 3. 

In this work, the DC microgrid is modeled as a Markov jump system that switches among different modes from time to time. Since the control laws may differ in terms of different modes of the DC microgrid, the mode-dependent controller is designed to help the control law fit the system mode and achieve better control performance.

3.2. Closed-Loop System Modeling

According to the fuzzy resilient controller design, the closed-loop system is obtained by substituting (10) into (9):

$$\dot{\overline{x}}\left(t\right)=\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j({\tilde{A}}_{i,a}\overline{x}\left(t\right)+\alpha \left(k\right)B_{es,a}\mathrm{\Delta }K\overline{x}\left(t\right))$$

(12)

where ${\tilde{A}}_{i,a}={\overline{A}}_{i,a}+B_{es,a}{K}_{j,a}$.

The goal of this work is to design appropriate injected current ${\overline{i}}_{es}\left(t\right)$, which is taken as the control input, to make sure system (12) is exponentially stable in the case of abrupt system changes and random gain perturbations.

Lemma 1. 

Given matrices X and Y of appropriate dimensions, and any positive definite matrix Z, the following inequality is true:

$$X{Y}^T+Y{X}^T\le XZ{X}^T+Y{Z}^{-1}{Y}^T$$

(13)

Definition 1 

([41]). For any solution x(t) of a system given the initial condition ${\mathcal{X}}_0$, system (12) is said to be mean square exponentially stable, if the following inequality holds

$$E\left\{\right||\overline{x}{\left(t\right)\left|\right|}^2\}\le {\mu }_1\left|\right|{\mathcal{X}}_0{\left|\right|}^2{\mu }_2^q,$$

(14)

where ${\mu }_1>0$, $0\le {\mu }_2<1$, and $q\ge 0$.

4. Stability Analysis Conditions

In this section, the exponential stability is analyzed for the system (12) by giving a set of linear matrix equalities (LMIs).

Theorem 1. 

Given the decay rate $\sigma >0$, upper bound of the controller uncertainty $\delta >0$, and the occurring probability $\overline{\alpha }>0$ of controller gain perturbations, if there exist positive symmetric matrices $P_a$, Q, matrices $K_{j,a}$, $j=1,2$, and a positive scalar $\overline{\lambda }$ that satisfies the following conditions for any $a\in M$ and $i,j=1,2$, system (12) is exponentially stable.

$$\begin{array}{cc}\hfill & {\Pi }_{i,i}^a\le 0,\hfill \end{array}$$

(15)

$$\begin{array}{cc}\hfill & {\Pi }_{i,j}^a+{\Pi }_{j,i}^a\le 0,i<j,\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill & Q\ge \overline{\lambda }I,\hfill \end{array}$$

(17)

where

$$\begin{array}{cc}\hfill & {\Pi }_{i,i}^a=\left[\begin{array}{ccc}\sigma P_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}+{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a& \delta I& \mathcal{M}\\ \ast & -\overline{\lambda }I& 0\\ \ast & \ast & -\mathcal{P}\end{array}\right],\hfill \\ \hfill & \mathcal{M}=[\sqrt{{\varphi }_{a1}}I,\cdots ,\sqrt{{\varphi }_{am}}I],\mathcal{P}=diag\{P_1^{-1},\cdots ,P_m^{-1}\}\hfill \end{array}$$

Proof. 

Take the Lyapunov function as follows:

$$V(\overline{x}\left(t\right),{\theta }_t)={\overline{x}}^T\left(t\right)P_{{\theta }_t}\overline{x}\left(t\right)$$

(18)

where $P_{{\theta }_t}=P_{{\theta }_t}^T$ is to be designed.

The infinitesimal generator $\mathcal{D}(·)$ of $V(\overline{x}\left(t\right),{\theta }_t)$ is defined as follows:

$$\mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)=li{m}_{\mathrm{\Delta }t\to 0}{\displaystyle \frac{E\{V(\overline{x}(t+\mathrm{\Delta }t),{\theta }_{t+\mathrm{\Delta }t})-V(\overline{x}\left(t\right),{\theta }_t)|x\left(t\right),{\theta }_t\}}{\mathrm{\Delta }t}}$$

(19)

Let ${\theta }_t=a\in M,{\theta }_{t+\mathrm{\Delta }t}=b\in M$. Based on the infinitesimal generator $\mathcal{D}(·)$, it follows from (18) that

$$\begin{array}{cc}\hfill & \mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)\hfill \\ \hfill =& li{m}_{\mathrm{\Delta }t\to 0}E\left\{{\displaystyle \frac{{\overline{x}}^T(t+\mathrm{\Delta }t)P_{{\theta }_{t+\mathrm{\Delta }t}}\overline{x}(t+\mathrm{\Delta }t)-{\overline{x}}^T\left(t\right)P_{{\theta }_t}\overline{x}\left(t\right)}{\mathrm{\Delta }t}}\right\}\hfill \\ \hfill =& li{m}_{\mathrm{\Delta }t\to 0}E\left\{{\displaystyle \frac{{\sum }_{b=1,b\ne a}^m{\phi }_{ab}\mathrm{\Delta }t{\overline{x}}^T(t+\mathrm{\Delta }t)P_b\overline{x}(t+\mathrm{\Delta }t)}{\mathrm{\Delta }t}}\right.\hfill \\ \hfill & \phantom{li{m}_{\mathrm{\Delta }t\to 0}}+\left.{\displaystyle \frac{{\widehat{\phi }}_a{\overline{x}}^T(t+\mathrm{\Delta }t)P_a\overline{x}(t+\mathrm{\Delta }t)-{\overline{x}}^T\left(t\right)P_a\overline{x}\left(t\right)\}}{\mathrm{\Delta }t}}\right\}\hfill \\ \hfill =& li{m}_{\mathrm{\Delta }t\to 0}E\left\{{\displaystyle \frac{{\sum }_{b=1}^m{\phi }_{ab}\mathrm{\Delta }t{\overline{x}}^T(t+\mathrm{\Delta }t)P_b\overline{x}(t+\mathrm{\Delta }t)}{\mathrm{\Delta }t}}\right.\hfill \\ \hfill & \phantom{li{m}_{\mathrm{\Delta }t\to 0}}+\left.{\displaystyle \frac{{\overline{x}}^T(t+\mathrm{\Delta }t)P_a\overline{x}(t+\mathrm{\Delta }t)-{\overline{x}}^T\left(t\right)P_a\overline{x}\left(t\right)}{\mathrm{\Delta }t}}\right\}\hfill \\ \hfill =& \sum _{b=1}^m{\phi }_{ab}{\overline{x}}^T\left(t\right)P_b\overline{x}\left(t\right)+E\left\{\mathrm{He}\left\{{\dot{\overline{x}}}^T\left(t\right)P_a\overline{x}\left(t\right)\right\}\right\}\hfill \end{array}$$

(20)

where ${\widehat{\phi }}_a=1+{\phi }_{aa}\mathrm{\Delta }t$.

Note that

$$\begin{array}{cc}\hfill & E\left\{{\dot{\overline{x}}}^T\left(t\right)P_a\overline{x}\left(t\right)\right\}\hfill \\ \hfill =& E\left\{\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j({\tilde{A}}_{i,a}\overline{x}\left(t\right)+\alpha \left(k\right)B_{es,a}\mathrm{\Delta }K\overline{x}\left(t\right))P_a\overline{x}\left(t\right)\right\}\hfill \end{array}$$

$$\begin{array}{cc}\hfill =& E\left\{\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j({\overline{A}}_{i,a}\overline{x}\left(t\right)+B_{es,a}(K_{j,a}+\alpha \left(k\right)\mathrm{\Delta }K)\overline{x}\left(t\right))P_a\overline{x}\left(t\right)\right\}\hfill \\ \hfill =& \sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j{\overline{x}}^T\left(t\right)[\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +\mathrm{He}\left\{\overline{\alpha }{\mathrm{\Delta }}^{T}K{B}_{es,a}^T{P}_a\right\}]\overline{x}\left(t\right)\hfill \end{array}$$

(21)

where ${\overline{P}}_a={\sum }_{b=1}^m{\phi }_{ab}{P}_b$.

According to Lemma 1, there exists a matrix $Q>0$ such that

$$\mathrm{He}\left\{\overline{\alpha }{\mathrm{\Delta }}^{T}K{B}_{es,a}^T{P}_a\right\}\le {\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\mathrm{\Delta }}^{T}K{Q}^{-1}\mathrm{\Delta }K$$

(22)

Due to (17), we have

$$Q^{-1}\le {\overline{\lambda }}^{-1}I$$

(23)

which, together with Assumption 1, yields

$${\mathrm{\Delta }}^{T}K{Q}^{-1}\mathrm{\Delta }K\le {\delta }^2{\overline{\lambda }}^{-1}I$$

(24)

Then, we further obtain from (22) that

$$\mathrm{He}\left\{\overline{\alpha }{\mathrm{\Delta }}^{T}K{B}_{es,a}^T{P}_a\right\}\le {\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I$$

(25)

Using (20), (21), and (25), we get

$$\begin{array}{cc}\hfill & \mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)\hfill \\ \hfill \le & \sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j{\overline{x}}^T\left(t\right)[{\overline{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I]\overline{x}\left(t\right)\hfill \end{array}$$

(26)

Next, we are about to calculate $\mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)+\sigma E\left\{V(\overline{x}\left(t\right),{\theta }_t)\right\}$. Based on (26) and (18), we have

$$\begin{array}{cc}\hfill & \mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)+\sigma E\left\{V(\overline{x}\left(t\right),{\theta }_t)\right\}\hfill \\ \hfill \le & \sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j{\overline{x}}^T\left(t\right)[{\widehat{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I]\overline{x}\left(t\right)\hfill \\ \hfill =& \sum _{i=1}^2{w}_i^2{\overline{x}}^T\left(t\right)[{\widehat{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I]\overline{x}\left(t\right)\hfill \\ \hfill & +\sum _{i=1}^2\sum _{j=i+1}^2{w}_i{w}_j{\overline{x}}^T\left(t\right)\left(\right[{\widehat{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I]\hfill \\ \hfill & +[{\widehat{P}}_a+\mathrm{He}\left\{{\overline{A}}_{j,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{i,a}^T{B}_{es,a}^T{P}_a\right\}\hfill \\ \hfill & +{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}I\left]\right)\overline{x}\left(t\right)\hfill \end{array}$$

(27)

where ${\widehat{P}}_a=\sigma P_a+{\overline{P}}_a$.

By Schur complementary [42], it is obtained from (15) that

$$\sigma P_a+{\overline{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{i,a}^T{B}_{es,a}^T{P}_a\right\}+{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}\le 0$$

(28)

Similarly, we obtain from (16) that

$$\begin{array}{cc}\hfill & \sigma P_a+{\overline{P}}_a+\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}+{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1}\hfill \\ \hfill & +(\sigma P_a+{\overline{P}}_a+\mathrm{He}\left\{{\overline{A}}_{j,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}+{\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a+{\delta }^2{\overline{\lambda }}^{-1})\le 0\hfill \end{array}$$

(29)

Using (27)–(29) yields

$$\mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)+\sigma E\left\{V(\overline{x}\left(t\right),{\theta }_t)\right\}\le 0$$

(30)

which further implies

$$E\left\{\right||\overline{x}{\left(t\right)\left|\right|}^2\}\le {\rho }_{min}^{-1}\left(P_a\right)E\left\{V(0,{\theta }_0)\right\}{exp}^{-\sigma t},\forall t\ge 0,{\theta }_0\in M$$

(31)

According to Definition 1, under the designed mode-dependent fuzzy resilient controller, the Markov jump DC microgrid (12) is mean square exponentially stable. Therefore, the proof is completed. □

Remark 4. 

The stability analysis conditions are developed to check the exponential stability of the closed-loop DC microgrid system under the proposed control law since stability is the basis of controller design in the subsequent. To achieve this goal, the Lyapunov stability theory is borrowed by constructing appropriate Lyapunov functions. Note that the developed stability analysis conditions rely on fuzzy rules, system modes, and controller gain perturbations, thus making a mode-dependent fuzzy resilient control law possible.

5. Controller Design Method

Theorem 2. 

Given the decay rate $\sigma >0$, the upper bound of the controller uncertainty $\delta >0$, and the occurring probability $\overline{\alpha }>0$ of controller gain perturbations, if there exist positive symmetric matrices $X_a$, Q, matrices ${\overline{K}}_{j,a}$, $j=1,2$, and a positive scalar $\overline{\lambda }$ that satisfy the following conditions for any $a\in M$ and $i,j=1,2$,

$$\begin{array}{cc}\hfill & {\widehat{\Pi }}_{i,i}^a\le 0,\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\widehat{\Pi }}_{i,j}^a+{\widehat{\Pi }}_{j,i}^a\le 0,i<j,\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill & Q\ge \overline{\lambda }I,\hfill \end{array}$$

(34)

where

$$\begin{array}{cc}\hfill & {\widehat{\Pi }}_{i,j}^a=\left[\begin{array}{ccc}\sigma X_a+He\left\{X_a{\overline{A}}_{i,a}^T\right\}+He\left\{{\overline{K}}_{j,a}^T{B}_{es,a}^T\right\}+{\overline{\alpha }}^2{B}_{es,a}Q{B}_{es,a}^T& \delta X_a& \widehat{\mathcal{M}}\\ \ast & -\overline{\lambda }I& 0\\ \ast & \ast & -\widehat{\mathcal{P}}\end{array}\right],\hfill \\ \hfill & \widehat{\mathcal{M}}=[\sqrt{{\varphi }_{a1}}{X}_a,\cdots ,\sqrt{{\varphi }_{am}}{X}_a],\widehat{\mathcal{P}}=diag\{X_1,\cdots ,X_m\}\hfill \end{array}$$

then system (12) is exponentially stable.

The fuzzy resilient controller gains are calculated by $K_{j,a}={\overline{K}}_{j,a}{X}_a^{-1}$.

Proof. 

Pre-and-post multiplying $diag\{P_a^{-1},I,{\underbrace{I,\cdots ,I}}_m\}$ on the left and right sides of (15), respectively, we have

$$\left[\begin{array}{ccc}\sigma P_a^{-1}+\mathrm{He}\left\{P_a^{-1}{\overline{A}}_{i,a}^T\right\}+\mathrm{He}\left\{P_a^{-1}{K}_{j,a}^T{B}_{es,a}^T\right\}+{\overline{\alpha }}^2{B}_{es,a}Q{B}_{es,a}^T& \delta P_a^{-1}& \tilde{\mathcal{M}}\\ \ast & -\overline{\lambda }I& 0\\ \ast & \ast & -\mathcal{P}\end{array}\right]\le 0$$

(35)

where $\tilde{\mathcal{M}}=[\sqrt{{\varphi }_{a1}}{P}_a^{-1},\cdots ,\sqrt{{\varphi }_{am}}{P}_a^{-1}]$, and $\mathcal{P}$ have been defined in Theorem 1.

Take the following notations:

$$X_a=P_a^{-1},{\overline{K}}_{j,a}=K_{j,a}{X}_a.$$

Substituting the above notations into (35) gives (32). In other words, (15) is ensured by (32). Using similar techniques, (33) is derived from (16). According to Theorem 1, the conditions listed in Theorem 2 guarantee the exponential stability of the Markov jump DC microgrid (12). The proof is then completed. □

Remark 5. 

The controller design method shown above is essentially to solve the non-convex stability conditions (15)–(17). It is rigorously proved in Theorem 1 that, once the selected controller gains $K_{j,a}$ satisfy the conditions shown in (15)–(17), the controller is capable of achieving the exponential stability of the Markov jump DC microgrid. However, due to the nonlinear terms in (15) and (16), such as $He\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}$ and ${\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a$, the conditions (15)–(17) cannot be easily resolved. By using congruent transformation, new variable substitution and Schur complementary, the traceable synthesis conditions are finally obtained in (32)–(34).

Remark 6. 

Note that the above results can also be extended to the case without controller gain perturbations after slight modifications. The following corollary presents the algorithm of controller gain computation.

Corollary 1. 

Given the decay rate $\sigma >0$, if there exist positive symmetric matrices $X_a$, Q, matrices ${\overline{K}}_{j,a}$, $j=1,2$, and a positive scalar $\overline{\lambda }$ that satisfy the following conditions for any $a\in M$ and $i,j=1,2$,

$$\begin{array}{cc}\hfill & {\check{\Pi }}_{i,i}^a\le 0,\hfill \end{array}$$

(36)

$$\begin{array}{cc}\hfill & {\check{\Pi }}_{i,j}^a+{\check{\Pi }}_{j,i}^a\le 0,i<j,\hfill \end{array}$$

(37)

where

$$\begin{array}{cc}\hfill & {\check{\Pi }}_{i,j}^a=\left[\begin{array}{cc}\sigma X_a+He\left\{X_a{\overline{A}}_{i,a}^T\right\}+He\left\{{\overline{K}}_{j,a}^T{B}_{es,a}^T\right\}& \widehat{\mathcal{M}}\\ \ast & -\widehat{\mathcal{P}}\end{array}\right],\hfill \end{array}$$

then system (12) is exponentially stable. The fuzzy resilient controller gains are calculated by $K_{j,a}={\overline{K}}_{j,a}{X}_a^{-1}$.

Proof. 

Without the controller gain perturbation, the closed-loop system becomes

$$\dot{\overline{x}}\left(t\right)=\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j{\tilde{A}}_{i,a}\overline{x}\left(t\right)$$

(38)

where ${\tilde{A}}_{i,a}={\overline{A}}_{i,a}+B_{es,a}{K}_{j,a}$.

Based on system (38) and using the Lyapunov function (18), we have

$$\begin{array}{c}\hfill \mathcal{D}V(\overline{x}\left(t\right),{\theta }_t)=\sum _{b=1}^m{\phi }_{ab}{\overline{x}}^T\left(t\right)P_b\overline{x}\left(t\right)+E\left\{\mathrm{He}\left\{{\dot{\overline{x}}}^T\left(t\right)P_a\overline{x}\left(t\right)\right\}\right\}\end{array}$$

Note that under this situation, it yields

$$\begin{array}{c}\hfill E\left\{{\dot{\overline{x}}}^T\left(t\right)P_a\overline{x}\left(t\right)\right\}=\sum _{i=1}^2\sum _{j=1}^2{w}_i{w}_j{\overline{x}}^T\left(t\right)[\mathrm{He}\left\{{\overline{A}}_{i,a}^T{P}_a\right\}+\mathrm{He}\left\{K_{j,a}^T{B}_{es,a}^T{P}_a\right\}]\end{array}$$

(39)

where ${\overline{P}}_a={\sum }_{b=1}^m{\phi }_{ab}{P}_b$.

Then, similar to Theorems 1 and 2, Corollary 1 is easily obtained. □

Remark 7. 

To make the design procedure more clear for the readers, we provide the following design flowchart (see Figure 2) to demonstrate the key steps during the fuzzy resilient controller design.

Remark 8. 

The main difficulties in deriving the main results are as follows. First, due to the introduction of the Markov jump model, both $P_a$ and ${\overline{P}}_a$ occur in the same condition. Thus, ${\overline{P}}_a$ must be separated appropriately. Second, the controller gain uncertainty is upper bounded instead of specific values. To obtain the design algorithm, the uncertainty has to be “eliminated”. Third, the randomness of controller gain perturbations brings nonlinear coupling terms ${\overline{\alpha }}^2{P}_a{B}_{es,a}Q{B}_{es,a}^T{P}_a$ that should be decoupled.

6. Simulation Example

To verify the effectiveness of the proposed results, the simulations are carried out where the system parameters of the Markov jump DC microgrid with two operation modes are listed as follows in

Table 2. System parameters.

$r_{L,1}$	1.1 $\mathrm{\Omega }$	$r_{L,2}$	1.0 $\mathrm{\Omega }$
$L_{1,1}$	39 mH	$L_{1,2}$	38 mH
$C_{1,1}$	500 $\mathsf{\mu }$F	$C_{1,2}$	503 $\mathsf{\mu }$F
$P_{1,1}$	300 W	$P_{1,2}$	302 W
$r_{s,1}$	1.1 $\mathrm{\Omega }$	$r_{s,2}$	1.15 $\mathrm{\Omega }$
$L_{s,1}$	39 mH	$L_{s,2}$	40 mH
$C_{s,1}$	500 $\mathsf{\mu }$F	$C_{s,2}$	498 $\mathsf{\mu }$F
$V_{dc}$	200 V	${\overline{v}}_{c,1,m}$	130.4 V
${\overline{v}}_{c,1,0}$	196.64 V

:

The rest of the simulation parameters are taken as $\sigma =10$, $\delta =0.1$, $\overline{\alpha }=0.01$, and the matrix of the mode transition rates is given as follows:

$$\mathrm{\Phi }=\left[\begin{array}{cc}-2& 2\\ 3& -3\end{array}\right]$$

The controller uncertainty is given as $\mathrm{\Delta }K=0.1sin\left(t\right)[1,1,1,1]$, which satisfies Assumption 1. The time interval is taken as $[0,0.5\mathrm{s}]$, and the initial value of the system is ${\overline{x}}_0=[14;-30;14;-30]$.

Based on these parameters and using Theorem 2, by virtue of the YALMIP toolbox in MATLAB 2019a, the mode-dependent fuzzy resilient controller gains are calculated as follows:

$$\begin{array}{cc}& K_{1,1}=[5.5694,0.4083,1.2005,0.3085],\\ & K_{1,2}=[5.5866,0.3968,1.1907,0.2846],\\ & K_{2,1}=[5.5007,0.3749,1.1912,0.3021],\\ & K_{2,2}=[5.5177,0.3612,1.1806,0.2781]\end{array}$$

Remark 9. 

As stated above, the simulation parameters mainly include the system matrices shown in Table 2, the decay rate σ, the expectation value $\overline{\alpha }$ of the random variable $\alpha \left(k\right)$, the upper bound δ of controller gain uncertainty, and the mode transition rates in the Φ matrix. Selecting different parameters results in different simulation results, and the effect of this parameter selection is reflected by the fuzzy resilient controller gains $K_{i,j}={\overline{K}}_{i,j}{X}_a^{-1}$, where ${\overline{K}}_{i,j}$ and $X_a^{-1}$ are adjusted according to the chosen simulation parameters. For example, the bigger the number of Markov jump modes, the less feasibility the LMI conditions of Theorem 2 have. The larger values of $\overline{\alpha }$ and δ may also bring a negative effect on the system performance.

Under the designed controller, the simulation results in Figure 3, Figure 4, Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 are obtained by executing 1000 Monte Carlo experiments. The jumping sequence of the system modes is described in Figure 3. To check the stability of the Markov jump DC microgrid without control, Figure 4 and Figure 5 are given from which one can see clearly that the open-loop system is not stable. By forcing the control law into the Markov jump DC microgrid, the current and voltage response in the CPL are, respectively, demonstrated in Figure 6 and Figure 7. Based on these two figures, it is seen that the designed controller works well in stabilizing the Markov jump DC microgrid. The curves of the fuzzy basis functions are shown in Figure 9.

Further, as shown in the remarks of the figure, the capacitor voltage $v_{C,1}$ arrives at its equilibrium point $v_{C,1,0}=196.6 \mathrm{V}$. In the meantime, the inductor current $i_{L,1}$ also reaches the operating point $1.526 \mathrm{A}$. The injection current (i.e., control input) is displayed in Figure 8, whose value becomes smaller as the system approaches the equilibrium point.

To further verify the effectiveness of the resilient control strategy, a comparative study is added. The voltage and current responses, with resilient control (named Case A) and without resilient control (named Case B), are respectively provided. The mode-dependent fuzzy controller gains without resilient control are calculated as follows:

$$\begin{array}{cc}& K_{1,1}=\left[\mathrm{4.18670.10831.03300.1122}\right],\\ & K_{1,2}=\left[\mathrm{4.38890.11641.03430.1112}\right],\\ & K_{2,1}=\left[\mathrm{4.11740.07451.02350.1058}\right],\\ & K_{2,2}=\left[\mathrm{4.31930.08031.02400.1047}\right]\end{array}$$

The simulation results in Case A are shown in Figure 6 and Figure 7, while the counterpart in Case B is added in Figure 10 and Figure 11. It is easily seen from Figure 10 and Figure 11 that the whole DC microgrid system is not stable without taking a resilient control strategy. While both the voltage and current trajectories are convergent under the proposed resilient control. From this point of view, the proposed method is effective in handling the controller gain uncertainties.

7. Conclusions

This paper focuses on the fuzzy resilient controller design for the DC microgrid subject to multiple operating modes and controller gain uncertainties. By developing the Markov jump DC microgrid model, the effect of abrupt parameter changes is well incorporated into the practical modeling of the DC microgrid. The derived exponential stability conditions provide the designers with an effective tool for analyzing the DC microgrid stability. Moreover, the effectiveness of the proposed control law can be checked beforehand via the proposed LMI-based controller gain computation algorithm. The simulations show that the proposed control method effectively stabilizes the Markov jump DC microgrid under sudden parameter changes, random perturbations of control law execution, and nonlinear dynamics. Note that the voltage and current states of the DC microgrid are transmitted to the controller by the communication network, which is of limited network resource. To alleviate the communication burden, in the future, the event-triggered mechanism [19,26,43] is to be investigated in the control of the DC microgrid to regulate the finite network resource and maintain the desired control performance. Also, the communication network is vulnerable to cyber-attacks [44,45]. Thus, secure control is another interesting topic for DC microgrids.