Output-feedback anti-disturbance consensus for multi-agent systems with application to secondary voltage regulation for microgrids

Abstract

This paper investigates a distributed finite-time consensus problem for second-order multi-agent systems (MASs) with disturbances and unmeasurable states. Output-feedback anti-disturbance consensus protocols are designed by finite-time observer (FTO) methods and the nonsingular integral terminal sliding-mode control (NITSMC). Some FTOs are first introduced for the agents to estimate the disturbances and unmeasurable states together. Then, the nonsingular integral terminal sliding-mode surfaces are proposed, based on which consensus protocols are proposed. Moreover, the consensus methods are also applied to microgrids (MGs). Based on MG systems, output voltages achieve consensus through secondary regulation. Simulations verify the effectiveness.

Keywords

Multi-agent systems output-feedback control consensus disturbance rejection control microgrids secondary voltage regulation

Introduction

Multi-agent systems (MASs) have received intense research interests, which are not only due to their practical applications (such as sensor distributed filtering (Ge et al., 2019), distributed control of power systems (Shi et al., 2019), and control of multiple quad-rotor aircraft (Du et al., 2019), etc.), but also their excellent efficiency when compared with conventional centralized control methods.

Consensus is a basic topic for distributed control, which means that the agent of a group achieves the agreement through implementation of suitable consensus controllers via local neighboring information. Many results have been reported on such topics (Babenko et al., 2018; Feng and Tu, 2016; Iqbal et al., 2018; Nowzari et al., 2019; Rehman et al., 2021; Seyboth et al., 2013; Shariati and Tavakoli, 2017). However, these results achieve uniformly asymptotic convergence or ultimately bounded. Taking convergence rates into account, it is satisfactory to convergence in finite time.

For MASs distributed finite-time control, some studies have been reported (Sharifi, 2022; Wang et al., 2017; Zuo et al., 2017). However, disturbances and unmeasurable states are not considered simultaneously in those results. On one hand, the disturbances are non-negligible in practical systems (Wang et al., 2016, 2020, 2021). On the other hand, the states of some control systems are unmeasurable, such as human–robot interaction forces (Yao et al., 2020) and motor speeds (Fan et al., 2021). Different from full-state feedback, such control design is more difficult. Cheng et al. (2015) studied the consensus problem for coupled harmonic oscillations with external disturbances and unmeasurable velocities. Ran et al. (2019) focused on a reliable and coordinated control problem of formation flying spacecraft, which considered unmeasurable velocities and external disturbances via observers. However, these results assume that the bounds of the disturbances are known, which are difficult to be obtained in practice.

In practice, there are broad applications for the consensus problem of MASs. An island alternating-current microgrid (MG) system is one of the typical applications where distributed generators (DGs) constitute a local grid Diaz et al. (2010). For an MG, the secondary regulation compensates for the deviations (Guerrero et al., 2011; Bidram and Davoudi, 2012). It can be divided by different ways of communicating, namely, centralized, distributed, and decentralized modes (Khayat et al., 2020; Li et al., 2021; Shafiee et al., 2014). In contrast to the other two modes, the distributed mode interacts signals with neighbors to facilitates plug and play, which has a certain research basis (Ning et al., 2021a; Pilloni et al., 2018; Sohrabzadi et al., 2022).

Disturbances are harmful to the coordinated operation of the whole MGs, which will cause device damage or even collapse (Ge et al., 2021). In Dehkordi et al. (2017) and Ning et al. (2021b), by using sliding-mode method, secondary regulation for MGs with disturbances was achieved. However, the derivatives of the output voltage are not accurate, caused by disturbances, and the available results are conservative in the assumption of disturbances.

The key challenges are dealing with disturbances and unmeasurable states simultaneously for second-order MASs and MGs in finite time. Meanwhile, the distributed consensus strategies should avoid using the inverse of Laplacian matrix directly, which is global information and inexistent in some communication topologies. This paper combines the nonsingular integral terminal sliding-mode control (NITSMC) and finite-time observer (FTO) techniques, proposes output-feedback anti-disturbance consensus (OFADC) protocols, achieves consensus for MASs with disturbances and unmeasurable states and are applied to secondary voltage regulation for MGs.

The main contributions are three aspects. First, the proposed consensus strategies avoiding the use of inverse of Laplacian matrix directly, and solve the disturbances and unmeasurable states for MASs. Second, applied to MG systems, the proposed secondary voltage regulation protocols deal with parametric perturbations and load variations. The unmeasurable derivatives of voltage are also considered. Last but not least, the proposed feedforward–feedback composite control algorithms relax the conservatism for the assumption of disturbances.

The rest part is as follows. Section “Preliminaries and problem formulation” introduces the main problems. Section “Consensus control design for MAS” discusses the consensus design for MASs and section “Application to secondary voltage regulation for MGs” applies to secondary regulation for MGs. Sections “Simulation results” and “Conclusion” are the simulations and conclusion, respectively.

Preliminaries and problem formulation

Notations

Denote sign function is $sgn (\cdot)$ . For $α \in R$ and $z = [z_{1}, \dots, z_{n}]^{T} \in R^{n}$ , denote $z^{γ} = [z_{1}^{γ}, \dots, z_{n}^{γ}]$ , $si g^{γ} (z) = [si g^{γ} (z_{1}), \dots, si g^{γ} (z_{n})]^{T}$ , where $si g^{γ} (w) = | w |^{γ} sgn (w)$ , $\forall w, γ \in R$ . For $z \in R^{n}$ , $sgn (z) = [sgn (z_{1}), \dots, sgn (z_{n})]^{T}$ . Denote $∥ z ∥_{2} = \sqrt{z^{T} z}$ as the Euclidean norm of $z \in R^{n}$ . The eigenvalues of symmetric matrix $Q \in R^{n \times n}$ exist $λ_{min} (Q) = λ_{1} (Q) \leq \dots \leq λ_{n} (Q) = λ_{max} (Q)$ .

Definitions and lemmas

Lemma 1. ( Huang et al., 2005 ) For $0 < p < 1$ and $z_{i} \in R^{m}, i = 1, \dots, m$ , there exists $(\sum_{i = 1}^{m} | z_{i} |)^{p} \leq \sum_{i = 1}^{m} | z_{i} |^{p}$ .

Lemma 2. ( Huang et al., 2005 ) For $p > 0, q > 0$ , $m, n \in R$ , if function $ϕ (m, n) > 0$ exists, then $| m |^{p} | n |^{q} \leq \frac{p ϕ (m, n) | m |^{p + q}}{p + q} + \frac{q ϕ^{- p / d} (m, n) | n |^{p + q}}{p + q}$ .

Consider the following system

\overset{\cdot}{φ} = g (φ), g (0_{m}) = 0_{m}, φ \in R^{m}

(1)

where $g : B \to R^{m}$ denotes a continuous function, $B \subseteq R^{n}$ is an open neighborhood.

Lemma 3. ( Bhat and Bernstein, 2005 ) There exists $ς_{1}, \dots, ς_{n} > 0$ such that $z^{n} + ς_{n} z^{n - 1} + \dots, + ς_{2} z + ς_{1}$ is a Hurwitz polynomial. Consider following system

{\overset{\cdot}{φ}}_{j} = φ_{j + 1}, j = 1, \dots, m - 1, {\overset{\cdot}{φ}}_{n} = μ

(2)

For the system equation (2), if $\forall ζ \in (0, 1)$ for $α \in (1 - ζ, 1)$ , a following controller

μ = - ς_{1} si g^{α_{1}} (φ_{1}) - \dots - ς_{m} si g^{α_{m}} (φ_{m})

(3)

realizes the system equation (2) globally finite-time stable, where $α_{j - 1} = \frac{α_{j} α_{j + 1}}{2 α_{j + 1} - α_{j}}$ , $j = 2, \dots, m$ , $α_{m} = α$ , $α_{m + 1} = 1$ .

Then consider the following system

\overset{\cdot}{φ} = μ + ϖ (t)

(4)

where $φ, ϖ (t), μ \in R$ are state, disturbance, and input, respectively.

Lemma 4. ( Shtessel et al., 2007 ) If $ϖ (t)$ is $h - 1 \geq 0$ times differentiable, there exists a known Lipschitz constant $L > 0$ for $ϖ^{(h - 1)} (t)$ , an observer is finite-time convergent as

{\begin{matrix} {\overset{\cdot}{ψ}}_{0} = μ + ϵ_{0} \\ ϵ_{0} = ψ_{1} - ρ_{0} L^{\frac{1}{h + 1}} si g^{\frac{h}{h + 1}} (ψ_{0} - φ) \\ {\overset{\cdot}{ψ}}_{l} = ϵ_{l}, \\ ϵ_{l} = ψ_{l + 1} - ρ_{l} L^{\frac{1}{h + 1 - l}} si g^{\frac{h - l}{h + 1 - l}} (ψ_{l} - θ_{l - 1}), l = 1, \dots, h - 1 \\ {\overset{\cdot}{ψ}}_{h} = - ρ_{h} L sgn (ψ_{h} - ϵ_{h - 1}) \end{matrix}

(5)

where $ρ_{0}, \dots, ρ_{h} > 0$ , and $ψ_{0} = \hat{φ}, ψ_{l} = \hat{ϖ^{(l - 1)}}, l = 1, \dots, h$ denote the estimates of $φ, ϖ^{(l - 1)}$ .

Graph theory notions

Directed graph $G = (V, E, A)$ contains a node set $V = {1, \dots, n}$ , an edge set $E \subseteq V \times V$ and a weighted adjacent matrix $A = [a_{ij}] \in R^{n \times n}$ . If the edge $(j, i) \in E$ , then $a_{ij} > 0$ , otherwise $a_{ij} = 0$ . Define the Laplacian matrix of $G$ as $L = [l_{ij}] \in R^{n \times n}$ , where $l_{ii} = \sum_{i \neq j} a_{ij}$ and $l_{ij} = - a_{ij}$ , $i \neq j$ . Label 0 for the leader, and the graph is $\bar{G} = (\bar{V}, \bar{E}, \bar{A})$ with the node set $\bar{V} = V \cup {0}$ . Denote $B = diag {b_{1}, \dots, b_{n}}$ with $b_{i} = 1$ , if the leader communicates with the followers, $b_{i} = 0$ , otherwise. The Laplacian matrix of $\bar{G}$ is $Q = B + L$ .

Assumption 1. At least a node has directed paths to the others in graph $\bar{G}$ .

Problem formulation

Consider the MAS with unmeasurable states, and the follower dynamics are

\begin{matrix} {\overset{\cdot}{x}}_{i, 1} = x_{i, 2} \\ {\overset{\cdot}{x}}_{i, 2} = u_{i} + ϖ_{i}, i \in V \end{matrix}

(6)

where $x_{i, 1}, x_{i, 2} \in R$ are states, $y_{i} = x_{i, 1}$ is output, $u_{i} \in R$ is controller, and $ϖ_{i} \in R$ is disturbance. The second state $x_{i, 2}$ is unmeasurable. The leader dynamics are

\begin{matrix} {\overset{\cdot}{x}}_{0, 1} = x_{0, 2} \\ {\overset{\cdot}{x}}_{0, 2} = u_{0} \end{matrix}

(7)

where $x_{0, 1}, x_{0, 2} \in R$ are leader states, $y_{0} = x_{0, 1}$ is output, and $u_{0} \in R$ is input which is predefined. The system is described by a directed graph $\bar{G}$ .

Assumption 2. The states $x_{0, 1}$ and $x_{0, 2}$ are bounded and there is an upper bound ${\bar{u}}_{0}$ such that $∥ u_{0} (t) ∥_{2} \leq {\bar{u}}_{0}$ .

For the MAS equations (6) and (7), the purpose is to achieve consensus for the agents’ outputs .

Consensus control design for MAS

The consensus protocols design for the MAS equations (6) and (7) include the following two parts.

FTO design

Assumption 3. $ϖ_{i}$ $(i \in V)$ is $h - 1$ times differentiable. $ϖ_{i}^{(h - 1)}$ exists a Lipschitz constant $L_{i}$ .

Remark 1. Assumptions 1–3 are reasonable for MASs. Assumptions 1 and 2 are used to ensure the connectivity of MASs, which are the most basic assumptions for the study of the MAS consensus problem. Several kinds of disturbances satisfy Assumption 3, so it is reasonable for this assumption.

For the follower $i$ ( $i \in V$ ), some FTOs are designed as

{\begin{matrix} {\overset{\cdot}{ψ}}_{i, 0} = ϵ_{i, 0} \\ ϵ_{i, 0} = - ρ_{i, 0} L_{i}^{\frac{1}{h + 1}} si g^{\frac{h}{h + 1}} (ψ_{i, 0} - x_{i, 1}) + ψ_{i, 1} \\ {\overset{\cdot}{ψ}}_{i, 1} = ϵ_{i, 1} + u_{i} \\ ϵ_{i, 1} = - ρ_{i, 1} L_{i}^{\frac{1}{h}} si g^{\frac{h - 1}{h}} (ψ_{i, 1} - ϵ_{i, 0}) + ψ_{i, 2} \\ {\overset{\cdot}{ψ}}_{i, l} = ϵ_{i, l} \\ ϵ_{i, l} = - ρ_{i, l} L_{i}^{\frac{1}{h + 1 - l}} si g^{\frac{h - l}{h + 1 - l}} (ψ_{i, l} - ϵ_{i, l - 1}) + ψ_{i, l + 1}, l = 2, \dots, h \\ {\overset{\cdot}{ψ}}_{i, h + 1} = - ρ_{i, h + 1} L_{i} sgn (ψ_{i, h + 1} - ϵ_{i, h}) \end{matrix}

(8)

where $ρ_{i, l}$ , $l = 2, \dots, m$ are observer gains, $ψ_{i, 0} = \hat{x_{i, 1}}$ , $ψ_{i, 1} = \hat{x_{i, 2}}$ , $ψ_{i, l} = \hat{ϖ_{i}^{(l - 2)}}$ , and $l = 2, \dots, h + 1$ represent the estimates of $x_{i, 1}$ , $x_{i, 2}$ and $ϖ_{i}^{(l - 2)}$ , respectively. From equations (6) and (8), the error dynamic of FTOs are

{\begin{matrix} {\overset{\cdot}{θ}}_{i, 0} = θ_{i, 1} - ρ_{i, 0} L_{i}^{\frac{1}{h + 1}} si g^{\frac{h}{h + 1}} (θ_{i, 0}) \\ {\overset{\cdot}{θ}}_{i, 1} = θ_{i, 2} - ρ_{i, 1} L_{i}^{\frac{1}{h}} si g^{\frac{h - 1}{h}} (θ_{i, 1} - {\overset{\cdot}{θ}}_{i, 0}) \\ {\overset{\cdot}{θ}}_{i, l} = θ_{i, l + 1} - ρ_{i, l} L_{i}^{\frac{1}{h + 1 - l}} si g^{\frac{h - l}{h + 1 - l}} (θ_{i, l} - {\overset{\cdot}{θ}}_{i, l - 1}), l = 2, \dots, h \\ {\overset{\cdot}{θ}}_{i, h + 1} \in [- L_{i}, L_{i}] - ρ_{i, h + 1} L_{i} sgn (θ_{i, h + 1} - {\overset{\cdot}{θ}}_{i, h}) \end{matrix}

(9)

where $θ_{i, 0} = x_{i, 1} - ψ_{i, 0}$ , $θ_{i, 1} = x_{i, 2} - ψ_{i, 1}$ , $θ_{i, l} = d_{i}^{(l - 2)} - ψ_{i, l}$ , and $l = 2, \dots, h + 1$ .

Proposition 1. Via choosing suitable $ρ_{i, l}$ , $i \in V, l = 0, \dots, h + 1$ , FTOs equation (8) are convergence in a finite-time $T_{1}$ , the errors satisfy $θ_{i, l} (t) = 0, i \in V, l = 0, \dots, h + 1, \forall t \geq T_{1}$ .

Proof. By error dynamic (9), Lemma 4, Proposition 1, et al

Composite finite-time consensus protocols design

Consensus errors are defined as

\begin{matrix} ϵ_{i, 1} = \sum_{j = 1}^{n} a_{ij} (x_{i, 1} - x_{j, 1}) + b_{i} (x_{i, 1} - x_{0, 1}) \\ ϵ_{i, 2} = \sum_{j = 1}^{n} a_{ij} (\hat{x_{i, 2}} - \hat{x_{j, 2}}) + b_{i} (\hat{x_{i, 2}} - x_{0, 2}) \end{matrix}

(10)

Denote $ϖ = [ϖ_{1}, \dots, ϖ_{n}]^{T}$ , $E_{c} = [ε_{1, c}, \dots, ε_{n, c}]^{T}$ , $u = [u_{1}, \dots, u_{n}]^{T}$ , $Λ_{l} = {[θ_{1, l}, \dots, θ_{n, l}]}^{T}$ , $c = 1, 2$ , and $l = 0, 1, 2 \dots, h + 1$ . The dynamics of tracking errors are

\begin{matrix} {\overset{\cdot}{E}}_{1} = E_{2} + {\tilde{Λ}}_{1} \\ {\overset{\cdot}{E}}_{2} = Q (u + ϖ - {\overset{\cdot}{Λ}}_{1} - 1_{n} u_{0}) \end{matrix}

(11)

where ${\overset{\cdot}{Λ}}_{1} = {[{\overset{\cdot}{Λ}}_{1, 1}, \dots, {\overset{\cdot}{Λ}}_{n, 1}]}^{T}$ and ${\tilde{Λ}}_{1} = Q Λ_{1}$ .

Theorem 1. For MAS equations (6) and (7), if Assumptions 1–3 hold, the OFADC protocols are presented $(i \in V)$

u_{i} = - γ_{i} sgn (\sum_{j = 1}^{n} a_{ij} (s_{i} - s_{j}) + b_{i} s_{i}) - \hat{ϖ_{i}}

(12)

where $γ_{i} \geq n (| \sum_{c = 1}^{2} ς_{c} si g^{α_{c}} (ε_{i, c}) | + b_{i} {\bar{u}}_{0} + ρ_{i}) / λ_{min} (Q)$ , $ϱ_{i} > 0$ . $\hat{ϖ_{i}}$ are the estimates of $ϖ_{i}$ from equation (8), and the nonsingular integral terminal sliding-mode surface functions $s_{i}$ are

s_{i} = \int_{0}^{t} (\sum_{c = 1}^{2} ς_{c} si g^{α_{c}} (ε_{i, c})) d ζ + ε_{i, 2}

(13)

where $ς_{1} > 0$ , $ς_{2} > 0$ , $0 < α_{1} < 1$ , $α_{2} = \frac{2 α_{1}}{1 + α_{1}}$ . Then $x_{i, 1} \to x_{i, 0}, i \in V$ is realized in finite time.

Proof. Denote $S = [s_{1}, \dots, s_{n}]^{T}$ , $\hat{ϖ} = [\hat{ϖ_{1}}, \dots, \hat{ϖ_{n}}]^{T}$ and $ϒ = diag {γ_{1}, \dots, γ_{n}}$ , then the sliding-mode surface functions and controllers are

S = E_{2} + \int_{0}^{t} [ς_{1} si g^{α_{1}} (E_{1}) + ς_{2} si g^{α_{2}} (E_{2})] d ζ

(14)

u = - ϒ sgn (Q S) - \hat{ϖ}

(15)

where $si g^{α_{c}} (E_{1}) = {[si g^{α_{c}} (e_{1, c}), \dots, si g^{α_{c}} (e_{n, c})]}^{T}$ , $c = 1, 2$ . By equations (14) and (15), the dynamics of tracking errors are

\begin{matrix} {\overset{\cdot}{E}}_{1} = E_{2} + {\tilde{Λ}}_{1} \\ {\overset{\cdot}{E}}_{2} = \overset{\cdot}{S} - ς_{1} si g^{α_{1}} (E_{1}) - ς_{2} si g^{α_{2}} (E_{2}) \end{matrix}

(16)

The following proof includes the proof of state boundedness for system equation (16) and global finite-time convergence.

Step 1. (State boundedness in $[0, T_{1})$ ) Define a energy function

W_{1} = \frac{1}{2} S^{T} S + \frac{1}{2} E_{1}^{T} E_{1} + \frac{1}{2} E_{2}^{T} E_{2}

(17)

${\overset{\cdot}{W}}_{1}$ along equations (14) and (16) is

\begin{matrix} {\overset{\cdot}{W}}_{1} = S^{T} Q (Λ_{2} - {\overset{\cdot}{Λ}}_{1} - 1_{n} u_{0}) - S^{T} Q ϒ sgn (Q S) \\ + S^{T} [ς_{1} si g^{α_{1}} (E_{1}) + ς_{2} si g^{α_{2}} (E_{2})] + E_{1}^{T} (E_{2} + {\tilde{Λ}}_{1}) \\ + E_{2}^{T} Q (- ϒ sgn (Q S) + Λ_{2} - {\overset{\cdot}{Λ}}_{1} - 1_{n} u_{0}) \end{matrix}

(18)

Consider $0 \geq - \underline{γ} λ_{min} (Q) n^{- \frac{1}{2}} \sum_{i = 1}^{n} | S_{i} | \geq - \underline{γ} ∥ Q S ∥_{2} \geq -$ $S^{T} Q ϒ sgn (Q S)$ , where $\underline{γ} = min_{i \in V} {γ_{i}}$ and $∥ si g^{α_{c}} (E_{c}) ∥_{2} \leq n ∥ E_{c} ∥_{2}^{α_{c}}$ , $c = 1, 2$ , then

\begin{matrix} {\overset{\cdot}{W}}_{1} \leq {∥ Q ∥_{2} [∥ {\overset{\cdot}{Λ}}_{1} ∥_{2} + ∥ Λ_{2} ∥_{2} + ({\bar{u}}_{0} + \bar{γ}) \sqrt{n}] + ∥ {\tilde{Λ}}_{1} ∥_{2}} \\ \times (∥ S ∥_{2} + \sum_{c = 1}^{2} ∥ E_{c} ∥_{2}) + \sum_{c = 1}^{2} \frac{n ς_{c} α_{c}}{1 + α_{c}} ∥ E_{c} ∥_{2}^{1 + α_{c}} \\ + \sum_{c = 1}^{2} \frac{n ς_{c}}{1 + α_{c}} ∥ S ∥_{2}^{1 + α_{c}} + \frac{1}{2} \sum_{c = 1}^{2} ∥ E_{c} ∥_{2}^{2} \end{matrix}

(19)

where $\bar{γ} = max_{i \in V} {γ_{i}}$ . From Lemma 2, $∥ E_{c} ∥_{2} \leq (∥ E_{c} ∥_{2}^{2} + 1) / 2$ , $∥ S ∥_{2}^{1 + α_{c}} \leq \frac{1 - α_{c}}{2} + \frac{1 + α_{c}}{2} ∥ S ∥_{2}^{2}$ , $∥ E_{c} ∥_{2}^{1 + α_{c}} \leq \frac{1 - α_{c}}{2} + \frac{1 + α_{c}}{2} ∥ E_{c} ∥_{2}^{2}$ , it is gained that

{\overset{\cdot}{W}}_{1} \leq δ_{1} W_{1} + δ_{2}

(20)

where $δ_{1} = ∥ {\tilde{Λ}}_{1} ∥_{2} + ∥ Q ∥_{2} [∥ {\dot{Λ}}_{1} ∥_{2} + ∥ Λ_{2} ∥_{2} + ({\bar{u}}_{0} + \bar{γ}) n^{\frac{1}{2}}] +$ $∥ Λ_{2} ∥_{2} + ({\bar{u}}_{0} + \bar{γ}) n^{\frac{1}{2}}] + 2 n \bar{ς} + \sum_{c = 1}^{2} n ς_{c} α_{c} + 1$ , $\bar{ς} = \max_{c = 1, 2} {ς_{c}}$ , and $δ_{2} = [∥ {\dot{Λ}}_{1} ∥_{2} + ∥ Λ_{2} ∥_{2} + ({\bar{u}}_{0} + \bar{γ}) n^{\frac{1}{2}}] ∥ Q ∥_{2} + ∥ {\tilde{Λ}}_{1} ∥_{2} +$ $\sum_{c = 1}^{2} (1 - α_{c}) n ς_{c} / 2$ . Since FTOs equation (8) are convergent, $Λ_{1} (t)$ , ${\overset{\cdot}{Λ}}_{1} (t)$ , $Λ_{2} (t)$ , ${\tilde{Λ}}_{1} (t)$ are bounded. Therefore, $S (t)$ , $E_{1} (t)$ , $E_{2} (t)$ are bounded.

Step 2. (Finite-time convergence) If $t \geq T_{1}$ , $Λ_{l} = 0$ , $l = 0, \dots, h + 1$ . Choose an energy function

W_{2} = \frac{1}{2} S^{T} S

(21)

${\overset{\cdot}{W}}_{2}$ along equation (14) is

\begin{matrix} {\overset{\cdot}{W}}_{2} \leq - \sum_{i = 1}^{n} [λ_{min} (Q) \underline{γ} / \sqrt{n} - b_{i} {\bar{u}}_{0} - | \sum_{c = 1}^{2} ς_{c} {sig}_{c}^{α} (ϵ_{i, c}) |] | s_{i} | \\ \leq - \sqrt{2} \underline{ϱ} (W_{2})^{\frac{1}{2}} \end{matrix}

(22)

where $\underline{γ} = min_{i \in V} {γ_{i}}$ , $ϱ_{i} > 0$ , and $\underline{ϱ} = min_{i \in V} {ϱ_{i}}$ , $i \in V$ . Once $S = 0$ in finite time, one obtains

\begin{matrix} {\overset{\cdot}{E}}_{1} = E_{2} \\ {\overset{\cdot}{E}}_{2} = - ς_{1} si g^{α_{1}} (E_{1}) - ς_{2} si g^{α_{2}} (E_{2}) \end{matrix}

(23)

According to Lemma 3, the proof is done.

Remark 2. For MAS equations (6) and (7), Theorem 1 still holds for the agents when the system states, disturbances and inputs are multidimensional, that is, $x_{i, 1}, x_{i, 2}, u_{i} \in R^{m}$ , where $m > 1$ is a constant.

Remark 3. Using the disturbances estimates and the states coming from the agent and its neighbors, the proposed OFADC protocols equation (12) have a feedforward-feedback composite form, which controls the states convergence and compensate the disturbance.

Remark 4. If there are no disturbances, to achieve consensus for MAS equations (6) and (7), it is still needed for some FTOs to estimate the unmeasurable states $x_{i, 2}$ , $i \in V$ . By Lemma 4, the observers for the follower $i$ are designed as ( $i \in V$ )

{\begin{matrix} {\overset{\cdot}{\bar{ψ}}}_{i, 0} = {\bar{ψ}}_{i, 1} - {\bar{ρ}}_{i, 0} {\bar{L}}_{i}^{\frac{1}{2}} si g^{\frac{1}{2}} ({\bar{ψ}}_{i, 0} - x_{i, 1}) = {\bar{ϵ}}_{i, 0} \\ {\overset{\cdot}{\bar{ψ}}}_{i, 1} = u_{i} - {\bar{ρ}}_{i, 1} {\bar{L}}_{i} sig ({\bar{ψ}}_{i, 1} - {\bar{ϵ}}_{i, 0}) \end{matrix}

(24)

where $ρ_{i, 0}$ and $ρ_{i, 1}$ are observer gains, ${\bar{ψ}}_{i, 0} = \hat{x_{i, 1}}$ , ${\bar{ψ}}_{i, 1} = \hat{x_{i, 2}}$ denote the estimates of $x_{i, 1}$ , $x_{i, 2}$ , respectively. The error dynamics are

{\begin{matrix} {\overset{\cdot}{\bar{θ}}}_{i, 0} = {\bar{θ}}_{i, 1} - {\bar{ρ}}_{i, 0} {\bar{L}}_{i}^{\frac{1}{2}} si g^{\frac{1}{2}} ({\bar{θ}}_{i, 0}) \\ {\overset{\cdot}{\bar{θ}}}_{i, 1} \in [- L_{i}, L_{i}] + {\bar{ρ}}_{i, 1} {\bar{L}}_{i} sgn ({\overset{\cdot}{\bar{θ}}}_{i, 0} - {\bar{θ}}_{i, 1}) \end{matrix}

(25)

where ${\bar{θ}}_{i, 0} = x_{i, 1} - ψ_{i, 0}$ , ${\bar{θ}}_{i, 1} = x_{i, 2} - ψ_{i, 1}$ . Consider Lemma 4, equation (24)convergence in finite time. The output-feedback reduced consensus (OFRC) protocols are ( $i \in V$ )

u_{i} = γ_{i} sgn (\sum_{j = 1}^{n} a_{ij} (s_{j} - s_{i}) - b_{i} s_{i})

(26)

where the sliding-mode surface functions are designed in equation (13) and the parameters are the same as those in OFADC protocols equation (12). Consensus is achieved under the OFRC protocols equation (26)if there are no disturbances. However, consensus cannot be realized in the case of disturbances by the OFRC protocols equation (26) as a result of the lack of disturbance compensations, which reflects the superiority of the proposed protocols in Theorem 1.

Application to secondary voltage regulation for MGs

MG systems modeling

An MG is formed from $n$ DGs, controllable loads and energy storages. Figure 1 displays the block diagram of an MG. Energy sources, voltage source converters (VSCs), LC filters, and output connectors make up DGs.

Angular frequency of the DG $i$ is $ω_{i}$ , and common reference frequency $ω_{com}$ is chosen from one DG. The angle frequency deviation $ν_{i}$ satisfies

\overset{\cdot}{ν_{i}} = ω_{i} - ω_{com}

(27)

The dynamics of the active power $P_{i}$ and reactive power $Q_{i}$ are

{\overset{\cdot}{P}}_{i} = (i_{odi} v_{odi} + i_{oqi} v_{oqi}) ω_{ci} - ω_{ci} P_{i}

(28)

{\overset{\cdot}{Q}}_{i} = (i_{odi} v_{oqi} - i_{oqi} v_{odi}) ω_{ci} - ω_{ci} Q_{i}

(29)

where $i_{oi} = [i_{odi}, i_{oqi}]^{T}$ and $v_{oi} = [v_{odi}, v_{oqi}]^{T}$ are output current and voltage and $ω_{ci}$ is the cutoff frequency. The frequency droop characteristic is $ω_{i} = ω_{ni} - m_{P_{i}} P_{i}$ , where $ω_{ni}$ is the primary control frequency reference and $m_{P_{i}}$ is the droop coefficient. The voltage droop characteristic is written as

v_{odi}^{*} = V_{ni} - n_{Qi} Q_{i}

v_{oqi}^{*} = 0

(30)

where $V_{ni}$ is the primary control voltage reference and $n_{Q_{i}}$ is the droop coefficient. The PI control of the voltage loop is

{\overset{\cdot}{χ}}_{di} = v_{odi}^{*} - v_{odi}

(31)

{\overset{\cdot}{χ}}_{qi} = v_{oqi}^{*} - v_{oqi}

(32)

i_{ldi}^{*} = i_{odi} + K_{PVi} (v_{odi}^{*} - v_{odi}) + K_{IVi} χ_{di} - ω_{b} C_{fi} v_{oqi}

(33)

i_{lqi}^{*} = i_{oqi} + K_{PVi} (v_{oqi}^{*} - v_{oqi}) + K_{IVi} χ_{qi} + ω_{b} C_{fi} v_{odi}

(34)

where $χ_{d}$ and $χ_{q}$ are auxiliary variables, $i_{li}^{*} = [i_{ldi}^{*}, i_{lqi}^{*}]^{T}$ is the reference current, $C_{fi}$ represents the filter capacitance value, $ω_{b}$ is the nominal angular frequency, and $K_{PVi}$ and $K_{IVi}$ are the positive parameters. The PI control of the current loop is

\overset{\cdot}{η_{di}} = i_{ldi}^{*} - i_{ldi}

(35)

\overset{\cdot}{η_{qi}} = i_{lqi}^{*} - i_{lqi}

(36)

v_{ldi}^{*} = K_{PCi} (i_{ldi}^{*} - i_{ldi}) + K_{ICi} η_{di} - ω_{b} L_{fi} i_{lqi}

(37)

v_{lqi}^{*} = K_{PCi} (i_{lqi}^{*} - i_{lqi}) + K_{ICi} η_{qi} + ω_{b} L_{fi} i_{ldi}

(38)

where $η_{d}$ and $η_{q}$ are auxiliary variables, $v_{li}^{*} = [v_{ldi}^{*}, v_{lqi}^{*}]^{T}$ is the control input for the inverter, $i_{li}^{*} = [i_{ldi}^{*}, i_{lqi}^{*}]^{T}$ is the filter inductance current, $L_{fi}$ represents the filter inductance value, and $K_{PCi}$ and $K_{ICi}$ are the positive parameters.

The dynamics of the filter are

{\overset{\cdot}{i}}_{ldi} = \frac{1}{L_{fi}} (v_{ldi} - R_{fi} i_{ldi} - v_{odi}) + ω_{i} i_{lqi}

(39)

{\overset{\cdot}{i}}_{lqi} = \frac{1}{L_{fi}} (v_{lqi} - R_{fi} i_{lqi} - v_{oqi}) - ω_{i} i_{ldi}

(40)

where $v_{li} = [v_{ldi}, v_{lqi}]^{T}$ is the inverter output voltage.

The dynamics of the output connector are

{\overset{\cdot}{v}}_{odi} = \frac{1}{C_{fi}} (i_{ldi} - i_{odi}) + ω_{i} v_{oqi}

(41)

{\overset{\cdot}{v}}_{oqi} = \frac{1}{C_{fi}} (i_{lqi} - i_{oqi}) - ω_{i} v_{odi}

(42)

{\overset{\cdot}{i}}_{odi} = \frac{1}{L_{ci}} (v_{odi} - R_{ci} i_{odi} - v_{bdi}) + ω_{i} i_{oqi}

(43)

{\overset{\cdot}{i}}_{oqi} = \frac{1}{L_{ci}} (v_{oqi} - R_{ci} i_{oqi} - v_{bqi}) - ω_{i} i_{odi}

(44)

where $v_{bi} = [v_{bdi}, v_{bqi}]^{T}$ is the bus voltage. The dynamic model of MGs is given by equations (27)–(44) as

{\begin{matrix} \begin{matrix} {\overset{\cdot}{x}}_{i} = g_{i} (x_{i}) u_{i} + k_{i} (x_{i}) f_{di} + f_{i} (x_{i}) \\ y_{i, 1} = h_{i, 1} (x_{i}) = v_{odi} \\ y_{i, 2} = - m_{Pi} P_{i} + ω_{ni} = h_{i, 2} (x_{i}) + u_{i, 2} = ω_{i} \end{matrix} \end{matrix}

(45)

where

f_{i} (x_{i}) = [\begin{matrix} - m_{pi} P_{i}, \\ ω_{ci} (v_{odi} i_{odi} + v_{oqi} i_{oqi} - P_{i}), \\ ω_{ci} (v_{oqi} i_{odi} + v_{odi} i_{oqi} - Q_{i}), \\ - n_{Qi} Q_{i} - v_{odi}, \\ - v_{oqi}, \\ i_{odi} - ω_{b} C_{fi} v_{oqi} + K_{PVi} (- n_{Qi} Q_{i} - v_{odi}) \\ + K_{IVi} χ_{di}, \\ i_{oqi} + ω_{b} C_{fi} v_{odi} - K_{PVi} v_{oqi} + K_{IVi} χ_{qi}, \\ - \frac{R_{fi}}{L_{fi}} i_{ldi} + ω_{i} i_{lqi} - ω_{b} i_{lqi} + \frac{K_{PCi}}{L_{fi}} (i_{odi} \\ - ω_{b} C_{fi} v_{oqi} - K_{PVi} n_{Qi} Q_{i} - K_{PVi} v_{odi} \\ + K_{IVi} χ_{di}) - \frac{1}{L_{fi}} v_{odi} + \frac{K_{ICi}}{L_{fi}} η_{qi}, \\ - \frac{R_{fi}}{L_{fi}} i_{lqi} - ω_{i} i_{lqi} + ω_{b} i_{ldi} + \frac{K_{PCi}}{L_{fi}} (i_{oqi} \\ + ω_{b} C_{fi} v_{odi} - K_{PVi} v_{oqi} + K_{IVi} χ_{qi} \\ - i_{lqi}) + \frac{K_{ICi}}{L_{fi}} η_{qi} - \frac{1}{L_{fi}} v_{oqi}, \\ ω_{i} v_{oqi} + \frac{1}{C_{fi}} i_{ldi} - \frac{1}{C_{fi}} i_{odi}, \\ ω_{i} v_{odi} + \frac{1}{C_{fi}} i_{lqi} - \frac{1}{C_{fi}} i_{oqi}, \\ - \frac{R_{ci}}{L_{ci}} i_{odi} + ω_{i} i_{oqi} + \frac{1}{L_{ci}} v_{odi}, \\ - \frac{R_{ci}}{L_{ci}} i_{oqi} + ω_{i} i_{odi} + \frac{1}{L_{ci}} v_{oqi} \end{matrix}]

k_{i} (x_{i}) = {[\begin{matrix} - 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & - 1 \end{matrix}]}^{T}

g_{i} (x_{i}) = {[\begin{matrix} 0 & 0 & 0 & 1 & 0 & K_{pvi} & 0 & \frac{K_{pci}}{L_{fi}} K_{pvi} & 0 & 0 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \end{matrix}]}^{T}

The state is $x_{i} = [ν_{i}, P_{i}, Q_{i}, χ_{di}, χ_{qi}, η_{di}, η_{qi}, i_{ldi}, i_{lqi}, v_{odi}, v_{oqi}, i_{odi}, i_{oqi}]^{T}$ . Denote $F_{i} (x_{i}) = k_{i} (x_{i}) + f_{di} f_{i} (x_{i})$ , where $f_{di} = [ω_{com}, v_{bdi}, v_{bqi}]^{T}$ . the input is $u_{i} = [V_{ni}, ω_{ni}]^{T}$ .

Feedback linearization

By feedback linearization, $v_{odi}$ and $V_{ni} = u_{i, 1}$ satisfy

{\overset{\cdot\cdot}{y}}_{i, 1} = ϖ_{i} + L_{g_{i}} L_{F_{i}} h_{i} (x_{i}) u_{i, 1} + L_{F_{i}}^{2} h_{i} (x_{i})

(46)

where $L_{F_{i}}^{2} h_{i} (x_{i})$ and $L_{g_{i}} L_{F_{i}} h_{i} (x_{i})$ are Lie derivatives and $ϖ_{i}$ denotes the disturbance. Denote $τ_{i} = L_{g_{i}} L_{F_{i}} h_{i} (x_{i}) u_{i, 1} + L_{F_{i}}^{2} h_{i} (x_{i})$ . Then $u_{i, 1}$ is written as

u_{i, 1} = \frac{τ_{i} - L_{F_{i}}^{2} h_{i} (x_{i})}{L_{g_{i}} L_{F_{i}} h_{i} (x_{i})}

(47)

where the terms of Lie derivative are

\begin{matrix} L_{F_{i}}^{2} h_{i} (x_{i}) = (- ω_{i}^{2} - \frac{K_{PCi} K_{PVi} + 1}{C_{fi} L_{fi}} - \frac{1}{C_{fi} L_{ci}}) v_{odi} \\ - \frac{K_{PCi} w_{b}}{L_{fi}} v_{oqi} + (\frac{K_{PCi}}{C_{fi} L_{fi}} + \frac{R_{ci}}{C_{fi} L_{ci}}) i_{od} \\ - \frac{2 ω_{i}}{C_{fi}} i_{oq} - \frac{K_{PCi} + R_{fi}}{C_{fi} L_{fi}} i_{ldi} + \frac{2 ω_{i} - ω_{b}}{C_{fi}} i_{lqi} \\ - \frac{K_{PCi} K_{PVi} n_{Qi}}{C_{fi} L_{fi}} Q_{i} + \frac{K_{PCi} K_{IVi}}{C_{fi} L_{fi}} φ_{di} \\ + \frac{K_{ICi}}{C_{fi} L_{fi}} η_{di} + \frac{1}{C_{fi} L_{ci}} v_{bdi} \end{matrix}

(48)

L_{g_{i}} L_{F_{i}} h_{i} (x_{i}) = \frac{K_{PCi} K_{PVi}}{C_{fi} L_{fi}}

(49)

Rewrite equation (46) as

{\overset{\cdot\cdot}{y}}_{i, 1} = ϖ_{i} + τ_{i}

(50)

Remark 5. Equation (50) can be rewritten as ${\overset{\cdot}{y}}_{i, 1} = {\overset{\cdot}{v}}_{odi} + ϖ_{i, 1}$ , ${\overset{\cdot\cdot}{v}}_{odi} = τ_{i} + ϖ_{i, 2}$ , where $ϖ_{i} = {\overset{\cdot}{ϖ}}_{i, 1} + ϖ_{i, 2}$ . Therefore, ${\overset{\cdot}{y}}_{i, 1}$ is unknown. There is a difference between stochastic noise and disturbance. In practice, disturbance usually refers to external disturbance, and noise is usually brought by internal components or electromagnetic radiation. Noise is generally random and irregular, and it can be processed by filter, which is not the main research issue in this paper.

Secondary voltage regulation

The MG system is assumed to satisfy Assumptions 1 and 3. For the DG $i$ , FTOs are designed as

{\begin{matrix} {\overset{\cdot}{ψ}}_{i, 0} = ϵ_{i, 0} \\ ϵ_{i, 0} = - ρ_{i, 0} L_{i}^{\frac{1}{h + 1}} si g^{\frac{h}{h + 1}} (ψ_{i, 0} - y_{i, 1}) + ψ_{i, 1} \\ {\overset{\cdot}{ψ}}_{i, 1} = τ_{i} + ϵ_{i, 1} \\ ϵ_{i, 1} = - ρ_{i, 1} L_{i}^{\frac{1}{h}} si g^{\frac{h - 1}{h}} (ψ_{i, 1} - ϵ_{i, 0}) + ψ_{i, 2} \\ {\overset{\cdot}{ψ}}_{i, l} = ϵ_{i, l} \\ ϵ_{i, l} = - ρ_{i, l} L_{i}^{\frac{1}{h + 1 - l}} si g^{\frac{h - l}{h + 1 - l}} (ψ_{i, l} - ϵ_{i, l - 1}) + ψ_{i, l + 1}, l = 2, \dots, h \\ {\overset{\cdot}{ψ}}_{i, h + 1} = - ϵ_{i, h + 1} L_{i} sgn (ψ_{i, h + 1} - ϵ_{i, h}) \end{matrix}

(51)

where $ρ_{i, l}$ , $l = 2, \dots, h$ are gains, $ψ_{i, 0} = \hat{y_{i, 1}}$ , $ψ_{i, 1} = \hat{{\overset{\cdot}{y}}_{i, 1}}$ , $ψ_{i, l} = \hat{ϖ_{i}^{(l - 2)}}$ , $l = 2, \dots, h + 1$ denote the estimates. By Lemma 5, the observers equation (51) are finite-time convergent.

Define consensus errors as

\begin{matrix} ϵ_{i, 1} = \sum_{j = 1}^{n} a_{ij} (y_{i, 1} - y_{j, 1}) + b_{i} (y_{i, 1} - y_{0, 1}) \\ ϵ_{i, 2} = \sum_{j = 1}^{n} a_{ij} (\hat{{\overset{\cdot}{y}}_{i, 1}} - \hat{{\overset{\cdot}{y}}_{j, 1}}) + b_{i} \hat{{\overset{\cdot}{y}}_{i, 1}} \end{matrix}

(52)

where $y_{0, 1} = v_{ref}$ is reference voltage.

Proposition 2. For an MG system equation (45) holding Assumptions 1 and 3, DFTSVR protocols are proposed as $(i \in V)$

\begin{matrix} τ_{i} = - \hat{ϖ_{i}} - γ_{i} sgn (\sum_{j = 1}^{n} a_{ij} (s_{i} - s_{j}) + b_{i} s_{i}) \end{matrix}

(53)

where $γ_{i} \geq n (ϵ_{i} + | \sum_{c = 1}^{2} ς_{c} si g^{α_{c}} (ϵ_{i, c}) |) / λ_{min} (Q)$ , $ϵ_{i} > 0$ . $\hat{ϖ_{i}}$ is from observers equation (51). The sliding-mode surface functions $s_{i}$ are

s_{i} = \int_{0}^{t} (\sum_{c = 1}^{2} ς_{c} si g^{α_{c}} (ε_{i, c})) d ζ + ε_{i, 2}

(54)

where $ς_{1} > 0$ , $ς_{2} > 0$ , $0 < α_{1} < 0$ , $α_{2} = \frac{2 α_{1}}{1 + α_{1}}$ , $c = 1, 2$ . Then $y_{i, 1} \to y_{0, 1}$ is realized in finite time.

Proof. The dynamic model of an MG equation (45) is transformed into model equation (50), which is similar to the model of second-order MAS equation (6). Therefore, Proposition 2 results from Theorem 1, whose proof is omitted.

Remark 6. The essence of droop control is a differential control. The load changes lead to a certain deviation for system voltage and its rated value. Therefore, it is hard to maintain the stability of the MGs for a long time only by relying on primary control, which requires the cooperation of primary control and secondary regulation. This section proposes DFTSVR protocols equation (53) for MGs achieve the voltage regulation under the disturbances.

Remark 7. The purpose of feedback linearization technique is using feedback to eliminate the nonlinear terms, establishing the dynamics between input and output states, and then simplifying the complex nonlinear systems. The disturbances are compensated for by designing FTOs equation (51). Comparing with the existing results, such as Dehkordi et al. (2017) and Ning et al. (2021b), the protocols equation (53)do not need to assume the bounds of the disturbances and their derivatives to be known, which relaxes the conservatism for the assumption of disturbances.

Similarly, contrasting with Cheng et al. (2015) and Ran et al. (2019), the OFADC protocols equation (12)for MAS do not need to assume the bounds of disturbances and derivatives being known. Moreover, the proposed distributed consensus protocols in equations (12) and ( 53 ) avoid using the inverse of Laplacian matrix directly.

Simulation results

Output-feedback anti-disturbance consensus protocols for MAS

The effectiveness of the proposed OFADC protocols equation (12) in Theorem 1 is illustrated by making comparisons with the OFRC protocols equation (26) and the consensus protocols in Ran et al., (2020) for a leader–follower MAS with four followers. Figure 2 shows the communication topology of the MAS.

The leader’s motion is $x_{0, 1} (t) = 3$ . Then $u_{0} (t) = 0$ and ${\bar{u}}_{0}$ is chosen as 0.2. When $t < 3 s$ , the disturbances are set as zeros and are suddenly acted on the agents when $t < 3 s$ as $ϖ_{1} = 0.8 \sin (t) + 0.02 t^{2} + 0.4 t + 3.6$ , $ϖ_{2} = 0.5 \sin (2.3 t + 0.2) - 0.1 t^{2} + 4.3$ , $ϖ_{3} = - 0.3 \sin (1.2 t - 0.6) - 0.3 t - 5.3$ , $ϖ_{4} = - 0.4 \sin (0.3 t + 0.2) - 0.08 t^{2} - 4.1$ , $\forall 3 s \leq t < 6 s$ . When $t \geq 6$ s, the disturbances are $ϖ_{1} = 0.8 \sin (t) + 6.72$ , $ϖ_{2} = 0.5 \sin (2.3 t + 0.2) + 0.7$ , $ϖ_{3} = - 0.3 \sin (1.2 t - 0.6) - 7.1$ , $ϖ_{4} = - 0.4 \sin (0.3 t + 0.2) - 6.98$ . The initial states for the follower are $x_{1} (0) = [0.5, 0]^{T}, x_{2} (0) = [1, 0]^{T}, x_{3} (0) = [1.5, 0]^{T}$ , $x_{4} (0) = [2, 0]^{T}$ . For fairness of comparison, the amplitudes of the proposed OFADC protocols equation (12) and the OFRC protocols equation (26) are limited within $\pm 30$ . Taking these constraints and conditions from Theorem 1 and Remark 4 into account, the parameters of protocols have been set. For the initial values of FTOs equation (8) and observers equation (24), the first order is set as $ψ_{1, 0} (0) = 0.5$ , $ψ_{2, 0} (0) = 1$ , $ψ_{3, 0} (0) = 1.5$ , $ψ_{4, 0} (0) = 2$ , ${\bar{ψ}}_{1, 0} (0) = 0.5$ , ${\bar{ψ}}_{2, 0} (0) = 1$ , ${\bar{ψ}}_{3, 0} (0) = 1.5$ , ${\bar{ψ}}_{4, 0} (0) = 2$ , and the other orders are set as zero. The control parameters of consensus protocols equation (12) are $ς_{1} = 12, ς_{2} = 7, α_{1} = 5 / 7, α_{2} = 5 / 6, ϱ_{1} = 1.2,$ $ϱ_{2} = 1.5, ϱ_{3} = 1.3, ϱ_{4} = 1.6$ . According to Lemma 4, the parameters setting of the FTOs equation (8) are $ρ_{1, 0} = 10, ρ_{1, 1} = ρ_{1, 2} = 6, ρ_{1, 3} = 9$ (for agent 1), $ρ_{2, 0} = 10, ρ_{2, 1} = 8, ρ_{2, 2} = 8, ρ_{2, 3} = 6$ (for agent 2), $ρ_{3, 0} = 11, ρ_{3, 1} = 6, ρ_{3, 2} = 5, ρ_{3, 3} = 8$ (for agent 3), $ρ_{4, 0} = 10, ρ_{4, 1} = 7, ρ_{4, 2} = 5, ρ_{4, 3} = 7$ (for agent 4). To obtain disturbance estimates and states quickly and accurately, the Lipschitz constants are selected as $L_{1} = 12$ , $L_{2} = 11$ , $L_{3} = 8$ , $L_{4} = 10$ . The parameters of equation (24) are the same as FTOs equation (8).

For further verification, a simulation case in Ran et al. (2020) is made for comparison. In order to observe the time-varying disturbances and unmeasurable states, observers are still selected as the FTOs equation (8). The controller is chosen as the formula equation (17) in Ran et al. (2020) $(i = 1, \dots, 4)$

u_{i} = \frac{1}{g_{i}^{o}} (cK {\hat{e}}_{i} - {\hat{x}}_{i, m + 1} + u_{0})

(55)

where ${\hat{x}}_{i, m + 1}$ is the estimation of disturbance, ${\hat{e}}_{i} = {[{\hat{e}}_{i, 1}, {\hat{e}}_{i, m}]}^{T}$ , and ${\hat{e}}_{i, j}$ is defined as

{\hat{e}}_{i, j} = \sum_{k \in N} a_{ik} ({\hat{x}}_{k, j} - {\hat{x}}_{i, j}) - b_{i} ({\hat{x}}_{i, j} - x_{0, j}), j = 1, 2

(56)

where ${\hat{x}}_{i, j}$ is the estimation of $x_{i, j}$ . Considering the fairness of comparison, the parameters are $g_{i}^{o} = 1, c = 1, K = [20, 20]$ .

The simulations of the different followers’ initial states and different leader’s motion are further conducted, respectively. For the simulation of different initial states, two groups of different initial states for the followers are selected as $x_{1} (0) = [1, 0.5]^{T}$ , $x_{2} (0) = [2, 1]^{T}$ , $x_{3} (0) = [4, 1.5]^{T}$ , $x_{4} (0) = [5, 2]^{T}$ (for the first group), $x_{1} (0) = [3, 1]^{T}$ , $x_{2} (0) = [4, 0.5]^{T}$ , $x_{3} (0) = [4.5, 2]^{T}$ , $x_{4} (0) = [5, 5]^{T}$ (for the second group). In this case, the leader’s motion, the disturbances, and the parameters setting are the same as the above simulation. For the simulation of different leader’s motion, the leader’s motion as chosen as $x_{0, 1} (t) = \sin (t) + 1.5$ , $x_{0, 2} (t) = \cos (t)$ , $u_{0} (t) = - \sin (t)$ . The followers’ initial states, the disturbances, and the parameters setting are the same as the above simulation.

From simulation Figures 3 and 4, the amplitudes of both the OFADC protocols equation (12) and the OFRC protocols equation (26) are within $\pm 30$ . From Figure 5, the disturbances and unmeasurable states are estimated by FTOs equation (8) quickly and accurately. Figure 6 shows the response curves of the followers’ outputs. From subfigures (a) and (b), it can be seen that the OFADC protocols equation (12) break through the impacts of disturbances. The convergence time for the followers’ states is 1.69 seconds and the convergence time for the FTO is 0.51 seconds. In subfigures (c) and (d), the followers’ outputs affected by the disturbances escape from the leader’s motion and do not return anymore under the OFRC protocols equation (26). Subfigures (e) and (f) show response curves under the consensus protocols equation (56), whose response speeds are slower and there are overshoots under the same control energy.

The simulations of different followers’ initial states and different leader’s motion are conducted under the OFADC protocols equation (12), the OFRC protocols equation (26) and the consensus protocols equation (56) in Ran et al. (2020), respectively. Figures 7 and 8 show the response curves of the followers’ outputs with two groups of different followers’ initial states, respectively. Figure 9 shows the response curves of the followers’ outputs with different leader’s motion. From Figures 7 –9, the effectiveness of the OFADC protocols equation (12) is further verified in the cases of different followers’ initial states and different leader’s motion. It can be seen that the follower’s outputs quickly, accurately, and stably track the leader’s trajectory under the OFADC protocols equation (12), while the system stability is hard to guarantee under the OFRC protocols equation (26). Under the proposed protocols equation (12), the system convergence speed and recovery ability under the disturbances are better than the ones under the consensus protocols equation (56) in Ran et al. (2020).

Voltage regulation protocols for MGs

The DFTSVR protocols for MGs in Proposition 2 are verified by simulation. Figure 10 displays the communication topology of 4 DGs.

The initial voltages $v_{odi} (0)$ are selected as $140 V$ , $200 V$ , $250 V$ , $300 V$ for $i = 1, 2, 3, 4$ and $v_{ref} = 311 V$ . Other initial states of DGs equation (45) and the FTOs equation (51) are 0. The MG parameters are shown in Table 1. Taking the parameter perturbation into consideration, it is assumed that the capacitance of LC filters have deviations $Δ C_{fi}, i = 1, 2, 3, 4$ , for each DG. The deviation values are set as $Δ C_{f 1} = 10 pF, Δ C_{f 2} = 20 pF, Δ C_{f 3} = 30 pF, Δ C_{f 4} = 40 pF$ .

Table 1.

Parameters of the MG system.

	$m_{P}$	$n_{Q}$	$R_{c}$	$R_{f}$	$L_{c}$	$L_{f}$	$C_{f}$	$K_{PV}$	$K_{PC}$	$K_{IV}$	$K_{IC}$
DGs 1-2	$9.4 \times 10^{- 5}$	$1.3 \times 10^{- 3}$	$4 Ω$	$0.1 Ω$	$0.35 mH$	$1.35 mH$	$50 μ F$	$0.1$	$15$	$420$	$20000$
DGs 3-4	$12.5 \times 10^{- 5}$	$1.5 \times 10^{- 3}$	$4 Ω$	$0.1 Ω$	$0.35 mH$	$1.35 mH$	$50 μ F$	$0.05$	$10.5$	$390$	$16000$

The control parameters of the DFTSVR protocols equation (53) are $ς_{1} = 15, ς_{2} = 6, α_{1} = 5 / 7, α_{2} = 5 / 6, ϱ_{1} = 1,$ $ς_{1} = 15, ς_{2} = 6, α_{1} = 5 / 7, α_{2} = 5 / 6, ϱ_{1} = 1,$ . The parameters of the FTOs equation (51) are $ρ_{1, 0} = 19, ρ_{1, 1} = 16, ρ_{1, 2} = 16, ρ_{1, 3} = 18$ (for DG 1), $ρ_{2, 0} = 18, ρ_{2, 1} = 15, ρ_{2, 2} = 14, ρ_{2, 3} = 16$ (for DG 2), $ρ_{3, 0} = 20, ρ_{3, 1} = 16, ρ_{3, 2} = 16, ρ_{3, 3} = 17$ (for DG 3), $ρ_{4, 0} = 19, ρ_{4, 1} = 13, ρ_{4, 2} = 15, ρ_{4, 3} = 14$ (for DG 4) and $L_{1} = 24$ , $L_{2} = 25$ , $L_{3} = 21$ , $L_{4} = 20$ . To further verify the influence of load power on secondary regulation, when $t = 4$ s, the load active power demand is increased by 1 kW.

The results are shown in Figures 11 –15. In Figure 13, $θ_{i, 0} = v_{odi} - \hat{v_{odi}}$ , $i = 1, 2, 3, 4 .$ The estimates $\hat{v_{odi}}$ converge to $v_{odi}$ , implying the disturbances are accurately estimated. From Figure 11, the DFTSVR protocols equation (53) realize the voltage consensus. Figure 12 displays the response curves of currents for the 4 DGs. It can be seen that with the change in load power demand, the output currents change correspondingly, but the output voltages remain stable. Figures 14 and 15 show the control inputs and disturbance estimation.

Conclusion

In this paper, by combination of the NITSMC and FTO methods, OFADC protocols have been proposed. Under the proposed protocols, the followers achieve output consensus in finite time. Moreover, the proposed scheme has also been applied to secondary regulation for MGs. Based on MG systems, DFTSVR protocols have been proposed and the output voltages achieve consensus. Simulations have validated the effectiveness. In the further work, the anti-disturbance control for MASs will be applied in more fields, and the combination of filtering and anti-disturbance control for MG system will be studied.

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the National Natural Science Foundation of China under Grant Nos 61873060, 62025302, and 62173221; the Key R & D plan of Jiangsu Province under Grant No. BE2020082-4; and the Postgraduate Research & Practice Innovation Program of Jiangsu Province under Grant No. KYCX22_0237.

ORCID iD

Xiangyu Wang

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