Admissible consensus of uncertain fractional-order singular multiagent systems with actuator fault

Abstract

Consensus of fractional-order singular multiagent systems is considered in this article. The model of uncertain systems with actuator fault is established, and its admissible consensus in the field of fractional-order systems is studied. First, the uncertain singular multiagent systems with actuator fault are proposed. Then, a state feedback controller and a state consensus protocol with an observer based on output information are designed. Then, new consensus criteria of uncertain fractional-order singular multiagent systems are proposed. Finally, for the sake of demonstrating the validity of proposed results, some relevant numerical examples are provided.

Keywords

Fractional-order systems singular systems multiagent systems consensus criteria fault-tolerant control

1. Introduction

In recent years, with the development of control and computer technology, distributed coordinated control of multiagent systems has gradually attracted more and more attention, and consensus research (through the design of consensus protocol, all agents converge to a uniform signal) which is a basic research issue also becomes a research hotspot and gradually penetrates into various fields, such as biology, intelligent robot, office automation, and other fields (Arezou et al., 2019; Fukudaet al., 2003; Ren et al., 2008; Wen et al., 2004). Many scholars have studied the multiagent systems from different directions such as the robustness, controllability, and consistency of multiagent systems (Liu et al., 2008; Xie and Wang, 2003; Xie et al., 2002; Wang and Yang, 2016).

However, it is noted that all of the above studies are based on integer-order systems, which cannot reflect their own performance when describing materials with memory and viscosity, so some scientists found that the stability and its own performance of fractional-order systems in describing such materials are well reflected (Abbas, 2014; Abbas, 2015; Yang and Fang, 2012). Fractional calculus is widely used in science, engineering, and biochemistry. Fractional derivative has a more accurate description of group behavior of multidynamic systems. In recent years, scientists have begun to study the consistency of fractional-order multiagent systems (FOMASs). For the research on the state consensus of FOMAS, the distributed coordination algorithm of FOMAS is given in the early literature (Cao et al., 2018; Chen et al., 2018a, 2018b) to make the systems achieve the state consensus. On this basis, external disturbance is added in Ren and Yu (2016), which proposes a pinning control input method such that the state consensus is achieved. Since then, input saturation and other related knowledge are added in Chen et al. (2018a, 2018b) so that the state consensus of the systems is proved by the Mittag-Leffler stability theory under the condition that input saturation and external disturbance exist at the same time. The issue of unknown nonlinearities is investigated to systems in Mo et al. (2019), the consensus analysis of the systems is completed by designing adaptive protocol, although unknown nonlinearities and unknown external disturbances exist simultaneously, a sliding mode observer is designed to verify the state consensus of the FOMAS on the basis of nonlinearities in Bai et al. (2017).

In the past two decades, more and more scholars have expended normal multiagent systems to singular systems in more extensive backgrounds, such as circuit systems, power systems, and economic networks, which describe systems with a larger range than nonsingular systems (Udwadia and Phohomsiri, 2006; Xi et al., 2016; Zheng et al., 2017), even some issues cannot be modeled by normal systems such as a type of transistor circuit presented in Newcomb (1981) and the three-link planar manipulator network studied in Yang and Liu (2012). Therefore, the importance of singular multiagent systems is obvious. With the development of research, some scholars have found that some circuits with super capacitors and super coils should be described by fractional-order singular linear systems Kaczorek (2011). FOSMAS is the extension of singular multiagent systems or FOMAS, which has potential application in the fields of swamp floating power station, robot systems, frequency synchronization of power grid equipped with super capacitor and super coil, and so on (Chen et al., 2018a, 2018b). However, there are few articles about FOSMAS. The consensus of FOSMAS is still an open question, and there are many unsolved problems in this domain. In practical application, many models have uncertainties, with the increasing complexity of the systems, the issues of faults are inevitable. During the past years, fault-tolerant control (FTC) research and their application to a wide range of industrial and commercial processes have been the subjects of intensive investigation (Jiang et al., 2010). When the sensor or other parts fail, it likely causes system instability. FTC technology can ensure the stability of the system and make the closed-loop system have better performance when some components fail. FTC technology has earned many achievements in practical application such as fault-tolerant control for wind turbine (Fan and Song, 2012) and attitude stability of spacecraft with input saturation (Hu et al., 2011). Therefore, based on the above motivation, the contributions of this article are summarized as follows:

Compared with the integer multiagent systems, there are few previous studies on fractional-order fields, especially in singular systems. On this basis, some research is done on the FOSMAS in this article;

Considering the influence of practical issues, uncertainties and actuator fault are added to the systems, and the issues of multiplication of uncertainties and actuator fault are dealt with;

A state feedback controller and an observer based on output information are designed to make the systems achieve admissible consensus.

The organization structure of the article is as follows: In Section 2, some graph theory notions and lemmas of fractional-order singular systems are introduced. Then, the corresponding model is established and transformed, and some related lemmas are presented in Section 3. New consensus criteria of FOMAS based on linear matrix inequalities are proposed in Section 4. At last, some examples are given to verify the effectiveness of the approach, and conclusions are drawn to end this article in Section 5.

Notations: I_N represents the N × N dimensional identity matrix, $ℝ^{m \times n}$ represents the set of m × n real matrix, ⊗ denotes the Kronecker product, diag(a₁, …, a_n) denotes the diagonal matrix with elements a₁, a₂, …, a_n, blockdiag(b₁, …, b_n) denotes a block diagonal matrix, and ⋆ represents the symmetric part of a matrix.

2. Preliminaries

In this section, some knowledge and lemmas about graph theory and fractional-order singular systems are introduced.

2.1. Graph theory notions

Graph theory knowledge of directed graph is used to represent the external communication relationship between agents in this article. Consider a multiagent system which consists of one leader and N followers. The directed graph is represented by $G = (V, ℰ, A)$ , $V = {v_{1}, v_{2}, \dots, v_{N}}$ is defined as the finite set of nonempty nodes representing individual agents, the finite set I = {1, 2, …, N} represents the index set of nodes, a set of edges or arcs of graph G represented by $ℰ \in (V, V)$ . A is defined as an adjacency matrix, this matrix takes a_ij as its element, where a_ij > 0, directed pair (v_i, v_j) represents a communication edge of graph G, where nodes v_i and v_j represent the starting position and the terminal node, respectively. Iff the ith node obtains communication information from the jth node, there will be $(v_{i}, v_{j}) \in ℰ$ , otherwise $(v_{i}, v_{j}) \notin ℰ$ . If edge $(v_{i}, v_{j}) \in ℰ$ , it means that v_i is the neighbor of v_j. Then all neighbors of v_i are written in the form of set $N_{i} = {v_{j} \in V | (v_{j}, v_{i}) \in ℰ}$ . For graph $G$ , the in-degree matrix $D$ is defined as $D = diag (d_{1}, d_{2}, \dots, d_{N})$ with d_i = ∑_j≠ia_ij. The Laplacian matrix is $L = D - A$ . Let the leader be associated with node 0, set $\bar{V} = V \cup {V_{0}}$ and the directed graph $\bar{G} = (\bar{V}, \bar{ℰ}, \bar{A})$ . Moreover, by setting a_i0 = 1, if there is an arc from 0 to i, else a_i0 = 0. $\bar{L} = L + A_{0}$ .

2.2. Caputo fractional operator and fractional-order singular systems

Definition 1

(Monje et al., 2007). The Caputo fractional derivative with order α of function x(t) is given by

D^{α} x (t) = \frac{1}{Γ (n - α)} \int_{0}^{t} {(t - τ)}^{n - α - 1} x^{(n)} (τ) d τ

where Γ(⋅) is the Euler gamma function and n is an integer satisfying n −1 < α ≤ n.

Lemma 1

(Marir et al., 2007). For the following linear fractional-order system

E D^{α} x (t) = A x (t) + B u (t)

(1)

when u(t) = 0, system (1) is called unforced system, which is written as triple (E, A, α), if the following three conditions are satisfied simultaneously:

The triple (E, A, α) is regular, if there exists a constant scalar $s \in ℂ$ such that det(s^αE − A) is not identically zero.

The triple (E, A, α) is impulse-free, if system (1) is regular and deg(det(λE − A)) = rank(E).

The triple (E, A, α) is stable, if $| arg (spec (E, A)) | > π α / 2$ , where $spec (E, A) = {λ | det (λ E - A) = 0, \forall λ \in ℂ}$ .

Then it can be concluded that unforced system (1) is admissible.

Lemma 2

(Zhang and Chen, 2018). The next two conditions are equivalent:

Unforced system (1) with order 0 < α < 1 is admissible.

There exist matrices $X, Y \in ℝ^{n \times n}$ , and $Q \in ℝ^{(n - m) \times n}$ such that (2) and (3) hold or (2) and (4) hold

[\begin{matrix} X & Y \\ - Y & X \end{matrix}] > 0

(2)

sym {A (a X E^{T} - b Y E^{T} + P Q)} < 0

(3)

sym {A^{T} (a X E - b Y E + \bar{P} Q)} < 0

(4)

where a and b denote

\sin (α π / 2)

and

\cos (α π / 2)

, respectively, and P and

\bar{P}

are any matrices with full column rank and satisfy EP = 0 and

E^{T} \bar{P} = 0

, respectively.

Lemma 3

(Xie, 1996). There exist two real appropriate dimension matrices X and Y such that Ω + X^TF(τ)Y + Y^TF^T(τ)X < 0, for all F^T(τ)F(τ) ≤ I, iff there exists an ɛ > 0 such that Ω + ɛX^TX + ɛ⁻¹Y^TY < 0.

3. Problem statements

Consider the uncertain fractional-order singular leader–follower multiagent systems consisted of one leader and N followers, where the dynamics models of leader are introduced as follows

\begin{array}{l} E D^{α} x_{0} (t) = (A + Δ A) x_{0} (t), \\ y_{0} (t) = C x_{0} (t) \end{array}

(5)

the dynamics of the ith agent are described as

\begin{array}{l} E D^{α} x_{i} (t) = (A + Δ A) x_{i} (t) + (B + Δ B) u_{i} (t), \\ y_{i} (t) = C x_{i} (t) \end{array}

(6)

where

x_{i} (t) \in ℝ^{n}

represents state vector of node i (i = 1, 2, …, N),

u_{i} (t) \in ℝ^{q}

represents the control input vector and

y_{i} (t) \in ℝ^{r}

denotes the measurement output vector,

x_{0} (t) \in ℝ^{n}

is the state and

y_{0} (t) \in ℝ^{r}

is the output of the leader,

A \in ℝ^{n \times n}

B \in ℝ^{n \times q}

, and

C \in ℝ^{r \times n}

are all known constant coefficient matrices,

E \in ℝ^{n \times n}

is a singular matrix with rank(E) = r < n,

Δ A \in ℝ^{n \times n} a n d Δ B \in ℝ^{n \times q}

denote uncertainties which are written as follows ΔA = D_AF_A(t)E_A and ΔB = D_BF_B(t)E_B, where D_A, D_B, E_A, and E_B are all known matrices, F_A(t) and F_B(t) are unknown matrices which satisfy

F_{A}^{T} (t) F_{A} (t) \leq I, F_{B}^{T} (t) F_{B} (t) \leq I

Definition 2

(Ni and Cheng, 2010). Systems (5) and (6) are said to be admissible consensus with protocol u_i(t) iff systems (5) and (6) are regular, stable, and impulse-free, the states of the agents satisfy the following equation for any initial values

lim_{t \to \infty} ‖ x_{i} (t) - x_{0} (t) ‖ = 0

(7)

for all i = 1, 2, …, N.

Assumption 1

(Su and Huang, 2012). The graph $\bar{G}$ has a spanning tree and the root node is $V_{0}$ . The pair (A, B) is controllable and the pair (A, C) is detectable.

To solve the FTC problem, the actuator fault model presented in Yang et al. (2001) is adopted as follows. For the control input u_i(t), set u_fi(t) denotes the signal such that

u_{f_{i}} (t) = F u_{i} (t)

(8)

with

F = diag {f_{1}, f_{2}, \dots, f_{S}}

which denotes the actuator faults function matrix, where 0 ≤ f_ih ≤ f_i ≤ f_iu < ∞ and f_ih < 1, f_iu ≥ 1, i = 1, 2, …, s represent known real constants, when f_i = 0, it is obvious that the actuator is in complete failure (the outage case) and when f_i = 1, the actuator is normal.

Now, actuator faults expressed as matrices are introduced as follows

F_{0} = diag (f_{01}, f_{02}, \dots, f_{0 s})

J = diag (j_{1}, j_{2}, \dots, j_{s})

| H | = diag (| h_{1} |, | h_{2} |, \dots, | h_{s} |)

where

F_{0 i} = \frac{f_{i h} + f_{i u}}{2}, h_{i} = \frac{f_{i} - f_{0 i}}{f_{0 i}}, j_{i} = \frac{f_{i u} - f_{i h}}{f_{i u} + f_{i h}}

(i = 1, 2, …, s) then it obtains

F = F_{0} (I + H), | H | \leq J \leq I

(9)

4. Main results

In this section, admissible consensus of uncertain FOSMAS with actuator fault is solved by state feedback control and output feedback control, respectively.

4.1. Consensus with state feedback control

In this subsection, a state feedback controller is designed to solve the admissible issue of the uncertain FOSMAS as follows

u_{i} (t) = K [\sum_{j = 1}^{N} a_{i j} (x_{i} (t) - x_{j} (t)) + a_{i 0} (x_{i} (t) - x_{0} (t))]

(10)

then system (6) with u_i(t) is rewritten as follows

\begin{array}{l} E D^{α} x_{i} (t) = (A + Δ A) x_{i} (t) \\ + (B + Δ B) F K [\sum_{j = 1}^{N} a_{i j} (x_{i} (t) - x_{j} (t)) \\ + a_{i 0} (x_{i} (t) - x_{0} (t))] . \end{array}

(11)

Letting $ϵ_{i} (t) = x_{i} (t) - x_{0} (t)$ , the following equation is concluded

\begin{array}{l} E D^{α} ϵ_{i} (t) = E D^{α} x_{i} (t) - E D^{α} x_{0} (t) \\ = (A + Δ A) ϵ_{i} (t) \\ + (B + Δ B) F K [\sum_{j = 1}^{N} a_{i j} (x_{i} (t) - x_{j} (t)) \\ + a_{i 0} (x_{i} (t) - x_{0} (t))] \end{array}

(12)

Combining equations (11) and (12) and letting

\begin{matrix} ϵ (t) = {[ϵ_{1}^{T} (t) ϵ_{2}^{T} (t) \dots ϵ_{N}^{T} (t)]}^{T} \end{matrix}

one gets the following equation

{\begin{cases} \bar{E} D^{α} x (t) = (I_{N} \otimes (A + Δ A)) x (t) \\ + (\bar{L} \otimes (B + Δ B) F K)) ϵ_{i} (t) \\ \bar{E} D^{α} ϵ (t) = ((I_{N} \otimes (A + Δ A)) \\ + (\bar{L} \otimes (B + Δ B) F K)) ϵ_{i} (t) \end{cases}

(13)

which is equivalent to the following equation

\bar{E} D^{α} η (t) = \bar{A} η (t),

(14)

where

η (t) = {[x^{T} (t) ϵ^{T} (t)]}^{T}

and

\bar{A} = [\begin{matrix} I_{N} \otimes (A + Δ A) & \bar{L} \otimes (B + Δ B) F K \\ 0 & I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K \end{matrix}]

\bar{E} = [\begin{matrix} I_{N} \otimes E & 0 \\ 0 & I_{N} \otimes E \end{matrix}]

Theorem 1

Systems (5) and (6) with fractional order 0 < α < 1 achieve admissible consensus with protocol (10) iff $(I_{N} \otimes E, I_{N} \otimes (A + Δ A), α) a n d (I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)$ are admissible.

Proof

According to Lemma 1, we first prove that the admissible of (Ē, Ā, α) is equivalent to the admissible of (I_N ⊗ E, I_N ⊗ (A + ΔA), α) and $(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)$ .

The following equation is concluded with the regular condition from Lemma 1

\begin{array}{l} det (s^{α} \bar{E} - \bar{A}) = det (s^{α} (I_{N} \otimes E) - I_{N} \otimes (A + Δ A)) \\ \times det (s^{α} (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) \\ + \bar{L} \otimes (B + Δ B) F K)) \end{array}

(15)

from (15), it is obvious that (Ē, Ā, α) is regular iff (I_N ⊗ E, I_N ⊗ (A + ΔA), α) and

(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)

are regular at the same time.

The impulse-free condition of (Ē, Ā, α) is $deg (det (λ \bar{E} - \bar{A})) = 2 N r$ , for $λ \in ℂ$ , from equation (15) one has

\begin{array}{l} (det (λ \bar{E} - \bar{A})) \\ = deg (det (λ (I_{N} \otimes E) - I_{N} \otimes (A + Δ A))) \\ + deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) \\ + \bar{L} \otimes (B + Δ B) F K))) \end{array}

then it obtains the following equation

\begin{array}{l} deg (det (λ (I_{N} \otimes E) - I_{N} \otimes (A + Δ A))) \\ = deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) \\ + \bar{L} \otimes (B + Δ B) F K))) = N r \end{array}

It is obvious that the degrees of

deg (det (λ (I_{N} \otimes E) - I_{N} \otimes (A + Δ A)))

and

deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K)))

are less than r.

The stability of (Ē, Ā, α) is equivalent to the stability of (I_N ⊗ E, I_N ⊗ (A + ΔA), α) and $(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)$ according to equation (15).

Then systems (5) and (6) with protocol (10) are achieved admissible consensus. □

According to Assumption 1, it is considered all the eigenvalues of $\bar{L}$ have positive real parts. By the Jordan matrix theorem, there exists a nonsingular matrix U ∈ R^N×N and

J (λ_{j}) = [\begin{matrix} λ_{j} & 1 \\ λ_{j} & ⋱ \\ ⋱ & 1 \\ λ_{j} \end{matrix}]

such that

\bar{L} = U^{- 1} \tilde{Δ} U

, where

\tilde{Δ}

is the Jordan matrix of

\bar{L}

, and

\tilde{Δ} = blockdiag {J (λ_{1}), \dots, J (λ_{k})}

, N₁ + N₂ + ⋯N_k = N. λ_j is the eigenvalue of

\bar{L}

. Then, the matrix Ā is transformed into the following form

\bar{A} = [\begin{matrix} A + Δ A & λ_{j} (B + Δ B) K \\ 0 & A + Δ A + λ_{j} (B + Δ B) F K \end{matrix}]

It is obvious that systems (5) and (6) with fractional order 0 < α < 1 are said to achieve admissible consensus via protocol (10) iff both (E, A + ΔA, α) and (E, A + ΔA + λ_j(B + ΔB)FK, α) are admissible, where j = 1, 2, …, N.

Theorem 2

Under Assumption 1, uncertain FOSMAS (5) and (6) with fractional order 0 < α < 1 are said to achieve admissible consensus by protocol (10) if the following conditions are satisfied: There exist matrices $X_{i}, Y_{i} \in ℝ^{n \times n}, Q_{1} \in ℝ^{(n - r) \times n}, i = 1,2$ , and $W_{1} \in ℝ^{q \times n}$ and some constants ɛ_k > 0, k = 1, 2, …, 6 such that

[\begin{matrix} X_{i} & Y_{i} \\ - Y_{i} & X_{i} \end{matrix}] > 0

(16)

[\begin{matrix} sym (A (a X_{1} E^{T} - b Y_{1} E^{T} + P_{1} Q_{1})) + ε_{1} D_{A} D_{A}^{T} & {(a X_{1} E^{T} - b Y_{1} E^{T} + P_{1} Q_{1}))}^{T} E_{A}^{T} \\ ⋆ & - ε_{1} I \end{matrix}] < 0

(17)

\begin{array}{l} [\begin{matrix} Φ_{1} & 0 & 0 & W_{1}^{T} \\ ⋆ & ε_{3} F_{0} J J^{T} F_{0}^{T} - 2 I & E_{B}^{T} & 0 \\ ⋆ & ⋆ & - ε_{2} I & 0 \\ \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} - ε_{3} I \\ ⋆ \\ ⋆ \\ ⋆ \end{array} \end{matrix} \begin{matrix} Φ_{2}^{T} E_{A}^{T} & W_{1}^{T} & W_{1}^{T} F_{0}^{T} E_{B}^{T} \\ 0 & 0 & 0 \\ 0 & 0 & 0 \\ \begin{array}{l} 0 \\ - ε_{4} I \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} 0 \\ 0 \\ - ε_{5} I \\ ⋆ \end{array} & \begin{array}{l} 0 \\ 0 \\ 0 \\ - ε_{6} I \end{array} \end{matrix}] <0 \end{array}

(18)

where P₁ is any matrix with full column rank and satisfies EP₁ = 0.

\begin{array}{l} Φ_{1} = sym (A Φ_{2} + λ_{j} B F_{0} W_{1}) + ε_{4} D_{A} D_{A}^{T} \\ + ε_{5} λ_{j}^{2} B F_{0} J J^{T} F_{0}^{T} B^{T} + (ε_{6} + ε_{2}) λ_{j}^{2} D_{B} D_{B}^{T} \\ Φ_{2} = a X_{2} E^{T} - b Y_{2} E^{T} + P_{2} Q_{2}, W_{1} = K Φ_{2} \end{array}

Proof

According to Lemma 2, we consider (A + ΔA) is admissible if two matrices $X_{1}, Y_{1} \in ℝ^{n \times n}$ satisfy equation (16) and

sym ((A + Δ A) (a X_{1} E^{T} - b Y_{1} E^{T} + P_{1} Q_{1})) < 0

(19)

under Lemma 3, there exists one scalar ɛ₁ > 0 such that inequality (19) is written as follows

\begin{array}{l} sym (A (a X_{1} E^{T} - b Y_{1} E^{T} + P_{1} Q_{1})) + ε_{1} D_{A} D_{A}^{T} \\ + ε_{1}^{- 1} {(a X_{1} E^{T} - b Y_{1} E^{T} + P_{1} Q_{1})}^{T} E_{A}^{T} E_{A} (a X_{1} E^{T} \\ - b Y_{1} E^{T} + P_{1} Q_{1}) < 0 \end{array}

(20)

Using the Schur complement in Boyd et al. (1994), equation (20) is equivalent to (17).

In the same way, A + ΔA + λ_i(B + ΔB)FK is admissible if two matrices $X_{2}, Y_{2} \in ℝ^{n \times n}$ satisfy equation (16) and

sym ([A + Δ A + λ_{i} (B + Δ B) F K Φ_{2}]) < 0

(21)

According to equation (9), inequality (21) is expanded as follows

\begin{array}{l} sym ((A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K + λ_{i} Δ B F_{0} K) Φ_{2} \\ + (λ_{i} Δ B F_{0} H K Φ_{2})) < 0 \end{array}

(22)

By using the Schur complement in Boyd et al. (1994), inequality (22) is deduced as follows

\begin{array}{l} sym ([\begin{matrix} (A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K + λ_{i} Δ B F_{0} K) Φ_{2} & λ_{i} Δ B \\ F_{0} H K Φ_{2} & - I \end{matrix}]) \\ - [\begin{matrix} λ_{i}^{2} Δ B Δ B^{T} + Φ_{2}^{T} K^{T} H^{T} F_{0}^{T} F_{0} H K Φ_{2} & 0 \\ 0 & 0 \end{matrix}] < 0 \end{array}

(23)

It is obvious that $λ_{i}^{2} Δ B Δ B^{T} + Φ_{2}^{T} K^{T} H^{T} F_{0}^{T} F_{0} H K Φ_{2} > 0$ , so it concludes that inequality (23) holds if the matrix

sym ([\begin{matrix} (A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K + λ_{i} Δ B F_{0} K) Φ_{2} & λ_{i} Δ B \\ F_{0} H K Φ_{2} & - I \end{matrix}]) < 0

(24)

For the convenience of the proof, inequality (24) is simplified to the following inequality

\begin{array}{l} sym ([\begin{matrix} (A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K + λ_{i} Δ B F_{0} K) Φ_{2} & 0 \\ 0 & - I \end{matrix}]) \\ + sym ([\begin{matrix} λ_{i} D_{B} \\ 0 \end{matrix}] F_{B} [\begin{matrix} 0 & E_{B} \end{matrix}]) \\ + sym ([\begin{matrix} 0 \\ F_{0} \end{matrix}] H [\begin{matrix} K Φ_{2} & 0 \end{matrix}]) < 0 \end{array}

(25)

According to Lemma 3 and equation (9), it is concluded that there exist two scalars ɛ₂ and ɛ₃ > 0 such that

\begin{array}{l} sym ([\begin{matrix} (A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K + λ_{i} Δ B F_{0} K) Φ_{2} & 0 \\ 0 & - I \end{matrix}]) \\ + ε_{2} [\begin{matrix} λ_{i} D_{B} \\ 0 \end{matrix}] {[\begin{matrix} λ_{i} D_{B} \\ 0 \end{matrix}]}^{T} + ε_{2}^{- 1} [\begin{matrix} 0 \\ E_{B}^{T} \end{matrix}] {[\begin{matrix} 0 \\ E_{B}^{T} \end{matrix}]}^{T} \\ + ε_{3} [\begin{matrix} 0 \\ F_{0} J \end{matrix}] {[\begin{matrix} 0 \\ F_{0} J \end{matrix}]}^{T} \\ + ε_{3}^{- 1} [\begin{matrix} {(K Φ_{2})}^{T} \\ 0 \end{matrix}] {[\begin{matrix} {(K Φ_{2})}^{T} \\ 0 \end{matrix}]}^{T} < 0 \end{array}

(26)

then by using the Schur complement in Boyd et al. (1994), it concludes that

[\begin{matrix} \begin{array}{l} sym ((A + Δ A + λ_{i} B F_{0} K + λ_{i} B F_{0} H K \\ + λ_{i} Δ B F_{0} K) Φ_{2}) + ε_{2} λ_{i}^{2} D_{B} D_{B}^{T} \end{array} & 0 & 0 & Φ_{2}^{T} K^{T} \\ 0 & ε_{3} F_{0} J J F_{0}^{T} - 2 I & E_{B}^{T} & 0 \\ ⋆ & ⋆ & - ε_{2} I & 0 \\ ⋆ & ⋆ & ⋆ & - ε_{3} I \end{matrix}] < 0

(27)

To deal with the uncertainties, inequality (27) is rewritten as follows

\begin{array}{l} [\begin{matrix} sym ((A + λ_{i} B F_{0} K) Φ_{2}) + ε_{2} λ_{i}^{2} D_{B} D_{B}^{T} \\ 0 \\ ⋆ \\ ⋆ \end{matrix} \\ \begin{matrix} 0 & 0 & Φ_{2}^{T} K^{T} \\ ε_{3} F_{0} J J^{T} F_{0}^{T} - 2 I & E_{B}^{T} & 0 \\ ⋆ & - ε_{2} I & 0 \\ ⋆ & ⋆ & - ε_{3} I \end{matrix}] \\ + sym ([\begin{matrix} D_{A} \\ 0 \\ 0 \\ 0 \end{matrix}] F_{A} [\begin{matrix} E_{A} Φ_{2} & 0 & 0 & 0 \end{matrix}]) \\ + sym ([\begin{matrix} λ_{i} B F_{0} \\ 0 \\ 0 \\ 0 \end{matrix}] H [\begin{matrix} K Φ_{2} & 0 & 0 & 0 \end{matrix}]) \\ + sym ([\begin{matrix} λ_{i} D_{B} \\ 0 \\ 0 \\ 0 \end{matrix}] F_{B} [\begin{matrix} E_{B} F_{0} K Φ_{2} & 0 & 0 & 0 \end{matrix}]) < 0 \end{array}

(28)

According to Lemma 3 and equation (9), it can be concluded that there exist three scalars ɛ₄ > 0, ɛ₅ > 0, and ɛ₆ > 0 such that

\begin{array}{l} [\begin{matrix} sym ((A + λ_{i} B F_{0} K) Φ_{2}) + ε_{2} λ_{i}^{2} D_{B} D_{B}^{T} \\ 0 \\ ⋆ \\ ⋆ \end{matrix} \\ \begin{matrix} 0 & 0 & Φ_{2}^{T} K^{T} \\ ε_{3} F_{0} J J^{T} F_{0}^{T} - 2 I & E_{B}^{T} & 0 \\ ⋆ & - ε_{2} I & 0 \\ ⋆ & ⋆ & - ε_{3} I \end{matrix}] \\ + ε_{4} [\begin{matrix} D_{A} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} D_{A} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + ε_{5} [\begin{matrix} λ_{i} B F_{0} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} λ_{i} B F_{0} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \\ + ε_{6} [\begin{matrix} λ_{i} D_{B} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} λ_{i} D_{B} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} + ε_{5}^{- 1} [\begin{matrix} {(K Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} {(K Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \\ + ε_{4}^{- 1} [\begin{matrix} {(E_{A} Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} {(E_{A} Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} \\ + ε_{6}^{- 1} [\begin{matrix} {(E_{B} F_{0} K Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}] {[\begin{matrix} {(E_{B} F_{0} K Φ_{2})}^{T} \\ 0 \\ 0 \\ 0 \end{matrix}]}^{T} < 0 \end{array}

(29)

Letting W₁ = KΦ₂ and using the Schur complement in Boyd et al. (1994), equation (29) is equivalent to (18).

The detailed explanations on how to construct the matrices X and Y are introduced in Appendix 1 (Zhang and Chen, 2018).

4.2. Consensus with output feedback control

In this subsection, a singular dynamic observer based on output information is constructed as follows

{\begin{array}{l} E D^{α} {\hat{x}}_{i} (t) = (A + Δ A) {\hat{x}}_{i} (t) + (B + Δ B) u_{i} (t) + v_{i} (t), \\ u_{i} (t) = K (\sum_{j = 1}^{N} a_{i j} ({\hat{x}}_{i} (t) - {\hat{x}}_{j} (t)) + a_{i 0} ({\hat{x}}_{i} (t) - x_{0} (t))), \\ v_{i} (t) = L (\sum_{j = 1}^{N} a_{i j} ({\tilde{y}}_{i} (t) - {\tilde{y}}_{j} (t)) + a_{i 0} {\tilde{y}}_{i} (t)) \end{array}

(30)

with

{\tilde{y}}_{i} (t) = y_{i} (t) - C {\hat{x}}_{i} (t)

and

L \in ℝ^{(n + p q) \times r}

denotes the observer gain matrix, which will be designed later.

Let $e_{i} (t) = {\hat{x}}_{i} (t) - x (t), ϵ_{i} (t) = x_{i} (t) - x_{0} (t)$ , then we will get the following formula

\begin{array}{l} E D^{α} e_{i} (t) = E D^{α} {\bar{x}}_{i} (t) - E D^{α} x_{i} (t) \\ = (A + Δ A) e_{i} (t) \\ + L C (\sum_{j = 1}^{N} a_{i j} (e_{i} (t) - e_{j} (t)) \\ + a_{i 0} e_{i} (t)) \end{array}

(31)

\begin{array}{l} E D^{α} ϵ (t) = (A + Δ A) ϵ (t) \\ + (B + Δ B) F K (\sum_{j = 1}^{N} a_{i j} (ϵ_{i} (t) - ϵ_{j} (t) \\ + e_{i} (t) - e_{j} (t)) + a_{i 0} (ϵ_{i} (t) + e_{i} (t))) \end{array}

(32)

denoting

σ (t) = {[ϵ_{i}^{T} (t), e_{i}^{T} (t)]}^{T}

with

e (t) = {[e_{1}^{T} (t), e_{2}^{T} (t), \dots, e_{N}^{T} (t)]}^{T}

and

ϵ (t) = {[ϵ_{1}^{T} (t), ϵ_{2}^{T} (t), \dots, ϵ_{N}^{T} (t)]}^{T}

, then equations (20) and (21) are described as follows

\bar{E} D^{α} σ (t) = \tilde{A} σ (t)

(33)

where

\tilde{A} = [\begin{matrix} I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K & s \bar{L} \otimes (B + Δ B) F K \\ 0 & I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C \end{matrix}]

and under Assumption 1, Ã is turned out as follows

\tilde{A} = [\begin{matrix} A + Δ A + λ_{j} (B + Δ B) F K & λ_{j} (B + Δ B) F K \\ 0 & A + Δ A + λ_{j} L C \end{matrix}]

Theorem 3

Systems (5) and (6) with fractional order 0 < α < 1 achieve admissible consensus with protocol (19) iff $(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α) a n d (I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C, α)$ are admissible.

Proof

The proof of Theorem 3 is similar to Theorem 1

\begin{array}{l} det (s^{α} \bar{E} - \tilde{A}) = det (s^{α} (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) \\ + \bar{L} \otimes (B + Δ B) F K)) \times det (s^{α} (I_{N} \otimes E) \\ - (I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C)) \end{array}

(34)

from (23), it is obvious that (Ē, Ã, α) is regular iff

(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)

and

(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C), α)

are regular simultaneously. From the proof of Theorem 1, we conclude that

\begin{array}{l} deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) \\ + \bar{L} \otimes (B + Δ B) F K))) \\ = deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C)))) \\ = N r \end{array}

and

deg (det (λ (I_{N} \otimes E) - (I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C))))

is no more than r, the stability of (Ē, Ã, α) is equivalent to that both of

(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes (B + Δ B) F K, α)

and

(I_{N} \otimes E, I_{N} \otimes (A + Δ A) + \bar{L} \otimes L C, α)

, which are stable from (23). □

Theorem 4

According to Assumption 1, uncertain FOSMAS (5) and (6) with fractional order 0 < α < 1 are said to achieve admissible consensus by protocol (30) if there exist matrices $X_{i}, Y_{i} \in ℝ^{n \times n}$ , i = 3, 4, $Q_{2}, Q_{3} \in ℝ^{(n - r) \times n}$ , and $W_{1}, W_{2} \in ℝ^{q \times n}$ and some scalars ɛ_k > 0, k = 2, 3, …, 7 such that

[\begin{matrix} X_{i} & Y_{i} \\ - Y_{i} & X_{i} \end{matrix}] > 0

(35)

\begin{array}{l} [\begin{matrix} Ψ_{1} & 0 & 0 & W_{1}^{T} \\ ⋆ & ε_{3} F_{0} J J^{T} F_{0}^{T} - 2 I & E_{B}^{T} & 0 \\ ⋆ & ⋆ & - ε_{2} I & 0 \\ \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} ⋆ \\ ⋆ \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} - ε_{3} I \\ ⋆ \\ ⋆ \\ ⋆ \end{array} \end{matrix} \\ \begin{matrix} Ψ_{2}^{T} E_{A}^{T} & W_{1}^{T} & W_{1}^{T} F_{0}^{T} E_{B}^{T} \\ 0 & 0 & 0 \\ \begin{array}{l} 0 \\ 0 \\ - ε_{4} I \\ ⋆ \\ ⋆ \end{array} & \begin{array}{l} 0 \\ 0 \\ 0 \\ - ε_{5} I \\ ⋆ \end{array} & \begin{array}{l} 0 \\ 0 \\ 0 \\ 0 \\ - ε_{6} I \end{array} \end{matrix}] <0 \end{array}

(36)

[\begin{matrix} sym (A^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) + λ_{i} C^{T} W_{2}) + ε_{7} E_{A}^{T} E_{A} & {(a X_{4} E - b Y_{4} E + \bar{P} Q_{3})}^{T} D_{A} \\ ⋆ & - ε_{7} I \end{matrix}] < 0

(37)

where

\bar{P}

is any matrix with full column rank and satisfies

E^{T} \bar{P} = 0

\begin{array}{l} Ψ_{1} = sym ((A Ψ_{2} + λ_{j} B F_{0} W_{1}) + ε_{4} D_{A} D_{A}^{T} \\ + ε_{5} λ_{j}^{2} B F_{0} J J^{T} F_{0}^{T} B^{T} + (ε_{6} + ε_{2}) λ_{j}^{2} D_{B} D_{B}^{T}, \\ Ψ_{2} = a X_{3} E^{T} - b Y_{3} E^{T} + P_{2} Q_{2}, W_{1} = K Ψ_{2}, \\ W_{2} = L^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) \end{array}

Proof

According to Lemma 2, we consider system (33) is admissible if A + ΔA + λ_j(B + ΔB)FK and A + ΔA + λ_jLC are admissible at the same time. The proof of the admissible of A + ΔA + λ_j(B + ΔB)FK is the same as Theorem 2.

A + ΔA + λ_jLC is admissible if there exist two matrices $X_{4}, Y_{4} \in ℝ^{n \times n}$ , which satisfy equation (35) and

sym ({(A + Δ A + λ_{i} L C)}^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3})) < 0

(38)

To facilitate analysis, equation (38) is expanded as follows

\begin{array}{l} sym (A^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) \\ + {(D_{A} F_{A} E_{A})}^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) \\ + λ_{i} C^{T} L^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3})) < 0 \end{array}

(39)

Denoting $W_{2} = L^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3})$ and according to Lemma 3, there exists a scalar ɛ₇ > 0 such that

\begin{array}{l} sym (A^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) + λ_{i} C^{T} W_{2}) + ε_{7} E_{A}^{T} E_{A} \\ + ε_{7}^{- 1} {(a X_{4} E - b Y_{4} E + \bar{P} Q_{3})}^{T} D_{A} \\ D_{A}^{T} (a X_{4} E - b Y_{4} E + \bar{P} Q_{3}) < 0 \end{array}

(40)

Using the Schur complement, equation (40) is equivalent to (37).

5. Numerical simulations

In this section, some numerical simulations are presented to expound our theoretical results.

Example 1

Consider a group of five agents with an interaction graph given by Figure 1. Uncertain FOSMAS (5) and (6) with α = 0.5 by protocol (10) are described by third-order integrators, where

A = [\begin{matrix} 0 & 1 & 2 \\ - 2 & 1 & 1 \\ - 1 & 2 & 1 \end{matrix}], B = [\begin{matrix} 1 & 1 \\ - 1 & 2 \\ - 2 & 0 \end{matrix}] C = [\begin{matrix} 1 & 0 & - 1 \end{matrix}], E = [\begin{matrix} 1 & 1 & 0 \\ 0 & 1 & 1 \\ 0 & 0 & 0 \end{matrix}] D_{A} = [\begin{matrix} - 0.2 & 0.1 & 0 \\ 0.1 & - 0.1 & 0.1 \\ 0 & 0.2 & 0 \end{matrix}], E_{A} = [\begin{matrix} 0 & 0 & 0.1 \\ 0.2 & 0 & 0.1 \\ 0 & 0.2 & 0.1 \end{matrix}] D_{B} = [\begin{matrix} 0.1 & 0.1 & 0 \\ 0.2 & - 0.1 & 0.1 \\ 0 & 0 & 0.2 \end{matrix}], E_{B} = [\begin{matrix} 0 & 0.1 \\ 0.2 & 0 \\ - 0.1 & 0.2 \end{matrix}] P_{1} = [\begin{matrix} - 1 \\ 1 \\ - 1 \end{matrix}], P_{2} = [\begin{matrix} 1 \\ - 1 \\ 1 \end{matrix}], f_{l} = [\begin{matrix} 0.6 & 0 \\ 0 & 0.4 \end{matrix}], f_{u} = [\begin{matrix} 0.9 & 0 \\ 0 & 0.6 \end{matrix}]

Matrices $L$ and ${\bar{A}}_{0}$ are determined as follows

L = [\begin{matrix} 1 & 0 & 0 & - 1 \\ - 1 & 2 & - 1 & 0 \\ - 1 & 0 & 1 & 0 \\ 0 & - 1 & 0 & 1 \end{matrix}], {\bar{A}}_{0} = [\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \end{matrix}]

the eigenvalues of

\bar{L}

are (0.2451, 1.8774 + 0.7449i, 1.8774 – 0.7449i, 2.0000). By solving the inequalities (16)–(18), some scalar constants ɛ_i > 0, i = 1, 2, …, 6 and the matrices X₁, Y₁, X₂, Y₂, Q₁, and Q₂ are obtained as follows

\begin{array}{l} ε_{1} = 13.5015, ε_{2} = 13.5784, ε_{3} = 13.3900 \\ ε_{4} = 13.5019, ε_{5} = 13.5765, ε_{6} = 13.5784 \end{array}

X_{1} = [\begin{matrix} 14.5768 & 0.2912 & - 0.7034 \\ 0.2912 & 11.1964 & - 2.6770 \\ - 0.7034 & - 2.6770 & 11.6086 \end{matrix}]

Y_{1} = [\begin{matrix} 0 & - 2.4291 & - 2.4291 \\ 2.4291 & 0 & - 2.4291 \\ 2.4291 & 2.4291 & 0 \end{matrix}]

X_{2} = [\begin{matrix} 14.5716 & 0.2740 & - 0.7153 \\ 0.2740 & 11.1803 & - 2.6759 \\ - 0.7153 & - 2.6759 & 11.6216 \end{matrix}]

Y_{2} = [\begin{matrix} 0 & - 2.4204 & - 2.4204 \\ 2.4204 & 0 & - 2.4204 \\ 2.4204 & 2.4204 & 0 \end{matrix}]

Q_{1} = [\begin{matrix} \begin{matrix} 1.0612 & - 5.9644 & - 3.0950 \end{matrix} \end{matrix}]

Q_{2} = [\begin{matrix} - 1.0519 & 6.0137 & 3.1045 \end{matrix}]

and the feedback gain matrix K is obtained

K = [\begin{matrix} 0.0193 & - 0.0525 & - 0.0200 \\ 0.0112 & - 0.0682 & - 0.0282 \end{matrix}]

Consider uncertain FOSMAS (5) and (6) with protocol (10), under the directed graph depicted in Figure 1, control signals u_i(t) are depicted in Figure 2, the admissible consensus tracking errors are depicted in Figure 3(a)–(c), it can obtain agent systems approach the leader systems infinitely. In other words, the admissible consensus of systems (5) and (6) can be achieved with state feedback controller (10).

Example 2

Consider uncertain FOSMAS (5) and (6) by protocol (30) with α = 0.5 and $\bar{P} = [\begin{matrix} 0 \\ 0 \\ 1 \end{matrix}]$ , by solving inequalities (35)–(37) we obtain

\begin{array}{l} ε_{2} = 25.9636, ε_{3} = 25.5736, ε_{4} = 25.8174 \\ ε_{5} = 25.9599, ε_{6} = 25.9636, ε_{7} = 25.7102 \end{array}

X_{1} = [\begin{matrix} 27.8627 & 0.5239 & - 1.3678 \\ 0.5239 & 21.3781 & - 5.1167 \\ - 1.3678 & - 5.1167 & 22.2220 \end{matrix}]

Y_{1} = [\begin{matrix} 0 & - 4.6280 & - 4.6280 \\ 4.6280 & 0 & - 4.6280 \\ 4.6280 & 4.6280 & 0 \end{matrix}]

X_{2} = [\begin{matrix} 15.9864 & - 2.7477 & 0 \\ - 2.7477 & 29.1944 & 0 \\ 0 & 0 & 25.9709 \end{matrix}]

Y_{2} = [\begin{matrix} 0 & 6.4779 & 0 \\ - 6.4779 & 0 & 0 \\ 0 & 0 & 0 \end{matrix}]

Q_{2} = [\begin{matrix} - 2.0114 & 11.4990 & 5.9362 \end{matrix}]

Q_{3} = [\begin{matrix} \begin{matrix} - 12.5898 & - 20.6314 & 5.3269 \end{matrix} \end{matrix}]

and the feedback gain matrices K and L are obtained

K = [\begin{matrix} 0.0193 & - 0.0525 & - 0.0200 \\ 0.0112 & - 0.0682 & - 0.0282 \end{matrix}]

L = {[\begin{matrix} - 17.4596 & 1.5129 & - 9.7215 \end{matrix}]}^{T}

Consider uncertain FOSMAS (5) and (6) with protocol (30) when α = 0.5, the state estimation error trajectories are depicted in Figure 4(a)–(c), which verify the presented estimator can estimate the state effectively.

The consensus tracking error trajectories with protocol (30) are shown in Figure 5(a)–(c), which shows that the consensus tracking errors are converged to zero when times tend to infinity, which means the admissible consensus can be achieved, which validates the availability of the new criterion in Theorem 4.

The actuator failure matrix F = diag{f₁, f₂}, where 0.5⩽f₁⩽1.1, 0.1⩽f₂⩽1.1.

When f₁ = 0.5 and f₂ = 0.1, the figure of matrix Ã is given in Figure 6. When f₁ = 1.1 and f₂ = 1.1, the figure of matrix Ã is depicted in the formulations below. To make the figures clear, parts of the whole figures are selected in Figures 6 and 7. It is obvious that the matrix Ã in closed-loop system (33) is stable under different actuator failure matrix F = diag{f₁, f₂} and feedback controller K and L.

Conclusion

In this article, the issue of uncertain FOSMAS with actuator fault is studied. A state feedback controller and an observer based on output information are designed for the systems. Then, distributed fault-tolerant consensus is designed and new criteria are proposed to achieve admissible consensus. Containment control of multiagent systems will be studied in the future. In the end, some numerical examples are provided and some figures are depicted to testify the availability of the consequences.

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 (NNSF) of China under Grant 61603055.

ORCID iD

Xuefeng Zhang

Appendix 1

n = size(B, 1); m = rank(E);

op = rref([E, eye(n)]); M = op(:, n + 1:2 ⋆ n);

y = op(:, 1:n)’; z = rref([y, eye(n)]);

N = z(:, n + 1:2 ⋆ n)’;

A = M ⋆ A ⋆ N; B = M ⋆ B;

setlmis([]);

[X1, n1, sX1] = lmivar(1, [m 1]);

[X2, n2, sX2] = lmivar(2, [n – m m]);

[X3, n2, sX3] = lmivar(1, [n – m 1]);

[X, n1, sX] = lmivar(3, [sX1, ...…

zeros(m(1), n − m(1)); sX2, sX3]);

Y = lmivar(2,[n(2),n(1)]);

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