Fixed time steps discrete-time sliding mode consensus protocols for two degree of freedom helicopter systems

Abstract

This paper presents analysis of number of steps required for the consensus in a leader-following discrete multi-agent system (DMAS) with discrete-time sliding mode protocols designed by Gao’s reaching law and power rate reaching laws. The DMAS is configured for communication with a fixed, undirected graph topology having one leader and other agents as followers. The sufficient condition for global stability is established using the Lyapunov function in both the cases. The efficacy of both the protocols is compared in simulation for number of steps required for the consensus of a homogeneous multiple two degree of freedom helicopter systems where the pitch angle and its velocity and yaw angle and its velocity are used for consensus. The simulation results reveal that the consensus performance due to protocol with power rate reaching law outperforms the protocol with Gao’s reaching law.

Keywords

Sliding mode control multi-agent system robotics robust control discrete-time sliding mode (DSM)

Introduction

In recent years, the research on distributed cooperative control has become very popular due to new age applications such as, automated highway systems (Quan and Jiang, 2014), power systems (Goldin, 2013; Yang et al., 2016), haptic collaboration over the internet (Huang and Lee, 2013), unmanned aerial vehicles sensor networks (Olfati-Saber et al., 2007), multi-vehicle systems (Kawakami and Namerikawa, 2009; Murray, 2007), smart grids (Loia et al., 2015), biological systems (Moon et al., 2016), robotic teams (Montijano et al., 2011; Tian et al., 2017), and many more different area of applications. In the distributed cooperative control of mutli-agent system (MAS) a local distributed controller is designed for each agent to achieve the common task by sharing the state information with each other (Zuo et al., 2018). To fulfill the objective various cooperative control algorithms such as rendezvous, flocking, consensus, formation, and so forth, are discussed in the literature (Lu et al., 2011; Yu et al., 2011). Among these algorithms, consensus algorithm is widely researched algorithm by researchers with different control techniques such as Proportional Integral (PI), Proportional Integral Derivative (PID), Linear Quadratic Regulator (LQR), $H_{\infty}$ , state feedback, observer-based control, Adaptive control and many more in recent years (Qin et al., 2017; Sun et al., 2017).

The consensus of MAS can be categorized into two classes (Liu et al., 2012; Pan et al., 2017; Yi et al., 2017): First one is leaderless consensus or simply average consensus algorithm in which all the agents in the network passing information to its neighbor and agree upon average consensus value. Another one is the leader-following consensus in which all the follower agents follow the leader. Many researchers have designed different consensus protocols for MAS, such as Duan et al. (2017) discussed the first-order consensus using an event-triggered approach with input delay constraint. Yating Wang et al. (2010) investigated the first- and second-order consensus problem using the Linear Matrix Inequality (LMI) technique. The consensus problem of a homogeneous system for a complex network using the PI control technique is proposed in Burbano and di Bernardo (2014). Saboori and Khorasani (2014) introduced the consensus problems with $H_{\infty}$ and weighted $H_{\infty}$ bounds for a homogeneous team of linear time-invariant (LTI) MAS with switching topology and directed communication network graph. Zhao and Huang (2018) designed adaptive leader-following consensus control of a high-order nonlinear system with time-varying constraint. Liu et al. (2013) proposed the decentralized state feedback consensus problem for MAS actuator saturation. Ren and Chen (2015), designed the sliding mode protocol for the consensus of higher-order nonlinear MAS and derived stability conditions for the consensus. Wang et al. (2017) proposed distributed active anti-disturbance consensus for leader-following higher-order multi-agent systems with unmatched disturbances by combining the non-singular terminal sliding-mode control (NTSMC) and disturbance. Qin et al. (2018) developed fault-tolerant cooperative tracking control using integral sliding mode control, in which they dealt with the cooperative tracking problem for a group of nonlinear systems with actuator fault and external disturbance/model uncertainty. Khoo et al. (2009) designed robust finite-time consensus for multiple robotic system using terminal sliding-mode.

All the above consensus protocols are proposed in continuous time. However, simultaneously, many authors and researchers have proposed protocols in discrete-time, which is more reliable for digital communication. Xu et al. (2013) discussed the observer-based discrete consensus protocol to solve the leader-following consensus problem where the observer estimates the leader states and tracking error using neighbor output information. Zhao and Lin (2017) introduced the consensus protocol for leader-following linear DMAS (discrete multi-agent system) using bounded state feedback control law and bounded output feedback control. Liu et al. (2017), developed consensus protocol for DMAS having external uncertainties using event triggering approach in which periodic protocol is transformed to aperiodic protocol and the protocol is applied to satellite system that does not require constant control effort. Li et al. (2018), designed a distributed adaptive control for a discrete-time MAS with the effect of nonlinearity and uncertainty.

Zhang et al. (2017), designed a leader-following consensus protocol using LQR methodology, in which all the follower agents follow the leader agent asymptotically.

Very recently, authors have proposed several discrete-time sliding mode (DSM) protocols for global consensus of DMAS. For example: (1) In Patel and Mehta (2018), a discrete-time sliding mode control using Gaos reaching law for global consensus of DMAS for a fixed graph topology is proposed and also a necessary condition for the global stability of the DMAS is established. (2) The protocol is further extended with power rate reaching law in Patel and Mehta (2019) to achieve global consensus of DMAS with switching graph topology. It is inferred that the chattering effect is significantly attenuated and also improves the convergence rate of global consensus of DMAS. However, in both the above papers, it is mentioned that the consensus in DMAS occur in finite time but the estimation of time required for the consensus is not exactly analyzed. So, in this paper, we have focused on the analysis of number of steps required for the global consensus with both the protocols namely with Gao’s reaching law and power rate reaching law. The main contributions of this paper, which is an extension work of the previous papers by the same authors, are given below:

Derived the topological DSM protocol with power rate reaching law using fixed graph topology for global consensus of DMAS.

Analyzed the number of steps required for the global consensus for both the protocols, namely DSM protocol, using Gaos rate reaching law and DSM protocol with power rate reaching law.

Derived the sufficient condition for global stability for both the protocols DSM protocols with Gaos reaching law and power rate reaching law.

Implementation and performance comparison of both the protocols on two degree of freedom (2-DOF) helicopter systems.

The paper is structured as follows. In the first section, introduction of graph theory and leader-following DMAS is presented. In the next two sections, analysis of number of steps required for the global leader-following consensus using Gao’s and power rate reaching law and global stability of DMAS is presented. Further, in the next section, the dynamic model of the 2-DOF helicopter system and its representation as DMAS are discussed. In the final section, the implementation results are discussed, followed by a conclusion.

Graph theory and representation of discrete-time leader-following MAS

Graph theory (Patel and Mehta, 2018)

Consider a weighted digraph $G = (V, E, A)$ with a non-empty finite set of $N$ nodes. $V = {v_{1}, v_{2}, \dots, v_{N}}$ , a set of edges $E \subset V \times V$ and the associated adjacency matrix $\tilde{A} = [a_{ij}] \in R^{N \times N}$ . An edge starting at node $j$ and ending at node $i$ is described by $(v_{j}, v_{i})$ , which means that the information transfer from node $j$ to node $i$ . The weight $a_{ij}$ of edge $(v_{j}, v_{i})$ is positive, i.e., $a_{ij} > 0$ otherwise, $a_{ij} = 0$ . Let’s define the in-degree matrix as $\tilde{D} = diag {d_{i}} \in R^{N \times N}$ with $d_{i} = \sum_{j \in N_{i}} a_{ij}$ and the Laplacian matrix as $\tilde{L} = \tilde{D} - \tilde{A}$ . In the directed graph agent $i$ can obtain information from the agent $j$ in a simplex mode of communication. However, in the undirected graph, agents $i$ and $j$ can share the information in the duplex mode of communication. For the transfer of information from one agent to other subsequent agents in particular group, at least one eigenvalue of Laplacian matrix $\tilde{L}$ must be zero. A DMAS consisting of $N + 1$ agents marked as $0, 1, 2, \dots N$ is considered in this paper. In this study, notation $0$ of DMAS is considered as a leader agent and the other agents $1, 2,, N$ are the follower agents. The communication amongst the follower agents is described using adjacency matrix and it’s weight must be $a_{ij} > 0$ if information shared between $i^{th}$ and $j^{th}$ follower agents, otherwise $a_{ij} = 0$ . The information linked from leader agent to specific follower agents in the same group can be represented using pinning gain matrix and the weight of this matrix is $a_{i 0} > 0$ , where $i = [1, 2, \dots N]$ if an information link between leader and follower agents otherwise $a_{i 0} = 0$ . This pinning gain matrix is diagonal matrix and it is defined as $\tilde{B} = diag (a_{10}, . ., a_{N 0})$ .

Assumption 1: This paper assumes that there exists a rooted spanning tree in $G$ and the information flow between followers is undirected.

Lemma 1: Xu et al. (2013). If G satisfies Assumption 1, then $(\tilde{L} + \tilde{B})$ is positive definite matrix.

Representation of discrete leader-following MAS

Consider the following identical linear DMAS

\begin{matrix} x_{i} (k + 1) = \tilde{E} x_{i} (k) + \tilde{F} (u_{i} (k) + {\tilde{D}}_{i} (k)) \forall_{i} \in N, \end{matrix}

(1)

where $i = 1, \dots, N$ , $\tilde{E} \in R^{n \times n}$ is the system matrix, $\tilde{F} \in R^{n \times m}$ is the input matrix of $i^{th}$ system respectively. State vector $x_{i} (k) \in R^{n}$ and the input vector $u_{i} (k) \in R^{m}$ , ${\tilde{D}}_{i} (k) \in R^{m}$ is external matched disturbance acting on $i^{th}$ agent system. Also, $∥ {\tilde{D}}_{i} (k) ∥$ $\leq α_{i}$ with a known upper bound $α_{i} > 0$

Assumption 2: $(\tilde{E}, \tilde{F})$ for $i^{th}$ agent system is controllable.

From equation (1), let us define a global DMAS as

X (k + 1) = (I_{N} \otimes \tilde{E}) X (k) + (I_{N} \otimes \tilde{F}) (u (k) + \tilde{D} (k)),

(2)

$X (k) = [x_{1} (k), x_{2} (k) \dots x_{N} (k)]^{T}$ ∈ $R^{nN}$ and the input vector $u (k) = [u_{1} (k), u_{2} (k) \dots u_{N} (k)]^{T}$ ∈ $R^{mN}$ , $\tilde{D} (k) = [{\tilde{D}}_{1} (k), {\tilde{D}}_{2} (k), {\tilde{D}}_{3} (k), \dots {\tilde{D}}_{N} (k)]^{T} \in R^{mN}$ matched disturbance vector. ⊗ represents the kronecker product.

From equation (2), we may rewritten as

X (k + 1) = \bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)),

(3)

where, $\bar{E} = (I_{N} \otimes \tilde{E})$ , $\bar{F} = (I_{N} \otimes \tilde{F})$ .

The dynamics of an autonomous leader in a leader-following DMAS is given as

x_{0} (k + 1) = {\tilde{E}}_{0} x_{0} (k) .

(4)

where $x_{0} (k)$ ∈ $R^{n}$ is the state vector of the leader.

Definition 1: The DMAS system defined in equation (3) achieve fixed-time steps consensus for $k \geq k^{★}$

lim_{k \to k^{★}} ∥ X (k) - 1_{N} \otimes x_{0} (k) ∥ = 0 .

(5)

where $k^{★}$ is positive integer.

Problem statement: To derive the number of steps required for global consensus of leader-following DMAS using the DSM protocols designed using Gao’s reaching law and power rate reaching law.

If there is an information linkage of $i^{th}$ agent to the leader then the weighted gain of the communication line of agents is $a_{i 0} > 0$ . The agent with $a_{i 0} > 0$ is known as the pinning gain matrix.

The local neighborhood leader-follower network error is defined as

{\bar{δ}}_{i} (k) = \sum_{j \in N} a_{ij} [x_{i} (k) - x_{j} (k)] + a_{i 0} [x_{i} (k) - x_{0} (k)] .

(6)

Considering Lemma 1, the global consensus error from equation (6) can be rewritten as (Patel and Mehta, 2018)

\bar{Λ} (k) = ((I_{N} + \tilde{D} + \tilde{B})^{- 1}) (\tilde{L} + \tilde{B})) \otimes I_{n}) \tilde{x},

(7)

where $\tilde{x} = X (k) - 1_{N} \otimes x_{0} (k) \in R^{nN}$ and $((I_{N} + \tilde{D} + \tilde{B})^{- 1})$ is non-singular matrix.The eigenvalues of the weighted matrix $\tilde{ξ} = (((I_{N} + \tilde{D} + \tilde{B})^{- 1}) (\tilde{L} + \tilde{B}))$ are strictly remain inside the unit circle as per the Gersgorin circle criteria (Varga, 2004).

Equation (7) may be written in compressed form as

\bar{Λ} (k) = (\tilde{ξ} \otimes I_{n}) \tilde{x} .

(8)

Consider $(\tilde{ξ} \otimes I_{n}) = \bar{ϒ}$ and injecting the value of $\tilde{x}$ , we can get

\bar{Λ} (k) = \bar{ϒ} (X (k) - 1_{N} \otimes x_{0} (k)),

(9)

\bar{Λ} (k) = \bar{ϒ} (X (k + 1) - 1_{N} \otimes x_{0} (k + 1)),

(10)

Analysis for number of steps required for global consensus of DMAS using the protocol with Gao’s reaching law

In this section, first we estimate the fixed time steps required to achieve the consensus and then derived the sufficient condition for global stability of DMAS with the DSM protocol using Gao’s reaching law.

Lemma 2: (Patel and Mehta, 2018): For the global DMAS defined in equation (3) is said to achieve the global consensus with the leader agent in equation (4) in fixed-time steps using protocol given as

\begin{matrix} u (k) = [- (σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1} [σ_{s}^{T} \bar{ϒ} \bar{E} X (k) + σ_{s}^{T} \bar{ϒ} (- 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ - (1 - \bar{λ} τ) \tilde{S} (k) + \bar{ϑ} τ sgn (\tilde{S} (k))]] - \tilde{D} (k), \end{matrix}

(11)

where $(σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1}$ is non-singular, $σ_{s}^{T} = I_{N} \otimes σ_{s_{i}}^{T}$ .

This protocol has been proposed for DMAS using fixed and switching graph topology in the papers (Patel and Mehta, 2018, 2019). However, this theorem is once again presented here for maintaining the continuity of the reader.

Proof: The sliding surface of $i^{th}$ agent is defined as

{\tilde{S}}_{i} (k) = σ_{s_{i}}^{T} {\bar{δ}}_{i} (k),

(12)

where $σ_{s_{i}}^{T}$ is the sliding mode gains for $i^{th}$ agent to be obtain using pole-placement approach (Nguyen et al., 2017), $τ$ is the sampling period of discrete-time system. From (12), the global sliding surface can be defined as

\tilde{S} (k) = σ_{s}^{T} \bar{Λ} (k) .

(13)

Using equation (9), we can get

\tilde{S} (k) = σ_{s}^{T} \bar{ϒ} (X (k) - 1_{N} \otimes x_{0} (k)) .

(14)

Further

\tilde{S} (k + 1) = σ_{s}^{T} \bar{ϒ} (X (k + 1) - 1_{N} \otimes x_{0} (k + 1)) .

(15)

Injecting the value of (3) and (4) into (15), we may get

\tilde{S} (k + 1) = σ_{s}^{T} \bar{ϒ} (\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) .

(16)

Now using Gao’s reaching law defined in Weibing Gao et al. (1995), reaching law of $i^{th}$ agent is defined as

{\tilde{S}}_{i} (k + 1) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} sgn ({\tilde{S}}_{i} (k)),

(17)

where $τ_{i}$ is sampling period, $λ_{i}, ϑ_{i} > 0$ are design parameters for the controller.

From equation (17), the global consensus reaching law for DMAS can be written as

\tilde{S} (k + 1) = (1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k)),

(18)

where, $\bar{ϑ} = [ϑ_{1}, ϑ_{2}, ϑ_{3}, \dots ϑ_{N}] \in R^{nN} > 0$ , $\bar{λ} = [λ_{1}, λ_{2}, λ_{3}, \dots λ_{N}] \in R^{nN} > 0$ , $0 < (1 - \bar{λ} τ) < 1$ .

Comparing equation (16) and equation (18), we may get

\begin{matrix} σ_{s}^{T} \bar{ϒ} (\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ = (1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k)) . \end{matrix}

(19)

Further, $u (k)$ is derived using equation (19) as

\begin{matrix} u (k) = [- (σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1} [σ_{s}^{T} \bar{ϒ} \bar{E} X (k) + σ_{s}^{T} \bar{ϒ} (- 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ - (1 - \bar{λ} τ) \tilde{S} (k) + \bar{ϑ} τ sgn (\tilde{S} (k))]] - \tilde{D} (k) . \end{matrix}

(20)

This completes the proof.

Using protocol in equation (11), the global consensus error defined in equation (9) cross the sliding surface, but no longer remain onto the surface rather move in a zig-zag style within a specified band called quasi-sliding mode band (QSMB) (Weibing Gao et al., 1995). To obtain the QSMB, let us consider reaching law defined in equation (17)

{\tilde{S}}_{i} (k + 1) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} sgn ({\tilde{S}}_{i} (k)),

(21)

where $τ_{i}$ is sampling period, $λ_{i}, ϑ_{i} > 0$ are design parameters for the controller.

The sign of the first right-side term of the above equation is the same as ${\tilde{S}}_{i} (k)$ , and the second term has the opposite sign of ${\tilde{S}}_{i} (k)$ . Hence, according to the definition of QSMB defined in Weibing Gao et al. (1995), the sgn of ${\tilde{S}}_{i} (k + 1)$ must be opposite to that of ${\tilde{S}}_{i} (k)$ . Thus, the above equation (21) can be written as

- {\tilde{S}}_{i} (k) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} sgn ({\tilde{S}}_{i} (k)) .

(22)

Further, evaluating equation (22)

2 {\tilde{S}}_{i} (k) = λ_{i} τ_{i} {\tilde{S}}_{i} (k) + ϑ_{i} τ_{i} sgn ({\tilde{S}}_{i} (k)) .

(23)

Once the consensus error trajectory reach inside the QSMB then ${\tilde{S}}_{i} (k)$ becomes ${\tilde{S}}_{i} (k) = Δ_{i}$ and $sgn ({\tilde{S}}_{i} (k)) = 1$ . Hence, equation (23) can be written as

Δ_{i} = \frac{ϑ_{i} τ_{i}}{2 - λ_{i} τ_{i}} .

(24)

Theorem 1: The no. of time steps required to achieve the global consensus of DMAS (3), with DSMC protocol (11) is $Max$ $(k^{★} + 1)$ where $k^{★}$ for the $i^{th}$ agent is given by

[k^{★}] = \log_{(1 - λ_{i} τ_{i})} (\frac{Δ_{i} λ_{i} + ϑ_{i}}{ϑ_{i} + λ_{i} | {\tilde{S}}_{i} (0) |}),

(25)

where, the notation $[k^{★}]$ denotes the maximum integer below the actual real number $k^{★}$ .

Proof : Based on initial state of $i^{th}$ agent, there are two cases: ${\tilde{S}}_{i} (0) > 0$ and ${\tilde{S}}_{i} (0) < 0$ .

$Case 1 :$ When ${\tilde{S}}_{i} (0) > 0$

From reaching law defined in equation (21), we may write following for $i^{th}$ agent

For $k = 0$ ,

{\tilde{S}}_{i} (1) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (0) - ϑ_{i} τ_{i}

and for $k = 1$ ,

{\tilde{S}}_{i} (2) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (1) - ϑ_{i} τ_{i} .

Substituting ${\tilde{S}}_{i} (1)$ into ${\tilde{S}}_{i} (2)$ , we may write

\begin{matrix} {\tilde{S}}_{i} (2) = (1 - λ_{i} τ_{i}) [(1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (0) - ϑ_{i} τ_{i}] - ϑ_{i} τ_{i} \\ ⋮ \end{matrix}

Similarly for $k = z$ , where $z$ is positive integer, we may write

\begin{matrix} {\tilde{S}}_{i} (z + 1) = (1 - λ_{i} τ_{i})^{z + 1} {\tilde{s}}_{i} (0) - (1 - λ_{i} τ_{i})^{z} ϑ_{i} τ_{i} \\ - (1 - λ_{i} τ_{i}) ϑ_{i} τ_{i} - (1 - λ_{i} τ_{i})^{0} ϑ_{i} τ_{i} \end{matrix}

(26)

In general

{\tilde{S}}_{i} (z + 1) = (1 - λ_{i} τ_{i})^{z + 1} {\tilde{s}}_{i} (0) - ϑ_{i} τ_{i} \times \sum_{β = 0}^{z} (1 - λ_{i} τ_{i})^{(z - β)}

(27)

{\tilde{S}}_{i} (z + 1) = (1 - λ_{i} τ_{i})^{z + 1} {\tilde{s}}_{i} (0) - h_{i},

(28)

where

h_{i} = ϑ_{i} τ_{i} \times \sum_{β = 0}^{m} (1 - λ_{i} τ_{i})^{(z - β)} .

(29)

Comparing equation (29) with geometric progression series defined as $a \frac{(1 - r^{d})}{1 - r}$ , where $a$ is the first term of the series and $r$ is the common ratio and $d$ is the number of terms in the series, we get

a = 1, r = (1 - λ_{i} τ_{i}), d = z + 1 .

Thus, (29) can be written as

h_{i} = ϑ_{i} τ_{i} \times \frac{1 - {(1 - λ_{i} τ_{i})}^{z + 1}}{λ_{i} τ_{i}} .

(30)

Substituting equation (30) into (28)

{\tilde{S}}_{i} (z + 1) = (1 - λ_{i} τ_{i})^{z + 1} {\tilde{S}}_{i} (0) - ϑ_{i} τ_{i} \times \frac{1 - {(1 - λ_{i} τ_{i})}^{z + 1}}{λ_{i} τ_{i}} .

(31)

Now defining $k^{★} = z + 1$ , we may write

{\tilde{S}}_{i} (k^{*}) = (1 - λ_{i} τ_{i})^{k^{★}} {\tilde{S}}_{i} (0) - ϑ_{i} τ_{i} \times \frac{1 - {(1 - λ_{i} τ_{i})}^{k^{★}}}{λ_{i} τ_{i}} .

(32)

From equation (32), we may infer that for $k \geq k^{★}$ steps the local neighborhood error trajectory of $i^{th}$ agent first time reaches to the QSMB and stays within a specified band (24).

Now for finding the $k^{★}$ , consider QSMB given in equation (24)

Δ_{i} = (1 - λ_{i} τ_{i})^{k^{★}} {\tilde{S}}_{i} (0) - ϑ_{i} τ_{i} \times \frac{1 - {(1 - λ_{i} τ_{i})}^{k^{★}}}{λ_{i} τ_{i}} .

(33)

From (33)

(1 - λ_{i} τ_{i})^{k^{★}} = \frac{Δ_{i} λ_{i} τ_{i} + ϑ_{i} τ_{i}}{(ϑ_{i} τ_{i}) + λ_{i} τ_{i} | {\tilde{S}}_{i} (0) |} .

(34)

Applying logarithm scale on both sides of equation (34), we get

[k^{★}] = \log_{(1 - λ_{i} τ_{i})} (\frac{Δ_{i} λ_{i} + ϑ_{i}}{ϑ_{i} + λ_{i} | {\tilde{S}}_{i} (0) |}) .

(35)

$Case 2 :$ ${\tilde{S}}_{i} (0) < 0$

In this case ${\tilde{S}}_{i} (k + 1)$ will be negative for all $k \geq k^{*} + 1$ . Hence, we may write

\begin{matrix} {\tilde{S}}_{i} (k^{★} + 1) \geq - (1 - λ_{i} τ_{i})^{k^{★} + 1} {\tilde{S}}_{i} (0) + ϑ_{i} τ_{i} \\ \times \frac{1 - {(1 - λ_{i} τ_{i})}^{k^{★} + 1}}{λ_{i} τ_{i}} . \end{matrix}

(36)

Above two cases show that $k^{★}$ is the least integer such that local neighbouring error trajectory of the $i^{th}$ agent is guaranteed to cross the sliding mode band defined in equation (24) and stay within it. Hence, the state consensus of $i^{th}$ agent with leader agent is begin within $k^{★} + 1$ steps. Here, base on initial condition of each agent states, the maximum $k^{★} + 1$ steps required of particular agent to reach the leader are considered as a required no. of steps for consensus.

This completes the proof.

Theorem 2 presents the global stability of leader-following DMAS with the proposed consensus protocol defined in Lemma 2 with fixed graph topology.

Theorem 2: The global stability of DMAS using the protocol (11) is guaranteed if the closed-loop global consensus error (9) for global DMAS (3) drives towards the global sliding surface and maintain on it for the gain $\bar{λ}$ , $\bar{ϑ} > 0$ and $0 < 1 - \bar{λ} τ < 1$ , provided the following conditions are true

0 \leq G < {\tilde{S}}^{T} (k) \tilde{S} (k) .

(37)

Proof: Let us consider the Lyapunov function as

\bar{V} (k) = {\tilde{S}}^{T} (k) \tilde{S} (k) .

(38)

Using forward difference, equation (38) is written as

\bar{Δ} V_{s} (k) = \bar{V} (k + 1) - \bar{V} (k),

(39)

\bar{Δ} V_{s} (k) = {\tilde{S}}^{T} (k + 1) \tilde{S} (k + 1) - {\tilde{S}}^{T} (k) \tilde{S} (k),

(40)

For guaranteed stability, $\bar{Δ} V_{s} (k) < 0$ .

Substituting $\tilde{S} (k + 1)$ (15) into (40), we can get

\begin{matrix} \bar{Δ} V_{s} (k) = [σ_{s}^{T} \bar{ϒ} (X (k + 1) - 1_{N} \otimes x_{0} (k + 1))]^{T} \\ [σ_{s}^{T} \bar{ϒ} (X (k + 1) - 1_{N} \otimes x_{0} (k + 1))] - {\tilde{S}}^{T} (k) \tilde{S} (k) . \end{matrix}

(41)

Using equations (3) and (4)

\begin{matrix} \bar{Δ} V_{s} (k) = [σ_{s}^{T} \bar{ϒ} [(\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) {]]}^{T} \\ [σ_{s}^{T} \bar{ϒ} [(\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) \\ - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k))]] - {\tilde{S}}^{T} (k) \tilde{S} (k) . \end{matrix}

(42)

Now substituting the global consensus protocol (11) into equation (42), we may get

\begin{matrix} \bar{Δ} V_{s} (k) = [(1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k))]^{T} * \\ [(1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k))] - {\tilde{S}}^{T} (k) \tilde{S} (k), \end{matrix}

(43)

denoting

\begin{matrix} G = [(1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k))]^{T} * \\ [(1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ sgn (\tilde{S} (k))] \\ \bar{Δ} V_{s} (k) = G - {\tilde{S}}^{T} (k) \tilde{S} (k) . \end{matrix}

(44)

From (44), the term $G$ can be tuned close to zero by selecting the proper value of $\bar{λ}$ and $\bar{ϑ}$ such that $\bar{Δ} V_{s} (k) < 0$ . Hence, the global stability of DMAS is assured.

The main detriment of the global consensus protocol (11) is chattering and also the constant control effort required during sliding phase for the consensus which consumes more energy. To overcome these drawbacks, the global topological consensus protocol inspired by power rate reaching law (Devika and Thomas, 2017) is derived. Hence, more effort is required when the global leader-following consensus error trajectory is away from the surface and decrease the effort when consensus error is close to the surface.

Analysis for number of steps required for global consensus of DMAS using the protocol with power rate reaching law

In this section, we propose analysis of number of steps required for the global consensus using DSM protocol with power rate reaching law.

Lemma 3: For the global DMAS (3) is said to achieve the global consensus with the leader agent in a fixed-time steps using protocol given as

\begin{matrix} u (k) = [- (σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1} [σ_{s}^{T} \bar{ϒ} \bar{E} X (k) + σ_{s}^{T} \bar{ϒ} (- 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ (1 - \bar{λ} τ) \tilde{S} (k) + \bar{ϑ} τ | \tilde{s} (k) |^{η} sgn (\tilde{S} (k))]] - \tilde{D} (k), \end{matrix}

(45)

where $(σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1}$ is non-singular.

This protocol has been proposed for DMAS with switching graph topology (Patel and Mehta, 2019). However, the proof of the Lemma is presented here for the clarity and continuity of the reader as given below.

Proof: First, defining the sliding surface of $i^{th}$ agent as

{\tilde{S}}_{i} (k) = σ_{s_{i}}^{T} {\bar{δ}}_{i} (k),

(46)

where $σ_{s_{i}}$ is the sliding mode gains to be obtain using pole-placement approach (Nguyen et al., 2017), $τ$ is the sampling period of discrete-time system. From (46), the global sliding surface can be defined as

\tilde{S} (k) = σ_{s}^{T} \bar{Λ} (k) .

(47)

Using equation (9), we can get

\tilde{S} (k) = σ_{s}^{T} \bar{ϒ} (X (k) - 1_{N} \otimes x_{0} (k)) .

(48)

Further

\tilde{S} (k + 1) = σ_{s}^{T} \bar{ϒ} (X (k + 1) - 1_{N} \otimes x_{0} (k + 1)) .

(49)

Substituting (3) and (4) into (49), we may get

\tilde{S} (k + 1) = σ_{s}^{T} \bar{ϒ} (\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) .

(50)

Now, motivated by Devika and Thomas (2017), reaching law of $i^{th}$ agent is defined as

{\tilde{S}}_{i} (k + 1) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} | {\tilde{S}}_{i} (k) |^{η_{i}} sgn ({\tilde{S}}_{i} (k)),

(51)

where $τ_{i}$ is sampling period, $λ_{i}, ϑ_{i} > 0$ , $η_{i}, 0 < η_{i} < 1$ are design parameters for the controller.

From equation (51), the global consensus reaching law for DMAS can be written as

\tilde{S} (k + 1) = (1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ | \tilde{S} (k) |^{η} sgn (\tilde{S} (k)),

(52)

where, $\bar{ϑ} = [ϑ_{1}, ϑ_{2}, ϑ_{3}, \dots ϑ_{N}] \in R^{nN} > 0$ , $\bar{λ} = [λ_{1}, λ_{2}, λ_{3}, \dots λ_{N}] \in R^{nN} > 0$ , $0 < (1 - \bar{λ} τ) < 1$ , $η = [η_{1}, η_{2}, η_{3} \dots η_{N}]$ .

Comparing equation (50) and equation (52), we may get

\begin{matrix} σ_{s}^{T} \bar{ϒ} (\bar{E} X (k) + \bar{F} (u (k) + \tilde{D} (k)) - 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ = (1 - \bar{λ} τ) \tilde{S} (k) - \bar{ϑ} τ | \tilde{S} (k) |^{η} sgn (\tilde{S} (k)) . \end{matrix}

(53)

Further, $u (k)$ is derived from equation (53) and expressed in terms of the global consensus protocol as

\begin{matrix} u (k) = [- (σ_{s}^{T} \bar{ϒ} \bar{F})^{- 1} [σ_{s}^{T} \bar{ϒ} \bar{E} X (k) + σ_{s}^{T} \bar{ϒ} (- 1_{N} \otimes {\tilde{E}}_{0} x_{0} (k)) \\ - (1 - \bar{λ} τ) \tilde{S} (k) + \bar{ϑ} τ | \tilde{S} (k) |^{η} sgn (\tilde{S} (k))]] - \tilde{D} (k) . \end{matrix}

(54)

This completes the proof.

To derived the width of sliding mode band (SMB) using power rate reaching law let us consider reaching law defined in equation (51)

{\tilde{S}}_{i} (k + 1) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} | {\tilde{S}}_{i} (k) |^{η_{i}} sgn ({\tilde{S}}_{i} (k)) .

The sign of the first right-side term of the above equation is the same as ${\tilde{S}}_{i} (k)$ , and the second term has the opposite sign of ${\tilde{S}}_{i} (k)$ . Hence, according to the definition of QSMB, the sgn of ${\tilde{S}}_{i} (k + 1)$ must be opposite to that of ${\tilde{S}}_{i} (k)$ . Thus, the above equation (55) can be written as

- {\tilde{S}}_{i} (k) = (1 - λ_{i} τ_{i}) {\tilde{S}}_{i} (k) - ϑ_{i} τ_{i} | {\tilde{S}}_{i} (k) |^{η_{i}} sgn ({\tilde{S}}_{i} (k)) .

(55)

Further, equation (55) is written as

2 {\tilde{S}}_{i} (k) = λ_{i} τ_{i} {\tilde{S}}_{i} (k) + ϑ_{i} τ_{i} | {\tilde{S}}_{i} (k) |^{η_{i}} sgn ({\tilde{S}}_{i} (k)) .

(56)

Once the consensus error trajectory defined in equation (6) inside the QSMB, then consider ${\tilde{S}}_{i} (k) = Φ_{i}$ , $sgn ({\tilde{S}}_{i} (k)) = 1$ for the $i^{th}$ agent and can be defined as

Φ_{i} = (\frac{ϑ_{i} τ_{i}}{2 - λ_{i} τ_{i}})^{\frac{1}{1 - η_{i}}} .

(57)

From equation (57), we may infer that an appropriate selection of $η_{i}$ reduces the chattering effect and increase the consensus convergence speed.

Theorem 3: The no. of time steps required to achieve the global consensus of DMAS (3) with DSM protocol (54) is Max $(k^{★} + 1)$ where $k^{★}$ for $i^{th}$ agent is given by

[k^{★}] = \frac{[\ln (| {\tilde{S}}_{i} {(0) |}^{1 - η_{i}} + \frac{ϑ_{i}}{λ_{i}}) - \ln (\frac{ϑ_{i}}{λ_{i}} + Φ_{i})]}{(1 - η_{i}) λ_{i} τ_{i}},

(58)

where the notation $[k^{★}]$ denotes the maximum integer bounded below the real number $k^{★}$ and ${\tilde{S}}_{i} (0)$ is the initial value of sliding-mode surface.

Proof: Consider the reaching law (54) for $i^{th}$ follower agent and assuming $τ_{i}$ is small enough, we may use continuous time derivative function

{\overset{\cdot}{\tilde{S}}}_{i} = lim_{δ \to 0} \frac{{\tilde{S}}_{i} (t + δ (t)) - {\tilde{S}}_{i} (t)}{δ (t)} .

(59)

Rewriting equation (59) as

{\overset{\cdot}{\tilde{S}}}_{i} = ({\tilde{S}}_{i} (k + 1) - {\tilde{S}}_{i} (k)) / τ_{i} = - λ_{i} {\tilde{S}}_{i} - ϑ_{i} {\tilde{S}}_{i}^{η_{i}} .

(60)

Calculating reaching time for error trajectories

\frac{d {\tilde{S}}_{i}}{dt} + λ_{i} {\tilde{S}}_{i} = - ϑ_{i} {\tilde{S}}_{i}^{η_{i}} .

(61)

Multiplying both sides with ${\tilde{s}}_{i}^{- η_{i}}$

{\tilde{S}}_{i}^{- η_{i}} \frac{d {\tilde{S}}_{i}}{dt} + λ_{i} {\tilde{S}}_{i}^{1 - η_{i}} = - ϑ_{i} .

(62)

Considering $z = {\tilde{s}}_{i}^{1 - η_{i}}$ and taking derivative of it, we may get

\frac{dz}{dt} = (1 - η_{i}) {\tilde{S}}_{i}^{- η_{i}} \frac{d {\tilde{S}}_{i}}{dt} .

(63)

Multiplying equation (62) by $(1 - η_{i})$ on both the sides

(1 - η_{i}) {\tilde{S}}_{i}^{- η} \frac{d {\tilde{S}}_{i}}{dt} + (1 - η_{i}) λ_{i} {\tilde{S}}_{i}^{1 - η_{i}} = - (1 - η_{i}) ϑ_{i} .

(64)

Substituting equation (63) into equation (64)

\frac{dz}{dt} + (1 - η_{i}) λ_{i} z = - (1 - η_{i}) ϑ_{i} .

(65)

Now from the first order non-homogeneous differential equation defined as

\frac{dz}{dt} + \tilde{p} (t) z = \tilde{Q} (t),

(66)

whose general solution is given as (Mondal and Roy, 2014)

z = c e^{- \int \tilde{p} (t) dt} + e^{- \int \tilde{p} (t) dt} \int \tilde{Q} (t) e^{- \int \tilde{p} (t) dt} dt,

(67)

\begin{matrix} z = (c e^{- \int (1 - η_{i}) λ_{i} dt} + e^{- \int (1 - η_{i}) λ_{i} dt}) \\ \times (\int [- (1 - η_{i}) ϑ_{i}] e^{- \int (1 - η_{i}) λ_{i} dt} dt) . \end{matrix}

(68)

Combining with equation (68), the solution of equation (66) is given by

\begin{matrix} z = c e^{- (1 - η_{i}) λ_{i} t} + e^{- (1 - η_{i}) λ_{i} t} \frac{([- (1 - η_{i}) ϑ_{i}] e^{(1 - η_{i}) λ_{i} t})}{(1 - η_{i}) λ_{i}}, \\ = c e^{- (1 - η_{i}) λ_{i} t} - \frac{ϑ_{i}}{λ_{i}} . \end{matrix}

(69)

Considering $z (t) = {\tilde{S}}_{i}^{1 - η_{i}} (t)$ , we may write

{\tilde{S}}_{i}^{1 - η_{i}} (t) = c e^{- (1 - η_{i}) λ_{i} t} - \frac{ϑ_{i}}{λ_{i}} .

(70)

When $t = 0$ , we may find the solution of $c$ from equation (70) as

c = | {\tilde{S}}_{i} (0) |^{(1 - η_{i})} + \frac{ϑ_{i}}{λ_{i}} .

(71)

Substituting equation (71) into equation (70), we get

{\tilde{S}}_{i}^{1 - η_{i}} (t) = [| {\tilde{S}}_{i} (0) |^{(1 - η_{i})} + \frac{ϑ_{i}}{λ_{i}}] e^{- (1 - η_{i}) λ_{i} t} - \frac{ϑ_{i}}{λ_{i}} .

(72)

From equation (72), the reaching time from any initial condition to sliding mode band (SMB) for $i^{th}$ agent is defined as

Φ_{i} = [| {\tilde{S}}_{i} (0) |^{(1 - η_{i})} + \frac{ϑ_{i}}{λ_{i}}] e^{- (1 - η_{i}) λ_{i} t} - \frac{ϑ_{i}}{λ_{i}} .

(73)

Further, using logarithmic scale function on the both side of equation (73), we may get

t = \frac{{\ln [| {\tilde{S}}_{i} (0) |^{(1 - η_{i})} + \frac{ϑ_{i}}{λ_{i}}] - \ln [Φ_{i} + \frac{ϑ_{i}}{λ_{i}}]}}{(1 - η_{i}) λ_{i}} .

(74)

Let us take $X = \ln [| {\tilde{S}}_{i} (0) |^{(1 - η_{i})} + \frac{ϑ_{i}}{λ_{i}}]$ , $Y = \ln [Φ_{i} + \frac{ϑ_{i}}{λ_{i}}]$ and equation (74), the discrete steps are calculated from any initial condition to SMB ${\tilde{S}}_{i} (k) = Φ_{i}$ is

k^{★} = t / τ_{i} = \frac{X - Y}{(1 - η_{i}) λ_{i} τ_{i}} .

(75)

From equation (75), it is noted that the parameters $λ_{i}$ , $ϑ_{i}$ , $η_{i}$ affect the convergence speed for the consensus.

This completes the proof.

The global stability of DMAS using the global consensus protocol (54) can be inferred similar to the earlier case present as Theorem 2.

DOF helicopter systems

The 2-DOF helicopter is a significant model to study from control engineering perspective due to Broad non-linear features, extremely cross-coupling impacts and open-loop instability. Figure 1 shows the 2-DOF helicopter model (fixed base) with two DC motor-driven propellers. The front propeller controls the elevation of the nose over the pitch axis and the back propeller controls the rotational motion around the yaw axis. The voltages are $\pm 24 V$ and $\pm 15 V$ , respectively, across the pitch and yaw motors (Patel and Mehta, 2019). The notations defined in Table-1 are used to derive the dynamics of 2-DOF helicopter system.

Figure 1.

2-DOF helicopter system.

Table 1.

Notation used for the modeling (Patel and Mehta, 2019).

$θ$	Pitch angle, deg
$ψ$	Yaw angle, deg
$K_{pp}$	Thrust force constant of yaw motor/propeller, N.m/V
$K_{yy}$	Thrust torque constant of yaw axis from yaw motor/ propeller, N.m/V
$K_{py}$	Thrust torque constant acting on pitch axis from yaw motor/propeller, N.m/V
$K_{yp}$	Thrust torque constant acting on yaw axis from pitch motor/propeller, N.m/V
$B_{p}$	Viscous damping about pitch axis, N/V
$B_{y}$	Viscous damping about yaw axis, N/V
$m_{h}$	Total moving mass of the helicopter (body, two propeller, assemblies, etc.), kg
$l$	Center of mass length along helicopter body from pitch axis, meter(m)
$J_{eq}, p$	Total moment of inertia about pitch axis, kg. $m^{2}$
$J_{eq}, y$	Total moment of inertia about yaw axis, kg. $m^{2}$
$V_{p}, V_{y}$	voltages applied to the front and rear motors
$g$	Gravitational constant, $9.8 m / s^{2}$

Consider 2-DOF helicopter dynamics acting as $i^{th}$ agent can be written as follows.

The dynamic equation of pitch axis $θ$ is written as

\begin{matrix} \overset{\cdot\cdot}{θ} = \frac{1}{J_{eq, p} + m_{h} l^{2}} [{K_{pp} V_{p} + K_{py} V_{y}} \\ - {B_{p} \overset{\cdot}{θ} + m_{h} glcos (θ) + m_{h} l^{2} \sin (θ) \cos (θ) {\overset{\cdot}{ψ}}^{2}}] . \end{matrix}

(76)

Consider $J_{tp} = J_{eq, p} + m_{h} l^{2}$ ,we may write equation (76) as

\begin{matrix} \overset{\cdot\cdot}{θ} = \frac{1}{J_{tp}} [{K_{pp} V_{p} + K_{py} V_{y}} \\ - {B_{p} \overset{\cdot}{θ} + m_{h} glcos (θ) + m_{h} l^{2} \sin (θ) \cos (θ) {\overset{\cdot}{ψ}}^{2}}] . \end{matrix}

(77)

Likewise, the yaw axis dynamics is defined as

\begin{matrix} \overset{\cdot\cdot}{ψ} = \frac{1}{J_{eq, y} + m_{h} l^{2} co s^{2} (θ)} \\ [{K_{yp} V_{p} + K_{yy} V_{y}} - {B_{y} \overset{\cdot}{ψ} + 2 m_{h} l^{2} \sin (θ) \cos (θ) \overset{\cdot}{ψ} {\overset{\cdot}{ψ}}^{2}}] . \end{matrix}

(78)

Consider $J_{ty} = J_{eq, y} + m_{h} l^{2} co s^{2} (θ)$ , and from the above equation, we can get

\overset{\cdot\cdot}{ψ} = \frac{1}{J_{ty}} [{K_{yp} V_{p} + K_{yy} V_{y}} - {B_{y} \overset{\cdot}{ψ} + 2 m_{h} l^{2} \sin (θ) \cos (θ) \overset{\cdot}{ψ} {\overset{\cdot}{ψ}}^{2}}] .

(79)

The $i^{th}$ agent system state space model is defined as

\begin{matrix} x_{i} (k + 1) = \tilde{E} x_{i} (k) + \tilde{F} (u_{i} (k) + {\tilde{D}}_{i} (k)) \forall_{i} \in N . \end{matrix}

(80)

Where, $\tilde{E} = [\begin{matrix} 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \\ 0 & 0 & - \frac{B_{p}}{J_{T_{p}}} & 0 \\ 0 & 0 & 0 & - \frac{B_{y}}{J_{T_{y}}} \end{matrix}]$ $\in R^{n \times n}$ and $\tilde{F} = [\begin{matrix} 0 & 0 \\ 0 & 0 \\ \frac{k_{p_{p}}}{J_{T_{p}}} & \frac{k_{p_{y}}}{J_{T_{p}}} \\ \frac{k_{y_{p}}}{J_{T_{y}}} & \frac{k_{y_{y}}}{J_{T_{y}}} \end{matrix}]$ $\in R^{n \times m}$ are the system matrix input matrix respectively. state vector $x_{i} (k) = [θ_{i} (k) ψ_{i} (k) {\overset{\cdot}{θ}}_{i} (k) {\overset{\cdot}{ψ}}_{i} (k)]$ ∈ $R^{n}$ , $u_{i} (k)$ ∈ $R^{m}$ and ${\tilde{D}}_{i}$ ∈ $R^{m}$ are state vector, input vector and matched disturbance acting on $i^{th}$ follower.

Simulation result discussion

Let us consider 4 no. of 2-DOF helicopter systems as agents in a leader-following network shown in Figure 2 to validate the effectiveness of the proposed global protocols for global DMAS. Here, in Figure 2, the vertex represents the physical system while the edge between agents represents the communication link either unidirectional or bidirectional. One helicopter act as a leader agent in graph topology and three others act as follower agents.

Figure 2.

Communication graph topology.

In this section, pitch position( $θ$ ) and its velocity ( $\overset{\cdot}{θ}$ ), yaw position( $ψ$ ) and its velocity ( $\overset{\cdot}{ψ}$ ) states of 2-DOF helicopter system are to be considered as a consensus single agent parameter. As mentioned, the node of the leader is thought to have the same dynamics as the node of the follower agents. Matlab R16 is used to perform the simulation.

Discretizing the continuous-time system defined in equation (80) at sampling period $τ = 0.03$ , we obtain the discrete-time system model as

\begin{matrix} x_{i} (k + 1) = \tilde{E} x_{i} (k) + \tilde{F} (u_{i} (k) + {\tilde{D}}_{i} (k)) \forall_{i} \in N . \end{matrix}

(81)

Where

\tilde{E} = [\begin{matrix} 1 & 0 & 0.0262 & 0 \\ 0 & 1 & 0 & 0.0285 \\ 0 & 0 & 0.7571 & 0 \\ 0 & 0 & 0 & 0.9004 \end{matrix}] \tilde{F} = [\begin{matrix} 0.0010 & 0 \\ 0.0001 & 0.0003 \\ 0.0620 & 0.0021 \\ 0.0069 & 0.0225 \end{matrix}]

As mentioned in section 3, consider MAS as an undirected graph topology with three follower nodes and one leader node. The notation of the leader node is given as 0, while the notation of the following nodes is given as 1, 2, 3, respectively. Their topology for communication is shown in Figure 2. Each follower agent and leader agent initial state is generated in the span $[- 1, 1]$ and the diagonal matrix $\tilde{D}$ , adjacency matrix $\tilde{A}$ pinning matrix gains matrix $\tilde{B}$ and Laplacian matrix $\tilde{L}$ are defined as

\begin{matrix} \tilde{D} = [\begin{matrix} 2 & 0 & 0 \\ 0 & 2 & 0 \\ 0 & 0 & 2 \end{matrix}], \end{matrix} \begin{matrix} \tilde{A} = [\begin{matrix} 0 & 1 & 1 \\ 1 & 0 & 1 \\ 1 & 1 & 0 \end{matrix}] \end{matrix}, \begin{matrix} \tilde{B} = diag {1, 1, 0} \end{matrix},

(82)

\begin{matrix} \tilde{L} = [\begin{matrix} 2 & - 1 & - 1 \\ - 1 & 2 & - 1 \\ - 1 & - 1 & 2 \end{matrix}] . \end{matrix}

(83)

The sliding mode gain for surface is calculated using pole-placement method that is obtain to be

σ_{s_{i}}^{T} = [\begin{matrix} 93.64 & 52.39 & 13.17 & - 0.0043 \\ 126.18 & 482.90 & - 1.5626 & 45.98 \end{matrix}] .

(84)

A slow varying disturbance is applied to each DMAS with magnitude ${\tilde{D}}_{i} (k) = 0.002 * \cos (0.86 k)$ to check the robustness of the global consensus protocol for the global DMAS. Gains for follower agents $ϑ_{i}$ , $λ_{i}$ are selected as $[2 2 2]$ and $[1 1 1]$ , respectively, for the global consensus protocol using Gao’s reaching law defined in equation (11). The value of $η_{i}$ for $i^{th}$ agent is considered to be 0.6. Proper choice of the $η_{i}$ value decreases the sliding mode band in power rate reaching law.

Figure 3 show the simulation results of the follower agents pitch position and velocity states consensus with leader position and velocity states using proposed Gao’s and power rate reaching law based global consensus protocols. It is observed that consensus of all the followers with leader has begun in fixed-time steps. However, the global consensus protocol using power rate reaching law gives faster consensus convergence speed compared with Gao’s reaching law. Figure 4 shows the simulation results of yaw position and velocity states consensus of follower with leader agent states and it is revealed that all the follower agents achieve the consensus with leader agent in fixed-time steps.

Figure 3.

Pitch position and velocity consensus.

Figure 4.

Yaw position and velocity consensus.

Figure 5 shows the follower agent’s individual consensus control effort (u), which has also been applied to individual agents in the same DMAS network. The magnified portion of the results for the individual follower agent shows the chattering effect.

Figure 5.

Consensus effort law of individual agent.

Figure 6 shows the sliding surface (consider as leader-following consensus error) of the individual agent of DMAS. It first crosses the sliding surface band called QSMB in fixed-time steps from any initial condition and remains within this band. The sliding-mode band changes from −0.03 to+0.03 $(O (τ))$ determined using (24) for the case of Gao’s reaching law defined in (57) and likewise sliding-mode band remains steadily in $O (τ^{2})$ for the case of power rate reaching law.

Figure 6.

Individual agent sliding surface.

Figure 7 shows the sliding variable of an individual agent which reaches to QSMB in $k^{★}$ steps which is defined in equation (25) and equation (58) using both the reaching laws and hence, the global consensus of follower agents states with leader state is achieved in $Max {(k^{★} + 1)}$ in the case of Gao’s reaching law. While in the case of power rate reaching law the global surface consensus has also achieved within $Max {(k^{★} + 1)}$ . Table 2 shows the performance comparison of number of steps taken for the global consensus of all the follower agents with leader agent using Gao’s and power rate reaching law. It is inferred that power rate reaching law outperform the Gao’s reaching law.

Figure 7.

Consensus of sliding surface.

Table 2.

Performance comparison of surface consensus of all the followers with leader.

	Gao’s reaching law		Power rate reaching law
	$S_{1}$	$S_{2}$	$S_{1}$	$S_{2}$
steps $(k^{★})$	134	173	117	148
Order of chattering magnitude	$O (τ)$	$O (τ)$	$O (τ^{2})$	$O (τ^{2})$

Figure 8 shows the error of pitch and yaw position of follower agents with leader agent using two different consensus reaching laws which subsequently converge to zero in fixed-time steps. Figure 9 express follower agents error of pitch and yaw velocity with the leader agent using two distinct reaching laws and noticed that reached within fixed-time steps. It can be noted that leader-following consensus error speedily reaches to zero in case of power rate reaching law compared with Gao’s reaching law.

Figure 8.

Position consensus tracking error.

Figure 9.

Velocity consensus tracking error.

Conclusion

In this paper, anayltical study of discrete steps required for the global consensus of follwer agents with leader using two distinct discrete-time sliding mode global consensus protocols based on Gao’s reaching law and power rate reaching law. A discrete multi-agent system which is combination of leader and follower agents is configured with a fixed, undirected graph topology. It is inferred that the time required for global consensus due to Gao’s reaching law takes more time in comparison with power rate reaching law. The results reveal that power rate reaching law has chattering magnitude $O (τ^{2})$ compared with Gao’s reaching law as $O (τ)$ . The global stability condition is also derived for the DMAS using the Lyapunov function. Finally, both the global consensus protocols are implemented in simulink environment for a homogeneous multi-agent system comprise of 2-DOF helicopter systems.

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) received no financial support for the research, authorship, and/or publication of this article.

ORCID iD

Axaykumar Mehta

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