Fractional-Order Multi-agent Formation Using Distributed NMPC Design with Obstacles and Collision Avoidance and Connectivity Maintenance

Abstract

This paper addresses distributed nonlinear model predictive controller design for formation control of agents with fractional-order dynamics (DNMPC-FCFO) in the presence of obstacles. By introducing new constraints, the collisions between non-neighboring agents are avoided while there is no need to use the information of non-neighboring agents. Moreover, by applying contractive constraints in our optimization problem the Lyapunov stability is guaranteed. Since in parallel DMPC method contraction occurred only on first two steps, the use of terminal components that are essential parts of conventional MPC to create stability is eliminated. These components usually complicate the design and are often used for low-end systems. Using fractional-order equations often leads to mathematical models capable of better describing experimental behavior, but due to memory effects, the controller design is usually more complex. The mathematical stability proof is provided in this regard. In the proposed scheme, considering limited communication range in mobile robots, the controller is designed to preserve the network connectivity. Simulation results show the effectiveness of the proposed method.

Appendices

Appendix

Graph

A graph is defined as a set of vertices and a group of edges, each of which connects a pair of vertices. In a mathematical representation, a digraph is represented as $G=\left(V,E\right)$, where represents the connection between vertices and edge, $E$ as $\left({v}_{i},{v}_{j}\right)$ represents the connection between vertices $i$ the tail of the edge, $j$ heat of the edge. Neighbors of agent $i$ are denoted by${N}_{i}=\left\{{v}_{i}:\left({v}_{i},{v}_{i}\right)\in E\right\}$.

Definition 1

(Iddrisu & Tetteh, 2017) The gamma function is defined as follows:

$$ \Gamma \left( n \right) = \mathop \smallint \limits_{0}^{\infty } t^{n - 1} e^{ - t} {\text{d}}t $$

(39)

For every non-negative integer n, the generalized factorial function can be written as follows:

$$ {\Gamma }\left( {n + 1} \right) = n{\Gamma }\left( n \right) = n\left( {n - 1} \right){\Gamma }\left( {n - 1} \right) = n! $$

(40)

Definition 2

(Mainardi, 2020) Mittag–Leffler function is a spectral function, which can depend on two parameters α and β be defined by the following series when the real part of α is strictly positive.

$$ {\rm E}_{\alpha ,\beta } \left( z \right) = \mathop \sum \limits_{m = 0}^{\infty } \frac{{z^{m} }}{{\Gamma \left( {\alpha m + \beta } \right)}} $$

(41)

When $\beta =\alpha =1$, Mittag–Leffler function becomes an exponential function:

$$ E_{1,1} \left( z \right) = \mathop \sum \limits_{m = 0}^{\infty } \frac{{z^{m} }}{{{\Gamma }\left( {m + 1} \right)}}{ } = \mathop \sum \limits_{m = 0}^{\infty } \frac{{z^{m} }}{{m!{ }}} = e^{z} $$

(42)

Considering $\rho$ is the shrinkage parameter and$0\le \rho <1$, we have ${e}^{\rho -1}\ge \rho ,$ thus for$k\in {\mathbb{N}}^{+}$, where ${\mathbb{N}}$ is a set of natural numbers; it can be written as:

$$ e^{{k\left( {\rho - 1} \right)}} \ge \rho^{k} $$

(43)

When $\beta = 1$,$0 < \alpha < 1$, and $z = \rho - 1$, we have:

$$ E_{\alpha ,1} \left( {\rho - 1} \right) = \mathop \sum \limits_{m = 0}^{\infty } \frac{{\left( {\rho - 1} \right)^{m} }}{{{\Gamma }\left( {\alpha m + 1} \right)}} $$

(44)

Using (43) and (44), it can be concluded:

$$ \left( {E_{\alpha ,1} \left( {\rho - 1} \right)} \right)^{k} \ge \rho^{k} $$

(45)

Distance from the line (Gore, 2017; Larson et al., 2007)

Given $ax+by+c=0$ on the plane and the nonzero $a$ and $b$, the distance of the point $({x}_{1},{y}_{1})$ from the line is defined as follows:

$$ {\text{Distance }}\left( {ax + by + c = 0,\left( {x_{1} ,y_{1} } \right)} \right) = \frac{{\left| {ax_{1} + by_{1} + c} \right|}}{{\sqrt {a^{2} + b^{2} } }}{ } $$

(46)

Now according to the definition of the distance of a point from a line:

$$ {\text{Distance }}\left( {ax + by + c = 0,\left( {x_{1} ,y_{1} } \right)} \right) \equiv \left\{ {\begin{array}{*{20}c} {0,} & {ax_{1} + by_{1} + c = 0} \\ { > 0,} & {{\text{otherwise}}} \qquad\quad\;\;\;\\ \end{array} } \right. $$

(47)