Full-State Prescribed Performance-Based Consensus of Double-Integrator Multi-Agent Systems with Jointly Connected Topologies

For convenience, we summarize some notations as follows. $\mathbb{R}^{N}$: the set of $N$ dimension real column vectors; $I_{N}$: the $N\times N$ identity matrix; $x^{{\sf\tiny T}}$: the transposition of $x$; ${\rm diag}\{d_{1},d_{2},\cdots,d_{n}\}$: diagonal matrix with entries $d_{1},d_{2},\cdots,d_{n}$.

An undirected graph $\mathcal{G}_{\sigma(t)}$ is composed of the node set $\mathcal{V}=\{1,2,\cdots,N\}$ and edge set $\mathcal{E}_{\sigma(t)}\subset\mathcal{V}\times\mathcal{V}$, where $\sigma(t):[0,+\infty)\to\mathcal{P}$, $\mathcal{P}=\{1,2,\cdots,n_{0}\}$ is a piecewise constant switching signal. Let $\mathcal{A}_{\sigma(t)}=[a_{ij}(t)]\in\mathbb{R}^{N\times N}$ be the adjacency matrix where $a_{ij}(t)=1$ if $(i,j)\in\mathcal{E}_{\sigma(t)}$ and $a_{ij}(t)=0$ otherwise. The neighbor set of node $i$ is $\mathcal{N}_{i}(t)=\{j\in\mathcal{V}:(i,j)\in\mathcal{E}_{\sigma(t)}\}$. The Laplacian matrix is defined as $\mathcal{L}_{\sigma(t)}=\mathcal{D}_{\sigma(t)}-\mathcal{A}_{\sigma(t)}$, where degree matrix $\mathcal{D}_{\sigma(t)}={\rm diag}\{d_{1}(t),d_{2}(t),\cdots,d_{N}(t)\}$ with diagonal entry $d_{i}(t)=\sum_{j\in\mathcal{N}_{i}(t)}a_{ij}(t)$. The incidence matrix is defined as $B_{\sigma(t)}=[b_{il}(t)]\in\mathbb{R}^{N\times M_{\sigma(t)}}$, $i\in\mathcal{V}$, $l\in\mathcal{H}_{\sigma(t)}$ where $\mathcal{H}_{\sigma(t)}=\{1,\cdots,M_{\sigma(t)}\}$ is a edge set. Besides, edge $(i,j)\in\mathcal{E}_{\sigma(t)}$, $j\in\mathcal{N}_{i}(t)$ is described as edge $l\in\mathcal{H}_{\sigma(t)}$. For the connected graph, the edge Laplacian matrix $\mathcal{L}_{\sigma(t)}^{e}=B_{\sigma(t)}^{{\sf\tiny T}}B_{\sigma(t)}$ is positive semidefinite [15, 16].

Suppose there exists an infinite, bounded and contiguous time interval $[t_{k},t_{k+1})$, $k\in\mathbb{N}$ where $t_{0}=0$ and $t_{k+1}-t_{k}\leq\nu$ for positive constant $\nu$. During each interval $[t_{k},t_{k+1})$, there is a sequence of nonoverlapping subintervals $[t_{k}^{0},t_{k}^{1}),\cdots,[t_{k}^{q},t_{k}^{q+1}),\cdots,[t_{k}^{m_{k}-1},t_{k}^{m_{k}})$, with $t_{k}^{0}=t_{k}$ and $t_{k}^{m_{k}}=t_{k+1}$, satisfying $t_{k}^{q+1}-t_{k}^{q}\geq\tau>0$, $0\leq q\leq m_{k}$, and $\tau$ is called dwell time. The graph $\mathcal{G}_{\sigma(t)}$ is fixed for $t\in[t_{k}^{q},t_{k}^{q+1})$. If the union graph $\cup_{q=0}^{m_{k}}\mathcal{G}_{\sigma(t_{k}^{q})}$ is connected, then $\mathcal{G}_{\sigma(t)}$ is jointly connected.

Partly referring to [9, 10], the prescribed performance is achieved if element $s_{l}(t)$ of tracking errors $s(t)\in\mathbb{R}^{M_{\sigma(t)}}$ evolves within the following performance bounds

where performance function $\rho_{s,l}(t)=(\rho_{s,l}^{0}-\rho_{s,l}^{\infty})e^{-\mathfrak{g}t}+\rho_{s,l}^{\infty}$ with positive constants $\rho_{s,l}^{0}$, $\rho_{s,l}^{\infty}$ and $\mathfrak{g}$. Next, we normalize $s_{l}(t)$ by $\hat{s}_{l}(t)=\frac{s_{l}(t)}{{\rho_{s,l}(t)}}$, and define the prescribed performance regions $D_{\hat{s}_{l}}\triangleq\left\{\hat{s}_{l}(t)\in{\mathbb{R}}:\hat{s}_{l}(t)\in(-1,1)\right\}$ with $D_{\hat{s}}\triangleq D_{\hat{s}_{1}}\times D_{\hat{s}_{2}}\times\cdots\times D_{\hat{s}_{M_{\sigma(t)}}}$. Let $\hat{s}=[\hat{s}_{1},\cdots,\hat{s}_{M_{\sigma(t)}}]^{{\sf\tiny T}}$ and $\rho_{s}(t)={\rm diag}\{\rho_{s,1}(t),\cdots,\rho_{s,{M_{\sigma(t)}}}(t)\}$. Thus $\hat{s}_{l}(t)\in D_{\hat{s}_{l}}$, $t\geq 0$ is equivalent to (1).

The normalized value $\hat{s}_{l}(t)$ is transformed by tracking error transformation functions that define a smooth and strictly increasing mapping $\varepsilon_{s,l}:D_{\hat{s}_{l}}\to(-\infty,\infty)$, with $\varepsilon_{s,l}(0)=0$. More specifically, one can select transformation function as

The time derivative of $\varepsilon_{s,l}(\hat{s}_{l})$ is

with $J_{s,l}(\hat{s}_{l},t)=\frac{2}{1-\hat{s}_{l}^{2}}>0$ and $\alpha_{s,l}(t)=-\frac{\dot{\rho}_{s,l}(t)}{\rho_{s,l}(t)}$. Let $\alpha_{s}(t)={\rm diag}\{\alpha_{s,1}(t),\cdots,\alpha_{s,{M_{\sigma(t)}}}(t)\}$. By calculating, the following inequality

holds for a positive constant $\bar{\alpha}_{s}$. In view of the properties of transformation function $\varepsilon_{s,l}(\hat{s}_{l})$, with constants $\zeta_{s,1},\zeta_{s,2}>0$, the following inequality

holds[12], where $s=[s_{1},\cdots,s_{M_{\sigma(t)}}]^{{\sf\tiny T}}$, $\hat{s}=[\hat{s}_{1},\cdots,\hat{s}_{M_{\sigma(t)}}]^{{\sf\tiny T}}$, $\varepsilon_{s}(\hat{s})=[\varepsilon_{s,1}(\hat{s}_{1}),\cdots,\varepsilon_{s,{M_{\sigma(t)}}}(\hat{s}_{M_{\sigma(t)}})]^{{\sf\tiny T}}$, and $J_{s}(\hat{s},t)={\rm diag}\{J_{s,1}(\hat{s}_{1},t),\cdots,J_{s,{M_{\sigma(t)}}}(\hat{s}_{M_{\sigma(t)}},t)\}$.

Consider a MAS with $N$ agents modeled by double-integrator dynamic

where ${x}_{i}(t),v_{i}(t),u_{i}(t)\in\mathbb{R}$ represent the position, velocity and control input of agent $i$, respectively.

Based on the matrix $B_{\sigma(t)}$, we define relative position

and relative velocity

where edge $l$ is connected between agents $i$ and $j$. Letting $x=[x_{1},\cdots,x_{N}]^{{\sf\tiny T}}$, $v=[v_{1},\cdots,v_{N}]^{{\sf\tiny T}}$, $y=[y_{1},\cdots,y_{M_{\sigma(t)}}]^{{\sf\tiny T}}$ and $z=[z_{1},\cdots,z_{M_{\sigma(t)}}]^{{\sf\tiny T}}$, we have $y=B_{\sigma(t)}^{{\sf\tiny T}}x$ and $z=B_{\sigma(t)}^{{\sf\tiny T}}v$.

In the following, the performance specifications are imposed on tracking errors $y_{l}(t)$ and $z_{l}(t)$. For convenience, the symbol ${\rm s}$ of section II-B is replaced with ${\rm y}$ and ${\rm q}$ respectively.

Objective: For the MAS (6) with switching topologies $\mathcal{G}_{\sigma(t)}$ and performance functions $\rho_{y,l}(t)$ and $\rho_{z,l}(t)$, design a distributed control protocol $u_{i}(t)$ to achieve consensus in the sense of

while transient performance $\hat{y}(t)\in D_{\hat{y}}$ and $\hat{z}(t)\in D_{\hat{z}}$, $t\geq 0$ are guaranteed.