Almost Output Synchronization for Multi-agent Systems with Non-identical Agents Under Time-Varying Topologies

Abstract

The synchronization problem of multi-agent systems with non-identical general agents under time-varying topologies and in the presence of external disturbances has not been solved for the introspective agents, i.e., agents have access to parts of their own states. This paper aims to tackle this problem. It is assumed that the time-varying topology switches among a certain amount of topologies with any a priori given dwell time and each topology contains a directed spanning tree. This paper proposes a family of distributed protocols for each agent only using relative information from its neighboring agents and some of its own states, such that synchronization can be achieved among agents while the effect of disturbances with finite power on the norm of all agents’ output disagreement can be suppressed as much as possible. It should be emphasized that agents’ controller states are exempted from the protocol design.

Proof of Lemma 1

Proof

Let $n_q\ge \max _{i=1,\ldots ,N}(n_{qi})$. Then, according to Sannuti et al. (2014, Theorem 1), there exists a pre-compensator to make agent $i\in \{1,\ldots ,N\}$ invertible and of equal rank $n_{q}$,

$$\begin{aligned} \left\{ \begin{array}{cl} {\dot{x}}_{p,i}^{1}&{}=A_{p,i}^{1}x_{p,i}^{1}+B_{p,i}^{1}u_{i}^{1},\\ {\bar{u}}_{i}&{}=C_{p,i}^{1} x_{p,i}^{1}x_{p,i}^{1}+D_{p,i}^{1}u_{i}^{1} , \end{array}\right. \end{aligned}$$

(1)

where $u_{i}^{1}\in {\mathbb {R}}^{p}$. Next, we concentrate on transforming different invertible system dynamics of equal rank to almost identical ones.

Let

$$\begin{aligned} {\tilde{x}}_{i}= \begin{pmatrix} {\bar{x}}_{i} \\ x_{p,i}^{1} \end{pmatrix}. \end{aligned}$$

There always exist nonsingular state transformation $\Gamma _{i,x}$ and input transformation $\Gamma _{i,u}$, see Sannuti and Saberi (1987), such that

$$\begin{aligned} {\tilde{x}}_{i}=\Gamma _{i,x}x_{i}, \qquad u_{i}^{1}=\Gamma _{i,u}u_{i}^{2}. \end{aligned}$$

(2)

where

$$\begin{aligned} x_{i}:=\begin{pmatrix} x_{i,a} \\ x_{i,d} \end{pmatrix}. \end{aligned}$$

Then, the interconnection of (1) and (1) can be written in the special form,

$$\begin{aligned} \left\{ \begin{array}{cl} {\dot{x}}_{i,a}&{}={\bar{A}}_{i,a}x_{i,a}+{\bar{L}}_{i,a}y_{i}+E_{i,a}{\bar{w}}_i,\\ {\dot{x}}_{i,d}&{}=Ax_{i,d}+B(u_{i}^{2} +D_{i,a}x_{i,a} +D_{i,d}x_{i,d})+E_{i,d}{\bar{w}}_i,\\ y_{i}&{}=Cx_{i,d}, \end{array}\right. \nonumber \\ \end{aligned}$$

(3)

where $A,\, B,\, C$ are as defined in (6). Note that there is an output injection for the zero dynamics. Therefore, we can have internal stability even if the system is not minimum phase.

Note that the information

$$\begin{aligned} {\bar{z}}_{m,i}:=\begin{pmatrix} z_{m,i} \\ x_{p,i}^{1} \end{pmatrix} \end{aligned}$$

is available for agent i, and ${\bar{z}}_{m,i}$ can be represented in terms of $x_{i,a},\,x_{i,d}$ as

$$\begin{aligned} {\bar{z}}_{m,i}={\bar{C}}_{m,i}\begin{pmatrix} x_{i,a}\\ x_{i,d}\end{pmatrix},\quad \text {where}\,\,{\bar{C}}_{m,i}=\begin{pmatrix} C_{m,i} &{}0\\ 0&{} I\end{pmatrix}\Gamma _{i,x}. \end{aligned}$$

We define that, for $i=1,\ldots ,N$,

$$\begin{aligned} {\bar{A}}_{i}=\begin{pmatrix} {\bar{A}}_{i,a} &{} {\bar{L}}_{i,a}C\\ BD_{i,a} &{} A+BD_{i,d}\end{pmatrix},\quad {\bar{B}}_{i}=\begin{pmatrix} 0\\ B\end{pmatrix}. \end{aligned}$$

Assumption 1 implies that $(C_{m,i},\,A_{i})$ is observable, which yields that $({\bar{C}}_{m,i},\,{\bar{A}}_{i})$ is observable. We then design an observer-based pre-compensator for the system (3) as

$$\begin{aligned} \left\{ \begin{array}{cl} \dot{\hat{{\bar{x}}}}_{i}&{}={\bar{A}}_{i}\hat{{\bar{x}}}_{i} +{\bar{B}}_{i}u_{i}^{2} -{\bar{K}}_{i}({\bar{z}}_{m,i}-{\bar{C}}_{m,i}\hat{{\bar{x}}}_{i}),\\ u_{i}^{2}&{}=\begin{pmatrix} -D_{i,a}\,\,\,&{} R-D_{i,d}\end{pmatrix} \hat{{\bar{x}}}_{i}+Mu_{i}, \end{array}\right. \end{aligned}$$

(4)

where $u_{i}\in {\mathbb {R}}^{p}$, ${\bar{K}}_{i}$ is chosen such that ${\bar{A}}_{i}+{\bar{K}}_{i}{\bar{C}}_{m,i}$ is Hurwitz stable, R is chosen such that $A+BR$ has desired eigenvalues in the open left half plane, and M is an arbitrary and non-singular matrix. Define the observer error ${\tilde{x}}_{i}=x_{i}-\hat{{\bar{x}}}_{i}$. Notice that the observer error dynamics is asymptotically stable. Moreover, the effect of $x_{i,a}$ on the dynamics $x_{i,d}$ is asymptotically canceled. Thus, the mapping from the new input $u_{i}$ to the output $y_{i}$ is given by

$$\begin{aligned} \left\{ \begin{array}{cl} {\dot{x}}_{i,d}&{}=(A + BR)x_{i,d}+B_{d}Mv_{i}+\rho _{i}+E_{i,d}{\bar{w}}_i,\\ y_{i}&{}=C {\bar{x}}_{i,d}, \end{array}\right. \end{aligned}$$

(5)

and the observer error dynamics is written as,

$$\begin{aligned} \left\{ \begin{array}{cl} \dot{{\tilde{x}}}_{i}&{}=({\bar{A}}_{i}+{\bar{K}}_{i}{\bar{C}}_{m,i}){\tilde{x}}_{i}+\begin{pmatrix}E_{i,a}\\ E_{i,d} \end{pmatrix}{\bar{w}}_i,\\ \rho _{i}&{}=B_d\begin{pmatrix} D_{i,a}\,\,\,&{} D_{i,d}-R\end{pmatrix}{\tilde{x}}_{i}. \end{array}\right. \end{aligned}$$

(6)

Then, let $x_i$ be $x_{i,d}$, which results in the dynamics in (8). Moreover, $H_i$, $E_{o,i}$, and $W_i$ can be found in (6). $\square$