# Event-Triggered $\ell_{2}$-Optimal Formation Control with
State-Estimation for Agents Modeled as LPV Systems

Gerald Gebhardt, Hamideh Saadabadi and Herbert Werner Herbert Werner is with
the Institute of Control Systems, Hamburg University of Technology,
Eissendorfer Str. 40, 21073 Hamburg, Germany<EMAIL_ADDRESS>Saadabadi
is with the Institute of Control Systems, Hamburg University of Technology,
Eissendorfer Str. 40, 21073 Hamburg, Germany<EMAIL_ADDRESS>Gebhardt is with the Institute of Control Systems, Hamburg University of
Technology, Eissendorfer Str. 40, 21073 Hamburg, Germany
<EMAIL_ADDRESS>
###### Abstract

This paper proposes a distributed scheme with different estimators for the
event-triggered formation control of polytopic homogeneously scheduled linear
parameter-varying (LPV) multi-agent systems (MAS). Each agent consists of a
time-triggered inner feedback loop and a larger event-triggered outer feedback
loop to track a formation reference signal and reject input and output noise.
If a local event-trigger condition is violated, the event-triggered outer
feedback loop is closed through the communication network. The event-trigger
condition is only based on locally available information. To design the
controller, a synthesis problem is formulated as a linear matrix inequality of
the size of a single agent under the assumption, that local estimators trigger
intercommunication events with neighboring agents if the event-trigger
condition is violated. The design procedure guarantees stability and bounded
${\ell_{2}}$-performance. Furthermore, the estimators are interchangeable for
a given controller. We compare in simulation zero-order hold, open-loop
estimation, and closed-loop estimation strategies. Simulation trials are
carried out with non-holonomic dynamic unicycles modeled as polytopic LPV
systems.

##  I Introduction

Cooperative control of MAS has been widely studied, and the many applications
include sensor networks, formation control of vehicles and swarm robotics
[1]-[2]. Autonomous vehicles or mobile robots are often subject to non-
holonomic constraints. The dynamics of such vehicles cannot be represented by
linear time-invariant systems, but they can be modeled as polytopic LPV
systems [3].

The number of intercommunication events for MAS can be unnecessarily high if
the agents communicate at fixed time intervals determined by the sampling rate
of a digital controller [4]-[5]. However, it can be reasonably reduced through
the application of distributed event-triggered control strategies. In [6]-[23]
the authors propose centralized and decentralized event-triggered formation
control or consensus for LTI MAS. In [24] a MAS control problem is modelled
via decomposable systems.

The event-triggered control of LPV systems is the subject of [25]-[40]. In
[27]-[29] the design of an output feedback and event-triggered state feedback
controller for discrete-time LPV systems with LMI conditions for bounded
$\ell_{2}$-performance is studied. In [30] event-triggered $H_{\infty}$ state-
feedback control for LPV systems with time triggered sensors is considered.
This approach is extended in [31] to event-triggered output feedback control.
The event-triggered control of continuous-time switched single loop LPV
systems is considered in [32]-[33]. Event-triggered fault detection schemes
are proposed in [34]-[35] for LPV systems.

In addition to event-triggered control, local state estimation can reduce the
number of intercommunication events rapidly and reasonably [41]-[46]. In [41]
sampled data is used for event-triggered state estimation with bounded
estimation error for a complex network. In [42] event-triggered, observer-
based estimation is applied to reject disturbances and track references for
discrete-time LTI systems. In [43] event-triggered distributed state
estimation is used for a class of uncertain stochastic systems, which are
subject to state-dependent noises and uncertainties. In [44] the optimal
trade-off between the expected number of transmissions and the mean square
estimation error is found over a finite horizon for an event-triggered
estimation strategy for LTI systems. In [45] moving horizon event-triggered
state estimation is used for an LTI MAS subject to noise and disturbances. In
[46] an observer-based optimal event-triggered control strategy for LTI
systems is proposed. In [47] event-triggered open-loop estimation is used for
a undirected continuous LTI MAS.

To the best of the authors’ knowledge, so far no results on distributed event-
triggered open-loop and fully connected closed-loop estimator-based control of
LPV agent networks have been reported.

The contribution of this paper is a method for co-designing an event-triggered
distributed LPV formation controller together with a trigger condition that
depends only on locally available information, with guaranteed stability,
$\ell_{2}$ performance, and interchangeable estimators. The gain-scheduled
state feedback matrices are found via solving LMIs of the size of a single
agent. The information transmitted to the neighbors depends on the applied
estimator. The estimators require a certain communication structure and signal
availability. Zero-order hold and open-loop estimation are considered for
arbitrary communication graphs which are assumed to be undirected and
connected, and closed-loop estimation is considered for a fully connected
graph, in contrast to [37], where only the zero-order hold estimation scheme
is investigated. The open-loop and closed-loop estimators also estimate the
scheduling parameters. For the proposed design procedure, it is assumed that
the scheduling is homogeneous. This assumption is approximately satisfied for
non-holonomic agents which move in formation. A simulation example suggests
that the proposed scheme will still work well even when this assumption is
violated through a locally acting disturbance.

The remainder of this paper is organized as follows: The preliminaries and
notation are given in section II. The problem is formulated, and the
estimators are proposed in section III. Section IV presents a sufficient LMI
condition for the considered problem. A non-holonomic system is introduced in
section V and the corresponding simulation results are given in section VI.
Finally in section VII conclusions are presented.

##  II Preliminaries and Notation

For a signal $x_{k}$ taking values in $\mathbb{R}^{n_{x}}$, the Euclidean
2-norm at time $k$ is $\left\lVert
x_{k}\right\rVert_{2}=\sqrt{x_{k}^{\top}x_{k}}$ , and the signal
${\ell_{2}}$-norm is defined via $\left\lVert
x_{k}\right\rVert_{\ell_{2}}=\sqrt{{\sum^{\infty}_{k=0}\left\lVert
x_{k}\right\rVert_{2}^{2}}.}$ An $m\times n$ matrix, which consists only of
zeros, is denoted by $0_{m\times n}$. If the dimensions of a zero matrix are
clear from context, the index is omitted. The $i^{\text{th}}$ unit vector
$q_{i}\in\mathbb{R}^{N}$ is a column vector, which consists of zeros, except
for the $i^{\text{th}}$ entry, which is equal to $1$. A column vector with $N$
rows, which entries are all equal to $1$, is denoted by $\mathbf{1}_{N}$. For
two matrices $A\in\mathbb{R}^{m\times n}:=[a_{ij}]$ and
$B\in\mathbb{R}^{p\times q}$ the Kronecker product between $A$ and $B$ is
defined as $A\otimes B:=[a_{ij}B]\in~{}\mathbb{R}^{mp\times nq}.$ Properties
of the Kronecker product are $(A\otimes B)^{\top}=A^{\top}\otimes B^{\top}$
and $(A\otimes B)(C\otimes D)=(AC)\otimes(BD).$ A Kronecker product between a
matrix $M\in\mathbb{R}^{N\times N}$ and the $n^{\text{th}}$ degree identity
matrix $I_{n}$ is expressed as $M_{(n)}=M\otimes I_{n}.$ Block diagonal
matrices can be represented as $D=\text{diag}(D_{1},\ldots,D_{N})$, where
$D_{i}$ is a scalar or a matrix.

The positive semi-definite and symmetric Laplacian $\mathcal{L}$ corresponds
to the undirected and connected graph $\mathcal{G}$. The column space
$\mathcal{R}(\mathbf{1}_{N})$, which is the null space of the Laplacian
$\mathcal{L}$, is the agreement space of the MAS and corresponds to the
Laplacian’s smallest and only non-positive eigenvalue
$0=\lambda_{1}(\mathcal{L})<\lambda_{2}(\mathcal{L})\leq\ldots\leq\lambda_{N}(\mathcal{L})$
[48].

##  III Problem Formulation

###  III-A Agent Model

Consider a group of $N$ agents with identical nonlinear dynamics, each modeled
as polytopic LPV system $P(\theta_{k}^{i})$ with state space realization

$\displaystyle x_{k+1}^{i}$
$\displaystyle=A(\theta_{k}^{i})x_{k}^{i}+B_{w}(\theta_{k}^{i})w_{k}^{i}+B_{u}(\theta_{k}^{i})u_{k}^{i},$
$\displaystyle y_{k}^{i}$
$\displaystyle=C_{y}(\theta_{k}^{i})x_{k}^{i},\;i=1,\ldots,N,$ (1)

where $x^{i}_{k}\in\mathbb{R}^{n_{x}}$, $w^{i}_{k}\in\mathbb{R}^{n_{w}}$,
$u^{i}_{k}\in\mathbb{R}^{n_{u}}$ and $y^{i}_{k}\in\mathbb{R}^{n_{y}}$, denote
the state, perturbation, input and transmitted output at time $k$. The system
$\check{P}(\theta_{k})$ containing the dynamics of all agents is expressed via

$\displaystyle x_{k+1}$
$\displaystyle=\check{A}(\theta_{k})x_{k}+\check{B}_{u}(\theta_{k})u_{k}+\check{B}_{w}(\theta_{k})w_{k},$
$\displaystyle y_{k}$ $\displaystyle=\check{C}_{y}(\theta_{k})x_{k},$ (2)

where stacked vectors, e.g.
$x_{k}=\left[(x_{k}^{1})^{\top}(x_{k}^{2})^{\top}\cdots(x_{k}^{N})^{\top}\right]^{\top},$
and diagonal matrices, e.g.

$\check{A}(\theta_{k})=\text{diag}\left(A(\theta^{1}_{k}),A(\theta^{2}_{k}),\ldots,A(\theta^{N}_{k})\right),$

are used. The model matrices, e.g. $A(\theta^{i}_{k})$ and
$B_{w}(\theta^{i}_{k})$, depend affinely on the time-varying vector of
scheduling variables $\theta^{i}_{k}$ of agent $i$. The scheduling variables
$\theta^{i}_{k}$ are restricted to a compact set $\Theta$ via
$\theta^{i}_{k}\in\Theta\subset\mathbb{R}^{n_{\theta}}$ at all times. We
assume that the parameter set

$\Theta=\left\\{\theta\in\mathbb{R}^{n_{\theta}}|\theta=\sum_{l=1}^{s}\alpha_{l}\theta_{l},\sum_{l=1}^{s}\alpha_{l}=1,\alpha_{l}\geq
0\right\\}$

is represented as a polytope in terms of vertex vectors
$\theta_{l}~{}\in~{}\mathbb{R}^{n_{\theta}}$. This restriction is represented
via $\theta\in\mathcal{F}_{\Theta}$. Defining vertex model matrices, e.g.
$A^{l}=A(\theta_{l})$, the LPV model matrices for agent $i$ can be expressed
in terms of the convex coordinates $\alpha_{l}(\theta_{k}^{i})$ as

$\left[\begin{array}[]{ccc}A(\theta^{i}_{k})&B_{w}(\theta^{i}_{k})&B_{u}(\theta^{i}_{k})\\\
C_{y}(\theta^{i}_{k})&0&0\\\
\end{array}\right]=\sum_{l=1}^{s}\alpha_{l}(\theta^{i}_{k})\left[\begin{array}[]{ccc}A^{l}&B_{w}^{l}&B_{u}^{l}\\\
C_{y}^{l}&0&0\end{array}\right].$ (3)

###  III-B Event-Triggered Formation Control

The group of $N$ agents, governed by (2), can communicate through an
undirected and connected graph $\mathcal{G}$. Every agent estimates the output
$\hat{y}^{j}_{k}$ of its neighbors $j\in\mathcal{N}_{i}$ and its own output
$\hat{y}^{i}_{k}$ in the same way as its neighbors do, to calculate the
estimated formation error

$\hat{\eta}^{i}_{k}=\sum_{j\in{\cal
N}_{i}}((r^{i}_{k}-\hat{y}_{k}^{i})-(r^{j}_{k}-\hat{y}_{k}^{j}))=(q_{i}^{\top}\mathcal{L})_{(n_{y})}(r_{k}-\hat{y}_{k})$
(4)

locally, where $r_{k}^{i}$ is the reference, which should be tracked. The
control structure for agent $i$ is shown in Figure 1.

ET
---
$\frac{1}{z-1}$
---
$\zeta^{i}_{k}$
---
$\hat{\eta}^{i}_{k}$
---
$F_{\zeta}(\theta_{k}^{i})$
---
$({q}_{i}^{\top}\mathcal{L})_{(n_{y})}$
---
$P(\theta_{k}^{i})$
---
$y^{i}_{k}$
---
$w^{i}_{k}$
---
$F_{x}(\theta_{k}^{i})$
---
$x_{k}^{i}$
---
$\hat{\eta}^{i}_{k}$
---
$r_{k}-\hat{y}_{k}$
---
$\kappa^{i}_{k}$
---
$u^{i}_{k}$
---
$\hat{y}_{k}$
---
Figure 1: Control structure for a single agent

The estimated formation error $\hat{\eta}_{k}$ is used to approximate the
formation error ${\eta}_{k}=\mathcal{L}_{(n_{y})}(r_{k}-{y}_{k}).$ Figure 2
shows the closed-loop network, where the cost $z_{k}$ and also estimators are
pictured.

$W_{z}\mathcal{L}_{(n_{y})}$
---
ET
---
$\zeta_{k}$
---
$\check{F}_{\zeta}(\theta_{k})$
---
$\check{P}(\theta_{k})$
---
$w_{k}$
---
$\check{F}_{x}(\theta_{k})$
---
$x_{k}$
---
$\hat{\eta}_{k}$
---
$u_{k}$
---
$\hat{y}_{k}$
---
Estimator
---
$\kappa_{k}$
---
$\hat{y}_{k}$
---
$y_{k}$
---
$-$
---
$r_{k}$
---
$\frac{1}{z-1}\mathcal{L}_{(n_{y})}$
---
$-$
---
$z_{k}$
---
Figure 2: Control structure of the closed-loop network with performance output
and estimator

The ”ET”-block in Figure 1 evaluates the local trigger condition

$(e^{i}_{k})^{\top}e^{i}_{k}\leq\sigma({\hat{\eta}_{k}^{i}})^{\top}\hat{\eta}_{k}^{i},$
(5)

where $e^{i}_{k}=\hat{y}^{i}_{k}-y^{i}_{k}$ is the estimation error and
$\sigma>0$ is the trigger level. If the current estimate $\hat{y}_{k}^{i}$
violates the event-trigger condition, a message $\kappa^{i}_{k}$ is sent. This
sets the estimation error $e_{k}^{i}$ to zero. A message consists of necessary
information to locally formulate new identical estimates $\hat{y}^{i}_{k+1}$,
$\hat{y}^{i}_{k+2}$, $\ldots$ by every agent $j\in\mathcal{N}_{i}\cup i$.
Three different estimators are presented in subsection III-C. The integrated
formation error

$\zeta^{i}_{k+1}=\zeta^{i}_{k}+\hat{\eta}^{i}_{k}$ (6)

is used together with the agent’s states $x^{i}_{k}$ for gain-scheduled state
feedback via

$u_{k}^{i}=F_{\zeta}(\theta_{k}^{i}){\zeta}_{k}^{i}+{F}_{x}(\theta_{k}^{i}){x}_{k}^{i}.$
(7)

Since the agents are affine in $\theta_{k}^{i}$, affine parameter dependence
is imposed on the controllers, too, i.e.

$F_{x}(\theta_{k}^{i})=\sum_{l=1}^{s}\alpha_{l}(\theta_{k}^{i})F_{x}^{l},\;F_{\zeta}(\theta_{k}^{i})=\sum_{l=1}^{s}\alpha_{l}(\theta_{k}^{i})F_{\zeta}^{l},$
(8)

where $F_{x}^{l}=F_{x}(\theta_{l})$ and $F_{\zeta}^{l}=F_{\zeta}(\theta_{l})$
denote the vertex controllers at vertex $l$. Thus, the input signal can be
expressed via

$u_{k}=\check{F}_{\zeta}(\theta_{k}){\zeta}_{k}+\check{F}_{x}(\theta_{k}){x}_{k}.$
(9)

Since the control objective is to design a controller which follows a
reference input $r_{k}$ and rejects the disturbance $w_{k}$ the control
performance can be measured with the performance output

$\displaystyle z_{k}$
$\displaystyle=w_{z}(k)\ast\mathcal{L}_{(n_{y})}(r_{k}-y_{k}),$ (10)

where $(\ast)$ denotes the discrete convolution and
$W_{z}(z)=\frac{\beta_{z}}{z+\alpha_{z}}$ is the $z$-transformed filter
$w_{z}(k)$ with variables $\alpha_{z}$ and $\beta_{z}$ as design parameters.
The parameter $\alpha_{z}$ is used to set a weight on the latest cost function
value. With (2) the performance output $z_{k}$ can be expressed as state space
model

$\displaystyle\tilde{z}_{k+1}$
$\displaystyle=-\alpha_{z}\tilde{z}_{k}+\beta_{z}r_{k}-\beta_{z}\check{C}_{y}(\theta_{k})x_{k},\;z_{k}=\mathcal{L}\tilde{z}_{k},$
(11)

with the local performance $\tilde{z}_{k}$ as state vector. (III-A), (4) and
(6) lead to

$\displaystyle\hat{\eta}_{k}$
$\displaystyle=\mathcal{L}_{(n_{y})}(r_{k}-\check{C}_{y}(\theta_{k})x_{k}-e_{k}),$
$\displaystyle\zeta_{k+1}$
$\displaystyle=\zeta_{k}+\mathcal{L}_{(n_{y})}(r_{k}-\check{C}_{y}(\theta_{k})x_{k}-e_{k}).$
(12)

Thus, (III-A), (9), (11) and (12) can be combined to form

$\displaystyle\left[\begin{array}[]{ccc|c|c}x_{k+1}^{\top}&\zeta_{k+1}^{\top}&\tilde{z}_{k+1}^{\top}&\hat{\eta}_{k}^{\top}&z_{k}^{\top}\end{array}\right]^{\top}=$
(14)
$\displaystyle\left[\begin{array}[]{ccc|c|ccc}\check{A}+\check{B}_{u}\check{F}_{x}&\check{B}_{u}\check{F}_{\zeta}&0&0&0&\check{B}_{w}\\\
-\mathcal{L}_{(n_{y})}\check{C}_{y}&I_{Nn_{y}}&0&-\mathcal{L}_{(n_{y})}&\mathcal{L}_{(n_{y})}&0\\\
-\beta_{z}\check{C}_{y}&0&-\alpha_{z}I_{Nn_{y}}&0&\beta_{z}I_{Nn_{y}}&0\\\
\hline\cr-\mathcal{L}_{(n_{y})}\check{C}_{y}&0&0&-\mathcal{L}_{(n_{y})}&\mathcal{L}_{(n_{y})}&0\\\
\hline\cr 0&0&\mathcal{L}_{(n_{y})}&0&0&0\end{array}\right]$ (20)
$\displaystyle\cdot$
$\displaystyle\left[\begin{array}[]{ccc|c|cc}x_{k}^{\top}&\zeta_{k}^{\top}&\tilde{z}_{k}^{\top}&e_{k}^{\top}&r_{k}^{\top}&w_{k}^{\top}\end{array}\right]^{\top},$
(22)

where the dependence on the scheduling parameter $\theta_{k}$ is omitted for
ease of notation. (22) can be abbreviated via

$\displaystyle\underbrace{\left[\begin{array}[]{c}\psi_{k+1}\\\
\hline\cr\hat{\eta}_{k}\\\ \hline\cr
z_{k}\end{array}\right]}_{=\nu_{k}}=\underbrace{\left[\begin{array}[]{c|c|c}\bar{A}_{\text{CL}}&\bar{B}_{e}&\bar{B}_{{{f}}}\\\
\hline\cr\bar{C}_{\hat{\eta}}&\bar{D}_{\hat{\eta}e}&\bar{D}_{\hat{\eta}{{f}}}\\\
\hline\cr\bar{C}_{z}&\bar{D}_{ze}&\bar{D}_{z{{f}}}\end{array}\right]}_{=\bar{H}(\theta_{k})}\underbrace{\left[\begin{array}[]{c}\psi_{k}\\\
\hline\cr e_{k}\\\ \hline\cr f_{k}\end{array}\right]}_{=\phi_{k}},$ (32)

where

$\psi_{k}=\left[x_{k}^{\top}\zeta_{k}^{\top}\tilde{z}_{k}^{\top}\right]^{\top}\in\mathbb{R}^{N\cdot
n_{\psi}},\;f_{k}=\left[r_{k}^{\top}w_{k}^{\top}\right]^{\top}\in\mathbb{R}^{N\cdot
n_{f}},$

$n_{\psi}=n_{x}+n_{y}+n_{z}$ and $n_{f}=n_{y}+n_{w}$. The horizontal $(-)$ and
vertical $(\,|\,)$ lines make the correlations between (22) and (32)
unambiguous. The closed-loop matrix

$\displaystyle\bar{A}_{\text{CL}}$
$\displaystyle=\underbrace{\left[\begin{array}[]{ccc}\check{A}&0&0\\\
-\mathcal{L}_{(n_{y})}\check{C}_{y}&I_{Nn_{y}}&0\\\
-\beta_{z}\check{C}_{y}&0&-\alpha_{z}I_{Nn_{y}}\end{array}\right]}_{=\bar{A}_{\text{OL}}}+\underbrace{\left[\begin{array}[]{c}\check{B}_{u}\\\
0\\\
0\end{array}\right]}_{=\bar{B}_{F}}\underbrace{\left[\begin{array}[]{c}\check{F}_{x}^{\top}\\\
\check{F}_{\zeta}^{\top}\\\ 0\end{array}\right]^{\top}}_{=\bar{F}}$ (42)

introduces feedback via $\bar{F}$. In equation (42), $I_{Nn_{y}}$ denotes the
identity matrix of degree $N\cdot n_{y}$.

###  III-C Estimation

The proposed zero-order hold estimation, open-loop estimation and closed-loop
estimation schemes require different information to formulate new estimates
locally.

####  III-C1 Zero-Order Hold Estimation

For zero-order hold estimation, the necessary information is the output
$\kappa^{i}_{k}=y^{i}_{k}$, since $\hat{y}^{i}_{k+1}=\hat{y}^{i}_{k}$ is the
estimate, if no event-triggering occurs at agent $i$. This estimate is updated
to $\hat{y}^{i}_{k}=y^{i}_{k}=\kappa^{i}_{k}$, if an event is triggered at
time $k$.

####  III-C2 Open-Loop Estimation

For open-loop estimation, agent $i$ sends a message
$\kappa^{i}_{k}=[(x_{k}^{i})^{\top}(\theta^{i}_{k})^{\top}]^{\top}$ to its
neighbors when the event-trigger condition is violated, since the local
estimation is based upon the prediction of the next scheduling parameter
$\hat{\theta}^{i}_{k+1}=f(\hat{\theta}^{i}_{k},\hat{x}^{i}_{k})$ and the
output prediction via

$\displaystyle\left[\begin{array}[]{c}\hat{x}_{k+1}\\\
\hline\cr\hat{y}_{k}\end{array}\right]=\left[\begin{array}[]{c}A(\hat{\theta}^{i}_{k})\\\
\hline\cr C_{y}(\hat{\theta}^{i}_{k})\end{array}\right]\hat{x}^{i}_{k}.$ (47)

The function $f(\hat{\theta}^{i}_{k},\hat{x}^{i}_{k})$ is a combination of the
locally available parameters $\hat{\theta}^{i}_{k}$ and $\hat{x}^{i}_{k}$ to
predict the next scheduling parameter $\hat{\theta}^{i}_{k+1}$.

####  III-C3 Closed-Loop Estimation for Fully Connected Network

For the proposed closed-loop estimator, the network needs to be fully
connected, i.e. $\mathcal{L}=NI_{N}-\mathbf{1}_{N}\mathbf{1}_{N}^{\top}$,
since every agent estimates the full network and is required to produce the
same estimate $\hat{y}_{k}$ as its neighbors. Furthermore, the necessary
information for this estimation scheme are $\hat{\theta}_{k},$ $\hat{x}_{k}$,
$\hat{\zeta}_{k}$ and $r_{k}$, since output estimates $\hat{y}_{k}$ are
generated via
$\hat{\theta}_{k+1}=f(\hat{\theta}_{k},\hat{x}_{k},\hat{\zeta}_{k},r_{k})$ and

$\displaystyle\left[\begin{array}[]{cc|c}\hat{x}_{k+1}^{\top}&\hat{\zeta}_{k+1}^{\top}&\hat{y}_{k}^{\top}\end{array}\right]^{\top}=$
(49)
$\displaystyle\left[\begin{array}[]{cc|ccccc}\check{A}(\hat{\theta}_{k})+\check{B}_{u}(\hat{\theta}_{k})\check{F}_{x}(\hat{\theta}_{k})&\check{B}_{u}(\hat{\theta}_{k})\check{F}_{\zeta}(\hat{\theta}_{k})&0\\\
-\mathcal{L}_{(n_{y})}\check{C}_{y}(\hat{\theta}_{k})&I_{Nn_{y}}&\mathcal{L}_{(n_{y})}\\\
\hline\cr\mathcal{L}_{(n_{y})}\check{C}_{y}(\hat{\theta}_{k})&0&0\end{array}\right]\left[\begin{array}[]{c}\hat{x}_{k}\\\
\hat{\zeta}_{k}\\\ \hline\cr r_{k}\end{array}\right].$ (56)

Thus, if an event is triggered at agent $i$ and time $k$, agent $i$ sends a
message
$\kappa^{i}_{k}=[({\theta}^{i}_{k})^{\top}({x}^{i}_{k})^{\top}({\zeta}^{i}_{k})^{\top}]^{\top}$
to the other agents.

###  III-D Problem Formulation

The problem addressed in this paper is the following: For a given positive
constant $\gamma$, a network with dynamics governed by (22) and (32) and a
given estimator, which ensures that (5) is true, find the scheduled state
feedback gain matrices $F_{x}(\theta_{k}^{i})$ and $F_{\zeta}(\theta_{k}^{i})$
in (7), such that the group of $N$ agents is stable and satisfies

$\left.\left\lVert
T_{z{{f}}}\right\rVert_{\ell_{2}}\right|_{x_{0}=0}=\sup_{\theta\in\mathcal{F}_{\Theta}}\sup_{{{f}}\neq
0}\left.\frac{\left\lVert
z_{k}\right\rVert_{\ell_{2}}}{\left\lVert{{f}}_{k}\right\rVert_{\ell_{2}}}\right|_{x_{0}=0}\leq\gamma.$
(57)

##  IV Controller Synthesis

Assumption 1: The scheduling in the group is homogeneous, i.e.
$\theta_{k}^{i}=\theta_{k}^{j},\;\forall\;1\leq i\leq N,1\leq j\leq N\text{
and }k\geq 1.$

The matrix inequality

$\displaystyle\left[\begin{array}[]{cccccc}{G^{l}}^{\top}+{G^{l}}-{S}\\\
0&I_{n_{y}}&&&\ast\\\ 0&0&tI_{n_{f}}\\\
{A}^{li}_{\text{OL}}{G}^{l}+B_{F}^{l}K^{l}&{B}^{i}_{e}&{B}^{li}_{{{f}}}&{S}\\\
{C}^{li}_{\hat{\eta}}{G}^{l}&{D}^{i}_{\hat{\eta}e}&{D}^{i}_{\hat{\eta}{{f}}}&0&\sigma_{x}I_{n_{y}}\\\
{C}^{i}_{z}{G}^{l}&{D}_{ze}&{D}_{z{{f}}}&0&0&I_{n_{y}}\end{array}\right]>0,$
(64)

where

$\displaystyle
G^{l}=\text{diag}(G^{l}_{1},G^{l}_{2}),\>G_{1}^{l}=\left[\begin{array}[]{cc}{G}_{x}^{l}&{G}_{x\zeta}^{l}\\\
{G}_{\zeta x}^{l}&{G}_{\zeta}^{l}\end{array}\right],$ (67) $\displaystyle
K^{l}=[K^{l}_{1}0_{n_{u}\times
n_{y}}],\>K_{1}^{l}=[{K}_{x}^{l}\>{K}_{x\zeta}^{l}],\>B_{F}^{l}=\left[\begin{array}[]{c}B_{u}^{l}\\\
0_{2\cdot n_{y}\times n_{u}}\\\ \end{array}\right],$ (70) $\displaystyle
n_{f}=n_{y}+n_{w},\>t=\gamma^{2},\>\sigma_{x}=\sigma^{-1},$ (71)
$\displaystyle\left[\begin{array}[]{c|c|c}{A}^{li}_{\text{OL}}&{B}^{i}_{e}&{B}^{li}_{{{f}}}\\\
\hline\cr{C}^{li}_{\hat{\eta}}&{D}^{i}_{\hat{\eta}e}&{D}^{i}_{\hat{\eta}{{f}}}\\\
\hline\cr{C}^{i}_{z}&{D}_{ze}&{D}_{z{{f}}}\end{array}\right]=$ (75)
$\displaystyle\left[\begin{array}[]{ccc|c|ccc}{A^{l}}&0&0&0&0&{B}_{w}^{l}\\\
-\lambda_{i}{C}^{l}_{y}&I_{n_{y}}&0&-\lambda_{i}I_{n_{y}}&\lambda_{i}I_{n_{y}}&0\\\
-\beta_{z}{C}^{l}_{y}&0&-\alpha_{z}I_{n_{y}}&0&\beta_{z}I_{n_{y}}&0\\\
\hline\cr-\lambda_{i}{C}^{l}_{y}&0&0&-\lambda_{i}I_{n_{y}}&\lambda_{i}I_{n_{y}}&0\\\
\hline\cr 0&0&\lambda_{i}I_{n_{y}}&0&0&0\end{array}\right]$ (81)

is linear in $S$, $G^{l}$, $K^{l}$, $t$ and $\sigma_{x}$. In (64), ($\ast$)
denotes a matrix block that can be inferred by symmetry.

###### Theorem 1

The closed-loop MAS defined by (22) and (32), with an estimator and event-
triggering mechanism, which ensures that (5) is true and distributed feedback
governed by (8), where $(F^{l}_{x}F^{l}_{\zeta})=K^{l}_{1}{G^{l}_{1}}^{-1}$ is
stable and satisfies (57), if there exist a symmetric positive-definite matrix
$S\in\mathbb{R}^{n_{\psi}\times n_{\psi}}$ and matrices
$K^{l}_{1}\in\mathbb{R}^{n_{u}\times(n_{y}+n_{x})}$,
$G^{l}_{1}\in\mathbb{R}^{(n_{y}+n_{x})\times(n_{y}+n_{x})}$ and
$G^{l}_{2}\in\mathbb{R}^{n_{y}\times n_{y}}$ that satisfy (64) for
$l=1,2,\ldots,s$ and $i=2,N$.

Proof: See appendix

Note that the assumption of homogeneous scheduling may be violated, especially
when a perturbation acts on an individual agent. The simulations shown below
suggest, that even then the proposed scheme performs well.

##  V LPV Representation of a Non-Holonomic Vehicle

To motivate the use of LPV models as agents in formation control problems,
unicycles are used as an example for non-holonomic systems. The position of
the agent in a plane is given by the Cartesian coordinates $x(t)$ and $y(t)$
and its orientation by the angle $\phi(t)$. The non-holonomic constraint on
the mobile robot shown in Figure 3 is that only acceleration via a force $f$
and steering via a torque $\tau$ is allowed.

###  V-A Non-holonomic system with handle point

The unicycle’s dynamic equations are

$\dot{x}=v\cos(\phi),\;\dot{y}=v\sin(\phi),\;\dot{v}=\frac{f}{m},\;\dot{\phi}=\omega,\dot{\omega}=\frac{\tau}{I},$

where $m$ is the unicycle’s mass and $I$ is its moment of inertia.

$x$
---
$y$
---

$(x,y)$
---

$v_{t}$
---

$v_{n}$
---

$\tilde{x}$
---
$\tilde{y}$
---

$\phi$
---

$\phi$
---

$(x^{d},y^{d})$
---

Figure 3: Dynamic unicycle with handle

Figure 3 shows the unicycle with a fictitious handle point at position
($x^{d},y^{d}$), which is a distance $d$ apart in $\tilde{x}$-direction from
the unicycles center of mass at position ($x,y$). The handle point is
introduced to improve the system’s controllability [49]. The transformation
matrix

$T_{\phi}=\left[\begin{array}[]{cc}\cos\phi&\sin\phi\\\ -\sin\phi&\cos\phi\\\
\end{array}\right]$ (82)

is used to transform from $xy$-coordinates to
$\tilde{x}\tilde{y}$-coordinates. An LPV model of the handle is then obtained
by writing the state space model as

$\displaystyle\left[(\dot{\tilde{x}}^{d})^{\top}(\dot{\tilde{y}}^{d})^{\top}\dot{v}_{n}^{\top}\dot{v}_{t}^{\top}\right]^{\top}=$
$\displaystyle\left[\begin{array}[]{cccc}0&\frac{v_{t}}{d}&1&0\\\
-\frac{v_{t}}{d}&0&0&1\\\ 0&0&0&0\\\
0&0&0&0\end{array}\right]\left[\begin{array}[]{c}\tilde{x}^{d}\\\
\tilde{y}^{d}\\\ v_{n}\\\
v_{t}\end{array}\right]+\left[\begin{array}[]{cc}0&0\\\ 0&0\\\
\frac{1}{m}&0\\\ 0&\frac{d}{I}\end{array}\right]\left[\begin{array}[]{c}f\\\
\tau\end{array}\right],$ (97)

where $v_{n}$ and $v_{t}$ are the normal and lateral velocities and the only
scheduling parameter is $v_{t}=\theta$. Note, that (97) has the form

$\dot{x}(t)=A(\theta(t))x(t)+B_{u}u(t)$ (98)

and that $A(\theta(t))$ is affine in $\theta$. Furthermore, the matrices
$B_{w}$, $B_{u}$ and $C_{y}$ are independent of the scheduling parameter.
Thus, the design procedure proposed in section IV is applicable. Euler’s
discretization is used to discretize the model with sampling time $T_{s}$ via

${x}_{k+1}=(I+T_{s}\cdot A(\theta_{k}))x_{k}+(T_{s}\cdot B_{u})u_{k}$ (99)

for the simulations.

##  VI Simulation Results

To show the effectiveness of the proposed method, a simulation example is
provided [50]. For the simulations a network of three agents with complete
communication graph and corresponding Laplacian
$\mathcal{L}=3I_{3}-\mathbf{1}_{3}\mathbf{1}_{3}^{\top}$ is used. The
reference is $r_{k}=[1,0,0,0.5,0.5,-0.5]^{\top},$ the trigger level ist
$\sigma=10^{-3}$ and the sampling time is $T_{s}=0.01$. Between the times
$t=4$ and $t=4.5$ an $\tilde{x}$-directional input disturbance $w_{k}^{1}$ is
applied to the agent 1. The simulation results are provided by Figures 4 and
5.

$-1$$-0.5$$0$$0.5$$1$$x(t)$Agent 1Agent 2Agent
3$0$$2$$4$$6$$8$$10$$-1$$-0.5$$0$$0.5$$1$$t$$y(t)$ Figure 4: Spatial response
for ZOH, OLE and CLE
$0$$1$$2$$3$$\left\lVert\eta_{k}\right\rVert$ZOHOLECLE$0$$1$$2$$3$ZOH$0$$1$$2$$3$OLE$0$$2$$4$$6$$8$$10$$0$$1$$2$$3$$t$CLE
Figure 5: Formation error and simultaneous trigger events for ZOH, OLE and CLE

Figure 4 shows the $xy$-directional time response of the three agents for
zero-order hold, open-loop and closed loop estimation. The spatial responses
are overlapping each other in Figure 4 for the three estimators. Figure 5
displays the Euclidean 2-norm of the formation error $\eta_{k}$ and the amount
of simultaneous trigger instants per time instant. The differences in the
spatial response are negligible, but the amount of trigger events varies
significantly. This result is also supported by Table I.

TABLE I: Estimator Performance Estimator | $\overline{\left\lVert\eta\right\rVert_{2}}$ | Trigger events | Trigger rate
---|---|---|---
ZOH | 0.3852 | 362 | 0.1205
OLE | 0.3995 | 131 | 0.0436
CLE | 0.3923 | 54 | 0.0180

The mean of the formation error norm
$\overline{\left\lVert\eta\right\rVert_{2}}=\sum_{k=1}^{\frac{30}{T_{s}}}\frac{T_{s}\left\lVert\eta_{k}\right\rVert_{2}}{30}$
in Table I does not differ significantly for the three proposed estimators.
However, Table I also shows the amount of trigger events and the trigger rate,
which is the amount of actual trigger events divided by the amount of possible
trigger events. The amount of trigger events and the trigger rate is almost
half as large for OLE compared to ZOH and more than seven times smaller for
CLE compared to ZOH in the considered simulation.

##  VII Conclusions

This paper proposes a distributed event-triggered formation control scheme for
polytopic LPV MAS, that guarantees stability and a level of control
performance for the network and is suitable for different local estimation
schemes. The synthesis problem is formulated as an LMI problem of the size of
a single agent. The estimators reduce the communication load and are
interchangeable for given control matrices. A simulation example illustrates
the practicality of the proposed method and compares the proposed estimators.

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Proof of Theorem 1:

Consider the Lyapunov function candidate

$V(x_{k},\zeta_{k},\tilde{z}_{k})=\left[\begin{array}[]{c}x_{k}\\\
\zeta_{k}\\\
\tilde{z}_{k}\end{array}\right]^{\top}\underbrace{\left[\begin{array}[]{ccc}\check{P}_{x}&\check{P}_{x\zeta}&\check{P}_{x\tilde{z}}\\\
\check{P}_{\zeta x}&\check{P}_{\zeta}&\check{P}_{\zeta\tilde{z}}\\\
\check{P}_{\tilde{z}x}&\check{P}_{\tilde{z}\zeta}&\check{P}_{\tilde{z}}\\\
\end{array}\right]}_{=\bar{P}}\left[\begin{array}[]{c}x_{k}\\\ \zeta_{k}\\\
\tilde{z}_{k}\end{array}\right],$

where $\bar{P}=\bar{P}^{\top}>0$. Stability, $\ell_{2}$ performance due to
(57) and the event-trigger condition via (5) are combined to

$\displaystyle\
{{\psi}}^{\top}_{k+1}\bar{P}{\psi}_{k+1}-{{\psi}}^{\top}_{k}\bar{P}{\psi}_{k}$
$\displaystyle\leq$
$\displaystyle-{{z}_{k}}^{\top}{{z}_{k}}+\gamma^{2}{f_{k}}^{\top}{f_{k}}-\sigma{\hat{{\eta}}^{\top}_{k}}\hat{{\eta}}_{k}+{e}_{k}^{\top}{e}_{k}$
. (100)

Applying (32) leads to

$\displaystyle 0$
$\displaystyle\geq\phi_{k}^{\top}\big{(}(\bar{H}^{\top}(\theta_{k})\bar{R}_{2}\bar{H}(\theta_{k})-\bar{R}_{1}\big{)}\phi_{k},$
$\displaystyle\bar{R}_{1}$
$\displaystyle=\text{diag}(\bar{P},I_{Nn_{y}},\gamma^{2}I_{Nn_{f}}),$
$\displaystyle\bar{R}_{2}$ $\displaystyle=\text{diag}(\bar{P},\sigma
I_{Nn_{y}},I_{Nn_{y}}),$ (101)

where $n_{f}=n_{y}+n_{w}$. Using a Schur complement results in

$\left[\begin{array}[]{cc}\bar{R}_{1}&\bar{H}^{\top}(\theta_{k})\bar{R}_{2}^{\top}\\\
\bar{R}_{2}\bar{H}(\theta_{k})&\bar{R}_{2}\end{array}\right]\geq 0,$ (102)

which is a non-linear matrix inequality. This inequality is linearized via
multiplying it from the right by
$\text{diag}(\bar{D}(\theta_{k}),\bar{R}_{2}^{-1}),$ where

$\bar{D}(\theta_{k})=\text{diag}(\bar{G}(\theta_{k}),I_{Nn_{y}+Nn_{w}})$

and

$\bar{G}(\theta_{k})=\text{diag}\left(\left[\begin{array}[]{cc}\check{G}_{x}(\theta_{k})&\check{G}_{x\zeta}(\theta_{k})\\\
\check{G}_{\zeta
x}(\theta_{k})&\check{G}_{\zeta}(\theta_{k})\end{array}\right],\check{G}_{{2}}(\theta_{k}\right)$

and from the left by its transpose, using the inequality
$\bar{S}\leq\bar{G}(\theta_{k})+\bar{G}(\theta_{k})^{\top}-\bar{S},$ where
$\bar{S}=\bar{P}^{-1}$, to get the more conservative but linear matrix
inequality

$\left[\begin{array}[]{cc}\bar{M}_{11}(\theta_{k})&\bar{M}_{21}^{\top}(\theta_{k})\\\
\bar{M}_{21}(\theta_{k})&\bar{M}_{22}\end{array}\right]\geq 0,$ (103)

where

$\displaystyle\bar{M}_{11}(\theta_{k})$
$\displaystyle=\text{diag}\big{(}\bar{G}(\theta_{k})+\bar{G}(\theta_{k})^{\top}-\bar{S},I_{Nn_{y}},tI_{Nn_{f}}\big{)},$
$\displaystyle\bar{M}_{21}(\theta_{k})$
$\displaystyle=\left[\begin{array}[]{ccc}\bar{A}_{\text{OL}}\bar{G}+\bar{B}_{F}\bar{K}&\bar{B}_{e}&\bar{B}_{{{f}}}\\\
\bar{C}_{\hat{\eta}}\bar{G}&\bar{D}_{\hat{\eta}e}&\bar{D}_{\hat{\eta}{{f}}}\\\
\bar{C}_{z}\bar{G}&\bar{D}_{ze}&\bar{D}_{z{{f}}}\end{array}\right](\theta_{k}),$
(107) $\displaystyle\bar{M}_{22}$
$\displaystyle=\text{diag}(\bar{S},\sigma_{x}I_{Nn_{y}},I_{Nn_{y}}),$
$\displaystyle\bar{K}(\theta_{k})$
$\displaystyle=[\check{K}_{x}(\theta_{k})\>\check{K}_{x\zeta}(\theta_{k})\>0_{Nn_{u}\times
Nn_{y}}],\>t=\gamma^{2},\>\sigma_{x}=\sigma^{-1}.$ (108)

Under the assumption of homogeneous scheduling (Assumption 1), the matrix
inequality can be diagonalized, since

$\theta_{k}=\mathbf{1}_{N}\otimes\theta_{k}^{1}$

implies applications of the Kronecker product, e.g.

$\check{A}(\theta_{k})=I_{N}\otimes A(\theta_{k}^{1}),$

and the transformation

$Z_{(n_{x})}^{\top}\mathcal{L}_{(n_{x})}Z_{(n_{x})}=\text{diag}(0,\lambda_{2},\ldots,\lambda_{N})\otimes
I_{n_{x}}$

has no effect on e.g. $\check{A}(\theta_{k})$ for orthonormal transformation
matrices $Z$, i.e.
$Z_{(n_{x})}^{\top}\check{A}(\theta_{k})Z_{(n_{x})}=\check{A}(\theta_{k})$,
due to properties of the Kronecker product. Thus, the transformation matrix

$\displaystyle\bar{T}=\text{diag}\big{(}Z_{(n_{x})},Z_{(n_{y})},Z_{(n_{y})},Z_{(n_{y})},Z_{(n_{y})},Z_{(n_{w})},$
$\displaystyle
Z_{(n_{x})},Z_{(n_{y})},Z_{(n_{y})},Z_{(n_{y})},Z_{(n_{y})}\big{)}$ (109)

brings (103) into diagonal form, where entries are dependent on $\lambda_{i}$
and ordered from $i=1$ to $i=N$, displayed for the vertex model matrices by
(64). The inequality (64) needs to be feasible for $l=1,\ldots,s$ for
$\lambda_{2}$ and $\lambda_{N}$, since $\theta^{i}_{k}\in\Theta$ and the
matrices are affine in $\lambda_{i}$ and $\lambda_{1}$ corresponds to the
agreement space.


#  Gröbner-Shirshov Bases for Commutative Algebras with Multiple Operators and
Free Commutative Rota-Baxter Algebras111Supported by the Young Project on
Natural Science Fund of Zhanjiang Normal University (No.QL0902).

Jianjun Qiu
Mathematics and Computational Science School
Zhanjiang Normal University
Zhanjiang 524048, China
<EMAIL_ADDRESS>
Abstract: In this paper, the Composition-Diamond lemma for commutative
algebras with multiple operators is established. As applications, the Gröbner-
Shirshov bases and linear bases of free commutative Rota-Baxter algebra, free
commutative $\lambda$-differential algebra and free commutative
$\lambda$-differential Rota-Baxter algebra are given, respectively.
Consequently, these three free algebras are constructed directly by
commutative $\Omega$-words.

Key words: commutative, Rota-Baxter algebras, $\lambda$-differential algebras,
$\lambda$-differential Rota-Baxter algebras, commutative algebras with
multiple operators, Gröbner-Shirshov bases.

AMS 2000 Subject Classification: 16S15, 13P10, 16W99, 17A50, 13XX

##  1 Introduction

Let $K$ be a unitary commutative ring and $\lambda\in K$. A Rota-Baxter
algebra of weight $\lambda$ is a $K$-algebra $R$ with a linear operator
$P:R\rightarrow R$ satisfying the Rota-Baxter relation:

$P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv),\forall u,v\in R.$

The Rota-Baxter operator first occurred in the paper of G. Baxter [1] to solve
an analytic problem, and the algebraic study of this operator was started by
G.-C. Rota [19].

There have been some constructions of free (commutative) Rota-Baxter algebras.
In this aspect, G.-C. Rota [19] and P. Cartier [9] gave the explicit
constructions of free commutative Rota-Baxter algebras of weight $\lambda=1$,
which they called shuffle Baxter and standard Baxter algebras, respectively.
Recently L. Guo and W. Keigher [13, 14] constructed the free commutative Rota-
Baxter algebras (with unit or without unit) for any $\lambda\in K$ using the
mixable shuffle product. These are called the mixable shuffle product
algebras, which generalize the classical construction of shuffle product
algebras. K. Ebrahimi-Fard and L. Guo [11] further constructed the free
associative Rota-Baxter algebras by using the Rota-Baxter words. In [12], K.
Ebrahimi-Fard and L. Guo use rooted trees and forests to give explicit
construction of free noncommutative Rota-Baxter algebras on modules and sets.

A differential algebra of weigh $\lambda$, also called $\lambda$-differential
algebra is a $K$-algebra $R$ with a linear operator $D:R\rightarrow R$ such
that

$D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\forall u,v\in R.$

Such an operator $D$ is called a $\lambda$-differential operator. E. Kolchin
[16] considered the differential algebra and constructed free differential
algebra of weight $\lambda=0$. L. Guo and W. Keigher [15] dealt with a
generalization of this algebra and using the same way to construct the free
differential algebra of weight $\lambda$ in both commutative and associative
case.

Similar to the relation of integral and differential operators, L. Guo and W.
Keigher [15] introduced the notion of $\lambda$-differential Rota-Baxter
algebra which is a $K$-algebra $R$ with a $\lambda$-differential operator $D$
and a Rota-Baxter operator $P$ such that $DP=Id_{R}$. In the same paper [15],
they construct the free $\lambda$-differential Rota-Baxter (commutative and
associative).

The Gröbner-Shirshov bases theory for Lie algebras was introduced by A. I.
Shirshov [4, 20]. Shirshov [20] defined the composition of two Lie
polynomials, and proved the Composition lemma for the Lie algebras. L. A.
Bokut [3] specialized the approach of Shirshov to associative algebras, see
also Bergman [2]. For commutative polynomials, this lemma is known as the
Buchberger’s Theorem in [7, 8].

The multi-operators algebras ($\Omega$-algebras) were introduced by A. G.
Kurosh [17] and the Gröbner-Shirshov bases for $\Omega$-algebras were given in
the paper by V. Drensky and R. Holtkamp [10]. Composition-Diamond lemma for
associative algebras with multiple linear operators (associative
$\Omega$-algebras) is established in a recent paper by L. A. Bokut, Y. Chen
and J. Qiu [5]. Also, the Gröbner-Shirshov bases for Rota-Baxter algebras is
established by L. A. Bokut, Y. Chen and X. Deng [6] and the Composition-
Diamond lemma for $\lambda$-differential associative algebras with multiple
operators is constructed by J. Qiu and Y. Chen [18].

In this paper, we deal with commutative algebras with multiple linear
operators. We construct free commutative algebras with multiple linear
operators and establish the Composition-Diamond lemma for such algebras. As
applications, we obtain Gröbner-Shirshov bases of free commutative Rota-Baxter
algebra, commutative $\lambda$-differential algebra and commutative
$\lambda$-differential Rota-Baxter algebra, respectively. Then, by using the
Composition-Diamond lemma, linear bases of these three free algebras are
obtained respectively.

The author would like to express his deepest gratitude to Professors L. A.
Bokut and Yuqun Chen for their kind guidance, useful discussions and
enthusiastic encouragements.

##  2  Free commutative algebras with multiple operators

In this section, we construct free commutative algebras with multiple linear
operators.

Let $K$ be a unitary commutative ring. A commutative algebra with multiple
operators is a commutative $K$-algebra $R$ with a set $\Omega$ of multi-linear
operators.

Let $X$ be a set, $CS(X)$ the free commutative semigroup on $X$ and

$\Omega=\bigcup_{t=1}^{\infty}\Omega_{t}$

where $\Omega_{t}$ is the set of $t$-ary operators.

Define

$\Gamma_{0}=X,\ \mathcal{Q}_{0}=CS(\Gamma_{0})$

and

$\Gamma_{1}=X\cup\Omega(\mathcal{Q}_{0}),\ \mathcal{Q}_{1}=CS(\Gamma_{1})$

where

$\Omega(\mathcal{Q}_{0})=\bigcup\limits_{t=1}^{\infty}\\{\theta(u_{1},u_{2},\dots,u_{t})|\theta\in\Omega_{t},u_{i}\in\mathcal{Q}_{0},\
i=1,2,\dots,t\\}.$

For $n>1$, define

$\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1}),\ \mathcal{Q}_{n}=CS(\Gamma_{n})$

where

$\Omega(\mathcal{Q}_{n-1})=\bigcup\limits_{t=1}^{\infty}\\{\theta(u_{1},u_{2},\dots,u_{t})|\theta\in\Omega_{t},u_{i}\in\mathcal{Q}_{n-1},\
i=1,2,\dots,t\\}.$

Let

$\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}.$

Then it is easy to see that $\mathcal{Q}(X)$ is a commutative semigroup such
that $\Omega(\mathcal{Q}(X))\subseteq\mathcal{Q}(X).$

Let $K[X;\Omega]$ be the commutative $K$-algebra spanned by $\mathcal{Q}(X)$.
Extend linearly each $\sigma\in\Omega_{t}$, $t\geq 1$,

$\sigma:\mathcal{Q}(X)^{t}\rightarrow\mathcal{Q}(X),\
(x_{1},x_{2},\dots,x_{n})\mapsto\sigma(x_{1},x_{2},\dots,x_{n})$

to

$K[X;\Omega]^{t}\rightarrow K[X;\Omega].$

Then, it is easy to see that $K[X;\Omega]$ is a free commutative algebra with
multiple operators $\Omega$ on set $X$.

##  3  Composition-Diamond lemma for commutative algebras with multiple
operators

In this section, we introduce the notions of Gröbner-Shirshov bases for the
commutative algebras with multiple operators and establish the Composition-
Diamond lemma for such algebras.

The element in $\mathcal{Q}(X)$ and $K[X;\Omega]$ are called commutative
$\Omega$-word and commutative $\Omega$-polynomial, respectively. For any
$u\in\mathcal{Q}(X)$, $u$ has a unique expression

$u=u_{1}u_{2}\cdots u_{n}$

where each $u_{i}\in X\cup\Omega(\mathcal{Q}(X))$. If this is the case, we
define $bre(u)=n$. Let $u\in\mathcal{Q}(X)$. Then

$dep(u)=\mbox{min}\\{n|u\in\mathcal{Q}_{n}\\}$

is called the depth of $u$.

Let $\star\notin X$. By a commutative $\star$-$\Omega$-word we mean any
expression in $\mathcal{Q}(X\cup\\{\star\\})$ with only one occurrence of
$\star$. We define the set of all commutative $\star$-$\Omega$-words on $X$ by
$\mathcal{Q}^{\star}(X)$. Let $u$ be a commutative $\star$-$\Omega$-word and
$s\in K[X;\Omega]$. Then we call

$u|_{s}=u|_{\star\ \mapsto s}\ \mbox{(replace $\star$ by $s$ in $u$)}$

is a commutative $s$-$\Omega$-word. For example, if

$u=x_{1}\theta_{3}(x_{2},\theta_{2}(\star
x_{4},x_{5}),x_{6})\in\mathcal{Q}^{\star}(X),$

then

$u|_{s}=u|_{\star\ \mapsto
s}=x_{1}\theta_{3}(x_{2},\theta_{2}(sx_{4},x_{5}),x_{6})$

is a commutative $s$-$\Omega$-word.

Similarly, we can define commutative $(\star_{1},\star_{2})$-$\Omega$-words as
expressions in $\mathcal{Q}(X\cup\\{\star_{1},\star_{2}\\})$ with only one
occurrence of $\star_{1}$ and only one occurrence of $\star_{2}$. Let us
denote by $\mathcal{Q}^{\star_{1},\star_{2}}(X)$ the set of all commutative
$(\star_{1},\star_{2})$-$\Omega$-words. Let
$u\in\mathcal{Q}^{\star_{1},\star_{2}}(X)$, $s_{1},s_{2}\in K[X;\Omega]$. Then
we call

$u|_{s_{1},\ s_{2}}=u|_{\star_{1}\mapsto s_{1},\star_{2}\mapsto s_{2}},$

a commutative $s_{1}$-$s_{2}$-$\Omega$-word.

Now, we assume that $\mathcal{Q}(X)$ is equipped with a monomial order $>$.
This means that $>$ is a well order on $\mathcal{Q}(X)$ such that for any
$w,v\in\mathcal{Q}(X),u\in\mathcal{Q}^{\star}(X)$,

$w>v\Rightarrow u|_{w}>u|_{v}.$

Note that such an order on $\mathcal{Q}(X)$ exists, for example, the order (1)
in the next section.

For every commutative $\Omega$-polynomial $f\in K[X;\Omega]$, let $\bar{f}$ be
the leading term of $f$. If the coefficient of $\bar{f}$ is $1$, then we call
$f$ monic.

Let $f,g$ be two monic $\Omega$-polynomials. Then there are two kinds of
compositions.

  1. (I)

If there exists a commutative $\Omega$-word $w=a\bar{f}=b\bar{g}$ for some
$a,b\in\mathcal{Q}(X)$ such that $bre(w)<bre(\bar{f})+bre(\bar{g})$, then we
call $(f,g)_{w}=af-bg$ the intersection composition of $f$ and $g$ with
respect to $w$.

  2. (II)

If there exists a commutative $\Omega$-word $w=\bar{f}=u|_{\bar{g}}$ for some
$u\in\mathcal{Q}^{\star}(X)$, then we call $(f,g)_{w}=f-u|_{g}$ the including
composition of $f$ and $g$ with respect to $w$.

In the above definition, $w$ is called the ambiguity of the composition.
Clearly,

$(f,g)_{w}\in Id(f,g)\ \ \ \mbox{ and }\ \ \ \overline{(f,g)_{w}}<w$

where $Id(f,g)$ is the ideal of $K[X;\Omega]$ generated by $f$ and $g$.

Let $f,g$ be commutative $\Omega$-polynomials and $g$ monic with
$\bar{f}=u|_{\bar{g}}$ for some $u\in\mathcal{Q}^{\star}(X)$. Then the
transformation

$f\rightarrow f-\alpha u|_{g}$

is called elimination of the leading commutative $\Omega$-word (ELW) of $f$ by
$g$, where $\alpha$ is the coefficient of the leading commutative
$\Omega$-word of $f$.

Let $S$ be a set of monic commutative $\Omega$-polynomials. Then the
composition $(f,g)_{w}$ is called trivial modulo $(S,w)$, if

$(f,g)_{w}=\sum\alpha_{i}u_{i}|_{s_{i}}$

where each $\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$, $s_{i}\in S$
and $u_{i}|_{\overline{s_{i}}}<w$. If this is the case, we write

$(f,g)_{w}\equiv 0\ \ mod(S,w).$

In general, for any two commutative $\Omega$-polynomials $p$ and $q$, $p\equiv
q\ \ mod(S,w)$ means $p-q=\sum\alpha_{i}u_{i}|_{s_{i}}$ where each
$\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$, $s_{i}\in S$ and
$u_{i}|_{\overline{s_{i}}}<w$.

Then $S$ is called a Gröbner-Shirshov basis in $K[X;\Omega]$ if any
composition $(f,g)_{w}$ of $f,g\in S$ is trivial modulo $(S,w)$.

###### Lemma 3.1

Let $S$ be a Gröbner-Shirshov basis in $K[X;\Omega]$ and $u_{1},\
u_{2}\in\mathcal{Q}^{\star}(X),\ s_{1},s_{2}\in S$. If
$w=u_{1}|_{\overline{s_{1}}}=u_{2}|_{\overline{s_{2}}}$, then

$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$

Proof: There are three cases to consider.

(I) The commutative $\Omega$-words $\overline{s_{1}}$ and $\overline{s_{2}}$
are disjoint. Then there exits a commutative
$(\star_{1},\star_{2})$-$\Omega$-words $\Pi$ such that

$\Pi|_{\overline{s_{1}},\
\overline{s_{2}}}=u_{1}|_{\overline{s_{1}}}=u_{2}|_{\overline{s_{2}}}.$

Then

$\displaystyle u_{2}|_{s_{2}}-u_{1}|_{s_{1}}$ $\displaystyle=$
$\displaystyle\Pi|_{\overline{s_{1}},\ s_{2}}-\Pi|_{s_{1},\ \overline{s_{2}}}$
$\displaystyle=$ $\displaystyle(-\Pi|_{s_{1}-\overline{s_{1}},\
s_{2}}+\Pi|_{s_{1},\ s_{2}})+(\Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}-\Pi|_{s_{1},\ s_{2}})$ $\displaystyle=$
$\displaystyle-\Pi|_{s_{1}-\overline{s_{1}},\ s_{2}}+\Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}$

Let

$-\Pi|_{s_{1}-\overline{s_{1}},\ s_{2}}=\sum\alpha_{2_{t}}u_{2_{t}}|_{s_{2}}\
\ \mbox{and}\ \ \Pi|_{s_{1},\
s_{2}-\overline{s_{2}}}=\sum\alpha_{1_{l}}u_{1_{l}}|_{s_{1}}.$

Since $\overline{s_{1}-\overline{s_{1}}}<\overline{s_{1}}$ and
$\overline{s_{2}-\overline{s_{2}}}<\overline{s_{2}}$, we have

$u_{2_{t}}|_{\overline{s_{2}}},\ u_{1_{l}}|_{\overline{s_{1}}}<w.$

Therefore

$u_{{}_{2}}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\alpha_{2_{t}}u_{2_{t}}|_{s_{2}}+\sum\alpha_{1_{l}}u_{1_{l}}|_{s_{1}}$

with each $u_{2_{t}}|_{\overline{s_{2}}},\ u_{1_{l}}|_{\overline{s_{1}}}<w$.
It follows that

$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$

(II) The commutative $\Omega$-words $\overline{s_{1}}$ and $\overline{s_{2}}$
have nonempty intersection but do not include each other. For example,

$a\overline{s_{1}}=b\overline{s_{2}}$

for some commutative $\Omega$-words $a,\ b$. Then there exists a commutative
$\star$-$\Omega$-word $\Pi$ such that

$\Pi|_{a\overline{s_{1}}}=u_{1}|_{\overline{s_{{}_{1}}}}=u_{2}|_{\overline{s_{2}}}=\Pi|_{b\overline{s_{2}}}.$

Then we have

$u_{{}_{2}}|_{s_{{}_{2}}}-u_{1}|_{s_{{}_{1}}}=\Pi|_{bs_{{}_{2}}}-\Pi|_{as_{{}_{1}}}=-\Pi|_{as_{{}_{1}}-bs_{{}_{2}}}.$

Since $S$ is a Gröbner-Shirshov basis in $K[X;\Omega]$, we have

$as_{1}-bs_{2}=\sum\alpha_{j}v_{j}|_{s_{j}}$

where each $\alpha_{j}\in K,\ v_{j}\in\mathcal{Q}^{\star}(X),\ s_{j}\in S$ and
$v_{j}|_{\overline{s_{j}}}<a\overline{s_{1}}$. Let

$\Pi|_{v_{j}|_{s_{j}}}=\Pi_{j}|_{s_{j}}.$

Then

$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\alpha_{j}\Pi_{j}|_{s_{j}}$

with

$\Pi_{j}|_{\overline{s_{j}}}<w.$

It follows that

$u_{1}|_{s_{1}}\equiv u_{2}|_{s_{2}}\ mod(S,w).$

(III) One of commutative $\Omega$-words $\overline{s_{1}}$, $\overline{s_{2}}$
is contained in the other. For example, let

$\overline{s_{1}}=u|_{\overline{s_{2}}}$

for some commutative $\star$-$\Omega$-word $u$. Then

$w=u_{2}|_{\overline{s_{2}}}=u_{1}|_{u|_{\overline{s_{2}}}}$

and

$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=u_{1}|_{u|_{s_{2}}}-u_{1}|_{s_{1}}=-u_{1}|_{s_{1}-u|_{s_{2}}}.$

Similarly to (II), we can obtain the result. $\blacksquare$

The following theorem is an analogy of Shirshov’s composition lemma for Lie
algebras [20], which was specialized to associative algebras by Bokut [3]. For
commutative algebras, this lemma is known as the Buchberger’s Theorem in [7,
8].

###### Theorem 3.2

(Composition-Diamond lemma) Let $S$ be a set of monic commutative
$\Omega$-polynomials in $K[X;\Omega]$ and $>$ a monomial order on
$\mathcal{Q}(X)$. Then the following statement are equivalent:

  1. (I)

$S$ is a Gröbner-Shirshov basis in $K[X;\Omega]$.

  2. (II)

$f\in Id(S)\Rightarrow\bar{f}=u|_{\overline{s}}$ for some
$u\in\mathcal{Q}^{\star}(X)$ and $s\in S$.

  3. (III)

$Irr(S)=\\{w\in\mathcal{Q}(X)|w\neq u|_{\overline{s}}\mbox{ for any}\
u\in\mathcal{Q}^{\star}(X)\ \mbox{and }s\in S\\}$ is a $K$-basis of
$K[X;\Omega]/Id(S)=K[X;\Omega|S]$.

Proof: (I) $\Longrightarrow$ (II) Let $0\neq f\in Id(S)$. Then

$f=\sum\limits_{i=1}^{n}\alpha_{i}u_{i}|_{s_{i}}$

where each $\alpha_{i}\in K$, $u_{i}\in\mathcal{Q}^{\star}(X)$ and $s_{i}\in
S$. Let $w_{i}=u_{i}|_{\overline{s_{i}}}$ and we arrange this leading
commutative $\Omega$-words in non-increasing order by

$w_{1}=w_{2}=\cdots=w_{m}>w_{m+1}\geq\cdots\geq w_{n}.$

We prove the result by induction on $m$.

If $m=1$, then $\bar{f}=u_{1}|_{\overline{s_{1}}}$.

Now we assume that $m\geq 2$. Then

$u_{1}|_{\overline{s_{1}}}=w_{1}=w_{2}=u_{2}|_{\overline{s_{2}}}.$

We prove the result by induction on $w_{1}$. If $\bar{f}=w_{1}$, there is
nothing to prove. Clearly, $w_{1}>\bar{f}$. Since $S$ is a Gröbner-Shirshov
basis in $K[X;\Omega]$, by Lemma 3.1, we have

$u_{2}|_{s_{2}}-u_{1}|_{s_{1}}=\sum\beta_{j}v_{j}|_{s_{j}}$

where $\beta_{j}\in K,\ s_{j}\in S,v_{j}\in\mathcal{Q}^{\star}(X)$ and
$v_{j}|_{\overline{s_{j}}}<w_{1}$. Therefore, since

$\alpha_{1}u_{1}|_{s_{1}}+\alpha_{2}u_{2}|_{s_{2}}=(\alpha_{1}+\alpha_{2})u_{1}|_{s_{1}}+\alpha_{2}(u_{2}|_{s_{2}}-u_{1}|_{s_{1}}),$

we have

$f=(\alpha_{1}+\alpha_{2})u_{1}|_{s_{1}}+\sum\alpha_{2}\beta_{j}v_{j}|_{s_{j}}+\sum\limits_{i=3}^{n}\alpha_{i}u_{i}|_{s_{i}}.$

If either $m>2$ or $\alpha_{1}+\alpha_{2}\neq 0$, then the result follows from
induction on $m$. If $m=2$ and $\alpha_{1}+\alpha_{2}=0$, then the result
follows from induction on $w_{1}$.

(II)$\Longrightarrow$ (III) For any $f\in K[X;\Omega]$, by ELWs, we can obtain
that $f+Id(S)$ can be expressed by the elements of $Irr(S)$. Now suppose
$\alpha_{1}u_{1}+\alpha_{2}u_{2}+\cdots\alpha_{n}u_{n}=0$ in $K[X;\Omega|S]$
with $u_{i}\in Irr(S)$, $u_{1}>u_{2}>\cdots>u_{n}$ and $\alpha_{i}\neq 0$.
Then, in $K[X;\Omega]$,

$g=\alpha_{1}u_{1}+\alpha_{2}u_{2}+\cdots+\alpha_{n}u_{n}\in Id(S).$

By (II), we have $u_{1}=\bar{g}\notin Irr(S)$, a contradiction. So $Irr(S)$ is
$K$-linearly independent. This shows that $Irr(S)$ is a $K$-basis of
$K[X;\Omega|S]$.

(III)$\Longrightarrow$(II) Let $0\neq f\in Id(S)$. Suppose that $\bar{f}\in
Irr(S)$. Then

$f+Id(S)=\alpha(\bar{f}+Id(S))+\sum\alpha_{i}(u_{i}+Id(S))$

where $u_{i}\in Irr(S)$ and $\bar{f}>u_{i}$. Therefore, $f+Id(S)\neq 0$, a
contradiction. So $\bar{f}=u|_{\bar{s}}$ for some $s\in S$ and
$u\in\mathcal{Q}^{\star}(X)$.

(II)$\Longrightarrow$(I) By the definition of the composition, we have
$(f,g)_{w}\in Id(S)$. If $(f,g)_{w}\neq 0$, then by (II),
$\overline{(f,g)_{w}}=u_{1}|_{\overline{s_{1}}}<w$ for some $s_{1}\in S$ and
$u_{1}\in\mathcal{Q}^{\star}(X)$. Let

$h=(f,g)_{w}-\alpha_{1}u_{1}|_{s_{1}}$

where $\alpha_{1}$ is the coefficient of $\overline{(f,g)_{w}}$. Then
$\bar{h}<w$ and $h\in Id(S)$. By induction on $w$, we can get the result.
$\blacksquare$

##  4  Gröbner-Shirshov bases for free commutative Rota-Baxter algebras

In this section, a Gröbner-Shirshov basis for free commutative Rota-Baxter
algebra was obtained. By using the Composition-Diamond lemma (Theorem 3.2), a
linear basis of such algebra was given and the free commutative Rota-Baxter
algebra was directly constructed by commutative $\Omega$-words.

First of all, we define an order on $\mathcal{Q}(X)$, which will be used in
this section. Let $X$ and $\Omega$ be well ordered. We define an order on
$\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by induction on $n$. For any
$u,v\in\mathcal{Q}_{0}=CS(X)$, we have

$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$

where each $x_{i}\in X,\ i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define

$u>v\Longleftrightarrow(bre(u),i_{1},i_{2},\dots,i_{t})>(bre(v),j_{1},j_{2},\dots,j_{t})\
\ \mbox{lexicographically}.$

Assume the order on $\mathcal{Q}_{n-1}$ has been defined. Now, we define an
order on $\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1})$. Let
$v_{1},v_{2}\in\Gamma_{n}$. Then $v_{1}>v_{2}$ means one of the following
holds:

(a) $v_{1},v_{2}\in X$ and $v_{1}>v_{2}$;

(b) $v_{1}\in\Omega(\mathcal{Q}_{n-1})$ and $v_{2}\in X$;

(c) $v_{1}=\theta_{k}(v_{1}^{\prime})$,
$v_{2}=\theta_{l}(v_{2}^{\prime}),v_{1}^{\prime}\in\mathcal{Q}_{n-1}^{k},v_{2}^{\prime}\in\mathcal{Q}_{n-1}^{l}$
with $\theta_{k}>\theta_{l}$ or $\theta_{k}=\theta_{l}$,
$v_{1}^{\prime}>v_{2}^{\prime}$ lexicographically.

For any $u,v\in\mathcal{Q}_{n}$, we have

$u=w_{1}^{i_{1}}w_{2}^{i_{2}}\cdots w_{t}^{i_{t}}\ \mbox{and}\
v=w_{1}^{j_{1}}w_{2}^{j_{2}}\cdots w_{t}^{j_{t}}$

where each $w_{i}\in\Gamma_{n}$, $i_{k},j_{l}\geq 0$ and $w_{i}>w_{i+1}$.
Here, $bre(u)=i_{1}+i_{2}+\cdots+i_{t}$. Define

$u>v\Longleftrightarrow(bre(u),i_{1},i_{2},\dots,i_{t})>(bre(v),j_{1},j_{2},\dots,j_{t})\
\ \mbox{lexicographically}$ (1)

Then the Order (1) is a monomial order on $\mathcal{Q}(X)$.

Let $K$ be a commutative ring with unit and $\lambda\in K$. A commutative
Rota-Baxter algebra of weight $\lambda$ ( see [1, 13, 19]) is a commutative
$K$-algebra $R$ with a linear operator $P:R\rightarrow R$ satisfying the Rota-
Baxter relation:

$P(u)P(v)=P(P(u)v)+P(uP(v))+\lambda P(uv),\ \forall u,v\in R.$

It is obvious that any commutative Rota-Baxter algebra is a commutative
algebra with multiple operators $\Omega$, where $\Omega=\\{P\\}$.

In this section, we assume that $\Omega=\\{P\\}$. Let $\mathcal{Q}(X)$ be
defined as before with $\Omega=\\{P\\}$ and $K[X;P]$ be the free commutative
algebra with operator $\Omega=\\{P\\}$ on set $X$.

###### Theorem 4.1

With the order (1) on $\mathcal{Q}(X)$,

$S=\\{P(u)P(v)-P(P(u)v)-P(uP(v))-\lambda P(uv)|\ u,v\in\mathcal{Q}(X)\\}$

is a Gröbner-Shirshov basis in $K[X;P]$.

Proof: The ambiguities of all possible compositions of the commutative
$\Omega$-polynomials in $S$ are:

$(i)\ \ \ \ P(u)P(v)P(w)\ \ \ \ \ \ \ \ (ii)\ \ \ \ P(z|_{P(v)P(w)})P(u)\ \ \
\ $

where $u,v,w\in\mathcal{Q}(X),z\in\mathcal{Q}^{\star}(X)$. It is easy to check
that all these compositions are trivial. Here, for example, we just check
$(i)$. For any $u,v\in\mathcal{Q}(X)$, let

$f(u,v)=P(u)P(v)-P(P(u)v)-P(uP(v))-\lambda P(uv).$

Then

$\displaystyle(f(u,v),f(v,w))_{P(u)P(v)P(w)}$
$\displaystyle=-P(P(u)v)P(w)-P(uP(v))P(w)-\lambda P(uv)P(w)$ $\displaystyle\ \
\ +P(u)P(P(v)w)+P(u)P(vP(w))+\lambda P(u)P(vw)$
$\displaystyle\equiv-P(P(P(u)v)w)-P(P(u)vP(w))-\lambda P(P(u)vw)$
$\displaystyle\ \ \ -P(P(uP(v))w)-P(uP(v)P(w))-\lambda P(uP(v)w)$
$\displaystyle\ \ \ -\lambda P(P(uv)w)-\lambda P(uvP(w))-\lambda^{2}P(uvw)$
$\displaystyle\ \ \ +P(P(u)P(v)w)+P(uP(P(v)w))+\lambda P(uP(v)w)$
$\displaystyle\ \ \ +P(P(u)vP(w))+P(uP(vP(w)))+\lambda P(uvP(w))$
$\displaystyle\ \ \ +\lambda P(P(u)vw)+\lambda P(uP(vw))+\lambda^{2}P(uvw)$
$\displaystyle\equiv-P(P(P(u)v)w)-P(P(uP(v))w)-\lambda P(P(uv)w)$
$\displaystyle\ \ \ -P(uP(P(v)w))-P(uP(vP(w)))-\lambda P(uP(vw))$
$\displaystyle\ \ \ +P(P(P(u)v)w)+P(P(uP(v))w)+\lambda P(P(uv)w)$
$\displaystyle\ \ \ +P(uP(P(v)w))+P(uP(vP(w)))+\lambda P(uP(vw))$
$\displaystyle\equiv 0\ \ mod(S,P(u)P(v)P(w)).\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \blacksquare$

Define

$\displaystyle\Phi_{0}$ $\displaystyle=$ $\displaystyle CS(X),$
$\displaystyle\Phi_{1}$ $\displaystyle=$ $\displaystyle\Phi_{0}\cup
P(\Phi_{0})\cup\Phi_{0}P(\Phi_{0}),$ $\displaystyle\vdots\ \ $ $\displaystyle\
\ \ \ \vdots$ $\displaystyle\Phi_{n}$ $\displaystyle=$
$\displaystyle\Phi_{0}\cup P(\Phi_{n-1})\cup\Phi_{0}P(\Phi_{n-1}),$
$\displaystyle\vdots\ \ $ $\displaystyle\ \ \ \ \vdots$

where

$\Phi_{0}P(\Phi_{n-1})=\\{uP(v)|u\in\Phi_{0},v\in\Phi_{n-1}\\}.$

Let

$\Phi(X)=\bigcup_{n\geq 0}\Phi_{n}.$

The elements in $\Phi(X)$ are called commutative Rota-Baxter words.

By Theorem 4.1 and Theorem 3.2, we have the following theorems.

###### Theorem 4.2

([6]) $Irr(S)=\Phi(X)$ is a basis of $K[X;\Omega|S]$.

###### Theorem 4.3

$K[X;\Omega|S]$ is a free commutative Rota-Baxter algebra of weight $\lambda$
on set $X$ with a basis $\Phi(X)$.

By using ELWs, we have the following algorithm. In fact, it is an algorithm to
compute the product of two commutative Rota-Baxter words in the free
commutative Rota-Baxter algebra $K[X;\Omega|S]$.

Algorithm Let $u,v\in\Phi(X)$. We define $u\diamond v$ by induction on
$n=dep(u)+dep(v)$.

(1) If $n=0$, then $u,v\in CS(X)$ and $u\diamond v=uv$.

(2) If $n\geq 1$, then $u\diamond v=$

$\left\\{\begin{array}[]{l@{\quad\quad}l}uv&\mbox{if}\ u\in CS(X)\ \mbox{or}\
v\in CS(X)\\\ u_{1}v_{1}(P(P(u^{{}^{\prime}})\diamond
v^{{}^{\prime}})+P(u^{{}^{\prime}}\diamond P(v^{{}^{\prime}}))+\lambda
P(u^{{}^{\prime}}\diamond v^{{}^{\prime}}))&\mbox{if}\
u=u_{1}P(u^{{}^{\prime}}),v=v_{1}P(v^{{}^{\prime}})\end{array}\right.$

##  5 Gröbner-Shirshov bases for free commutative $\lambda$-differential
algebras

In this section, we give a Gröbner-Shirshov basis for a free commutative
$\lambda$-differential algebra and then by using the Composition-Diamond lemma
(Theorem 3.2), we obtain a linear basis of such an algebra, which is the same
as the one in [15]. Consequently, we construct the free $\lambda$-differential
algebra on set $X$ directly by commutative $\Omega$-words.

Let $K$ be a commutative unitary ring and $\lambda\in K$. A commutative
$\lambda$-differential algebra ([15, 16]) over $K$ is a commutative
$K$-algebra $R$ together with a linear operator $D:R\rightarrow R$ such that

$D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),\ \forall u,v\in R.$

It is obvious that any commutative $\lambda$-differential algebra is a
commutative algebra with multiple operators $\Omega$ on $X$, where
$\Omega=\\{D\\}$.

In this section, we assume that $\Omega=\\{D\\}$. Let $K[X;D]$ be the free
commutative algebra with operator $\Omega=\\{D\\}$. Let $X$ be well ordered.
For any $u\in\mathcal{Q}(X)$, $u$ has a unique expression

$u=u_{1}u_{2}\cdots u_{n}$

where $n\geq 1$ and each $u_{i}\in X\cup D(\mathcal{Q}(X))$. Define $deg(u)$
the number of $x\in X$ in $u$. For example, if $u=x_{1}D(D(x_{2}))D(x_{3})$,
then $deg(u)=3$.

We define an order on $\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by
induction on $n$. For any $u,v\in\mathcal{Q}_{0}=CS(X)$, we have

$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$

where each $x_{i}\in X$, $i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define

$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \mbox{lexicographically}.$

Assume the order on $\mathcal{Q}_{n-1}$ has been defined. Now, we define an
order on $\Gamma_{n}=X\cup\Omega(\mathcal{Q}_{n-1})$ firstly. Let
$v_{1},v_{2}\in\Gamma_{n}$. Then $v_{1}>v_{2}$ means one of the following
holds:

(a) $v_{1},v_{2}\in X$ and $v_{1}>v_{2}$;

(b) $v_{1}\in D(\mathcal{Q}_{n-1})$ and $v_{2}\in X$;

(c) $v_{1}=D(v_{1}^{\prime})$, $v_{2}=D(v_{2}^{\prime})$ and
$v_{1}^{\prime}>v_{2}^{\prime}$.

For any $u,v\in\mathcal{Q}_{n}$, we have

$u=w_{1}^{i_{1}}w_{2}^{i_{2}}\cdots w_{t}^{i_{t}}\ \mbox{and}\
v=w_{1}^{j_{1}}w_{2}^{j_{2}}\cdots w_{t}^{j_{t}}$

where each $w_{i}\in\Gamma_{n}$, $i_{k},j_{l}\geq 0$ and $w_{i}>w_{i+1}$.
Define

$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \ \mbox{lexicographically}$ (2)

Then the Order (2) is a monomial order on $\mathcal{Q}(X)$.

###### Theorem 5.1

With the Order (2) on $\mathcal{Q}(X)$,

$S=\\{D(uv)-D(u)v-uD(v)-\lambda D(u)D(v)|\ u,v\in\mathcal{Q}(X)\\}$

is a Gröbner-Shirshov basis in $K[X;D]$.

Proof: The ambiguities of all possible compositions of the commutative
$\Omega$-polynomials in $S$ are

$D(u|_{{}_{D(xy)}}v)$

where $x,y,v\in\mathcal{Q}(X),u\in\mathcal{Q}^{\star}(X)$. Let

$g(u,v)=D(uv)-D(u)v-uD(v)-\lambda D(u)D(v).$

Then

$\displaystyle(g(u|_{{}_{D(xy)}},v),g(x,y))_{D(u|_{{}_{D(xy)}}v)}$
$\displaystyle=-D(u|_{{}_{D(xy)}})v-u|_{{}_{D(xy)}}D(v)+\lambda
D(u|_{{}_{D(xy)}})D(v)$ $\displaystyle\ \ \
+D(u|_{{}_{D(x)y}}v)+D(u|_{{}_{xD(y)}}v)+\lambda D(u|_{{}_{D(x)D(y)}}v)$
$\displaystyle\equiv-D(u|_{{}_{D(x)y}})v-D(u|_{{}_{xD(y)}})v-\lambda
D(u|_{{}_{D(x)D(y)}})v$ $\displaystyle\ \ \
-u|_{{}_{D(x)y}}D(v)-u|_{{}_{xD(y)}}D(v)-\lambda u|_{{}_{D(x)D(y)}}D(v)$
$\displaystyle\ \ \ -\lambda D(u|_{{}_{D(x)y}})D(v)-\lambda
D(u|_{{}_{xD(y)}})D(v)-\lambda^{2}D(u|_{{}_{D(x)D(y)}})D(v)$ $\displaystyle\ \
\ +D(u|_{{}_{D(x)y}})v+u|_{{}_{D(x)y}}D(v)+\lambda D(u|_{{}_{D(x)y}})D(v)$
$\displaystyle\ \ \ +D(u|_{{}_{xD(y)}})v+u|_{{}_{xD(y)}}D(v)+\lambda
D(u|_{{}_{xD(y)}})D(v)$ $\displaystyle\ \ \ +\lambda
D(u|_{{}_{D(x)D(y)}})v+\lambda
u|_{{}_{D(x)D(y)}}D(v)+\lambda^{2}D(u|_{{}_{D(x)D(y)}})D(v)$
$\displaystyle\equiv 0\ \ mod(S,D(u|_{{}_{D(xy)}}v).\ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \blacksquare$

Let $D^{\omega}(X)=\\{D^{i}(x)|i\geq 0,x\in X\\}$, where $D^{0}(x)=x$ and
$CS(D^{\omega}(X))$ the free commutative semigroup generated by
$D^{\omega}(X)$.

###### Theorem 5.2

([15]) $Irr(S)=CS(D^{\omega}(X))$ is a $K$-basis of $K[X;D|S]$.

Proof: By Theorem 3.2 and Theorem 5.1.

###### Theorem 5.3

([15]) $K[X;D|S]$ is a free commutative $\lambda$-differential algebra on set
$X$ with a basis $CS(D^{\omega}(X))$.

Proof: By Theorem 5.2.

By using ELWs, we have the following algorithm.

Algorithm ([15]) Let $u=u_{1}u_{2}\cdots u_{n}$, where each $u_{k}\in
D^{\omega}(X),n>0$. Define $D(u)$ by induction on $n$.

(1) If $n=1$, i.e., $u=D^{i}(x)$ for some $i\geq 0,x\in X$, then
$D(u)=D^{(i+1)}(x)$.

(2) If $n\geq 1$, then

$D(u)=D(u_{1}u_{2}\cdots u_{n})=D(u_{1})(u_{2}\cdots u_{n})+u_{1}D(u_{2}\cdots
u_{n})+\lambda D(u_{1})D(u_{2}\cdots u_{n}).$

##  6  Gröbner-Shirshov bases for free commutative $\lambda$-differential
Rota-Baxter algebras

In this section, we give a Gröbner-Shirshov basis for a free commutative
$\lambda$-differential Rota-Baxter algebra on a set. By using the Composition-
Diamond lemma for commutative algebras with multiple operators (Theorem 3.2),
we obtain a linear basis of a free commutative $\lambda$-differential Rota-
Baxter algebra on a set. Also, we construct the free commutative
$\lambda$-differential Rota-Baxter algebra on a set directly by commutative
$\Omega$-words.

Let $K$ be a unitary commutative ring and $\lambda\in K$. A commutative
$\lambda$-differential Rota-Baxter algebra ([15]) is a commutative $K$-algebra
$R$ with two linear operators $P,D:R\rightarrow R$ such that, for any $u,v\in
R$,

(I) (Rota-Baxter relation) $P(u)P(v)=P(uP(v))+P(P(u)v)+\lambda P(uv),$

(II) ($\lambda$-differential relation) $D(uv)=D(u)v+uD(v)+\lambda D(u)D(v),$

(III) $D(P(u))=u$.

It is obvious that any commutative $\lambda$-differential Rota-Baxter algebra
is a commutative algebra with multiple operators $\Omega$ where
$\Omega=\\{P,D\\}$.

In this section, we assume that $\Omega=\\{P,D\\}$. Let $K[X;\Omega]$ be the
free commutative algebra with multiple operators $\Omega$ on $X$, where
$\Omega=\\{P,D\\}$.

Let $X$ be well ordered and $D>P$. For any $u\in\mathcal{Q}(X)$, define
$deg_{{}_{P}}(u)$ the number of $P$ in $u$.

Define an order on $\mathcal{Q}(X)=\bigcup_{n\geq 0}\mathcal{Q}_{n}$ by
induction on $n$. For any $u,v\in\mathcal{Q}_{0}=CS(X)$, we have

$u=x_{1}^{i_{1}}x_{2}^{i_{2}}\cdots x_{t}^{i_{t}}\ \mbox{and}\
v=x_{1}^{j_{1}}x_{2}^{j_{2}}\cdots x_{t}^{j_{t}}$

where each $x_{i}\in X$, $i_{k},j_{l}\geq 0$ and $x_{i}>x_{i+1}$. Then we
define

$u>v\Longleftrightarrow(deg(u),i_{1},i_{2},\cdots,i_{t})>(deg(v),j_{1},j_{2},\cdots,j_{t})\
\ \ \ \mbox{lexicographically}.$

Assume the order on $\mathcal{Q}_{n-1}$ has been defined. For any
$u,v\in\mathcal{Q}_{n}$, we have

$u=u_{1}^{k_{1}}\cdots u_{s}^{k_{s}}\ \mbox{and}\ v=u_{1}^{l_{1}}\cdots
u_{s}^{l_{s}}$

where each $u_{i}\in X\cup\Omega(\mathcal{Q}_{n-1})$, $k_{i},l_{i}\geq 0$ and
$u_{i}>u_{i+1}$. Here, $u_{i}>u_{i+1}$ means one of the following holds:

(a) $u_{i},u_{i+1}\in X$ and $u_{i}>u_{i+1}$,

(b) $u_{i}\in D(\mathcal{Q}_{n-1})$ or $u_{i}\in P(\mathcal{Q}_{n-1})$ and
$u_{i+1}\in X$,

(c) $u_{i}=\theta_{1}(u_{i}^{\prime})$, $u_{i+1}=\theta_{2}(u_{i+1}^{\prime})$
where $\theta_{1},\theta_{2}\in\\{D,P\\}$ and

$(deg(u_{i}),deg_{{}_{P}}(u_{i}),\theta_{1},u_{i}^{\prime})>(deg(u_{i}),deg_{{}_{P}}(u_{i}),\theta_{2},u_{i+1}^{\prime})\
\ \ \mbox{lexicographically}$

Then we define $u>v$ if and only if

$(deg(u),deg_{{}_{P}}(u),bre(u),k_{1},\dots,k_{s})>(deg(v),deg_{{}_{P}}(v),bre(v),l_{1},\dots,l_{s})\
\ \mbox{lexicographically}$ (3)

Then the Order (3) is a monomial order on $\mathcal{Q}(X)$.

Let $S$ be a set consisting of the following commutative $\Omega$-polynomials:

  1. 1

$P(u)P(v)-P(uP(v))-P(P(u)v)-\lambda P(uv),\ \ u,v\in\mathcal{Q}(X)$;

  2. 2

$D(uv)-D(u)v-uD(v)-\lambda D(u)D(v),\ \ \ u,v\in\mathcal{Q}(X)$;

  3. 3

$D(P(u))-u,\ \ \ u\in\mathcal{Q}(X).$

###### Theorem 6.1

With the Order (3) on $\mathcal{Q}(X)$, $S$ is a Gröbner-Shirshov basis in
$K[X;\Omega]$.

Proof. Denote by $i\wedge j$ the composition of $\Omega$-polynomials of type
$i$ and type $j$. The ambiguities of all possible compositions of commutative
$\Omega$-polynomials in $S$ are only as below. In the following list, $i\wedge
j\ \ \ w$ means $w$ is the ambiguity of the composition $i\wedge j$.

$3\wedge 3\ \ \ D(P(u|_{{}_{D(P(v))}}))$, | $3\wedge 2\ \ \ D(P(u|_{{}_{D(vw)}}))$
---|---
$3\wedge 1\ \ \ D(P(u|_{{}_{P(v)P(w)}}))$ |  $2\wedge 3\ \ \ D(u|_{{}_{D(P(v))}}w)$
$2\wedge 2\ \ \ D(u|_{{}_{D(vw)}}z)$, | $2\wedge 1\ \ \ D(u|_{{}_{P(v)P(w)}}z)$
$1\wedge 3\ \ \ P(u|_{{}_{D(P(v))}})P(w)$ | $1\wedge 2\ \ \ P(u|_{{}_{D(vw)}})P(z)$
$1\wedge 1\ \ \ P(z)P(v)P(w)$, |  $1\wedge 1\ \ \ P(v)P(u|_{{}_{P(w)P(z)}})$

where $u\in\mathcal{Q}^{\star}(X),v,w,z\in\mathcal{Q}(X)$. It is easy to check
that all these compositions are trivial. Similar to the proofs in Theorem 4.1
and Theorem 5.1, $1\wedge 1$ and $2\wedge 2$ are trivial. Others are also
easily checked. Here we just check one for examples.

$\displaystyle 2\wedge 3$ $\displaystyle=$
$\displaystyle-D(u|_{{}_{D(P(v))}})w-u|_{{}_{D(P(v))}}D(w)-\lambda
D(u|_{{}_{D(P(v))}})D(w)+D(u|_{{}_{v}}w)$ $\displaystyle\equiv$
$\displaystyle-D(u|_{{}_{v}})w-u|_{{}_{v}}D(w)-\lambda D(u|_{{}_{v}})D(w)$
$\displaystyle+D(u|_{{}_{v}})w+u|_{{}_{v}}D(w)+\lambda D(u|_{{}_{v}})D(w)$
$\displaystyle\equiv$ $\displaystyle 0\ mod(S,D(u|_{{}_{D(P(v))}}w)).\ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \blacksquare$

Let $D^{\omega}(X)=\\{D^{i}(x)|i\geq 0,x\in X\\}$, where $D^{0}(x)=x$. Define

$\displaystyle\Upsilon_{0}$ $\displaystyle=$ $\displaystyle
CS(D^{\omega}(X)),$ $\displaystyle\Upsilon_{1}$ $\displaystyle=$
$\displaystyle\Upsilon_{0}\cup
P(\Upsilon_{0})\cup\Upsilon_{0}P(\Upsilon_{0}),$ $\displaystyle\vdots\ \ $
$\displaystyle\ \ \ \ \vdots$ $\displaystyle\Upsilon_{n}$ $\displaystyle=$
$\displaystyle\Upsilon_{0}\cup
P(\Upsilon_{n-1})\cup\Upsilon_{0}P(\Upsilon_{n-1}),$ $\displaystyle\vdots\ \ $
$\displaystyle\ \ \ \ \vdots$

and

$\Upsilon(D^{\omega}(X))=\bigcup_{n\geq 0}\Upsilon_{n}.$

###### Theorem 6.2

$Irr(S)=\Upsilon(D^{\omega}(X))$ is a basis of $K[X;\Omega|S]$.

Proof: By Theorem 3.2 and Theorem 6.1, we can obtain the result easily.
$\blacksquare$

###### Theorem 6.3

$K[X;\Omega|S]$ is a free commutative $\lambda$-differential Rota-Baxter
algebra on set $X$ with a basis $\Upsilon(D^{\omega}(X))$.

Proof: By Theorem 6.2.

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