# Distributed Consensus of Heterogeneous Multi-Agent Systems Based on
Feedforward Control

Liping Zhang, Huanshui Zhang This work was supported by the National Natural
Science Foundation of China (62103240), Youth Foundation of Natural Science
Foundation of Shandong Province (ZR2021QF147), Foundation for Innovative
Research Groups of the National Natural Science Foundation of China(61821004),
Major Basic Research of Natural Science Foundation of Shandong Province
(ZR2021ZD14), High-level Talent Team Project of Qingdao West Coast New Area
(RCTD-JC-2019-05).L. Zhang and H. Zhang are with the College of Electrical
Engineering and Automation, Shandong University of Science and Technology,
Qingdao 266590, China<EMAIL_ADDRESS>hszhang@sdu.edu.cn).

###### Abstract

This paper studies the consensus problem of heterogeneous multi-agent systems
by the feedforward control and linear quadratic (LQ) optimal control theory.
Different from the existing consensus control algorithms, which require to
design an additional distributed observer for estimating the leader’s
information and to solve a set of regulator equations. In this paper, by
designing a distributed feedforward controller, a non-standard neighbor error
system is transformed into a standard linear system, and then an optimal
consensus controller is designed by minimizing a combined state error with
neighbour agents. The proposed optimal controller is obtained by solving
Riccati equations, and it is shown that the corresponding cost function under
the proposed distributed controllers is asymptotically optimal. The proposed
consensus algorithm can be directly applicable to solve the consensus problem
of homogeneous systems. Simulation example indicates the effectiveness of the
proposed scheme and a much faster convergence speed than the existing
algorithm.

###### Index Terms:

Heterogeneous multi-agent system, Distributed feedforward control, Optimal
control.

##  I Introduction

In recent years, multi-agent systems has attracted considerable attentions for
its extensive engineering applications in unmanned aerial vehicles and
satellite formation, distributed robotics and wireless sensor networks [1, 2,
3]. Consensus problem of multi-agent system is an important and fundamental
problem, whose essential task is to design a distributed controller based on
the interaction with neighbor agents such that all agents achieve an
agreement. A general framework of the consensus problem for first-order
integrator networks is addressed with a fixed topology in [4]. [5] further
proposes leader-following consensus algorithms for second-order integrator
agents. [6] and [7] develop necessary and sufficient conditions for
consensusability of discrete-time and continuous-time multi-agent systems,
respectively. Variants of these algorithms have been designed with different
communication topologies, system dynamics and performance requirements, see
the works [8, 9, 10, 11] and the references therein. It is noted that the
aforementioned works focus on homogeneous multi-agent systems with the
identical dynamics for each agent.

However, in practice, many systems are heterogeneous with different system
dynamics and even state space dimension. Under the circumstances, the
traditional state consensus algorithm based on Kronecker product method is not
applicable, so the consensus problem of heterogenous multi-agent system is
more challenging than the homogeneous case. In recent years, the distributed
feedforward control approach [12, 13] has been widely used in solving the
consensus problem of heterogeneous multi-agent systems. Based on the
feedforward control theory framework, researchers also further studied the
robust output regulation problem [14], leader-follower consensus for uncertain
nonlinear system subject to communication delays and switching networks [15],
and bipartite output consensus over cooperative-competition networks [16], to
name just a few. The core of this method includes two points: one significant
point is the design of distributed observer or compensator to estimate the
leader’s state; another necessary part is the solving of output regulation
equations. It’s should be pointed that the chosen of the coupling gain to
ensure the stability of distributed observer is dependent on the eigenvalues
of the communication topology. Besides, in light with the distributed internal
model method [17], some effort has also been done to the consensus of
heterogeneous systems [18], while an internal model requirement is necessary
and sufficient for synchronizability of heterogeneous agents, and a
transmission zero condition never holds when the dimension of the system
output is greater than that of the system input. Moreover, to our knowledge,
the existing consensus algorithms for heterogeneous multi-agent systems may be
hard directly reduced to that of the homogeneous case.

Inspired by the above analyses, in this paper, we study the consensus problem
of heterogeneous multi-agent systems with a novel consensus control protocol
based on the feedforward control and LQ optimal control theory. Compared with
the existing results, the main contributions of this work are: 1) Since each
agent is of different dynamics, we first design a distributed feedforward
controller based on the state and input from neighbor agents, a non-standard
error system with neighbour agents can be converted into a standard linear
system. Then, an optimal feedback controller based on the observer
incorporating each agent’s historical state information is designed by
minimizing the state error of different neighbour agents; 2) The proposed
feedback controller is obtained by solving Riccati equations, and the
corresponding global cost function under the proposed controllers are
asymptotically optimal; 3) No pre-design an additional distributed observer
for the reference system, and remove the solvability assumption of output
regulation equations, the proposed consensus control algorithm can be directly
reduced to solve the consensus problem of the homogeneous systems, which is
entirely different from the existing consensus algorithms.

The following notations will be used throughout this paper:
$\mathbb{R}^{n\times m}$ represents the set of $n\times m$-dimensional real
matrices. $I_{p}$ is the identity matrix with a dimension $p\times p$
$\mbox{diag}\\{a_{1},a_{2},\cdots,a_{N}\\}$ denotes the diagonal matrix with
diagonal elements being $a_{1},\cdots,a_{N}$. $\|x\|$ is the 2-norm of a
vector $x$. $\rho(A)$ is the spectral radius of matrix $A$. $A^{{\dagger}}$
and $A^{T}$ denote the transpose and the Moore-Penrose inverse of a matrix
$A$. $Rang(A)$ denotes the range of $A$.

Let the interaction among $N+1$ agents be described by a directed graph
$\mathcal{G}=\\{\mathcal{V},\mathcal{E},\mathcal{A}\\}$, where
$\mathcal{V}=\\{0,1,2,\cdots,N\\}$ is the set of vertices (nodes),
$\mathcal{E}\subseteq\mathcal{V}\times\mathcal{V}$ is the set of edges. Here,
the node $0$ is associated with the leader system, and the node $i=1,\cdots,N$
is associated with the followers systems.
$\mathcal{A}=[a_{ij}]\in\mathbb{R}^{(N+1)\times(N+1)}$ is the weight matrix of
$\mathcal{G}$, $a_{ij}\neq 0$ if and only if the edge $(i,j)\in\mathcal{E}$,
i.e., agent $j$ can use the state or the input of agent $i$ for control. The
neighbor of agent $i$ is denoted by
$\mathcal{N}_{i}=\\{j|(i,j)\in\mathcal{E}\\}$. A directed graph has a directed
spanning tree if there exists a node called root such that there exists a
directed path from this node to every other node.

##  II Problem Formulation

We consider a heterogeneous discrete-time multi-agent system consisting of $N$
agents over a directed graph $\mathcal{G}$ with the dynamics of each agent
given by

$\displaystyle x_{i}(k+1)=A_{i}x_{i}(k)+B_{i}u_{i}(k),i=1,2,\cdots,N$ (1)

where $x_{i}(k)\in\mathbb{R}^{n}$ and $u_{i}(k)\in\mathbb{R}^{m_{i}}$ are the
state and the input of each agent. $A_{i}\in\mathbb{R}^{n\times n}$ and
$B_{i}\in\mathbb{R}^{n\times m_{i}}$ are the coefficient matrices,
respectively.

The dynamic of the leader is given by

$\displaystyle x_{0}(k+1)=A_{0}x_{0}(k),$ (2)

where $x_{0}\in\mathbb{R}^{n\times n}$ is the state of the leader, and
$A_{0}\in\mathbb{R}^{n\times n}$ is the system matrix.

The cost function of multi-agent systems (1) is given by

$\displaystyle J(s,\infty)$
$\displaystyle=\sum_{k=s}^{\infty}\left(\sum_{i=1}^{N}\sum_{j\in\mathcal{N}_{i}}(x_{i}(k)-x_{j}(k))^{T}Q(x_{i}(k)-x_{j}(k))\right.$
$\displaystyle\quad\left.+\sum_{i=1}^{N}u_{i}^{T}(k)R_{i}u_{i}(k)\right)$ (3)

where $Q\geq 0$ and $R_{i}>0$ are weighting matrices.

We aim to design a distributed control protocol $u_{i}(k)$ based on the
available information from neighbors for (1) to minimize the performance (II).
In this case, heterogeneous multi-agent systems (1) and (2) can achieve the
leader-follower consensus, i.e., for any initial conditions $x_{i0}$,

$\displaystyle\lim_{k\to\infty}\|x_{i}(k)-x_{0}(k)\|=0,\quad i=1,\cdots,N.$

######  Assumption 1.

The directed graph $\mathcal{G}$ has a spanning tree with the node $0$ as the
root, i.e., $a_{i0}>0$.

##  III Main Results

###  III-A State consensus of heterogeneous multi-agent systems

Define the relative state error variables as:

$\displaystyle e_{ij}(k)=x_{i}(k)-x_{j}(k),$

then, the state error system is given by

$\displaystyle e_{ij}(k+1)$
$\displaystyle=A_{i}e_{ij}(k)+(A_{i}-A_{j})x_{j}(k)$
$\displaystyle\quad+B_{i}u_{i}(k)-B_{j}u_{j}(k)$ (4)

with $i=1,\cdots,N$, $j\in\mathcal{N}_{i}$ and $B_{0}=0$.

Design the controllers as

$\displaystyle u_{i}(k)=\tilde{u}_{i}(k)+\bar{u}_{i}(k),$ (5)

where the feedforward control $\tilde{u}_{i}(k)$ is designed as:

$\displaystyle\tilde{u}_{i}(k)=-B_{i}^{-1}(A_{i}-A_{j})x_{j}(k)+B_{i}^{-1}B_{j}u_{j}(k),$
(6)

which only needs the state and input information from the neighbor agent $j$,
so $\tilde{u}_{i}(k)$ in (6) is distributed. Under the feedforward control
(6), the relative neighbor error system (III-A) is rewritten as the following
standard linear system:

$\displaystyle e_{ij}(k+1)$
$\displaystyle=A_{i}e_{ij}(k)+B_{i}\bar{u}_{i}(k).$ (7)

Then, the corresponding global error system is

$\displaystyle E(k+1)=\tilde{A}E(k)+\tilde{B}\bar{u}(k),$ (8)

where
$E(k)=\begin{bmatrix}e_{10}^{T}(k)&\cdots&e_{ij}^{T}(k)\cdots&e_{N,N-1}^{T}(k)\end{bmatrix}^{T}$,
$\bar{u}(k)=\begin{bmatrix}\bar{u}_{1}^{T}(k)&\bar{u}_{2}^{T}(k)\cdots&\bar{u}_{N}^{T}(k)\end{bmatrix}^{T}$,
$\tilde{A}=diag\\{A_{1},A_{2},\cdots,A_{N}\\}$, and
$\tilde{B}=diag\\{B_{1},B_{2},\cdots,B_{N}\\}$.

In this case, the cost function (II) is reformulated as

$\displaystyle J(s,\infty)$
$\displaystyle=\sum_{k=s}^{\infty}\left(\sum_{i=1}^{N}\sum_{j\in\mathcal{N}_{i}}e_{ij}^{T}(k)Qe_{ij}(k)+\sum_{i=1}^{N}\bar{u}_{i}^{T}(k)R_{i}\bar{u}_{i}(k)\right)$
$\displaystyle=\sum_{k=s}^{\infty}[E^{T}(k)\mathcal{Q}E(k)+\bar{u}^{T}(k)R\bar{u}(k)],$
(9)

with $\mathcal{Q}=I_{N}\otimes Q$ and $R=diag\\{R_{1},\cdots,R_{N}\\}$.

######  Remark 1.

The designed distributed feedforward controller $\tilde{u}_{i}(k)$ in (6)
needs the invertibility of the matrix $B_{i},i=1,\cdots,N$. When $B_{i}$ is
irreversible, for any $x_{i}(k)\in\mathbb{R}^{n}$, the equation
$B_{i}\tilde{u}_{i}(k)=(A_{i}-A_{j})x_{i}(k)$ has a solution if and only if
$Rang(A_{i}-A_{j})\subseteq Rang(B_{i})$, then, for $\forall
z(k)\in\mathbb{R}^{n}$, the feedforward controller is given by

$\displaystyle\tilde{u}_{i}(k)=-B_{i}^{\dagger}(A_{i}-A_{j})x_{i}(k)+(I-B_{i}^{\dagger}B_{i})z(k).$

Besides, when the system $(A_{i},B_{i})$ is uncontrollable, the solution for
$\tilde{u}_{i}(k)$ can also be obtained by mismatch control method proposed in
[19].

The following key task is to determine the feedback controller
$\bar{u}_{i}(k)$ based on LQ optimal control theory. In the ideal (complete
graph) case, the error information $E(k)$ is available for all agents, the
solvability of the optimal control problem for system (8) with the cost
function (III-A) is equivalent to the following standard LQ optimal control
problem [20].

######  Lemma 1.

[20] Suppose that $E(k)$ is available for all agents, the optimal controller
is given

$\displaystyle\bar{u}^{*}(k)=KE(k)$ (10)

where the feedback gain $K$ is given by

$\displaystyle K=-(R+\tilde{B}^{T}P\tilde{B})^{-1}\tilde{B}^{T}P\tilde{A}$
(11)

and $P$ is the solution of the following ARE

$\displaystyle
P=\tilde{A}^{T}P\tilde{A}+\mathcal{Q}-\tilde{A}^{T}P\tilde{B}(R+\tilde{B}^{T}P\tilde{B})^{-1}\tilde{B}^{T}P\tilde{A}.$
(12)

The corresponding optimal cost function is

$\displaystyle J^{*}(s,\infty)=E^{T}(s)PE(s).$ (13)

Moreover, if $P$ is the unique positive definite solution to (12), then
$\tilde{A}+\tilde{B}K$ is stable.

It’s obvious that multi-agent systems (1) and (2) can achieve leader-follower
consensus under the centralized optimal controller (10). However, when $E(k)$
is not available for all agents, we next will design the decentralized
controller based on observers incorporating agent’s historical state
information.

To proceed further, the system (8) is rewritten as

$\displaystyle E(k+1)$
$\displaystyle=\tilde{A}E(k)+\sum_{i=1}^{N}\tilde{B}_{i}\bar{u}_{i}(k),$ (14)
$\displaystyle Y_{i}(k)$ $\displaystyle=H_{i}E(k),i=1,\cdots,N$ (15)

where
$\tilde{B}_{i}=\begin{bmatrix}0&\cdots&B^{T}_{i}&0\cdots&0\end{bmatrix}^{T}$,
$Y_{i}(k)$ is measurement, and $H_{i}$ is composed of $0$ and $I_{n}$, whose
specific forms depend on the interaction among agents.

Next, we design a new distributed feedback controllers as

$\displaystyle\bar{u}_{i}^{*}(k)=K_{i}\hat{E}_{i}(k)$ (16)

where $\hat{E}_{i}(k)$ is a distributed observer based on the available
information of agent $i$ to estimate the global error $E(k)$,

$\displaystyle\hat{E}_{1}(k+1)$
$\displaystyle=\tilde{A}\hat{E}_{1}(k)+\tilde{B}_{1}\bar{u}_{1}^{*}(k)+\tilde{B}_{2}K_{2}\hat{E}_{1}(k)+\cdots$
$\displaystyle\quad+\tilde{B}_{N}K_{N}\hat{E}_{1}(k)+L_{1}(Y_{1}(k)-H_{1}\hat{E}_{1}(k)),$
(17a) $\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\hat{E}_{i}(k+1)$
$\displaystyle=\tilde{A}\hat{E}_{i}(k)+\tilde{B}_{1}K_{1}\hat{E}_{i}(k)+\cdots$
$\displaystyle\quad+\tilde{B}_{i-1}K_{i-1}\hat{E}_{i}(k)+\tilde{B}_{i}\bar{u}_{i}^{*}(k)$
$\displaystyle\quad+\tilde{B}_{i+1}K_{i+1}\hat{E}_{i}(k)+\cdots+\tilde{B}_{N}K_{N}\hat{E}_{i}(k)$
$\displaystyle\quad+L_{i}(Y_{i}(k)-H_{i}\hat{E}_{i}(k))$ (17b)
$\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\hat{E}_{N}(k+1)$
$\displaystyle=\tilde{A}\hat{E}_{N}(k)+\tilde{B}_{1}K_{1}\hat{E}_{N}(k)+\cdots$
$\displaystyle\quad+\tilde{B}_{N-1}K_{N-1}\hat{E}_{N}(k)+\tilde{B}_{N}\bar{u}_{N}^{*}(k)$
$\displaystyle\quad+L_{N}(Y_{N}(k)-H_{N}\hat{E}_{N}(k))$ (17c)

where $K_{i}=\begin{bmatrix}0&\cdots&I&0\cdots&0\end{bmatrix}K$, which is
obtained by solving ARE (12), $L_{i}$ is to be determined later to ensure the
stability of the observers.

######  Theorem 1.

Assumption 1 holds. Consider the global error system (14), and the distributed
control laws (16) with (17). If there exist observer gains
$L_{i},i=1,\cdots,N$ such that the matrix

$\displaystyle
A_{c}=\begin{bmatrix}\Theta_{1}&-\tilde{B}_{2}K_{2}&\cdots&-\tilde{B}_{N}K_{N}\\\
-\tilde{B}_{1}K_{1}&\Theta_{2}&\cdots&-\tilde{B}_{N}K_{N}\\\
\vdots&\vdots&\ddots&\vdots\\\
-\tilde{B}_{1}K_{1}&\cdots&-\tilde{B}_{N-1}K_{N-1}&\Theta_{N}\end{bmatrix}$
(18)

is stable, where
$\Theta_{i}=\tilde{A}+\tilde{B}K-\bar{B}_{i}K_{i}-L_{i}H_{i}$. Then the
observers (17) are stable, i.e.,

$\displaystyle\lim_{k\to\infty}\|\hat{E}_{i}(k)-E(k)\|=0.$ (19)

Moreover, if the Riccati equation (12) has a positive definite solution $P$,
under the distributed controllers (5) with (6) and (16), the multi-agent
systems (1) and (2) can achieve leader-follower consensus.

###### Proof.

Define the observer error vector $\tilde{E}_{i}(k)=E(k)-\hat{E}_{i}(k)$. Then,
combining system (14) with observers (17), one obtains

$\displaystyle E(k+1)$
$\displaystyle=(\tilde{A}+\tilde{B}K)E(k)-\tilde{B}_{1}K_{1}\tilde{E}_{1}(k)$
$\displaystyle-\tilde{B}_{2}K_{2}\tilde{E}_{2}(k)-\cdots-\tilde{B}_{N}K_{N}\tilde{E}_{N}(k)$
(20) $\displaystyle\tilde{E}_{1}(k+1)$
$\displaystyle=(\tilde{A}+\tilde{B}K-\tilde{B}_{1}K_{1}-L_{1}H_{1})\tilde{E}_{1}(k)$
$\displaystyle-\tilde{B}_{2}K_{2}\tilde{E}_{2}(k)-\cdots-\tilde{B}_{N}K_{N}\tilde{E}_{N}(k)$
(21a) $\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\tilde{E}_{i}(k+1)$
$\displaystyle=(\tilde{A}+\tilde{B}K-\tilde{B}_{i}K_{i}-L_{i}H_{i})\tilde{E}_{i}(k)$
$\displaystyle\quad-\tilde{B}_{1}K_{1}\tilde{E}_{1}(k)-\cdots-\tilde{B}_{i-1}K_{i-1}\tilde{E}_{i-1}(k)$
$\displaystyle\quad-\tilde{B}_{i+1}K_{i+1}\tilde{E}_{i+1}(k)-\cdots-\tilde{B}_{N}K_{N}\tilde{E}_{N}(k)$
(21b) $\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\tilde{E}_{N}(k+1)$
$\displaystyle=(\tilde{A}+\tilde{B}K-\tilde{B}_{N}K_{N}-L_{N}H_{N})\tilde{E}_{N}(k)$
$\displaystyle-\tilde{B}_{1}K_{1}\tilde{E}_{1}(k)-\cdots-\tilde{B}_{N-1}K_{N-1}\tilde{E}_{N-1}(k)$
(21c)

According to (21), we have

$\displaystyle\tilde{E}(k+1)=\tilde{A}_{c}\tilde{E}(k),$ (22)

where
$\tilde{E}(k)=\begin{bmatrix}\tilde{E}_{1}^{T}(k),\tilde{E}_{2}^{T}(k),\cdots,\tilde{E}_{N}^{T}(k)\end{bmatrix}^{T}$.
Obviously, if there exist matrices $L_{i}$ such that $A_{c}$ is stable, then
observer errors $\tilde{E}(k)$ converge to zero as $k\to\infty$, i.e., Eq.
(19) holds. Furthermore, it follows from (III-A) and (21) that

$\displaystyle\begin{bmatrix}E(k+1)\\\
\tilde{E}(k+1)\end{bmatrix}=\bar{A}_{c}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}$ (23)

where $\bar{A}_{c}=\begin{bmatrix}\tilde{A}+\tilde{B}K&\Psi\\\
0&A_{c}\end{bmatrix}$ and
$\Psi=\begin{bmatrix}-\tilde{B}_{1}K_{1}&\cdots&-\tilde{B}_{N}K_{N}\end{bmatrix}$.
Since $P$ is the positive definite solution to Riccati equation (12), then
$\tilde{A}+\tilde{B}K$ is stable, based on the LQ control theory, the leader-
follower consensus of multi-agent system (1) can be achieved, i.e.,
$\lim_{k\to\infty}\|x_{i}(k)-x_{0}(k)\|=0$ for $i=1,2,\cdots,N$.

The proof is completed.

∎

Observe from (23) that since $K$ has been given in (11), the consensus error
system relies on $A_{c}$ which is determined by the observer gains $L_{i}$.
Thus, to speed up the convergence of consensus, the remaining problem is to
choose the matrix $L_{i},i=1,2,\cdots,N$ such that the spectral radius of
$A_{c}$ is as small as possible. To this end, we can solve the following
optimization problem: the minimization problem of the maximum singular value
for $A_{c}$.

######  Lemma 2.

Let $\alpha>0$ such that

$\displaystyle A_{c}^{T}A_{c}\leq\alpha I,$

or

$\displaystyle\begin{bmatrix}-\alpha I&A_{c}^{T}\\\ A_{c}&-I\end{bmatrix}\leq
0,$ (24)

where $A_{c}$ is as in (18). Then the optimal gain matrices $L_{i}$ are chosen
by:

$\displaystyle\min_{L_{i}}\alpha\quad\quad\mbox{s.t.}\quad\eqref{Lmi},\quad
i=1,\cdots,N.$ (25)

######  Remark 2.

Lemma 2 provides an approach to choose gain matrices $L_{i}$ such that the
spectral radius of $A_{c}$ is as small as possible. But, sometimes this method
may not guarantee the spectral radius of $A_{c}$ is less than $1$. In that
case, we can determine $L_{i}$ by alternative linear matrix inequality
methods.

Next, we will derive the cost difference between the proposed distributed
controller (16), and the centralized optimal control (10), and then analyze
the asymptotical optimal property of the corresponding cost function.

For the convenience of analysis, denote that

$\displaystyle\mathcal{M}_{1}$
$\displaystyle=(\tilde{A}+\tilde{B}K)^{T}P\Psi-\begin{bmatrix}K_{1}^{T}R_{1}K_{1}&\cdots&K_{N}^{T}R_{N}K_{N}\end{bmatrix},$
$\displaystyle\mathcal{M}_{2}$
$\displaystyle=\begin{bmatrix}K_{1}^{T}R_{1}K_{1}&\cdots&0\\\ 0&\ddots&0\\\
\vdots&\vdots&\vdots\\\
0&\cdots&K_{N}^{T}R_{N}K_{N}\end{bmatrix}+\Psi^{T}P\Psi.$

######  Theorem 2.

Under the proposed distributed controllers (16) and (17) with
$L_{i},i=1,2,\cdots,N$ chosen from the optimization in Lemma 2, the
corresponding cost function is given by

$\displaystyle J^{\star}(s,\infty)$
$\displaystyle=E^{T}(s)PE(s)+\sum_{k=s}^{\infty}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}^{T}\begin{bmatrix}0&\mathcal{M}_{1}\\\
\mathcal{M}_{1}^{T}&\mathcal{M}_{2}\end{bmatrix}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}$ (26)

Moreover, the cost difference between the cost function (2) and the cost under
the centralized optimal control is given by

$\displaystyle\Delta J(s,\infty)$
$\displaystyle=J^{\star}(s,\infty)-J^{*}(s,\infty)$
$\displaystyle=\sum_{k=s}^{\infty}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}^{T}\begin{bmatrix}0&\mathcal{M}_{1}\\\
\mathcal{M}_{1}^{T}&\mathcal{M}_{2}\end{bmatrix}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}.$ (27)

In particular, the optimal cost function difference will approach to zero as
$s$ is sufficiently large. That is to say, the proposed consensus controller
can achieve the optimal cost (asymptotically).

###### Proof.

According to (12) and (III-A), we have

$\displaystyle E^{T}(k)PE(k)-E^{T}(k+1)PE(k+1)$
$\displaystyle=E^{T}(k)(\mathcal{Q}+K^{T}RK)E(k)$
$\displaystyle\quad-\tilde{E}^{T}(k)\Psi^{T}P(\tilde{A}+\tilde{B}K)E(k)$
$\displaystyle\quad-E^{T}(k)(\tilde{A}+\tilde{B}K)^{T}P\Psi\tilde{E}(k)-\tilde{E}^{T}(k)\Psi^{T}P\Psi\tilde{E}(k)$

Based on the cost function (II) under the centralized optimal control, by
performing summation on $k$ from $s$ to $\infty$ and applying algebraic
calculations yields

$\displaystyle E^{T}(s)PE(s)-E^{T}(\infty)PE(\infty)$
$\displaystyle=J(s,\infty)-\sum_{k=s}^{\infty}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}^{T}\begin{bmatrix}0&\mathcal{M}_{1}\\\
\mathcal{M}_{1}^{T}&\mathcal{M}_{2}\end{bmatrix}\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}$ (28)

It follows from Theorem 1 that $\lim_{k\to\infty}E^{T}(k)PE(k)=0$. Therefore,
the corresponding optimal cost function $J^{\star}(s,\infty)$ under the
proposed distributed observer-based controller (16) is derived in (2). In line
with (13), the optimal cost difference (2) holds. Furthermore, we can analyze
the asymptotically optimality for the distributed controller (16), which is
similar to that in [21]. To avoid repetition, it’s not described in this
paper. This proof is completed. ∎

###  III-B Comparison with traditional consensus algorithms

Firstly, the consensus with the new controller has a faster convergence speed
than the traditional consensus algorithms.

In fact, from the closed-loop system (23), one has

$\displaystyle\left\|\begin{bmatrix}E(k)\\\
\tilde{E}(k)\end{bmatrix}\right\|\leq\rho(\bar{A}_{c})\left\|\begin{bmatrix}E(k-1)\\\
\tilde{E}(k-1)\end{bmatrix}\right\|,$ (29)

where $\rho(\bar{A}_{c})$ is the spectra radius of $\tilde{A}+\tilde{B}K$ and
$A_{c}$. In particularly, $\tilde{A}+\tilde{B}K$ is the closed-loop system
matrix obtained by the optimal feedback control (11), that is,
$E(k+1)=(\tilde{A}+\tilde{B}K)E(k)$ while $E(k+1)^{T}\mathcal{Q}E(k+1)$ is
minimized as in (II), so the modulus of the eigenvalues for
$\tilde{A}+\tilde{B}K$ is minimized in certain sense. Besides, based on the
optimization in Lemma 2, we can appropriately select $L_{i}$ such that the
upper bound of the spectral radius $\rho(A_{c})$ is as small as possible. From
these perspectives, $\rho(\bar{A}_{c})$ is made more to be small. This is in
comparison with the conventional consensus algorithms where the maximum
eigenvalue of the matrix $\bar{A}_{c}$ is not minimized and determined by the
eigenvalues of the Laplacian matrix $\mathcal{L}$. Therefore, it can be
expected that the proposed approach can achieve a faster convergence than the
conventional algorithms as demonstrated in the simulation examples in Section
IV.

Second, the cost difference $\Delta J(s,\infty)$ between the new distributed
controller (16) and the centralized optimal control (10) is provided in
Theorem 2, and it is equal to zero as $s\to\infty$. That is to say, the
corresponding cost function under the proposed distributed controllers (16) is
asymptotically optimal.

######  Remark 3.

The proposed distributed consensus algorithms (5) with (6),(16) and (17) can
be directly reduced to the homogeneous case [21], which is totally different
from the relevant research works [13, 22, 17]. In the current works, the
solvability assumption of output regulation equations and an additional
distributed observer to estimate the leader’s state have been removed A kind
of novel distributed controller based on the distributed feedforward control
and observer containing decentralized information is developed to solve the
output consensus problem of heterogeneous systems (1) and (2). Moreover, the
feedback gain matrices $K_{i}$ are obtained by solving Riccati equations,
which never require the calculation of eigenvalues for communication topology.

###  III-C Output consensus of heterogeneous multi-agent systems

Different from the cases discussed in Section III-A, where the state dimension
of each subsystem is identical, we further study the output consensus of
heterogeneous systems with different state space dimensions.

Consider the heterogeneous multi-agent systems consisting of $N$ followers,
whose dynamics are given by

$\displaystyle x_{i}(k+1)$ $\displaystyle=A_{i}x_{i}(k)+B_{i}u_{i}(k),$
$\displaystyle y_{i}(k)$ $\displaystyle=C_{i}x_{i}(k),i=1,2,\cdots,N$ (30)

where $x_{i}(k)\in\mathbb{R}^{n_{i}}$ and $u_{i}(k)\in\mathbb{R}^{m_{i}}$ are
the state and the input of each agent. $A_{i}\in\mathbb{R}^{n_{i}\times
n_{i}}$ ,$B_{i}\in\mathbb{R}^{n_{i}\times m_{i}}$ and
$C_{i}\in\mathbb{R}^{q\times n_{i}}$ are the coefficient matrices,
respectively.

The dynamic of the leader is given by

$\displaystyle x_{0}(k+1)$ $\displaystyle=A_{0}x_{0}(k),$ $\displaystyle
y_{0}(k)$ $\displaystyle=C_{0}x_{0}(k)$ (31)

where $x_{0}\in\mathbb{R}^{p\times p}$ and $y_{0}(k)\in\mathbb{R}^{q}$ is the
state of the leader, $A_{0}\in\mathbb{R}^{p\times p}$ and
$C_{0}\in\mathbb{R}^{q\times p}$ are the coefficient matrices.

The cost function of multi-agent systems (III-C) is given by

$\displaystyle J(s,\infty)$
$\displaystyle=\sum_{k=s}^{\infty}\left(\sum_{i=1}^{N}\sum_{j\in\mathcal{N}_{i}}(y_{i}(k)-y_{j}(k))^{T}Q(y_{i}(k)-y_{j}(k))\right.$
$\displaystyle\quad\left.+\sum_{i=1}^{N}u_{i}^{T}(k)R_{i}u_{i}(k)\right)$ (32)

where $Q\geq 0$ and $R_{i}>0$ are weighting matrices.

Because the state space dimension of each subsystem is not identical, the
above state consensus is meaningless. We need to design a distribute
controller $u_{i}(k)$ to ensure the output of the followers synchronize to the
leader’s output,i.e.,

$\displaystyle\lim_{k\to\infty}\|y_{i}(k)-y_{0}(k)\|=0,\quad i=1,2,\cdots,N.$

To this end, we define relative output error variables:

$\displaystyle\varepsilon_{ij}(k)=y_{i}(k)-y_{j}(k),\quad
j\in\mathcal{N}_{i}.$ (33)

Then,

$\displaystyle\varepsilon_{ij}(k+1)$
$\displaystyle=\varepsilon_{ij}(k)+C_{i}(A_{i}-I_{n_{i}})x_{i}(k)+C_{i}B_{i}u_{i}(k)$
$\displaystyle\quad-C_{j}(A_{j}-I_{n_{j}})x_{j}(k)-C_{j}B_{j}u_{j}(k)$ (34)

with $B_{0}=0,u_{0}(k)=0$ and $j\in\mathcal{N}_{i}$.

Similarly, we design the controllers $u_{i}(k)$ as

$\displaystyle u_{i}(k)=\tilde{u}_{i}(k)+\bar{u}_{i}(k),$ (35)

where the feedforward controller $\tilde{u}_{i}(k)$ is:

$\displaystyle\tilde{u}_{i}(k)$
$\displaystyle=-B_{i}^{-1}(A_{i}-I_{n_{i}})x_{i}(k)+(C_{i}B_{i})^{-1}(C_{j}B_{j})u_{j}(k)$
$\displaystyle\quad+(C_{i}B_{i})^{-1}C_{j}(A_{j}-I_{n_{j}})x_{j}(k).$ (36)

Obvious the distributed feedforward controller $\tilde{u}_{i}(k)$ is designed
by using only the state information of itself and its neighbor agent $j$ and
the control input of neighbor agent $j$. Under the feedforward controller
(III-C), the output error system (III-C) is converted into a standard linear
system

$\displaystyle\varepsilon_{ij}(k+1)=\varepsilon_{ij}(k)+C_{i}B_{i}\bar{u}_{i}(k),$
(37)

the corresponding global form is

$\displaystyle\Upsilon(k+1)=\mathcal{A}\Upsilon(k)+\mathcal{B}\bar{u}(k),$
(38)

where
$\Upsilon(k)=\begin{bmatrix}\varepsilon_{10}^{T}(k)&\cdots&\varepsilon_{ij}^{T}(k)\cdots&\varepsilon_{N,N-1}^{T}(k)\end{bmatrix}^{T}$,
$\bar{u}(k)=\begin{bmatrix}\bar{u}_{1}^{T}(k)&\bar{u}_{2}^{T}(k)\cdots&\bar{u}_{N}^{T}(k)\end{bmatrix}^{T}$,
$\mathcal{A}=I_{N}\otimes I_{q}$, and
$\mathcal{B}=diag\\{C_{1}B_{1},C_{2}B_{2},\cdots,C_{N}B_{N}\\}$.

Then, the cost function (III-C) is reformulated as

$\displaystyle J(s,\infty)$
$\displaystyle=\sum_{k=s}^{\infty}\left(\sum_{i=1}^{N}\sum_{j\in\mathcal{N}_{i}}\varepsilon_{ij}^{T}(k)Q\varepsilon_{ij}(k)+\sum_{i=1}^{N}\bar{u}_{i}^{T}(k)R_{i}\bar{u}_{i}(k)\right)$
$\displaystyle=\sum_{k=s}^{\infty}[\Upsilon^{T}(k)\mathcal{Q}\Upsilon(k)+\bar{u}^{T}(k)R\bar{u}(k)],$
(39)

with $\mathcal{Q}=I_{N}\otimes Q$ and $R=diag\\{R_{1},\cdots,R_{N}\\}$.

Next, we will focus on designing the feedback control $\bar{u}_{i}(k)$ by
minimizing the performance (III-C).

We rewrite the system (38) as

$\displaystyle\Upsilon(k+1)$
$\displaystyle=\mathcal{A}\Upsilon(k)+\sum_{i=1}^{N}\mathcal{B}_{i}\bar{u}_{i}(k),$
(40) $\displaystyle\mathcal{Y}_{i}(k)$
$\displaystyle=F_{i}\Upsilon(k),i=1,\cdots,N$ (41)

where
$\mathcal{B}_{i}=\begin{bmatrix}0&\cdots&(C_{i}B_{i})^{T}&0\cdots&0\end{bmatrix}^{T}$,
$\mathcal{Y}_{i}(k)$ is measurement, and $F_{i}$ is composed of $0$ and
$I_{n}$, whose specific forms depend on the interaction among agents.

Next, we design a new distributed feedback controllers as

$\displaystyle\bar{u}_{i}^{*}(k)=\mathcal{K}_{i}\hat{\Upsilon}_{i}(k)$ (42)

where $\hat{\Upsilon}_{i}(k)$ is a distributed observer based on the available
information of agent $i$ to estimate the global error $\Upsilon(k)$,

$\displaystyle\hat{\Upsilon}_{1}(k+1)$
$\displaystyle=\mathcal{A}\hat{\Upsilon}_{1}(k)+\mathcal{B}_{1}\bar{u}_{1}^{*}(k)+\mathcal{B}_{2}\mathcal{K}_{2}\hat{\Upsilon}_{1}(k)+\cdots$
$\displaystyle\quad+\mathcal{B}_{N}\mathcal{K}_{N}\hat{\Upsilon}_{1}(k)+\Phi_{1}(\mathcal{Y}_{1}(k)-F_{1}\hat{\Upsilon}_{1}(k)),$
(43a) $\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\hat{\Upsilon}_{i}(k+1)$
$\displaystyle=\mathcal{A}\hat{\Upsilon}_{i}(k)+\mathcal{B}_{1}\mathcal{K}_{1}\hat{\Upsilon}_{i}(k)+\cdots$
$\displaystyle\quad+\mathcal{B}_{i-1}\mathcal{K}_{i-1}\hat{\Upsilon}_{i}(k)+\mathcal{B}_{i}\bar{u}_{i}^{*}(k)$
$\displaystyle\quad+\mathcal{B}_{i+1}\mathcal{K}_{i+1}\hat{\Upsilon}_{i}(k)+\cdots+\mathcal{B}_{N}\mathcal{K}_{N}\hat{\Upsilon}_{i}(k)$
$\displaystyle\quad+\Phi_{i}(\mathcal{Y}_{i}(k)-F_{i}\hat{\Upsilon}_{i}(k))$
(43b) $\displaystyle\quad\cdots\quad\cdots\quad\cdots$
$\displaystyle\hat{\Upsilon}_{N}(k+1)$
$\displaystyle=\mathcal{A}\hat{\Upsilon}_{N}(k)+\mathcal{B}_{1}\mathcal{K}_{1}\hat{\Upsilon}_{N}(k)+\cdots$
$\displaystyle\quad+\mathcal{B}_{N-1}\mathcal{K}_{N-1}\hat{\Upsilon}_{N}(k)+\mathcal{B}_{N}\bar{u}_{N}^{*}(k)$
$\displaystyle\quad+\Phi_{N}(\mathcal{Y}_{N}(k)-F_{N}\hat{\Upsilon}_{N}(k))$
(43c)

where
$\mathcal{K}_{i}=\begin{bmatrix}0&\cdots&I&0\cdots&0\end{bmatrix}\mathcal{K}$,
which is obtained by solving a ARE

$\displaystyle\mathcal{P}=\mathcal{A}^{T}\mathcal{P}\mathcal{A}+\mathcal{Q}-\mathcal{A}^{T}P\mathcal{B}(R+\mathcal{B}^{T}\mathcal{P}\mathcal{B})^{-1}\mathcal{B}^{T}\mathcal{P}\mathcal{A}$
(44)

and $\Phi_{i}$ is observer gain matrix to be determined later.

######  Theorem 3.

Under the distributed control laws (42) with (43), consider the global error
system (40), If there exist observer gains $\Phi_{i},i=1,\cdots,N$ such that
the matrix

$\displaystyle
H_{c}=\begin{bmatrix}\Sigma_{1}&-\mathcal{B}_{2}\mathcal{K}_{2}&\cdots&-\mathcal{B}_{N}\mathcal{K}_{N}\\\
-\mathcal{B}_{1}\mathcal{K}_{1}&\Sigma_{2}&\cdots&-\mathcal{B}_{N}\mathcal{K}_{N}\\\
\vdots&\vdots&\ddots&\vdots\\\
-\mathcal{B}_{1}\mathcal{K}_{1}&\cdots&-\mathcal{B}_{N-1}\mathcal{K}_{N-1}&\Sigma_{N}\end{bmatrix}$
(45)

is stable, where
$\Sigma_{i}=\mathcal{A}+\mathcal{B}\mathcal{K}-\mathcal{B}_{i}\mathcal{K}_{i}-\Phi_{i}F_{i}$.
Then the observers (43) are stable under the controller (42), i.e.,

$\displaystyle\lim_{k\to\infty}\|\hat{\Upsilon}_{i}(k)-\Upsilon(k)\|=0.$ (46)

Moreover, if the Riccati equation (44) has a positive definite solution
$\mathcal{P}$, under the distributed controllers (III-C) and (42) with (43),
the heterogeneous multi-agent systems (III-C) and (III-C) can achieve output
consensus. The corresponding cost function is asymptotically optimal.

###### Proof.

This proof is similar to that in Theorem 1 and Theorem 2. So the details are
omitted. ∎

##  IV Numerical Simulation

In this section, we validate the the proposed theoretical results through the
following numerical example.

######  Example 1.

Figure 1: Communication topology among four agents

Consider the multi-agent system consisting of three heterogeneous follower
agents with the system matrices:

$\displaystyle A_{1}$ $\displaystyle=\begin{bmatrix}1.1&1\\\
-2&1\end{bmatrix},A_{2}=\begin{bmatrix}-2&1\\\
1&-1\end{bmatrix},A_{3}=\begin{bmatrix}0.5&0\\\ -2&-0.6\end{bmatrix}$
$\displaystyle B_{1}$ $\displaystyle=\begin{bmatrix}0&1\\\
0.4&0.2\end{bmatrix},B_{2}=\begin{bmatrix}0.1&0.2\\\
1&0.5\end{bmatrix},B_{3}=\begin{bmatrix}0.2&0\\\ 0.1&1\end{bmatrix}$

The leader’s state trajectory is generated by a sinusoidal trajectory
generator, its dynamics is given by:

$\displaystyle A_{0}=\begin{bmatrix}cos(0.5)&sin(0.5)\\\
-sin(0.5)&cos(0.5)\end{bmatrix}.$

The interactions of agents are given in Fig.1, which satisfies Assumption 1.
Each agent only receives neighbor error information, so we can determine
$H_{i}$ as

$\displaystyle
H_{1}=\begin{bmatrix}I_{2}&0&0\end{bmatrix},H_{2}=\begin{bmatrix}0&I_{2}&0\end{bmatrix},H_{3}=\begin{bmatrix}0&0&I_{2}\end{bmatrix}$

We choose

$\displaystyle Q=R_{1}=R_{2}=R_{3}=I_{2}.$

According to ARE (12) and the Lemma (2), the feedback gains $K_{i}$ and the
observer gain $L_{i}$ can be obtained, respectively. Fig.2 displays that the
observer error vectors $\tilde{E}_{i}(k)$ under the proposed controller (16)
converge to zero. The state trajectories of three followers and the leader are
shown in Fig.4, which implies that the state of followers can synchronize with
the state generated by the leader after 16 steps. With the same initial
conditions, applying the existing consensus algorithm in [13], Fig.5 and Fig.6
show the first state trajectories for four agents and the state error
trajectories between agents and the leader, respectively, To further compare
the consensus speed with the traditional method, we calculate the spectral
radius as $\rho(A_{c})=0.7860$ by our method, and according to the existing
method, the spectral radius is $\rho(A_{c})=0.9651$. Therefore, Therefore, the
proposed distributed controller (5) with (6),(16) and (17) can make all agents
tack to the leader with a faster convergence speed.

Figure 2: Observer error trajectories $\tilde{E}_{i}(k)$. Figure 3: The first
state trajectories of each agent $x_{i}(k),i=0,1,2,3$. Figure 4: The first
state error trajectories by the proposed method. Figure 5: The first state
trajectories of each agent by the existing method. Figure 6: The first state
error trajectories by the existing method.

##  V Conclusions

In this paper, we have studied the consensus problem for heterogeneous multi-
agent systems by a new feedforward control approach and optimal control
theory. The proposed new distributed controller has contained two parts: a
distributed feedforward controller has been designed based on the information
of agents and its neighbors, and then an optimal consensus controller based on
observers involving agent’s historical state information has been obtained by
solving Riccati equations. Different from the existing consensus control
algorithms, the current algorithm has removed the requirements for designing
an additional distributed observer and for solving a set of regulator
equations. It is shown that the corresponding cost function under the proposed
distributed controllers was asymptotically optimal. The proposed consensus
algorithm can be directly reduced to the homogeneous systems. Simulation
example indicated the effectiveness of the proposed scheme, the proposed
approach can achieve a faster convergence than the conventional algorithms.

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# Floquet Engineering to Overcome No-Go Theorem of Noisy Quantum Metrology

Si-Yuan Bai* — Key Laboratory of Quantum Theory and Applications of MoE,
Lanzhou Center for Theoretical Physics, and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China  Jun-Hong
An* —<EMAIL_ADDRESS>Key Laboratory of Quantum Theory and Applications of
MoE, Lanzhou Center for Theoretical Physics, and Key Laboratory of Theoretical
Physics of Gansu Province, Lanzhou University, Lanzhou 730000, China

###### Abstract

Permitting a more precise measurement to physical quantities than the
classical limit by using quantum resources, quantum metrology holds a promise
in developing many revolutionary technologies. However, the noise-induced
decoherence forces its superiority to disappear, which is called no-go theorem
of noisy quantum metrology and constrains its application. We propose a scheme
to overcome the no-go theorem by Floquet engineering. It is found that, by
applying a periodic driving on the atoms of the Ramsey spectroscopy, the
ultimate sensitivity to measure their frequency characterized by quantum
Fisher information returns to the ideal $t^{2}$ scaling with the encoding time
whenever a Floquet bound state is formed by the system consisting of each
driven atom and its local noise. Combining with the optimal control, this
mechanism also allows us to retrieve the ideal Heisenberg-limit scaling with
the atom number $N$. Our result gives an efficient way to avoid the no-go
theorem of noisy quantum metrology and to realize high-precision measurements.

##  I Introduction

Aiming at precise measurements to physical quantities, metrology plays a vital
role in developing highly advanced technologies. Restricted by classical
statistics, the metrology precision achievable in classical physics is bounded
by the shot-noise limit (SNL) $N^{-1/2}$ with $N$ being the number of employed
resource in measurements. The advent of quantum metrology reveals that the SNL
can be beaten by quantum effects [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12]. It
has been found that, by using entanglement of quantum probes, one can achieve
a frequency measurement in Ramsey spectroscopy with a precision of Heisenberg
limit (HL) $N^{-1}$. Quantum metrology has versatile applications in next-
generation gyroscope [13, 14, 15, 16], atomic clock [17, 18, 19, 20, 21],
magnetometers [22, 23, 24], and gravimetry [25].

The realization of quantum metrology is challenged by the noise-induced
decoherence in its stability and scalability [26, 27, 28, 29, 30, 31, 32]. It
was found that the Markovian dephasing noise forces the HL in Ramsey
spectroscopy not only to return to the SNL at an optimal encoding time but
also to become divergent in the long-time condition [33]. Being universal for
any Markovian noise [34, 35, 36, 37, 38, 39, 40], these two destructive
consequences are called no-go theorem of noisy quantum metrology [41, 42].
Further studies showed that the non-Markovian effect of the dephasing noise
can reduce the HL to the Zeno limit $N^{-3/4}$ at an optimal time [43, 44, 45,
42, 46, 47]. Many efforts, e.g., adaptive [17, 48] and nondemolition [49]
measurements, correlated decoherence [50], purification [51], error correction
[52, 53, 54, 55, 56, 57, 58], and dynamical control [59, 60], have been
proposed to restore the HL. Although partially recovering the quantum
superiority in its scaling with $N$, the divergence fate of the precision in
the long-time condition does not change. A mechanism to solve the divergence
problem was proposed in [16, 61], but the recovery of the HL for arbitrary $N$
is unavailable. Hence, how to retrieve the HL and overcome the precision
divergence with time simultaneously is still an open question.

Inspired by the advance that Floquet engineering has become a versatile tool
in quantum control [62, 63, 64, 65] and generating novel quantum phases [66,
67, 68], see a review in Ref. [69], we propose a scheme to overcome the no-go
theorem of noisy quantum metrology by Floquet engineering. We discover that,
via applying a periodic driving on the atoms of the Ramsey-spectroscopy-based
quantum metrology under local dissipative noises, the precision divergence in
the long-time condition is avoided as long as a Floquet bound state (FBS) is
formed in the quasienergy spectrum of the system formed by each driven atom
and its local noise. Further using this mechanism, we can completely recover
the HL by optimally designing the driving amplitude. Retrieving the ideal
scaling of the precision with $N$ and $t$ simultaneously, our result paves the
way to realize quantum metrology in practice.

Figure 1: Schematic illustration of noisy quantum metrology under periodic
control. $\hat{\mathcal{U}}_{j}(t)$ is the evolution operator of the $j$th
periodically driven atom interacting with its environment.

##  II Noisy quantum metrology

Quantum metrology pursues a highly precise measurement of physical quantities
by using quantum resources of a probe. To measure a quantity $\theta$ of a
system, we first prepare a probe in a state $\rho_{\text{in}}$ and couple it
to the system to encode $\theta$ into the probe state $\rho_{\theta}$. Then we
measure a certain observable $\hat{O}$ of the probe and infer the value of
$\theta$ from the result. The inevitable errors make us unable to estimate
$\theta$ precisely. According to quantum parameter estimation theory [70, 71],
the ultimate precision of $\theta$ optimizing all possible observable
$\hat{O}$ is constrained by the quantum Cramér-Rao bound
$\delta\theta=1/\sqrt{\upsilon\mathcal{F}_{\theta}}$, where $\delta\theta$ is
the standard error of the estimate, $\upsilon$ is the number of repeated
measurements, and
$\mathcal{F}_{\theta}=\text{Tr}(\hat{L}_{\theta}^{2}\rho_{\theta})$ is the
quantum Fisher information (QFI) characterizing the most information of
$\theta$ extractable from $\rho_{\theta}$. $\hat{L}_{\theta}$ called symmetric
logarithmic derivative is defined as
$\partial_{\theta}\rho(\theta)=(\hat{L}_{\theta}\rho_{\theta}+\rho_{\theta}\hat{L}_{\theta})/2$.
If $\delta\theta\propto N^{-1/2}$, with $N$ being the number of the used
resource, then the precision is called the SNL, which is the achievable limit
of any classical measurement. The SNL can be beaten by using quantum
protocols.

Figure 2: (a) Quasienergy spectrum in different driving amplitude $A$. The
shaded backgrounds mark the regions with the FBS. (b) Evolution of
$\mathcal{F}_{\omega_{0}}/t^{2}$ in different $A$ via numerically solving Eq.
(5). The red line shows the result evaluated from Eq. (7) at $t=31T$. (c)
Evolution of $\mathcal{F}_{\omega_{0}}$ in the absence of the FBS when $A=16h$
(magenta dot-dashed line), $20h$ (green dashed line), and $0$ (black line).
(d) Evolution of $\mathcal{F}_{\omega_{0}}/t$ in the presence of the FBS when
$A=11h$ (light-red line), whose stroboscopic values are given by the red line
marked by the red squares. The lines marked by the orange triangle and the
cyan diamond are the stroboscopic values of $\mathcal{F}_{\omega_{0}}/t$ when
$A=8h$ and $32h$, respectively. The blue dashed lines are the QFI evaluated
from Eq. (6), which match with the stroboscopic values in the long-time limit.
The insert is the comparison of the numerical result (light-red line) with the
one evaluated from the FBS (black dot-dashed line). We use $\omega_{T}=12h$,
$g=\omega_{0}=h$, $\omega_{c}=0$, and $N=20$.

In a Ramsey spectroscopy to measure the atomic frequency $\omega_{0}$ [33],
one chooses $N$ atoms themselves as the probe and prepares their state in a
Greenberger-Horne-Zeilinger (GHZ) type entangled state
$|\psi_{\text{in}}\rangle=(|g\rangle^{\otimes N}+|e\rangle^{\otimes
N})/\sqrt{2}$, where $|g\rangle$ and $|e\rangle$ are the atomic ground and
excited states. Then the free evolution governed by the $j$th atomic
Hamiltonian
$\hat{H}_{0,j}=\omega_{0}\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j}$, with
$\hbar=1$ and $\hat{\sigma}_{j}=|g_{j}\rangle\langle e_{j}|$, encodes
$\omega_{0}$ into the state
$|\psi(t)\rangle=e^{-i\sum_{j}\hat{H}_{0,j}t}|\psi_{\text{in}}\rangle$. The
ultimate precision described by the QFI is evaluated as
$\mathcal{F}_{\omega_{0}}(t)=N^{2}t^{2}$, which scales with $N$ as the HL and
beats the classical SNL $N$ times due to the utilized entanglement [18, 72,
73]. It exhibits the quantum superiority of the metrology scheme. Another
feature is that $\mathcal{F}_{\omega_{0}}(t)$ scales with the encoding time as
$t^{2}$, which also acts as a resource to increase the precision [74].

In practice, the encoding process is influenced by the dissipative
environments, which cause the energy loss and the entanglement degradation of
the probe. The encoding is governed by $\hat{H}=\sum_{j}\hat{H}_{j}$ with

$\hat{H}_{j}=\hat{H}_{0,j}+\sum_{k}[\omega_{k}\hat{a}_{j,k}^{\dagger}\hat{a}_{j,k}+g_{j,k}(\hat{a}_{j,k}\hat{\sigma}_{j}^{\dagger}+\text{H.c.})],$
(1)

where $\hat{a}_{j,k}$ is the annihilation operator of the $k$th mode with
frequency $\omega_{k}$ of the environment felt by the $j$th atom and $g_{j,k}$
is their coupling strength. The coupling is further described by the spectral
density $J(\omega)=\sum_{k}|g_{k}|^{2}\delta(\omega-\omega_{k})$. After
tracing out the environmental degrees of freedom from the dynamics of the
total system, we obtain the non-Markovian master equation [61]

$\dot{\rho}(t)=\sum_{j=1}^{N}\\{-i\omega(t)[\hat{\sigma}_{j}^{\dagger}\hat{\sigma}_{j},\rho(t)]+\gamma(t)\check{\mathcal{L}}_{j}\rho(t)\\},$
(2)

where $\check{\mathcal{L}}_{j}\rho(t)\equiv
2\hat{\sigma}_{j}\rho(t)\hat{\sigma}_{j}^{{\dagger}}-\\{\rho(t),\hat{\sigma}_{j}^{{\dagger}}\hat{\sigma}_{j}\\}$,
$\gamma(t)\equiv-\mathrm{Re}[\dot{c}(t)/c(t)]$ is the dissipation rate, and
$\omega(t)\equiv-\mathrm{Im}[\dot{c}(t)/c(t)]$ is the renormalized frequency.
Here, $c(t)$ is determined by the integro-differential equation

$~{}\dot{c}(t)+i\omega_{0}c(t)+\int_{0}^{t}\nu(t-\tau)c(\tau)d\tau=0,$ (3)

where $\nu(x)\equiv\int_{0}^{\infty}J(\omega)e^{-i\omega x}d\omega$ is the
environmental correlation function and $c(0)=1$. Solving Eq. (2) under the
initial state $|\psi_{\text{in}}\rangle$, we can calculate the corresponding
QFI according to [75, 70]

$\mathcal{F}_{\omega_{0}}=\sum_{i,j}[\lambda^{-1}_{i}{\lambda}_{i}^{\prime
2}+4\lambda_{i}\langle\lambda^{\prime}_{i}|\lambda^{\prime}_{i}\rangle-\frac{8\lambda_{i}\lambda_{j}}{\lambda_{i}+\lambda_{j}}|\langle\lambda_{i}|\lambda^{\prime}_{j}\rangle|^{2}],~{}~{}~{}$
(4)

where the prime denotes the derivative with respect to $\omega_{0}$ and
$\rho(t)=\sum_{i}\lambda_{i}|\lambda_{i}\rangle\langle\lambda_{i}|$.

In the special case when the probe-environment coupling is weak and the
characteristic timescale of the environment is smaller than that of the probe,
we can apply the Markovian approximation to Eq. (3). The approximate solution
reads $c_{\text{MA}}(t)=\exp\\{-\kappa t-i[\omega_{0}+\Delta(\omega_{0})]t\\}$
[76, 77], where $\kappa=\pi J(\omega_{0})$ and
$\Delta(\omega_{0})=\mathcal{P}\int_{0}^{\infty}\frac{J(\omega)}{\omega_{0}-\omega}d\omega$.
Here, $\mathcal{P}$ denotes the Cauchy principal value. The QFI is
$\mathcal{F}^{\text{MA}}_{\omega_{0}}(t)=\frac{2N^{2}t^{2}}{1+(e^{2\kappa
t}-1)^{N}+e^{2N\kappa t}}$. First, we find that
$\mathcal{F}^{\text{MA}}_{\omega_{0}}(t)$ has a maximum
$\max_{t}\mathcal{F}^{\text{MA}}_{\omega_{0}}=0.24/\kappa^{2}$ when
$t_{\text{opt}}=1.11/(\kappa N)$ in the large-$N$ condition. In the Ramsey
spectroscopy, one generally repeats the measurement within a time duration
$T_{R}$. Thus, we have the repeating times $\upsilon=T_{R}/t_{\text{opt}}$.
According to quantum Cramér-Rao bound, the ultimate precision is
$\min_{t}\delta\omega^{\text{MA}}_{0}=(0.22T_{R}N/\kappa)^{-1/2}$, which is
the SNL [33]. Second, we see
$\lim_{t\rightarrow\infty}\mathcal{F}^{\text{MA}}_{\omega_{0}}(t)=0$ and
$\lim_{t\rightarrow\infty}\delta\omega_{0}^{\text{MA}}(t)=\infty$, which
implies the breakdown of the scheme in the long-time condition. Thus, the
noise washes out the quantum superiority. This is called the no-go theorem of
noisy quantum metrology [42, 41].

##  III Floquet engineering

To overcome the no-go theorem, we apply a periodic driving
$\hat{H}_{c,j}(t)=f(t)\hat{\sigma}^{\dagger}_{j}\hat{\sigma}_{j}$, with
$f(t)={A\over 2}[1-\cos(\omega_{T}t)]$ and $\omega_{T}$ being the driving
frequency, on the probe. This is realizable by a time-dependent Zeeman
magnetic field [78, 79, 80, 81]. It is interesting to find that the master
equation in this case takes the same form as Eq. (2), but Eq. (3) becomes

$\dot{c}(t)+i[\omega_{0}+f(t)]c(t)+\int_{0}^{t}d\tau\nu(t-\tau)c(\tau)=0.$ (5)

Because of the dynamical independence of each atom, Eq. (5) is equivalent to
$c(t)=\langle
e_{j},\\{0_{k}\\}|\hat{\mathcal{U}}_{j}(t)|e_{j},\\{0_{k}\\}\rangle$ for each
atom and its local environment, where
$\hat{\mathcal{U}}_{j}(t)=\hat{\mathcal{T}}e^{-i\int_{0}^{t}[\hat{H}_{j}+\hat{H}_{c,j}(\tau)]d\tau}$
is the evolution operator, $|\\{0_{k}\\}\rangle$ is the environmental vacuum
state, and $\hat{\mathcal{T}}$ is the time-ordering operator. According to the
Floquet theorem, $\hat{\mathcal{U}}_{j}(t)$ for the periodic system
$\hat{H}_{j}(t)=\hat{H}_{j}(t+T)$, with $T=2\pi/\omega_{T}$, is expanded as
$\hat{\mathcal{U}}_{j}(t)=\sum_{\alpha}e^{-i\epsilon_{\alpha}t}|u_{\alpha}(t)\rangle\langle
u_{\alpha}(0)|$. Here $\epsilon_{\alpha}$ and
$|u_{\alpha}(t)\rangle=|u_{\alpha}(t+T)\rangle$, being called quasienergies
and quasistationary states, respectively, are determined by the Floquet eigen
equation
$[\hat{H}_{j}(t)-i\partial_{t}]|u_{\alpha}(t)\rangle=\epsilon_{\alpha}|u_{\alpha}(t)\rangle$
[82, 83]. The quasienergy spectrum of a periodically driven two-level system
coupled to an environment constitutes a finite-width quasienergy band and
possibly formed discrete quasienergy levels, which we call FBSs. The
contribution of the quasistationary states from the quasienergy band to the
long-time $c(t)$ tends to zero due to the out-of-phase interference [62, 63].
Only the ones from FBSs survive. Then the long-time solution of Eq. (5) under
the stroboscopic dynamics, i.e., $t=nT$, is [77]

$\displaystyle\lim_{t\rightarrow\infty}c(t)=\sum_{l=1}^{M}Z_{l}e^{-i\epsilon^{\text{b}}_{l}t}$
(6)

when $M$ FBSs are formed, where $Z_{l}=|\langle
u^{\text{b}}_{l}(0)|e_{j},\\{0_{k}\\}\rangle|^{2}$ and
$|u^{\text{b}}_{l}(t)\rangle$ is the $l$th FBS with quasienergy
$\epsilon^{\text{b}}_{l}$.

In the absence of the FBS, it is natural to expect that
$\mathcal{F}_{\omega_{0}}(t)$ tends to zero because $c(t)$ asymptotically
approaches zero. Consistent with the Markovian result, the metrology scheme in
this case is also broken by the dissipation. We focus on the case that one FBS
is formed. Substituting Eq. (6) into Eqs. (2) and (4) under the initial GHZ
state, we obtain [77]

$\lim_{N,t\rightarrow\infty}\mathcal{F}_{\omega_{0}}(t)\simeq
y_{N}(Z^{2})(Nt\partial_{\omega_{0}}\epsilon^{\text{b}})^{2}+{N(\partial_{\omega_{0}}Z^{2})^{2}\over
Z^{2}(1-Z^{2})},$ (7)

where $y_{N}(x)=2x^{N}/[1+x^{N}+(1-x)^{N}]$. We find that Eq. (7) returns to
its ideal $t^{2}$ scaling with the encoding time up to a time-independent
shift of the second term. Furthermore, its scaling relation with $N$ would
return to the HL if $y_{N}(Z^{2})$ is managed to be recast into a
$N$-independent function. It is realizable by assuming
$\text{opt}Z^{2}=e^{-a/N}$, which results in
$y_{N}(Z^{2})\backsimeq\frac{2}{e^{a}+1}$ and

$\text{opt}_{A}\lim_{N,t\rightarrow\infty}\mathcal{F}_{\omega_{0}}(t)=\frac{2}{e^{a}+1}(Nt\partial_{\omega_{0}}\epsilon^{\text{b}})^{2},$
(8)

with $a$ being a positive constant. Then we use $\text{opt}Z^{2}$ to inversely
design the optimal driving amplitude $A$ for given $N$. Thus, it is remarkable
to find that the ideal $(Nt)^{2}$ scaling of $\mathcal{F}_{\omega_{0}}(t)$ is
recovered and the no-go theorem is overcome via Floquet engineering and
optimal control. Note that our result goes beyond the ones in [16, 61], where,
although the error divergence with time was avoided, the sensitivity was even
worse than the SNL for large $N$.

Figure 3: Stroboscopic values of $\mathcal{F}_{\omega_{0}}/N$ at $t=50T$ as a
function of the atom number $N$ in (a) for $A=8h$ (the orange triangle line),
$11h$ (the red square line), and $32h$ (the cyan diamond line), and in (b) for
all $A\in[0,100]h$ supporting the FBS (the gray lines). The inset of (a) shows
the large-$N$ behavior for $A=32h$, where the blue dashed and green dot-dashed
lines are from the first and second terms of Eq. (7), respectively. The cyan
in (b) is the ideal HL. The red line is the numerical result under the
optimized $A$ such that $\text{opt}_{A}y_{N}(Z^{2})=0.5$. The black dashed
line is the analytic result evaluated from Eq. (7). (c) $f_{N}(Z^{2})$ as a
function of $Z^{2}$ for $N$ ranging from $1$ to $19$ with a fixed interval $2$
by the lines from left to the right. The dependence of $Z^{2}$ on $N$ to make
$y_{N}(Z^{2})=0.5$ is denoted by the blue dots in the inset, which are
numerically fitted as $Z^{2}=e^{-1.1/N}$. (d) Optimized $A$ in different $N$
such that $y_{N}(Z^{2})=0.5$, which causes
$\lim_{t\rightarrow\infty}\mathcal{F}_{\omega_{0}}\propto N^{2}$ shown by the
red line in (b). The inset shows $Z^{2}$ in different $A$. The shaded
backgrounds mark the regions where the FBS is formed. Other parameters are the
same as Fig. 2.

##  IV Numerical results

We consider that the environment consists of a two-dimensional structured
reservoir, with a finite bandwidth $8h$, whose dispersion relation reads
$\omega_{\bf k}=\omega_{c}-2h(\cos k_{x}+\cos k_{y})$ [84, 85]. It may be
realized by a $L\times L$ coupled resonator array or optical lattice with
$\omega_{c}$ being the eigen frequency of the resonator or the lattice and $h$
being the nearest-neighbor hopping rate. The atom-environment coupling
strength is $g_{j,{\bf k}}=ge^{-i{\bf k}\cdot{\bf r}_{j}}/L$. A prerequisite
for forming the FBS is the existence of band gaps in the quasienergy spectrum,
which occurs when $\omega_{T}>8h$ [77]. The quasienergy spectrum of the total
system formed by each driven atom and its environment in Fig. 2(a) shows that,
with increasing the driving amplitude $A$, the two branches of FBS
quasienergies residing in the band gaps divide the spectrum into two regions:
without the FBS when $A\in(0,4.9)h\cup(13.9,28.3)h$ and one FBS when
$A\in(4.9,13.9)h\cup(28.3,35.2)h$. Via numerically solving Eq. (5), we obtain
the evolution of $\mathcal{F}_{\omega_{0}}/t^{2}$ in different $A$ [see Fig.
2(b)]. In the region without the FBS, $\mathcal{F}_{\omega_{0}}/t^{2}$
exclusively decays to zero, which exhibits the destructive effect of the no-go
theorem. It is interesting to observe that, as long as the FBS is formed,
$\mathcal{F}_{\omega_{0}}/t^{2}$ tends to a nonzero constant with a persistent
tiny-amplitude oscillation. The red line in Fig. 2(b) from Eq. (7) indicates
that the peaks of the oscillation match with the stroboscopic dynamics. This
confirms that the $t^{2}$ scaling of $\mathcal{F}_{\omega_{0}}$ in the ideal
case is recovered in the long-time condition. A detailed analysis of the FBS-
absence cases in Fig. 2(c) shows that, although being dramatically enhanced
over the one without the driving (black solid line),
$\mathcal{F}_{\omega_{0}}$ under the driving decays to zero finally after a
transient increasing. On the contrary, the stroboscopic values of
$\mathcal{F}_{\omega_{0}}/t$ in the presence of the FBS when $A=11h$ [see the
light-red line in Fig. 2(d)], which match the analytical results in Eq. (6),
show a linear increasing with $t$. This confirms the $t^{2}$ scaling of
$\mathcal{F}_{\omega_{0}}$ in the long-time dynamics. The similar behavior is
observed for $A=8h$ and $32h$ in Fig. 2(d). The matching of the numerical
result with the one from the FBS in the inset of Fig. 2(d) verifies the
dominant role of the FBS in the long-time dynamics. All these results indicate
that the problem of the error divergence with time has been avoided due to the
formation of the FBS.

To reveal whether the HL is retrievable or not when the FBS is formed, we plot
in Fig. 3(a) the long-stroboscopic time $\mathcal{F}_{\omega_{0}}/N$ as a
function of atom number $N$. It indicates a near-linear increasing of
$\mathcal{F}_{\omega_{0}}/N$ with $N$, i.e., $\mathcal{F}_{\omega_{0}}\propto
N^{2}$, only for small $N$. For large $N$, $\mathcal{F}_{\omega_{0}}/N$
exhibits an unwanted exponential decay to a constant governed by the second
term of Eq. (7) [see the inset of Fig. 3(a)]. Being similar to the result in
the absence of the periodic driving [16, 61], such a $N$-dependent behavior is
universal to all the driving amplitude $A$ supporting the FBS [see Fig. 3(b)].
We find that $\mathcal{F}_{\omega_{0}}/N$ in different $A$ fill up the region
below the ideal HL. This implies the possibility to avoid the decay of
$\mathcal{F}_{\omega_{0}}$ with $N$ and thus to retrieve the $N^{2}$ scaling
of $\mathcal{F}_{\omega_{0}}$ by optimizing $A$. For this purpose, we plot in
Fig. 3(c) $y_{N}(Z^{2})$ as a function of $Z^{2}$ in different $N$. It
exponentially decreases with increasing $N$ for a definite $Z^{2}$. Because
all the curves of $y_{N}(Z^{2})$ for different $N$ have two same endpoints,
$(0,0)$ and $(1,1)$, we can always find an optimal $Z^{2}$ for a given $N$ to
make $y_{N}(Z^{2})$ being a constant. By requesting $y_{N}(Z^{2})=0.5$, we
plot in the inset of Fig. 3(c) the needed $Z^{2}$ as a function of $N$. The
numerical fitting reveals $\text{opt}Z^{2}|_{y_{N}(Z^{2})=0.5}=e^{-1.1/N}$,
which permits us to design the optimal $A$ from the numerical correspondence
between $Z^{2}$ and $A$ in the inset of Fig. 3(d). Figure 3(d) shows the
obtained optimal $A$ in different $N$. The corresponding
$\text{opt}\lim_{t\rightarrow\infty}\mathcal{F}_{\omega_{0}}(t)$ is plotted by
the red line in Fig. 3(b), which matches with our analytical result in Eq.
(8). It clearly depicts that the decay with $N$ is avoided and the HL
$\mathcal{F}_{\omega_{0}}\propto N^{2}$ is retrieved by the optimally designed
periodic driving. The results reveal that, given an atom number $N$, we always
have an experimentally realizable optimal periodic driving such that the no-go
theorem of noisy quantum metrology is completely overcome.

##  V Discussion and conclusion

Going beyond the widely used Markovian approximation [17, 48, 49, 50, 51, 52,
53, 54, 55, 56, 57, 58], our scheme is an exact characterization to remove the
noise effects on quantum metrology via periodic driving. This enables us
retrieve the ideal HL scaling of the precision in the Ramsey spectroscopy with
respect to not only the atom number but also the encoding time. It succeeds in
overcoming the challenge set by decoherence on the scalability and stability
of quantum metrology. Although only the GHZ state is studied, our FBS
mechanism is applicable to other initial states, e.g., the spin squeezing
state [5, 86]. Because of the building block role of Ramsey spectroscopy in
developing quantum gyroscope, atomic clock, magnetometers, and gravimetry [13,
14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25], our result has a practical
meaning in realizing these technique innovations. Floquet engineering is
widely used in quantum technologies [62, 63, 64, 65, 66, 67, 68, 69]. Without
resorting to the well designed control pulses [59, 60, 87, 88], our always-on
sinusoidal periodic driving meets the experimental accessibility. The system-
environment bound state in the static case has been observed in circuit QED
[89] and cold atom [90] systems, which provides a support to realize our FBS
in periodically driven systems.

In summary, we have proposed a Floquet engineering scheme to eliminate the
detrimental effects of dissipative noises on quantum metrology. Via applying a
periodic driving on the atoms of the Ramsey-spectroscopy-based quantum
metrology in the presence of local noises, we have found that its precision
characterized by the QFI is essentially determined by the quasienergy-spectrum
feature of the system formed by each driven atom and its noise. Whenever a FBS
is formed in the spectrum, the ideal $t^{2}$ scaling of the QFI with the
encoding time is recovered. Further combining the optimal control to the
driving field, the HL scaling of the QFI with arbitrary atom number $N$ is
also retrieved. Giving a flexible way to solve the long-standing no-go theorem
of noisy quantum metrology, our scheme supplies a guideline to realize the
ultrasensitive measurement in practical situation.

##  VI Acknowledgments

The work is supported by the National Natural Science Foundation (Grants No.
12205128, No. 12275109, No. 12247101, and No. 11834005), China Postdoctoral
Science Foundation (Grants No. BX20220138 and No. 2022M710063), and the
Supercomputing Center of Lanzhou University.

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