\section{Introduction}
In this paper, a robust tracking control scheme is designed for a class of uncertain nonlinear systems.  The design is composed of two steps. In the first step a linearized uncertainty model for the uncertain nonlinear system is developed using a robust feedback linearization approach.
The feedback linearization approach has many applications in the process control and aerospace industries. Using this method, a large class of nonlinear systems can be made to exhibit linear input-output behavior using a nonlinear state feedback control law. Once the input-output map is linearized, any linear controller design method can be used to design a desired controller. One of the limitations of the standard feedback linearization method is that the model of the system must be exactly known. In the presence of uncertainty in the system, exact feedback linearization is not possible and uncertain nonlinear terms remain in the system.

In order to resolve the issue of  uncertainty after canceling the nominal nonlinear terms using the feedback linearization method, several approaches have been considered in the literature \cite{FBRC,NBSS,FBLetter,FBchemical,FBRC,FBbound,FB_robust_Friedovich,FBobserver1}. Most of these approaches use adaptive or related design procedures to estimate the uncertainty in the system. In these methods,  mismatched uncertainties are decomposed into matched and mismatched parts. These methods typically require the mismatched parts not to exceed some maximum allowable  bound \cite{FBoutput}. The existing results that are based on mismatched uncertainties either do not guarantee stability or require some stringent conditions on the nonlinearities and uncertainties to guarantee the stability of the system \cite{FBchemical,FBRC}.

In this paper, we approach the uncertainty issue in a different way and represent the uncertain nonlinear system in an uncertain linearized form. We use a nominal feedback linearization method to cancel the nominal nonlinear terms, and use a generalized mean value theorem to linearize the nonlinear uncertain terms. In our previous work \cite{Rehman_ASJC,CDC02}, the uncertain nonlinear terms are linearized using a Taylor expansion at a steady state operating point by considering a structured representation of the uncertainties. This linearization approach approximates the actual nonlinear uncertainty by considering only the first order terms and neglecting all of the higher order terms.  In \cite{CDC02}, the uncertain nonlinear terms are linearized using a Taylor expansion but an unstructured representation of uncertainty is considered. In both of these methods, the linearized uncertainty model was obtained by ignoring higher order terms.  In \cite{Rehman_ASCC01}, we introduced the linearization of nonlinear terms using using a generalized mean value theorem \cite{Lin_algebra,Mcleod_Mean_value} approach. This method exactly linearizes the uncertain nonlinear terms at an arbitrary point and therefore, no higher order terms exist. In \cite{Rehman_ASCC01}, the upper bound on the uncertainties is obtained by using unstructured uncertainty representations. The bound obtained using an unstructured uncertainty representation may be conservative which may degrade the performance of the closed loop system. In order to reduce conservatism  and to obtain an uncertain linearized model with a structured uncertainty representation, a different approach for obtaining an upper bound is presented here. In contrast to \cite{Rehman_ASCC01}, here we propose a minimax linear quadratic regulation (LQR) \cite{IP} controller which combines with a standard feedback linearization law and gives a stable closed loop system in the presence of uncertainty. Here, we assume that the uncertainty satisfies a certain integral quadratic constraint (IQC).

The paper is organized as follows. Section \ref{sec:PSlqr} presents a description of the considered class of uncertain nonlinear systems. Our approach to robust feedback linearization is given in Section \ref{sec:FBlqr}. Derivation of the linearized uncertainty model and tracking controller for an air-breathing hypersonic flight vehicle (AHFV) along with simulation results are presented in Section \ref{sec:VM}. The paper is concluded in Section \ref{sec:concl} with some final remarks on the proposed scheme.

\section{System Definition}\label{sec:PSlqr}
Consider a multi-input multi-output (MIMO) uncertain nonlinear system with the same number of inputs and outputs as follows:
\begin{align}
\label{eqsystemlqr}
\dot{x}(t)&=f(x,\hat{p})+\sum\limits_{k=1}^m{g_k(x,\hat{p})u_k(t)}+\epsilon \bar{g}(\hat{p},x,u),\nonumber\\
y_i(t)&=\nu_i(x),\quad i=1,2,\cdots,m
\end{align}
where $x(t)\in \mathbb{R}^n$, $u(t)=[u_1.....u_m]^T\in\mathbb{R}^m$, $y(t)=[y_1....y_m]^T\in\mathbb{R}^m$ and $\epsilon\neq0$.
Furthermore, the system has norm bounded uncertain parameters lumped in the vector $\hat{p}\in \mathbb{R}^q$. Also, $f(x,\hat{p})$,  $g_i(x,\hat{p})$, and  $\nu_i(x,\hat{p})$ for $i=1,\cdots,m$ are assumed to be infinitely differentiable (or differentiable to a sufficiently large degree) functions of their arguments. The term $\bar{g}(\hat{p},x,u)$ in (\ref{eqsystemlqr}) is a nonlinear function which represents the couplings in the system.  The full state vector $x$ is assumed to be available for measurement.

\section{Feedback Linearization}\label{sec:FBlqr}
In this section, we first simplify the model (\ref{eqsystemlqr}) so that the term involving $\epsilon$ vanishes. Here we assume that $\vert\epsilon\vert$ is sufficiently small and hence indicates weak coupling. In general, $\epsilon$ depends on the physical parameters of the system and may be available through measurement or known in advance. Instead of neglecting this coupling in the control design, we model $\epsilon \bar{g}(\hat{p},x,u)$ as an uncertainty function $\tilde{g}(\hat{p},\bar{p},x)$ with certain bound, where, $\bar{p}$ denotes a new uncertainty parameter whose magnitude is bounded. The parameter $\bar{p}$ appears due to the removal of coupling terms which depend on the input $u$. Now we can write (\ref{eqsystemlqr}) as follows:
\begin{equation}
\label{eqSSsystem}
\begin{split}
\dot{x}(t)&=\bar{f}(x,p)+\sum\limits_{k=1}^m{g_k(x,\hat{p})u_k(t)}\\
y_i(t)&=\nu_i(x),\quad i=1,2,\cdots,m\\
\end{split}
\end{equation}
where $\bar{f}(x,p)=f(x,\hat{p})+\tilde{g}(\hat{p},\bar{p},x)$ is an infinitely differentiable function and $p=[\hat{p} \quad \bar{p} ]^T$. Also, note that in equation (\ref{eqSSsystem}), which includes the new uncertain parameter $\bar{p}$, we can write the system in terms of a new uncertainty  vector $p=p_0+\Delta p \in \mathbb{R}^{\bar{q}}$, where  $\bar{q}=q+1$. Here, $p_0$ is the vector of the nominal values of the parameter vector $p$ and $\Delta p$ is the vector of uncertainties in the corresponding parameters as follows:
\begin{align*}
&p_{0}=\left[\begin{array}{ccccc}
p_{10} & p_{20} & \cdots &  p_{({\bar{q}}-1)0} & p_{{\bar{q}}0}
\end{array}\right],\\
&\Delta p=\left[\begin{array}{ccccc}
\Delta p_1 & \Delta p_2 & \cdots &  \Delta p_{({\bar{q}}-1)} & \Delta p_{\bar{q}}
\end{array}\right].
\end{align*}
We assume that a bound on $|\Delta p_s|$ is known for each $s\in \{1,\cdots,\bar{q}\}$. We also assume that the functions in the system (\ref{eqSSsystem}) are differentiable. The standard feedback linearization method can be used on the nominal model (without uncertainties) by differentiating each individual element $y_i$ of the output vector $y$ a sufficient number of times until a term containing the control element $u$ appears explicitly. The number of differentiations needed is equal to the relative degree $r_i$ of the system with respect to each output for $i=1,2,~\cdots~m$. Note that a nonlinear system of the form (\ref{eqSSsystem}) with $m$ output channels has a vector relative degree $r=[r_1~r_2~\cdots~r_m]$ \cite{IS}. We assume that the nonlinear system (\ref{eqSSsystem}) has full relative degree; i.e. $\sum\limits_{i=1}^m{r_i}=n$, where $n$ is the order of the system.

It is shown in \cite{CDC01} that in the presence of uncertainties exact cancellation of the nonlinearities is not possible because only an upper bound on the uncertainties is known: Indeed, we obtain
\begin{equation}
\label{eqSsystem}
\begin{split}
\dot{x}(t)&=\underbrace{\bar{f}_0(x,p_0)+\sum\limits_{k=1}^m{g_{k0}(x,\hat{p_0})u_k(t)}}_\text{Nominal part}\\
&+\underbrace{\Delta \bar{f}(x,p)+\sum\limits_{k=1}^m{\Delta g_k(x,p)u_k(t)}}_\text{Uncertain part}\\
y_i(t)&=\nu_i(x),\quad i=1,2,\cdots,m
\end{split}
\end{equation}
where, $\Delta \bar{f}$, and $\Delta g_k$ are the uncertain parts of their respective functions. After taking the Lie derivative of the regulated outputs a sufficient number of times, the system (\ref{eqSsystem})  can be written as follows:
\begin{align}
\label{eqdiffoutputlqr}
\left[\begin{array}{c}
y_1^{r_1}\\
\vdots \\
y_m^{r_m}
\end{array}\right]
&=f_*(x)+g_*(x)u \nonumber\\
&+\left[\begin{array}{c}
L_{\Delta\bar{f}}^{r_1}(\nu_1)+\sum\limits_{k=1}^m L^{r_1-1}_{ \Delta g_k}[L_{ \Delta \bar{f}}( \nu_1)]u_k\\
\vdots\\
L_{\Delta\bar{f}}^{r_m}(\nu_m)+\sum\limits_{k=1}^m L^{r_m-1}_{\Delta g_k}[L_{\Delta \bar{f}}(\nu_m)]u_k
\end{array}\right],
\end{align}
where,
\begin{footnotesize}
\begin{align*}
f_*(x)&=[L_{\bar{f}_0}^{r_1}(\nu_1) \cdots L_{\bar{f}_0}^{r_m}(\nu_m)]^T,\\
g_*(x)&=\left[
\begin{array}{cc}
L_{g_{10}}L_{\bar{f}_0}^{r_1-1}(\nu_1)& L_{g_{20}}L_{\bar{f}_0}^{r_1-1}(\nu_1)\dots L_{g_{m0}}L_{\bar{f}_0}^{r_1-1}(\nu_1)\\
L_{g_{10}}L_{\bar{f}_0}^{r_2-1}(\nu_2) & L_{g_{20}}L_{\bar{f}_0}^{r_2-1}(\nu_2)\dots L_{g_{m0}}L_{\bar{f}_0}^{r_2-1}(\nu_2)\\
\vdots &\vdots\quad\quad\quad\quad\quad\quad\vdots \\
L_{g_{10}}L_{\bar{f}_0}^{r_m-1}(\nu_m)&L_{g_{20}}L_{\bar{f}_0}^{r_m-1}(\nu_m)\dots L_{g_{m0}}L_{\bar{f}_0}^{r_m-1}(\nu_m)
\end{array}\right],
\end{align*}
\end{footnotesize}
and the Lie derivative of the functions $\nu_i$ with respect to the vector fields $\bar{f}$ and $g_k$ are given by
\begin{footnotesize}
\begin{align*}
L_{\bar{f}} \nu_i=&\frac{\partial \nu_i (x)}{\partial x} \bar{f} ,~ L_{\bar{f}}^j\nu_i=L_{\bar{f}} (L_{\bar{f}}^{j-1}\nu_i (x))
,~ L_{g_k }(\nu_i)=\frac{\partial \nu_i (x)}{\partial x}g_k.
\end{align*}
\end{footnotesize}
Note that in equation (\ref{eqdiffoutputlqr}), we have deliberately lumped the uncertainties at the end of a chain of integrators. This is because the uncertainties in $y^{1}_i,\cdots,y^{r_i-1}_i$ will be included in the diffeomorphism, which will be defined in the sequel. This definition of the diffeomorphism is in contrast to \cite{CDC01}, where the uncertainties in $y^{1}_i,\cdots,y^{r_i-1}_i$ are assumed to be zero; i.e. they satisfy a generalized matching condition\cite{NBSS}.

The feedback control law
\begin{equation}
\label{equ}
u=-g_*(x)^{-1}f_*(x)+g_*(x)^{-1}v,
\end{equation}
partially linearizes the input-output map (\ref{eqdiffoutputlqr}) in the presence of uncertainties as follows:

\begin{align}
\label{eqsdiffoutputm}
&y^{r_i}_*=\underbrace{\left[
\begin{array}{c}
v_1\\
\vdots\\
v_m
\end{array}\right]}_\text{Nominal part}+
\underbrace{\left[
\begin{array}{c}
\Delta W_1^{r_1}(x,u,p)\\
\vdots\\
\Delta W_m^{r_m}(x,u,p)
\end{array}\right]}_\text{Uncertainty part},
\end{align}
where
$\Delta W_i^{r_i}(x,u,p)=L_{\Delta \bar{f}}^{r_i}( \nu_i)+\sum\limits_{k=1}^m L^{r_i-1}_{\Delta g_k}[L_{\bar{f}}( \nu_i)]u_k$,  $y_*=[y_1^{r_1} .... y_m^{r_m}]^T$, and $v=[v_1 .... v_m]^T$ is the new control input vector. Furthermore, we define an uncertainty vector \begin{small}$\Delta W_i$\end{small} which represents the uncertainty in each derivative of the $i^{th}$ regulated output as
\begin{equation}
\label{eqWi}
\begin{split}
\Delta W_i(x,u,p)
&=\left[\begin{array}{c}
L_{\Delta \bar{f}}(\nu_i)\\
L_{\Delta \bar{f}}^{2}(\nu_i)\\
\vdots\\
L_{\Delta \bar{f}}^{r_i}( \nu_i)+\sum\limits_{k=1}^m L^{r_i-1}_{\Delta g_k}[L_{\bar{f}}( \nu_i)]u_k
\end{array}\right]\\&=\left[\begin{array}{c}
\Delta W_i^{1}(x,u,p)\\
\Delta W_i^{2}(x,u,p)\\
\vdots\\
\Delta W_i^{r_i}(x,u,p)
\end{array}\right],
\end{split}
\end{equation}
and write $y_i$ for $i=1,2,\cdots,m$ as given below.
\begin{small}
\begin{equation}
\label{eqyi}
\left[\begin{array}{c}
y_i^1\\
y_i^2\\
\vdots\\
y_i^{r_i}
\end{array}\right]=\left[\begin{array}{c}
0\\
0\\
\vdots\\
v_i
\end{array}\right]+\left[\begin{array}{c}
\Delta W_i^{1}(x,u,p)\\
\Delta W_i^{2}(x,u,p)\\
\vdots\\
\Delta W_i^{r_i}(x,u,p)
\end{array}\right].
\end{equation}
\end{small}
Let us define a nominal diffeomorphism similar to the one defined in \cite{Rehman_ASCC01} for each partially linearized system in (\ref{eqyi}) for $i=1,\cdots, m$ as given below:
\begin{equation}
\label{eqdiffm}
\chi_i=T_i(x,p_0)=\left[\begin{array}{ccccc}
\int y_i-y_{ic} & y_i-y_{ic} & y^{1}_i & .. & y^{r_i-1}_i\end{array}\right]^{T}.
\end{equation}
Using the diffeomorphism (\ref{eqdiffm}) and system (\ref{eqyi}), we obtain the following:
\begin{equation}
\label{eqfullBvsky}
\dot{\chi}=A{\chi}+B v+\Delta \bar{W}(\chi,v,p),\quad\quad
\end{equation}
where $\chi(t)=[\chi_1^T(t),\cdots,\chi_m^T(t)]^T\in \mathbb{R}^{\bar{n}}$ ($\bar{n}=n+m$), $v(t)=[v_1~v_2 \cdots v_m]^T\in\mathbb{R}^m$ is the new control input vector, $ \Delta \bar{W}(\chi,v,p)=[\Delta \bar{W}_1^T(.), \Delta \bar{W}_2^T(.),~ \cdots ~,  \Delta \bar{W}_m^T(.)]^T$ is a transformed version of $\Delta W (x,u,p)$ using (\ref{eqdiffm}) and $\Delta\bar{W}_i(.)=[w_i^{(1)}(.), w_i^{(2)}(.),~ \cdots,~ w_i^{(r_i)}(.)]^T$ for $i=1,2,~\cdots,~m$. Also,
\[
A=\left[
\begin{array}{ccc}
A_1&  \dots  & 0\\
\vdots   & \ddots & \vdots \\
0 & \dots & A_m
\end{array}\right];
~~B=\left[
\begin{array}{cccc}
\bar{B}_1&  \dots  & 0\\
\vdots   & \ddots & \vdots \\
0 &  \dots & \bar{B}_m
\end{array}\right],
\]
where
\[
A_i=\left[\begin{array}{cccccc}
0 & 1 & 0 & \cdots & 0 & 0\\
0 & 0 & 1 & \cdots & 0 & 0\\
\vdots & &\vdots & &\vdots &\\
0 & 0 & 0 & 0 & 0 & 0\\
\end{array}\right],\quad
\bar{B}_i=\left[\begin{array}{c}
0\\
\vdots
\\
0\\
1\\
\end{array}\right].
\]

In our previous work, these uncertainty terms are linearized at a steady state operating point and all the higher order terms in states, control and parameters are ignored in order to obtain a fully linearized form for (\ref{eqfullBvsky}). In this paper, we adopt a different approach to the linearization of the uncertain nonlinear terms in (\ref{eqfullBvsky}). Here, we perform the linearization of $\Delta \bar{W}(\chi,v,p)$ using a generalized mean value theorem \cite{Lin_algebra, vector_valued_mean} such that no higher order terms exist.
\begin{theorem}\label{th_mean}
Let $\bar{w}_i^{(j)}:\mathbb{R}^{\bar{n}} \rightarrow \mathbb{R}$ be a differentiable mapping on $\mathbb{R}^{\bar{n}}$ with a Lipschitz continuous gradient $\nabla \bar{w}_i^{(j)}$. Then for given $\chi$ and $\chi(0)$ in $\mathbb{R}^{\bar{n}}$, there exists a vector $c_i=\chi+\bar{t}(\chi-\chi(0))$ with $\bar{t}\in [0,1]$, such that
\begin{equation}
\label{eqvect_mean}
\bar{w}_i^{(j)}(\chi)-\bar{w}_i^{(j)}(\chi(0)))=\nabla \bar{w}_i^{(j)}(c_i).(\chi-\chi(0)).
\end{equation}
\end{theorem}
\startproof
For proof, see \cite{vector_valued_mean}.
\finishproof

We can apply Theorem \ref{th_mean} to the nonlinear uncertain part of (\ref{eqfullBvsky}). Let us define a hyper rectangle
\begin{equation}
\mathfrak{B}=\{\left[
\begin{array}{c}
\chi\\
v
 \end{array}\right]:\begin{array}{c}
\underline{\chi_i}\leq\chi_i\leq\bar{\chi_i}\\
\underline{v_i}\leq v_i\leq\bar{v_i}
 \end{array}\},
\end{equation}
where $\underline{\chi_i}$, and $\underline{v_i}$  denote the lower bounds and $\bar{\chi_i}$, and $\bar{v_i}$ denote the upper bounds on the new states and inputs respectively. For this purpose, the gradient of $w_i^{(j)}(.)$ is found by differentiating it with respect to $\chi$ and $v$ at an arbitrary operating point $c_{ij}=(\tilde{\chi}, ~ \tilde{v}, ~ \tilde{p})$ for $i=1,2,\cdots,m$ and $j=1,2,\cdots, r_i$ where, $\left[\tilde{\chi}^T~~ \tilde{v}^T\right]^{T}\in\mathfrak{B}$, and $\tilde{p}\in\Theta$. We assume $w_i^{(j)}(\chi_0,v_0,p_0)=0 $, $\chi(0)=0$, and $v(0)=0$ and write $w_i^{(j)}(.)$ as follows:
\begin{equation}
w_i^{(j)}(\chi,v,p)  = {w'}_i^{(j)}(c_{ij}).[\chi^T \quad v^T]^T.
\end{equation}
 Then $\Delta \bar{W}(.)$ can be written as
\begin{equation}
\label{eqbarW}
\Delta\bar{W}(.)= \Phi \left[
\begin{array}{c}
\chi\\
v
 \end{array}\right],
\end{equation}
where
\[
\Phi=\left[
\begin{array}{c}
{w'}_{1}^{(1)}(c_{11}) \\
\vdots\\
 {w'}_{1}^{(r_1)}(c_{1r_1}) \\
\vdots\\
 {w'}_{m}^{(1)}(c_{m1}) \\
\vdots\\
 {w'}_{m}^{(r_m)}(c_{m r_m})
\end{array}\right].
\]
\subsection{Linearized model with structured uncertainty representation}\label{sec:structure_model}
The equation (\ref{eqfullBvsky}) can be written in a linearized form using (\ref{eqbarW}). Note that the matrix $\Phi$ is unknown. However, it is possible to write bounds on each term ${w'}_{i}^{(j)}(c_{ij})$ in $\Phi$ individually and represent them in a structured form. For this purpose, we define each individual bound as follows:
\begin{equation}
\label{eqrhos}
\begin{split}
\hat{\rho}_{1}&=\max_{c_{11}}(\Vert {w'}_{1}^{(1)}(c_{11}) \Vert),\\
\hat{\rho}_{2}&=\max_{c_{12}}(\Vert {w'}_{1}^{(2)}(c_{12}) \Vert), \\
&\vdots\\
\hat{\rho}_{r_1}&=\max_{c_{1r_1}}(\Vert {w'}_{1}^{(r_1)}(c_{1r_1})\Vert), \\
\hat{\rho}_{r_{1+1}}&=\max_{c_{21}}(\Vert {w'}_{2}^{(1)}(c_{21}) \Vert), \\
&\vdots\\
\hat{\rho}_{r_1+r_2+\cdots+r_m}&=\max_{c_{m r_m }}(\Vert {w'}_{m}^{(r_m)}(c_{m r_m }) \Vert).
\end{split}
\end{equation}
Using the definition in (\ref{eqrhos}) and (\ref{eqbarW}), the model (\ref{eqfullBvsky}) can be rewritten as
\begin{footnotesize}
\begin{equation}
\label{eqgforms}
\begin{split}
\dot {\chi}(t) &=(A+\tilde{C}_{1} \Delta_{1} \tilde{K}_{1}+\tilde{C}_{2} \Delta_{2} \tilde{K}_{2}+\cdots +\tilde{C}_{r_1} \Delta_{r_1} \tilde{K}_{r_1}+\cdots\\
&+\tilde{C}_{\bar{n}} \Delta_{\bar{n}} \tilde{K}_{\bar{n}})\chi(t)
+(B+\tilde{C}_{1} \Delta_{1} \tilde{G}_{1}+\tilde{C}_{2} \Delta_{2} \tilde{G}_{2}+\cdots \\
&+\tilde{C}_{r_1} \Delta_{r_1} \tilde{G}_{r_1}+\cdots
+\tilde{C}_{\bar{n}} \Delta_{\bar{n}} \tilde{G}_{\bar{n}}) v(t),
\end{split}
\end{equation}
\end{footnotesize}
where $\tilde{C}_{k}$ for $k=1,2,\cdots,\bar{n}$ is a $\bar{n} \times 1$  vector  whose $k$th entry is one and the other entries are zeros, $\tilde{K}_{k}$ for $k=1,2,\cdots,\bar{n}$ is a $1 \times \bar{n}$  vector whose $k$th entry is $\hat{\rho}_{k}$ and the other entries are zero,
$\tilde{G}_{1},\cdots,\tilde{G}_{r_{i-1}},\tilde{G}_{r_{i+1}},\cdots, \tilde{G}_{\bar{n}-1}=0$ for $i=1,2,\cdots,m$ and $\tilde{G}_{(.)}$ is a $1 \times m$  vector as defined below:
\begin{footnotesize}
\begin{equation}
\label{eqGs}
\begin{array}{c}
\tilde{G}_{r_1}=[\hat{\rho}_{r_1}~ \hat{\rho}_{r_1} \cdots \hat{\rho}_{r_1}],\\
\tilde{G}_{r_1+r_2}=[\hat{\rho}_{r_1+r_2}~ \hat{\rho}_{r_1+r_2} \cdots \hat{\rho}_{r_1+r_2}],\\
\vdots\\
\tilde{G}_{r_1+r_2+\cdots+r_m}=[\hat{\rho}_{r_1+r_2+\cdots+r_m}~ \hat{\rho}_{r_1+r_2+\cdots+r_m} \cdots \hat{\rho}_{r_1+r_2+\cdots+r_m}],
\end{array}
\end{equation}
\end{footnotesize}
and $\Vert \Delta_{k} \Vert < 1$.
Using the above definitions of variables, we will write the system in a general MIMO form as given below:
\begin{equation}
\label{eqggforms}
\begin{split}
\dot {\chi}(t) &=A\chi(t)+Bv(t)+\sum^{\bar{n}}_{k=1} \tilde{C}_k\zeta_k(t);\\
z_k(t)&=\tilde{K}_k{\chi(t)}+\tilde{G}_kv(t);\quad k=1,2,\cdots,\bar{n},\\
\end{split}
\end{equation}
where $\chi(t)\in \mathbb{R}^{\bar{n}}$ is the state; $\zeta_k (t)=\Delta_{k}[\tilde{K}_k\chi+\tilde{G}_1 v] \in \mathbb{R}$ is the uncertainty input, $z_k(t)\in \mathbb{R}$ is the uncertainty output, $v(t)\in\mathbb{R}^m$ is the new control input vector and $y(t)\in \mathbb{R}^m$ is the measured output vector.

\begin{theorem}
\label{th1m}
Consider the nonlinear uncertain system (\ref{eqsystemlqr}) with vector relative degree $\{r_1,\cdots,r_m\}$ at $x=x(0)$. Suppose also that $\bar{f}(x(0),p)=0$ and $\nu(x(0))=0$. There exist a feedback control law of the form (\ref{equ}) and coordinate transformation (\ref{eqdiffm}), defined locally around $x(0)$ transforming the nonlinear system into an equivalent linear controllable system (\ref{eqggforms}) with  uncertainty norm bound $\hat{\rho}$ for (\ref{eqggforms}) in a certain domain of attraction if
\begin{enumerate}
\item $L_{\Delta \bar{f}}L_{\bar{f}}^jh\neq0$ and $L_{\Delta g}L_{\bar{f}}^jh\neq0$ for all $0\leq i \leq r_i-1$,
\item $r_1+r_2+\cdots+r_m=n$,
\item the matrix $\beta =[g_*(x,p_0)]^{-1}$ exists,
\item The uncertainty satisfies $\vert \Delta \vert \leq 1 \quad \forall {\chi}\in \mathfrak{B}, {v}\in \mathfrak{B} \text{, and}~ p \in \Theta$.
\end{enumerate}
\end{theorem}
\startproof
The proof directly follows from the form of the feedback control law (\ref{equ}) which cancels all the nominal nonlinearities and the linearize the remaining uncertain nonlinear terms using the generalized mean value theorem at $\left[\tilde{\chi}~ \tilde{v}\right]^T\in\mathfrak{B}$, and $\tilde{p}\in \Theta$. Since the generalized mean value theorem allows us to write any nonlinear function as an equivalent linear function, which will be tangent to the nonlinear function at some given points, we can linearize the remaining uncertain nonlinear terms using the generalized mean value theorem. Finally, it is straight forward to write the entire uncertain nonlinear system (\ref{eqsystemlqr}) in the linear controllable form (\ref{eqggforms}) by finding the maximum norm bound  $\hat{\rho}$ in (\ref{eqrhos}) on the linearized uncertain terms in the region being considered, $\left[\tilde{\chi}~ \tilde{v}\right]^T\in\mathfrak{B}$, and ${p}\in \Theta$.
\finishproof

\section{AHFV Example}\label{sec:VM}
\subsection{Vehicle Model}
The nonlinear equations of motion of an AHFV used in this study are taken from the work of Bolender et al \cite{BD01} and the description of the coefficients are taken from Sigthorsson and Serrani \cite{LPV01}. The equations of motion are given below:
\begin{small}
\begin{equation*}
\label{eqn1}
\dot {V}=\frac{T \cos \alpha -D}{m}- g \sin\gamma,~~
\dot {\gamma }=\frac{L+T\sin \alpha }{mV}-\frac{g \cos \gamma }{V}
\end{equation*}
\begin{equation}
\label{eqn2}
\dot {h}=V\sin \gamma,\quad
\dot {\alpha }=Q-\dot {\gamma },~~
\dot {Q}=M_{yy} /I_{yy}
\end{equation}
\begin{equation*}
\label{eqn3}
\ddot{\eta_i}=-2\zeta_{m} w_{m,i} \dot{\eta_i}-w_{m,i}^2 \eta_i+N_i,\qquad i=1,2,3
\end{equation*}
\end{small}
See \cite{LPV01,BD01} for a full description of the variables in this model. The forces and moments in actual nonlinear model are intractable and do not give a closed representation of the relationship between control inputs and controlled outputs. In order to obtain tractable expressions for the purpose of control design, these forces and moments are replaced with curve-fitted approximations in \cite{LPV01} which leads to  a curve-fitted model (CFM). The CFM has been derived by assuming a flat earth and unit vehicle depth and retains the relevant dynamical features of the actual model and also offers the advantage of being analytically tractable \cite{LPV01}. The approximations of the forces and moments are given as follows in \cite{LPV01}:
\begin{small}
\begin{equation}
L\approx \bar{q} S C_L (\alpha ,\delta_e,\delta_c,\Delta \tau_1,\Delta \tau_2 ),
\end{equation}
\begin{equation}
D\approx \bar{q} S C_D (\alpha ,\delta_e,\delta_c,\Delta \tau_1,\Delta \tau_2 ),
\end{equation}
\begin{equation}
M_{yy} \approx z_T T+\bar{q} S \bar {c} C_M(\alpha ,\delta_e,\delta_c,\Delta \tau_1,\Delta \tau_2 ),
\end{equation}
\begin{equation}
T\approx \bar{q}  [\phi C_{T,\phi}(\alpha,\Delta \tau_1,M_{\infty})+C_T(\alpha,\Delta \tau_1,M_{\infty},A_d)],
\end{equation}
\begin{equation}
N_i\approx\bar{q} C_{N_i} [\alpha ,\delta_e,\delta_c,\Delta \tau_1,\Delta \tau_2],\quad i=1,2,3.
\end{equation}
\end{small}
The coefficients obtained from fitting the curves are given as follows. These coefficients are obtained by assuming states and inputs are bounded and only valid for the given range. Here, we remove the function arguments for the sake of brevity:
\begin{small}
\[
C_L=C_L^\alpha \alpha+C_L^{\delta_e} \delta_e+C_L^{\delta_c} \delta_c+C_L^{\Delta\tau_1}\Delta\tau_1+C_L^{\Delta\tau_2}\Delta\tau_2+C_L^0,\]
\[
C_M=C_M^\alpha \alpha+C_M^{\delta_e} \delta_e+C_M^{\delta_c} \delta_c+C_M^{\Delta\tau_1}\Delta\tau_1+C_M^{\Delta\tau_2}\Delta\tau_2+C_M^0,
\]
\begin{align*}
C_D =& C_D^{(\alpha+\Delta\tau_1)^2} (\alpha+\Delta\tau_1)^2+C_D^{(\alpha+\Delta\tau_1)} (\alpha+\Delta\tau_1)\\
&+C_D^{\delta_e^2} \delta_e^2+C_D^{\delta_e} \delta_e+C_D^{\delta_c^2} \delta_c^2+C_D^{\delta_c} \delta_c
+C_D^{\alpha\delta_e}\alpha\delta_e\\
&+C_D^{\alpha\delta_c}\alpha\delta_c+C_D^{\delta\tau_1} \delta\tau_1+C_D^0,
\end{align*}
\begin{align*}
C_{T,\phi}&=C_{T,\phi}^\alpha \alpha+C_{T,\phi}^{\alpha M_{\infty}^{-2}} \alpha M_{\infty}^{-2}+ C_{T,\phi}^{\alpha \Delta\tau_1} \alpha \Delta\tau_1\\
&+C_{T,\phi}^{M_{\infty}^{-2}} M_{\infty}^{-2}+C_{T,\phi}^{{\Delta\tau_1}^2} {\Delta\tau_1}^2+C_{T,\phi}^{{\Delta\tau_1}}{\Delta\tau_1}+C_{T,\phi}^0,
\end{align*}
\[
C_T=C_T^{A_d}A_d+C_T^\alpha \alpha+C_{T}^{M_{\infty}^{-2}} M_{\infty}^{-2}+C_{T}^{{\Delta\tau_1}}{\Delta\tau_1}+C_T^0,
\]
\[
C_{N_i}=C_{N_i}^\alpha \alpha+C_{N_i}^{\delta_e} \delta_e+C_{N_i}^{\delta_c} \delta_c+C_{N_i}^{\Delta\tau_1}\Delta\tau_1+C_{N_i}^{\Delta\tau_2}\Delta\tau_2+C_{N_i}^0.
\]
\end{small}
Here, $M_{\infty}$ is the free-stream Mach number, and $\bar{q}$ is the dynamic pressure, which are defined as follows:
\begin{small}
\begin{equation}
\bar{q}=\frac{\rho(h) V^2}{2},\quad
M_{\infty}=\frac{V}{M_0}.
\end{equation}
\end{small}
Also, $\rho(h)$ is the altitude dependent air-density and $M_0$ is the speed of sound at a given altitude and temperature.
The nonlinear equations of motion have five rigid body states; i.e., velocity $V$, altitude $h$, angle of attack $\alpha$, flight path angle $\gamma$, and pitch rate $Q$. The CFM also has $6$ vibrational modes and they are represented by generalized modal coordinates $\eta_i$. There are four inputs and they are the diffuser-area-ratio $A_d$, throttle setting or fuel equivalence ratio $\phi$, elevator deflection $\delta_e$, and canard deflection $\delta_c$. In this example, tracking of velocity and altitude will be considered.
\subsection{Feedback linearization of the AHFV nonlinear model}\label{sec:CLUM}
\subsubsection{Simplification of the CFM}\label{sec:SM}
The CFM contains input coupling terms in the lift and drag coefficients. We simplify the CFM in a robust way as presented in Section \ref{sec:FBlqr} so that the simplified model approximates the real model and the input term  vanishes in the low order derivatives during feedback linearization.
In the simplification process, we will first remove the flexible states as they have stable dynamics. A canard is introduced in the AHFV model by Bolender and Doman \cite{BD02} to cancel the elevator-lift coupling using an ideal interconnect gain $k_{ec}=-\frac{C_L^{\delta_e}}{C_L^{\delta_c}}$  which relates the canard deflection to elevator deflection ($\delta_c=k_{ec} \delta_e$). In practice, an ideal interconnect gain is hard to achieve and thus exact cancellation of the lift-elevator coupling is not possible. In the simplified model, we assume that the interconnect gain is uncertain with a bound on its magnitude and it  also satisfies an IQC \cite{IP}. The drag coefficient is also affected due to the presence of elevator and canard coupling terms in the corresponding expression. We also model this coupling as uncertainty.
The simplified expressions for lift, moment, drag, and  thrust coefficients now can be written as follows:
\begin{footnotesize}
\begin{equation}
\begin{split}
C_L&=C_L^\alpha \alpha+C_L^0+\Delta C_l,\\
C_M&=C_M^\alpha \alpha+[C_M^{\delta_e}-C_M^{\delta_c}(\frac{C_L^{\delta_e}}{C_L^{\delta_c}})] \delta_e+C_M^0,\\
C_D &= C_D^{(\alpha+\Delta\tau_1)^2} (\alpha)^2+C_D^{(\alpha+\Delta\tau_1)} (\alpha)+C_D^0+\Delta C_d,\\
C_{T,\phi}&=C_{T,\phi}^\alpha \alpha+C_{T,\phi}^{\alpha M_{\infty}^{-2}} \alpha M_{\infty}^{-2}\alpha+C_{T,\phi}^{M_{\infty}^{-2}} M_{\infty}^{-2}+C_{T,\phi}^0,\\
C_T&=C_T^{A_d}A_d+C_T^\alpha \alpha+C_{T}^{M_{\infty}^{-2}} M_{\infty}^{-2}+C_T^0,
\end{split}
\end{equation}
\end{footnotesize}
where $\Delta C_l$ is the uncertainty in the lift coefficient $C_L(\alpha)$ due to the uncertain interconnect gain and $\Delta C_d$ is the uncertainty in the drag coefficient $C_D(\alpha)$ due to the input coupling terms. Furthermore, in order to obtain full relative degree for the purpose of feedback linearization, we dynamically extend the system by introducing second order actuator dynamics into the fuel equivalence ratio input as follows:
\begin{small}
\begin{equation}
\label{eqfuel}
\ddot {\phi }=-2\zeta \omega _n \dot {\phi }-\omega _n^2 \phi +\omega
_n^2 \phi _c.
\end{equation}
\end{small}
After this extension we have two more states $\phi$ and $\dot{\phi}$, and thus the sum of the vector relative degree is equal to the order of the system $n$; i.e. $n=7$ and thus satisfying one of the conditions for feedback linearization\cite{IS}.
\begin{table}
\caption{States, Inputs and Selected Physical Parameters at the Trim Condition}
\label{tab01}
\centering
\begin{tabular}{l c}
\hline\hline
Vehicle Velocity $V$ & $7702.08$ ft/sec\\
Altitude $h$& 85000 ft \\
Fuel-to-air ratio $\phi$ & $0.4388$  \\
Pitch Rate $Q$ & $0$  \\
Angle of Attack $\alpha$ & $-0.0134$ rad \\
Flight Path Angle $\gamma$ & $0$\\
\hline \hline
Elevator Deflection $\delta_e$& $2.005$ deg\\
Canard Deflection $\delta_c$& $2.79$ deg \\
Diffuser Area ratio $A_d$ & $1$ \\
\hline \hline
Reference Area $S$ & $17$ sq-ft.ft$^{-1}$\\
Mean Aerodynamic Chord  $c$ & $17$ ft\\
Air Density $\rho$ & $6.6\times 10^{-5}$~slugs/ft$^3$\\
Mass with 50\% fuel level $m$ & $147.9$ slug. ft$^{-1}$ \\
Moment of Inertia $I_{yy}$ &  $5.0\times 10^{5}$~slugs/ft$^2$/(rad . ft)\\
\hline \hline
\end{tabular}
\end{table}
We use Theorem \ref{th1m} to linearize the AHFV dynamics. The outputs to be regulated are selected as velocity $V$ and altitude $h$ using two inputs, elevator deflection $\delta_e$, and fuel equivalence ratio $\phi_c$. Note that the canard deflection $\delta_c$ is a function of the elevator deflection $\delta_e$ and they are related via an interconnect gain. Also, we fix the diffuser area ratio $A_d$ to unity.
The new simplified model consists of seven rigid states $x=\left[\begin{array}{ccccccccc}
 V & h & \gamma & \alpha & \phi & \dot{\phi} & Q
\end{array}\right]^T$ and can be represented by a general form as follows:
\begin{small}
\begin{align}
\label{equsystemm}
\dot{x}(t)&=\bar{f}(x,p)+\sum\limits_{k=1}^2{g_k(x,p)u_k};\quad
y_i(t)=\nu_i(x),~i=1,2.
\end{align}
\end{small}
where the control vector $u$ and output vector $y$ are defined as
\begin{small}
\[
u=[u_1, u_2]^T=\left[ {\delta _e, \phi _c } \right]^T, y=[y_1, y_2]^T=\left[ {V,h}
\right]^T.
\]
\end{small}
The following set of uncertain parameters are considered for the development of a linearized uncertainty model:
\begin{small}
\begin{align}
p=[
&C_L^\alpha \quad C_M^{\delta_c} \quad C_{T,\phi}^{\alpha M_{\infty}^{-2}} \quad C_{T,\phi}^{M_{\infty}^{-2}} \Delta C_l \quad \Delta C_d \quad \Delta C_T \nonumber\\
&\Delta C_M \quad \Delta C_{T,\phi}]^T \in \mathbb{R}^{9}.
\end{align}
\end{small}
We assume that  $p \in \Theta$, where $\Theta=\{p\in \mathbb{R}^{9}~ |~ 0.9 p_0 \leq p_i \leq 1.1 p_0~~ \text{for}\quad i=1,\cdots,9\}$.
In order to get the linearized uncertain model for the uncertain nonlinear AHFV model, the output $V$ and the output $h$ are differentiated three times and four times respectively using the Lie derivative:
\begin{small}
\begin{equation}
\label{equv3dotm}
\left[
\begin{array}{c}
\stackrel{\dots}{V} \\
h^{4}
\end{array}\right]\underbrace{
=f_*(x,p_0)+
g_*(x,p_0)
\left[
\begin{array}{c}
\delta _e \\
\phi _c
\end{array}\right]}_\text{Nominal nonlinear part}+\underbrace{
\left[\begin{array}{c}
\Delta\stackrel{\dots}{V}(x,u,p) \\
\Delta {h}^4(x,u,p)\end{array}\right]}_\text{Uncertain nonlinear part},
\end{equation}
\end{small}
where
\begin{footnotesize}
\begin{equation*}
f_*(x,p_0)=\left[
\begin{array}{c}
L_f^3 V\\
L_f^4 h
\end{array}\right], \quad
g_*(x,p_0)=\left[\begin{array}{cc}
{L_{g_1}(L_f^2 V)} & {L_{g_2}(L_f^2 V)}\\
{L_{g_1}(L_f^3 h)} & {L_{g_2}(L_f^3 h)}
\end{array}\right],
\end{equation*}
\end{footnotesize}
and $\Delta\stackrel{\dots}{V}(x,u,p_0,\Delta p)$ and $\Delta {h}^4(x,u,p_0,\Delta p)$ are the uncertainties in $\stackrel{\dots}{V}$ and ${h}^4$ respectively.
The application of the control law (\ref{equ}) yields the following:
\begin{small}
\begin{equation}
\label{equfblin}
\left[\begin{array}{c}
\stackrel{\dots}{V} \\
h^{4}
\end{array}\right]=\underbrace{\left[\begin{array}{c}
v_1 \\
v_2
\end{array}\right]}_\text{Nominal linear part}+\underbrace{
\left[\begin{array}{c}
\Delta\stackrel{\dots}{V}(x,u,p) \\
\Delta {h}^4(x,u,p)\end{array}\right]}_\text{Uncertain nonlinear part}.
\end{equation}
\end{small}
Also, by using the fact that there are no uncertainty terms in $\dot{V}$, and $\dot{h}$, we can write linearized input-output map for the AHFV model using (\ref{eqyi}) as follows:
\begin{small}
\begin{equation}
\label{equyfblin}
\left[\begin{array}{c}
\dot{V}\\
\ddot{V}\\
\stackrel{\dots}{V} \\
\dot{h}\\
\ddot{h}\\
\stackrel{\dots}{h} \\
h^{4}
\end{array}\right]=\left[\begin{array}{c}
0\\
0\\
v_1 \\
0\\
0\\
0\\
v_2
\end{array}\right]+
\left[\begin{array}{c}
0\\
\Delta\ddot{V}\\
\Delta\stackrel{\dots}{V}(x,u,p) \\
0\\
\Delta\ddot{h}\\
\Delta\stackrel{\dots}{h}(x,u,p) \\
\Delta {h}^4(x,u,p)\end{array}\right].
\end{equation}
\end{small}
Let us define a diffeomorphism for each system as in (\ref{eqdiffm}) which maps the new vectors $\xi$ and $\eta$ respectively to the original vector $x$ as follows:
\begin{small}
\begin{equation}
\xi=T_1(x,p_0,V_c)
,\quad
\eta=T_2(x,p_0,h_c),
\end{equation}
where
\begin{align*}
&T_1(x,p,V_c)=\left[
\begin{array}{cccc}
\int_0^{t}{(V(\tau)-V_c)}d\tau & V-V_c & \dot{V} & \ddot{V}\end{array}\right]^T,\\
&T_2(x,p,h_c)=\left[
\begin{array}{ccccc}
\int_0^{t}{ (h(\tau)-h_c)}d\tau & h-h_c & \dot{h} & \ddot{h} & \stackrel{\dots}{h}\end{array}\right]^T,
\end{align*}
\end{small}
and $V_c$ and $h_c$ are the desired command values for the velocity and altitude respectively.
We write each diffeomorphism as follows:
\begin{small}
\begin{equation}
\label{eqtransformx}
\chi = T(x,p_0,V_c,h_c),
\end{equation}
where $\chi= \left[\begin{array}{ccccccccc}
\xi_1 & \xi_2 & \xi_3 &\xi_4 & \eta_1& \eta_2& \eta_3& \eta_4& \eta_5 \end{array}\right]^T$, and
$T(x,p,V_c,h_c)=\left[\begin{array}{cc}
T_1(x,p,V_c) \\
T_2(x,p,h_c) \end{array}\right]$.
\end{small}
Now, we can transform the nominal part of (\ref{equyfblin}) into the new states using the transformation (\ref{eqtransformx}) and linearize the uncertainty parts of (\ref{equfblin}) using the generalized mean value theorem  as follows:
\begin{small}
\begin{align}
\label{eqlfinalm}
\left[\begin{array}{c}
\dot{\xi}_1 \\
\dot{\xi}_2 \\
\dot{\xi}_3 \\
\dot{\xi}_4 \\
\dot{\eta_1}\\
\dot{\eta_2}\\
\dot{\eta_3}\\
\dot{\eta_4}\\
\dot{\eta}_5
\end{array}\right]=\left[\begin{array}{c}
0\\
0\\
0\\
v_1 \\
0\\
0\\
0\\
v_2
\end{array}\right]
+
\left[\begin{array}{c}
0\\
0\\
\Delta w_{1}(\tilde{\chi},\tilde{v},p) \\
\Delta w_{2}(\tilde{\chi},\tilde{v},p) \\
0\\
0\\
\Delta w_{3}(\tilde{\chi},\tilde{v},p) \\
\Delta w_{4}(\tilde{\chi},\tilde{v},p) \\
\Delta w_{5}(\tilde{\chi},\tilde{v},p)
\end{array}\right]\chi+
\left[\begin{array}{c}
0\\
0\\
0\\
\Delta \tilde{w}_{1}(\tilde{\chi},\tilde{v},p) \\
0\\
0\\
0\\
0\\
\Delta \tilde{w}_{2}(\tilde{\chi},\tilde{v},p)
\end{array}\right]v.
\end{align}
\end{small}
In this section, we write the equation (\ref{eqlfinalm}) in a structured form as presented in Subsection \ref{sec:structure_model}. Using (\ref{eqrhos}), (\ref{eqgforms}), and (\ref{eqGs}) we can write (\ref{eqlfinalm}) as given below:
\begin{small}
\begin{equation}
\label{eqgforms_ex}
\begin{split}
\dot {\chi}(t) &=(A+\tilde{C}_{1} \Delta_{1} \tilde{K}_{1}+\tilde{C}_{2} \Delta_{2} \tilde{K}_{2}+\cdots +\tilde{C}_{9} \Delta_{9} \tilde{K}_{9})\chi(t)\\
&+(B+\tilde{C}_{1} \Delta_{1} \tilde{G}_{1}+\tilde{C}_{2} \Delta_{2} \tilde{G}_{2}+\cdots +\tilde{C}_{9} \Delta_{9} \tilde{G}_{9}) v(t)
\end{split}
\end{equation}
\end{small}
where
\begin{footnotesize}
\[
\tilde{C}_3=\left[
\begin{array}{ccccccccc}
0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right],\]
\[\tilde{C}_4=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0
\end{array}\right],
\]
\[
\tilde{C}_7=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0
\end{array}\right],\]
\[\tilde{C}_8=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0
\end{array}\right],
\]
\[
\tilde{C}_9=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1
\end{array}\right],\]\[
\tilde{C}_1=\tilde{C}_2=\tilde{C}_5=\tilde{C}_6=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right];
\]
\[
\tilde{K}_3=\left[
\begin{array}{ccccccccc}
\hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3 & \hat{\rho}_3
\end{array}\right],\]
\[\tilde{K}_4=\left[
\begin{array}{ccccccccc}
\hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4 & \hat{\rho}_4
\end{array}\right],
\]
\[
\tilde{K}_7=\left[
\begin{array}{ccccccccc}
\hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7 & \hat{\rho}_7
\end{array}\right],\]
\[\tilde{K}_8=\left[
\begin{array}{ccccccccc}
\hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8 & \hat{\rho}_8
\end{array}\right],
\]
\[
\tilde{K}_9=\left[
\begin{array}{ccccccccc}
\hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9 & \hat{\rho}_9
\end{array}\right],
\]
\[
\tilde{K}_1=\tilde{K}_2=\tilde{K}_5=\tilde{K}_6=\left[
\begin{array}{ccccccccc}
0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0
\end{array}\right];
\]
\[
\tilde{G}_4=\left[
\begin{array}{cc}
\hat{\rho}_4 & \hat{\rho}_4
\end{array}\right],\]
\[\tilde{G}_9=\left[
\begin{array}{cc}
\hat{\rho}_9 & \hat{\rho}_9
\end{array}\right],
\]
\[
\tilde{G}_1=\tilde{G}_2=\tilde{G}_3=\tilde{G}_5=\tilde{G}_6=\tilde{G}_7=\tilde{G}_8=\left[
\begin{array}{cc}
0 & 0
\end{array}\right].
\]
\end{footnotesize}
Using the above definition of the variables, we can write the system in the general MIMO form given in (\ref{eqggforms}), where $\chi(t)\in \mathbb{R}^9$ is the state, $\zeta_k (t)=\Delta_{k}[\tilde{K}_k\chi+\tilde{G}_1 v] \in \mathbb{R}$ is the uncertainty input, $z_k(t)\in \mathbb{R}$ is the uncertainty output, and $v(t)\in\mathbb{R}^2$ is the new control input vector.
\subsection{Minimax LQR Control Design}
The linearized model (\ref{eqggforms}) corresponding to the AHFV uncertain nonlinear model (\ref{eqn2}) allows for the design of a minimax LQR controller for the velocity and altitude reference tracking problem. The method of designing a minimax LQR controller is given in \cite{IP}. Here, we follow the same method and proposed a minimax LQR controller for the linearized system (\ref{eqggforms}). We assume the uncertainty in the system (\ref{eqggforms}) satisfies following IQC and the original state vector $x$ is available for measurement.
\begin{small}
\begin{equation}
\label{eqIQC}
\int_0^{\infty}(\parallel z_j(t)\parallel^2-\parallel \xi_j(t)\parallel^2)dt\geq-\chi^T(0)D_j\chi(0),
\end{equation}
\end{small}
where $D_j>0$ for each $j=1,\cdots,m$ is a given positive definite matrix. The cost function selected is as given below:
\begin{equation}
\label{eqfcost}
F=\int_0^{\infty}{[\chi(t)^T Q \chi(t)+v(t)^T R v(t)]dt},
\end{equation}
where $Q=Q^T>0$ and $R=R^T>0$ are the state and control weighting matrices respectively. A minimax optimal controller can be designed by solving a game type Riccati equation
\begin{small}
\begin{align}
\label{eqARE}
&(A-BE^{-1}G^T K)^T X_\tau +X_\tau(A-BE^{-1}G^T K)\nonumber\\&+X_\tau(CC^T-BE^{-1}B^T)X_\tau
+K^T(I-GE^{-1}G^T)K=0,
\end{align}
where
\[
K=\left[\begin{array}{c}
Q^{1/2} \\
0 \\
\sqrt{\tau_1}\tilde{K}_1 \\
\vdots\\
\sqrt{\tau_9}\tilde{K}_9
\end{array}\right]
,\quad
G=\left[\begin{array}{c}
0 \\
R^{1/2} \\
\sqrt{\tau_1}\tilde{G}_1 \\
\vdots\\
\sqrt{\tau_9}\tilde{G}_9
\end{array}\right]
,\quad
E=GG^T,
\]
\[
C=\left[\begin{array}{ccc}
\frac{1}{\sqrt{\tau_1}}C_1 & \dots & \frac{1}{\sqrt{\tau_9}}C_9
\end{array}\right]
\]
\end{small}
The weighting matrices $Q$ and $R$, and parameters $\tau_k$, for $k=1,2,\cdots,9$ are selected such that they give the minimum bound
\begin{small}
\begin{equation}
\label{eqfbond}
\text{min}[\chi^T(0)X_\tau \chi(0)+\sum_{j=1}^{9}\tau_j \chi^T(0)D_j\chi(0)],
\end{equation}
\end{small}
on the cost function (\ref{eqfcost}). The minimax LQR control law can be obtained by solving the ARE (\ref{eqARE}) for given values of the parameters as given below:
\begin{small}
\begin{equation}
\label{eqcontrllaw_mv}
v(t)=-G_\tau\chi(t),
\end{equation}
where
\[
G_\tau=E^{-1}[B^T X_\tau+G^T K]
\]
\end{small}
is the controller gain matrix. The parameters $Q$ and $R$ are selected intuitively so that required performance can be obtained and $\tau_k$ for $k=1,2,\cdots,9$ correspond to the minimum bound on (\ref{eqfcost}). These parameters are given as follows:
\begin{small}
\begin{equation*}
\label{eqSW_mv}
\mathbf{Q}=\textbf{diag}\left[
\begin{array}{c}
1000, 500, 500, 100, 0.001, 100, 100, 500, 500
\end{array}\right],
\end{equation*}
\begin{equation*}
\label{eqCW_mv}
\mathbf{R}=\left[
\begin{array}{cc}
3.0 & 0\\
0 & 3.0
\end{array}\right],\quad \tau_3=547.9,
\tau_4=8.0,~~ \tau_7=4935.7,
\end{equation*}
\[
\tau_8=4935.3,~~\tau_9=3768.0.
\]
\end{small}
\subsection{Simulation Results} \label{sec:resultslqr_mv}
The closed loop nonlinear AHFV system with the minimax LQR controller (\ref{eqcontrllaw_mv}) is simulated using different sizes and combinations of uncertainty. For the sake of brevity, here we evaluate the performance of the proposed controller by using step input commands for the following three cases:
\begin{itemize}
\item[1.] Uncertain parameters equal to their nominal values, with no uncertainty.
\item[2.] Uncertain parameters $20\%$ lower than their nominal values.
\item[3.] Uncertain parameters $20\%$ larger than their nominal values.
\end{itemize}
The responses of the closed loop system given in Fig. \ref{fig:vstep_mv1} -- Fig. \ref{fig:hstep_flex_mv} show that the minimax LQR controller along with the feedback linearization law gives satisfactory performance.
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=Vstep_lqrp20.eps, scale=0.38}
\caption{System response to a step change in velocity reference.}
\label{fig:vstep_mv1}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=Vstep_control_lqrp20.eps, scale=0.38}
\caption{Control input responses corresponding to a step change in velocity reference.}
\label{fig:vstep_mv}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=Vstep_states_lqrp20.eps, scale=0.38}
\caption{Responses of $\alpha$ and $\gamma$ for a step change in velocity reference.}
\label{fig:vstep_states_mv1}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=Vstep_flex_lqrp20.eps, scale=0.38}
\caption{Flexible states during velocity reference tracking.}
\label{fig:vstep_flex_mv}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=hstep_lqrp20.eps, scale=0.38}
\caption{System response to a step change in altitude reference.}
\label{fig:hstep_mv2}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=hstep_control_lqrp20.eps, scale=0.38}
\caption{Control input responses corresponding to a step change in altitude reference.}
\label{fig:hstep_mv}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=Vstep_states_lqrp20.eps, scale=0.38}
\caption{Responses of $\alpha$ and $\gamma$ for a step change in altitude reference.}
\label{fig:hstep_states_mv3}
\end{center}
\end{figure}
\begin{figure}[htbp]
\hfill
\begin{center}
\epsfig{file=hstep_flex_lqrp20.eps, scale=0.38}
\caption{Flexible states during altitude reference tracking.}
\label{fig:hstep_flex_mv}
\end{center}
\end{figure}
\section{Conclusion}\label{sec:concl}
In this paper, a robust nonlinear tracking control scheme for a class of uncertain nonlinear systems has been proposed. The proposed method uses a robust feedback linearization approach and the generalized mean value theorem to obtain an uncertain linear model for the corresponding uncertain nonlinear system. The scheme allows for a structured uncertainty representation.  In order to demonstrate the applicability of the proposed method to a real world problem, the method is applied to a tracking control problem for an air-breathing hypersonic flight vehicle. Simulation results for step changes in the velocity and altitude reference commands show that the proposed scheme works very well in this example and the tracking of velocity and altitude is achieved effectively even in the presence of uncertainties.


\section{Introduction}
One of the major goals of relativistic heavy-ion collisions is to explore the QCD phase diagram and phase boundary~\cite{StephanovPD,adams2005experimental,conservecharge0,conservecharge1}. Theoretical studies have shown the existence of a critical point (CP) at finite baryon chemical potential and temperature~\cite{CEP1, CEP2}. This CP is proposed to be characterized by a second-order phase transition, which becomes a unique property of strongly interacting matter~\cite{searchCEP1,searchCEP2,searchCEP3,searchCEP4,STARPRLMoment}.

In the thermodynamic limit, the correlation length diverges at the CP and the system becomes scale invariant and fractal~\cite{invariant1,invariant2,invariant3}. It has been shown that the density fluctuations near the QCD critical point form a distinct pattern of power-law or intermittency behavior in the matter produced in high energy heavy-ion collisions~\cite{Antoniou2006PRL,Antoniou2010PRC,Antoniou2016PRC,Antoniou2018PRD}. Intermittency is a manifestation of the scale invariance and fractality of  physical process and the stochastic nature of the underlying scaling law~\cite{invariant3}. It can be revealed in transverse momentum spectrum as a power-law behavior of scaled factorial moment~\cite{invariant1, Antoniou2006PRL}. In current high energy heavy-ion experiments at CERN SPS~\cite{NA49SFM,NA61universe}, the NA49 and NA61 collaborations have explored the critical fluctuations by employing an intermittency analysis with various sizes of colliding nuclei. The power-law behavior has been observed in Si + Si collisions at 158A GeV~\cite{NA49SFM}. At RHIC BES-I energies, the preliminary result of a critical exponent extracted from intermittency index shows a minimum around $\sqrt{s_\mathrm{NN}}$ = 20 - 30 GeV in central collisions~\cite{STARintermittency}. In the mean time, several model studies have made efforts to investigate the unique behaviors of intermittency under various underlying mechanisms~\cite{CMCPLB,UrQMDLi,IJMPE,amptNPA,epos}.

For a self-similar system with intermittency, it is expected that the multiplicity distribution in momentum space is associated with a strong clustering effect, which indicating a remarkably structured phase-space density~\cite{invariant3,densityF}. However, the inclusive single-particle multiplicity spectra in finite space of high energy collisions are significantly influenced by background effects. The multiplicity distribution is changed or distorted by conservation law, resonance decay and statistical fluctuations, etc. It has been studied that the statistical fluctuations due to finite number of particles~\cite{SFM1} or the choices of the size in momentum space~\cite{APPB} will influence the measured SFMs. Therefore, it is necessary to estimate and remove these trivial effects in order to get a clean power-law exponent and then one can compare the measured intermittency with theoretical predictions. For this purpose, Ochs~\cite{OchsCumulative} and Bialas~\cite{BialasCumulative} have proposed to study intermittency in terms of variables, called as cumulative variable, in which the single-particle density is a constant. It is shown that the cumulative variable very effectively reduces the distortions of the simple scaling law caused by a non-uniform single-particle spectrum and there is no bias from the shape of the inclusive distribution. We will study how to remove the trivial background effects by the cumulative variable method in the measurement of SFMs in heavy-ion collisions.

In the study of intermittency in high energy experiments, particle detector has a finite detection efficiency, which simply results from the limited capability of the detector to register the finite-state particles~\cite{STAREfficiency,EfficiencyLuo}. This efficiency effect will lead to the loss of particle multiplicity in an event, which will change the shape of the original event-by-event multiplicity distributions in momentum space~\cite{EfficiencyLuo,MomentOverview}. Since the calculation of SFMs is determined by the particle distributions, its value could be significantly modified by the detector efficiency, which will distort the original signal possibly induced by the CP. We should recover the SFM of the original true multiplicity distributions from the experimentally measured one by applying a proper efficiency correction technique. Then the true intermittency index can be extracted from the efficiency corrected SFM.

The paper is organized as follows. A brief introduction to the UrQMD model is given in Sec. II. In sec. III, we introduce the method of intermittency analysis by using SFMs. Then, the collision energy and centrality dependence of SFMs are investigated by UrQMD model in Au + Au collisions from $\sqrt{s_\mathrm{NN}}$ = 7.7 to 200 GeV. In Sec. V, we discuss the estimation and subtraction of background in the calculations of SFMs. In Sec. VI, the efficiency correction formula is deduced, followed by a check of the validity of the method by UrQMD model. Finally, we give a summary and outlook of this work.

\section{UrQMD MODEL}
The UrQMD (Ultra Relativistic Quantum Molecular Dynamics) model is a microscopic many-body model and has been extensively applied to simulate p + p, p + A, and A + A interactions in the ultra-relativistic heavy-ion collisions~\cite{MBUrQMD,SABassUrQMD,HPUrQMD}. It provides phase space descriptions of different reaction mechanisms based on the covariant propagation of all hadrons with stochastic binary scattering, color string formation and resonance decay~\cite{SABassUrQMD}. This model includes baryon-baryon, meson-baryon and meson-meson interactions with more than 50 baryon and 45 meson species. It preserves the conservation of electric charge, baryon number, and strangeness number as expected for the evolution of the QCD matter. It models the phenomena of baryon stopping which is an essential feature encountered in heavy ions at low beam energies. It is a well-designed transport model~\cite{MBUrQMD} for simulations with the entire available range of energies from SIS energy ($\sqrt{s_\mathrm{NN}}=2$ GeV) to the top RHIC energy ($\sqrt{s_\mathrm{NN}}=200$ GeV). More details about the UrQMD model can be found in Refs~\cite{MBUrQMD,SABassUrQMD,HPUrQMD}.

The UrQMD model is a suitable simulator to estimate the non-critical contribution and other trivial background effects in the measurement of correlations and fluctuations in heavy-ion collisions. In this work, we use the UrQMD model with version 2.3 to generate Monte-Carlo event samples in Au + Au collisions at RHIC energies. The corresponding statistics are 72.5, 105, 106, 81, 133, 38, 56 millions at $\sqrt{s_\mathrm{NN}}$ = 7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV, respectively.

 \begin{figure*}
     \centering
     \includegraphics[scale=0.85]{pdf/CFM-FM-UrQMD-7-200GeV.pdf}
      \caption{The second-order scaled factorial moment (black circles) , $F_{2}(M)$, as a function of number of partitioned cells in a double-logarithmic scale at $\sqrt{s_\mathrm{NN}}$ = 7.7 - 200 GeV from UrQMD model. The black lines are the power-law fitting. The corresponding red ones represent the SFMs calculated by the cumulative variable method.}
     \label{Fig:F2E}
     \end{figure*}

     \begin{figure*}
     \centering
     \includegraphics[scale=0.85]{pdf/Centrality-FM-CFM.pdf}
      \caption{The second-order scaled factorial moment (black circles) as a function of number of partitioned cells from the most central ($0-5\%$) to the most peripheral ($60-80\%$) collisions at $\sqrt{s_\mathrm{NN}}$ = 19.6 GeV. The corresponding red ones represent the SFMs calculated by the cumulative variable method.}
     \label{Fig:F2C}
     \end{figure*}

\section{Method of Analysis}
It is argued that in heavy-ion collisions large baryon density fluctuation may provide a unique signal to the phase transition in the QCD phase diagram. It is expected to observe the critical density fluctuations as a power-law pattern on available phase space resolution if system freezes out right in the vicinity of the critical point~\cite{Antoniou2006PRL,NA61universe}.

In high energy experiments, the power-law or intermittency behavior can be measured by calculations of SFMs of baryon number density~\cite{HattaPRL,Antoniou2006PRL, NA61universe}. For this purpose, an available region of momentum space is partitioned into $M^{D}$ equal-size bins, $F_{q}(M)$ is defined as:\\
\begin{equation}
F_{q}(M)=\frac{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}(n_{i}-1)\cdots(n_{i}-q+1)\rangle}{\langle\frac{1}{M^{D}}\sum_{i=1}^{M^{D}}n_{i}\rangle^{q}},
 \label{Eq:FM}
\end{equation}

\noindent with $M^{D}$ the number of cells in the D-dimensional partitioned momentum space, $n_{i}$ the measured multiplicity in the $i$-th cell, and $q$ the order of the moment.

If the system exhibits critical fluctuations, $F_{q}(M)$ is expected to follow a scaling function:~\cite{Antoniou2006PRL,NA49SFM}\\
\begin{equation}
F_{q}(M)\sim (M^{D})^{\phi_{q}}, M\rightarrow\infty.
 \label{Eq:PowerLaw}
\end{equation}

\noindent A power-law behavior of $F_{q}$ on the partitioned number $M^{D}$ when $M$ is large enough is referred as intermittency. The scaling exponent, $\phi_{q}$, is called the intermittency index that characterizes the strength of the intermittency behavior. By using a critical equation of state for a 3D Ising system, the second-order intermittency index in a two-dimensional transverse momentum space is predicted to be $\phi_{2}=\frac{5}{6}$~\cite{Antoniou2006PRL} for baryon density and $\phi_{2}=\frac{2}{3}$ for sigma condensate~\cite{NGNPA2005}. The search of multiplicity fluctuations in increasing number of partition intervals using the SFMs method was first proposed several years ago~\cite{SFM1,SFM2}. Recent studies show that one can probe QCD critical fluctuations~\cite{CMCPLB} and estimate the possible critical region~\cite{Antoniou2018PRD} from intermittency analysis in relativistic heavy-ion collisions.

In the following section, we will calculate the second-order SFM of proton density in transverse momentum space by using event samples from UrQMD model in Au + Au collisions at $\sqrt{s_\mathrm{NN}}$ = 7.7 - 200 GeV. Then, the intermittency index, $\phi_{2}$, can be extract by fitting from Eq.~\eqref{Eq:PowerLaw}.

\section{Energy and centrality dependence of Scaled Factorial moments}
By using the UrQMD model, we generate event samples at various centralities in Au + Au collisions at $\sqrt{s_\mathrm{NN}}$ = 7.7, 11.5, 19.6, 27, 39, 62.4, 200 GeV. In the model calculations, we apply the same kinematic cuts and technical analysis methods as those used in the RHIC/STAR experiment data~\cite{net_proton2014}. The protons are measured at mid-rapidity($|y|<0.5$) within the transverse momentum $0.4<p_{T}<2.0$ GeV/c. The centrality is defined by the charged pion and kaon multiplicities within pseudo-rapidity $|\eta|<1.0$, which can effectively avoid auto-correlation effect in the measurement of SFMs. In our analysis, we focus on proton multiplicities in two dimensional transverse momentum spaces of $p_{x}$ and $p_{y}$. The available 2D region of transverse momentum is partitioned into $M^2$ equal-size bins to calculate SFMs in various sizes of cells. The statistical error is estimated by the Bootstrap method~\cite{BEboostrap}.

In Fig.~1, the black circles represent the SFMs as a function of number of partitioned bins, directly calculated in transverse momenta for proton numbers in $0-5\%$ most central Au + Au collisions at $\sqrt{s_\mathrm{NN}}$ = 7.7 - 200 GeV. It is observed that $F_{2}(M)$ increases slowly with increasing number of dividing bins. The black lines show the power-law fitting of $F_{2}(M)$ according to Eq.~\eqref{Eq:PowerLaw}. The slopes of the fitting, $i.e.$ the intermittency indices $\phi_{2}$, are found to be small at all energies. And they are much less than the theoretical prediction $\phi_{2}=5/6$ for a critical system of the 3D Ising universality class~\cite{Antoniou2006PRL}.

The $F_{2}(M)$ measured at various collision centralities in Au + Au collisions at $\sqrt{s_\mathrm{NN}}$ = 19.6 GeV are shown as the black circles in Fig.~2. And the black lines are the fitting according to Eq.~\eqref{Eq:PowerLaw}. Again, we find that the directly calculated SFMs can be fitted with a small intermittency index. The values of $\phi_{2}$ increase slightly from the most central ($0-5\%$) to the most peripheral ($60-80\%$) collisions.

Therefore, we observe that the intermittency indices from the directly calculated SFMs are small but non-zero in Au + Au collisions from UrQMD model. But we know that the UrQMD model is a transport model which does not include any critical related self-similar fluctuations~\cite{MBUrQMD} and thus $\phi_{2}$ should be zero. There must exist some trivial non-critical contributions from background. We will investigate how to remove the background effects from the directly measured SFMs in the next section.

\section{Background Subtraction}
\begin{figure}[!htp]
     \centering
     \includegraphics[scale=0.45]{pdf/CFM-FM-CMCMergeGaus0p5.pdf}
     \label{Fig:GasandCMC}
     \caption{The black symbols represent the second-order scaled factorial moment as a function of number of partitioned cells (a) in pure CMC events and (b) in CMC events contaminated with Gaussian background fluctuations. The corresponding red ones are the SFMs calculated by the cumulative variable method.}
\end{figure}

To extract the signature of critical fluctuations, it is essential to understand the non-critical effects or background contributions in heavy-ion collisions on the experimental observables. The background effects will change the inclusive single-particle multiplicity distributions in the measured finite momentum space and thus will significantly influence the value of calculated SFMs and the intermittency index thereby. In this purpose, NA49 and NA61 use the mixed event method to estimate and subtract background by assuming that the particle multiplicity in each cell can be simply divided into background and critical contributions~\cite{NA49SFM}. In this paper, we pursue the cumulative variable method, which has been proved to drastically reduce distortions of intermittency due to non-uniform single-particle density from background contributions~\cite{OchsCumulative,BialasCumulative,na22}, to understand and remove the background effects.

Following Ochs and Bialas~\cite{BialasCumulative,OchsCumulative}, the cumulative variable $\large X(x)$ is related to the single-particle density distribution $\rho(x)$ through:

\begin{equation}
  \large X(x)=\frac{\int_{x_{min}}^{x} \rho(x)dx}{\int_{x_{min}}^{x_{max}}\rho(x)dx}.
 \label{Eq:cvariable}
\end{equation}

\noindent Here $x$ represents the original measured variable, {\it e.g.,} $p_{x}$ or $p_{y}$. $\rho(x)$ is the density function of $x$. $x_{min}$ and $x_{max}$ are the lower and upper phase space limits of the chosen variable $x$.

The cumulative variable $X(x)$ is determined by the shape of density distribution $\rho(x)$. And the distribution of the new variable $X(x)$ is uniform in the interval from 0 to 1. It has been proved that the cumulative variable could remove the dependence of the intermittency parameters on the shape of particle density distributions and give a new way to compare measurements from different experiments~\cite{BialasCumulative}. In order to use the cumulative variable, the two-dimensional momentum space $p_{x}p_{y}$ which is partitioned into $M^{2}$ cells will transfer to be $p_{X}p_{Y}$ space. And the SFM directly calculated in $p_{x}p_{y}$ space ($F_{2}(M)$) will transfer to be $CF_{2}(M)$, which is now calculated in $p_{X}p_{Y}$ space. The process of fitting $\phi_{2}^{c}$ from $CF_{2}(M)$ is similar to that of $\phi_{2}$ from $F_{2}(M)$ according to Eq.~\eqref{Eq:PowerLaw}.

In order to test the validity of the cumulative variable method in the calculations of SFMs, we use a Critical Monte-Carlo (CMC) model~\cite{Antoniou2006PRL,CMCPLB} of the 3D Ising universality class to generate critical event sample. The CMC model involves self-similar or intermittency nature of particle correlations and leads to an intermittency index of $\phi_{2}=\frac{5}{6}$~\cite{Antoniou2006PRL}. In Fig.~3(a), both $F_{2}(M)$ and $CF_{2}(M)$ are shown in the same pad. We observe that $CF_{2}(M)$ follows a good power-law behavior as $F_{2}(M)$ with increasing $M^{2}$. Within statistical errors, the intermittency index $\phi_{2}^{c}$ fitted from $CF_{2}(M)$ equals to $\phi_{2}$ obtained from $F_{2}(M)$. It means that the cumulative variable method does not change the intermittency behavior for a pure critical signal event sample. In Fig.~3(b), the CMC event sample is contaminated by hand with a statistical Gaussian background contribution, with the mixed probability $\lambda=95\%$. The chosen value of $\lambda$ is close to the one used in the simulations of background in NA49 experiment~\cite{NA49SFM}. In this case, one finds that the directly calculated $F_{2}(M)$ deviates substantially from the linear dependence, i.e. violation of the scaling law because of the Gaussian background contribution. However, the trend of $CF_{2}(M)$, which is calculated by the cumulative variable method, still obey similar power-law dependence on $M^{2}$ as that in Fig.~3(a). Furthermore, the intermittency index $\phi_{2}^{c}$ calculated from $CF_{2}(M)$ keeps unchanged when comparing to the one in original CMC sample shown in Fig.~3(a). We feel that these results are encouraging. It confirms that, in the intermittency analysis, the cumulative variable method can efficiently remove the distortions or effects caused by background contribution.

Let us go back to the problem of background effect in the UrQMD model remained in Sec. IV. We calculate SFMs in the same event sample by the proposed cumulative variable method and then get the intermittency index from $CF_{2}(M)$. The results are shown as red triangles and red lines in Fig.~1 and Fig.~2. The $CF_{2}(M)$ is found to be nearly flat with increasing number of cells in all measured energies and centralities. Furthermore, we note that the intermittency index, with the value nearly around $0$, is much smaller than the one directly calculated from $F_{2}(M)$. It verifies that the background of non-critical effect can be efficiently removed by the cumulative variable method in the calculation of SFMs in UrQMD model. This method could also be used for the intermittency analysis in the ongoing experimental at RHIC/STAR or further heavy-ion experiments in search of the QCD critical point.

\section{Efficiency correction}
One of the difficulties of measuring SFMs and intermittency in experiment is efficiency correction. It is known that the values of SFMs are changed from the true ones due to the fact that detectors miss some particles with a probability named efficiency. To understand the underlying physics associated with this measurement, one needs to perform a careful study on the efficiency effect. Generally, the efficiencies in experiments are obtained by Monte Carlo (MC) embedding technique~\cite{embedding}. This allows for the determination of the efficiency, which is the ratio of the matched MC tracks number and the number of input tracks. It contains the effects of tracking efficiency, detector acceptance and interaction losses.

We denote the number of produced particles as $N$ and the number of measured ones as $n$ with a detection efficiency $\epsilon$. The true factorial moment is calculated as $f_{q}^{true}=\langle N(N-1)...(N-q+1)\rangle$. It can be recovered by dividing the measured factorial moment, $f_{q}^{measured}=\langle n(n-1)...(n-q+1)\rangle$, with appropriate powers of the detection efficiency~\cite{EfficiencyFM, EfficiencyLuo,EfficiencyKoch}:

\begin{equation}
 f_{q}^{corrected}=\frac{f_{q}^{measured}}{\epsilon^{q}}=\frac{\langle n(n-1)...(n-q+1)\rangle}{\epsilon^{q}}.
 \label{Eq:f2correction}
\end{equation}
 This strategy has been used for the efficiency correction in the high-order cumulant analysis~\cite{EfficiencyLuo,MomentOverview, EfficiencyFM,EfficiencyToshihiro, EfficiencyKoch}. Consider the probability to detect a particle is governed by a binomial distribution, both cumulants~\cite{EfficiencyLuo,MomentOverview} and off-diagonal cumulants~\cite{EfficiencyKoch} can be expressed in term of factorial moments and then could be corrected by using Eq.~\eqref{Eq:f2correction}.

We apply the strategy to the efficiency correction for SFMs defined in Eq.~\eqref{Eq:FM}. Since the available region of phase space is partitioned into a lattice of $M^{2}$ equal-size cells, every element, $f_{q,i}^{measured}=n_{i}\times(n_{i}-1)...(n_{i}-q+1)$, of measured $F_{q}(M)$ should be corrected one by one. In this way, the efficiency corrected $F_{q}(M)$ is:

\begin{eqnarray}
  F_{q}^{corrected}(M) &=&\large \frac{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}} \frac{f_{q,i}^{measured}}{\bar{\epsilon_{i}}^{q}}\rangle}{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}} \frac{f_{1,i}^{measured}}{\bar{\epsilon_{i}}}\rangle^{q}} \nonumber\\
 &=& \frac{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{n_{i}(n_{i}-1)\cdots(n_{i}-q+1)}{\bar{\epsilon_{i}}^{q}}\rangle}{\langle\frac{1}{M^{2}}\sum_{i=1}^{M^{2}}\frac{n_{i}}{\bar{\epsilon_{i}}}\rangle^{q}}.
 \label{Eq:FMcorrection}
 \end{eqnarray}

\noindent Here, $n_{i}$ denotes the number of measured particles located in the $i$-th cell with detection efficiency $\epsilon_{i}$. The mean $\bar{\epsilon_{i}}$, is calculated by $(\epsilon_{1}+\epsilon_{2}\cdots \epsilon_{n_i})/n_i$, representing the average efficiency in the $i$-th cell. We call the efficiency correction technique of Eq.~\eqref{Eq:FMcorrection} the cell-by-cell method.

In order to demonstrate the validity of the cell-by-cell method, we employ the UrQMD model with the particle detection efficiencies used in real experiment. It is simulated by injecting particle tracks from UrQMD events into the RHIC/STAR detector acceptance with the experimental efficiencies. In the STAR experiment, the detection efficiency is not a constant but depends on the momentum of particles~\cite{STAREfficiency,STARMoment,LuoCPOD}. The particle identification method is different between low and high $p_{T}$ regions. The main particle detector at STAR, the Time Projection Chamber (TPC), is used to obtain momentum of charged particles at low $p_{T}$ region of $0.4<p_{T}<0.8$ GeV/c~\cite{STAREfficiency}. And the Time-Of-Flight (TOF) detector is used to do the particle identification at relatively high $p_{T}$ region of $0.8<p_{T}<2$ GeV/c~\cite{STARMoment,LuoCPOD}. In this case, particles need to be counted separately for the two $p_{T}$ regions, in which the values of the efficiency are different.

\begin{figure}[!htp]
\hspace{-0.8cm}
     \includegraphics[scale=0.45]{pdf/FM-TPC-Efficiency.pdf}

      \caption{(a) Experimental tracking efficiencies as a function of $p_{T}$ in TPC detector at mid-rapidity ($|y|<0.5$) for protons in $0 - 5\%$ Au + Au collisions. (b) The second-order SFM as a function of number of partitioned cells from UrQMD calculations. The black circles represent the true $F_{2}(M)$, the blue solid triangles are the measured $F_{2}(M)$ after applying the TPC efficiency, and the red stars show the efficiency corrected SFMs by using the cell-by-cell method.}
 \label{Fig:F2TPC}
 \end{figure}

In Fig.~4(a), we firstly show the $p_{T}$ dependence of the experimental efficiency in only TPC detector at mid-rapidity ($|y|<0.5$) region for protons in the most central Au + Au collisions at $\sqrt{s_\mathrm{NN}}$ = 19.6 GeV~\cite{STAREfficiency}. It firstly increases with increasing $p_{T}$, and then gets saturated in higher $p_{T}$ regions. After employing this $p_{T}$ dependent efficiency into the UrQMD event sample, the measured $F_{2}(M)$ is calculated. Then, we apply the correction formula of Eq.~\eqref{Eq:FMcorrection} to do the efficiency correction on the measured $F_{2}(M)$. In Fig.~4(b), the black circles represent the original true $F_{2}(M)$, the blue triangles are the measured $F_{2}(M)$ with STAR TPC efficiency, and the red stars show the efficiency corrected $F_{2}(M)$. It is observed that the measured SFMs with efficiency are systematically smaller than the true ones, especially in the large number of partitioned cells. However, the efficiency corrected SFMs are found to be well consistent with the original true ones.

\begin{figure}[hbtp]
\hspace{-0.8cm}
 \includegraphics[scale=0.45]{pdf/FM-TPCTOF-Efficiency.pdf}
  \caption{(a) Experimental tracking efficiencies as a function of $p_{T}$ in TPC+TOF detectors at mid-rapidity ($|y|<0.5$) for protons in $0 - 5\%$ Au + Au collisions. (b) The second-order SFM as a function of number of partitioned cells from UrQMD calculations. The black circles represent the true $F_{2}(M)$, the blue solid triangles are the measured $F_{2}(M)$ after applying the TPC+TOF efficiency, and the red stars show the efficiency corrected SFMs by using the cell-by-cell method.}
    \label{Fig:F2TPTOF}
\end{figure}

For the case of TPC+TOF efficiencies, Fig.~5(a) shows the tracking efficiencies as a function of $p_{T}$ in TPC and TOF at STAR~\cite{LuoCPOD, EfficiencyToshihiro, STARMoment}. One notes that there is a steplike dependence of the efficiencies on $p_{T}$. The reason is that the particle identification method is different between TPC and TOF detectors at STAR experiment. We apply the TPC+TOF efficiency to the UrQMD event sample at $\sqrt{s_\mathrm{NN}}$ = 19.6 GeV and then correct the measured SFMs by Eq.~\eqref{Eq:FMcorrection}. The results are shown in Fig.~5(b). Again, the SFMs corrected by the proposed cell-by-cell method are verified to be coincide with the original true ones.

In this section, we have demonstrated that the cell-by-cell method could be served as a precise and effective way of efficiency correction on SFMs. It can be easily applied to current studies at STAR~\cite{STARintermittency}, NA49~\cite{NA49SFM}, NA61~\cite{NA61universe} and other heavy-ion experiments in the intermittency analysis. It should also be noted that one needs to consider how to treat the momentum resolution in different experiments. Since we use the $p_{T}$ of individual particles to get the efficiency, the momentum resolution might directly affect the calculation of SFMs. This effect could be studied by smearing the $p_{T}$ for each particle with the known value of the momentum resolution.

\section{summary and outlook}
In summary, we investigate collision energy and centrality dependence of the SFMs in Au + Au collisions at $\sqrt{s_\mathrm{NN}}=7.7-200$ GeV by using the UrQMD model. The second-order intermittency index is found to be small but non-zero in the transport model without implementing any critical related self-similar fluctuations. A cumulative variable method is then proposed to be used in the calculations of SFMs to remove background in the intermittency analysis. It has been verified that this method can successfully reduce the distortion of a Gaussian background contribution from a pure self-similar event sample generated by the CMC model. After applying the method to the UrQMD event sample, it confirms that the non-critical background effect can be efficiently removed and the value of the intermittency index is close to zero.

In the experimental measurements of intermittency, the measured SFMs should be corrected for detecting efficiencies. We derive a cell-by-cell formula in the calculation of SFMs in heavy-ion collisions. The validity of the method has been checked in the UrQMD event sample which is employed with both the TPC and TPC+TOF tracking efficiencies used in the RHIC/STAR experiment. It is demonstrated that the cell-by-cell method provides a precise and effective way for the efficiency correction on SFMs. The correction method is universal and can be applied to the ongoing studies of intermittency in heavy-ion experiments.

 \begin{figure}
     \includegraphics[scale=0.35]{pdf/UrQMD-phi2-energy-dependence-vsData.pdf}
     \caption{The second-order intermittency index measured at NA49~\cite{NA49SFM,NA49C+C} (solid blue symbols) and NA61~\cite{NA61Ar+Sc} (open blue circles). The results from the UrQMD model in central Au + Au collisions are plotted as black circles. The red arrow represents the theoretic expectation from a critical QCD model~\cite{Antoniou2006PRL}.}
     \label{Figphi2-energy}
   \end{figure}

In current experimental explorations of the intermittency in heavy-ion collisions, the NA49 and NA61 collaborations have directly measured $\phi_{2}$ at various sizes of colliding nuclei~\cite{NA49SFM,NA49C+C,NA61Ar+Sc}, which are represented as blue symbols in Fig.~6. The intermittency parameter at $\sqrt{s_\mathrm{NN}}$ = 17.3 GeV for the Si + Si system at NA49 experiment approaches the theoretic expectation value, shown as red arrow in the figure, in the second-order phase transition in a critical QCD model~\cite{Antoniou2006PRL}. The black circles of the UrQMD results give a flat trend with the value around 0 at all energies due to no critical mechanisms are implemented in the transport model.

The RHIC/STAR experiment has been running the second phase of beam energy scan (BES-II) program in 2018 - 2021~\cite{BESII}. With significant improved statistics and particle identification in BES-II, it would be interesting that STAR could measure intermittency to explore the CP in the QCD phase diagram. Our work provides a non-critical baseline and give a guidance of background subtraction and efficiency correction for the calculations of intermittency in heavy-ion collisions.

\section* {Acknowledgements}
This work is supported by the Ministry of Science and Technology (MoST) under
grant No. 2016YFE0104800, the National Key Research and Development Program of China (Grant No. 2020YFE0202002 and 2018YFE0205201), the National Natural Science Foundation of China (Grant No. 11828500,11828501, 11575069, 11890711 and 11861131009). The first two authors contribute equally to this work.